Problems

Let \(\mathbf{n}\) be a vector of unit length in three dimensions. For each vector \(\mathbf{r}\), \(\mathrm{f}(\mathbf{r})\) is defined by \[ \mathrm{f}(\mathbf{r}) = \mathbf{n} \times \mathbf{r}\,. \]

1. Given that \[ \mathbf{n} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \] show that the \(x\)-component of \(\mathrm{f}(\mathrm{f}(\mathbf{r}))\) is \(-x(b^2+c^2)+aby+acz\). Show further that \[ \mathrm{f}(\mathrm{f}(\mathbf{r})) = (\mathbf{n}.\mathbf{r})\mathbf{n} - \mathbf{r}\,. \] Explain, by means of a diagram, how \(\mathrm{f}(\mathrm{f}(\mathbf{r}))\) is related to \(\mathbf{n}\) and \(\mathbf{r}\).
2. Let \(R\) be the point with position vector \(\mathbf{r}\) and \(P\) be the point with position vector \(\mathrm{g}(\mathbf{r})\), where \(\mathrm{g}\) is defined by \[ \mathrm{g}(\mathbf{s}) = \mathbf{s} + \sin\theta\,\mathrm{f}(\mathbf{s}) + (1-\cos\theta)\,\mathrm{f}(\mathrm{f}(\mathbf{s}))\,. \] By considering \(\mathrm{g}(\mathbf{n})\) and \(\mathrm{g}(\mathbf{r})\) when \(\mathbf{r}\) is perpendicular to \(\mathbf{n}\), state, with justification, the geometric transformation which maps \(R\) onto \(P\).
3. Let \(R\) be the point with position vector \(\mathbf{r}\) and \(Q\) be the point with position vector \(\mathrm{h}(\mathbf{r})\), where \(\mathrm{h}\) is defined by \[ \mathrm{h}(\mathbf{s}) = -\mathbf{s} - 2\,\mathrm{f}(\mathrm{f}(\mathbf{s}))\,. \] State, with justification, the geometric transformation which maps \(R\) onto \(Q\).

1. Use De Moivre's theorem to prove that for any positive integer \(k > 1\), \[ \sin(k\theta) = \sin\theta\cos^{k-1}\theta \left( k - \binom{k}{3}(\sec^2\theta - 1) + \binom{k}{5}(\sec^2\theta - 1)^2 - \cdots \right) \] and find a similar expression for \(\cos(k\theta)\).
2. Let \(\theta = \cos^{-1}(\frac{1}{a})\), where \(\theta\) is measured in degrees, and \(a\) is an odd integer greater than \(1\). Suppose that there is a positive integer \(k\) such that \(\sin(k\theta) = 0\) and \(\sin(m\theta) \neq 0\) for all integers \(m\) with \(0 < m < k\). Show that it would be necessary to have \(k\) even and \(\cos(\frac{1}{2}k\theta) = 0\). Deduce that \(\theta\) is irrational.
3. Show that if \(\phi = \cot^{-1}(\frac{1}{b})\), where \(\phi\) is measured in degrees, and \(b\) is an even integer greater than \(1\), then \(\phi\) is irrational.

1. Two particles \(A\) and \(B\), of masses \(m\) and \(km\) respectively, lie at rest on a smooth horizontal surface. The coefficient of restitution between the particles is \(e\), where \(0 < e < 1\). Particle \(A\) is then projected directly towards particle \(B\) with speed \(u\). Let \(v_1\) and \(v_2\) be the velocities of particles \(A\) and \(B\), respectively, after the collision, in the direction of the initial velocity of \(A\). Show that \(v_1 = \alpha u\) and \(v_2 = \beta u\), where \(\alpha = \dfrac{1 - ke}{k+1}\) and \(\beta = \dfrac{1+e}{k+1}\). Particle \(B\) strikes a vertical wall which is perpendicular to its direction of motion and a distance \(D\) from the point of collision with \(A\), and rebounds. The coefficient of restitution between particle \(B\) and the wall is also \(e\). Show that, if \(A\) and \(B\) collide for a second time at a point \(\frac{1}{2}D\) from the wall, then \[ k = \frac{1+e-e^2}{e(2e+1)}\,. \]
2. Three particles \(A\), \(B\) and \(C\), of masses \(m\), \(km\) and \(k^2m\) respectively, lie at rest on a smooth horizontal surface in a straight line, with \(B\) between \(A\) and \(C\). A vertical wall is perpendicular to this line and lies on the side of \(C\) away from \(A\) and \(B\). The distance between \(B\) and \(C\) is equal to \(d\) and the distance between \(C\) and the wall is equal to \(3d\). The coefficient of restitution between each pair of particles, and between particle \(C\) and the wall, is \(e\), where \(0 < e < 1\). Particle \(A\) is then projected directly towards particle \(B\) with speed \(u\). Show that, if all three particles collide simultaneously at a point \(\frac{3}{2}d\) from the wall, then \(e = \frac{1}{2}\).

