Problems

1. Show that making the substitution \(x = \frac{1}{t}\) in the integral \[\int_a^b \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,,\] where \(b > a > 0\), gives the integral \[\int_{b^{-1}}^{a^{-1}} \frac{-t}{(1+t^2)^{\frac{3}{2}}}\,\mathrm{d}t\,.\]
2. Evaluate:
   1. \(\displaystyle\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,;\)
   2. \(\displaystyle\int_{-2}^{2} \frac{1}{(1+x^2)^{\frac{3}{2}}}\,\mathrm{d}x\,.\)
3. 1. Show that \[\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^2}\,\mathrm{d}x = \int_{\frac{1}{2}}^{2} \frac{x^2}{(1+x^2)^2}\,\mathrm{d}x = \frac{1}{2}\int_{\frac{1}{2}}^{2} \frac{1}{1+x^2}\,\mathrm{d}x\,,\] and hence evaluate \[\int_{\frac{1}{2}}^{2} \frac{1}{(1+x^2)^2}\,\mathrm{d}x\,.\]
   2. Evaluate \[\int_{\frac{1}{2}}^{2} \frac{1-x}{x(1+x^2)^{\frac{1}{2}}}\,\mathrm{d}x\,.\]

1. The real numbers \(x\), \(y\) and \(z\) satisfy the equations \[y = \frac{2x}{1-x^2}\,,\qquad z = \frac{2y}{1-y^2}\,,\qquad x = \frac{2z}{1-z^2}\,.\] Let \(x = \tan\alpha\). Deduce that \(y = \tan 2\alpha\) and show that \(\tan\alpha = \tan 8\alpha\). Find all solutions of the equations, giving each value of \(x\), \(y\) and \(z\) in the form \(\tan\theta\) where \(-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi\).
2. Determine the number of real solutions of the simultaneous equations \[y = \frac{3x - x^3}{1-3x^2}\,,\qquad z = \frac{3y - y^3}{1-3y^2}\,,\qquad x = \frac{3z - z^3}{1-3z^2}\,.\]
3. Consider the simultaneous equations \[y = 2x^2 - 1\,,\qquad z = 2y^2 - 1\,,\qquad x = 2z^2 - 1\,.\]
   1. Determine the number of real solutions of these simultaneous equations with \(|x| \leqslant 1\), \(|y| \leqslant 1\), \(|z| \leqslant 1\).
   2. By finding the degree of a single polynomial equation which is satisfied by \(x\), show that all solutions of these simultaneous equations have \(|x| \leqslant 1\), \(|y| \leqslant 1\), \(|z| \leqslant 1\).

Let \(\mathrm{p}(x)\) be a polynomial of degree \(n\) with \(\mathrm{p}(x) > 0\) for all \(x\) and let \[\mathrm{q}(x) = \sum_{k=0}^{n} \mathrm{p}^{(k)}(x)\,,\] where \(\mathrm{p}^{(k)}(x) \equiv \dfrac{\mathrm{d}^k \mathrm{p}(x)}{\mathrm{d}x^k}\) for \(k \geqslant 1\) and \(\mathrm{p}^{(0)}(x) \equiv \mathrm{p}(x)\).

1. 1. Explain why \(n\) must be even and show that \(\mathrm{q}(x)\) takes positive values for some values of \(x\).
   2. Show that \(\mathrm{q}'(x) = \mathrm{q}(x) - \mathrm{p}(x)\).
2. In this part you will be asked to show the same result in three different ways.
   1. Show that the curves \(y = \mathrm{p}(x)\) and \(y = \mathrm{q}(x)\) meet at every stationary point of \(y = \mathrm{q}(x)\). Hence show that \(\mathrm{q}(x) > 0\) for all \(x\).
   2. Show that \(\mathrm{e}^{-x}\mathrm{q}(x)\) is a decreasing function. Hence show that \(\mathrm{q}(x) > 0\) for all \(x\).
   3. Show that \[\int_0^{\infty} \mathrm{p}(x+t)\mathrm{e}^{-t}\,\mathrm{d}t = \mathrm{p}(x) + \int_0^{\infty} \mathrm{p}^{(1)}(x+t)\mathrm{e}^{-t}\,\mathrm{d}t\,.\] Show further that \[\int_0^{\infty} \mathrm{p}(x+t)\mathrm{e}^{-t}\,\mathrm{d}t = \mathrm{q}(x)\,.\] Hence show that \(\mathrm{q}(x) > 0\) for all \(x\).

