LFM Stats And Pure

Prove that:

1. if \(a+2b+3c=7x\), then \[ a^{2}+b^{2}+c^{2}=\left(x-a\right)^{2}+\left(2x-b\right)^{2}+\left(3x-c\right)^{2}; \]
2. if \(2a+3b+3c=11x\), then \[ a^{2}+b^{2}+c^{2}=\left(2x-a\right)^{2}+\left(3x-b\right)^{2}+\left(3x-c\right)^{2}. \]

Let \[ y=\dfrac{x^2+x\sin\theta+1}{x^2+x\cos\theta+1} \,.\]

1. Given that \(x\) is real, show that \[ (y\cos\theta -\sin\theta)^2 \ge 4 (y-1)^2 \,. \] Deduce that \[ y^2+1 \ge 4(y-1)^2 \,, \] and hence that \[ \dfrac {4-\sqrt7}3 \le y \le \dfrac {4+\sqrt7}3 \,. \]
2. In the case $y= \dfrac {4+\sqrt7}3 \,$, show that \[\sqrt{y^2+1}=2(y-1)\] and find the corresponding values of \(x\) and \(\tan\theta\).

1. Use the substitution \(\sqrt x = y\) (where \(y\ge0\)) to find the real root of the equation \[ x + 3\, \sqrt x - \tfrac12 =0\,. \]
2. Find all real roots of the following equations:
   • \(x+10\,\sqrt{x+2\, }\, -22 =0\,\);
   • \(x^2 -4x + \sqrt{2x^2 -8x-3 \,}\, -9 =0\,\).

1. By considering the equation \(x^2+x-a=0\,\), show that the equation \(x={(a-x)\vphantom M}^{\frac12}\) has one real solution when \(a\ge0\) and no real solutions when \(a<0\,\). Find the number of distinct real solutions of the equation \[ x={\big((1+a)x-a\big)}^{\!\frac13} \] in the cases that arise according to the value of \(a\).
2. Find the number of distinct real solutions of the equation \[ x={(b+x)\vphantom M}^{\frac12} \] in the cases that arise according to the value of \(b\,\).

Let \(\f(x)=x^2+px+q\) and \(\g(x)=x^2+rx+s\,\). Find an expression for \(\f ( \g (x))\) and hence find a necessary and sufficient condition on \(a\), \(b\) and \(c\) for it to be possible to write the quartic expression \(x^4+ax^3+bx^2+cx+d\) in the form \(\f ( \g (x))\), for some choice of values of \(p\), \(q\), \(r\) and~\(s\). Show further that this condition holds if and only if it is possible to write the quartic expression \(x^4+ax^3+bx^2+cx+d\) in the form \((x^2+vx+w)^2-k\), for some choice of values of \(v\), \(w\) and \(k\). Find the roots of the quartic equation \(x^4-4x^3+10x^2-12x+4=0\,\).

In this question \(a\) and \(b\) are distinct, non-zero real numbers, and \(c\) is a real number.

1. Show that, if \(a\) and \(b\) are either both positive or both negative, then the equation \[ \displaystyle \frac {x }{ x-a} + \frac{x }{ x-b} = 1 \] has two distinct real solutions.
2. Show that, if \(c\ne1\), the equation \[\displaystyle \frac x { x-a} + \frac{x}{ x-b} = 1 + c\] has exactly one real solution if $ \displaystyle c^2 = - \frac {4ab}{\l a - b \r ^2}\;. \;\; $ Show that this condition can be written $\displaystyle c^2= 1 - \l \frac {a+b}{a-b} \r ^2 $ and deduce that it can only hold if \(0 < c^2 \le 1\,\).

Prove that, if \(\vert \alpha\vert < 2\sqrt{2},\) then there is no value of \(x\) for which \begin{equation} x^2 -{\alpha}\vert x \vert + 2 < 0\;. \tag{\(*\)} \end{equation} Find the solution set of \((*)\) for \({\alpha}=3\,\). For \({\alpha} > 2\sqrt{2}\,\), the sum of the lengths of the intervals in which \(x\) satisfies \((*)\) is denoted by \(S\,\). Find \(S\) in terms of \({\alpha}\) and deduce that \(S < 2{\alpha}\,\). Sketch the graph of \(S\,\) against \(\alpha \,\).

Show Solution

There are two cases to consider by they are equivalent to \(x^2 \pm \alpha x + 2 < 0\), which has no solution solutions if \(\Delta < 0\), ie if \(\alpha^2 - 4\cdot1\cdot2 < 0 \Leftrightarrow |\alpha| < 2\sqrt{2}\). If \(\alpha = 3\), we have \begin{align*} && 0 & > x^2-3x+2 \\ &&&= (x-2)(x-1) \\ \Rightarrow && x & \in (1,2) \\ \\ && 0 &> x^2+3x+2 \\ &&& = (x+2)(x+1) \\ \Rightarrow && x &\in (-2,-1) \end{align*} Both cases work here, so \(x \in (-2, -1) \cup (1,2)\). \begin{align*} && 0 &> x^2 \pm \alpha x + 2 \\ &&&= (x \pm \tfrac{\alpha}{2})^2 -\frac{\alpha^2-8}{4} \end{align*} The potential intervals therefore are \((\frac{\alpha -\sqrt{\alpha^2-8}}{2}, \frac{\alpha +\sqrt{\alpha^2-8}}{2})\) and \((\frac{-\alpha -\sqrt{\alpha^2-8}}{2}, \frac{-\alpha +\sqrt{\alpha^2-8}}{2})\). Neither of these intervals overlap with \(0\), since \(\alpha^2 > \alpha^2-8\), and their lengths are both \(\sqrt{\alpha^2-8}\), therefore \(S = 2\sqrt{\alpha^2-8} < 2\alpha\)

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Find all real values of \(x\) that satisfy:

1. \( \ds \sqrt{3x^2+1} + \sqrt{x} -2x-1=0 \;;\)
2. \( \ds \sqrt{3x^2+1} - 2\sqrt{x} +x-1=0 \;;\)
3. \( \ds \sqrt{3x^2+1} - 2\sqrt{x} -x+1=0 \;.\)

1. Express \(\left(3+2\sqrt{5} \, \right)^3\) in the form \(a+b\sqrt{5}\) where \(a\) and \(b\) are integers.
2. Find the positive integers \(c\) and \(d\) such that \(\sqrt[3]{99-70\sqrt{2}\;}\) = \(c - d\sqrt{2} \,\).
3. Find the two real solutions of \(x^6 - 198 x^3 + 1 = 0 \,\).

Give a condition that must be satisfied by \(p\), \(q\) and \(r\) for it to be possible to write the quadratic polynomial \(px^2 + qx + r\) in the form \(p \l x + h \r^2\), for some \(h\). Obtain an equation, which you need not simplify, that must be satisfied by \(t\) if it is possible to write \[ \l x^2 + \textstyle{{1 \over 2}} bx + t \r^2 - \l x^4 + bx^3 + cx^2 +dx +e \r \] in the form \(k \l x + h \r^2\), for some \(k\) and \(h\). Hence, or otherwise, write \(x^4 + 6x^3 + 9x^2 -2x -7\) as a product of two quadratic factors.

Show that if \(x\) and \(y\) are positive and \(x^3 + x^2 = y^3 - y^2\) then \(x < y\,\). Show further that if \(0 < x \le y - 1\), then \(x^3 + x^2 < y^3 - y^2\). Prove that there does not exist a pair of {\sl positive} integers such that the difference of their cubes is equal to the sum of their squares. Find all the pairs of integers such that the difference of their cubes is equal to the sum of their squares.

Let \[\f(x) = a \sqrt{x} - \sqrt{x - b}\;, \] where \(x\ge b >0\) and \(a>1\,\). Sketch the graph of \(\f(x)\,\). Hence show that the equation \(\f(x) = c\), where \(c>0\), has no solution when \(c^2 < b \l a^2 - 1 \r\,\). Find conditions on \(c^2\) in terms of \(a\) and \(b\) for the equation to have exactly one or exactly two solutions. Solve the equations

1. \(3 \sqrt{x} - \sqrt{x - 2} = 4\, ,\)
2. \(3 \sqrt{x} - \sqrt{x - 3} = 5\;\).

Show Solution

\begin{align*} && f'(x) &= \frac12 ax^{-1/2}-\frac12(x-b)^{-1/2} \\ \Rightarrow f'(x) = 0: && 0 &= \frac{a\sqrt{x-b}-\sqrt{x}}{\sqrt{x(x-b)}} \\ \Rightarrow && x &= a^2(x-b)\\ \Rightarrow && x &= \frac{a^2b}{a^2-1} \\ && f(x) &= a^2 \sqrt{\frac{b}{a^2-1}} - \sqrt{\frac{a^2b}{a^2-1}-b} \\ &&&= a^2 \sqrt{\frac{b}{a^2-1}} - \sqrt{\frac{b}{a^2-1}} \\ &&&= \sqrt{b(a^2-1)} \end{align*}

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If \(c\) is below the turning point, ie \(c^2 < b(a^2-1)\) there is no solution. If \(c^2 = b(a^2-1)\) there is exactly one solution. If \(b(a^2-1) < c^2 < (f(b))^2 = a^2b\) then there are two solutions, otherwise there is exactly one solution.

1. \(c^2 = 16\), \(2 \cdot (3^2-1) = 16\), so we should have exactly one solution at \(x = \frac{3^2 \cdot 2}{3^2 -1 } = \frac{9}{4}\)
2. \(c^2 = 25\) and \(3 \cdot (3^2 - 1) = 24, 3 \cdot (3^2) = 27\), so we look for two solutions. \begin{align*} && 5 & = 3 \sqrt{x} - \sqrt{x-3} \\ \Rightarrow && 25 &= 9x+x-3-6\sqrt{x(x-3)} \\ \Rightarrow && 3\sqrt{x(x-3)} &= 5x-14 \\ \Rightarrow && 9x(x-3) &= 25x^2-140x+196 \\ \Rightarrow && 0 &= 16x^2-113x+196 \\ &&&= (x-4)(16x-49) \\ \Rightarrow && x &= 4, \frac{49}{16} \end{align*}

Show that setting \(z - z^{-1}=w\) in the quartic equation \[ z^4 +5z^3 +4z^2 -5z +1=0 \] results in the quadratic equation \(w^2+5w+6=0\). Hence solve the above quartic equation. Solve similarly the equation \[ 2z^8 -3z^7-12z^6 +12z^5 +22z^4-12z^3 -12 z^2 +3z +2=0 \;. \]

Consider the equation \[ x^2 - b x + c = 0 \;, \] where \(b\) and \(c\) are real numbers.

1. Show that the roots of the equation are real and positive if and only if \(b>0\) and \phantom{} \(b^2\ge4c>0\), and sketch the region of the \(b\,\)-\(c\) plane in which these conditions hold.
2. Sketch the region of the \(b\,\)-\(c\) plane in which the roots of the equation are real and less than \(1\) in magnitude.

Solve the inequalities

1. \(1+2x-x^2 >2/x \quad (x\ne0)\) ,
2. \(\sqrt{3x+10} > 2+\sqrt{x+4} \quad (x\ge -10/3)\).

Show that \[ x^2-y^2 +x+3y-2 = (x-y+2)(x+y-1) \] and hence, or otherwise, indicate by means of a sketch the region of the \(x\)-\(y\) plane for which $$ x^2-y^2 +x+3y>2. $$ Sketch also the region of the \(x\)-\(y\) plane for which $$ x^2-4y^2 +3x-2y<-2. $$ Give the coordinates of a point for which both inequalities are satisfied or explain why no such point exists.

Show Solution

\begin{align*} && (x-y+2)(x+y-1) &= (x-y)(x+y)-(x-y)+2(x+y)-2 \\ &&&= x^2-y^2+x+3y-2 \end{align*}

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\begin{align*} x^2-4y^2 +3x-2y+2 &= (x - 2 y + 1) (x + 2 y + 2) \end{align*}

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Consider the point \(x = 0, y = \frac32\), then \(\frac92 - \frac94 = \frac94 > 2\) and \(-4\cdot\frac94-2\cdot \frac32 = -12 < -2\) so this is an example of a point in both regions

Consider the quadratic equation $$ nx^2+2x \sqrt{pn^2+q} + rn + s = 0, \tag{*} $$ where \(p>0\), \(p\neq r\) and \(n=1\), \(2\), \(3\), \(\ldots\) .

1. For the case where \(p=3\), \(q=50\), \(r=2\), \(s=15\), find the set of values of \(n\) for which equation \((*)\) has no real roots.
2. Prove that if \(p < r\) and \(4q(p-r) > s^2\), then \((*)\) has no real roots for any value of \(n\).
3. If \(n=1\), \(p-r=1\) and \(q={s^2}/8\), show that \((*)\) has real roots if, and only if, \(s \le 4-2\sqrt{2}\ \) or \(\ s \ge 4+2\sqrt{2}\).

1. Find the greatest and least values of \(bx+a\) for \(-10\leqslant x \leqslant 10\), distinguishing carefully between the cases \(b>0\), \(b=0\) and \(b<0\).
2. Find the greatest and least values of \(cx^{2}+bx+a\), where \(c\ge0\), for \(-10\leqslant x \leqslant 10\), distinguishing carefully between the cases that can arise for different values of \(b\) and \(c\).

The \(n\) positive numbers \(x_{1},x_{2},\dots,x_{n}\), where \(n\ge3\), satisfy $$ x_{1}=1+\frac{1}{x_{2}}\, ,\ \ \ x_{2}=1+\frac{1}{x_{3}}\, , \ \ \ \dots\; , \ \ \ x_{n-1}=1+\frac{1}{x_{n}}\, , $$ and also $$ \ x_{n}=1+\frac{1}{x_{1}}\, . $$ Show that

1. \(x_{1},x_{2},\dots,x_{n}>1\),
2. \({\displaystyle x_{1}-x_{2}=-\frac{x_{2}-x_{3}}{x_{2}x_{3}}}\),
3. \(x_{1}=x_{2}=\cdots=x_{n}\).

1. Find the real values of \(x\) for which \[ x^{3}-4x^{2}-x+4\geqslant0. \]
2. Find the three lines in the \((x,y)\) plane on which \[ x^{3}-4x^{2}y-xy^{2}+4y^{3}=0. \]
3. On a sketch shade the regions of the \((x,y)\) plane for which \[ x^{3}-4x^{2}y-xy^{2}+4y^{3}\geqslant0. \]

1. By setting \(y=x+x^{-1},\) find the solutions of \[ x^{4}+10x^{3}+26x^{2}+10x+1=0. \]
2. Solve \[ x^{4}+x^{3}-10x^{2}-4x+16=0. \]

Let \(a,b,c,d,p\) and \(q\) be positive integers. Prove that:

1. if \(b > a\) and \(c > 1,\) then \(bc\geqslant2c\geqslant2+c\);
2. if \(a < b\) and \(d < c\), then \(bc-ad\geqslant a+c\);
3. if \({\displaystyle \frac{a}{b} < p < \frac{c}{d}}\), then \(\left(bc-ad\right)p\geqslant a+c\);
4. if \({\displaystyle \frac{a}{b} < \frac{p}{q} < \frac{c}{d}},\) then \({\displaystyle p\geqslant\frac{a+c}{bc-ad}}\) and \({\displaystyle q\geqslant\frac{b+d}{bc-ad}}\).

Prove that both \(x^{4}-2x^{3}+x^{2}\) and \(x^{2}-8x+17\) are non-negative for all real \(x\). By considering the intervals \(x\leqslant0\), \(0 < x\leqslant2\) and \(x > 2\) separately, or otherwise, prove that the equation \[ x^{4}-2x^{3}+x^{2}-8x+17=0 \] has no real roots. Prove that the equation \(x^{4}-x^{3}+x^{2}-4x+4=0\) has no real roots.

The function \(\mathrm{Min}\) is defined as \[ \mathrm{Min}(a, b) = \begin{cases} a & \text{if } a \leq b \\ b & \text{if } a > b \end{cases} \]

1. Sketch the graph \(y = \mathrm{Min}(x^2, 2x)\).
2. Solve the equation \(2\mathrm{Min}(x^2, 2x) = 5x - 3\).
3. Solve the equation \(\mathrm{Min}(x^2, 2x) + \mathrm{Min}(x^3, 4x) = mx\) in the cases \(m = 2\) and \(m = 6\).
4. Show that \((1, -3)\) is a local maximum point on the curve \(y = 2\mathrm{Min}(x^2, x^3) - 5x\) and find the other three local maxima and minima on this curve. Sketch the curve.

The Devil's Curve is given by $$y^2(y^2 - b^2) = x^2(x^2 - a^2),$$ where \(a\) and \(b\) are positive constants.

1. In the case \(a = b\), sketch the Devil's Curve.
2. Now consider the case \(a = 2\) and \(b = \sqrt{5}\), and \(x \geq 0\), \(y \geq 0\).
   1. Show by considering a quadratic equation in \(x^2\) that either \(0 \leq y \leq 1\) or \(y \geq 2\).
   2. Describe the curve very close to and very far from the origin.
   3. Find the points at which the tangent to the curve is parallel to the \(x\)-axis and the point at which the tangent to the curve is parallel to the \(y\)-axis.
   Sketch the Devil's Curve in this case.
3. Sketch the Devil's Curve in the case \(a = 2\) and \(b = \sqrt{5}\) again, but with \(-\infty < x < \infty\) and \(-\infty < y < \infty\).

In this question, \(\lfloor x \rfloor\) denotes the greatest integer that is less than or equal to \(x\), so that (for example) \(\lfloor 2.9 \rfloor = 2\), \(\lfloor 2\rfloor = 2\) and \(\lfloor -1.5 \rfloor = -2\). On separate diagrams draw the graphs, for \(-\pi \le x \le \pi\), of:

1. Sketch the curve \(y = \e^x (2x^2 -5x+ 2)\,.\) Hence determine how many real values of \(x\) satisfy the equation \(\e^x (2x^2 -5x+ 2)= k\) in the different cases that arise according to the value of \(k\). {\em You may assume that \(x^n \e^x\to 0\) as \(x\to-\infty\) for any integer \(n\).}
2. Sketch the curve \(\displaystyle y = \e^{x^2} (2x^4 -5x^2+ 2)\,\).

In this question, \(\lfloor x \rfloor\) denotes the greatest integer that is less than or equal to \(x\), so that \(\lfloor 2.9 \rfloor = 2 = \lfloor 2.0 \rfloor\) and \(\lfloor -1.5 \rfloor = -2\). The function \(\f\) is defined, for \(x\ne0\), by \(\f(x) = \dfrac{\lfloor x \rfloor}{x}\,\).

1. Sketch the graph of \(y=\f(x)\) for \(-3\le x \le 3\) (with \(x\ne0\)).
2. By considering the line \(y= \frac7{12}\) on your graph, or otherwise, solve the equation \(\f(x) = \frac7 {12}\,\). Solve also the equations \(\f(x) =\frac{17}{24}\) and \(\f(x) = \frac{4 }{3 }\,\).
3. Find the largest root of the equation \(\f(x) =\frac9{10}\,\).

1. Sketch the curve \(y=\sqrt{1-x} + \sqrt{3+x}\;\). Use your sketch to show that only one real value of \(x\) satisfies \[ \sqrt{1-x} + \sqrt{3+x} = x+1\,, \] and give this value.
2. Determine graphically the number of real values of \(x\) that satisfy \[ 2\sqrt{1-x} = \sqrt{3+x} + \sqrt{3-x}\;. \] Solve this equation.

A curve has equation \(y=2x^3-bx^2+cx\). It has a maximum point at \((p,m)\) and a minimum point at \((q,n)\) where \(p>0\) and \(n>0\). Let \(R\) be the region enclosed by the curve, the line \(x=p\) and the line \(y=n\).

1. Express \(b\) and \(c\) in terms of \(p\) and \(q\).
2. Sketch the curve. Mark on your sketch the point of inflection and shade the region \(R\). Describe the symmetry of the curve.
3. Show that \(m-n=(q-p)^3\).
4. Show that the area of \(R\) is \(\frac12 (q-p)^4\).

Sketch the curve with cartesian equation \[ y = \frac{2x(x^2-5)}{x^2-4} \] and give the equations of the asymptotes and of the tangent to the curve at the origin. Hence determine the number of real roots of the following equations:

1. \(3x(x^2-5)= (x^2-4)(x+3)\,\);
2. \(4x(x^2-5)= (x^2-4)(5x-2)\,\);
3. \(4x^2(x^2-5)^2= (x^2-4)^2(x^2+1)\,\).

Show Solution

\begin{align*} && y &= \frac{2x(x^2-5)}{x^2-4} \\ &&&= 2x(x^2-5)(-\tfrac14)(1-\tfrac14x^2)^{-1} \\ &&&= \tfrac52x + \cdots \\ &&&= \frac{2x(x^2-4)-2x}{x^2-4} \\ &&&= 2x - \frac{2x}{x^2-4} \end{align*}

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1. We are looking for the intersections of \(y = \frac23(x+3)\) and \(y = f(x)\)
   

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   Therefore 3 real roots.
2. We are looking for intersections of \(y = \frac12(5x-2)\) and \(y = f(x)\)
   

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   so one solution.
3. We are looking for intersections of \(y = f(x)^2\) and \(y = x^2+1\), or \(y = \sqrt{x^2+1}\) and \(y = f(x)\) where \(f(x) \geq 0\)
   

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   So \(3\) solutions.

The notation \({\lfloor } x \rfloor\) denotes the greatest integer less than or equal to the real number \(x\). Thus, for example, \(\lfloor \pi\rfloor =3\,\), \(\lfloor 18\rfloor =18\,\) and \(\lfloor-4.2\rfloor = -5\,\).

1. Two curves are given by \(y= x^2+3x-1\) and \(y=x^2 +3\lfloor x\rfloor -1\,\). Sketch the curves, for \(1\le x \le 3\,\), on the same axes. Find the area between the two curves for \(1\le x \le n\), where \(n\) is a positive integer.
2. Two curves are given by \(y= x^2+3x-1\) and \(y=\lfloor x\rfloor ^2+3\lfloor x\rfloor -1\,\). Sketch the curves, for \(1\le x \le 3\,\), on the same axes. Show that the area between the two curves for \(1\le x \le n\), where \(n\) is a positive integer, is \[ \tfrac 16 (n-1)(3n+11)\,. \]

The equation of a curve is \(y=\f ( x )\) where \[ \f ( x ) = x-4-\frac{16 \l 2x+1 \r^2}{x^2 \l x - 4 \r} \;. \]

1. Write down the equations of the vertical and oblique asymptotes to the curve and show that the oblique asymptote is a tangent to the curve.
2. Show that the equation \(\f ( x ) =0\) has a double root.
3. Sketch the curve.

The curve \(C\) has equation $$ y = x(x+1)(x-2)^4. $$ Determine the coordinates of all the stationary points of \(C\) and the nature of each. Sketch \(C\). In separate diagrams draw sketches of the curves whose equations are:

1. \( y^2 = x(x+1)(x-2)^4\;\);
2. \(y = x^2(x^2+1)(x^2-2)^4\,\).

Show Solution

\begin{align*} && y &= x(x+1)(x-2)^4 \\ \Rightarrow && y' &= (x+1)(x-2)^4+x(x-2)^4+4x(x+1)(x-2)^3 \\ &&&= (x-2)^3 \left ( (2x+1)(x-2)+4x(x+1) \right) \\ &&&= (x-2)^3 \left (2x^2-3x-2+4x^2+4x \right) \\ &&&=(x-2)^3(6x^2+x-2) \\ &&&=(x-2)^3(2x-1)(3x+2) \end{align*} Therefore there are stationary points at \((2,0), (\frac12, -\frac{625}{64}), (-\frac23, -\frac{4078}{81})\) \((0,2)\) is a minimum by considering the sign of \(y'\) either side. \( (-\frac23, \frac{2560}{729})\) is a minimum, since it's the first stationary point. \( (\frac12, \frac{243}{64})\) is a maximum since you can't have consecutive minima and the second derivative is clearly non-zero.

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1. \(\,\)
   

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2. \(\,\)
   

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Find the coordinates of the turning point on the curve \(y = x^2 - 2bx + c\,\). Sketch the curve in the case that the equation \(x^2 - 2bx + c=0\) has two distinct real roots. Use your sketch to determine necessary and sufficient conditions on \(b\) and \(c\) for the equation \(x^2 - 2bx + c = 0\) to have two distinct real roots. Determine necessary and sufficient conditions on \(b\) and \(c\) for this equation to have two distinct positive roots. Find the coordinates of the turning points on the curve \(y = x^3 - 3b^2x + c\) (with \(b>0\)) and hence determine necessary and sufficient conditions on \(b\) and \(c\) for the equation \(x^3 - 3b^2x + c = 0\) to have three distinct real roots. Determine necessary and sufficient conditions on \(a\,\), \(b\) and \(c\) for the equation \(\l x - a \r^3 - 3b^2 \l x - a \r + c = 0\) to have three distinct positive roots. Show that the equation \(2x^3 - 9x^2 + 7x - 1 = 0\) has three distinct positive roots.

Show Solution

\begin{align*} y &= x^2-2bx+c \\ &= (x-b)^2+c-b^2 \end{align*} Therefore the turning point is at \((b,c-b^2)\)

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Therefore there will be two distinct roots if \(c -b^2 < 0 \Rightarrow c < b^2\). For the equation to have two distinct positive roots it needs to have two distinct roots (ie the condition above) and \(y(0) = c\) needs to be positive, ie \(c > 0\). Therefore the curve has two distinct positive roots if \(0 < c < b^2\). The turning points on \(y = x^3-3b^2x+c\) will have \(0 = y' = 3x^2-3b^2 \Rightarrow x = \pm b\) Therefore for the cubic to have three distinct real root we must have a root between the turning points, \(y(-b) > 0 > y(b)\) \(b^3-3b^3+c = c-2b^3 < 0\) \((-b)^3+3b^3+c = c+2b^3 > 0\) ie \(-2b^3 < c < 2b^3\). The equation \(\l x - a \r^3 - 3b^2 \l x - a \r + c = 0\) will have 3 distinct roots if \(-2b^3 < c < 2b^3\), they will all be positive if the \(y(0) < 0\) and \(a+b > 0\) (ie the first turning point is in the first quadrant, ie \(-a^3+3b^2a+c < 0, a+b>0\). \begin{align*} 2x^3 - 9x^2 + 7x - 1 &= 2(x^3-\frac92x^2+\frac72 x-\frac12) \\ &= 2((x-\frac{3}{2})^3-\frac{27}{4}x+\frac{27}{8}+\frac72x-\frac12) \\ &= 2((x-\frac{3}{2})^3-\frac{13}{4}x+\frac{23}{8}) \\ &= 2((x-\frac{3}{2})^3-\frac{13}{4}(x-\frac{3}{2})-\frac{39}{8}+\frac{23}{8}) \\ &= 2((x-\frac{3}{2})^3-\frac{13}{4}(x-\frac{3}{2})-2) \\ \end{align*} Therefore in our notation \(a = \frac32, b = \sqrt{13/12}, c = 2\). Clearly \(a+b > 0\), so we need to check \(|c| < 2b^3\) which is clearly true as \(b > 1\). Finally we need to check \(y(0) = -1\), so all conditions are satisfied and there are 3 distinct roots.

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Sketch the graph of the function \([x/N]\), for \(0 < x < 2N\), where the notation \([y]\) means the integer part of \(y\). (Thus \([2.9] = 2\), \ \([4]=4\).)

1. Prove that \[ \sum_{k=1}^{2N} (-1)^{[k/N]} k = 2N-N^2. \]
2. Let \[ S_N = \sum_{k=1}^{2N} (-1)^{[k/N]} 2^{-k}. \] Find \(S_N\) in terms of \(N\) and determine the limit of \(S_N\) as \(N\to\infty\).

The function \(\f(x)\) is defined by $$ \f(x) = \frac{x( x - 2 )(x-a)}{ x^2 - 1}. $$ Prove algebraically that the line \(y = x + c\) intersects the curve \(y = \f ( x )\) if \(\vert a \vert \ge1\), but there are values of \(c\) for which there are no points of intersection if \(\vert a \vert <1\). Find the equation of the oblique asymptote of the curve \(y=\f(x)\). Sketch the graph in the two cases

1. \(a<-1\)
2. \(-1 < a < -\frac12\)

The curve \(C\) has equation $$ y = \frac x {\sqrt{x^2-2x+a}}\; , $$ where the square root is positive. Show that, if \(a>1\), then \(C\) has exactly one stationary point. Sketch \(C\) when (i) \(a=2\) and (ii) \(a=1\).

Show Solution

\begin{align*} && y &= \frac x {\sqrt{x^2-2x+a}} \\ && y' &= \frac{\sqrt{x^2-2x+a} - \frac{x(x-1)}{\sqrt{x^2-2x+a}}}{x^2-2x+a} \\ &&&= \frac{-x+a}{(x^2-2x+a)^{3/2}} \end{align*} Since the denominator is always positive, the only stationary point is when \(x = a\)

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Let $$y^2=x^2(a^2-x^2),$$ where \(a\) is a real constant. Find, in terms of \(a\), the maximum and minimum values of \(y\). Sketch carefully on the same axes the graphs of \(y\) in the cases \(a=1\) and \(a=2\).

Show Solution

\begin{align*} && y^2 &= x^2a^2-x^2 \\ &&&= \frac{a^4}{4} -\left ( x^2 -\frac{a^2}{2} \right)^2 \end{align*} Therefore the maximum and minimum values of \(y\) are \(\pm \frac{a^2}2\)

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`24 Hour Spares' stocks a small, widely used and cheap component. Every \(T\) hours \(X\) units arrive by lorry from the wholesaler, for which the owner pays a total \(\pounds (a+qX)\). It costs the owner \(\pounds b\) per hour to store one unit. If she has the units in stock she expects to sell \(r\) units per hour at \(\pounds(p+q)\) per unit. The other running costs of her business remain at \(\pounds c\) pounds an hour irrespective of whether she has stock or not. (All of the quantities \(T,X,a,b,r,q,p\) and \(c\) are greater than 0.) Explain why she should take \(X\leqslant rT\). Given that the process may be assumed continuous (the items are very small and she sells many each hour), sketch \(S(t)\) the amount of stock remaining as a function of \(t\) the time from the last delivery. Compute the total profit over each period of \(T\) hours. Show that, if \(T\) is fixed with \(T\geqslant p/b\), the business can be made profitable if \[ p^{2}>2\frac{(a+cT)b}{r}. \]

1. Show that the equation \[ (x-1)^{4}+(x+1)^{4}=c \] has exactly two real roots if \(c>2,\) one root if \(c=2\) and no roots if \(c<2\).
2. How many real roots does the equation \(\left(x-3\right)^{4}+\left(x-1\right)^{4}=c\) have?
3. How many real roots does the equation \(\left|x-3\right|+\left|x-1\right|=c\) have?
4. How many real roots does the equation \(\left(x-3\right)^{3}+\left(x-1\right)^{3}=c\) have?

Sketch the curve \[ \mathrm{f}(x)=x^{3}+Ax^{2}+B \] first in the case \(A>0\) and \(B>0\), and then in the case \(A<0\) and \(B>0.\) Show that the equation \[ x^{3}+ax^{2}+b=0, \] where \(a\) and \(b\) are real, will have three distinct real roots if \[ 27b^{2}+3a^{3}b<0, \] but will have fewer than three if \[ 27b^{2}+4a^{3}b<0. \]

Let \(N=10^{100}.\) The graph of \[ \mathrm{f}(x)=\frac{x^{N}}{1+x^{N}}+2 \] for \(-3\leqslant x\leqslant3\) is sketched in the following diagram. \noindent

Sketch the curves given by \[ y=x^{3}-2bx^{2}+c^{2}x, \] where \(b\) and \(c\) are non-negative, in the cases: \begin{questionparts} \item \(2b < c\sqrt{3}\) \item \(2b=c\sqrt{3}\neq0\) \item \(c\sqrt{3} < 2b < 2c\), \item \(b=c\neq0\) \item \(b > c > 0\), \item \(c=0,b\neq0\) \item \(c=b=0\). \end{questionpart} Sketch also the curves given by \(y^{2}=x^{3}-2bx^{2}+c^{2}x\) in the cases \((i), (v)\) and \((vii)\).

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\((i)\)

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\((v)\)

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\((vii)\)

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Sketch the curve whose cartesian equation is \[ y=\frac{2x(x^{2}-5)}{x^{2}-4}, \] and give the equations of the asymptotes and of the tangent to the curve at the origin. Hence, or otherwise, determine (giving reasons) the number of real roots of the following equations:

1. \(4x^{2}(x^{2}-5)=(5x-2)(x^{2}-4)\);
2. \(4x^{2}(x^{2}-5)^{2}=(x^{2}-4)^{2}(x^{2}+1)\);
3. \(4z^{2}(z-5)^{2}=(z-4)^{2}(z+1)\).

