Problems

A cube of uniform density \(\rho\) is placed on a horizontal plane and a second cube, also of uniform density \(\rho\), is placed on top of it. The lower cube has side length \(1\) and the upper cube has side length \(a\), with \(a \leqslant 1\). The centre of mass of the upper cube is vertically above the centre of mass of the lower cube and all the edges of the upper cube are parallel to the corresponding edges of the lower cube. The contacts between the two cubes, and between the lower cube and the plane, are rough, with the same coefficient of friction \(\mu < 1\) in each case. The midpoint of the base of the upper cube is \(X\) and the midpoint of the base of the lower cube is \(Y\). A horizontal force \(P\) is exerted, perpendicular to one of the vertical faces of the upper cube, at a point halfway between the two vertical edges of this face, and a distance \(h\), with \(h < a\), above the lower edge of this face.

1. Show that, if the two cubes remain in equilibrium, the normal reaction of the plane on the lower cube acts at a point which is a distance \[\frac{P(1+h)}{(1+a^3)\rho g}\] from \(Y\), and find a similar expression for the distance from \(X\) of the point at which the normal reaction of the lower cube on the upper cube acts.

The force \(P\) is now gradually increased from zero.

2. Show that, if neither cube topples, equilibrium will be broken by the slipping of the upper cube on the lower cube, and not by the slipping of the lower cube on the ground.
3. Show that, if \(a = 1\), then equilibrium will be broken by the slipping of the upper cube on the lower cube if \(\mu(1+h) < 1\) and by the toppling of the lower and upper cube together if \(\mu(1+h) > 1\).
4. Show that, in a situation where \(a < 1\) and \(h\bigl(1 + a^3(1-a)\bigr) > a^4\), and no slipping occurs, equilibrium will be broken by the toppling of the upper cube.
5. Show, by considering \(a = \frac{1}{2}\) and choosing suitable values of \(h\) and \(\mu\), that the situation described in (iv) can in fact occur.

In this question, you may use without proof the results \[\sum_{r=0}^{n} \binom{n}{r} = 2^n \quad \text{and} \quad \sum_{r=0}^{n} r\binom{n}{r} = n\,2^{n-1}.\]

1. Show that \[r\binom{2n}{r} = (2n+1-r)\binom{2n}{2n+1-r}\] for \(1 \leqslant r \leqslant 2n\). Hence show that \[\sum_{r=0}^{2n} r\binom{2n}{r} = 2\sum_{r=n+1}^{2n} r\binom{2n}{r}.\]
2. A fair coin is tossed \(2n\) times. The value of the random variable \(X\) is whichever is the larger of the number of heads and the number of tails shown. If \(n\) heads and \(n\) tails are shown, then \(X = n\). Show that \[\mathrm{E}(X) = n\left(1 + \frac{1}{2^{2n}}\binom{2n}{n}\right).\]
3. Show that \(\dfrac{1}{2^{2n}}\dbinom{2n}{n}\) decreases as \(n\) increases.
4. In a game, you choose a value of \(n\) and pay \(\pounds n\); then a fair coin is tossed \(2n\) times. You win an amount in pounds equal to the larger of the number of heads and the number of tails shown. If \(n\) heads and \(n\) tails are shown, then you win \(\pounds n\). How should you choose \(n\) to maximise your expected winnings per pound paid?

1. A point is chosen at random in the square \(0 \leqslant x \leqslant 1\), \(0 \leqslant y \leqslant 1\), so that the probability that a point lies in any region is equal to the area of that region. \(R\) is the random variable giving the distance of the point from the origin. Show that the cumulative distribution function of \(R\) is given by \[\mathrm{P}(R \leqslant r) = \sqrt{r^2 - 1} + \tfrac{1}{4}\pi r^2 - r^2 \cos^{-1}(r^{-1}),\] when \(1 \leqslant r \leqslant \sqrt{2}\). What is the cumulative distribution function when \(0 \leqslant r \leqslant 1\)?
2. Show that \(\displaystyle\mathrm{E}(R) = \frac{2}{3}\int_1^{\sqrt{2}} \frac{r^2}{\sqrt{r^2-1}}\,\mathrm{d}r\).
3. Show further that \(\mathrm{E}(R) = \frac{1}{3}\Bigl(\sqrt{2} + \ln\bigl(\sqrt{2}+1\bigr)\Bigr)\).

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 \(\mu(2) < m(2)\).