Two light elastic springs each have natural length \(a\). One end of each spring is attached to a particle \(P\) of weight \(W\). The other ends of the springs are attached to the end-points, \(B\) and \(C\), of a fixed horizontal bar \(BC\) of length \(2a\). The moduli of elasticity of the springs \(PB\) and \(PC\) are \(s_1 W\) and \(s_2 W\) respectively; these values are such that the particle \(P\) hangs in equilibrium with angle \(BPC\) equal to \(90^\circ\).

1. Let angle \(PBC = \theta\). Show that \(s_1 = \dfrac{\sin\theta}{2\cos\theta - 1}\) and find \(s_2\) in terms of \(\theta\).
2. Take the zero level of gravitational potential energy to be the horizontal bar \(BC\) and let the total potential energy of the system be \(-paW\). Show that \(p\) satisfies \[ \frac{1}{2}\sqrt{2} \geqslant p > \frac{1}{4}(1+\sqrt{3}) \] and hence that \(p = 0.7\), correct to one significant figure.

A fair coin is tossed \(N\) times and the random variable \(X\) records the number of heads. The mean deviation, \(\delta\), of \(X\) is defined by \[ \delta = \mathrm{E}\big(|X - \mu|\big) \] where \(\mu\) is the mean of \(X\).

1. Let \(N = 2n\) where \(n\) is a positive integer.
   1. Show that \(\mathrm{P}(X \leqslant n-1) = \frac{1}{2}\big(1 - \mathrm{P}(X=n)\big)\).
   2. Show that \[ \delta = \sum_{r=0}^{n-1}(n-r)\binom{2n}{r}\frac{1}{2^{2n-1}}\,. \]
   3. Show that for \(r > 0\), \[ r\binom{2n}{r} = 2n\binom{2n-1}{r-1}\,. \] Hence show that \[ \delta = \frac{n}{2^{2n}}\binom{2n}{n}\,. \]
2. Find a similar expression for \(\delta\) in the case \(N = 2n+1\).

1. The point \(A\) lies on the circumference of a circle of radius \(a\) and centre \(O\). The point \(B\) is chosen at random on the circumference, so that the angle \(AOB\) has a uniform distribution on \([0, 2\pi]\). Find the expected length of the chord \(AB\).
2. The point \(C\) is chosen at random in the interior of a circle of radius \(a\) and centre \(O\), so that the probability that it lies in any given region is proportional to the area of the region. The random variable \(R\) is defined as the distance between \(C\) and \(O\). Find the probability density function of \(R\). Obtain a formula in terms of \(a\), \(R\) and \(t\) for the length of a chord through \(C\) that makes an acute angle of \(t\) with \(OC\). Show that as \(C\) varies (with \(t\) fixed), the expected length \(\mathrm{L}(t)\) of such chords is given by \[ \mathrm{L}(t) = \frac{4a(1-\cos^3 t)}{3\sin^2 t}\,. \] Show further that \[ \mathrm{L}(t) = \frac{4a}{3}\left(\cos t + \tfrac{1}{2}\sec^2(\tfrac{1}{2}t)\right). \]
3. The random variable \(T\) is uniformly distributed on \([0, \frac{1}{2}\pi]\). Find the expected value of \(\mathrm{L}(T)\).

Prove, from the identities for \(\cos(A \pm B)\), that \[ \cos a \cos 3a \equiv \tfrac{1}{2}(\cos 4a + \cos 2a). \] Find a similar identity for \(\sin a \cos 3a\).

1. Solve the equation \[ 4\cos x \cos 2x \cos 3x = 1 \] for \(0 \leqslant x \leqslant \pi\).
2. Prove that if \[ \tan x = \tan 2x \tan 3x \tan 4x \qquad (\dagger) \] then \(\cos 6x = \tfrac{1}{2}\) or \(\sin 4x = 0\). Hence determine the solutions of equation \((\dagger)\) with \(0 \leqslant x \leqslant \pi\).

In this question, the numbers \(a\), \(b\) and \(c\) may be complex.