Show Solution

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The gradient at the origin is \(\frac{5}{2}\) which we can observe by looking at the taylor series

1. \begin{align*} && 4x^{2}(x^{2}-5)&=(5x-2)(x^{2}-4) \\ && \frac{2x(x^2-5)}{x^2-4} &= \frac{5x-2}{2x} = \frac52 -\frac1x \end{align*} Therefore it will have two roots as the hyperbola will intersect our graph in two places. (In the upper left and lower right quadrants).
2. \begin{align*} && 4x^{2}(x^{2}-5)^{2}&=(x^{2}-4)^{2}(x^{2}+1) \\ && \frac{2x(x^2-5)}{x^2-4} &= \frac{(x^2-4)(x^2+1)}{2x} \end{align*}
   

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   No solutions \begin{align*} && 4z^{2}(z-5)^{2}&=(z-4)^{2}(z+1) \\ && \frac{2z(z^2-5)}{z^2-4} &= \frac{(z+5)(z-4)(z+1)}{(z-5)(z+4)} \end{align*}
   

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   5 solutions

The function \(\mathrm{f}\) is defined by \[ \mathrm{f}(x)=\frac{\left(x-a\right)\left(x-b\right)}{\left(x-c\right)\left(x-d\right)}\qquad\left(x\neq c,\ x\neq d\right), \] where \(a,b,c\) and \(d\) are real and distinct, and \(a+b\neq c+d\). Show that \[ \frac{x\mathrm{f}'(x)}{\mathrm{f}(x)}=\left(1-\frac{a}{x}\right)^{-1}+\left(1-\frac{b}{x}\right)^{-1}-\left(1-\frac{c}{x}\right)^{-1}-\left(1-\frac{d}{x}\right)^{-1}, \] \((x\neq0,x\neq a,x\neq b)\) and deduce that when \(\left|x\right|\) is much larger than each of \(\left|a\right|,\left|b\right|,\left|c\right|\) and \(\left|d\right|,\) the gradient of \(\mathrm{f}(x)\) has the same sign as \((a+b-c-d).\) It is given that there is a real value of real value of \(x\) for which \(\mathrm{f}(x)\) takes the real value \(z\) if and only if \[ [\left(c-d\right)^{2}z+\left(a-c\right)\left(b-d\right)+\left(a-d\right)\left(b-c\right)]^{2}\geqslant4\left(a-c\right)\left(b-d\right)\left(a-d\right)\left(b-c\right). \] Describe briefly a method by which this result could be proved, but do not attempt to prove it. Given that \(a < b\) and \(a < c < d\), make sketches of the graph of \(\mathrm{f}\) in the four distinct cases which arise, indicating the cases for which the range of \(\mathrm{f}\) is not the whole of \(\mathbb{R}.\)

Show Solution

Notice that \(\ln f(x) = \ln (x - a) + \ln (x-b) - \ln (x-c) - \ln (x-d)\) therefore: \begin{align*} \frac{\d}{\d x}: && \frac{f'(x)}{f(x)} &= (x-a)^{-1}+(x-b)^{-1}-(x-c)^{-1} - (x-d)^{-1} \\ &&&= \frac{1}{x} \left ( (1-\frac{a}{x})^{-1}+(1-\frac{b}{x})^{-1}-(1-\frac{c}{x})^{-1} - (1-\frac{d}{x})^{-1}\right) \end{align*} Multiplying by \(x\) gives the desired result. When \(|x|\) is very large then: \begin{align*} \frac{x f'(x)}{f(x)} &\approx 1 + \frac{a}{x} + o(\frac{1}{x^2})+ 1 + \frac{b}{x} + o(\frac{1}{x^2})-(1 + \frac{c}{x} + o(\frac{1}{x^2}))-(1 + \frac{d}{x} + o(\frac{1}{x^2})) \\ &= \frac{a+b-c-d}{x} + o(x^{-2}) \end{align*} Dividing by \(x\) we obtain \(\frac{f'(x)}{f(x)} \approx \frac{a+b-c-d}{x^2} + o(x^{-3})\) if \(|x|\) is sufficiently large this will be dominated by the \(\frac{a+b-c-d}{x^2}\) term which will have the same sign as \((a+b-c-d)\). When \(|x|\) is very large all of the brackets will have the same sign, and therefore \(f(x)\) will be positive, and so \(f'(x)\) must have the same sign as \(a+b-c-d\). To prove this result, we could set \(f(x) = k\) and rearrange to form a quadratic in \(x\). We could then check the discriminant is non-zero. Case 1: \(a < c < d < b\) and \(a+b > c+d \Rightarrow\) not all values reached and approx asymtope from below on the right and above on the left.

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Case 2: \(a < c < d < b\) and \(a + b < c + d \Rightarrow\) not all values hit, but approach from above on the right and below on the left

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Case 2: \(a < c < b < d \Rightarrow\) all values hit

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Case 3: \(a < b < c < d \Rightarrow \) not all values hit

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Sketch the graph of \(8y=x^{3}-12x\) for \(-4\leqslant x\leqslant4\), marking the coordinates of the turning points. Similarly marking the turning points, sketch the corresponding graphs in the \((X,Y)\)-plane, if \begin{alignat*}{3} \rm{(a)} & \quad & & X=\tfrac{1}{2}x, & \qquad & Y=y,\\ \rm{(b)} & & & X=x, & & Y=\tfrac{1}{2}y,\\ \rm{(c)} & & & X=\tfrac{1}{2}x+1, & & Y=y,\\ \rm{(d)} & & & X=x, & & Y=\tfrac{1}{2}y+1. \end{alignat*} Find values for \(a,b,c,d\) such that, if \(X=ax+b,\) \(Y=cy+d\), then the graph in the \((X,Y)\)-plane corresponding to \(8y=x^{3}-12x\) has turning points at \((X,Y)=(0,0)\) and \((X,Y)=(1,1)\).

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\(8\frac{\d y}{\d x} = 3(x^2-4)\) so the turning points are at \((\pm 2, \mp 2)\)

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We need either \begin{align*} && \begin{cases} -2a+b &= 0 \\ 2c+d &= 0 \\ 2a+b &= 1 \\ -2c+d &= 1 \end{cases} && \text{ or } && \begin{cases} -2a+b &= 1 \\ 2c+d &= 1 \\ 2a+b &= 0 \\ -2c+d &= 0 \end{cases} \\ \Rightarrow && \begin{cases} -2a+b &= 0 \\ 2a+b &= 1 \\ 2c+d &= 0 \\ -2c+d &= 1 \end{cases} && \text{ or } && \begin{cases} -2a+b &= 1 \\ 2a+b &= 0 \\ 2c+d &= 1 \\ -2c+d &= 0 \end{cases}\\ \Rightarrow && \begin{cases} (a,b) = (\frac14,\frac12) \\ (c,d) = (-\frac14, \frac12)\end{cases} && \text{ or } && \begin{cases} (a,b) = (-\frac14,\frac12) \\ (c,d) = (\frac14, \frac12)\end{cases} \end{align*} So either \(X = \frac14 x + \frac12, Y = -\frac14 y + \frac12\) or \(X = -\frac14x + \frac12, Y = \frac14y + \frac12\)

For any two real numbers \(x_1\) and \(x_2\), show that $$|x_1 + x_2| \leq |x_1| + |x_2|.$$ Show further that, for any real numbers \(x_1, x_2, \ldots, x_n\), $$|x_1 + x_2 + \cdots + x_n| \leq |x_1| + |x_2| + \cdots + |x_n|.$$

1. The polynomial f is defined by $$f(x) = 1 + a_1 x + a_2 x^2 + \cdots + a_{n-1} x^{n-1} + x^n$$ where the coefficients are real and satisfy \(|a_i| \leq A\) for \(i = 1, 2, \ldots, n-1\), where \(A \geq 1\).
   1. If \(|x| < 1\), show that $$|f(x) - 1| \leq \frac{A|x|}{1 - |x|}.$$
   2. Let \(\omega\) be a real root of f, so that \(f(\omega) = 0\). In the case \(|\omega| < 1\), show that $$\frac{1}{1 + A} \leq |\omega| \leq 1 + A. \quad (*)$$
   3. Show further that the inequalities \((*)\) also hold if \(|\omega| \geq 1\).
2. Find the integer root or roots of the quintic equation $$135x^5 - 135x^4 - 100x^3 - 91x^2 - 126x + 135 = 0.$$

1. Find integers \(m\) and \(n\) such that $$\sqrt{3+2\sqrt{2}} = m + n\sqrt{2}.$$
2. Let \(f(x) = x^4 - 10x^2 + 12x - 2\). Given that the equation \(f(x) = 0\) has four real roots, explain why \(f(x)\) can be written in the form $$f(x)=(x^2 + sx + p)(x^2 - sx + q)$$ for some real constants \(s\), \(p\) and \(q\), and find three equations for \(s\), \(p\) and \(q\). Show that $$s^2(s^2 - 10)^2 + 8s^2 - 144 = 0$$ and find the three possible values of \(s^2\). Use the smallest of these values of \(s^2\) to solve completely the equation \(f(x) = 0\), simplifying your answers as far as you can.

Show that, if \(k\) is a root of the quartic equation \[ x^4 + ax^3 + bx^2 + ax + 1 = 0\,, \tag{\(*\)} \] then \(k^{-1}\) is a root. You are now given that \(a\) and \(b\) in \((*)\) are both real and are such that the roots are all real.

1. Write down all the values of \(a\) and \(b\) for which \((*)\) has only one distinct root.
2. Given that \((*)\) has exactly three distinct roots, show that either \(b=2a-2\) or \(b=-2a-2\,\).
3. Solve \((*)\) in the case \(b= 2 a -2\,\), giving your solutions in terms of \(a\).

1. Write down the most general polynomial of degree 4 that leaves a remainder of 1 when divided by any of \(x-1\,\), \(x-2\,\), \(x-3\,\) or \(x-4\,\).
2. The polynomial \(\P(x)\) has degree \(N\), where \(N\ge1\,\), and satisfies \[ \P(1) = \P(2) = \cdots = \P(N) =1\,. \] Show that \(\P(N+1) \ne 1\,\). Given that \(\P(N+1)= 2\,\), find \(\P(N+r)\) where \(r\) is a positive integer. Find a positive integer \(r\), independent of \(N\), such that \(\P(N+r) = N+r\,\).
3. The polynomial \({\rm S}(x)\) has degree 4. It has integer coefficients and the coefficient of \(x^4\) is 1. It satisfies \[ {\rm S}(a) = {\rm S}(b) = {\rm S}(c) = {\rm S}(d) = 2001\,, \] where \(a\), \(b\), \(c\) and \(d\) are distinct (not necessarily positive) integers.
   • [{\bf (a)}] Show that there is no integer \(e\) such that \({\rm S}(e) = 2018\,\). \medskip
   • [{\bf (b)}] Find the number of ways the (distinct) integers \(a\), \(b\), \(c\) and \(d\) can be chosen such that \({\rm S}(0) = 2017\) and \(a < b< c< d\,\).

Use the factor theorem to show that \(a+b-c\) is a factor of \[ (a+b+c)^3 -6(a+b+c)(a^2+b^2+c^2) +8(a^3+b^3+c^3) \,. \tag{\(*\)} \] Hence factorise (\(*\)) completely.

1. Use the result above to solve the equation \[ (x+1)^3 -3 (x+1)(2x^2 +5) +2(4x^3+13)=0\,. \]
2. By setting \(d+e=c\), or otherwise, show that \((a+b-d-e)\) is a factor of \[ (a+b+d+e)^3 -6(a+b+d+e)(a^2+b^2+d^2+e^2) +8(a^3+b^3+d^3+e^3) \, \] and factorise this expression completely. Hence solve the equation \[ (x+6)^3 - 6(x+6)(x^2+14) +8(x^3+36)=0\,. \]

1. For \(n=1\), \(2\), \(3\) and \(4\), the functions \(\p_n\) and \(\q_n\) are defined by \[ \p_n(x) = (x+1)^{2n} - (2n+1)x (x^2+x+1)^{n-1} \] and \[ \q_n(x) = \frac{x^{2n+1}+1}{x+1} \ \ \ \ \ \ \ \ \ \ \ \ (x\ne -1) \,. \ \ \ \ \ \ \ \ \ \ \] Show that \(\p_n(x)\equiv \q_n(x)\) (for \(x\ne-1\)) in the cases \(n=1\), \(n=2\) and \(n=3\). Show also that this does not hold in the case \(n=4\).
2. Using results from part (i):
   • [\bf (a)] express \( \ \dfrac {300^3 +1}{301}\,\) as the product of two factors (neither of which is 1);
   • [\bf (b)] express \( \ \dfrac {7^{49}+1}{7^7+1}\,\) as the product of two factors (neither of which is 1), each written in terms of various powers of 7 which you should not attempt to calculate explicitly.

1. Given that the cubic equation \(x^3+3ax^2 + 3bx +c=0\) has three distinct real roots and \(c<0\), show with the help of sketches that either exactly one of the roots is positive or all three of the roots are positive.
2. Given that the equation \(x^3 +3ax^2+3bx+c=0\) has three distinct real positive roots show that \begin{equation*} a^2>b>0, \ \ \ \ a<0, \ \ \ \ c<0\,. \tag{\(*\)} \end{equation*} \noindent[{\bf Hint}: Consider the turning points.]
3. Given that the equation \(x^3 +3ax^2+3bx+c=0\) has three distinct real roots and that \begin{equation*} ab<0, \ \ \ \ c>0\,, \end{equation*} determine, with the help of sketches, the signs of the roots.
4. Show by means of an explicit example (giving values for \(a\), \(b\) and \(c\)) that it is possible for the conditions (\(*\)) to be satisfied even though the corresponding cubic equation has only one real root.

If \(\p(x)\) and \(\q(x)\) are polynomials of degree \(m\) and \(n\), respectively, what is the degree of \(\p(\q(x))\)?

1. The polynomial \(\p(x)\) satisfies \[ \p(\p(\p(x)))- 3 \p(x)= -2x\, \] for all \(x\). Explain carefully why \(\p(x)\) must be of degree 1, and find all polynomials that satisfy this equation.
2. Find all polynomials that satisfy \[ 2\p(\p(x)) +3 [\p(x)]^2 -4\p(x) =x^4 \] for all \(x\).

The polynomial \(\f(x)\) is defined by \[ \f(x) = x^n + a_{\low{n-1}}x^{n-1} + \cdots + a_{\low2} x^2+ a_{\low1} x + a_{\low0}\,, \] where \(n\ge2\) and the coefficients \(a_{\low0}\), \(\ldots,\) \(a_{\low{n-1}}\) are integers, with \(a_0\ne0\). Suppose that the equation \(\f(x)=0\) has a rational root \(p/q\), where \(p\) and \(q\) are integers with no common factor greater than \(1\), and \(q>0\). By considering \(q^{n-1}\f(p/q)\), find the value of \(q\) and deduce that any rational root of the equation \(\f(x)=0\) must be an integer.

1. Show that the \(n\)th root of \(2\) is irrational for \(n\ge2\).
2. Show that the cubic equation \[ x^3- x +1 =0 \] has no rational roots.
3. Show that the polynomial equation \[ x^n- 5x +7 =0 \] has no rational roots for \(n\ge2\).

1. The number \(\alpha\) is a common root of the equations \(x^2 +ax +b=0\) and \(x^2+cx+d=0\) (that is, \(\alpha\) satisfies both equations). Given that \(a\ne c\), show that \[ \alpha =- \frac{b-d}{a-c}\,. \] Hence, or otherwise, show that the equations have at least one common root if and only if \[ (b-d)^2 -a(b-d)(a-c) + b(a-c)^2 =0\,. \] Does this result still hold if the condition \(a\ne c\) is not imposed?
2. Show that the equations \(x^2+ax+b=0\) and \(x^3+(a+1)x^2+qx+r=0\) have at least one common root if and only if \[ (b-r)^2-a(b-r)(a+b-q) +b(a+b-q)^2=0\,. \] Hence, or otherwise, find the values of \(b\) for which the equations \(2x^2+5 x+2 b=0\) and \(2x^3+7x^2+5x+1=0\) have at least one common root.

1. By considering the positions of its turning points, show that the curve with equation \[ y=x^3-3qx-q(1+q)\,, \] where \(q>0\) and \(q\ne1\), crosses the \(x\)-axis once only.
2. Given that \(x\) satisfies the cubic equation \[ x^3-3qx-q(1+q)=0\,, \] and that \[ x=u+q/u\,, \] obtain a quadratic equation satisfied by \(u^3\). Hence find the real root of the cubic equation in the case \(q>0\), \(q\ne1\).
3. The quadratic equation \[ t^2 -pt +q =0\, \] has roots \(\alpha \) and \(\beta\). Show that \[ \alpha^3+\beta^3 = p^3 -3qp\,. \] It is given that one of these roots is the square of the other. By considering the expression \((\alpha^2 -\beta)(\beta^2-\alpha)\), find a relationship between \(p\) and \(q\). Given further that \(q>0\), \(q\ne1\) and \(p\) is real, determine the value of \(p\) in terms of \(q\).

The points \(S\), \(T\), \(U\) and \(V\) have coordinates \((s,ms)\), \((t,mt)\), \((u,nu)\) and \((v,nv)\), respectively. The lines \(SV\) and \(UT\) meet the line \(y=0\) at the points with coordinates \((p,0)\) and \((q,0)\), respectively. Show that \[ p = \frac{(m-n)sv}{ms-nv}\,, \] and write down a similar expression for \(q\). Given that \(S\) and \(T\) lie on the circle \(x^2 + (y-c)^2 = r^2\), find a quadratic equation satisfied by \(s\) and by \(t\), and hence determine \(st\) and \(s+t\) in terms of \(m\), \(c\) and \(r\). Given that \(S\), \(T\), \(U\) and \(V\) lie on the above circle, show that \(p+q=0\).

The polynomial \(\p(x)\) is of degree 9 and \(\p(x)-1\) is exactly divisible by \((x-1)^5\).

1. Find the value of \(\p(1)\).
2. Show that \(\p'(x)\) is exactly divisible by \((x-1)^4\).
3. Given also that \(\p(x)+1\) is exactly divisible by \((x+1)^5\), find \(\p(x)\).

A curve is given by the equation \[ y = ax^3 - 6ax^2+ \left( 12a + 12 \right)x - \left( 8a + 16 \right)\,, \tag{\(*\)} \] where \(a\) is a real number. Show that this curve touches the curve with equation \[ y=x^3 \tag{\(**\)} \] at \(\left( 2 \, , \, 8 \right)\). Determine the coordinates of any other point of intersection of the two curves.

1. Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = 2\).
2. Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = 1\).
3. Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = -2\).

1. Given that \(x^2 - y^2 = \left( x - y \right)^3\) and that \(x-y = d\) (where \(d \neq 0\)), express each of \(x\) and \(y\) in terms of \(d\). Hence find a pair of integers \(m\) and \(n\) satisfying \(m-n = \left( \sqrt {m} - \sqrt{n} \right)^3\) where \(m > n > 100\).
2. Given that \(x^3 - y^3 = \left( x - y \right)^4\) and that \(x-y = d\) (where \(d \neq 0\)), show that \(3xy = d^3 - d^2\). Hence show that \[ 2x = d \pm d \sqrt {\frac{4d-1 }{3}} \] and determine a pair of distinct positive integers \(m\) and \(n\) such that \(m^3 - n^3 = \left( m - n \right)^4\).

Show that \(x^3-3xbc + b^3 + c^3\) can be written in the form \(\left( x+ b+ c \right) {\rm Q}( x)\), where \({\rm Q}( x )\) is a quadratic expression. Show that \(2{\rm Q }( x )\) can be written as the sum of three expressions, each of which is a perfect square. It is given that the equations \(ay^2 + by + c =0\) and \(by^2 + cy + a = 0\) have a common root \(k\). The coefficients \(a\), \(b\) and \(c\) are real, \(a\) and \(b\) are both non-zero, and \(ac \neq b^2\). Show that \[ \left( ac - b^2 \right) k = bc - a^2 \] and determine a similar expression involving \(k^2\). Hence show that \[ \left( ac - b^2 \right) \left(ab-c^2 \right) = \left( bc - a^2 \right)^2 \] and that \( a^3 -3abc + b^3 +c^3 = 0\,\). Deduce that either \(k=1\) or the two equations are identical.

1. Show that \(x-3\) is a factor of \begin{equation} x^3-5x^2+2x^2y+xy^2-8xy-3y^2+6x+6y \;. \tag{\(*\)} \end{equation} Express (\( * \)) in the form \((x-3)(x+ay+b)(x+cy+d)\) where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.
2. Factorise \(6y^3-y^2-21y+2x^2+12x-4xy+x^2y-5xy^2+10\) into three linear factors.

Let \[ \f(x) = x^n + a_1 x^{n-1} + \cdots + a_n\;, \] where \(a_1\), \(a_2\), \(\ldots\), \(a_n\) are given numbers. It is given that \(\f(x)\) can be written in the form \[ \f(x) = (x+k_1)(x+k_2)\cdots(x+k_n)\;. \] By considering \(\f(0)\), or otherwise, show that \(k_1k_2 \ldots k_n =a_n\). Show also that $$(k_1+1)(k_2+1)\cdots(k_n+1)= 1+a_1+a_2+\cdots+a_n$$ and give a corresponding result for \((k_1-1)(k_2-1)\cdots(k_n-1)\). Find the roots of the equation \[ x^4 +22x^3 +172x^2 +552x+576=0\;, \] given that they are all integers.

Sketch, without calculating the stationary points, the graph of the function \(\f(x)\) given by \[ \f(x) = (x-p)(x-q)(x-r)\;, \] where \(p < q < r\). By considering the quadratic equation \(\f'(x)=0\), or otherwise, show that \[ (p+q+r)^2 > 3(qr+rp+pq)\;. \] By considering \((x^2+gx+h)(x-k)\), or otherwise, show that \(g^2>4h\,\) is a sufficient condition but not a necessary condition for the inequality \[ (g-k)^2>3(h-gk) \] to hold.

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Since there are two turning points the derivative (a quadratic) has two distinct real roots. \begin{align*} && f'(x) &= 3x^2-2(p+q+r)x+(pq+qr+rp) \\ && 0 &< \Delta = 4(p+q+r)^2 - 4\cdot 3(pq+qr+rp) \\ \Rightarrow && (p+q+r)^2 &> 3(pq+qr+rp) \end{align*} If \(g^2 > 4h\) then \(p(x) = (x^2+gx+h)(x-k)\) has at least 2 real roots (possibly one repeated, and in particular it has two turning point, ie \begin{align*} && p'(x) &= (2x+g)(x-k)+(x^2+gx+h) \\ &&&= 3x^2+(2g-2k)x + (h-kg) \\ && 0 &< \Delta = 4(g-k)^2 - 4\cdot 3 (h-gk) \\ \Rightarrow && (g-k)^2 &> 3(h-gk) \end{align*} Pick \(g = h = 1\) and \(k = 1000\) then \((-999)^2 > 0 > 3(1-1000)\) so it is sufficient but not necessary.

Given that \[ x^4 + p x^2 + q x + r = ( x^2 - a x + b ) ( x^2 + a x + c ) , \] express \(p\), \(q\) and \(r\) in terms of \(a\), \(b\) and \(c\). Show also that \( a^2\) is a root of the cubic equation $$ u^3 + 2 p u^2 + ( p^2 - 4 r ) u - q^2 = 0 . $$ Explain why this equation always has a non-negative root, and verify that \(u = 9\) is a root in the case \(p = -1\), \(q = -6\), \(r = 15\) . Hence, or otherwise, express $$y^4 - 8 y^3 + 23 y^2 - 34 y + 39$$ as a product of two quadratic factors.

Prove that if \({(x-a)^{2}}\) is a factor of the polynomial \(\p(x)\), then \(\p'(a)=0\). Prove a corresponding result if \((x-a)^4\) is a factor of \(\p(x).\) Given that the polynomial $$ x^6+4x^5-5x^4-40x^3-40x^2+32x+k $$ has a factor of the form \({(x-a)}^4\), find \(k\).

Consider the cubic equation \[ x^3-px^2+qx-r=0\;, \] where \(p\ne0\) and \(r\ne 0\).

1. If the three roots can be written in the form \(ak^{-1}\), \(a\) and \(ak\) for some constants \(a\) and \(k\), show that one root is \(q/p\) and that \(q^3 -rp^3=0\;.\)
2. If \(r=q^3/p^3\;\), show that \(q/p\) is a root and that the product of the other two roots is \((q/p)^2\). Deduce that the roots are in geometric progression.
3. Find a necessary and sufficient condition involving \(p\), \(q\) and \(r\) for the roots to be in arithmetic progression.

Show that, when the polynomial \({\rm p} (x)\) is divided by \((x-a)\), where \(a\) is a real number, the remainder is \({\rm p}(a)\).

1. When the polynomial \({\rm p}(x)\) is divided by \(x-1,\,x-2,\,x-3\) the remainders are 3,1,5 respectively. Given that $${\rm p}(x)=(x-1)(x-2)(x-3){\rm q}(x)+{\rm r} (x),$$ where \({\rm q}(x)\) and \({\rm r}(x)\) are polynomials with \({\rm r}(x)\) having degree less than three, find \({\rm r}(x)\).
2. Find a polynomial \({\rm P}(x)\) of degree \(n+1\), where \(n\) is a given positive integer, such that for each integer \(a\) satisfying \(0\le a\le n\), the remainder when \({\rm P}_n(x)\) is divided by \(x-a\) is \(a\).

Let \(\mathrm{p}_{0}(x)=(1-x)(1-x^{2})(1-x^{4}).\) Show that \((1-x)^{3}\) is a factor of \(\mathrm{p}_{0}(x).\) If \(\mathrm{p}_{1}(x)=x\mathrm{p}_{0}'(x)\) show, by considering factors of the polynomials involved, that \(\mathrm{p}_{0}'(1)=0\) and \(\mathrm{p}_{1}'(1)=0.\) By writing \(\mathrm{p}_{0}(x)\) in the form \[ \mathrm{p}_{0}(x)=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+c_{5}x^{5}+c_{6}x^{6}+c_{7}x^{7}, \] deduce that \begin{alignat*}{2} 1+2+4+7 & \quad=\quad & & 3+5+6\\ 1^{2}+2^{2}+4^{2}+7^{2} & \quad=\quad & & 3^{2}+5^{2}+6^{2}. \end{alignat*} Show that we can write the integers \(1,2,\ldots,15\) in some order as \(a_{1},a_{2},\ldots,a_{15}\) in such a way that \[ a_{1}^{r}+a_{2}^{r}+\cdots+a_{8}^{r}=a_{9}^{r}+a_{10}^{r}+\cdots+a_{15}^{r} \] for \(r=1,2,3.\)

1. By considering the binomial expansion of \((1+x)^{2m+1}\), prove that \[ \binom{ 2m \! +\! 1}{ m} < 2^{2m}\,, \] for any positive integer \(m\).
2. For any positive integers \(r\) and \(s\) with \(r< s\), \(P_{r,s}\) is defined as follows: \(P_{r,s}\) is the product of all the prime numbers greater than \(r\) and less than or equal to \(s\), if there are any such primes numbers; if there are no such primes numbers, then \(P_{r,s}=1\,\). For example, \(P_{3,7}=35\), \(P_{7,10}=1\) and \(P_{14,18}=17\). Show that, for any positive integer \(m\), \(P_{m+1\,,\, 2m+1} \) divides \(\displaystyle \binom{ 2m \! +\! 1}{ m} \,,\) and deduce that \[ P_{m+1\,,\, 2m+1} < 2^{2m} \,. \]
3. Show that, if \(P_{1,k} < 4^k\) for \(k = 2\), \(3\), \(\ldots\), \(2m\), then \( P_{1,2m+1} < 4^{2m+1}\,\).
4. Prove that \(\P_{1,n} < 4^n\) for \(n\ge2\).

For positive integers \(n\), \(a\) and \(b\), the integer \(c_r\) (\(0\le r\le n\)) is defined to be the coefficient of~\(x^r\) in the expansion in powers of \(x\) of \((a+bx)^n\). Write down an expression for \(c_r\) in terms of \(r\), \(n\), \(a\) and~\(b\). For given \(n\), \(a\) and \(b\), let \(m\)~denote a value of~\(r\) for which \(c_r\)~is greatest (that is, \(c_m \ge c_r\) for \(0\le r\le n\)). Show that \[ \frac{b(n+1)}{a+b} - 1 \le m \le \frac {b(n+1)}{a+b} \,. \] Deduce that \(m\) is either a unique integer or one of two consecutive integers. Let \(\.G(n,a,b)\) denote the unique value of~\(m\) (if there is one) or the larger of the two possible values of~\(m\).

1. Evaluate \(\.G(9,1,3)\) and \(\.G(9,2,3)\).
2. For any positive integer \(k\), find \(\.G(2k,a,a)\) and \(\.G(2k-1,a,a)\) in terms of~\(k\).
3. For fixed \(n\) and \(b\), determine a value of~\(a\) for which \(\.G(n,a,b)\) is greatest.
4. For fixed~\(n\), find the greatest possible value of \(\.G(n,1,b)\). For which values of~\(b\) is this greatest value achieved?

By considering the coefficient of \(x^r\) in the series for \((1+x)(1+x)^n\), or otherwise, obtain the following relation between binomial coefficients: \[ \binom n r + \binom n {r-1} = \binom {n+1} r \qquad (1\le r\le n). \] The sequence of numbers \(B_0\), \(B_1\), \(B_2\), \(\ldots\) is defined by \[ B_{2m} = \sum_{j=0}^m \binom{2m-j}j \text{ and } B_{2m+1} = \sum_{k=0}^m \binom{2m+1-k}k . \] Show that \(B_{n+2} - B_{n+1} = B_{n}\,\) (\(n=0\), \(1\), \(2\), \(\ldots\,\)). What is the relation between the sequence \(B_0\), \(B_1\), \(B_2\), \(\ldots\) and the Fibonacci sequence \(F_0\), \(F_1\), \(F_2\), \(\ldots\) defined by \(F_0=0\), \(F_1=1\) and \(F_n = F_{n-1}+F_{n-2}\) for \(n\ge2\)?

By considering the expansion of \(\left(1+x\right)^{n}\) where \(n\) is a positive integer, or otherwise, show that:

1. \[\binom{n}{0}+\binom{n}1+\binom{n}2 +\cdots +\binom{n}n=2^{n} \]
2. \[\binom{n}{1}+2\binom{n}2+3\binom{n}3 +\cdots +n\binom{n}n=n2^{n-1} \]
3. \[\binom{n}{0}+\frac12\binom{n}1+\frac13\binom{n}2 +\cdots +\frac1{n+1}\binom{n}n=\frac1{n+1}(2^{n+1}-1) \]
4. \[\binom{n}{1}+2^2\binom{n}2+3^2\binom{n}3 +\cdots +n^2\binom{n}n=n(n+1)2^{n-2} \]

Show that the coefficient of \(x^{-12}\) in the expansion of \[ \left(x^{4}-\frac{1}{x^{2}}\right)^{5} \left(x-\frac{1}{x}\right)^{6} \] is \(-15\), and calculate the coefficient of \(x^2\). Hence, or otherwise, calculate the coefficients of \(x^4\) and \(x^{38}\) in the expansion of \[ (x^2-1)^{11}(x^4+x^2+1)^5. \]

By considering the expansions in powers of \(x\) of both sides of the identity $$ {(1+x)^n}{(1+x)^n}\equiv{(1+x)^{2n}}, $$ show that $$ \sum_{s=0}^n {n\choose s}^2 = {2n\choose n}, $$ where \(\displaystyle {n\choose s}= \frac{n!}{s!\,(n-s)!}\). By considering similar identities, or otherwise, show also that:

1. if \(n\) is an even integer, then \(\displaystyle \sum_{s=0}^n {{(-1)}^s}{n \choose s}^2= (-1)^{n/2}{n \choose n/2};\)
2. \(\displaystyle \sum\limits_{t=1}^ n 2t { n \choose t}^2 = n {2n\choose n} .\)

By considering the coefficient of \(x^{n}\) in the identity \((1-x)^{n}(1+x)^{n}=(1-x^{2})^{n},\) or otherwise, simplify \[ \binom{n}{0}^{2}-\binom{n}{1}^{2}+\binom{n}{2}^{2}-\binom{n}{3}^{2}+\cdots+(-1)^{n}\binom{n}{n}^{2} \] in the cases (i) when \(n\) is even, (ii) when \(n\) is odd.

Write down the binomial expansion of \((1+x)^{n}\), where \(n\) is a positive integer.