1. Show that, if \((x-\sqrt{2})^2 = 3\), then \(x^4 - 10x^2 + 1 = 0\). Deduce that, if \(\mathrm{f}(x) = x^4 - 10x^2 + 1\), then \(\mathrm{f}(\sqrt{2}+\sqrt{3}) = 0\).
2. Find a polynomial \(\mathrm{g}\) of degree 8 with integer coefficients such that \(\mathrm{g}(\sqrt{2}+\sqrt{3}+\sqrt{5}) = 0\). Write your answer in a form without brackets.
3. Let \(a\), \(b\) and \(c\) be the three roots of \(t^3 - 3t + 1 = 0\). Find a polynomial \(\mathrm{h}\) of degree 6 with integer coefficients such that \(\mathrm{h}(a+\sqrt{2}) = 0\), \(\mathrm{h}(b+\sqrt{2}) = 0\) and \(\mathrm{h}(c+\sqrt{2}) = 0\). Write your answer in a form without brackets.
4. Find a polynomial \(\mathrm{k}\) with integer coefficients such that \(\mathrm{k}(\sqrt[3]{2}+\sqrt[3]{3}) = 0\). Write your answer in a form without brackets.

1. The sequence \(x_n\) for \(n = 0, 1, 2, \ldots\) is defined by \(x_0 = 1\) and by \[x_{n+1} = \frac{x_n + 2}{x_n + 1}\] for \(n \geqslant 0\).
   1. Explain briefly why \(x_n \geqslant 1\) for all \(n\).
   2. Show that \(x_{n+1}^2 - 2\) and \(x_n^2 - 2\) have opposite sign, and that \[\left|x_{n+1}^2 - 2\right| \leqslant \tfrac{1}{4}\left|x_n^2 - 2\right|\,.\]
   3. Show that \[2 - 10^{-6} \leqslant x_{10}^2 \leqslant 2\,.\]
2. The sequence \(y_n\) for \(n = 0, 1, 2, \ldots\) is defined by \(y_0 = 1\) and by \[y_{n+1} = \frac{y_n^2 + 2}{2y_n}\] for \(n \geqslant 0\).
   1. Show that, for \(n \geqslant 0\), \[y_{n+1} - \sqrt{2} = \frac{(y_n - \sqrt{2})^2}{2y_n}\] and deduce that \(y_n \geqslant 1\) for \(n \geqslant 0\).
   2. Show that \[y_n - \sqrt{2} \leqslant 2\left(\frac{\sqrt{2}-1}{2}\right)^{2^n}\] for \(n \geqslant 1\).
   3. Using the fact that \[\sqrt{2} - 1 < \tfrac{1}{2}\,,\] or otherwise, show that \[\sqrt{2} \leqslant y_{10} \leqslant \sqrt{2} + 10^{-600}\,.\]

The sequence \(F_n\), for \(n = 0, 1, 2, \ldots\), is defined by \(F_0 = 0\), \(F_1 = 1\) and by \(F_{n+2} = F_{n+1} + F_n\) for \(n \geqslant 0\). Prove by induction that, for all positive integers \(n\), \[\begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix} = \mathbf{Q}^n,\] where the matrix \(\mathbf{Q}\) is given by \[\mathbf{Q} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}.\]

