Let \(I(m, n) = \int_{0}^{\frac{1}{2}\pi} \cos^m x \sin^n x\, dx\). Using integration by parts, or otherwise, show that \begin{equation*} I(m, n) = \frac{n-1}{m+n}I(m, n-2) \end{equation*} if \(m \geq 0\), \(n \geq 2\). Let \begin{equation*} C = \int_{0}^{\frac{1}{2}\pi} \frac{\cos^2 x}{\cos x + \sin x}\, dx, \quad S = \int_{0}^{\frac{1}{2}\pi} \frac{\sin^2 x}{\cos x + \sin x}\, dx. \end{equation*} By considering \(C+S\), or otherwise, show that \begin{equation*} C = \frac{1}{32}(7\pi - 8). \end{equation*}
In an examination taken by a class of \(m\) pupils, the number of marks obtained by each one may be assumed to be random, with probability \(N^{-1}\) of taking any of the values \(1, 2, \ldots, N\). If different pupils' scores are independent, find an expression for the probability that the top mark in the class is \(k\), and for the probability that the difference between the top mark and the bottom mark is \(r\). [Note first that the probability that all the marks lie between \(x\) and \(y\) is \(N^{-m}(y-x+1)^m\).]
A bag contains \(B\) black balls and \(W\) white balls. If balls are drawn randomly from the bag one at a time without replacement, what is the probability that exactly \(j\) black balls come before the first white one? By considering this result, or otherwise, prove the identity \begin{equation*} \sum_{j=0}^{M} \binom{M}{j}\binom{M+N-j}{j} = \frac{M+N+1}{N+1} \end{equation*} for non-negative integers \(M\) and \(N\), where \begin{equation*} \binom{M}{j} = \frac{M!}{j!(M-j)!}. \end{equation*}
Every morning I walk to the bus stop and must then decide whether to catch my journey on foot, taking 10 minutes, or wait for the bus which covers the same distance in 2 minutes. Since I have no watch I have no idea when the next bus will arrive, though I know that on this route buses run exactly 10 minutes apart. My observations suggest that other people arrive randomly at the bus stop in such a way that (since after the departure of a bus, the probability that there is no-one waiting is \(e^{-t/5}\)) that, if I adopt the strategy 'wait for the bus if there is already someone at the bus, walk if not', my mean journey time will be about 6 seconds less than if I always walk.
Two particles, of masses \(M\) and \(m\), lie in contact and at rest on a smooth horizontal table. They are connected together by a light elastic string of natural length \(l\) and modulus \(\lambda\). If the particle of mass \(m\) is set in motion with a horizontal velocity \(v\), show that the particles will collide after a time \begin{equation*} \frac{2l}{v} + \pi\sqrt{\frac{Mm}{\lambda(M+m)}}. \end{equation*} Find their distance, at the instant of collision, from their initial position.
A motor car of mass \(M\) kg has an engine which, at full throttle, will supply a power \(A\omega(a-\omega)\) watts, where \(A\), \(a\) are constants and \(\omega\) is the speed of the engine in radians/sec. The speed \(v\) of the car, in m/sec, is related to the engine speed by \(v = r\omega\), where the constant \(r\) can be varied by changing gear. Find the time it would take, without changing gear, to accelerate in a straight line from rest to a speed \(V\), where \(V < ar\). Show that this is least, for fixed \(V\), when the chosen gear has \begin{equation*} r = \frac{V}{a}\left(\frac{x}{x-1}\right), \end{equation*} where \(x\) is the root of \begin{equation*} 2\log x = x-1 \end{equation*} which satisfies \(x > 2\). You may neglect air resistance, and assume that the engine power is transmitted to the car with perfect efficiency.
A straight river of width \(d\) flows with uniform speed \(u\). A man, who can swim with constant speed \(v\) (\(v > u\)) and run with constant speed \(w\), starts from a point \(P\) on one bank of the river. He wishes to reach the point \(Q\) on the other bank directly opposite to \(P\). Show that, if he swims across in a straight line, he will take a time \(T_0 = d(v^2-u^2)^{-\frac{1}{2}}\). Find the total time \(T(t)\) that he would take if he first runs upstream for a time \(t > 0\) and then swims to \(Q\) in a straight line. Hence show, by considering \(dT/dt\) or otherwise, that if \(uw > v^2-u^2\) then \(T(t) < T_0\) for sufficiently small values of \(t\).
A sequence of functions \(P_n(x)\), \(n = 0, 1, 2, \ldots\), is defined by setting \begin{align*} P_0(x) &= 1, \quad P_1(x) = x,\\ nP_n(x) &= (2n-1)xP_{n-1}(x) - (n-1)P_{n-2}(x) \end{align*} and requiring \begin{equation*} P_n(x) = \sum_{r=0}^{n} A(n, r)x^r. \end{equation*} if \(n \geq 2\). Show that \(P_n(x)\) is a polynomial of degree \(n\), say Construct a flow diagram for the evaluation of the coefficients in \(P_N(x)\) for a given value of \(N \geq 2\).
The atmosphere at a height \(z\) above ground level is in equilibrium with density \(\rho(z)\). Neglecting the curvature of the earth, show that the pressure \(p(z)\) is given by \begin{equation*} \frac{dp}{dz} = -\rho g, \end{equation*} where \(g(z)\) is the acceleration due to gravity at a height \(z\). If the earth is now assumed to be spherical, it can be shown that the above still holds and that \(g\) is inversely proportional to the square of the distance from the centre of the earth. Assuming also that \(p\), \(\rho\), \(T\) are connected by the relations \begin{equation*} p = k\rho^\gamma, \quad p = R\rho T, \end{equation*} where \(T(z)\) is the temperature of the atmosphere at a height \(z\) and where \(k\), \(R\) are constants with \(\gamma > 1\), show that \begin{equation*} T = T_0\left(1 - \frac{(\gamma-1)a\rho_0g_0z}{(\gamma R\rho_0a+z)}\right), \end{equation*} where \(a\) is the radius of the earth and \(p_0\), \(\rho_0\), \(T_0\), \(g_0\) denote the values of \(p\), \(\rho\), \(T\), \(g\) at \(z = 0\).
A point \(P\) with position vector \(\mathbf{p}(t)\) at time \(t\) moves in a plane in such a way that \begin{equation*} \mathbf{p}\cdot\dot{\mathbf{p}} = 0 \quad \text{and} \quad \ddot{\mathbf{p}} = -\lambda(t)\mathbf{p}, \quad \text{for all} \, t, \end{equation*} where dots denote differentiation with respect to \(t\). If \(P\) is initially at unit distance from the origin, describe its subsequent motion and show that \(\lambda(t)\) is constant. Points \(Q\) and \(R\), with position vectors \(\mathbf{q}(t)\) and \(\mathbf{r}(t)\), move in the same plane so that \begin{equation*} \dot{\mathbf{q}} = \mathbf{k} \quad \text{and} \quad \mathbf{r} = \mathbf{p} + \mathbf{q}, \end{equation*} where \(\mathbf{k}\) is a constant unit vector. Find the conditions required to make \(\mathbf{r}(t)\) vanish at some \(t\), and describe the possible motions of \(R\).