1. By substituting particular values of \(x\) in the above expression, or otherwise, show that, if \(n\) is an even positive integer, \[ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{n}=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\cdots+\binom{n}{n-1}=2^{n-1}. \]
2. Show that, if \(n\) is any positive integer, then \[ \binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\cdots+n\binom{n}{n}=n2^{n-1}. \]

1. The functions \(\mathrm{a, b, c}\) and \(\mathrm{d}\) are defined by
   • \({\rm a}(x) =x^2 \ \ \ \ (-\infty < x < \infty),\)
   • \({\rm b}(x) = \ln x \ \ \ \ (x > 0),\)
   • \({\rm c}(x) =2x \ \ \ \ (-\infty < x < \infty),\)
   • \({\rm d}(x)= \sqrt x \ \ \ \ (x\ge0) \,.\)
   Write down the following composite functions, giving the domain and range of each: \[ \rm cb, \quad ab, \quad da, \quad ad. \]
2. The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by
   • \(\f(x)= \sqrt{x^2-1\,} \ \ \ \ (\vert x \vert \ge 1),\)
   • \(\g(x) = \sqrt{x^2+1\,} \ \ \ \ (-\infty < x < \infty).\)
   Determine the composite functions \(\mathrm{fg}\) and \(\mathrm{gf}\), giving the domain and range of each.
3. Sketch the graphs of the functions \(\h\) and \({\rm k}\) defined by
   • \(\h(x) = x+\sqrt{x^2-1\,}\, \ \ \ \ ( x \ge1)\),
   • \({\rm k}(x) = x-\sqrt{x^2-1\,}\, \ \ \ \ (\vert x\vert \ge1),\)
   justifying the main features of the graphs, and giving the equations of any asymptotes. Determine the domain and range of the composite function \(\mathrm{kh}\).

The point \(P\) has coordinates \((x,y)\) which satisfy \[ x^2+y^2 + kxy +3x +y =0\,. \]

1. Sketch the locus of \(P\) in the case \(k=0\), giving the points of intersection with the coordinate axes.
2. By factorising \(3x^2 +3y^2 +10xy\), or otherwise, sketch the locus of \(P\) in the case \(k=\frac{10}{3}\,\), giving the points of intersection with the coordinate axes.
3. In the case \(k=2\), let \(Q\) be the point obtained by rotating \(P\) clockwise about the origin by an angle~\(\theta\), so that the coordinates \((X,Y)\) of \(Q\) are given by \[ X=x\cos\theta +y\sin\theta\,, \ \ \ \ Y= -x\sin\theta + y\cos\theta\,. \] Show that, for \(\theta =45^\circ\), the locus of \(Q\) is \( \sqrt2 Y= (\sqrt2 X+1 )^2 - 1 .\) Hence, or otherwise, sketch the locus of \(P\) in the case \(k=2\), giving the equation of the line of symmetry.

Let \(a_n\) be the coefficient of \(x^n\) in the series expansion, in ascending powers of \(x\), of \[\displaystyle \frac{1+x}{(1-x)^2(1+x^2)} \,, \] where \(\vert x \vert <1\,\). Show, using partial fractions, that either \(a_n =n+1\) or \(a_n = n+2\) according to the value of \(n\). Hence find a decimal approximation, to nine significant figures, for the fraction \( \displaystyle \frac{11\,000}{8181}\). \newline [You are not required to justify the accuracy of your approximation.]

The function \(\mathrm{f}\) is defined, for \(x\neq1\) and \(x\neq2\) by \[ \mathrm{f}(x)=\frac{1}{\left(x-1\right)\left(x-2\right)} \] Show that for \(\left|x\right|<1\) \[ \mathrm{f}(x)=\sum_{n=0}^{\infty}x^{n}-\frac{1}{2}\sum_{n=0}^{\infty}\left(\frac{x}{2}\right)^{n} \] and that for \(1<\left|x\right|<2\) \[ \mathrm{f}(x)=-\sum_{n=1}^{\infty}x^{-n}-\frac{1}{2}\sum_{n=0}^{\infty}\left(\frac{x}{2}\right)^{n} \] Find an expression for \(\mbox{f}(x)\) which is valid for \(\left|x\right|>2\).

In this question, the definition of \(\displaystyle\binom pq\) is taken to be \[ \binom pq = \begin{cases} \dfrac{p!}{q!(p-q)!} & \text{ if } p\ge q\ge0 \,,\\[4mm] 0 & \text{ otherwise } . \end{cases} \]

1. Write down the coefficient of \(x^n\) in the binomial expansion for \((1-x)^{-N}\), where \(N\) is a positive integer, and write down the expansion using the \(\Sigma\) summation notation. By considering $ (1-x)^{-1} (1-x)^{-N} \, ,$ where \(N\) is a positive integer, show that \[ \sum_{j=0}^n \binom { N+j -1}{j} = \binom{N+n}{n}\,. \]
2. Show that, for any positive integers \(m\), \(n\) and \(r\) with \(r\le m+n\), \[ \binom{m+n} r = \sum _{j=0}^r \binom m j \binom n {r-j} \,. \]
3. Show that, for any positive integers \(m\) and \(N\), \[ \sum_{j=0}^n(-1)^{j} \binom {N+m} {n-j} \binom {m+j-1}{j } = \displaystyle \binom N n . \]

Given an infinite sequence of numbers \(u_0\), \(u_1\), \(u_2\), \(\ldots\,\), we define the {\em generating function}, \(\f\), for the sequence by \[ \f(x) = u_0 + u_1x +u_2 x^2 +u_3 x^3 + \cdots \,. \] Issues of convergence can be ignored in this question.

1. Using the binomial series, show that the sequence given by \(u_n=n\,\) has generating function \(x(1-x)^{-2}\), and find the sequence that has generating function \(x(1-x)^{-3}\). Hence, or otherwise, find the generating function for the sequence \(u_n =n^2\). You should simplify your answer.
2. • [\bf (a)] The sequence \(u_0\), \(u_1\), \(u_2\), \(\ldots\,\) is determined by \(u_{n} = ku_{n-1}\) (\(n\ge1\)), where \(k\) is independent of \(n\), and \(u_0=a\). By summing the identity \(u_{n}x^n \equiv ku_{n-1}x^n\), or otherwise, show that the generating function, f, satisfies \[ \f(x) = a + kx \f(x) \,. \] Write down an expression for \(\f(x)\). \vspace{3mm}
   • [\bf (b)] The sequence \(u_0\), \(u_1\), \(u_2\), \(\ldots\,\) is determined by \(u_{n} = u_{n-1}+ u_{n-2}\) (\(n\ge2\)) and \(u_0=0\), \(u_1=1\). Obtain the generating function.

Write down the general term in the expansion in powers of \(x\) of \((1-x^6)^{-2}\,\).

1. Find the coefficient of \(x^{24}\) in the expansion in powers of \(x\) of \[ (1-x^6)^{-2} (1-x^3)^{-1}\,.\] Obtain also, and simplify, formulae for the coefficient of \(x^n\) in the different cases that arise.
2. Show that the coefficient of \(x^{24}\) in the expansion in powers of \(x\) of \[ (1-x^6)^{-2} (1-x^3)^{-1} (1-x)^{-1}\,\] is \(55\), and find the coefficients of \(x^{25}\) and \(x^{66}\).

Use the binomial expansion to show that the coefficient of \(x^r\) in the expansion of \((1-x)^{-3}\) is \(\frac12 (r+1)(r+2)\,\).

1. Show that the coefficient of \(x^r\) in the expansion of \[ \frac{1-x+2x^2}{(1-x)^3} \] is \(r^2+1\) and hence find the sum of the series \[ 1+\frac22+\frac54+\frac{10}8+\frac{17}{16}+\frac{26}{32}+\frac{37}{64} +\frac{50}{128}+ \cdots \,. \]
2. Find the sum of the series \[ 1+2+\frac94+2+\frac{25}{16}+\frac{9}{8}+\frac{49}{64} + \cdots \,. \]

1. Show that \(1.3.5.7. \;\ldots \;.(2n-1)=\dfrac {(2n)!}{2^n n!}\;\) and that, for $\vert x \vert < \frac14$, \[ \frac{1}{\sqrt{1-4x\;}\;} =1+\sum_{n=1}^\infty \frac {(2n)!}{(n!)^2} \, x^n \,. \]
2. By differentiating the above result, deduce that \[ \sum _{n=1}^\infty \frac{(2n)!}{n!\,(n-1)!} \left(\frac6{25}\right)^{\!\!n} = 60 \,. \]
3. Show that \[ \sum _{n=1}^\infty \frac{2^{n+1}(2n)!}{3^{2n}(n+1)!\,n!} = 1 \,. \]

In this question, you are not required to justify the accuracy of the approximations.

1. Write down the binomial expansion of \(\displaystyle \left( 1+\frac k {100} \right)^{\!\frac12}\)in ascending powers of \(k\), up to and including the \(k^3\) term.
   1. Use the value \(k=8\) to find an approximation to five decimal places for \(\sqrt{3}\,\).
   2. By choosing a suitable integer value of \(k\), find an approximation to five decimal places for \(\sqrt6\,\).
2. By considering the first two terms of the binomial expansion of \(\displaystyle \left( 1+\frac k {1000} \right)^{\!\frac13}\), show that \(\dfrac{3029}{2100}\) is an approximation to \(\sqrt[3]{3}\).

1. Write down the general term in the expansion in powers of \(x\) of \((1-x)^{-1}\), \((1-x)^{-2}\) and \((1-x)^{-3}\), where \(|x| <1\). Evaluate \(\displaystyle \sum_{n=1}^\infty n 2^{-n}\) and \(\displaystyle \sum_{n=1}^\infty n^22^{-n}\).
2. Show that $\displaystyle (1-x)^{-\frac12} = \sum_{n=0}^\infty \frac{(2n)!}{(n!)^2} \frac{x^n}{2^{2n}}\( , for \)|x|<1$. Evaluate \(\displaystyle \sum_{n=0}^\infty \frac{(2n)!} {(n!)^2 2^{2n}3^{n}} \) and \(\displaystyle \sum_{n=1}^\infty \frac{n(2n)!} {(n!)^2 2^{2n}3^{n}}\).

1. The function \(\f(x)\) is defined for \(\vert x \vert < \frac15\) by \[ \f(x) = \sum_{n=0}^\infty a_n x^n\;, \] where \(a_0=2\), \(a_1=7\) and \(a_n =7a_{n-1} - 10a_{n-2}\) for \(n\ge{2}\,\). Simplify \(\f(x) - 7x\f(x) + 10x^2\f(x)\,\), and hence show that \(\displaystyle\f(x) = {1\over 1-2x} + {1 \over 1-5x} \;\). Hence show that \(a_n=2^n + 5^n\,\).
2. The function \(\g(x)\) is defined for \(\vert x \vert < \frac13\) by \[ \g(x) = \sum_{n=0}^\infty b_n x^n \;, \] where \(b_0=5\,\), \(b_1 =10 \,\), \(b_2=40\,\), \(b_3=100\) and \(b_n = pb_{n-1} + qb_{n-2}\) for \(n\ge{2}\,\). Obtain an expression for \(\g(x)\) as the sum of two algebraic fractions and determine \(b_n\) in terms of \(n\).

Show that $\ds ^{2r} \! {\rm C}_r =\frac{1\times3\times\dots\times (2r-1)}{r!} \, \times 2^r \;, $ for \(r\ge1\,\).

1. Give the first four terms of the binomial series for \(\l 1 - p \r^{-\frac12}\). By choosing a suitable value for \(p\) in this series, or otherwise, show that $$ \displaystyle \sum_{r=0}^\infty \frac{ {\vphantom {\A}}^{2r} \! {\rm C}_r }{ 8^r} = \sqrt 2 \; .$$
2. Show that $$ \displaystyle \sum_{r=0}^\infty \frac{\l 2r + 1 \r \; {\vphantom{A}}^{2r} \! {\rm C} _r }{ 5^r} =\big( \sqrt 5\big)^3 \;. $$

1. In the binomial expansion of \((2x+1/x^{2})^{6}\;\) for \(x\ne0\), show that the term which is independent of \(x\) is \(240\). Find the term which is independent of \(x\) in the binomial expansion of \((ax^3+b/x^{2})^{5n}\,\).
2. Let \(\f(x) =(x^6+3x^5)^{1/2}\,\). By considering the expansion of \((1+3/x)^{1/2}\,\) show that the term which is independent of \(x\) in the expansion of \(\f(x)\) in powers of \(1/x\,\), for \( \vert x\vert>3\,\), is \(27/16\,\). Show that there is no term independent of \(x\,\) in the expansion of \(\f(x)\) in powers of \(x\,\), for \( \vert x\vert<3\,\).

Use the binomial expansion to obtain a polynomial of degree \(2\) which is a good approximation to \(\sqrt{1-x}\) when \(x\) is small.

1. By taking \(x=1/100\), show that \(\sqrt{11}\approx79599/24000\), and estimate, correct to 1 significant figure, the error in this approximation. (You may assume that the error is given approximately by the first neglected term in the binomial expansion.)
2. Find a rational number which approximates \(\sqrt{1111}\) with an error of about \(2 \times {10}^{-12}\).

Use the first four terms of the binomial expansion of \((1-1/50)^{1/2}\), writing \(1/50 = 2/100\) to simplify the calculation, to derive the approximation \(\sqrt 2 \approx 1.414214\). Calculate similarly an approximation to the cube root of 2 to six decimal places by considering \((1+N/125)^a\), where \(a\) and \(N\) are suitable numbers. [You need not justify the accuracy of your approximations.]

1. Find the coefficient of \(x^{6}\) in \[(1-2x+3x^{2}-4x^{3}+5x^{4})^{3}.\] You should set out your working clearly.
2. By considering the binomial expansions of \((1+x)^{-2}\) and \((1+x)^{-6}\), or otherwise, find the coefficient of \(x^{6}\) in \[(1-2x+3x^{2}-4x^{3}+5x^{4}-6x^{5}+7x^{6})^{3}.\]

Given that \(b>a>0\), find, by using the binomial theorem, coefficients \(c_{m}\) (\(m=0,1,2,\ldots\)) such that \[ \frac{1}{\left(1-ax\right)\left(1-bx\right)}=c_{0}+c_{1}x+c_{2}x^{2}+\ldots+c_{m}x^{m}+\cdots \] for \(b\left|x\right|<1\). Show that \[ c_{m}^{2}=\frac{a^{2m+2}-2(ab)^{m+1}+b^{2m+2}}{(a-b)^{2}}\,. \] Hence, or otherwise, show that \[ c_{0}^{2}+c_{1}^{2}x+c_{2}^{2}x^{2}+\cdots+c_{m}^{2}x^{m}+\cdots=\frac{1+abx}{\left(1-abx\right)\left(1-a^{2}x\right)\left(1-b^{2}x\right)}\,, \] for \(x\) in a suitable interval which you should determine.

1. 1. Show that if the complex number \(z\) satisfies the equation \[z^2 + |z + b| = a,\] where \(a\) and \(b\) are real numbers, then \(z\) must be either purely real or purely imaginary.
   2. Show that the equation \[z^2 + \left|z + \frac{5}{2}\right| = \frac{7}{2}\] has no purely imaginary roots.
   3. Show that the equation \[z^2 + \left|z + \frac{7}{2}\right| = \frac{5}{2}\] has no purely real roots.
   4. Show that, when \(\frac{1}{2} < b < \frac{3}{4}\), the equation \[z^2 + |z + b| = \frac{1}{2}\] will have at least one purely imaginary root and at least one purely real root.
2. Solve the equation \[z^3 + |z + 2|^2 = 4.\]

The point \(P\) in the Argand diagram is represented by the the complex number \(z\), which satisfies $$zz^* - az^* - a^*z + aa^* - r^2 = 0.$$ Here, \(r\) is a positive real number and \(r^2 \neq a^*a\). By writing \(|z - a|^2\) as \((z - a)(z - a)^*\), show that the locus of \(P\) is a circle, \(C\), the radius and the centre of which you should give.

1. The point \(Q\) is represented by \(\omega\), and is related to \(P\) by \(\omega = \frac{1}{z}\). Let \(C'\) be the locus of \(Q\). Show that \(C'\) is also a circle, and give its radius and centre. If \(C\) and \(C'\) are the same circle, show that $$(|a|^2 - r^2)^2 = 1$$ and that either \(a\) is real or \(a\) is imaginary. Give sketches to indicate the position of \(C\) in these two cases.
2. Suppose instead that the point \(Q\) is represented by \(\omega\), where \(\omega = \frac{1}{z^*}\). If the locus of \(Q\) is \(C\), is it the case that either \(a\) is real or \(a\) is imaginary?

1. The distinct points \(A\), \(Q\) and \(C\) lie on a straight line in the Argand diagram, and represent the distinct complex numbers \(a\), \(q\) and \(c\), respectively. Show that \(\dfrac {q-a}{c-a}\) is real and hence that \((c-a)(q^*-a^*) = (c^*-a^*)(q-a)\,\). Given that \(aa^* = cc^* = 1\), show further that \[ q+ ac q^* = a+c \,. \]
2. The distinct points \(A\), \(B\), \(C\) and \(D\) lie, in anticlockwise order, on the circle of unit radius with centre at the origin (so that, for example, \(aa^* =1\)). The lines \(AC\) and \(BD\) meet at \(Q\). Show that \[ (ac-bd)q^* = (a+c)-(b+d) \,, \] where \(b\) and \(d\) are complex numbers represented by the points \(B\) and \(D\) respectively, and show further that \[ (ac-bd) (q+q^*) = (a-b)(1+cd) +(c-d)(1+ab) \,. \]
3. The lines \(AB\) and \(CD\) meet at \(P\), which represents the complex number \(p\). Given that \(p\) is real, show that \(p(1+ab)=a+b\,\). Given further that \(ac-bd \ne 0\,\), show that \[ p(q+q^*) = 2 \,. \]

A quadrilateral drawn in the complex plane has vertices \(A\), \(B\), \(C\) and~\(D\), labelled anticlockwise. These vertices are represented, respectively, by the complex numbers \(a\), \(b\), \(c\) and~\(d\). Show that \(ABCD\) is a parallelogram (defined as a quadrilateral in which opposite sides are parallel and equal in length) if and only if \(a+c =b+d\,\). Show further that, in this case, \(ABCD\) is a square if and only if \({\rm i}(a-c)=b-d\). Let \(PQRS\) be a quadrilateral in the complex plane, with vertices labelled anticlockwise, the internal angles of which are all less than~\(180^\circ\). Squares with centres \(X\), \(Y\), \(Z\) and~\(T\) are constructed externally to the quadrilateral on the sides \(PQ\), \(QR\), \(RS\) and~\(SP\), respectively.

1. If \(P\) and \(Q\) are represented by the complex numbers \(p\) and~\(q\), respectively, show that \(X\) can be represented by \[ \tfrac 12 \big( p(1+{\rm i} ) + q (1-{\rm i})\big) \,. \]
2. Show that \(XY\!ZT\) is a square if and only if \(PQRS\) is a parallelogram.

Let \(z\) and \(w\) be complex numbers. Use a diagram to show that \(\vert z-w \vert \le \vert z\vert + \vert w \vert\,.\) For any complex numbers \(z\) and \(w\), \(E\) is defined by \[ E = zw^* + z^*w +2 \vert zw \vert\,. \]

1. Show that \(\vert z-w\vert^2 = \left( \vert z \vert + \vert w\vert\right)^2 -E\,\), and deduce that \(E\) is real and non-negative.
2. Show that \(\vert 1-zw^*\vert^2 = \left ( 1 +\vert zw \vert \right)^2 -E\,\).

Let \(x+{\rm i} y\) be a root of the quadratic equation \(z^2 + pz +1=0\), where \(p\) is a real number. Show that \(x^2-y^2 +px+1=0\) and \((2x+p)y=0\). Show further that \[ \text{\ either \ } p=-2x \text{ \ \ or \ \ } p=-(x^2+1)/x \text{ \ with \ } x\ne0 \,. \] Hence show that the set of points in the Argand diagram that can (as \(p\) varies) represent roots of the quadratic equation consists of the real axis with one point missing and a circle. This set of points is called the {\em root locus} of the quadratic equation. Obtain and sketch in the Argand diagram the root locus of the equation \[ pz^2 +z+1=0\, \] and the root locus of the equation \[ pz^2 + p^2z +2=0\,. \]

The complex numbers \(z\) and \(w\) are related by \[ w= \frac{1+\mathrm{i}z}{\mathrm{i}+z}\,. \] Let \(z=x+\mathrm{i}y\) and \(w=u+\mathrm{i}v\), where \(x\), \(y\), \(u\) and \(v\) are real. Express \(u\) and \(v\) in terms of \(x\) and \(y\).

1. By setting \(x=\tan(\theta/2)\), or otherwise, show that if the locus of \(z\) is the real axis \(y=0\), \(-\infty < x < \infty\), then the locus of \(w\) is the circle \(u^2+v^2=1\) with one point omitted.
2. Find the locus of \(w\) when the locus of \(z\) is the line segment \(y=0\), \(-1 < x < 1\,\).
3. Find the locus of \(w\) when the locus of \(z\) is the line segment \(x=0\), \(-1 < y < 1\,\).
4. Find the locus of \(w\) when the locus of \(z\) is the line \(y=1\), \(-\infty < x < \infty\,\).

The points \(A\), \(B\) and \(C\) in the Argand diagram are the vertices of an equilateral triangle described anticlockwise. Show that the complex numbers \(a\), \(b\) and \(c\) representing \(A\), \(B\) and \(C\) satisfy \[2c= (a+b) +\mathrm{i}\sqrt3(b-a).\] Find a similar relation in the case that \(A\), \(B\) and \(C\) are the vertices of an equilateral triangle described clockwise.

1. The quadrilateral \(DEFG\) lies in the Argand diagram. Show that points \(P\), \(Q\), \(R\) and \(S\) can be chosen so that \(PDE\), \(QEF\), \(RFG\) and \(SGD\) are equilateral triangles and \(PQRS\) is a parallelogram.
2. The triangle \(LMN\) lies in the Argand diagram. Show that the centroids \(U\), \(V\) and \(W\) of the equilateral triangles drawn externally on the sides of \(LMN\) are the vertices of an equilateral triangle. \noindent [{\bf Note:} The {\em centroid} of a triangle with vertices represented by the complex numbers \(x\),~\(y\) and~\(z\) is the point represented by \(\frac13(x+y+z)\,\).]

The distinct points \(P\), \(Q\), \(R\) and \(S\) in the Argand diagram lie on a circle of radius \(a\) centred at the origin and are represented by the complex numbers \(p\), \(q\), \(r\) and \(s\), respectively. Show that \[ pq = -a^2 \frac {p-q}{p^*-q^*}\,. \] Deduce that, if the chords \(PQ\) and \(RS\) are perpendicular, then \(pq+rs=0\). The distinct points \(A_1\), \(A_2\), \(\ldots\), \(A_n\) (where \(n\ge3\)) lie on a circle. The points \hbox{\(B_1\), \(B_2\), \(\ldots\), \(B_{n}\)} lie on the same circle and are chosen so that the chords \(B_1B_2\), \(B_2B_3\), \(\ldots\), \(B_nB_{1}\) are perpendicular, respectively, to the chords \(A_1A_2\), \(A_2A_3\), \(\ldots\), \(A_nA_1\). Show that, for \(n=3\), there are only two choices of \(B_1\) for which this is possible. What is the corresponding result for \(n=4\)? State the corresponding results for values of \(n\) greater than 4.

Show that the distinct complex numbers \(\alpha\), \(\beta\) and \(\gamma\) represent the vertices of an equilateral triangle (in clockwise or anti-clockwise order) if and only if \[ \alpha^2 + \beta^2 +\gamma^2 -\beta\gamma - \gamma \alpha -\alpha\beta =0\,. \] Show that the roots of the equation \begin{equation*} z^3 +az^2 +bz +c=0 \tag{\(*\)} \end{equation*} represent the vertices of an equilateral triangle if and only if \(a^2=3b\). Under the transformation \(z=pw+q\), where \(p\) and \(q\) are given complex numbers with \(p\ne0\), the equation (\(*\)) becomes \[ w^3 +Aw^2 +Bw +C=0\,. \tag{\(**\)} \] Show that if the roots of equation \((*)\) represent the vertices of an equilateral triangle, then the roots of equation \((**)\) also represent the vertices of an equilateral triangle.

In this question, \(a\) and \(c\) are distinct non-zero complex numbers. The complex conjugate of any complex number \(z\) is denoted by \(z^*\). Show that \[ |a - c|^2 = aa^* + cc^* -ac^* - ca^* \] and hence prove that the triangle \(OAC\) in the Argand diagram, whose vertices are represented by \(0\), \(a\) and \(c\) respectively, is right angled at \(A\) if and only if \(2aa^* = ac^*+ca^*\,\). Points \(P\) and \(P'\) in the Argand diagram are represented by the complex numbers \(ab\) and \(\ds \frac{a}{b^*}\,\), where \(b\) is a non-zero complex number. A circle in the Argand diagram has centre \(C\) and passes through the point \(A\), and is such that \(OA\) is a tangent to the circle. Show that the point \(P\) lies on the circle if and only if the point \(P'\) lies on the circle. Conversely, show that if the points represented by the complex numbers \(ab\) and \(\ds \frac{a}{b^*}\), for some non-zero complex number \(b\) with \(bb^* \ne 1\,\), both lie on a circle centre \(C\) in the Argand diagram which passes through \(A\), then \(OA\) is a tangent to the circle.

Four complex numbers \(u_1\), \(u_2\), \(u_3\) and \(u_4\) have unit modulus, and arguments \(\theta_1\), \(\theta_2\), \(\theta_3\) and \(\theta_4\), respectively, with \(-\pi < \theta_1 < \theta_2 < \theta_3 < \theta_4 < \pi\). Show that \[ \arg \l u_1 - u_2 \r = \tfrac{1}{2} \l \theta_1 + \theta_2 -\pi \r + 2n\pi \] where \(n = 0 \hspace{4 pt} \mbox{or} \hspace{4 pt} 1\,\). Deduce that \[ \arg \l \l u_1 - u_2 \r \l u_4 - u_3 \r \r = \arg \l \l u_1 - u_4 \r \l u_3 - u_2 \r \r + 2n\pi \] for some integer \(n\). Prove that \[ | \l u_1 - u_2 \r \l u_4 - u_3 \r | + | \l u_1 - u_4 \r \l u_3 - u_2 \r | = | \l u_1 - u_3 \r \l u_4 - u_2 \r |\;. \]

1. Prove that the equations $$ \left|z - (1 + \mathrm{i}) \right|^2 = 2 \eqno(*) $$ and $$ \qquad \quad \ \left|z - (1 - \mathrm{i}) \right|^2 = 2 \left|z - 1 \right|^2 $$ describe the same locus in the complex \(z\)--plane. Sketch this locus.
2. Prove that the equation $$ \arg \l {z - 2 \over z} \r = {\pi \over 4} \eqno(**) $$ describes part of this same locus, and show on your sketch which part.
3. The complex number \(w\) is related to \(z\) by \[ w = {2 \over z}\;. \] Determine the locus produced in the complex \(w\)--plane if \(z\) satisfies \((*)\). Sketch this locus and indicate the part of this locus that corresponds to \((**)\).

In an Argand diagram, \(O\) is the origin and \(P\) is the point \(2+0\mathrm{i}\). The points \(Q\), \(R\) and \(S\) are such that the lengths \(OP\), \(PQ\), \(QR\) and \(RS\) are all equal, and the angles \(OPQ\), \(PQR\) and \(QRS\) are all equal to \({5{\pi}}/6\), so that the points \(O\), \(P\), \(Q\), \(R\) and \(S\) are five vertices of a regular 12-sided polygon lying in the upper half of the Argand diagram. Show that \(Q\) is the point \(2 + \sqrt 3 + \mathrm{i}\) and find \(S\). The point \(C\) is the centre of the circle that passes through the points \(O\), \(P\) and \(Q\). Show that, if the polygon is rotated anticlockwise about \(O\) until \(C\) first lies on the real axis, the new position of \(S\) is $$ - \tfrac{1}{2} (3\sqrt 2+ \sqrt6)(\sqrt3-\mathrm{i})\;. $$

Define the modulus of a complex number \(z\) and give the geometric interpretation of \(\vert\,z_1-z_2\,\vert\) for two complex numbers \(z_1\) and \(z_2\). On the basis of this interpretation establish the inequality $$\vert\,z_1+z_2\,\vert\le \vert\,z_1\,\vert+\vert\,z_2\,\vert.$$ Use this result to prove, by induction, the corresponding inequality for \(\vert\,z_1+\cdots+z_n\,\vert\). The complex numbers \(a_1,\,a_2,\,\ldots,\,a_n\) satisfy \(|a_i|\le 3\) (\(i=1, 2, \ldots , n\)). Prove that the equation $$a_1z+a_2z^2\cdots +a_nz^n=1$$ has no solution \(z\) with \(\vert\,z\,\vert\le 1/4\).

1. In the Argand diagram, the points \(Q\) and \(A\) represent the complex numbers \(4+6i\) and \(10+2i\). If \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) are the vertices, taken in clockwise order, of a regular hexagon (regular six-sided polygon) with centre \(Q\), find the complex number which represents \(B\).
2. Let \(a\), \(b\) and \(c\) be real numbers. Find a condition of the form \(Aa+Bb+Cc=0\), where \(A\), \(B\) and \(C\) are integers, which ensures that \[\frac{a}{1+i}+\frac{b}{1+2i}+\frac{c}{1+3i}\] is real.

The complex numbers \(w=u+\mathrm{i}v\) and \(z=x+\mathrm{i}y\) are related by the equation $$z= (\cos v+\mathrm{i}\sin v)\mathrm{e}^u.$$ Find all \(w\) which correspond to \(z=\mathrm{i\,e}\). Find the loci in the \(x\)--\(y\) plane corresponding to the lines \(u=\) constant in the \(u\)--\(v\) plane. Find also the loci corresponding to the lines \(v=\) constant. Illustrate your answers with clearly labelled sketches. Identify two subsets \(W_1\) and \(W_2\) of the \(u\)--\(v\) plane each of which is in one-to-one correspondence with the first quadrant \(\{(x,\,y):\,x>0,\,y>0\}\) of the \(x\)--\(y\) plane. Identify also two subsets \(W_3\) and \(W_4\) each of which is in one-to-one correspondence with the set \(\{z\,:0<\,\vert z\vert\,<1\}\). \noindent[{\bf NB} `one-to-one' means here that to each value of \(w\) there is only one corresponding value of \(z\), and vice-versa.]

If $$ z^{4}+z^{3}+z^{2}+z+1=0\tag{*} $$ and \(u=z+z^{-1}\), find the possible values of \(u\). Hence find the possible values of \(z\). [Do not try to simplify your answers.] Show that, if \(z\) satisfies \((*)\), then \[z^{5}-1=0.\] Hence write the solutions of \((*)\) in the form \(z=r(\cos\theta+i\sin\theta)\) for suitable real \(r\) and \(\theta\). Deduce that \[\sin\frac{2\pi}{5}=\frac{\surd(10+2\surd 5)}{4} \ \ \hbox{and}\ \ \cos\frac{2\pi}{5}=\frac{-1+\surd 5}{4}.\]

1. Find all rational numbers \(r\) and \(s\) which satisfy \[ (r+s\sqrt{3})^{2}=4-2\sqrt{3}. \]
2. Find all real numbers \(p\) and \(q\) which satisfy \[ (p+q\mathrm{i})^{2}=(3-2\sqrt{3})+2(1-\sqrt{3})\mathrm{i}. \]
3. Solve the equation \[ (1+\mathrm{i})z^{2}-2z+2\sqrt{3}-2=0, \] writing your solutions in as simple a form as possible.

The variable non-zero complex number \(z\) is such that \[ \left|z-\mathrm{i}\right|=1. \] Find the modulus of \(z\) when its argument is \(\theta.\) Find also the modulus and argument of \(1/z\) in terms of \(\theta\) and show in an Argand diagram the loci of points which represent \(z\) and \(1/z\). Find the locus \(C\) in the Argand diagram such that \(w\in C\) if, and only if, the real part of \((1/w)\) is \(-1\).