1. By considering the matrix \(\mathbf{Q}^n\), show that \(F_{n+1}F_{n-1} - F_n^2 = (-1)^n\) for all positive integers \(n\).
2. By considering the matrix \(\mathbf{Q}^{m+n}\), show that \(F_{m+n} = F_{m+1}F_n + F_m F_{n-1}\) for all positive integers \(m\) and \(n\).
3. Show that \(\mathbf{Q}^2 = \mathbf{I} + \mathbf{Q}\). In the following parts, you may use without proof the Binomial Theorem for matrices: \[(\mathbf{I} + \mathbf{A})^n = \sum_{k=0}^{n} \binom{n}{k} \mathbf{A}^k.\]
   1. Show that, for all positive integers \(n\), \[F_{2n} = \sum_{k=0}^{n} \binom{n}{k} F_k\,.\]
   2. Show that, for all positive integers \(n\), \[F_{3n} = \sum_{k=0}^{n} \binom{n}{k} 2^k F_k\] and also that \[F_{3n} = \sum_{k=0}^{n} \binom{n}{k} F_{n+k}\,.\]
   3. Show that, for all positive integers \(n\), \[\sum_{k=0}^{n} (-1)^{n+k} \binom{n}{k} F_{n+k} = 0\,.\]

1. The complex numbers \(z\) and \(w\) have real and imaginary parts given by \(z = a + \mathrm{i}b\) and \(w = c + \mathrm{i}d\). Prove that \(|zw| = |z||w|\).
2. By considering the complex numbers \(2 + \mathrm{i}\) and \(10 + 11\mathrm{i}\), find positive integers \(h\) and \(k\) such that \(h^2 + k^2 = 5 \times 221\).
3. Find positive integers \(m\) and \(n\) such that \(m^2 + n^2 = 8045\).
4. You are given that \(102^2 + 201^2 = 50805\). Find positive integers \(p\) and \(q\) such that \(p^2 + q^2 = 36 \times 50805\).
5. Find three distinct pairs of positive integers \(r\) and \(s\) such that \(r^2 + s^2 = 25 \times 1002082\) and \(r < s\).
6. You are given that \(109 \times 9193 = 1002037\). Find positive integers \(t\) and \(u\) such that \(t^2 + u^2 = 9193\).

A tetrahedron is called isosceles if each pair of edges which do not share a vertex have equal length.

1. Prove that a tetrahedron is isosceles if and only if all four faces have the same perimeter.

Let \(OABC\) be an isosceles tetrahedron and let \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\).

2. By considering the lengths of \(OA\) and \(BC\), show that \[2\mathbf{b}.\mathbf{c} = |\mathbf{b}|^2 + |\mathbf{c}|^2 - |\mathbf{a}|^2.\] Show that \[\mathbf{a}.(\mathbf{b}+\mathbf{c}) = |\mathbf{a}|^2.\]
3. Let \(G\) be the centroid of the tetrahedron, defined by \(\overrightarrow{OG} = \frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c})\). Show that \(G\) is equidistant from all four vertices of the tetrahedron.
4. By considering the length of the vector \(\mathbf{a}-\mathbf{b}-\mathbf{c}\), or otherwise, show that, in an isosceles tetrahedron, none of the angles between pairs of edges which share a vertex can be obtuse. Can any of them be right angles?

A truck of mass \(M\) is connected by a light, rigid tow-bar, which is parallel to the ground, to a trailer of mass \(kM\). A constant driving force \(D\) which is parallel to the ground acts on the truck, and the only resistance to motion is a frictional force acting on the trailer, with coefficient of friction \(\mu\).

• When the truck pulls the trailer up a slope which makes an angle \(\alpha\) to the horizontal, the acceleration is \(a_1\) and there is a tension \(T_1\) in the tow-bar.
• When the truck pulls the trailer on horizontal ground, the acceleration is \(a_2\) and there is a tension \(T_2\) in the tow-bar.
• When the truck pulls the trailer down a slope which makes an angle \(\alpha\) to the horizontal, the acceleration is \(a_3\) and there is a tension \(T_3\) in the tow-bar.

All accelerations are taken to be positive when in the direction of motion of the truck.

1. Show that \(T_1 = T_3\) and that \(M(a_1 + a_3 - 2a_2) = 2(T_2 - T_1)\).
2. It is given that \(\mu < 1\).
   1. Show that \[a_2 < \tfrac{1}{2}(a_1 + a_3) < a_3\,.\]
   2. Show further that \[a_1 < a_2\,.\]

In this question, the \(x\)- and \(y\)-axes are horizontal and the \(z\)-axis is vertically upwards.

