Prove that if \(u\) and \(v\) are functions of \(x\) and if \(n\) is a positive integer then \begin{equation*} \frac{d^n}{dx^n}(uv) = \sum_{r=0}^{n} \binom{n}{r} \frac{d^r u}{dx^r} \frac{d^{n-r} v}{dx^{n-r}} = \frac{d^n u}{dx^n}v + \binom{n}{1} \frac{d^{n-1} u}{dx^{n-1}} \frac{dv}{dx} + \cdots + \binom{n}{r} \frac{d^{n-r} u}{dx^{n-r}} \frac{d^r v}{dx^r} + \cdots + u\frac{d^n v}{dx^n}, \end{equation*} where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). The functions \(L_n\) are defined by \begin{equation*} L_n(x) = e^x \frac{d^n}{dx^n}(e^{-x}x^n). \end{equation*} Show that \(L_n\) is a polynomial of degree \(n\). By considering \begin{equation*} \frac{d^{n+1}}{dx^{n+1}}(e^{-x}x^n), \end{equation*} prove that \begin{equation*} L_{n+1}(x) = (n+1-x)L_n(x) + x\frac{d}{dx}L_n(x). \end{equation*}
A square \(ABCD\) is made of stiff cardboard, and has sides of length \(2a\). Points \(P\), \(Q\), \(R\), \(S\) are taken inside the square, each at a distance \(xa\) from the centre; they are so placed that when the triangles \(APB\), \(BQC\), \(CRD\), \(DSA\) are cut away a single piece of cardboard remains, which can be folded about \(PQ\), \(QR\), \(RS\), \(SP\) so as to form the surface of a pyramid with \(A\), \(B\), \(C\), \(D\) coinciding at its apex. Show that the volume of the pyramid cannot exceed \begin{equation*} \frac{32\sqrt{2}}{75\sqrt{3}}a^3. \end{equation*}
A curve is given parametrically by \begin{align*} x &= a(\cos\theta + \log\tan\tfrac{1}{2}\theta)\\ y &= a\sin\theta, \end{align*} where \(0 < \theta < \frac{1}{2}\pi\) and \(a\) is constant. The points with parameters \(\theta, \frac{1}{2}\pi\) are denoted by \(P, A\) respectively; the tangent at \(P\) meets the \(x\)-axis at \(Q\). Prove that \(PQ = a\). Let \(C\) be the centre of curvature at \(P\) and let \(s\) be the arc length from \(A\) to \(P\). By considering \(ds/d\theta\), or otherwise, show that \(CQ\) is parallel to the \(y\)-axis.
Suppose that \(f\) is defined for \(a < x < b\), that \(a < c < b\), and that \(f'(c) = 0\). Show how one may, in general, determine whether \(f\) has a maximum, a minimum or neither at \(c\) by considering the sign of \(f'(x)\) in the neighbourhood of \(c\). Let \begin{equation*} f(x) = x^p(1-x)^q, \end{equation*} where \(p > 1\), \(q > 1\). Sketch the graph of \(f(x)\) for \(0 \leq x \leq 1\). Show by means of sketches how \(f\) behaves in this interval for other positive values of \(p\) and \(q\), distinguishing between different ranges of values of \(p\) and \(q\) so as to indicate the different types of curve that may occur.
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.