Show Solution

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\(z\) is a point on the circle shown: Therefore using the cosine rule \(|z|^2 = 1^2 + 1^2 - 2\cdot 1 \cdot 1 \cdot \cos (2 \theta) = 2 -2\cos 2\theta = 2\sin^2 \theta \Rightarrow |z| = \sqrt{2}|\sin \theta|\) \(\frac{1}{z}\) has modulus \(\frac{1}{\sqrt{2}|\sin \theta|}\) and argument \(-\theta\). \(|\frac{1}{z} - i| = 1 \Rightarrow |1-iz| = |z| \Rightarrow |-i-z| = |z|\) ie we're looking for the points on the perpendicular bisector of \(0\) and \(-i\). \(\textrm{Re}\left (\frac{1}{w}\right) = -1 \Rightarrow -1 = \textrm{Re} \left (\frac{1}{a+ib} \right) = \frac{a-ib}{a^2+b^2} = \frac{a}{a^2+b^2} \Rightarrow a^2+b^2 = -a \Rightarrow (a+\tfrac12)^2+b^2 = \tfrac14\) so we are looking at a circle radius \(\tfrac12\) centre \(-\frac12\)

Let \(a,b,c,d,p,q,r\) and \(s\) be real numbers. By considering the determinant of the matrix product \[ \begin{pmatrix}z_{1} & z_{2}\\ -z_{2}^{*} & z_{1}^{*} \end{pmatrix}\begin{pmatrix}z_{3} & z_{4}\\ -z_{4}^{*} & z_{3}^{*} \end{pmatrix}, \] where \(z_{1},z_{2},z_{3}\) and \(z_{4}\) are suitably chosen complex numbers, find expressions \(L_{1},L_{2},L_{3}\) and \(L_{4},\) each of which is linear in \(a,b,c\) and \(d\) and also linear in \(p,q,r\) and \(s,\) such that \[ (a^{2}+b^{2}+c^{2}+d^{2})(p^{2}+q^{2}+r^{2}+s^{2})=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2}. \]

The four points \(A,B,C,D\) in the Argand diagram (complex plane) correspond to the complex numbers \(a,b,c,d\) respectively. The point \(P_{1}\) is mapped to \(P_{2}\) by rotating about \(A\) through \(\pi/2\) radians. Then \(P_{2}\) is mapped to \(P_{3}\) by rotating about \(B\) through \(\pi/2\) radians, \(P_{3}\) is mapped to \(P_{4}\) by rotating about \(C\) through \(\pi/2\) radians and \(P_{4}\) is mapped to \(P_{5}\) by rotating about \(D\) through \(\pi/2\) radians, each rotation being in the positive sense. If \(z_{i}\) is the complex number corresponding to \(P_{i},\) find \(z_{5}\) in terms of \(a,b,c,d\) and \(z_{1}.\) Show that \(P_{5}\) will coincide with \(P_{1},\) irrespective of the choice of the latter if, and only if \[a-c=\mathrm{i}(b-d)\] and interpret this condition geometrically. The points \(A,B\) and \(C\) are now chosen to be distinct points on the unit circle and the angle of rotation is changed to \(\theta,\) where \(\theta\neq0,\) on each occasion. Find the necessary and sufficient condition on \(\theta\) and the points \(A,B\) and \(C\) for \(P_{4}\) always to coincide with \(P_{1}.\)

The function \(\mathrm{f}\) is defined, for any complex number \(z\), by \[ \mathrm{f}(z)=\frac{\mathrm{i}z-1}{\mathrm{i}z+1}. \] Suppose throughout that \(x\) is a real number.

1. Show that \[ \mathrm{Re}\,\mathrm{f}(x)=\frac{x^{2}-1}{x^{2}+1}\qquad\mbox{ and }\qquad\mathrm{Im}\,\mathrm{f}(x)=\frac{2x}{x^{2}+1}. \]
2. Show that \(\mathrm{f}(x)\mathrm{f}(x)^{*}=1,\) where \(\mathrm{f}(x)^{*}\) is the complex conjugate of \(\mathrm{f}(x)\).
3. Find expressions for \(\mathrm{Re}\,\mathrm{f}(\mathrm{f}(x))\) and \(\mathrm{Im}\,\mathrm{f}(\mathrm{f}(x)).\)
4. Find \(\mathrm{f}(\mathrm{f}(\mathrm{f}(x))).\)

The point in the Argand diagram representing the complex number \(z\) lies on the circle with centre \(K\) and radius \(r\), where \(K\) represents the complex number \(k\). Show that $$ zz^* -kz^* -k^*z +kk^* -r^2 =0. $$ The points \(P\), \(Q_1\) and \(Q_2\) represent the complex numbers \(z\), \(w_1\) and \(w_2\) respectively. The point \(P\) lies on the circle with \(OA\) as diameter, where \(O\) and \(A\) represent \(0\) and \(2i\) respectively. Given that \(w_1=z/(z-1)\), find the equation of the locus \(L\) of \(Q_1\) in terms of \(w_1\) and describe the geometrical form of \(L\). Given that \(w_2=z^*\), show that the locus of \(Q_2\) is also \(L\). Determine the positions of \(P\) for which \(Q_1\) coincides with \(Q_2\).

\textit{In this question, the argument of a complex number is chosen to satisfy \(0\leqslant\arg z<2\pi.\)} Let \(z\) be a complex number whose imaginary part is positive. What can you say about \(\arg z\)? The complex numbers \(z_{1},z_{2}\) and \(z_{3}\) all have positive imaginary part and \(\arg z_{1}<\arg z_{2}<\arg z_{3}.\) Draw a diagram that shows why \[ \arg z_{1}<\arg(z_{1}+z_{2}+z_{3})<\arg z_{3}. \] Prove that \(\arg(z_{1}z_{2}z_{3})\) is never equal to \(\arg(z_{1}+z_{2}+z_{3}).\)

Three rods have lengths \(a\), \(b\) and \(c\), where \(a< b< c\). The three rods can be made into a triangle (possibly of zero area) if \(a+b\ge c\). Let \(T_{n}\) be the number of triangles that can be made with three rods chosen from \(n\) rods of lengths \(1\), \(2\), \(3\), \(\ldots\) , \(n\) (where \(n\ge3\)). Show that \(T_8-T_7 = 2+4+6\) and evaluate \(T_8 -T_6\). Write down expressions for \(T_{2m}-T_{2m-1}\) and \(T_{2m} - T_{2m-2}\). Prove by induction that \(T_{2m}=\frac 16 m (m-1)(4m+1)\,\), and find the corresponding result for an odd number of rods.

A positive integer with \(2n\) digits (the first of which must not be \(0\)) is called a balanced number if the sum of the first \(n\) digits equals the sum of the last \(n\) digits. For example, \(1634\) is a \(4\)-digit balanced number, but \(123401\) is not a balanced number.

1. Show that seventy \(4\)-digit balanced numbers can be made using the digits \(0, 1, 2, 3\) and \(4\).
2. Show that \(\frac16 {k \left( k+1 \right) \left( 4k+5 \right) }\) \(4\)-digit balanced numbers can be made using the digits \(0\) to \(k\). You may use the identity $\displaystyle \sum _{r=0}^{n} r^2 \equiv \tfrac{1}{6} {n \left( n+1 \right) \left( 2n+1 \right) } \;$.

\(47231\) is a five-digit number whose digits sum to \(4+7+2+3+1 = 17\,\).

1. Show that there are \(15\) five-digit numbers whose digits sum to \(43\). You should explain your reasoning clearly.
2. How many five-digit numbers are there whose digits sum to \(39\)?

How many integers between \(10\,000\) and \(100\,000\) (inclusive) contain exactly two different digits? (\(23\,332\) contains exactly two different digits but neither of \(33\,333\) and \(12\,331\) does.)

Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins. In how many ways can you make up 20 pence using 20p, 10p, 5p, 2p and 1p coins? You are reminded that no credit will be given for unexplained answers.

I have two dice whose faces are all painted different colours. I number the faces of one of them \(1,2,2,3,3,6\) and the other \(1,3,3,4,5,6.\) I can now throw a total of 3 in two different ways using the two number \(2\)'s on the first die once each. Show that there are seven different ways of throwing a total of 6. I now renumber the dice (again only using integers in the range 1 to 6) with the results shown in the following table \noindent

A \(3\times3\) magic square is a \(3\times3\) array \[ \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h & k \end{array} \] whose entries are the nine distinct integers \(1,2,3,4,5,6,7,8,9\) and which has the property that all its rows, columns and main diagonals add up to the same number \(n\). (Thus \(a+b+c=d+e+f=g+h+k=a+d+g=b+e+h=c+f+k=a+e+k=c+e+g=n.)\)

1. Show that \(n=15.\)
2. Show that \(e=5.\)
3. Show that one of \(b,d,h\) or \(f\) must have value \(9\).
4. Find all \(3\times3\) magic squares with \(b=9.\)
5. How many different \(3\times3\) magic squares are there? Why?

Let \(X\) be a random variable which takes only the finite number of different possible real values \(x_{1},x_{2},\ldots,x_{n}.\) Define the expectation \(\mathrm{E}(X)\) and the variance \(\mathrm{var}(X)\) of \(X\). Show that , if \(a\) and \(b\) are real numbers, then \(\mathrm{E}(aX+b)=a\mathrm{E}(X)+b\) and express \(\mathrm{var}(aX+b)\) similarly in terms of \(\mathrm{var}(X)\). Let \(\lambda\) be a positive real number. By considering the contribution to \(\mathrm{var}(X)\) of those \(x_{i}\) for which \(\left|x_{i}-\mathrm{E}(X)\right|\geqslant\lambda,\) or otherwise, show that \[ \mathrm{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right)\leqslant\frac{\mathrm{var}(X)}{\lambda^{2}}\,. \] Let \(k\) be a real number satisfying \(k\geqslant\lambda.\) If \(\left|x_{i}-\mathrm{E}(X)\right|\leqslant k\) for all \(i\), show that \[ \mathrm{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right)\geqslant\frac{\mathrm{var}(X)-\lambda^{2}}{k^{2}-\lambda^{2}}\,. \]

1. Alice tosses a fair coin twice and Bob tosses a fair coin three times. Calculate the probability that Bob gets more heads than Alice.
2. Alice tosses a fair coin three times and Bob tosses a fair coin four times. Calculate the probability that Bob gets more heads than Alice.
3. Let \(p_1\) be the probability that Bob gets the same number of heads as Alice, and let~\(p_2\) be the probability that Bob gets more heads than Alice, when Alice and Bob each toss a fair coin \(n\) times. Alice tosses a fair coin \(n\) times and Bob tosses a fair coin \(n+1\) times. Express the probability that Bob gets more heads than Alice in terms of \(p_1\) and \(p_2\), and hence obtain a generalisation of the results of parts (i) and (ii).

From the integers \(1, 2, \ldots , 52\), I choose seven (distinct) integers at random, all choices being equally likely. From these seven, I discard any pair that sum to 53. Let \(X\) be the random variable the value of which is the number of discarded pairs. Find the probability distribution of \(X\) and show that \(\E (X) = \frac 7 {17}\). Note: \(7\times 17 \times 47 =5593\).

Each day, I have to take \(k\) different types of medicine, one tablet of each. The tablets are identical in appearance. When I go on holiday for \(n\) days, I put \(n\) tablets of each type in a container and on each day of the holiday I select \(k\) tablets at random from the container.

1. In the case \(k=3\), show that the probability that I will select one tablet of each type on the first day of a three-day holiday is \(\frac9{28}\). Write down the probability that I will be left with one tablet of each type on the last day (irrespective of the tablets I select on the first day).
2. In the case \(k=3\), find the probability that I will select one tablet of each type on the first day of an \(n\)-day holiday.
3. In the case \(k=2\), find the probability that I will select one tablet of each type on each day of an \(n\)-day holiday, and use Stirling's approximation \[ n!\approx \sqrt{2n\pi} \left(\frac n\e\right)^n \] to show that this probability is approximately \(2^{-n} \sqrt{n\pi\;}\).

I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616.

I am selling raffle tickets for \(\pounds1\) per ticket. In the queue for tickets, there are \(m\) people each with a single \(\pounds1\) coin and \(n\) people each with a single \(\pounds2\) coin. Each person in the queue wants to buy a single raffle ticket and each arrangement of people in the queue is equally likely to occur. Initially, I have no coins and a large supply of tickets. I stop selling tickets if I cannot give the required change.

1. In the case \(n=1\) and \(m\ge1\), find the probability that I am able to sell one ticket to each person in the queue.
2. By considering the first three people in the queue, show that the probability that I am able to sell one ticket to each person in the queue in the case \(n=2\) and \(m\ge2\) is \(\dfrac{m-1}{m+1}\,\).
3. Show that the probability that I am able to sell one ticket to each person in the queue in the case \(n=3\) and \(m\ge3\) is \(\dfrac{m-2}{m+1}\,\).

Three married couples sit down at a round table at which there are six chairs. All of the possible seating arrangements of the six people are equally likely.

1. Show that the probability that each husband sits next to his wife is \(\frac{2}{15}\).
2. Find the probability that exactly two husbands sit next to their wives.
3. Find the probability that no husband sits next to his wife.

1. A bag contains \(N\) sweets (where \(N \ge 2\)), of which \(a\) are red. Two sweets are drawn from the bag without replacement. Show that the probability that the first sweet is red is equal to the probability that the second sweet is red.
2. There are two bags, each containing \(N\) sweets (where \(N \ge 2\)). The first bag contains \(a\) red sweets, and the second bag contains \(b\) red sweets. There is also a biased coin, showing Heads with probability \(p\) and Tails with probability \(q\), where \(p+q = 1\). The coin is tossed. If it shows Heads then a sweet is chosen from the first bag and transferred to the second bag; if it shows Tails then a sweet is chosen from the second bag and transferred to the first bag. The coin is then tossed a second time: if it shows Heads then a sweet is chosen from the first bag, and if it shows Tails then a sweet is chosen from the second bag. Show that the probability that the first sweet is red is equal to the probability that the second sweet is red.

Three pirates are sharing out the contents of a treasure chest containing \(n\) gold coins and \(2\) lead coins. The first pirate takes out coins one at a time until he takes out one of the lead coins. The second pirate then takes out coins one at a time until she draws the second lead coin. The third pirate takes out all the gold coins remaining in the chest. Find:

1. the probability that the first pirate will have some gold coins;
2. the probability that the second pirate will have some gold coins;
3. the probability that all three pirates will have some gold coins.

On the basis of an interview, the \(N\) candidates for admission to a college are ranked in order according to their mathematical potential. The candidates are interviewed in random order (that is, each possible order is equally likely).

1. Find the probability that the best amongst the first \(n\) candidates interviewed is the best overall.
2. Find the probability that the best amongst the first \(n\) candidates interviewed is the best or second best overall.

A school has \(n\) pupils, of whom \(r\) play hocket, where \(n\geqslant r\geqslant2.\) All \(n\) pupils are arranged in a row at random.

1. What is the probability that there is a hockey player at each end of the row?
2. What is the probability that all the hockey players are standing together?
3. By considering the gaps between the non-hockey-players, find the probability that no two hockey players are standing together, distinguishing between cases when the probability is zero and when it is non-zero.

A set of \(2N+1\) rods consists of one of each length \(1,2,\ldots,2N,2N+1\), where \(N\) is an integer greater than 1. Three different rods are selected from the set. Suppose their lengths are \(a,b\) and \(c\), where \(a>b>c\). Given that \(a\) is even and fixed, show, by considering the possible values of \(b\), that the number of selections in which a triangle can then be formed from the three rods is \[ 1+3+5+\cdots+(a-3), \] where we allow only non-degenerate triangles (i.e. triangles with non-zero area). Similarly obtain the number of selections in which a triangle may be formed when \(a\) takes some fixed odd value. Write down a formula for the number of ways of forming a non-degenerate triangle and verify it for \(N=3\). Hence show that, if three rods are drawn at random without replacement, then the probability that they can form a non-degenerate triangle is \[ \frac{(N-1)(4N+1)}{2(4N^{2}-1)}. \]

Four children, \(A\), \(B\), \(C\) and \(D\), are playing a version of the game `pass the parcel'. They stand in a circle, so that \(ABCDA\) is the clockwise order. Each time a whistle is blown, the child holding the parcel is supposed to pass the parcel immediately exactly one place clockwise. In fact each child, independently of any other past event, passes the parcel clockwise with probability~\(\frac{1}{4}\), passes it anticlockwise with probability \(\frac{1}{4}\) and fails to pass it at all with~probability \(\frac{1}{2}\). At the start of the game, child~\(A\) is holding the parcel. The probability that child \(A\) is holding the parcel just after the whistle has been blown for the \(n\)th time is \(A_n\), and \(B_n\), \(C_n\) and \(D_n\) are defined similarly.

1. Find \(A_1\), \(B_1\), \(C_1\) and \(D_1\). Find also \(A_2\), \(B_2\), \(C_2\) and \(D_2\).
2. %Write down %expressions for \(A_{n+1}\), \(B_{n+1}\), \(C_{n+1}\) and \(D_{n+1}\) in %terms of \(A_n\), \(B_n\), \(C_n\) and~\(D_n\). By first considering \(B_{n+1}+D_{n+1}\), or otherwise, find \(B_n\) and \(D_n\). Find also expressions for \(A_n\) and \(C_n\) in terms of \(n\).

A bag contains three coins. The probabilities of their showing heads when tossed are \(p_1\), \(p_2\) and \(p_3\).

1. A coin is taken at random from the bag and tossed. What is the probability that it shows a head?
2. A coin is taken at random from the bag (containing three coins) and tossed; the coin is returned to the bag and again a coin is taken at random from the bag and tossed. Let \(N_1\) be the random variable whose value is the number of heads shown on the two tosses. Find the expectation of \(N_1\) in terms of \(p\), where \(p = \frac{1}{3}(p_1+p_2+p_3)\,\), and show that \(\var(N_1) =2p(1-p)\,\).
3. Two of the coins are taken at random from the bag (containing three coins) and tossed. Let \(N_2\) be the random variable whose value is the number of heads showing on the two coins. Find \(\E(N_2)\) and \(\var(N_2)\).
4. Show that \(\var(N_2)\le \var(N_1)\), with equality if and only if \(p_1=p_2=p_3\,\).

I have a sliced loaf which initially contains \(n\) slices of bread. Each time I finish setting a STEP question, I make myself a snack: either toast, using one slice of bread; or a sandwich, using two slices of bread. I make toast with probability~\(p\) and I make a sandwich with probability \(q\), where \(p+q=1\), unless there is only one slice left in which case I must, of course, make toast. Let \(s_r\) (\(1 \le r \le n\)) be the probability that the \(r\){th} slice of bread is the second of two slices used to make a sandwich and let \(t_r\) (\(1 \le r \le n\)) be the probability that the \(r\){th} slice of bread is used to make toast. What is the value of \(s_1\)? Explain why the following equations hold: \begin{align*} \phantom{\hspace{2cm} (2\le r \le n-1)} t_r &= (s_{r-1}+ t_{r-1})\,p \hspace{2cm} (2\le r \le n-1)\,; \\ \phantom{\hspace{1.53cm} (2\le r \le n) } s_r &= 1- (s_{r-1} + t_{r-1}) \hspace{1.53cm} ( 2\le r \le n )\,. \end{align*} Hence, or otherwise, show that \(s_{r} = q(1-s_{r-1})\) for \(2\le r\le n-1\)\,. Show further that \[ \phantom{\hspace{2.7cm} (1\le r\le n)\,,} s_r = \frac{q+(-q)^r}{1+q} \hspace{2.7cm} (1\le r\le n-1)\,, \, \hspace{0.14cm} \] and find the corresponding expression for \(t_r\). Find also expressions for \(s_n\) and \(t_n\) in terms of \(q\).

Four players \(A\), \(B\), \(C\) and \(D\) play a coin-tossing game with a fair coin. Each player chooses a sequence of heads and tails, as follows: \noindent Player A: HHT; \ \ Player B: THH; \ \ Player C: TTH; \ \ Player D: HTT. \noindent The coin is then tossed until one of these sequences occurs, in which case the corresponding player is the winner.

1. Show that, if only \(A\) and \(B\) play, then \(A\) has a probability of \(\frac14\) of winning.
2. If all four players play together, find the probabilities of each one winning.
3. Only \(B\) and \(C\) play. What is the probability of \(C\) winning if the first two tosses are~TT? Let the probabilities of \(C\) winning if the first two tosses are HT, TH and HH be \(p\), \(q\) and \(r\), respectively. Show that \(p=\frac12 +\frac12q\). Find the probability that \(C\) wins.

A modern villa has complicated lighting controls. In order for the light in the swimming pool to be on, a particular switch in the hallway must be on and a particular switch in the kitchen must be on. There are four identical switches in the hallway and four identical switches in the kitchen. Guests cannot tell whether the switches are on or off, or what they control. Each Monday morning a guest arrives, and the switches in the hallway are either all on or all off. The probability that they are all on is \(p\) and the probability that they are all off is \(1-p\). The switches in the kitchen are each on or off, independently, with probability \(\frac12\).

1. On the first Monday, a guest presses one switch in the hallway at random and one switch in the kitchen at random. Find the probability that the swimming pool light is on at the end of this process. Show that the probability that the guest has pressed the swimming pool light switch in the hallway, given that the light is on at the end of the process, is \(\displaystyle \frac{1-p}{1+2p}\).
2. On each of seven Mondays, guests go through the above process independently of each other, and each time the swimming pool light is found to be on at the end of the process. Given that the most likely number of days on which the swimming pool light switch in the hallway was pressed is 3, show that \(\frac14 < p < \frac{5}{14}\).

Xavier and Younis are playing a match. The match consists of a series of games and each game consists of three points. Xavier has probability \(p\) and Younis has probability \(1-p\) of winning the first point of any game. In the second and third points of each game, the player who won the previous point has probability \(p\) and the player who lost the previous point has probability \(1-p\) of winning the point. If a player wins two consecutive points in a single game, the match ends and that player has won; otherwise the match continues with another game.

1. Let \(w\) be the probability that Younis wins the match. Show that, for \(p\ne0\), \[ w = \frac{1-p^2}{2-p}. \] Show that \(w>\frac12\) if \(p<\frac12\), and \(w<\frac12\) if \(p>\frac12\). Does \(w\) increase whenever \(p\) decreases?
2. If Xavier wins the match, Younis gives him \(\pounds1\); if Younis wins the match, Xavier gives him \(\pounds k\). Find the value of \(k\) for which the game is `fair' in the case when \(p =\frac23\).
3. What happens when \(p = 0\)?

Rosalind wants to join the Stepney Chess Club. In order to be accepted, she must play a challenge match consisting of several games against Pardeep (the Club champion) and Quentin (the Club secretary), in which she must win at least one game against each of Pardeep and Quentin. From past experience, she knows that the probability of her winning a single game against Pardeep is \(p\) and the probability of her winning a single game against Quentin is \(q\), where \(0 < p < q < 1\).

1. The challenge match consists of three games. Before the match begins, Rosalind must choose either to play Pardeep twice and Quentin once or to play Quentin twice and Pardeep once. Show that she should choose to play Pardeep twice.
2. In order to ease the entry requirements, it is decided instead that the challenge match will consist of four games. Now, before the match begins, Rosalind must choose whether to play Pardeep three times and Quentin once (strategy 1), or to play Pardeep twice and Quentin twice (strategy 2) or to play Pardeep once and Quentin three times (strategy~3). Show that, if \(q-p > \frac 12\), Rosalind should choose strategy 1. If \(q-p<\frac12\), give examples of values of \(p\) and \(q\) to show that strategy 2 can be better or worse than strategy 1.

Satellites are launched using two different types of rocket: the Andover and the Basingstoke. The Andover has four engines and the Basingstoke has six. Each engine has a probability~\(p\) of failing during any given launch. After the launch, the rockets are retrieved and repaired by replacing some or all of the engines. The cost of replacing each engine is \(K\). For the Andover, if more than one engine fails, all four engines are replaced. Otherwise, only the failed engine (if there is one) is replaced. Show that the expected repair cost for a single launch using the Andover is \[ 4Kp(1+q+q^2-2q^3) \ \ \ \ \ \ \ \ \ \ \ \ \ (q=1-p) \tag{*} \] For the Basingstoke, if more than two engines fail, all six engines are replaced. Otherwise only the failed engines (if there are any) are replaced. Find, in a form similar to \((*)\), the expected repair cost for a single launch using the Basingstoke. Find the values of \(p\) for which the expected repair cost for the Andover is \(\frac23\) of the expected repair cost for the Basingstoke.

Prove that, for any real numbers \(x\) and \(y\), \(x^2+y^2\ge2xy\,\).

1. Carol has two bags of sweets. The first bag contains \(a\) red sweets and \(b\) blue sweets, whereas the second bag contains \(b\) red sweets and \(a\) blue sweets, where \(a\) and \(b\) are positive integers. Carol shakes the bags and picks one sweet from each bag without looking. Prove that the probability that the sweets are of the same colour cannot exceed the probability that they are of different colours.
2. Simon has three bags of sweets. The first bag contains \(a\) red sweets, \(b\) white sweets and \(c\) yellow sweets, where \(a\), \(b\) and \(c\) are positive integers. The second bag contains \(b\) red sweets, \(c\) white sweets and \(a\) yellow sweets. The third bag contains \(c\) red sweets, \(a\) white sweets and \(b\) yellow sweets. Simon shakes the bags and picks one sweet from each bag without looking. Show that the probability that exactly two of the sweets are of the same colour is \[ \frac {3(a^2b+b^2c+c^2a+ab^2 + bc^2 +ca^2)}{(a+b+c)^3}\,, \] and find the probability that the sweets are all of the same colour. Deduce that the probability that exactly two of the sweets are of the same colour is at least 6 times the probability that the sweets are all of the same colour.

Bag \(P\) and bag \(Q\) each contain \(n\) counters, where \(n\ge2\). The counters are identical in shape and size, but coloured either black or white. First, \(k\) counters (\(0\le k\le n\)) are drawn at random from bag \(P\) and placed in bag \(Q\). Then, \(k\) counters are drawn at random from bag \(Q\) and placed in bag \(P\).

1. If initially \(n-1\) counters in bag \(P\) are white and one is black, and all \(n\) counters in bag \(Q\) are white, find the probability in terms of \(n\) and \(k\) that the black counter ends up in bag \(P\). Find the value or values of \(k\) for which this probability is maximised.
2. If initially \(n-1\) counters in bag \(P\) are white and one is black, and \(n-1\) counters in bag \(Q\) are white and one is black, find the probability in terms of \(n\) and \(k\) that the black counters end up in the same bag. Find the value or values of \(k\) for which this probability is maximised.

In the High Court of Farnia, the outcome of each case is determined by three judges: the ass, the beaver and the centaur. Each judge decides its verdict independently. Being simple creatures, they make their decisions entirely at random. Past verdicts show that the ass gives a guilty verdict with probability \(p\), the beaver gives a guilty verdict with probability \(p/3\) and the centaur gives a guilty verdict with probability \(p^2\). Let \(X\) be the number of guilty verdicts given by the three judges in a case. Given that \(\E(X)= 4/3\), find the value of \(p\). The probability that a defendant brought to trial is guilty is \(t\). The King pronounces that the defendant is guilty if at least two of the judges give a guilty verdict; otherwise, he pronounces the defendant not guilty. Find the value of \(t\) such that the probability that the King pronounces correctly is \(1/2\).

Four students, Arthur, Bertha, Chandra and Delilah, exchange gossip. When Arthur hears a rumour, he tells it to one of the other three without saying who told it to him. He decides whom to tell by choosing at random amongst the other three, omitting the ones that he knows have already heard the rumour. When Bertha, Chandra or Delilah hear a rumour, they behave in exactly the same way (even if they have already heard it themselves). The rumour stops being passed round when it is heard by a student who knows that the other three have already heard it. Arthur starts a rumour and tells it to Chandra. By means of a tree diagram, or otherwise, show that the probability that Arthur rehears it is \(3/4\). Find also the probability that Bertha hears it twice and the probability that Chandra hears it twice.

Show Solution

Without loss of generality, \(C\) will tell \(B\) about the rumour. If \(B\) tells \(D\) then \(D\) can either tell \(A\) or \(C\) at which point either \(A\) is told or the rumour stops spreading.

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Therefore \(\mathbb{P}(\text{Arthur rehears}) = 3/4\) For the chances Chandra hears it twice, still WLOG, assume she tells B:

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So her chance of hearing it twice is \(\frac12\) The person who hears it 3rd has a \(\frac12\) chance of hearing it twice, but the person who hears if 4th has no chance. Therefore Bertha has a \(\frac14\) chance of hearing it twice.

1. Prove that if \(x>0\) then \(x+x^{-1}\ge2.\;\) I have a pair of six-faced dice, each with faces numbered from 1 to 6. The probability of throwing \(i\) with the first die is \(q_{i}\) and the probability of throwing \(j\) with the second die is \(r_{j}\) (\(1\le i,j \le 6\)). The two dice are thrown independently and the sum noted. By considering the probabilities of throwing 2, 12 and 7, show the sums \(2, 3, \dots, 12\) are not equally likely.
2. The first die described above is thrown twice and the two numbers on the die noted. Is it possible to find values of \(q_{j}\) so that the probability that the numbers are the same is less than \(1/36\)?

A pack of \(2n\) (where \(n\geqslant4\)) cards consists of two each of \(n\) different sorts. If four cards are drawn from the pack without replacement show that the probability that no pairs of identical cards have been drawn is \[ \frac{4(n-2)(n-3)}{(2n-1)(2n-3)}. \] Find the probability that exactly one pair of identical cards is included in the four. If \(k\) cards are drawn without replacement and \(2 < k < 2n,\) find an expression for the probability that there are exactly \(r\) pairs of identical cards included when \(r < \frac{1}{2}k.\) For even values of \(k\) show that the probability that the drawn cards consist of \(\frac{1}{2}k\) pairs is \[ \frac{1\times3\times5\times\cdots\times(k-1)}{(2n-1)(2n-3)\cdots(2n-k+1)}. \]

Starting with the result \(\P(A\cup B) = \P(A)+P(B) - \P(A\cap B)\), prove that \[ \P(A\cup B\cup C) = \P(A)+\P(B)+\P(C) - \P(A\cap B) - \P(B\cap C) - \P(C \cap A) + \P(A\cap B\cap C) \,. \] Write down, without proof, the corresponding result for four events \(A\), \(B\), \(C\) and \(D\). A pack of \(n\) cards, numbered \(1, 2, \ldots, n\), is shuffled and laid out in a row. The result of the shuffle is that each card is equally likely to be in any position in the row. Let \(E_i\) be the event that the card bearing the number \(i\) is in the \(i\)th position in the row. Write down the following probabilities:

1. \(\P(E_i)\);
2. \(\P(E_i\cap E_j)\), where \(i\ne j\);
3. \(\P(E_i\cap E_j\cap E_k)\), where \(i\ne j\), \(j\ne k\) and \(k\ne i\).

1. The probability that a hobbit smokes a pipe is 0.7 and the probability that a hobbit wears a hat is 0.4. The probability that a hobbit smokes a pipe but does not wear a hat is \(p\). Determine the range of values of \(p\) consistent with this information.
2. The probability that a wizard wears a hat is 0.7; the probability that a wizard wears a cloak is 0.8; and the probability that a wizard wears a ring is 0.4. The probability that a wizard does not wear a hat, does not wear a cloak and does not wear a ring is 0.05. The probability that a wizard wears a hat, a cloak and also a ring is 0.1. Determine the probability that a wizard wears exactly two of a hat, a cloak, and a ring. The probability that a wizard wears a hat but not a ring, given that he wears a cloak, is \(q\). Determine the range of values of \(q\) consistent with this information.

Explain why, if \(\mathrm{A, B}\) and \(\mathrm{C}\) are three events, \[ \mathrm{P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) +P(A \cap B \cap C)}, \] where \(\mathrm{P(X)}\) denotes the probability of event \(\mathrm{X}\). A cook makes three plum puddings for Christmas. He stirs \(r\) silver sixpences thoroughly into the pudding mixture before dividing it into three equal portions. Find an expression for the probability that each pudding contains at least one sixpence. Show that the cook must stir 6 or more sixpences into the mixture if there is to be less than \({1 \over 3}\) chance that at least one of the puddings contains no sixpence. Given that the cook stirs 6 sixpences into the mixture and that each pudding contains at least one sixpence, find the probability that there are two sixpences in each pudding.