1. A particle \(P_\alpha\) is projected from the origin with speed \(u\) at an acute angle \(\alpha\) above the positive \(x\)-axis. The curve \(E\) is given by \(z = A - Bx^2\) and \(y = 0\). If \(E\) and the trajectory of \(P_\alpha\) touch exactly once, show that \[u^2 - 2gA = u^2(1 - 4AB)\cos^2\alpha\,.\] \(E\) and the trajectory of \(P_\alpha\) touch exactly once for all \(\alpha\) with \(0 < \alpha < \frac{1}{2}\pi\). Write down the values of \(A\) and \(B\) in terms of \(u\) and \(g\).

An explosion takes place at the origin and results in a large number of particles being simultaneously projected with speed \(u\) in different directions. You may assume that all the particles move freely under gravity for \(t \geqslant 0\).

2. Describe the set of points which can be hit by particles from the explosion, explaining your answer.
3. Show that, at a time \(t\) after the explosion, the particles lie on a sphere whose centre and radius you should find.
4. Another particle \(Q\) is projected horizontally from the point \((0, 0, A)\) with speed \(u\) in the positive \(x\) direction. Show that, at all times, \(Q\) lies on the curve \(E\).
5. Show that for particles \(Q\) and \(P_\alpha\) to collide, \(Q\) must be projected a time \(\dfrac{u(1-\cos\alpha)}{g\sin\alpha}\) after the explosion.

1. \(X_1\) and \(X_2\) are both random variables which take values \(x_1, x_2, \ldots, x_n\), with probabilities \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\) respectively. The value of random variable \(Y\) is defined to be that of \(X_1\) with probability \(p\) and that of \(X_2\) with probability \(q = 1-p\). If \(X_1\) has mean \(\mu_1\) and variance \(\sigma_1^2\), and \(X_2\) has mean \(\mu_2\) and variance \(\sigma_2^2\), find the mean of \(Y\) and show that the variance of \(Y\) is \(p\sigma_1^2 + q\sigma_2^2 + pq(\mu_1 - \mu_2)^2\).
2. To find the value of random variable \(B\), a fair coin is tossed and a fair six-sided die is rolled. If the coin shows heads, then \(B = 1\) if the die shows a six and \(B = 0\) otherwise; if the coin shows tails, then \(B = 1\) if the die does not show a six and \(B = 0\) if it does. The value of \(Z_1\) is the sum of \(n\) independent values of \(B\), where \(n\) is large. Show that \(Z_1\) is a Binomial random variable with probability of success \(\frac{1}{2}\). Using a Normal approximation, show that the probability that \(Z_1\) is within \(10\%\) of its mean tends to \(1\) as \(n \longrightarrow \infty\).
3. To find the value of random variable \(Z_2\), a fair coin is tossed and \(n\) fair six-sided dice are rolled, where \(n\) is large. If the coin shows heads, then the value of \(Z_2\) is the number of dice showing a six; if the coin shows tails, then the value of \(Z_2\) is the number of dice not showing a six. Use part (i) to write down the mean and variance of \(Z_2\). Explain why a Normal distribution with this mean and variance will not be a good approximation to the distribution of \(Z_2\). Show that the probability that \(Z_2\) is within \(10\%\) of its mean tends to \(0\) as \(n \longrightarrow \infty\).

Each of the independent random variables \(X_1, X_2, \ldots, X_n\) has the probability density function \(\mathrm{f}(x) = \frac{1}{2}\sin x\) for \(0 \leqslant x \leqslant \pi\) (and zero otherwise). Let \(Y\) be the random variable whose value is the maximum of the values of \(X_1, X_2, \ldots, X_n\).

1. Explain why \(\mathrm{P}(Y \leqslant t) = \big[\mathrm{P}(X_1 \leqslant t)\big]^n\) and hence, or otherwise, find the probability density function of \(Y\).

Let \(m(n)\) be the median of \(Y\) and \(\mu(n)\) be the mean of \(Y\).