1. Let \(p\), \(q\) and \(r\) be real numbers. Given that there are numbers \(a\) and \(b\) such that \[ a + b = p, \quad a^2 + b^2 = q \quad \text{and} \quad a^3 + b^3 = r, \qquad (*) \] show that \(3pq - p^3 = 2r\).
2. Conversely, you are given that the real numbers \(p\), \(q\) and \(r\) satisfy \(3pq - p^3 = 2r\). By considering the equation \(2x^2 - 2px + (p^2 - q) = 0\), show that there exist numbers \(a\) and \(b\) such that the three equations \((*)\) hold.
3. Let \(s\), \(t\), \(u\) and \(v\) be real numbers. Given that there are distinct numbers \(a\), \(b\) and \(c\) such that \[ a + b + c = s, \quad a^2 + b^2 + c^2 = t, \quad a^3 + b^3 + c^3 = u \quad \text{and} \quad abc = v, \] show, using part~(i), that \(c\) is a root of the equation \[ 6x^3 - 6sx^2 + 3(s^2 - t)x + 3st - s^3 - 2u = 0 \] and write down the other two roots. Deduce that \(s^3 - 3st + 2u = 6v\).
4. Find numbers \(a\), \(b\) and \(c\) such that \[ a + b + c = 3, \quad a^2 + b^2 + c^2 = 1, \quad a^3 + b^3 + c^3 = -3 \quad \text{and} \quad abc = 2, \qquad (**) \] and verify that your solution satisfies the four equations \((**)\).

In this question, \(x\), \(y\) and \(z\) are real numbers. Let \(\lfloor x \rfloor\) denote the largest integer that satisfies \(\lfloor x \rfloor \leqslant x\) and let \(\{x\}\) denote the fractional part of~\(x\), so that \(x = \lfloor x \rfloor + \{x\}\) and \(0 \leqslant \{x\} < 1\). For example, if \(x = 4.2\), then \(\lfloor x \rfloor = 4\) and \(\{x\} = 0.2\) and if \(x = -4.2\), then \(\lfloor x \rfloor = -5\) and \(\{x\} = 0.8\).

1. Solve the simultaneous equations \begin{align*} \lfloor x \rfloor + \{y\} &= 4.9, \\ \{x\} + \lfloor y \rfloor &= -1.4. \end{align*}
2. Given that \(x\), \(y\) and \(z\) satisfy the simultaneous equations \begin{align*} \lfloor x \rfloor + y + \{z\} &= 3.9, \\ \{x\} + \lfloor y \rfloor + z &= 5.3, \\ x + \{y\} + \lfloor z \rfloor &= 5, \end{align*} show that \(\{y\} + z = 3.2\) and solve the equations.
3. Solve the simultaneous equations \begin{align*} \lfloor x \rfloor + 2y + \{z\} &= 3.9, \\ \{x\} + 2\lfloor y \rfloor + z &= 5.3, \\ x + 2\{y\} + \lfloor z \rfloor &= 5. \end{align*}

1. Sketch the curve \(y = xe^x\), giving the coordinates of any stationary points.
2. The function \(f\) is defined by \(f(x) = xe^x\) for \(x \geqslant a\), where \(a\) is the minimum possible value such that \(f\) has an inverse function. What is the value of~\(a\)? Let \(g\) be the inverse of \(f\). Sketch the curve \(y = g(x)\).
3. For each of the following equations, find a real root in terms of a value of the function~\(g\), or demonstrate that the equation has no real root. If the equation has two real roots, determine whether the root you have found is greater than or less than the other root.
   1. \(e^{-x} = 5x\)
   2. \(2x \ln x + 1 = 0\)
   3. \(3x \ln x + 1 = 0\)
   4. \(x = 3\ln x\)
4. Given that the equation \(x^x = 10\) has a unique positive root, find this root in terms of a value of the function~\(g\).

1. Use the substitution \(y = (x - a)u\), where \(u\) is a function of \(x\), to solve the differential equation \[ (x - a)\frac{dy}{dx} = y - x, \] where \(a\) is a constant.
2. The curve \(C\) with equation \(y = f(x)\) has the property that, for all values of \(t\) except \(t = 1\), the tangent at the point \(\bigl(t,\, f(t)\bigr)\) passes through the point \((1, t)\).
   1. Given that \(f(0) = 0\), find \(f(x)\) for \(x < 1\). Sketch \(C\) for \(x < 1\). You should find the coordinates of any stationary points and consider the gradient of \(C\) as \(x \to 1\). You may assume that \(z\ln|z| \to 0\) as \(z \to 0\).
   2. Given that \(f(2) = 2\), sketch \(C\) for \(x > 1\), giving the coordinates of any stationary points.

A plane circular road is bounded by two concentric circles with centres at point~\(O\). The inner circle has radius \(R\) and the outer circle has radius \(R + w\). The points \(A\) and \(B\) lie on the outer circle, as shown in the diagram, with \(\angle AOB = 2\alpha\), \(\tfrac{1}{3}\pi \leqslant \alpha \leqslant \tfrac{1}{2}\pi\) and \(0 < w < R\).