Show Solution

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When we add everything in \(A\),\(B\), \(C\) we overcount the overlaps. When we remove the overlaps we remove the centre section too many times, so we have to add it back on in the end. Let \(X_i\) be the probability that the \(i\)th pudding contains a sixpence. \begin{align*} && \mathbb{P}(X_1^c \cup X_2^c \cup X_3^c) &=\mathbb{P}(X_1^c \cap X_2^c \cap X_3^c) + \mathbb{P}(X_1^c)+\mathbb{P}(X_2^c)-\mathbb{P}(X_3^c)+\\ &&&\quad\quad-\mathbb{P}(X_1^c \cap X_2^c )-\mathbb{P}( X_2^c \cap X_3^c)-\mathbb{P}(X_1^c \cap X_3^c) \\ &&&= 0 + (\tfrac23)^r+ (\tfrac23)^r+ (\tfrac23)^r + \\ &&&\quad\quad - (\tfrac13)^r- (\tfrac13)^r- (\tfrac13)^r \\ &&&= \frac{3\cdot2^r-3}{3^{r}} \\ \Rightarrow && \mathbb{P}(\text{all contain a sixpence}) &= 1 - \frac{3\cdot2^r-3}{3^{r}} \\ &&&= \frac{3^r-3\cdot2^r+3}{3^r} \end{align*} When \(r = 5\) we have \(\frac{3 \cdot 32-3}{3^5} = \frac{31}{81} > \frac13\) When \(r = 6\) we have \(\frac{3 \cdot 64-3}{3^6} = \frac{7}{27} < \frac13\) Therefore, the chef must stir in at least \(6\). \begin{align*} && \mathbb{P}(\text{two in each}|\text{at least 1 in each}) &= \frac{ \mathbb{P}(\text{two in each} \cap \text{at least 1 in each}) }{ \mathbb{P}(\text{at least 1 in each}) } \\ &&&= \frac{ \mathbb{P}(\text{two in each}) }{ \mathbb{P}(\text{at least 1 in each}) } \\ &&&= \frac{90/3^6}{20/27} \\ &&&= \frac{1/3}{2} = \frac16 \end{align*}

The set \(S\) is the set of all integers from 1 to \(n\). The set \(T\) is the set of all distinct subsets of \(S\), including the empty set \(\emptyset\) and \(S\) itself. Show that \(T\) contains exactly \(2^n\) sets. The sets \(A_1, A_2, \ldots, A_m\), which are not necessarily distinct, are chosen randomly and independently from \(T\), and for each \(k\) \((1 \leq k \leq m)\), the set \(A_k\) is equally likely to be any of the sets in \(T\).

1. Write down the value of \(P(1 \in A_1)\).
2. By considering each integer separately, show that \(P(A_1 \cap A_2 = \emptyset) = \left(\frac{3}{4}\right)^n\). Find \(P(A_1 \cap A_2 \cap A_3 = \emptyset)\) and \(P(A_1 \cap A_2 \cap \cdots \cap A_m = \emptyset)\).
3. Find \(P(A_1 \subseteq A_2)\), \(P(A_1 \subseteq A_2 \subseteq A_3)\) and \(P(A_1 \subseteq A_2 \subseteq \cdots \subseteq A_m)\).

Given that \(0 < r < n\) and \(r\) is much smaller than \(n\), show that \(\dfrac {n-r}n \approx \e^{-r/n}\). There are \(k\) guests at a party. Assuming that there are exactly 365 days in the year, and that the birthday of any guest is equally likely to fall on any of these days, show that the probability that there are at least two guests with the same birthday is approximately \(1-\e^{-k(k-1)/730}\). Using the approximation \( \frac{253}{365} \approx \ln 2\), find the smallest value of \(k\) such that the probability that at least two guests share the same birthday is at least \(\frac12\). How many guests must there be at the party for the probability that at least one guest has the same birthday as the host to be at least \(\frac12\)?

The mountain villages \(A,B,C\) and \(D\) lie at the vertices of a tetrahedron, and each pair of villages is joined by a road. After a snowfall the probability that any road is blocked is \(p\), and is independent of the conditions of any other road. The probability that, after a snowfall, it is possible to travel from any village to any other village by some route is \(P\). Show that $$ P =1- p^2(6p^3-12p^2+3p+4). $$ %In the case \(p={1\over 3}\) show that this probability is \({208 \over 243}\).

Suppose that a solution \((X,Y,Z)\) of the equation \[X+Y+Z=20,\] with \(X\), \(Y\) and \(Z\) non-negative integers, is chosen at random (each such solution being equally likely). Are \(X\) and \(Y\) independent? Justify your answer. Show that the probability that \(X\) is divisible by \(5\) is \(5/21\). What is the probability that \(XYZ\) is divisible by 5?

In certain forms of Tennis two players \(A\) and \(B\) serve alternate games. Player \(A\) has probability \(p\low_{A}\) of winning a game in which she serves and \(p\low_{B}\) of winning a game in which player \(B\) serves. Player \(B\) has probability \(q\low_{B}=1-p\low_{B}\) of winning a game in which she serves and probability \(q\low_{A}=1-p\low_{A}\) of winning a game in which player \(A\) serves. In Shortened Tennis the first player to lead by 2 games wins the match. Find the probability \(P_{\text{short}}\) that \(A\) wins a Shortened Tennis match in which she serves first and show that it is the same as if \(B\) serves first. In Standard Tennis the first player to lead by 2 or more games after 4 or more games have been played wins the match. Show that the probability that the match is decided in 4 games is \[ p^{2}_Ap_{B}^{2}+q_{A}^{2}q_{B}^{2}+2(p\low_{A}p\low_{B}+q\low_{A}q\low_{B})(p\low_{A}q\low_{B}+q\low_{A}p\low_{B}). \] If \(p\low_{A}=p\low_{B}=p\) and \(q\low_{A}=q\low_{B}=q,\) find the probability \(P_{\text{stan}}\) that \(A\) wins a Standard Tennis match in which she serves first. Show that \[ P_{\text{stan}}-P_{\text{short}}=\frac{p^{2}q^{2}(p-q)}{p^{2}+q^{2}}. \]

A point moves in unit steps on the \(x\)-axis starting from the origin. At each step the point is equally likely to move in the positive or negative direction. The probability that after \(s\) steps it is at one of the points \(x=2,x=3,x=4\) or \(x=5\) is \(\mathrm{P}(s).\) Show that \(\mathrm{P}(5)=\frac{3}{16},\) \(\mathrm{P}(6)=\frac{21}{64}\) and \[ \mathrm{P}(2k)=\binom{2k+1}{k-1}\left(\frac{1}{2}\right)^{2k} \] where \(k\) is a positive integer. Find a similar expression for \(\mathrm{P}(2k+1).\) Determine the values of \(s\) for which \(\mathrm{P}(s)\) has its greatest value.

The four towns \(A,B,C\) and \(D\) are linked by roads \(AB,AC,CB,BD\) and \(CD.\) The probability that any one road will be blocked by snow on the 1st of January is \(p\), independent of what happens to any other \([0 < p < 1]\). Show that the probability that any open route from \(A\) to \(D\) is \(ABCD\) is \[ p^{2}(1-p)^{3}. \] In order to increase the probability that it is possible to get from \(A\) to \(D\) by a sequence of unblocked roads the government proposes either to snow-proof the road \(AB\) (so that it can never be blocked) or to snow-proof the road \(CB.\) Because of the high cost it cannot do both. Which road should it choose (or are both choices equally advantageous)? In fact, \(p=\frac{1}{10}\) and the government decides that it is only worth going ahead if the present probability of \(A\) being cut off from \(D\) is greater than \(\frac{1}{100}.\) Will it go ahead?

A patient arrives with blue thumbs at the doctor's surgery. With probability \(p\) the patient is suffering from Fenland fever and requires treatment costing \(\pounds 100.\) With probability \(1-p\) he is suffering from Steppe syndrome and will get better anyway. A test exists which infallibly gives positive results if the patient is suffering from Fenland fever but also has probability \(q\) of giving positive results if the patient is not. The test cost \(\pounds 10.\) The doctor decides to proceed as follows. She will give the test repeatedly until either the last test is negative, in which case she dismisses the patient with kind words, or she has given the test \(n\) times with positive results each time, in which case she gives the treatment. In the case \(n=0,\) she treats the patient at once. She wishes to minimise the expected cost \(\pounds E_{n}\) to the National Health Service.

1. Show that \[ E_{n+1}-E_{n}=10p-10(1-p)q^{n}(9-10q), \] and deduce that if \(p=10^{-4},q=10^{-2},\) she should choose \(n=3.\)
2. Show that if \(q\) is larger than some fixed value \(q_{0},\) to be determined explicitly, then whatever the value of \(p,\) she should choose \(n=0.\)

My two friends, who shall remain nameless, but whom I shall refer to as \(P\) and \(Q\), both told me this afternoon that there is a body in my fridge. I'm not sure what to make of this, because \(P\) tells the truth with a probability of only \(p\), while \(Q\) (independently) tells the truth with probability \(q\). I haven't looked in the fridge for some time, so if you had asked me this morning, I would have said that there was just as likely to be a body in it as not. Clearly, in view of what \(P\) and \(Q\) told me, I must revise this estimate. Explain carefully why my new estimate of the probability of there being a body in the fridge should be \[ \frac{pq}{1-p-q+2pq}. \] I have now been to look in the fridge, and there is indeed a body in it; perhaps more than one. It seems to me that only my enemy \(A\), or my enemy \(B\), or (with a bit of luck) both \(A\) and \(B\) could be in my fridge, and this morning I would have judged these three possibilities to be equally likely. But tonight I asked \(P\) and \(Q\) separately whether or not \(A\) was in the fridge, and they each said that he was. What should be my new estimate of the probability that both \(A\) and \(B\) are in my fridge? Of course, I tell the truth always.

A multiple-choice test consists of five questions. For each question, \(n\) answers are given (\(n\ge2\)) only one of which is correct and candidates either attempt the question by choosing one of the \(n\) given answers or do not attempt it. For each question attempted, candidates receive two marks for the correct answer and lose one mark for an incorrect answer. No marks are gained or lost for questions that are not attempted. The pass mark is five. Candidates A, B and C don't understand any of the questions so, for any question which they attempt, they each choose one of the \(n\) given answers at random, independently of their choices for any other question.

1. Candidate A chooses in advance to attempt exactly \(k\) of the five questions, where \(k=0\), 1, 2, 3, 4~or~5. Show that, in order to have the greatest probability of passing the test, she should choose \(k=4\,\).
2. Candidate B chooses at random the number of questions he will attempt, the six possibilities being equally likely. Given that Candidate B passed the test find, in terms of~\(n\), the probability that he attempted exactly four questions. %Show that this probability is an increasing function of \(n\).
3. For each of the five questions Candidate C decides whether to attempt the question by tossing a biased coin. The coin has a probability of \(\frac n{n+1}\) of showing a head, and she attempts the question if it shows a head. Find the probability, in terms of \(n\), that Candidate C passes the test.

A bag contains eleven small discs, which are identical except that six of the discs are blank and five of the discs are numbered, using the numbers 1, 2, 3, 4 and 5. The bag is shaken, and four discs are taken one at a time without replacement. Calculate the probability that:

1. all four discs taken are numbered;
2. all four discs taken are numbered, given that the disc numbered ``3'' is taken first;
3. exactly two numbered discs are taken, given that the disc numbered ``3'' is taken first;
4. exactly two numbered discs are taken, given that the disc numbered ``3'' is taken;
5. exactly two numbered discs are taken, given that a numbered disc is taken first;
6. exactly two numbered discs are taken, given that a numbered disc is taken.

I know that ice-creams come in \(n\) different sizes, but I don't know what the sizes are. I am offered one of each in succession, in random order. I am certainly going to choose one - the bigger the better - but I am not allowed more than one. My strategy is to reject the first ice-cream I am offered and choose the first one thereafter that is bigger than the first one I was offered; if the first ice-cream offered is in fact the biggest one, then I have to put up with the last one, however small. Let \(\P_n(k)\) be the probability that I choose the \(k\)th biggest ice-cream, where \(k=1\) is the biggest and \(k=n\) is the smallest.

1. Show that \(\P_4(1) = \frac{11}{24}\) and find \(\P_4(2)\), \(\P_4(3)\) and \(\P_4(4)\).
2. Find an expression for \(\P_n(1)\).

Oxtown and Camville are connected by three roads, which are at risk of being blocked by flooding. On two of the three roads there are two sections which may be blocked. On the third road there is only one section which may be blocked. The probability that each section is blocked is \(p\). Each section is blocked independently of the other four sections. Show that the probability that Oxtown is cut off from Camville is \(p^3 \l 2-p \r^2\). I want to travel from Oxtown to Camville. I choose one of the three roads at random and find that my road is not blocked. Find the probability that I would not have reached Camville if I had chosen either of the other two roads. You should factorise your answer as fully as possible. Comment briefly on the value of this probability in the limit \(p\to1\).

The twins Anna and Bella share a computer and never sign their e-mails. When I e-mail them, only the twin currently online responds. The probability that it is Anna who is online is \(p\) and she answers each question I ask her truthfully with probability \(a\), independently of all her other answers, even if a question is repeated. The probability that it is Bella who is online is~\(q\), where \(q=1-p\), and she answers each question truthfully with probability \(b\), independently of all her other answers, even if a question is repeated.

1. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. Show that the probability that the coin did come down heads is \(\frac{1}{2}\) if and only if \(2(ap+bq)=1\).
2. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `no'. Show that the probability (taking into account these answers) that the coin did come down heads is \(\frac{1}{2}\,\).
3. I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `yes'. Show that, if \(2(ap+bq)=1\), the probability (taking into account these answers) that the coin did come down heads is \(\frac{1}{2}\,\).

In a rabbit warren, underground chambers \(A, B, C\) and \(D\) are at the vertices of a square, and burrows join \(A\) to \(B\), \ \(B\) to \(C\), \ \(C\) to \(D\) and \(D\) to \(A\). Each of the chambers also has a tunnel to the surface. A rabbit finding itself in any chamber runs along one of the two burrows to a neighbouring chamber, or leaves the burrow through the tunnel to the surface. Each of these three possibilities is equally likely. Let \(p_A\,\), \(p_B\,\), \(p_C\) and \(p_D\) be the probabilities of a rabbit leaving the burrow through the tunnel from chamber \(A\), given that it is currently in chamber \(A, B, C\) or \(D\), respectively.

1. Explain why \(p_A = \frac13 + \frac13p_B + \frac13 p_D\).
2. Determine \(p_A\,\).
3. Find the probability that a rabbit which starts in chamber \(A\) does not visit chamber~\(C\), given that it eventually leaves the burrow through the tunnel in chamber \(A\).

In a game, a player tosses a biased coin repeatedly until two successive tails occur, when the game terminates. For each head which occurs the player wins \(\pounds 1\). If \(E\) is the expected number of tosses of the coin in the course of a game, and \(p\) is the probability of a head, explain why \[ E = p \l 1 + E \r + \l 1 - p \r p \l 2 + E \r + 2 \l 1 - p \r ^2\,, \] and hence determine \(E\) in terms of \(p\). Find also, in terms of \(p\), the expected winnings in the course of a game. A second game is played, with the same rules, except that the player continues to toss the coin until \(r\) successive tails occur. Show that the expected number of tosses in the course of a game is given by the expression \(\displaystyle {1 - q^r \over p q^r}\,\), where \(q = 1 - p\).

The random variable \(U\) takes the values \(+1\), \(0\) and \(-1\,\), each with probability \(\frac13\,\). The random variable \(V\) takes the values \(+1\) and \(-1\) as follows:

1. Show that the probability that both roots of the equation \(x^2+Ux+V=0\) are real is \(\frac12\;\).
2. Find the expected value of the larger root of the equation \(x^2+Ux+V=0\,\), given that both roots are real.
3. Find the probability that the roots of the equation $$x^3+(U-2V)x^2+(1-2UV)x + U=0$$ are all positive.

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence of heads and tails and the player whose sequence appears first wins. Four players, \(A\), \(B\), \(C\) and \(D\) take turns to play the game. Each time they play, \(A\) is assigned the sequence TTH (i.e.~Tail then Tail then Head), \(B\) is assigned THH, \(C\) is assigned HHT and \(D\) is assigned~HTT.

1. \(A\) and \(B\) play the game. Let \(p_{\mathstrut\mbox{\tiny HH}}\), \(p_{\mathstrut\mbox{\tiny HT}}\), \(p_{\mathstrut\mbox{\tiny TH}}\) and \(p_{\mathstrut\mbox{\tiny TT}}\) be the probabilities of \(A\) winning the game given that the first two tosses of the coin show HH, HT, TH and TT, respectively. Explain why \(p_{\mathstrut\mbox{\tiny TT}} = 1\,\), and why $p_{\mathstrut\mbox{\tiny HT}} = {1 \over 2} \, p_{\mathstrut\mbox{\tiny TH}} + {1\over 2} \, p_{\mathstrut\mbox{\tiny TT}}\,$. Show that $p_{\mathstrut\mbox{\tiny HH}} = p_{\mathstrut\mbox{\tiny HT}} = {2 \over 3}$ and that \(p_{\mathstrut\mbox{\tiny TH}} = {1\over 3}\,\). Deduce that the probability that A wins the game is \({2\over 3}\,\).
2. \(B\) and \(C\) play the game. Find the probability that \(B\) wins.
3. Show that if \(C\) plays \(D\), then \(C\) is more likely to win than \(D\), but that if \(D\) plays \(A\), then \(D\) is more likely to win than \(A\).

Every person carries two genes which can each be either of type \(A\) or of type \(B\). It is known that \(81\%\) of the population are \(AA\) (i.e. both genes are of type \(A\)), \(18\%\) are \(AB\) (i.e. there is one gene of type \(A\) and one of type \(B\)) and \(1\%\) are \(BB\). A child inherits one gene from each of its parents. If one parent is \(AA\), the child inherits a gene of type \(A\) from that parent; if the parent is \(BB\), the child inherits a gene of type \(B\) from that parent; if the parent is \(AB\), the inherited gene is equally likely to be \(A\) or \(B\).

1. Given that two \(AB\) parents have four children, show that the probability that two of them are \(AA\) and two of them are \(BB\) is \(3/128\).
2. My mother is \(AB\) and I am \(AA\). Find the probability that my father is \(AB\).

It is known that there are three manufacturers \(A, B, C,\) who can produce micro chip MB666. The probability that a randomly selected MB666 is produced by \(A\) is \(2p\), and the corresponding probabilities for \(B\) and \(C\) are \(p\) and \(1 - 3p\), respectively, where \({{0} \le p \le {1 \over 3}}.\) It is also known that \(70\%\) of MB666 micro chips from \(A\) are sound and that the corresponding percentages for \(B\) and \(C\) are \(80\%\) and \(90\%\), respectively. Find in terms of \(p\), the conditional probability, \(\P(A {\vert} S)\), that if a randomly selected MB666 chip is found to be sound then it came from \(A\), and also the conditional probability, \(\P(C {\vert} S)\), that if it is sound then it came from \(C\). A quality inspector took a random sample of one MB666 micro chip and found it to be sound. She then traced its place of manufacture to be \(A\), and so estimated \(p\) by calculating the value of \(p\) that corresponds to the greatest value of \(\P(A {\vert} S)\). A second quality inspector also a took random sample of one MB666 chip and found it to be sound. Later he traced its place of manufacture to be \(C\) and so estimated \(p\) by applying the procedure of his colleague to \(\P(C {\vert} S)\). Determine the values of the two estimates and comment briefly on the results obtained.

Write down the probability of obtaining \(k\) heads in \(n\) tosses of a fair coin. Now suppose that \(k\) is known but \(n\) is unknown. A maximum likelihood estimator (MLE) of \(n\) is defined to be a value (which must be an integer) of \(n\) which maximizes the probability of \(k\) heads. A friend has thrown a fair coin a number of times. She tells you that she has observed one head. Show that in this case there are two MLEs of the number of tosses she has made. She now tells you that in a repeat of the exercise she has observed \(k\) heads. Find the two MLEs of the number of tosses she has made. She next uses a coin biased with probability \(p\) (known) of showing a head, and again tells you that she has observed \(k\) heads. Find the MLEs of the number of tosses made. What is the condition for the MLE to be unique?

The diagnostic test AL has a probability 0.9 of giving a positive result when applied to a person suffering from the rare disease mathematitis. It also has a probability 1/11 of giving a false positive result when applied to a non-sufferer. It is known that only \(1\%\) of the population suffer from the disease. Given that the test AL is positive when applied to Frankie, who is chosen at random from the population, what is the probability that Frankie is a sufferer? In an attempt to identify sufferers more accurately, a second diagnostic test STEP is given to those for whom the test AL gave a positive result. The probablility of STEP giving a positive result on a sufferer is 0.9, and the probability that it gives a false positive result on a non-sufferer is \(p\). Half of those for whom AL was positive and on whom STEP then also gives a positive result are sufferers. Find \(p\).

I have a bag initially containing \(r\) red fruit pastilles (my favourites) and \(b\) fruit pastilles of other colours. From time to time I shake the bag thoroughly and remove a pastille at random. (It may be assumed that all pastilles have an equal chance of being selected.) If the pastille is red I eat it but otherwise I replace it in the bag. After \(n\) such drawings, I find that I have only eaten one pastille. Show that the probability that I ate it on my last drawing is \[\frac{(r+b-1)^{n-1}}{(r+b)^{n}-(r+b-1)^{n}}.\]

Mr Blond returns to his flat to find it in complete darkness. He knows that this means that one of four assassins Mr 1, Mr 2, Mr 3 or Mr 4 has set a trap for him. His trained instinct tells him that the probability that Mr \(i\) has set the trap is \(i/10\). His knowledge of their habits tells him that Mr \(i\) uses a deadly trained silent anaconda with probability \((i+1)/10\), a bomb with probability \(i/10\) and a vicious attack canary with probability \((9-2i)/10\) \([i=1,2,3,4]\). He now listens carefully and, hearing no singing, concludes correctly that no canary is involved. If he switches on the light and the trap is a bomb he has probability \(1/2\) of being killed but if the trap is an anaconda he has probability \(2/3\) of survival. If he does not switch on the light and the trap is a bomb he is certain to survive but, if the trap is an anaconda, he has a probability \(1/2\) of being killed. His professional pride means that he must enter the flat. Advise Mr Blond, giving reasons for your advice.

It has been observed that Professor Ecks proves three types of theorems: 1, those that are correct and new; 2, those that are correct, but already known; 3, those that are false. It has also been observed that, if a certain of her theorems is of type \(i\), then her next theorem is of type \(j\) with probability \(p\low_{ij},\) where \(p\low_{ij}\) is the entry in the \(i\)th row and \(j\)th column of the following array: \[ \begin{pmatrix}0.3 & 0.3 & 0.4\\ 0.2 & 0.4 & 0.4\\ 0.1 & 0.3 & 0.6 \end{pmatrix}\,. \] Let \(a_{i},\) \(i=1,2,3\), be the probability that a given theorem is of type \(i\), and let \(b_{j}\) be the consequent probability that the next theorem is of type \(j\).

1. Explain why \(b_{j}=a\low_{1}p\low_{1j}+a\low_{2}p\low_{2j}+a\low_{3}p\low_{3j}\,.\)
2. Find values of \(a\low_{1},a\low_{2}\) and \(a\low_{3}\) such that \(b_{i}=a_{i}\) for \(i=1,2,3.\)
3. For these values of the \(a_{i}\) find the probabilities \(q\low_{ij}\) that, if a particular theorem is of type \(j\), then the \textit{preceding }theorem was of type \(i\).

A biased coin, with a probability \(p\) of coming up heads and a probability \(q=1-p\) of coming up tails, is tossed repeatedly. Let \(A\) be the event that the first run of \(r\) successive heads occurs before the first run of \(s\) successive tails. If \(H\) is the even that on the first toss the coin comes up heads and \(T\) is the event that it comes up tails, show that \begin{alignat*}{1} \mathrm{P}(A|H) & =p^{\alpha}+(1-p^{\alpha})\mathrm{P}(A|T),\\ \mathrm{P}(A|T) & =(1-q^{\beta})\mathrm{P}(A|H), \end{alignat*} where \(\alpha\) and \(\beta\) are to be determined. Use these two equations to find \(\mathrm{P}(A|H),\) \(\mathrm{P}(A|T),\) and hence \(\mathrm{P}(A).\)

A message of \(10^{k}\) binary digits is sent along a fibre optic cable with high probabilities \(p_{0}\) and \(p_{1}\) that the digits 0 and 1, respectively, are received correctly. If the probability of a digit in the original message being a 1 is \(\alpha,\) find the probability that the entire message is received correctly. Find the probability \(\beta\) that a randomly chosen digit in the message is received as a 1 and show that \(\beta=\alpha\) if, and only if \[ \alpha=\frac{q_{0}}{q_{1}+q_{0}}, \] where \(q_{0}=1-p_{0}\) and \(q_{1}=1-p_{1}.\) If this condition is satisfied and the received message consists entirely of zeros, what is the probability that it is correct? If now \(q_{0}=q_{1}=q\) and \(\alpha=\frac{1}{2},\) find the approximate value of \(q\) which will ensure that a message of one million binary digits has a fifty-fifty chance of being received entirely correctly. The probability of error \(q\) is proportional to the square of the length of the cable. Initially the length is such that the probability of a message of one million binary bits, among which 0 and 1 are equally likely, being received correctly is \(\frac{1}{2}.\) What would this probability become if a booster station were installed at its mid-point, assuming that the booster station re-transmits the received version of the message, and assuming that terms of order \(q^{2}\) may be ignored?

Bread roll throwing duels at the Drones' Club are governed by a strict etiquette. The two duellists throw alternatively until one is hit, when the other is declared the winner. If Percy has probability \(p>0\) of hitting his target and Rodney has probability \(r>0\) of hitting his, show that, if Percy throws first, the probability that he beats Rodney is \[ \frac{p}{p+r-pr}. \] Algernon, Bertie and Cuthbert decide to have a three sided duel in which they throw in order \(\mathrm{A,B,C,A,B,C,}\ldots\) except that anyone who is hit must leave the game. Cuthbert always his target, Bertie hits his target with probability \(3/5\) and Algernon hits his target with probability \(2/5.\) Bertie and Cuthbert will always aim at each other if they are both still in the duel. Otherwise they aim at Algernon. With his first shot Algernon may aim at either Bertie or Cuthbert or deliberately miss both. Faced with only one opponent Algernon will aim at him. What are Algernon's changes of winning if he:

• sep}{3mm}
• [\bf (i)] hits Cuthbert with his first shot?
• [\bf (ii)] hits Bertie with his first shot?
• [\bf (iii)] misses with his first shot?

Calamity Jane sits down to play the game of craps with Buffalo Bill. In this game she rolls two fair dice. If, on the first throw, the sum of the dice is \(2,3\) or \(12\) she loses, while if it is \(7\) or \(11\) she wins. Otherwise Calamity continues to roll the dice until either the first sum is repeated, in which case she wins, or the sum is \(7\), in which case she loses. Find the probability that she wins on the first throw. Given that she throws more than once, show that the probability that she wins on the \(n\)th throw is \[ \frac{1}{48}\left(\frac{3}{4}\right)^{n-2}+\frac{1}{27}\left(\frac{13}{18}\right)^{n-2}+\frac{25}{432}\left(\frac{25}{36}\right)^{n-2}. \] Given that she throws more than \(m\) times, where \(m>1,\) what is the probability that she wins on the \(n\)th throw?

There are 28 colleges in Cambridge, of which two (New Hall and Newnham) are for women only; the others admit both men and women. Seven women, Anya, Betty, Celia, Doreen, Emily, Fariza and Georgina, are all applying to Cambridge. Each has picked three colleges at random to enter on her application form.

1. What is the probability that Anya's first choice college is single-sex?
2. What is the probability that Betty has picked Newnham?
3. What is the probability that Celia has picked at least one single-sex college?
4. Doreen's first choice is Newnham. What is the probability that one of her other two choices is New Hall?
5. Emily has picked Newnham. What is the probability that she has also picked New Hall?
6. Fariza's first choice college is single-sex. What is the probability that she has also chosen the other single-sex college?
7. One of Georgina's choices is a single-sex college. What is the probability that she has also picked the other single-sex college?

Captain Spalding is on a visit to the idyllic island of Gambriced. The population of the island consists of the two lost tribes of Frodox and the latest census shows that \(11/16\) of the population belong to the Ascii who tell the truth \(3/4\) of the time and \(5/16\) to the Biscii who always lie. The answers of an Ascii to each question (even if it is the same as one before) are independent. Show that the probability that an Ascii gives the same answer twice in succession to the same question is \(5/8\). Show that the probability that an Ascii gives the same answer twice is telling the truth is \(9/10.\) Captain Spalding addresses one of the natives as follows. \hspace{1.5em} \textsl{Spalding: }My good man, I'm afraid I'm lost. Should I go left or right to reach the nearest town?\nolinebreak \hspace{1.5em}\textsl{Native: }Left. \hspace{1.5em}\textsl{Spalding: }I am a little deaf. Should I go left or right to reach the nearest town? \hspace{1.5em}\textsl{Native (patiently): }Left. Show that, on the basis of this conversation, Captain Spalding should go left to try and reach the nearest town and that there is a probability \(99/190\) that this is the correct direction. The conversation resumes as follows. \hspace{1.5em}\textsl{Spalding: }I'm sorry I didn't quite hear that. Should I go left or right to reach the nearest town? \hspace{1.5em}\textsl{Native (loudly and clearly): }Left. Shouls Captain Spalding go left or right and why? Show that if he follows your advice the probability that this is the correct direction is \(331/628\).

A taxi driver keeps a packet of toffees and a packet of mints in her taxi. From time to time she takes either a toffee (with probability \(p\)) or mint (with probability \(q=1-p\)). At the beginning of the week she has \(n\) toffees and \(m\) mints in the packets. On the \(N\)th occasion that she reaches for a sweet, she discovers (for the first time) that she has run out of that kind of sweet. What is the probability that she was reaching for a toffee?

1. Show that, for any functions \(f\) and \(g\), and for any \(m \geq 0\), $$\sum_{r=1}^{m+1} f(r)\sum_{s=r-1}^m g(s) = \sum_{s=0}^m g(s)\sum_{r=1}^{s+1} f(r)$$
2. The random variables \(X_0, X_1, X_2, \ldots\) are defined as follows:
   • \(X_0\) takes the value \(0\) with probability \(1\);
   • \(X_{n+1}\) takes the values \(0, 1, \ldots, X_n + 1\) with equal probability, for \(n = 0, 1, \ldots\)
   1. Write down \(E(X_1)\). Find \(P(X_2 = 0)\) and \(P(X_2 = 1)\) and show that \(P(X_2 = 2) = \frac{1}{6}\). Hence calculate \(E(X_2)\).
   2. For \(n \geq 1\), show that $$P(X_n = 0) = \sum_{s=0}^{n-1} \frac{P(X_{n-1} = s)}{s+2}$$ and find a similar expression for \(P(X_n = r)\), for \(r = 1, 2, \ldots, n\).
   3. Hence show that \(E(X_n) = \frac{1}{2}(1 + E(X_{n-1}))\). Find an expression for \(E(X_n)\) in terms of \(n\), for \(n = 1, 2, \ldots\)

A game in a casino is played with a fair coin and an unbiased cubical die whose faces are labelled \(1, 1, 1, 2, 2\) and \(3.\) In each round of the game, the die is rolled once and the coin is tossed once. The outcome of the round is a random variable \(X\). The value, \(x\), of \(X\) is determined as follows. If the result of the toss is heads then \(x= \vert ks -1\vert\), and if the result of the toss is tails then \(x=\vert k-s\vert\), where \(s\) is the number on the die and \(k\) is a given number. Show that \(\mathbb{E}(X^2) = k +13(k-1)^2 /6\). Given that both \(\mathbb{E}(X^2)\) and \(\mathbb{E}(X)\) are positive integers, and that \(k\) is a single-digit positive integer, determine the value of \(k\), and write down the probability distribution of \(X\). A gambler pays \(\pounds 1\) to play the game, which consists of two rounds. The gambler is paid: \begin{enumerate} \item \(\pounds w\), where \(w\) is an integer, if the sum of the outcomes of the two rounds exceeds \(25\); \item \(\pounds 1\) if the sum of the outcomes equals \(25\); \item nothing if the sum of the outcomes is less that \(25\). \end{itemize} Find, in terms of \(w\), an expression for the amount the gambler expects to be paid in a game, and deduce the maximum possible value of \(w\), given that the casino's owners choose \(w\) so that the game is in their favour.

A discrete random variable \(X\) takes only positive integer values. Define \(\E(X)\) for this case, and show that \[\E(X) =\sum^{\infty}_{n=1}\P\left(X\ge n \right).\] I am collecting toy penguins from cereal boxes. Each box contains either one daddy penguin or one mummy penguin. The probability that a given box contains a daddy penguin is \(p\) and the probability that a given box contains a mummy penguin is \(q\), where \(p\ne0\), \(q\ne0\) and \(p+q=1\,\). Let \(X\) be the number of boxes that I need to open to get at least one of each kind of penguin. Show that \(\P(X\ge 4)= p^{3}+q^{3}\), and that \[ \E(X)=\frac{1}{pq}-1.\, \] Hence show that \(\E(X)\ge 3\,\).

