Problems

1. Find the value of \(m\) for which the line \(y = mx\) touches the curve \(y = \ln x\,\). If instead the line intersects the curve when \(x = a\) and \(x = b\), where \(a < b\), show that \(a^b = b^a\). Show by means of a sketch that \(a < \e < b\).
2. The line \(y=mx+c\), where \(c>0\), intersects the curve \(y=\ln x\) when \(x=p\) and \(x=q\), where \(p < q\). Show by means of a sketch, or otherwise, that \(p^q > q^p\).
3. Show by means of a sketch that the straight line through the points \((p, \ln p)\) and \((q, \ln q)\), where \(\e\le p < q\,\), intersects the \(y\)-axis at a positive value of \(y\). Which is greater, \(\pi^\e\) or \(\e^\pi\)?
4. Show, using a sketch or otherwise, that if \(0 < p < q\) and \(\dfrac{\ln q - \ln p}{q-p} = \e^{-1}\), then \(q^p > p^q\).

For \(n\ge 0\), let \[ I_n = \int_0^1 x^n(1-x)^n\d x\,. \]

1. For \(n\ge 1\), show by means of a substitution that \[ \int_0^1 x^{n-1}(1-x)^n\d x = \int_0^1 x^n(1-x)^{n-1}\d x\, \] and deduce that \[ 2 \int_0^1 x^{n-1}(1-x)^n\d x = I_{n-1}\,. \] Show also, for \(n\ge1\), that \[ I_n = \frac n {n+1} \int_0^1 x^{n-1} (1-x)^{n+1} \d x \] and hence that \(I_n = \dfrac{n}{2(2n+1)} I_{n-1}\,.\)
2. When \(n\) is a positive integer, show that \[ I_n = \frac{(n!)^2}{(2n+1)!}\,. \]
3. Use the substitution \(x= \sin^2 \theta\) to show that \(I_{\frac12}= \frac \pi 8\), and evaluate \(I_{\frac32}\).

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.

The line passing through the point \((a,0)\) with gradient \(b\) intersects the circle of unit radius centred at the origin at \(P\) and \(Q\), and \(M\) is the midpoint of the chord \(PQ\). Find the coordinates of \(M\) in terms of \(a\) and \(b\).

1. Suppose \(b\) is fixed and positive. As \(a\) varies, \(M\) traces out a curve (the locus of \(M\)). Show that \(x=- by\) on this curve. Given that \(a\) varies with \(-1\le a \le 1\), show that the locus is a line segment of length \(2b/(1+b^2)^\frac12\). Give a sketch showing the locus and the unit circle.
2. Find the locus of \(M\) in the following cases, giving in each case its cartesian equation, describing it geometrically and sketching it in relation to the unit circle:
   • \(a\) is fixed with \(0 < a < 1\), and \(b\) varies with \(-\infty < b < \infty\);
   • \(ab=1\), and \(b\) varies with \(0< b\le1\).

1. A function \(\f(x)\) satisfies \(\f(x) = \f(1-x)\) for all \(x\). Show, by differentiating with respect to \(x\), that \(\f'(\frac12) =0\,\). If, in addition, \(\f(x) = \f(\frac1x)\) for all (non-zero) \(x\), show that \(\f'(-1)=0\) and that \(\f'(2)=0\).
2. The function \(\f\) is defined, for \(x\ne0\) and \(x\ne1\), by \[ \f(x) = \frac {(x^2-x+1)^3}{(x^2-x)^2} \,. \] Show that \(\f(x)= \f(\frac 1 x)\) and \(\f(x) = \f(1-x)\). Given that it has exactly three stationary points, sketch the curve \(y=\f(x)\).
3. Hence, or otherwise, find all the roots of the equation \(\f(x) = \dfrac {27} 4\,\) and state the ranges of values of \(x\) for which \(\f(x) > \dfrac{27} 4\,\). Find also all the roots of the equation \(\f(x) = \dfrac{343}{36}\,\) and state the ranges of values of \(x\) for which \(\f(x) > \dfrac{343}{36}\).