+ - ↺

1. Show that I cannot cycle from \(A\) to \(B\) in a straight line, while remaining on the road.
2. I take a path from \(A\) to \(B\) that is an arc of a circle. This circle is tangent to the inner edge of the road, and has radius \(R + d\) (where \(d > w\)) and centre~\(O'\). My path is represented by the dashed arc in the above diagram. Let \(\angle AO'B = 2\theta\).
   1. Use the cosine rule to find \(d\) in terms of \(w\), \(R\) and \(\cos\alpha\).
   2. Find also an expression for \(\sin(\alpha - \theta)\) in terms of \(R\), \(d\) and \(\sin\alpha\).
   You are now given that \(\dfrac{w}{R}\) is much less than \(1\).
3. Show that \(\dfrac{d}{R}\) and \(\alpha - \theta\) are also both much less than \(1\).
4. My friend cycles from \(A\) to \(B\) along the outer edge of the road. Let my path be shorter than my friend's path by distance~\(S\). Show that \[ S = 2(R+d)(\alpha - \theta) + 2\alpha(w - d). \] Hence show that \(S\) is approximately a fraction \[ \frac{\sin\alpha - \alpha\cos\alpha}{\alpha(1 - \cos\alpha)} \cdot \frac{w}{R} \] of the length of my friend's path.

1. The matrix \(\mathbf{R}\) represents an anticlockwise rotation through angle \(\varphi\) (\(0^\circ \leqslant \varphi < 360^\circ\)) in two dimensions, and the matrix \(\mathbf{R} + \mathbf{I}\) also represents a rotation in two dimensions. Determine the possible values of \(\varphi\) and deduce that \(\mathbf{R}^3 = \mathbf{I}\).
2. Let \(\mathbf{S}\) be a real matrix with \(\mathbf{S}^3 = \mathbf{I}\), but \(\mathbf{S} \neq \mathbf{I}\). Show that \(\det(\mathbf{S}) = 1\). Given that \[ \mathbf{S} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] show that \(\mathbf{S}^2 = (a+d)\mathbf{S} - \mathbf{I}\). Hence prove that \(a + d = -1\).
3. Let \(\mathbf{S}\) be a real \(2 \times 2\) matrix. Show that if \(\mathbf{S}^3 = \mathbf{I}\) and \(\mathbf{S} + \mathbf{I}\) represents a rotation, then \(\mathbf{S}\) also represents a rotation. What are the possible angles of the rotation represented by \(\mathbf{S}\)?

1. Show that, for \(n = 2, 3, 4, \ldots\), \[ \frac{d^2}{dt^2}\bigl[t^n(1-t)^n\bigr] = n\,t^{n-2}(1-t)^{n-2}\bigl[(n-1) - 2(2n-1)t(1-t)\bigr]. \]
2. The sequence \(T_0, T_1, \ldots\) is defined by \[ T_n = \int_0^1 \frac{t^n(1-t)^n}{n!}\,e^t\,dt. \] Show that, for \(n \geqslant 2\), \[ T_n = T_{n-2} - 2(2n-1)T_{n-1}. \]
3. Evaluate \(T_0\) and \(T_1\) and deduce that, for \(n \geqslant 0\), \(T_n\) can be written in the form \[ T_n = a_n + b_n e, \] where \(a_n\) and \(b_n\) are integers (which you should not attempt to evaluate).
4. Show that \(0 < T_n < \dfrac{e}{n!}\) for \(n \geqslant 0\). Given that \(b_n\) is non-zero for all~\(n\), deduce that \(\dfrac{-a_n}{b_n}\) tends to \(e\) as \(n\) tends to infinity.

Two particles, of masses \(m_1\) and \(m_2\) where \(m_1 > m_2\), are attached to the ends of a light, inextensible string. A particle of mass \(M\) is fixed to a point \(P\) on the string. The string passes over two small, smooth pulleys at \(Q\) and \(R\), where \(QR\) is horizontal, so that the particle of mass \(m_1\) hangs vertically below \(Q\) and the particle of mass \(m_2\) hangs vertically below~\(R\). The particle of mass \(M\) hangs between the two pulleys with the section of the string \(PQ\) making an acute angle of \(\theta_1\) with the upward vertical and the section of the string \(PR\) making an acute angle of \(\theta_2\) with the upward vertical. \(S\) is the point on \(QR\) vertically above~\(P\). The system is in equilibrium.

1. Using a triangle of forces, or otherwise, show that:
   1. \(\sqrt{m_1^2 - m_2^2} < M < m_1 + m_2\)\,;
   2. \(S\) divides \(QR\) in the ratio \(r : 1\), where \[ r = \frac{M^2 - m_1^2 + m_2^2}{M^2 - m_2^2 + m_1^2}. \]
2. You are now given that \(M^2 = m_1^2 + m_2^2\). Show that \(\theta_1 + \theta_2 = 90^\circ\) and determine the ratio of \(QR\) to \(SP\) in terms of the masses only.