I seat \(n\) boys and \(3\) girls in a line at random, so that each order of the \(n+3\) children is as likely to occur as any other. Let \(K\) be the maximum number of consecutive girls in the line so, for example, \(K=1\) if there is at least one boy between each pair of girls.

1. Find \(\P(K=3)\).
2. Show that \[\P(K=1)= \frac{n(n-1)}{(n+2)(n+3)}\,. \]
3. Find \(\E(K)\).

A box contains \(n\) pieces of string, each of which has two ends. I select two string ends at random and tie them together. This creates either a ring (if the two ends are from the same string) or a longer piece of string. I repeat the process of tying together string ends chosen at random until there are none left. Find the expected number of rings created at the first step and hence obtain an expression for the expected number of rings created by the end of the process. Find also an expression for the variance of the number of rings created. Given that \(\ln 20 \approx 3\) and that \(1+ \frac12 + \cdots + \frac 1n \approx \ln n\) for large \(n\), determine approximately the expected number of rings created in the case \(n=40\,000\).

A frog jumps towards a large pond. Each jump takes the frog either \(1\,\)m or \(2\,\)m nearer to the pond. The probability of a \(1\,\)m jump is \(p\) and the probability of a \(2\,\)m jump is \(q\), where \(p+q=1\), the occurence of long and short jumps being independent.

1. Let \(p_n(j)\) be the probability that the frog, starting at a point \((n-\frac12)\,\)m away from the edge of the pond, lands in the pond for the first time on its \(j\)th jump. Show that \(p_2(2)=p\).
2. Let \(u_n\) be the expected number of jumps, starting at a point \((n-\frac12)\,\)m away from the edge of the pond, required to land in the pond for the first time. Write down the value of \(u_1\). By finding first the relevant values of \(p_n(m)\), calculate \(u_2\) and show that $u_3= 3-2q+q^2\(.
3. Given that \)u_n\( can be expressed in the form \)u_n= A(-q)^{n-1} +B +Cn$, where \(A\), \(B\) and \(C\) are constants (independent of \(n\)), show that \(C= (1+q)^{-1}\) and find \(A\) and \(B\) in terms of \(q\). Hence show that, for large \(n\), \(u_n \approx \dfrac n{p+2q}\) and explain carefully why this result is to be expected.

A bag contains \(b\) balls, \(r\) of them red and the rest white. In a game the player must remove balls one at a time from the bag (without replacement). She may remove as many balls as she wishes, but if she removes any red ball, she loses and gets no reward at all. If she does not remove a red ball, she is rewarded with \pounds 1 for each white ball she has removed. If she removes \(n\) white balls on her first \(n\) draws, calculate her expected gain on the next draw and show that %her expected total reward would be the same as before it is zero if \(\ds n = {b-r \over r+1}\,\). Hence, or otherwise, show that she will maximise her expected total reward if she aims to remove \(n\) balls, where \[ n = \mbox{ the integer part of } \ds {b + 1 \over r + 1}\;. \] With this value of \(n\), show that in the case \(r=1\) and \(b\) even, her expected total reward is \(\pounds {1 \over 4}b\,\), and find her expected total reward in the case \(r=1\) and \(b\) odd.

In a bag are \(n\) balls numbered 1, 2, \(\ldots\,\), \(n\,\). When a ball is taken out of the bag, each ball is equally likely to be taken.

1. A ball is taken out of the bag. The number on the ball is noted and the ball is replaced in the bag. The process is repeated once. Explain why the expected value of the product of the numbers on the two balls is \[ \frac 1 {n^2} \sum_{r=1}^n\sum_{s=1}^n rs \] and simplify this expression.
2. A ball is taken out of the bag. The number on the ball is noted and the ball is {\sl not} replaced in the bag. Another ball is taken out of the bag and the number on this ball is noted. Show that the expected value of the product of the two numbers is \[ \frac{(n+1)(3n+2)}{12}\;. \]

You play the following game. You throw a six-sided fair die repeatedly. You may choose to stop after any throw, except that you must stop if you throw a 1. Your score is the number obtained on your last throw. Determine the strategy that you should adopt in order to maximize your expected score, explaining your reasoning carefully.

A bag contains 5 white balls, 3 red balls and 2 black balls. In the game of Blackball, a player draws a ball at random from the bag, looks at it and replaces it. If he has drawn a white ball, he scores one point, while for a red ball he scores two points, these scores being added to his total score before he drew the ball. If he has drawn a black ball, the game is over and his final score is zero. After drawing a red or white ball, he can either decide to stop, when his final score for the game is the total so far, or he may elect to draw another ball. The starting score is zero. Juggins' strategy is to continue drawing until either he draws a black ball (when of course he must stop, with final score zero), or until he has drawn three (non-black) balls, when he elects to stop. Find the probability that in any game he achieves a final score of zero by employing this strategy. Find also his expected final score. Muggins has so far scored \(N\) points, and is deciding whether to draw another ball. Find the expected score if another ball is drawn, and suggest a strategy to achieve the greatest possible average final score in each game.

1. The three integers \(n_1\), \(n_2\) and \(n_3\) satisfy \(0 < n_1 < n_2 < n_3\) and \(n_1 + n_2 > n_3\). Find the number of ways of choosing the pair of numbers \(n_1\) and \(n_2\) in the cases \(n_3 = 9\) and \(n_3 = 10\). Given that \(n_3 = 2n + 1\), where \(n\) is a positive integer, write down an expression (which you need not prove is correct) for the number of ways of choosing the pair of numbers \(n_1\) and \(n_2\). Simplify your expression. Write down and simplify the corresponding expression when \(n_3 = 2n\), where \(n\) is a positive integer.
2. You have \(N\) rods, of lengths \(1, 2, 3, \ldots, N\) (one rod of each length). You take the rod of length \(N\), and choose two more rods at random from the remainder, each choice of two being equally likely. Show that, in the case \(N = 2n + 1\) where \(n\) is a positive integer, the probability that these three rods can form a triangle (of non-zero area) is $$\frac{n - 1}{2n - 1}.$$ Find the corresponding probability in the case \(N = 2n\), where \(n\) is a positive integer.
3. You have \(2M + 1\) rods, of lengths \(1, 2, 3, \ldots, 2M + 1\) (one rod of each length), where \(M\) is a positive integer. You choose three at random, each choice of three being equally likely. Show that the probability that the rods can form a triangle (of non-zero area) is $$\frac{(4M + 1)(M - 1)}{2(2M + 1)(2M - 1)}.$$ Note: \(\sum_{k=1}^{K} k^2 = \frac{1}{6}K(K + 1)(2K + 1)\).

1. \(X_{1},X_{2},\ldots,X_{n}\) are independent identically distributed random variables drawn from a uniform distribution on \([0,1].\) The random variables \(A\) and \(B\) are defined by \[ A=\min(X_{1},\ldots,X_{n}),\qquad B=\max(X_{1},\ldots,X_{n}). \] For any fixed \(k\), such that \(0< k< \frac{1}{2},\) let \[ p_{n}=\mathrm{P}(A\leqslant k\mbox{ and }B\geqslant1-k). \] What happens to \(p_{n}\) as \(n\rightarrow\infty\)? Comment briefly on this result.
2. Lord Copper, the celebrated and imperious newspaper proprietor, has decided to run a lottery in which each of the \(4,000,000\) readers of his newspaper will have an equal probability \(p\) of winning \(\pounds 1,000,000\) and their changes of winning will be independent. He has fixed all the details leaving to you, his subordinate, only the task of choosing \(p\). If nobody wins \(\pounds 1,000,000\), you will be sacked, and if more than two readers win \(\pounds 1,000,000,\) you will also be sacked. Explaining your reasoning, show that however you choose \(p,\) you will have less than a 60\% change of keeping your job.

In a lottery, each of the \(N\) participants pays \(\pounds c\) to the organiser and picks a number from \(1\) to \(N\). The organiser picks at random the winning number from \(1\) to \(N\) and all those participants who picked this number receive an equal share of the prize, \(\pounds J\).

1. The participants pick their numbers independently and with equal probability. Obtain an expression for the probability that no participant picks the winning number, and hence determine the organiser's expected profit. Use the approximation \[ \left( 1 - \frac{a}{N} \right)^N \approx \e^{-a} \tag{\(*\)} \] to show that if \(2Nc = J\) then the organiser will expect to make a loss. {\bf Note}: \(\e > 2\).
2. Instead of the numbers being equally popular, a fraction \(\gamma\) of the numbers are popular and the rest are unpopular. For each participant, the probability of picking any given popular number is \(\dfrac{a}{N}\) and the probability of picking any given unpopular number is~\(\dfrac{b}{N}\,\). Find a relationship between \(a\), \(b\) and \(\gamma\). Show that, using the approximation \((*)\), the organiser's expected profit can be \mbox{expressed} in the form \[ A\e^{-a} + B\e^{-b} +C \,, \] where \(A\), \(B\) and \(C\) can be written in terms of \(J\), \(c\), \(N\) and \(\gamma\). In the case \(\gamma = \frac18\) and \(a=9b\), find \(a\) and \(b\). Show that, if \(2Nc = J\), then the organiser will expect to make a profit. {\bf Note}: \(\e < 3\).

A random number generator prints out a sequence of integers \(I_1, I_2, I_3, \dots\). Each integer is independently equally likely to be any one of \(1, 2, \dots, n\), where \(n\) is fixed. The random variable \(X\) takes the value \(r\), where \(I_r\) is the first integer which is a repeat of some earlier integer. Write down an expression for \(\mathbb{P}(X=4)\).

1. Find an expression for \(\mathbb{P}(X=r)\), where \(2\le r\le n+1\). Hence show that, for any positive integer \(n\), \[ \frac 1n + \left(1-\frac1n\right) \frac 2 n + \left(1-\frac1n\right)\left(1-\frac2n\right) \frac3 n + \cdots \ = \ 1 \,. \]
2. Write down an expression for \(\mathbb{E}(X)\). (You do not need to simplify it.)
3. Write down an expression for \(\mathbb{P}(X\ge k)\).
4. Show that, for any discrete random variable \(Y\) taking the values \(1, 2, \dots, N\), \[ \mathbb{E}(Y) = \sum_{k=1}^N \mathbb{P}(Y\ge k)\,. \] Hence show that, for any positive integer \(n\), \[ \left(1-\frac{1^2}n\right) + \left(1-\frac1n\right)\left(1-\frac{2^2}n\right) + \left(1-\frac1n\right)\left(1-\frac{2}n\right)\left(1-\frac{3^2}n\right) + \cdots \ = \ 0. \]

Integers \(n_{1},n_{2},\ldots,n_{r}\) (possibly the same) are chosen independently at random from the integers \(1,2,3,\ldots,m\). Show that the probability that \(\left|n_{1}-n_{2}\right|=k\), where \(1\leqslant k\leqslant m-1\), is \(2(m-k)/m^{2}\) and show that the expectation of \(\left|n_{1}-n_{2}\right|\) is \((m^{2}-1)/(3m)\). Verify, for the case \(m=2\), the result that the expection of \(\left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|\) is \(2(m^{2}-1)/(3m).\) Write down the expectation, for general \(m\), of \[ \left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|+\cdots+\left|n_{r-1}-n_{r}\right|. \] Desks in an examination hall are placed a distance \(d\) apart in straight lines. Each invigilator looks after one line of \(m\) desks. When called by a candidate, the invigilator walks to that candidate's desk, and stays there until called again. He or she is equally likely to be called by any of the \(m\) candidates in the line but candidates never call simultaneously or while the invigilator is attending to another call. At the beginning of the examination the invigilator stands by the first desk. Show that the expected distance walked by the invigilator in dealing with \(N+1\) calls is \[ \frac{d(m-1)}{6m}[2N(m+1)+3m]. \]

An examination consists of several papers, which are marked independently. The mark given for each paper can be an integer from \(0\) to \(m\) inclusive, and the total mark for the examination is the sum of the marks on the individual papers. In order to make the examination completely fair, the examiners decide to allocate the mark for each paper at random, so that the probability that any given candidate will be allocated \(k\) marks \((0\leqslant k\leqslant m)\) for a given paper is \((m+1)^{-1}\). If there are just two papers, show that the probability that a given candidate will receive a total of \(n\) marks is \[ \frac{2m-n+1}{\left(m+1\right)^{2}} \] for \(m< n\leqslant2m\), and find the corresponding result for \(0\leqslant n\leqslant m\). If the examination consists of three papers, show that the probability that a given candidate will receive a total of \(n\) marks is \[ \frac{6mn-4m^{2}-2n^{2}+3m+2}{2\left(m+1\right)^{2}} \] in the case \(m< n\leqslant2m\). Find the corresponding result for \(0\leqslant n\leqslant m\), and deduce the result for \(2m< n\leqslant3m\).

A cricket team has only three bowlers, Arthur, Betty and Cuba, each of whom bowls 30 balls in any match. Past performance reveals that, on average, Arthur takes one wicket for every 36 balls bowled, Betty takes one wicket for every 25 balls bowled, and Cuba takes one wicket for every 41 balls bowled.

1. In one match, the team took exactly one wicket, but the name of the bowler was not recorded. Using a binomial model, find the probability that Arthur was the bowler.
2. Show that the average number of wickets taken by the team in a match is approximately 3. Give with brief justification a suitable model for the number of wickets taken by the team in a match and show that the probability of the team taking at least five wickets in a given match is approximately \(\frac15\). [You may use the approximation \(\e^3 = 20\).]

Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of opposite polarities facing each other, they join together to form a single unit. If they have ends of the same polarity facing each other, they stand apart. Find the expectation and variance of the number of separate units in terms of the total number \(N\) of magnets.

By considering the coefficients of \(t^{n}\) in the equation \[(1+t)^{n}(1+t)^{n}=(1+t)^{2n},\] or otherwise, show that \[\binom{n}{0}\binom{n}{n}+\binom{n}{1}\binom{n}{n-1}+\cdots +\binom{n}{r}\binom{n}{n-r}+\cdots+\binom{n}{n}\binom{n}{0} =\binom{2n}{n}.\] The large American city of Triposville is laid out in a square grid with equally spaced streets running east-west and avenues running north-south. My friend is staying at a hotel \(n\) avenues west and \(n\) streets north of my hotel. Both hotels are at intersections. We set out from our own hotels at the same time. We walk at the same speed, taking 1 minute to go from one intersection to the next. Every time I reach an intersection I go north with probability \(1/2\) or west with probability \(1/2\). Every time my friend reaches an intersection she goes south with probability \(1/2\) or east with probability \(1/2\). Our choices are independent of each other and of our previous decisions. Indicate by a sketch or by a brief description the set of points where we could meet. Find the probability that we meet. Suppose that I oversleep and leave my hotel \(2k\) minutes later than my friend leaves hers, where \(k\) is an integer and \(0\leqslant 2k\leqslant n\). Find the probability that we meet. Have you any comment? If \(n=1\) and I leave my hotel \(1\) minute later than my friend leaves hers, what is the probability that we meet and why?

Show Solution

\begin{align*} && (1+t)^{n}(1+t)^{n}&=(1+t)^{2n} \\ [t^n]: &&\sum_{k=0}^n \underbrace{\binom{n}{k}}_{t^k\text{ from left bracket}} \underbrace{\binom{n}{n-k}}_{t^{n-k}\text{ from right bracket}} &= \binom{2n}{n} \end{align*}

+ - ↺

From each point, we can get to the diagonal ahead of us, so each move only takes us one diagonal closer together. Therefore we can only meet on the diagonal. The number of routes we can meet at is \begin{align*} && R &= \sum_{k=0}^n \underbrace{\binom{n}{k}}_{\text{I go up } k}\underbrace{\binom{n}{n-k}}_{\text{she goes down }n-k} \\ &&&= \binom{2n}{n} \end{align*} Therefore the probability is \(\displaystyle \frac1{2^{2n}} \binom{2n}n\). If I leave \(2k\) minutes late, then we will be attempting meet on a diagonal which is \(2k\) closer to me. The probability this occurs is \begin{align*} && \frac{1}{2^{2n}}\sum_{j=0}^{n-k}\binom{n-k}{j}\binom{n+k}{n-j} &= \frac{1}{2^{2n}}\binom{2n}{n} \end{align*} (by considering the coefficient of \(t^n\) in \((1+t)^{n+k}(1+t)^{n-k} =(1+t)^{2n}\)) This probability is unchanged, because you can consider the two paths as one path by one random person, conditional on them meeting and the delay doesn't change anything. If \(n = 1\) and I leave late, the only way we meet is if we end up walking towards each other down the same street (not at an intersection). This means I need to walk towards the intersection she reaches after the first minute \(\frac12\) and she needs to walk towards me \(\frac12\) so we have probability \(\frac14\)

Widgets are manufactured in batches of size \((n+N)\). Any widget has a probability \(p\) of being faulty, independent of faults in other widgets. The batches go through a quality control procedure in which a sample of size \(n\), where \(n\geqslant2\), is taken from each batch and tested. If two or more widgets in the sample are found to be faulty, all widgets in the batch are tested and all faults corrected. If fewer than two widgets in the sample are found to be faulty, the sample is replaced in the batch and no faults are corrected. Show that the probability that the batch contains exactly \(k\), where \(k\leqslant N\), faulty widgets after quality control is \[ \frac{\left[N+1+k\left(n-1\right)\right]N!}{\left(N-k+1\right)!k!}p^{k}\left(1-p\right)^{N+n-k}, \] and verify that this formula also gives the correct answer for \(k=N+1\). Show that the expected number of faulty widgets in a batch after quality control is \[ \left[N+n+pN(n-1)\right]p(1-p)^{n-1}. \]

1. By considering the sum of a geometric series, or otherwise, show that \[\sum_{r=1}^{\infty} rx^{r-1} = \frac{1}{(1-x)^2} \quad \text{for } |x| < 1.\]
2. Ali plays a game with a fair \(2k\)-sided die. He rolls the die until the first \(2k\) appears. Ali wins if all the numbers he rolls are even.
   1. Find the probability that Ali wins the game. If Ali wins the game, he earns £1 for each roll, including the final one. If he loses, he earns nothing.
   2. Find Ali's expected earnings from playing the game.
3. Find a simplified expression for \[1 + 2\binom{n}{1}x + 3\binom{n}{2}x^2 + \ldots + (n+1)x^n,\] where \(n\) is a positive integer.
4. Zen plays a different game with a fair \(2k\)-sided die. She rolls the die until the first \(2k\) appears, and wins if the numbers rolled are strictly increasing in size. For example, if \(k = 3\), she wins if she rolls 2, 6 or 1, 4, 5, 6, but not if she rolls 1, 4, 2, 6 or 1, 3, 3, 6. If Zen wins the game, she earns £1 for each roll, including the final one. If she loses, she earns nothing. Find Zen's expected earnings from playing the game.
5. Using the approximation \[\left(1 + \frac{1}{n}\right)^n \approx e \quad \text{for large } n,\] show that, when \(k\) is large, Zen's expected earnings are a little over 35\% more than Ali's expected earnings.

1. Two people adopt the following procedure for deciding where to go for a cup of tea: either to a hotel or to a tea shop. Each person has a coin which has a probability \(p\) of showing heads and \(q\) of showing tails (where \(p+q = 1\)). In each round of the procedure, both people toss their coins once. If both coins show heads, then both people go to the hotel; if both coins show tails, then both people go to the tea shop; otherwise, they continue to the next round. This process is repeated until a decision is made. Show that the probability that they make a decision on the \(n\)th round is $$(q^2 + p^2)(2qp)^{n-1}.$$ Show also that the probability that they make a decision on or before the \(n\)th round is at least $$1 - \frac{1}{2^n}$$ whatever the value of \(p\).
2. Three people adopt the following procedure for deciding where to go for a cup of tea: either to a hotel or to a tea shop. Each person has a coin which has a probability \(p\) of showing heads and \(q\) of showing tails (where \(p + q = 1\)). In the first round of the procedure, all three people toss their coins once. If all three coins show heads, then all three people go to the hotel; if all three coins show tails, then all three people go to the tea shop; otherwise, they continue to the next round. In the next round the two people whose coins showed the same face toss again, but the third person just turns over his or her coin. If all three coins show heads, then all three people go to the hotel; if all three coins show tails, then all three people go to the tea shop; otherwise, they go to the third round. Show that the probability that they make a decision on or before the second round is at least \(\frac{7}{16}\), whatever the value of \(p\).

\(A,B\) and \(C\) play a table tennis tournament. The winner of the tournament will be the first person to win two games in a row. In any game, whoever is not playing acts as a referee, and each playerhas equal chance of winning the game. The first game of the tournament is played between \(A\) and \(B\), with \(C\) as referee. Thereafter, if the tournament is still undecided at the end of any game, the winner and referee of that game play the next game. The tournament is recorded by listing in order the winners of each game, so that, for example, \(ACC\) records a three-game tournament won by \(C\), the first game having been won by \(A\). Determine which of the following sequences of letters could be the record of a complete tournament, giving brief reasons for your answers:

1. \(ACB\),
2. \(ABB\),
3. \(ACBB\).

In a game, I toss a coin repeatedly. The probability, \(p\), that the coin shows Heads on any given toss is given by \[ p= \frac N{N+1} \,, \] where \(N\) is a positive integer. The outcomes of any two tosses are independent. The game has two versions. In each version, I can choose to stop playing after any number of tosses, in which case I win \pounds\(H\), where \(H\) is the number of Heads I have tossed. However, the game may end before that, in which case I win nothing.

1. In version 1, the game ends when the coin first shows Tails (if I haven't stopped playing before that). I decide from the start to toss the coin until a total of \(h\) Heads have been shown, unless the game ends before then. Find, in terms of \(h\) and \(p\), an expression for my expected winnings and show that I can maximise my expected winnings by choosing \(h=N\).
2. In version 2, the game ends when the coin shows Tails on two {\em consecutive} tosses (if I haven't stopped playing before that). I decide from the start to toss the coin until a total of \(h\) Heads have been shown, unless the game ends before then. Show that my expected winnings are \[ \frac{ hN^h (N+2)^h}{(N+1)^{2h}} \,. \] In the case \(N=2\,\), use the approximation \(\log_3 2 \approx 0.63\) to show that the maximum value of my expected winnings is approximately \pounds 3.

A biased coin has probability \(p\) of showing a head and probability \(q\) of showing a tail, where \(p\ne0\), \(q\ne0\) and \(p\ne q\). When the coin is tossed repeatedly, runs occur. A {\em straight run} of length \(n\) is a sequence of \(n\) consecutive heads or \(n\) consecutive tails. An {\em alternating run} of length \(n\) is a sequence of length \(n\) alternating between heads and tails. An alternating run can start with either a head or a tail. Let \(S\) be the length of the longest straight run beginning with the first toss and let \(A\) be the length of the longest alternating run beginning with the first toss.

1. Explain why \(\P(A=1)=p^2+q^2\) and find \(\P(S=1)\). Show that \(\P(S=1)<\P(A=1)\).
2. Show that \(\P(S=2)= \P(A=2)\) and determine the relationship between \(\P(S=3)\) and \( \P(A=3)\).
3. Show that, for \(n>1\), \(\P(S=2n)>\P(A=2n)\) and determine the corresponding relationship between \(\P(S=2n+1)\) and \(\P(A=2n+1)\). [You are advised {\em not} to use \(p+q=1\) in this part.]

The infinite series \(S\) is given by \[ S = 1 + (1 + d)r + (1 + 2d)r^2 + \cdots + (1+nd)r^n +\cdots\; ,\] for \(\vert r \vert <1\,\). By considering \(S - rS\), or otherwise, prove that \[ S = \frac 1{1-r} + \frac {rd}{(1-r)^2} \,.\] Arthur and Boadicea shoot arrows at a target. The probability that an arrow shot by Arthur hits the target is \(a\); the probability that an arrow shot by Boadicea hits the target is \(b\). Each shot is independent of all others. Prove that the expected number of shots it takes Arthur to hit the target is \(1/a\). Arthur and Boadicea now have a contest. They take alternate shots, with Arthur going first. The winner is the one who hits the target first. The probability that Arthur wins the contest is \(\alpha\) and the probability that Boadicea wins is \(\beta\). Show that \[ \alpha = \frac a {1-a'b'}\,, \] where \(a' = 1-a\) and \(b'=1-b\), and find \(\beta\). Show that the expected number of shots in the contest is \(\displaystyle \frac \alpha a + \frac \beta b\,.\)

I have two identical dice. When I throw either one of them, the probability of it showing a 6 is \(p\) and the probability of it not showing a 6 is \(q\), where \(p+q=1\). As an experiment to determine \(p\), I throw the dice simultaneously until at least one die shows a 6. If both dice show a six on this throw, I stop. If just one die shows a six, I throw the other die until it shows a 6 and then stop.

1. Show that the probability that I stop after \(r\) throws is \(pq^{r-1}(2-q^{r-1}-q^r)\), and find an expression for the expected number of throws. [{\bf Note:} You may use the result $\ds \sum_{r=0}^\infty rx^r = x(1-x)^{-2}$.]
2. In a large number of such experiments, the mean number of throws was \(m\). Find an estimate for \(p\) in terms of \(m\).

A men's endurance competition has an unlimited number of rounds. In each round, a competitor has, independently, a probability \(p\) of making it through the round; otherwise, he fails the round. Once a competitor fails a round, he drops out of the competition; before he drops out, he takes part in every round. The grand prize is awarded to any competitor who makes it through a round which all the other remaining competitors fail; if all the remaining competitors fail at the same round the grand prize is not awarded. If the competition begins with three competitors, find the probability that:

1. all three drop out in the same round;
2. two of them drop out in round \(r\) (with \(r \ge 2\)) and the third in an earlier round;
3. the grand prize is awarded.

The life of a certain species of elementary particles can be described as follows. Each particle has a life time of \(T\) seconds, after which it disintegrates into \(X\) particles of the same species, where \(X\) is a random variable with binomial distribution \(\mathrm{B}(2,p)\,\). A population of these particles starts with the creation of a single such particle at \(t=0\,\). Let \(X_n\) be the number of particles in existence in the time interval \(nT < t < (n+1)T\,\), where \(n=1\,\), \(2\,\), \(\ldots\). Show that \(\P(X_1=2 \mbox { and } X_2=2) = 6p^4q^2\;\), where \(q=1-p\,\). Find the possible values of \(p\) if it is known that \(\P(X_1=2 \vert X_2=2) =9/25\,\). Explain briefly why \(\E(X_n) =2p\E(X_{n-1})\) and hence determine \(\E(X_n)\) in terms of \(p\). Show that for one of the values of \(p\) found above \(\lim_{n \to \infty}\E(X_n) = 0\) and that for the other \(\lim_{n \to \infty}\E(X_n) = + \infty\,\).

If a football match ends in a draw, there may be a "penalty shoot-out". Initially the teams each take 5 shots at goal. If one team scores more times than the other, then that team wins. If the scores are level, the teams take shots alternately until one team scores and the other team does not score, both teams having taken the same number of shots. The team that scores wins. Two teams, Team A and Team B, take part in a penalty shoot-out. Their probabilities of scoring when they take a single shot are \(p_A\) and \(p_B\) respectively. Explain why the probability \(\alpha\) of neither side having won at the end of the initial \(10\)-shot period is given by $$\alpha =\sum_{i=0}^5\binom{5}{i}^2(1-p_A)^i(1-p_B)^i\,p_A^{5-i}p_B^{5-i}.$$ Show that the expected number of shots taken is \(\ds10+ \frac{2\alpha}\beta\;,\) where \(\beta=p_A+p_B-2p_Ap_B\,.\)

I have \(k\) different keys on my key ring. When I come home at night I try one key after another until I find the key that fits my front door. What is the probability that I find the correct key in exactly \(n\) attempts in each of the following three cases?

1. At each attempt, I choose a key that I have not tried before but otherwise each choice is equally likely.
2. At each attempt, I choose a key from all my keys and each of the \(k\) choices is equally likely.
3. At the first attempt, I choose from all my keys and each of the \(k\) choices is equally likely. Thereafter, I choose from the keys that I did not try the previous time but otherwise each choice is equally likely.

The game of Cambridge Whispers starts with the first participant Albert flipping an un-biased coin and whispering to his neighbour Bertha whether it fell `heads' or `tails'. Bertha then whispers this information to her neighbour, and so on. The game ends when the final player Zebedee whispers to Albert and the game is won, by all players, if what Albert hears is correct. The acoustics are such that the listeners have, independently at each stage, only a probability of 2/3 of hearing correctly what is said. Find the probability that the game is won when there are just three players. By considering the binomial expansion of \((a+b)^n+(a-b)^n\), or otherwise, find a concise expression for the probability \(P\) that the game is won when is it played by \(n\) players each having a probability \(p\) of hearing correctly. % Show in particular that, if \(n\) is even, %\(P(n,1/10) = P(n,9/10)\).% How do you explain this apparent anomaly? To avoid the trauma of a lost game, the rules are now modified to require Albert to whisper to Bertha what he hears from Zebedee, and so keep the game going, if what he hears from Zebedee is not correct. Find the expected total number of times that Albert whispers to Bertha before the modified game ends. \noindent [You may use without proof the fact that \(\sum_1^\infty kx^{k-1}=(1-x)^{-2}\) for \(\vert x\vert<1\).]

The makers of Cruncho (`The Cereal Which Cares') are giving away a series of cards depicting \(n\) great mathematicians. Each packet of Cruncho contains one picture chosen at random. Show that when I have collected \(r\) different cards the expected number of packets I must open to find a new card is \(n/(n-r)\) where \(0\leqslant r\leqslant n-1.\) Show by means of a diagram, or otherwise, that \[ \frac{1}{r+1}\leqslant\int_{r}^{r+1}\frac{1}{x}\,\mathrm{d}x\leqslant\frac{1}{r} \] and deduce that \[ \sum_{r=2}^{n}\frac{1}{r}\leqslant\ln n\leqslant\sum_{r=1}^{n-1}\frac{1}{r} \] for all \(n\geqslant2.\) My children will give me no peace until we have the complete set of cards, but I am the only person in our household prepared to eat Cruncho and my spouse will only buy the stuff if I eat it. If \(n\) is large, roughly how many packets must I expect to consume before we have the set?

An unbiased twelve-sided die has its faces marked \(A,A,A,B,B,B,B,B,B,B,B,B.\) In a series of throws of the die the first \(M\) throws show \(A,\) the next \(N\) throws show \(B\) and the \((M+N+1)\)th throw shows \(A\). Write down the probability that \(M=m\) and \(N=n\), where \(m\geqslant0\) and \(n\geqslant1.\) Find

1. the marginal distributions of \(M\) and \(N\),
2. the mean values of \(M\) and \(N\).

1. A bag of sweets contains one red sweet and \(n\) blue sweets. I take a sweet from the bag, note its colour, return it to the bag, then shake the bag. I repeat this until the sweet I take is the red one. Find an expression for the probability that I take the red sweet on the \(r\)th attempt. What value of \(n\) maximises this probability?
2. Instead, I take sweets from the bag, without replacing them in the bag, until I take the red sweet. Find an expression for the probability that I take the red sweet on the \(r\)th attempt. What value of \(n\) maximises this probability?

A pack of cards consists of \(n+1\) cards, which are printed with the integers from \(0\) to \(n\). A~game consists of drawing cards repeatedly at random from the pack until the card printed with 0 is drawn, at which point the game ends. After each draw, the player receives \(\pounds 1\) if the card drawn shows any of the integers from \(1\) to \(w\) inclusive but receives nothing if the card drawn shows any of the integers from \(w+1\) to \(n\) inclusive.

1. [\bf (i)] In one version of the game, each card drawn is replaced immediately and randomly in the pack. Explain clearly why the probability that the player wins a total of exactly \(\pounds 3\) is equal to the probability of the following event occurring: out of the first four cards drawn which show numbers in the range \(0\) to \(w\), the numbers on the first three are non-zero and the number on the fourth is zero. Hence show that the probability that the player wins a total of exactly \(\pounds 3\) is equal to \(\ds \frac{w^3}{(w+1)^4}\). Write down the probability that the player wins a total of exactly \(\pounds r\) and hence find the expected total win.
2. [\bf (ii)] In another version of the game, each card drawn is removed from the pack. Show that the expected total win in this version is half of the expected total win in the other version.

A bag contains \(b\) black balls and \(w\) white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside.

1. If the black balls drawn are also put aside, find an expression for the expected number of black balls that have been drawn when the last white ball is removed.
2. If instead the black balls drawn are put back into the bag, prove that the expected number of times a black ball has been drawn when the first white ball is removed is \(b/w\,\). Hence write down, in the form of a sum, an expression for the expected number of times a black ball has been drawn when the last white ball is removed.