In this question, the following theorem may be used.\newline {\sl Let \(u_1\), \(u_2\), \(\ldots\) be a sequence of (real) numbers. If the sequence is bounded above (that is, \(u_n\le b\) for all \(n\), where \(b\) is some fixed number) and increasing (that is, \(u_n\ge u_{n-1}\) for all \(n\)), then the sequence tends to a limit (that is, converges).} The sequence \(u_1\), \(u_2\), \(\ldots\) is defined by \(u_1=1\) and \[ u_{n+1} = 1+\frac 1{u_n} \ \ \ \ \ \ \ \ \ \ (n\ge1)\,. \tag{\(*\)} \]

1. Show that, for \(n\ge3\), \[ u_{n+2}-u_n = \frac{u_{n} - u_{n-2}}{(1+u_n)(1+u_{n-2})} . \]
2. Prove, by induction or otherwise, that \(1\le u_n \le 2\) for all \(n\).
3. Show that the sequence \(u_1\), \(u_3\), \(u_5\), \(\ldots\) tends to a limit, and that the sequence \(u_2\), \(u_4\), \(u_6\), \(\ldots\) tends to a limit. Find these limits and deduce that the sequence \(u_1\), \(u_2\), \(u_3\), \(\ldots\,\) tends to a limit. Would this conclusion change if the sequence were defined by \((*)\) and \(u_1=3\)?

1. Write down a solution of the equation \[ x^2-2y^2 =1\,, \tag{\(*\)} \] for which \(x\) and \(y\) are non-negative integers. Show that, if \(x=p\), \(y=q\) is a solution of (\(*\)), then so also is \(x=3p+4q\), \(y=2p+3q\). Hence find two solutions of \((*)\) for which \(x\) is a positive odd integer and \(y\) is a positive even integer.
2. Show that, if \(x\) is an odd integer and \(y\) is an even integer, \((*)\) can be written in the form \[ n^2 = \tfrac12 m(m+1)\,, \] where \(m\) and \(n\) are integers.
3. The positive integers \(a\), \(b\) and \(c\) satisfy \[ b^3=c^4-a^2\,, \] where \(b\) is a prime number. Express \(a\) and \(c^2\) in terms of \(b\) in the two cases that arise. Find a solution of \(a^2+b^3=c^4\), where \(a\), \(b\) and \(c\) are positive integers but \(b\) is not prime.

The function \(\f\) satisfies \(\f(x)>0\) for \(x\ge0\) and is strictly decreasing (which means that \(\f(b)<\f(a)\) for \(b>a\)).

1. For \(t\ge0\), let \(A_0(t)\) be the area of the largest rectangle with sides parallel to the coordinate axes that can fit in the region bounded by the curve \(y=\f(x)\), the \(y\)-axis and the line \(y=\f(t)\). Show that \(A_0(t)\) can be written in the form \[ A_0(t) =x_0\left( \f(x_0) -\f(t)\right), \] where \(x_0\) satisfies \(x_0 \f'(x_0) +\f(x_0) = \f(t)\,\).
2. The function g is defined, for \(t> 0\), by \[ \g(t) =\frac 1t \int_0^t \f(x) \d x\,. \] Show that \(t \g'(t) = \f(t) -\g(t)\,\). Making use of a sketch show that, for \(t>0\), \[ \int_0^t \left( \f(x) - \f(t)\right) \d x > A_0(t) \] and deduce that \(-t^2 \g'(t)> A_0(t)\).
3. In the case \(\f(x)= \dfrac 1 {1+x}\,\), use the above to establish the inequality \[ \ln \sqrt{1+t} > 1 - \frac 1 {\sqrt{1+t}} \,, \] for \(t>0\).

The diagram shows three identical discs in equilibrium in a vertical plane. Two discs rest, not in contact with each other, on a horizontal surface and the third disc rests on the other two. The angle at the upper vertex of the triangle joining the centres of the discs is \(2\theta\).

1. Show that the normal reaction between the horizontal surface and a disc in contact with the surface is \(\frac32 W\,\).
2. Find the normal reaction between two discs in contact and show that the magnitude of the frictional force between two discs in contact is \(\dfrac{W\sin\theta}{2(1+\cos\theta)}\,\).
3. Show that if \(\mu <2- \surd3\,\) there is no value of \(\theta\) for which equilibrium is possible.

A particle is projected at an angle of elevation \(\alpha\) (where \(\alpha>0\)) from a point \(A\) on horizontal ground. At a general point in its trajectory the angle of elevation of the particle from \(A\) is \(\theta\) and its direction of motion is at an angle \(\phi\) above the horizontal (with \(\phi\ge0\) for the first half of the trajectory and \(\phi\le0\) for the second half). Let \(B\) denote the point on the trajectory at which \(\theta = \frac12 \alpha\) and let \(C\) denote the point on the trajectory at which \(\phi = -\frac12\alpha\).

1. Show that, at a general point on the trajectory, \(2\tan\theta = \tan \alpha + \tan\phi\,\).
2. Show that, if \(B\) and \(C\) are the same point, then \( \alpha = 60^\circ\,\).
3. Given that \(\alpha < 60^\circ\,\), determine whether the particle reaches the point \(B\) first or the point \(C\) first.