2. Find an expression for \(m(n)\) in terms of \(n\). How does \(m(n)\) change as \(n\) increases?
3. Show that \[\mu(n) = \pi - \frac{1}{2^n}\int_0^{\pi} (1-\cos x)^n\,\mathrm{d}x\,.\]
   1. Show that \(\mu(n)\) increases with \(n\).
   2. Show that \(\mu(2) < m(2)\).

The distinct points \(P(2ap,\, ap^2)\) and \(Q(2aq,\, aq^2)\) lie on the curve \(x^2 = 4ay\), where \(a > 0\).

1. Given that \[(p+q)^2 = p^2q^2 + 6pq + 5\,,\tag{\(*\)}\] show that the line through \(P\) and \(Q\) is a tangent to the circle with centre \((0,\, 3a)\) and radius \(2a\).
2. Show that, for any given value of \(p\) with \(p^2 \neq 1\), there are two distinct real values of \(q\) that satisfy equation \((*)\). Let these values be \(q_1\) and \(q_2\). Find expressions, in terms of \(p\), for \(q_1 + q_2\) and \(q_1 q_2\).
3. Show that, for any given value of \(p\) with \(p^2 \neq 1\), there is a triangle with one vertex at \(P\) such that all three vertices lie on the curve \(x^2 = 4ay\) and all three sides are tangents to the circle with centre \((0,\, 3a)\) and radius \(2a\).

The polar curves \(C_1\) and \(C_2\) are defined for \(0 \leqslant \theta \leqslant \pi\) by \[r = k(1 + \sin\theta)\] \[r = k + \cos\theta\] respectively, where \(k\) is a constant greater than \(1\).

1. Sketch the curves on the same diagram. Show that if \(\theta = \alpha\) at the point where the curves intersect, \(\tan\alpha = \dfrac{1}{k}\).
2. The region A is defined by the inequalities \[0 \leqslant \theta \leqslant \alpha \quad \text{and} \quad r \leqslant k(1+\sin\theta)\,.\] Show that the area of A can be written as \[\frac{k^2}{4}(3\alpha - \sin\alpha\cos\alpha) + k^2(1 - \cos\alpha)\,.\]
3. The region B is defined by the inequalities \[\alpha \leqslant \theta \leqslant \pi \quad \text{and} \quad r \leqslant k + \cos\theta\,.\] Find an expression in terms of \(k\) and \(\alpha\) for the area of B.
4. The total area of regions A and B is denoted by \(R\). The area of the region enclosed by \(C_1\) and the lines \(\theta = 0\) and \(\theta = \pi\) is denoted by \(S\). The area of the region enclosed by \(C_2\) and the lines \(\theta = 0\) and \(\theta = \pi\) is denoted by \(T\). Show that as \(k \to \infty\), \[\frac{R}{T} \to 1\] and find the limit of \[\frac{R}{S}\] as \(k \to \infty\).

1. Show that, if \(a\) and \(b\) are complex numbers, with \(b \neq 0\), and \(s\) is a positive real number, then the points in the Argand diagram representing the complex numbers \(a + sbi\), \(a - sbi\) and \(a + b\) form an isosceles triangle. Given three points which form an isosceles triangle in the Argand diagram, explain with the aid of a diagram how to determine the values of \(a\), \(b\) and \(s\) so that the vertices of the triangle represent complex numbers \(a + sbi\), \(a - sbi\) and \(a + b\).
2. Show that, if the roots of the equation \(z^3 + pz + q = 0\), where \(p\) and \(q\) are complex numbers, are represented in the Argand diagram by the vertices of an isosceles triangle, then there is a non-zero real number \(s\) such that \[\frac{p^3}{q^2} = \frac{27(3s^2 - 1)^3}{4(9s^2 + 1)^2}\,.\]
3. Sketch the graph \(y = \dfrac{(3x-1)^3}{(9x+1)^2}\), identifying any stationary points.
4. Show that if the roots of the equation \(z^3 + pz + q = 0\) are represented in the Argand diagram by the vertices of an isosceles triangle then \(\dfrac{p^3}{q^2}\) is a real number and \(\dfrac{p^3}{q^2} > -\dfrac{27}{4}\).