A group of biologists attempts to estimate the magnitude, \(N\), of an island population of voles ({\it Microtus agrestis}). Accordingly, the biologists capture a random sample of 200 voles, mark them and release them. A second random sample of 200 voles is then taken of which 11 are found to be marked. Show that the probability, \(p_N\), of this occurrence is given by $$ p_N = k{{{\big((N-200)!\big)}^2} \over {N!(N-389)!}}, $$ where \(k\) is independent of \(N\). The biologists then estimate \(N\) by calculating the value of \(N\) for which \(p_N\) is a maximum. Find this estimate. All unmarked voles in the second sample are marked and then the entire sample is released. Subsequently a third random sample of 200 voles is taken. Write down the probability that this sample contains exactly \(j\) marked voles, leaving your answer in terms of binomial coefficients. Deduce that $$ \sum_{j=0}^{200}{389 \choose j}{3247 \choose {200-j}} = {3636 \choose 200}. $$

An examiner has to assign a mark between 1 and \(m\) inclusive to each of \(n\) examination scripts (\(n\leqslant m\)). He does this randomly, but never assigns the same mark twice. If \(K\) is the highest mark that he assigns, explain why \[ \mathrm{P}(K=k)=\left.\binom{k-1}{n-1}\right/\binom{m}{n} \] for \(n\leqslant k\leqslant m,\) and deduce that \[ \sum_{k=n}^{m}\binom{k-1}{n-1}=\binom{m}{n}\,. \] Find the expected value of \(K\).

I have a bag containing \(M\) tokens, \(m\) of which are red. I remove \(n\) tokens from the bag at random without replacement. Let \[ X_{i}=\begin{cases} 1 & \mbox{ if the \ensuremath{i}th token I remove is red;}\\ 0 & \mbox{ otherwise.} \end{cases} \] Let \(X\) be the total number of red tokens I remove.

1. Explain briefly why \(X=X_{1}+X_{2}+\cdots+X_{n}.\)
2. Find the expectation \(\mathrm{E(}X_{i}).\)
3. Show that \(\mathrm{E}(X)=mn/M\).
4. Find \(\mathrm{P}(X=k)\) for \(k=0,1,2,\ldots,n\).
5. Deduce that \[ \sum_{k=1}^{n}k\binom{m}{k}\binom{M-m}{n-k}=m\binom{M-1}{n-1}. \]

Balls are chosen at random without replacement from an urn originally containing \(m\) red balls and \(M-m\) green balls. Find the probability that exactly \(k\) red balls will be chosen in \(n\) choices \((0\leqslant k\leqslant m,0\leqslant n\leqslant M).\) The random variables \(X_{i}\) \((i=1,2,\ldots,n)\) are defined for \(n\leqslant M\) by \[ X_{i}=\begin{cases} 0 & \mbox{ if the \(i\)th ball chosen is green}\\ 1 & \mbox{ if the \(i\)th ball chosen is red. } \end{cases} \] Show that

1. \(\mathrm{P}(X_{i}=1)=\dfrac{m}{M}.\)
2. \(\mathrm{P}(X_{i}=1\mbox{ and }X_{j}=1)=\dfrac{m(m-1)}{M(M-1)}\), for \(i\neq j\).

In a television game show, a contestant has to open a door using a key. The contestant is given a bag containing \(n\) keys, where \(n\ge2\). Only one key in the bag will open the door. There are three versions of the game. In each version, the contestant starts by choosing a key at random from the bag.

1. In version 1, after each failed attempt at opening the door the key that has been tried is put back into the bag and the contestant again selects a key at random from the bag. By considering the binomial expansion of \(( 1 - q)^{-2}\), or otherwise, find the expected number of attempts required to open the door.
2. In version 2, after each failed attempt at opening the door the key that has been tried is put aside and the contestant selects another key at random from the bag. Find the expected number of attempts required to open the door.
3. In version 3, after each failed attempt at opening the door the key that has been~tried is put back into the bag and another incorrect key is added to the bag. The contestant then selects a key at random from the bag. Show that the probability that the contestant draws the correct key at the \(k\)th attempt is \[ \frac{n-1}{(n+k-1)(n+k-2)} \,. \] Show also, using partial fractions, that the expected number of attempts required to open the door is infinite. You may use without proof the result that $ \displaystyle \sum_{m=1}^N \dfrac 1 m \to \infty \, $ as \(N\to \infty\,\).

A fair die with faces numbered \(1\), \(\ldots\,\), \(6\) is thrown repeatedly. The events \(A\), \(B\), \(C\), \(D\) and \(E\) are defined as follows.\\ \hspace*{20pt} \(A\): \ the first 6 arises on the \(n\)th throw.\\ \hspace*{20pt} \(B\): \ at least one 5 arises before the first 6. \\ \hspace*{20pt} \(C\): \ at least one 4 arises before the first 6. \\ \hspace*{20pt} \(D\): \ exactly one 5 arises before the first 6. \\ \hspace*{20pt} \(E\): \ exactly one 4 arises before the first 6. Evaluate the following probabilities: ({\bf i}) \ \(\P(A)\) \ \ \ \ ({\bf ii}) \ \(\P(B)\) \ \ \ \ ({\bf iii}) \ \(\P(B\cap C)\) \ \ \ \ ({\bf iv}) \ \(\P(D)\) \ \ \ \ ({\bf v}) \ \(\P(D\cup E)\) \,. \noindent For some parts of this question, you may want to make use of the binomial expansion in the form: \[ (1-x)^{-n} = 1 +nx +\frac {n(n+1)}2 x^2 + \cdots + \frac {(n+r-1)!}{r! (n-1)!}x^r +\cdots \ . \]

In this question, the notation \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\), so for example \(\lfloor \pi\rfloor = 3\) and \(\lfloor 3 \rfloor =3\).

1. A bag contains \(n\) balls, of which \(b\) are black. A sample of \(k\) balls is drawn, one after another, at random {\sl with} replacement. The random variable \(X\) denotes the number of black balls in the sample. By considering \[ \frac{\P(X=r+1)}{\P(X=r)}\,, \] show that, in the case that it is unique, the most probable number of black balls in the sample is \[ \left\lfloor \frac{(k+1)b}{n}\right\rfloor. \] Under what circumstances is the answer not unique?
2. A bag contains \(n\) balls, of which \(b\) are black. A sample of \(k\) balls (where \(k\le b\)) is drawn, one after another, at random {\sl without} replacement. Find, in the case that it is unique, the most probable number of black balls in the sample. Under what circumstances is the answer not unique?

The probability of throwing a head with a certain coin is \(p\) and the probability of throwing a tail is \(q=1-p\). The coin is thrown until at least two heads and at least two tails have been thrown; this happens when the coin has been thrown \(N\) times. Write down an expression for the probability that \(N=n\). Show that the expectation of \(N\) is $$ 2\bigg({1\over pq} -1-pq\bigg). $$

A coin has probability \(p\) (\(0 < p < 1\)) of showing a head when tossed. Give a careful argument to show that the \(k\)th head in a series of consecutive tosses is achieved after exactly \(n\) tosses with probability \[ \binom{n-1}{k-1}p^{k}(1-p)^{n-k}\qquad(n\geqslant k). \] Given that it took an even number of tosses to achieve exactly \(k-1\) heads, find the probability that exactly \(k\) heads are achieved after an even number of tosses. If this coin is tossed until exactly 3 heads are obtained, what is the probability that exactly 2 of the heads occur on even-numbered tosses?

In a certain factory, microchips are made by two machines. Machine A makes a proportion~\(\lambda\) of the chips, where \(0 < \lambda < 1\), and machine B makes the rest. A proportion \(p\) of the chips made by machine A are perfect, and a proportion \(q\) of those made by machine B are perfect, where \(0 < p < 1\) and \(0 < q < 1\). The chips are sorted into two groups: group 1 contains those that are perfect and group 2 contains those that are imperfect. In a large random sample taken from group 1, it is found that \(\frac 2 5\) were made by machine A. Show that \(\lambda\) can estimated as \[ {2q \over 3p + 2q}\;. \] Subsequently, it is discovered that the sorting process is faulty: there is a probability of \(\frac 14\) that a perfect chip is assigned to group 2 and a probability of \(\frac 14\) that an imperfect chip is assigned to group 1. Taking into account this additional information, obtain a new estimate of \(\lambda\,\).

The probability of throwing a 6 with a biased die is \(p\,\). It is known that \(p\) is equal to one or other of the numbers \(A\) and \(B\) where \(0 < A < B < 1 \,\). Accordingly the following statistical test of the hypothesis \(H_0: \,p=B\) against the alternative hypothesis \(H_1: \,p=A\) is performed. The die is thrown repeatedly until a 6 is obtained. Then if \(X\) is the total number of throws, \(H_0\) is accepted if \(X \le M\,\), where \(M\) is a given positive integer; otherwise \(H_1\) is accepted. Let \({\alpha}\) be the probability that \(H_1\) is accepted if \(H_0\) is true, and let \({\beta}\) be the probability that \(H_0\) is accepted if \(H_1\) is true. Show that \({\beta} = 1- {\alpha}^K,\) where \(K\) is independent of \(M\) and is to be determined in terms of \(A\) and \(B\,\). Sketch the graph of \({\beta}\) against \({\alpha}\,\).

Show Solution

\(X \sim Geo(p)\). \(\alpha = \mathbb{P}(X > M | p = B) = (1-B)^{M}\) \(\beta = \mathbb{P}(X \leq M | p = A) = 1 - \mathbb{P}(X > M | p = A) = 1 - (1-A)^{M}\) \begin{align*} \ln \alpha &= M \ln(1-B) \\ \ln (1-\beta) &= M \ln(1-A) \\ \frac{\ln \alpha}{\ln (1-\beta)} &= \frac{\ln(1-B)}{\ln(1-A)} \\ \ln(1-\beta) &= \ln \alpha \frac{\ln (1-A)}{\ln(1-B)} \\ \beta &= 1- \alpha^{ \frac{\ln (1-A)}{\ln(1-B)} } \end{align*} and \(K = \frac{\ln (1-A)}{\ln(1-B)} \) Since \(0 < A < B < 1\) we must have that \(0 < 1 - B < 1-A < 1\) and \(\ln(1-B) < \ln(1-A) < 0\) so \(0 < K < 1\)

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1. Find the maximum value of \(\sqrt{p(1-p)}\) as \(p\) varies between \(0\) and \(1\).
2. Suppose that a proportion \(p\) of the population is female. In order to estimate \(p\) we pick a sample of \(n\) people at random and find the proportion of them who are female. Find the value of \(n\) which ensures that the chance of our estimate of \(p\) being more than \(0.01\) in error is less than 1\%.
3. Discuss how the required value of \(n\) would be affected if (a) \(p\) were the proportion of people in the population who are left-handed; (b) \(p\) were the proportion of people in the population who are millionaires.

In Fridge football, each team scores two points for a goal and one point for a foul committed by the opposing team. In each game, for each team, the probability that the team scores \(n\) goals is \(\left(3-\left|2-n\right|\right)/9\) for \(0\leqslant n\leqslant4\) and zero otherwise, while the number of fouls committed against it will with equal probability be one of the numbers from \(0\) to \(9\) inclusive. The numbers of goals and fouls of each team are mutually independent. What is the probability that in some game a particular team gains more than half its points from fouls? In response to criticisms that the game is boring and violent, the ruling body increases the number of penalty points awarded for a foul, in the hope that this will cause large numbers of fouls to be less probable. During the season following the rule change, 150 games are played and on 12 occasions (out of 300) a team committed 9 fouls. Is this good evidence of a change in the probability distribution of the number of fouls? Justify your answer.

The random variable \(X\) has the probability density function on the interval \([0, 1]\): $$f(x) = \begin{cases} nx^{n-1} & 0 \leq x \leq 1, \\ 0 & \text{elsewhere}, \end{cases}$$ where \(n\) is an integer greater than 1.

1. Let \(\mu = E(X)\). Find an expression for \(\mu\) in terms of \(n\), and show that the variance, \(\sigma^2\), of \(X\) is given by $$\sigma^2 = \frac{n}{(n + 1)^2(n + 2)}.$$
2. In the case \(n = 2\), show without using decimal approximations that the interquartile range is less than \(2\sigma\).
3. Write down the first three terms and the \((k + 1)\)th term (where \(0 \leq k \leq n\)) of the binomial expansion of \((1 + x)^n\) in ascending powers of \(x\). By setting \(x = \frac{1}{n}\), show that \(\mu\) is less than the median and greater than the lower quartile. Note: You may assume that $$1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots < 4.$$

A continuous random variable \(X\) has a triangular distribution, which means that it has a probability density function of the form \[ \f(x) = \begin{cases} \g(x) & \text{for \(a< x \le c\)} \\ \h(x) & \text{for \(c\le x < b\)} \\ 0 & \text{otherwise,} \end{cases} \] where \(\g(x)\) is an increasing linear function with \(\g(a)=0\), \(\h(x)\) is a decreasing linear function with \(\h(b) =0\), and \(\g(c)=\h(c)\). Show that \(\g(x) = \dfrac{2(x-a)}{(b-a)(c-a)}\) and find a similar expression for \(\h(x)\).

1. Show that the mean of the distribution is \(\frac13(a+b+c)\).
2. Find the median of the distribution in the different cases that arise.

Fire extinguishers may become faulty at any time after manufacture and are tested annually on the anniversary of manufacture. The time \(T\) years after manufacture until a fire extinguisher becomes faulty is modelled by the continuous probability density function \[ f(t) = \begin{cases} \frac{2t}{(1+t^2)^2}& \text{for \(t\ge0\)}\,,\\[4mm] \ \ \ \ 0& \text{otherwise}. \end{cases} \] A faulty fire extinguisher will fail an annual test with probability \(p\), in which case it is destroyed immediately. A non-faulty fire extinguisher will always pass the test. All of the annual tests are independent. Show that the probability that a randomly chosen fire extinguisher will be destroyed exactly three years after its manufacture is \(p(5p^2-13p +9)/10\). Find the probability that a randomly chosen fire extinguisher that was destroyed exactly three years after its manufacture was faulty 18 months after its manufacture.

In this question, you may use without proof the following result: \[ \int \sqrt{4-x^2}\, \d x = 2 \arcsin (\tfrac12 x ) + \tfrac 12 x \sqrt{4-x^2} +c\,. \] A random variable \(X\) has probability density function \(\f\) given by \[ \f(x) = \begin{cases} 2k & -a\le x <0 \\[3mm] k\sqrt{4-x^2} & \phantom{-} 0\le x \le 2 \\[3mm] 0 & \phantom{-}\text{otherwise}, \end{cases} \] where \(k\) and \(a\) are positive constants.

1. Find, in terms of \(a\), the mean of \(X\).
2. Let \(d\) be the value of \(X\) such that \(\P(X> d)=\frac1 {10}\,\). Show that \(d<0\) if \(2a> 9\pi\) and find an expression for \(d\) in terms of \(a\) in this case.
3. Given that \(d=\sqrt 2\), find \(a\).

Sketch the graph of \[ y= \dfrac1 { x \ln x} \text{ for \(x>0\), \(x\ne1\)}.\] You may assume that \(x\ln x \to 0\) as \(x\to 0\). The continuous random variable \(X\) has probability density function \[ \f(x) = \begin{cases} \dfrac \lambda {x\ln x}& \text{for \(a\le x \le b\)}\;, \\[3mm] \ \ \ 0 & \text{otherwise }, \end{cases} \] where \(a\), \(b\) and \(\lambda\) are suitably chosen constants.

1. In the case \(a=1/4\) and \(b=1/2\), find \(\lambda\,\).
2. In the case \(\lambda=1\) and \(a>1\), show that \(b=a^\e\).
3. In the case \(\lambda =1\) and \(a=\e\), show that \(\P(\e^{3/2}\le X \le \e^2)\approx \frac {31}{108}\,\).
4. In the case \(\lambda =1\) and \(a=\e^{1/2}\), find \(\P(\e^{3/2}\le X \le \e^2)\;\).

The random variable \(X\) can take the value \mbox{\(X=-1\)}, and also any value in the range \mbox{\(0\le X <\infty\,\)}. The distribution of \(X\) is given by \[ \P(X=-1) =m \,, \ \ \ \ \ \ \ \P(0\le X\le x) = k(1-\e^{-x})\,, \] for any non-negative number \(x\), where \(k\) and \(m\) are constants, and \(m <\frac12\,\).

1. Find \(k\) in terms of \(m\).
2. Show that \(\E(X)= 1-2m\,\).
3. Find, in terms of \(m\), \(\var (X)\) and the median value of~\(X\).
4. Given that \[ \int_0^\infty y^2 \e^{-y^2} \d y = \tfrac14 \sqrt{ \pi}\;,\] find \(\E\big(\vert X \vert^{\frac12}\big)\,\) in terms of \(m\).

The life times of a large batch of electric light bulbs are independently and identically distributed. The probability that the life time, \(T\) hours, of a given light bulb is greater than \(t\) hours is given by \[ \P(T>t) \; = \; \frac{1}{(1+kt)^\alpha}\;, \] where \(\alpha\) and \(k\) are constants, and \(\alpha >1\). Find the median \(M\) and the mean \(m\) of \(T\) in terms of \(\alpha\) and \(k\). Nine randomly selected bulbs are switched on simultaneously and are left until all have failed. The fifth failure occurs at 1000 hours and the mean life time of all the bulbs is found to be 2400 hours. Show that \(\alpha\approx2\) and find the approximate value of \(k\). Hence estimate the probability that, if a randomly selected bulb is found to last \(M\) hours, it will last a further \(m-M\) hours.

A stick is broken at a point, chosen at random, along its length. Find the probability that the ratio, \(R\), of the length of the shorter piece to the length of the longer piece is less than \(r\). Find the probability density function for \(R\), and calculate the mean and variance of \(R\).

Each of my \(n\) students has to hand in an essay to me. Let \(T_{i}\) be the time at which the \(i\)th essay is handed in and suppose that \(T_{1},T_{2},\ldots,T_{n}\) are independent, each with probability density function \(\lambda\mathrm{e}^{-\lambda t}\) (\(t\geqslant0\)). Let \(T\) be the time I receive the first essay to be handed in and let \(U\) be the time I receive the last one.

1. Find the mean and variance of \(T_{i}.\)
2. Show that \(\mathrm{P}(U\leqslant u)=(1-\mathrm{e}^{-\lambda u})^{n}\) for \(u\geqslant0,\) and hence find the probability density function of \(U\).
3. Obtain \(\mathrm{P}(T>t),\) and hence find the probability density function of \(T\).
4. Write down the mean and variance of \(T\).

When he sets out on a drive Mr Toad selects a speed \(V\) kilometres per minute where \(V\) is a random variable with probability density \[ \alpha v^{-2}\mathrm{e}^{-\alpha v^{-1}} \] and \(\alpha\) is a strictly positive constant. He then drives at constant speed, regardless of other drivers, road conditions and the Highway Code. The traffic lights at the Wild Wood cross-roads change from red to green when Mr Toad is exactly 1 kilometre away in his journey towards them. If the traffic light is green for \(g\) minutes, then red for \(r\) minutes, then green for \(g\) minutes, and so on, show that the probability that he passes them after \(n(g+r)\) minutes but before \(n(g+r)+g\) minutes, where \(n\) is a positive integer, is \[ \mathrm{e}^{-\alpha n(g+r)}-\mathrm{e}^{-\alpha\left(n(g+r)\right)+g}. \] Find the probability \(\mathrm{P}(\alpha)\) that he passes the traffic lights when they are green. Show that \(\mathrm{P}(\alpha)\rightarrow1\) as \(\alpha\rightarrow\infty\) and, by noting that \((\mathrm{e}^{x}-1)/x\rightarrow1\) as \(x\rightarrow0\), or otherwise, show that \[ \mathrm{P}(\alpha)\rightarrow\frac{g}{r+g}\quad\mbox{ as }\alpha\rightarrow0. \] {[}NB: the traffic light show only green and red - not amber.{]}

The prevailing winds blow in a constant southerly direction from an enchanted castle. Each year, according to an ancient tradition, a princess releases 96 magic seeds from the castle, which are carried south by the wind before falling to rest. South of the castle lies one league of grassy parkland, then one league of lake, then one league of farmland, and finally the sea. If a seed falls on land it will immediately grow into a fever tree. (Fever trees do not grow in water). Seeds are blown independently of each other. The random variable \(L\) is the distance in leagues south of the castle at which a seed falls to rest (either on land or water). It is known that the probability density function \(\mathrm{f}\) of \(L\) is given by \[ \mathrm{f}(x)=\begin{cases} \frac{1}{2}-\frac{1}{8}x & \mbox{ for }0\leqslant x\leqslant4,\\ 0 & \mbox{ otherwise.} \end{cases} \] What is the mean number of fever trees which begin to grow each year?

1. The random variable \(Y\) is defined as the distance in leagues south of the castle at which a new fever tree grows from a seed carried by the wind. Sketch the probability density function of \(Y\), and find the mean of \(Y\).
2. One year messengers bring the king the news that 23 new fever trees have grown in the farmland. The wind never varies, and so the king suspects that the ancient tradition have not been followed properly. Is he justified in his suspicions?

Wondergoo is applied to all new cars. It protects them completely against rust for three years, but thereafter the probability density of the time of onset of rust is proportional to \(t^{2}/(1+t^{2})^{2}\) for a car of age \(3+t\) years \((t\geqslant0)\). Find the probability that a car becomes rusty before it is \(3+t\) years old. Every car is tested for rust annually on the anniversary of its manufacture. If a car is not rusty, it will certainly pass; if it is rusty, it will pass with probability \(\frac{1}{2}.\) Cars which do not pass are immediately taken off the road and destroyed. What is the probability that a randomly selected new car subsequently fails a test taken on the fifth anniversary of its manufacture? Find also the probability that a car which was destroyed immediately after its fifth anniversary test was rusty when it passed its fourth anniversary test.

The random variable \(X\) has mean \(\mu\) and variance \(\sigma^2\), and the function \({\rm V}\) is defined, for \(-\infty < x < \infty\), by \[ {\rm V}(x) = \E \big( (X-x)^2\big) . \] Express \({\rm V}(x)\) in terms of \(x\), \( \mu\) and \(\sigma\). The random variable \(Y\) is defined by \(Y={\rm V}(X)\). Show that \[ \E(Y) = 2 \sigma^2 %\text{ \ \ and \ \ } %\Var(Y) = \E(X-\mu)^4 -\sigma^4 . \tag{\(*\)} \] Now suppose that \(X\) is uniformly distributed on the interval \(0\le x \le1\,\). Find \({\rm V}(x)\,\). Find also the probability density function of \(Y\!\) and use it to verify that \((*)\) holds in this case.

In this question, you may use without proof the results: \[ \sum_{r=1}^n r = \tfrac12 n(n+1) \qquad\text{and}\qquad \sum_{r=1}^n r^2 = \tfrac1 6 n(n+1)(2n+1)\,. \] The independent random variables \(X_1\) and \(X_2\) each take values \(1\), \(2\), \(\ldots\), \(N\), each value being equally likely. The random variable \(X\) is defined by \[ X= \begin{cases} X_1 & \text { if } X_1\ge X_2\\ X_2 & \text { if } X_2\ge X_1\;. \end{cases} \]

1. Show that \(\P(X=r) = \dfrac{2r-1}{N^2}\,\) for \(r=1\), \(2\), \(\ldots\), \(N\).
2. Find an expression for the expectation, \(\mu\), of \(X\) and show that \(\mu=67.165\) in the case \(N=100\).
3. The median, \(m\), of \(X\) is defined to be the integer such that \(\P(X\ge m) \ge \frac 12\) and \(\P(X\le m)\ge \frac12\). Find an expression for \(m\) in terms of \(N\) and give an explicit value for \(m\) in the case \(N=100\).
4. Show that when \(N\) is very large, \[ \frac \mu m \approx \frac {2\sqrt2}3\,. \]

1. Three real numbers are drawn independently from the continuous rectangular distribution on \([ 0, 1 ]\,\). The random variable \(X\) is the maximum of the three numbers. Show that the probability that \(X \le 0.8\) is \(0.512\,\), and calculate the expectation of \(X\).
2. \(N\) real numbers are drawn independently from a continuous rectangular distribution on \([ 0, a ]\,\). The random variable \(X\) is the maximum of the \(N\) numbers. A hypothesis test with a significance level of 5\% is carried out using the value, \(x\), of \(X \). The null hypothesis is that \(a=1\) and the alternative hypothesis is that \(a<1 \,\). The form of the test is such that \(H_0\) is rejected if \(x < c\,\), for some chosen number \(c\,\). Using the approximation \(2^{10} \approx 10^3\,\), determine the smallest integer value of \(N\) such that if \(x \le 0.8\) the null hypothesis will be rejected. With this value of \(N\), write down the probability that the null hypothesis is rejected if \(a = 0.8\,\), and find the probability that the null hypothesis is rejected if \(a = 0.9\,\).

The random variable \(X\) is uniformly distributed on the interval \([-1,1]\). Find \(\E(X^2)\) and \(\var (X^2)\). A second random variable \(Y\), independent of \(X\), is also uniformly distributed on \([-1,1]\), and \(Z=Y-X\). Find \(\E(Z^2)\) and show that \(\var (Z^2) = 7 \var (X^2)\).

Two computers, LEP and VOZ are programmed to add numbers after first approximating each number by an integer. LEP approximates the numbers by rounding: that is, it replaces each number by the nearest integer. VOZ approximates by truncation: that is, it replaces each number by the largest integer less than or equal to the number. The fractional parts of the numbers to be added are uniformly and independently distributed. (The fractional part of a number \(a\) is \(a-\left\lfloor a\right\rfloor ,\) where \(\left\lfloor a\right\rfloor \) is the largest integer less than or equal to \(a\).) Both computers approximate and add 1500 numbers. For each computer, find the probability that the magnitude of error in the answer will exceed 15. How many additions can LEP perform before the probability that the magnitude of error is less than 10 drops below 0.9?

Trains leave Barchester Station for London at 12 minutes past the hour, taking 60 minutes to complete the journey and at 48 minutes past the hour taking 75 minutes to complete the journey. The arrival times of passengers for London at Barchester Station are uniformly distributed over the day and all passengers take the first available train. Show that their average journey time from arrival at Barchester Station to arrival in London is 84.6 minutes. Suppose that British Rail decide to retime the fast 60 minute train so that it leaves at \(x\) minutes past the hour. What choice of \(x\) will minimise the average journey time?

A train of length \(l_{1}\) and a lorry of length \(l_{2}\) are heading for a level crossing at speeds \(u_{1}\) and \(u_{2}\) respectively. Initially the front of the train and the front of the lorry are at distances \(d_{1}\) and \(d_{2}\) from the crossing. Find conditions on \(u_{1}\) and \(u_{2}\) under which a collision will occur. On a diagram with \(u_{1}\) and \(u_{2}\) measured along the \(x\) and \(y\) axes respectively, shade in the region which represents collision. Hence show that if \(u_{1}\) and \(u_{2}\) are two independent random variables, both uniformly distributed on \((0,V)\), then the probability of a collision in the case when initially the back of the train is nearer to the crossing than the front of the lorry is \[ \frac{l_{1}l_{2}+l_{2}d_{1}+l_{1}d_{2}}{2d_{2}\left(l_{2}+d_{2}\right)}. \] Find the probability of a collision in each of the other two possible cases.

A point \(P\) is chosen at random (with uniform distribution) on the circle \(x^{2}+y^{2}=1\). The random variable \(X\) denotes the distance of \(P\) from \((1,0)\). Find the mean and variance of \(X\). Find also the probability that \(X\) is greater than its mean.

Show Solution

Consider the angle from the origin, then \(P = (\cos \theta, \sin \theta)\) where \(\theta \sim U(0, 2\pi)\), and \(X = \sqrt{(\cos \theta - 1)^2 + \sin^2 \theta}\) \begin{align*} \mathbb{E}[X] &= \int_0^{2\pi} \sqrt{(\cos \theta - 1)^2 + \sin^2 \theta} \frac1{2\pi} \d \theta \\ &= \frac1{2\pi}\int_0^{2\pi} \sqrt{2 - 2\cos \theta} \d \theta \\ &= \frac{1}{2\pi}\int_0^{2\pi} \sqrt{4\sin^2 \frac{\theta}{2}} \d \theta \\ &= \frac{1}{\pi}\int_0^{2\pi} \left |\sin \frac{\theta}{2} \right| \d \theta \\ &= \frac{1}{\pi} \left [ -2\cos \frac{\theta}{2} \right]_0^{2\pi} \\ &= \frac1{\pi} \l 2 + 2\r \\ &= \frac{4}{\pi} \end{align*} \begin{align*} \mathbb{E}(X^2) &= \frac1{2\pi}\int_0^{2\pi} (\cos \theta - 1)^2 + \sin^2 \theta \d \theta \\ &= \frac1{2\pi}\int_0^{2\pi} 2 - 2 \cos \theta \d \theta \\ &= \frac{4\pi}{2\pi} \\ &= 2 \\ \end{align*} \(\Rightarrow\) \(\mathrm{Var}(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2 = 2 - \frac{16}{\pi^2} = \frac{2\pi^2 - 16}{\pi^2}\).

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Where the line makes a length longer than \(\frac{4}{\pi}\) it will make an angle at the origin of \(2\sin^{-1} \frac{2}{\pi}\). Therefore the probability of being larger than this is \(\frac{2\pi - 2 \times 2\sin^{-1} \frac{2}{\pi}}{2 \pi} = 1 - \frac{2}{\pi} \sin^{-1} \frac{2}{\pi} \approx 0.560\)

1. A point \(P\) lies in an equilateral triangle \(ABC\) of height 1. The perpendicular distances from \(P\) to the sides \(AB\), \(BC\) and \(CA\) are \(x_1\), \(x_2\) and \(x_3\), respectively. By considering the areas of triangles with one vertex at \(P\), show that \(x_1+x_2+x_3=1\). Suppose now that \(P\) is placed at random in the equilateral triangle (so that the probability of it lying in any given region of the triangle is proportional to the area of that region). The perpendicular distances from \(P\) to the sides \(AB\), \(BC\) and \(CA\) are random variables \(X_1\), \(X_2\) and \(X_3\), respectively. In the case \(X_1= \min(X_1,X_2,X_3)\), give a sketch showing the region of the triangle in which \(P\) lies. Let \(X= \min(X_1,X_2,X_3)\). Show that the probability density function for \(X\) is given by \[ \f(x) = \begin{cases} 6(1-3x) & 0 \le x \le \frac13\,, \\ 0 & \text{otherwise}\,. \end{cases} \] Find the expected value of \(X\).
2. A point is chosen at random in a regular tetrahedron of height 1. Find the expected value of the distance from the point to the closest face. \newline [The volume of a tetrahedron is \(\frac13 \times \text{area of base}\times\text{height}\) and its centroid is a distance \(\frac14\times \text{height}\) from the base.]

1. My favourite dartboard is a disc of unit radius and centre \(O\). I never miss the board, and the probability of my hitting any given area of the dartboard is proportional to the area. Each throw is independent of any other throw. I throw a dart \(n\) times (where \(n>1\)). Find the expected area of the smallest circle, with centre \(O\), that encloses all the \(n\) holes made by my dart. Find also the expected area of the smallest circle, with centre \(O\), that encloses all the \((n-1)\) holes nearest to \(O\).
2. My other dartboard is a square of side 2 units, with centre \(Q\). I never miss the board, and the probability of my hitting any given area of the dartboard is proportional to the area. Each throw is independent of any other throw. I throw a dart \(n\) times (where \(n>1\)). Find the expected area of the smallest square, with centre \(Q\), that encloses all the \(n\) holes made by my dart.
3. Determine, without detailed calculations, whether the expected area of the smallest circle, with centre \(Q\), on my square dartboard that encloses all the \(n\) holes made by my darts is larger or smaller than that for my circular dartboard.

Two points are chosen independently at random on the perimeter (including the diameter) of a semicircle of unit radius. The area of the triangle whose vertices are these two points and the midpoint of the diameter is denoted by the random variable \(A\). Show that the expected value of \(A\) is \((2+\pi)^{-1}\).

Show Solution

There are \(3\) possible numbers of points on the curved part of the perimeter. \(0\): The area is \(0\) \(1\):

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The area of the triangle is \(\frac12 |x| \sin \theta\) Where \(X\) is the point along the diameter which is \(U[-1,1]\) and \(\theta \sim U(0, \pi)\) Therefore \begin{align*} \mathbb{E}(A|\text{one on diameter}) &= \int_{0}^\pi \frac{1}{\pi} \int_{-1}^1\frac{1}{2}\frac12 |x| \sin \theta \d x \d \theta \\ &= \frac{1}{2\pi}\frac12 \int_{0}^\pi \sin \theta \d \theta \cdot 2\int_{0}^1 x\d x \\ &=\frac{1}{2\pi}\cdot 2 \cdot \frac12 = \frac{1}{2\pi} \end{align*} \(2\): If both are on the curved section

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Then the area is \(\frac12 \sin \theta\) where \(\theta = |\theta_1 - \theta_2|\) and \(\theta_i \sim U[0, \pi]\) Therefore the area is \begin{align*} \mathbb{E}(A|\text{none on diameter}) &= \int_{0}^\pi\frac{1}{\pi} \int_{0}^\pi\frac{1}{\pi} \frac12 \sin |\theta_1 - \theta_2| \d \theta_1 \d \theta_2 \\ &= \frac{1}{\pi^2}\frac12 \int_{0}^\pi \left (\int_{0}^{\theta_2} \sin (\theta_2 - \theta_1) \d \theta_1-\int_{\theta_2}^{\pi} \sin (\theta_2 - \theta_1) \d \theta_1 \right)\d \theta_2 \\ &= \frac{1}{\pi^2}\frac12 \int_{0}^\pi \left [2\cos(\theta_2 - \theta_2)-\cos(\theta_2 - 0)-\cos(\theta_2 - \pi) \right]\d \theta_2 \\ &= \frac{1}{\pi} \end{align*} Therefore the expected area is: \begin{align*} \mathbb{E}(A ) &= \mathbb{E}(A|\text{one on diameter})\cdot \mathbb{P}(\text{one on diameter}) + \mathbb{E}(A|\text{none on diameter})\cdot \mathbb{P}(\text{none on diameter}) \\ &= \frac{1}{2\pi}\mathbb{P}(\text{one on diameter}) + \frac{1}{\pi}\cdot \mathbb{P}(\text{none on diameter}) \\ &= \frac{1}{2\pi} \cdot 2 \cdot \frac{\pi}{\pi + 2} \cdot \frac{2}{\pi + 2} + \frac1{\pi} \cdot \frac{\pi}{\pi + 2} \cdot \frac{\pi}{\pi+2} \\ &= \frac{2 + \pi}{(\pi+2)^2} \\ &= \frac{1}{\pi+2} \end{align*}

A densely populated circular island is divided into \(N\) concentric regions \(R_1\), \(R_2\), \(\ldots\,\), \(R_N\), such that the inner and outer radii of \(R_n\) are \(n-1\) km and \(n\) km, respectively. The average number of road accidents that occur in any one day in \(R_n\) is \(2-n/N\,\), independently of the number of accidents in any other region. Each day an observer selects a region at random, with a probability that is proportional to the area of the region, and records the number of road accidents, \(X\), that occur in it. Show that, in the long term, the average number of recorded accidents per day will be \[ 2-\frac16\left(1+\frac1N\right)\left(4-\frac1N\right)\;. \] [Note: \(\sum\limits_{n=1}^N n^2 = \frac16 N(N+1)(2N+1) \;\).] Show also that \[ \P(X=k) = \frac{\e^{-2}N^{-k-2}}{k!}\sum_{n=1}^N (2n-1)(2N-n)^k\e^{n/N} \;. \] Suppose now that \(N=3\) and that, on a particular day, two accidents were recorded. Show that the probability that \(R_2\) had been selected is \[ \frac{48}{48 + 45\e^{1/3} +25 \e^{-1/3}}\;. \]

Harry the Calculating Horse will do any mathematical problem I set him, providing the answer is 1, 2, 3 or 4. When I set him a problem, he places a hoof on a large grid consisting of unit squares and his answer is the number of squares partly covered by his hoof. Harry has circular hoofs, of radius \(1/4\) unit. After many years of collaboration, I suspect that Harry no longer bothers to do the calculations, instead merely placing his hoof on the grid completely at random. I often ask him to divide 4 by 4, but only about \(1/4\) of his answers are right; I often ask him to add 2 and 2, but disappointingly only about \(\pi/16\) of his answers are right. Is this consistent with my suspicions? I decide to investigate further by setting Harry many problems, the answers to which are 1, 2, 3, or 4 with equal frequency. If Harry is placing his hoof at random, find the expected value of his answers. The average of Harry's answers turns out to be 2. Should I get a new horse?

The cakes in our canteen each contain exactly four currants, each currant being randomly placed in the cake. I take a proportion \(X\) of a cake where \(X\) is a random variable with density function \[{\mathrm f}(x)=Ax\] for \(0\leqslant x\leqslant 1\) where \(A\) is a constant.

1. What is the expected number of currants in my portion?
2. If I find all four currants in my portion, what is the probability that I took more than half the cake?

When I throw a dart at a target, the probability that it lands a distance \(X\) from the centre is a random variable with density function \[ \mathrm{f}(x)=\begin{cases} 2x & \text{ if }0\leqslant x\leqslant1;\\ 0 & \text{ otherwise.} \end{cases} \] I score points according to the position of the dart as follows: %

1. Show that my expected score from one dart is 15/8.
2. I play a game with the following rules. I start off with a total score 0, and each time~I throw a dart my score on that throw is added to my total. Then: \newline \hspace*{10mm} if my new total is greater than 3, I have lost and the game ends; \newline \hspace*{10mm} if my new total is 3, I have won and the game ends; \newline \hspace*{10mm} if my new total is less than 3, I throw again. Show that, if I have won such a game, the probability that I threw the dart three times is 343/2231.

To celebrate the opening of the financial year the finance minister of Genland flings a Slihing, a circular coin of radius \(a\) cm, where \(0 < a < 1\), onto a large board divided into squares by two sets of parallel lines 2 cm apart. If the coin does not cross any line, or if the coin covers an intersection, the tax on yaks remains unchanged. Otherwise the tax is doubled. Show that, in order to raise most tax, the value of \(a\) should be \[\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}.\] If, indeed, \(a=\left(1+{\displaystyle \frac{\pi}{4}}\right)^{-1}\) and the tax on yaks is 1 Slihing per yak this year, show that its expected value after \(n\) years will have passed is \[ \left(\frac{8+\pi}{4+\pi}\right)^{n}.\]

\item A needle of length two cm is dropped at random onto a large piece of paper ruled with parallel lines two cm apart.

1. By considering the angle which the needle makes with the lines, find the probability that the needle crosses the nearest line given that its centre is \(x\) cm from it, where \(0 < x < 1\).
2. Given that the centre of the needle is \(x\) cm from the nearest line and that the needle crosses that line, find the cumulative distribution function for the length of the shorter segment of the needle cut off by the line.
3. Find the probability that the needle misses all the lines.

Three points, \(P,Q\) and \(R\), are independently randomly chosen on the perimeter of a circle. Prove that the probability that at least one of the angles of the triangle \(PQR\) will exceed \(k\pi\) is \(3(1-k)^{2}\) if \(\frac{1}{2}\leqslant k\leqslant1.\) Find the probability if \(\frac{1}{3}\leqslant k\leqslant\frac{1}{2}.\)

When Septimus Moneybags throws darts at a dart board they are certain to end on the board (a disc of radius \(a\)) but, it must be admitted, otherwise are uniformly randomly distributed over the board.

1. Show that the distance \(R\) that his shot lands from the centre of the board is a random variable with variance \(a^{2}/18.\)
2. At a charity fete he can buy \(m\) throws for \(\pounds(12+m)\), but he must choose \(m\) before he starts to throw. If at least one of his throws lands with \(a/\sqrt{10}\) of the centre he wins back \(\pounds 12\). In order to show that a good sport he is, he is determined to play but, being a careful man, he wishes to choose \(m\) so as to minimise his expected loss. What values of \(m\) should he choose?

By making the substitution \(y=\cos^{-1}t,\) or otherwise, show that \[ \int_{0}^{1}\cos^{-1}t\,\mathrm{d}t=1. \] A pin of length \(2a\) is thrown onto a floor ruled with parallel lines equally spaced at a distance \(2b\) apart. The distance \(X\) of its centre from the nearest line is a uniformly distributed random variable taking values between \(0\) and \(b\) and the acute angle \(Y\) the pin makes with a direction perpendicular to the line is a uniformly distributed random variable taking values between \(0\) and \(\pi/2\). \(X\) and \(Y\) are independent. If \(X=x\) what is the probability that the pin crosses the line? If \(a < b\) show that the probability that the pin crosses a line for a general throw is \(\dfrac{2a}{\pi b}.\)

Two points are chosen independently at random on the perimeter (including the diameter) of a semicircle of unit radius. What is the probability that exactly one of them lies on the diameter? Let the area of the triangle formed by the two points and the midpoint of the diameter be denoted by the random variable \(A\).

1. Given that exactly one point lies on the diameter, show that the expected value of \(A\) is \(\left(2\pi\right)^{-1}\).
2. Given that neither point lies on the diameter, show that the expected value of \(A\) is \(\pi^{-1}\). [You may assume that if two points are chosen at random on a line of length \(\pi\) units, the probability density function for the distance \(X\) between the two points is \(2\left(\pi-x\right)/\pi^{2}\) for \(0\leqslant x\leqslant\pi.\)]

1. The random variable \(Z\) has a Normal distribution with mean \(0\) and variance \(1\). Show that the expectation of \(Z\) given that \(a < Z < b\) is \[ \frac{\exp(- \frac12 a^2) - \exp(- \frac12 b^2) } {\sqrt{2\pi\,} \,\big(\Phi(b) - \Phi(a)\big)}, \] where \(\Phi\) denotes the cumulative distribution function for \(Z\).
2. The random variable \(X\) has a Normal distribution with mean \(\mu\) and variance \(\sigma^2\). Show that \[ \E(X \,\vert\, X>0) = \mu + \sigma \E(Z \,\vert\,Z > -\mu/\sigma). \] Hence, or otherwise, show that the expectation, \(m\), of \(\vert X\vert \) is given by \[ m= \mu \big(1 - 2 \Phi(- \mu / \sigma)\big) + \sigma \sqrt{2 / \pi}\; \exp(- \tfrac12 \mu^2 / \sigma^2) \,. \] Obtain an expression for the variance of \(\vert X \vert\) in terms of \(\mu \), \(\sigma \) and \(m\).

A continuous random variable \(X\) has probability density function given by \[ \f(x) = \begin{cases} 0 & \mbox{for } x<0 \\ k\e^{-2 x^2} & \mbox{for } 0\le x< \infty \;,\\ \end{cases} \] where \(k\) is a constant.

1. Sketch the graph of \(\f(x)\).
2. Find the value of \(k\).
3. Determine \(\E(X)\) and \(\var(X)\).
4. Use statistical tables to find, to three significant figures, the median value of \(X\).

The probability density function \(\f(x)\) of the random variable \(X\) is given by $$ \f(x) = k\left[{\phi}(x) + {\lambda}\g(x)\right],\,\,\,\, $$ where \({\phi}(x)\) is the probability density function of a normal variate with mean~0 and variance~1, \(\lambda \) is a positive constant, and \(\g(x)\) is a probability density function defined by \[ \g(x)= \begin{cases} 1/\lambda & \mbox{for \(0 \le x \le {\lambda}\)}\,;\\ 0& \mbox{otherwise} . \end{cases} \] Find \(\mu\), the mean of \(X\), in terms of \(\lambda\), and prove that \(\sigma\), the standard deviation of \(X\), satisfies. $$ \sigma^2 = \frac{\lambda^4 +4{\lambda}^3+12{\lambda}+12} {12(1 + \lambda )^2}\;. $$ In the case \(\lambda=2\):

1. draw a sketch of the curve \(y=\f(x)\);
2. express the cumulative distribution function of \(X\) in terms of \(\Phi(x)\), the cumulative distribution function corresponding to \(\phi(x)\);
3. evaluate \(\P(0

The random variable \(X\) has mean \(\mu\) and standard deviation \(\sigma\). The distribution of \(X\) is symmetrical about \(\mu\) and satisfies: \[\P \l X \le \mu + \sigma \r = a \mbox{ and } \P \l X \le \mu + \tfrac{1}{ 2}\sigma \r = b\,,\] where \(a\) and \(b\) are fixed numbers. Do not assume that \(X\) is Normally distributed.

1. Determine expressions (in terms of \(a\) and \(b\)) for \[ \P \l \mu-\tfrac12 \sigma \le X \le \mu + \sigma \r \mbox{ and } \P \l X \le \mu +\tfrac12 \sigma \; \vert \; X \ge \mu - \tfrac12 \sigma \r.\]
2. My local supermarket sells cartons of skimmed milk and cartons of full-fat milk: \(60\%\) of the cartons it sells contain skimmed milk, and the rest contain full-fat milk. The volume of skimmed milk in a carton is modelled by \(X\) ml, with \(\mu = 500\) and \(\sigma =10\,\). The volume of full-fat milk in a carton is modelled by \(X\) ml, with \(\mu = 495\) and \(\sigma = 10\,\).
   1. Today, I bought one carton of milk, chosen at random, from this supermarket. When I get home, I find that it contains less than 505 ml. Determine an expression (in terms of \(a\) and \(b\)) for the probability that this carton of milk contains more than 500 ml.
   2. Over the years, I have bought a very large number of cartons of milk, all chosen at random, from this supermarket. \(70\%\) of the cartons I have bought have contained at most 505 ml of milk. Of all the cartons that have contained at least 495 ml of milk, one third of them have contained full-fat milk. Use this information to estimate the values of \(a\) and \(b\).

{\it Tabulated values of \({\Phi}(\cdot)\), the cumulative distribution function of a standard normal variable, should not be used in this question.} Henry the commuter lives in Cambridge and his working day starts at his office in London at 0900. He catches the 0715 train to King's Cross with probability \(p\), or the 0720 to Liverpool Street with probability \(1-p\). Measured in minutes, journey times for the first train are \(N(55,25)\) and for the second are \(N(65,16)\). Journey times from King's Cross and Liverpool Street to his office are \(N(30,144)\) and \(N(25,9)\), respectively. Show that Henry is more likely to be late for work if he catches the first train. \par Henry makes \(M\) journeys, where \(M\) is large. Writing \(A\) for \(1-{\Phi}(20/13)\) and \(B\) for \(1-{\Phi}(2)\), find, in terms of \(A\), \(B\), \(M\) and \(p\), the expected number, \(L\), of times that Henry will be late and show that for all possible values of \(p\), $$ BM \le L \le AM. $$ Henry noted that in 3/5 of the occasions when he was late, he had caught the King's Cross train. Obtain an estimate of \(p\) in terms of \(A\) and \(B\). [A random variable is said to be \(N\left({{\mu}, {\sigma}^2}\right)\) if it has a normal distribution with mean \({\mu}\) and variance \({\sigma}^2\).]

Suppose \(X\) is a random variable with probability density \[ \mathrm{f}(x)=Ax^{2}\exp(-x^{2}/2) \] for \(-\infty < x < \infty.\) Find \(A\). You belong to a group of scientists who believe that the outcome of a certain experiment is a random variable with the probability density just given, while other scientists believe that the probability density is the same except with different mean (i.e. the probability density is \(\mathrm{f}(x-\mu)\) with \(\mu\neq0\)). In each of the following two cases decide whether the result given would shake your faith in your hypothesis, and justify your answer.

1. A single trial produces the result 87.3.
2. 1000 independent trials produce results having a mean value \(0.23.\)

The time taken for me to set an acceptable examination question it \(T\) hours. The distribution of \(T\) is a truncated normal distribution with probability density \(\f\) where \[ \mathrm{f}(t)=\begin{cases} \dfrac{1}{k\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{t-\sigma}{\sigma}\right)^{2}\right) & \mbox{ for }t\geqslant0\\ 0 & \mbox{ for }t<0. \end{cases} \] Sketch the graph of \(\f(t)\). Show that \(k\) is approximately \(0.841\) and obtain the mean of \(T\) as a multiple of \(\sigma\). Over a period of years, I find that the mean setting time is 3 hours.

1. Find the approximate probability that none of the 16 questions on next year's paper will take more than 4 hours to set.
2. Given that a particular question is unsatisfactory after 2 hours work, find the probability that it will still be unacceptable after a further 2 hours work.

Find the probability that the quadratic equation \[ X^{2}+2BX+1=0 \] has real roots when \(B\) is normally distributed with zero mean and unit variance. Given that the two roots \(X_{1}\) and \(X_{2}\) are real, find:

1. the probability that both \(X_{1}\) and \(X_{2}\) are greater than \(\frac{1}{5}\);
2. the expected value of \(\left|X_{1}+X_{2}\right|\);

The parliament of Laputa consists of 60 Preservatives and 40 Progressives. Preservatives never change their mind, always voting the same way on any given issue. Progressives vote at random on any given issue.

1. A randomly selected member is known to have voted the same way twice on a given issue. Find the probability that the member will vote the same way a third time on that issue.
2. Following a policy change, a proportion \(\alpha\) of Preservatives now consistently votes against Preservative policy. The Preservative leader decides that an election must be called if the value of \(\alpha\) is such that, at any vote on an item of Preservative policy, the chance of a simple majority would be less than 80\%. By making a suitable normal approximation, estimate the least value of \(\alpha\) which will result in an election being called.

Fifty times a year, 1024 tourists disembark from a cruise liner at a port. From there they must travel to the city centre either by bus or by taxi. Tourists are equally likely to be directed to the bus station or to the taxi rank. Each bus of the bus company holds 32 passengers, and the company currently runs 15 buses. The company makes a profit of \(\pounds\)1 for each passenger carried. It carries as many passengers as it can, with any excess being (eventually) transported by taxi. Show that the largest annual licence fee, in pounds, that the company should consider paying to be allowed to run an extra bus is approximately \[ 1600 \Phi(2) - \frac{800}{\sqrt{2\pi}}\big(1- \e^{-2}\big)\,, \] where \(\displaystyle \Phi(x) =\dfrac1{\sqrt{2\pi}} \int_{-\infty}^x \e^{-\frac12t^2}\d t\,\). You should not consider continuity corrections.

On \(K\) consecutive days each of \(L\) identical coins is thrown \(M\) times. For each coin, the probability of throwing a head in any one throw is \(p\) (where \(0 < p < 1\)). Show that the probability that on exactly \(k\) of these days more than \(l\) of the coins will each produce fewer than \(m\) heads can be approximated by \[ {K \choose k}q^k(1-q)^{K-k}, \] where \[ q=\Phi\left( \frac{2h-2l-1}{2\sqrt{h} }\right), \ \ \ \ \ \ h=L\Phi\left( \frac{2m-1-2Mp}{2\sqrt{ Mp(1-p)}}\right) \] and \(\Phi(\cdot)\) is the cumulative distribution function of a standard normal variate. Would you expect this approximation to be accurate in the case \(K=7\), \(k=2\), \(L=500\), \(l=4\), \(M=100\), \(m=48\) and \(p=0.6\;\)?

Show Solution

Let \(H_i\) be the random variable of how many heads the \(i\)th coin throws on a given day. Then \(H_i \sim B(M,p)\), and the probability that a given coin produces fewer than \(m\) heads is \(p_h = \P(H_i < m)\) Let \(C\) be the random variable the number of coins producing fewer than \(m\) heads, then \(C \sim B(L, p_h)\). The probability that more than \(l\) of the coins produce fewer than \(m\) heads is therefore \(\P(C > l)\). Finally, the probability that on exactly \(k\) days more than \(l\) of the coins will produce fewer than \(m\) heads is: \[ \binom{K}{k} \cdot \P(C > l)^k \cdot (1-\P(C > l))^{K-k} \] Let's start by assuming that all our Binomials can be approximated by a normal distribution. \(B(M,p) \approx N(Mp, Mp(1-p))\) and so: \begin{align*} p_h &= \P(H_i < m) \\ &\approx \P( \sqrt{Mp(1-p)}Z+Mp < m-\frac12) \\ &= \P \l Z < \frac{2m-2Mp-1}{2\sqrt{Mp(1-p)}} \r \\ &= \Phi\l\frac{2m-2Mp-1}{2\sqrt{Mp(1-p)}} \r \end{align*} \(B(L, p_h) \approx B \l L, \P \l Z < \frac{2m-2Mp-1}{2\sqrt{Mp(1-p)}} \r\r = B(L, \frac{h}{L}) \approx N(h, \frac{h(L-h)}{L})\) Therefore \begin{align*} \P(C > l) &= 1-\P(C \leq l) \\ &\approx 1- \P \l \sqrt{\frac{h(L-h)}{L}} Z + h \leq l+\frac12 \r \\ &= 1 - \P \l Z \leq \frac{2l-2h+1}{2\sqrt{\frac{h(L-h)}{L}}}\r \\ &= 1- \Phi\l \frac{2l-2h+1}{2\sqrt{\frac{h(L-h)}{L}}} \r \\ &= \Phi\l \frac{2h-2l-1}{2\sqrt{\frac{h(L-h)}{L}}} \r \end{align*} If we can approximate \(\sqrt{1-\frac{h}{L}}\) by \(1\) then we obtain the approximation in the question. Alternatively, \(B(L, \frac{h}{L}) \approx Po(h)\) and \(Po(h) \approx N(h,h)\) so we obtain: \begin{align*} \P(C > l) &= 1-\P(C \leq l) \\ &\approx 1 - \P(\sqrt{h} Z +h < l + \frac12) \\ &= 1 - \P \l Z < \frac{2l-2h+1}{2\sqrt{h}} \r \\ &= \Phi \l \frac{2h - 2l -1}{2\sqrt{h}}\r \end{align*} as required. [I think this is what the examiners expected]. Considering the case \(K=7\), \(k=2\), \(L=500\), \(l=4\), \(M=100\), \(m=48\) and \(p=0.6\), we have the first normal approximation depends on \(Mp\) and \(M(1-p)\) being large. They are \(60\) and \(40\) respectively, so this is likely a good approximation. The first approximation finds that \begin{align*} h &= 500 \cdot \Phi \l \frac{2 \cdot 48 - 2 \cdot 60 - 1}{2\sqrt{24}} \r \\ &= 500 \cdot \Phi \l \frac{2 \cdot 48 - 2 \cdot 60 - 1}{2\sqrt{24}} \r \\ &= 500 \cdot \Phi \l \frac{-25}{2 \sqrt{24}} \r \\ &\approx 500 \cdot \Phi (-2.5) \\ &= 500 \cdot 0.0062 \\ &\approx 3.1 \end{align*} The second binomial approximation will be good if \(500 \cdot \frac{3.1}{500} = 3.1\) is large, but this is quite small. Therefore, we shouldn't expect this to be a good approximation. However, since \(m = 48\) is far from the mean (in a normalised sense), we might expect the percentage error to be large. [Alternatively, using what I expect the desired approach] The approximation of \(B(L, \frac{h}{L}) \approx Po(h)\) is acceptable since \(n>50\) and \(h < 5\). The approximation of \(Po(h) \sim N(h,h)\) is not acceptable since \(h\) is small (in particular \(h < 15\)) Finally, we can compute all these values exactly using a modern calculator. \begin{array}{l|cc} & \text{correct} & \text{approx} \\ \hline p_h & 0.005760\ldots & 0.005362\ldots \\ \P(C > l) & 0.164522\ldots & 0.133319\ldots \\ \text{ans} & 0.231389\ldots & 0.182516\ldots \end{array} We can also see how the errors propagate, by doing the calculations assuming the previous steps are correct, and also including the Poisson step. \begin{array}{lccc} & \text{correct} & \text{approx} & \text{using approx } p_h \\ \hline p_h & 0.005760\ldots & 0.005362\ldots & - \\ \P(C > l)\quad [Po(h)] & 0.164522\ldots & 0.165044\ldots & 0.134293\ldots \\ \P(C > l)\quad [N(h,h)] & 0.164522\ldots & 0.169953\ldots & 0.133319\ldots \\ \P(C > l)\quad [N(h,h(1-\frac{h}{L})] & 0.164522\ldots & 0.169255\ldots & 0.132677\ldots \\ \text{ans} & 0.231389\ldots & 0.231389\ldots \end{array} By doing this, we discover that the largest errors are actually coming not from approximating the second approximation but from the small absolute (but large relative error) in the first approximation. This is, in fact, a coincidence; we can observe it by investigating the specific values being used. The first approximation looks as follows:

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You might not be able to tell, but there's actually two plots on this chart. However, let's zoom in on the area we are worried about:

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We can see there are small differences, which could be large in percentage terms. (As we found when we computed them directly).

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First, we can immediately see that if we just look at the distribution of \(B(L, p_h)\) and \(B(L, p_{h_\text{approx}})\) we get quite different results, even before we do any approximations.

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If we plot the probability distribution of \(B(L, p_h)\) vs \(N(Lp_h, Lp_h(1-p_h))\) we find that it is not a great approximation.

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However, the CDF happens to be a very good approximation *just* for the value we care about. Very lucky, but not possible for someone sitting STEP to know at the time!

Each time it rains over the Cabbibo dam, a volume \(V\) of water is deposited, almost instanetaneously, in the reservoir. Each day (midnight to midnight) water flows from the reservoir at a constant rate \(u\) units of volume per day. An engineer, if present, may choose to alter the value of \(u\) at any midnight.

1. Suppose that it rains at most once in any day, that there is a probability \(p\) that it will rain on any given day and that, if it does, the rain is equally likely to fall at any time in the 24 hours (i.e. the time at which the rain falls is a random variable uniform on the interval \([0,24]\)). The engineers decides to take two days' holiday starting at midnight. If at this time the volume of water in the reservoir is \(V\) below the top of the dam, find an expression for \(u\) such that the probability of overflow in the two days is \(Q\), where \(Q < p^{2}.\)
2. For the engineer's summer holidays, which last 18 days, the reservoir is drained to a volume \(kV\) below the top of the dam and the rate of outflow \(u\) is set to zero. The engineer wants to drain off as little as possible, consistent with the requirement that the probability that the dam will overflow is less than \(\frac{1}{10}.\) In the case \(p=\frac{1}{3},\) find by means of a suitable approximation the required value of \(k\).
3. Suppose instead that it may rain at most once before noon and at most once after noon each day, that the probability of rain in any given half-day is \(\frac{1}{6}\) and that it is equally likely to rain at any time in each half-day. Is the required value of \(k\) lower or higher?

It is believed that the population of Ruritania can be described as follows:

1. \(25\%\) are fair-haired and the rest are dark-haired;
2. \(20\%\) are green-eyed and the rest hazel-eyed;
3. the population can also be divided into narrow-headed and broad-headed;
4. no narrow-headed person has green eyes and fair hair;
5. those who are green-eyed are as likely to be narrow-headed as broad-headed;
6. those who are green-eyed and broad-headed are as likely to be fair-headed as dark-haired;
7. half of the population is broad-headed and dark-haired;
8. a hazel-eyed person is as likely to be fair-haired and broad-headed as dark-haired and narrow-headed.

Show Solution

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Conditions tell us: \begin{align*} && a+b+d+e &= 0.25 \\ && b+c+e+f &= 0.2 \\ && e &= 0 \\ && b+c &= e + f \\ && b &= c \\ && c+h &= 0.5 \\ && a &= g \\ \end{align*}

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So \(4b = 0.2 \Rightarrow b = 0.05\)

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And \begin{align*} && 0.25 &= a + d + 0.05 \\ && 1 &= 2a + d + 0.65 \\ \Rightarrow && a &= 0.15 \\ && d &= 0.05 \end{align*}

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So the proportion who are narrow-headed is \(30\%\). It's obviously relatively unlikely for your six Ruritanian friends to all be broad-headed if it's a random sample, but friendship groups are are likely to be biased so it's not too surprising. Assuming there is a sufficiently large number of Ruritanians, we might model the number of narrow-headed Ruritanians from a sample of \(200\) as \(X \sim B(200, 0.3)\). Computing \(\mathbb{P}(X \leq 50)\) by hand is tricky, so let's use a binomial approximation to obtain: \(X \approx N(60, 42)\) and \begin{align*} \mathbb{P}(X \leq 50) &\approx \mathbb{P} \left (Z \leq \frac{50 - 60+0.5}{\sqrt{42}} \right) \\ &\approx \mathbb{P} \left (Z \leq -\frac{9.5}{6.5} \right) \\ &\approx \mathbb{P} \left (Z \leq -\frac{3}{2} \right) \\ &\approx 5\% \end{align*} (actually this approximation gives \(7.1\%\) and the binomial value gives \(7.0\%\)). This also seems somewhat surprising

Let \(X\) be a standard normal random variable. If \(M\) is any real number, the random variable \(X_{M}\) is defined in terms of \(X\) by \[ X_{M}=\begin{cases} X & \mbox{if }X < M,\\ M & \mbox{if }X\geqslant M. \end{cases} \] Show that the expectation of \(X_{M}\) is given by \[ \mathrm{E}(X_{M})=-\phi(M)+M(1-\Phi(M)), \] where \(\phi\) is the probability density function, and \(\Phi\) is the cumulative distribution function of \(X\). Fifty times a year, 1024 tourists disembark from a cruise liner at the port of Slaka. From there they must travel to the capital either by taxi or by bus. Officials of HOGPo are equally likely to direct a tourist to the bus station or to the taxi rank. Each bus of the bus coorperative holds 31 passengers, and the coorperative currently runs 16 buses. The bus coorperative makes a profit of 1 vloska for each passenger carried. It carries all the passengers it can, with any excess being (eventually) transported by taxi. What is the largest annual bribe the bus coorperative should consider paying to HOGPo in order to be allowed to run an extra bus?

The function f is defined by $$\f(x) = 2\sin x - x\,.$$ Show graphically that the equation \(\f(x)=0\) has exactly one root in the interval \([\frac12\pi ,\,{\pi}]\,\). This interval is denoted \(I_0\). In order to determine the root, a sequence of intervals \(I_1\), \(I_2, \,\ldots\) is generated in the following way. If the interval \(I_n=[a_n,b_n]\,\), and \(c_n=(a_n+b_n)/2\,\), then \begin{equation*} I_{n+1}= \begin{cases} [a_n,c_n] & \text{if \(\; \f(a_n)\f(c_n)<0 \,\)}; \\[5pt] [c_n,b_n] & \text{if \(\; \f(c_n)\f(b_n)<0 \,\)}. \end{cases} \end{equation*} By using the approximations \(\ds \frac 1{\sqrt{2}} \approx 0.7\) and \({\pi} \approx \sqrt{10} \,\), show that \(I_2=[\frac12{\pi},\,\frac58{\pi}]\) and find \(I_3\,\).

Sketch the graphs of \(y=\sec x\) and \(y=\ln(2\sec x)\) for \(0\leqslant x\leqslant\frac{1}{2}\pi\). Show graphically that the equation \[ kx=\ln(2\sec x) \] has no solution with \(0\leqslant x<\frac{1}{2}\pi\) if \(k\) is a small positive number but two solutions if \(k\) is large. Explain why there is a number \(k_{0}\) such that \[ k_{0}x=\ln(2\sec x) \] has exactly one solution with \(0\leqslant x<\frac{1}{2}\pi\). Let \(x_{0}\) be this solution, so that \(0\leqslant x_{0}<\frac{1}{2}\pi\) and \(k_{0}x_{0}=\ln(2\sec x_0)\). Show that \[ x_{0}=\cot x_{0}\ln(2\sec x_{0}). \] Use any appropriate method to find \(x_{0}\) correct to two decimal places. Hence find an approximate value for \(k_{0}\).

Show Solution

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The red line is \(y = \ln (2 \sec x)\), blue is \(y = \sec x\). We can see that if the gradient is too small it never touches the red line. If it is large it will cross the red line twice in that interval. For some value it will be perfectly tangent. Since the line is tangent we must have \begin{align*} && y &= \ln (2 \sec x) \\ \Rightarrow && \frac{\d y}{ \d x} &= \frac{1}{2 \sec x} \cdot 2\sec x \tan x \\ &&&= \tan x \\ \Rightarrow && k_0 &=\tan x_0 \\ \Rightarrow && k_0 x_0 &= \ln(2 \sec x_0 ) \\ \Rightarrow && x_0 &= \cot x_0 \ln (2 \sec x_0) \end{align*} If \(f(x) =x- \cot x \ln (2 \sec x)\), then \(f'(x) =1 - 1+\ln(2\sec x) \cosec^2x = \ln(2 \sec x)\cosec^2x \) so we should look at \begin{align*} x_{n+1} &= x_n - \frac{f(x_n)}{f'(x_n)} \\ &= x_n - \frac{x_n- \cot x_n \ln (2 \sec x_n)}{\ln(2 \sec x_n)\cosec^2x_n } \\ &= x_n \left (1 - \frac{\sin^2 x_n}{\ln (2 \sec x_n)}\right) +\sin x_n \cos x_n \end{align*} \begin{array}{c|c} n & x_n \\ \hline 1 & \frac{\pi}{4} \\ 2 & 0.907701\ldots \\ 3 & 0.91439340\ldots \\ 4 & 0.914403867\ldots \\ 5 & 0.91440386\ldots \\ 6 & 0.91440386\ldots \\ \end{array} The sign change test shows that \(x_0 \approx 0.91\) and \(k_0 = \tan(x_0) \approx 1.30\)