Interpret geometrically the statement that, if \(f(x) \geq 0\) when \(a \leq x \leq b\), then \[\int_a^b f(x) dx \geq 0.\] Express \[\int_{2n\pi}^{(2n+2)\pi} \frac{\sin x}{x+\pi} dx\] in terms of an integral over the interval \(2n\pi \leq x \leq (2n+1)\pi\), where \(n\) is a positive integer. Deduce, or prove otherwise, that \[\int_0^T \frac{\sin x}{x+\pi} dx \geq 0 \quad \text{for all } T > 0.\] Given that \(g(x) > 0\) for all \(x \geq 0\), suggest a further condition on \(g(x)\) which ensures that \[\int_0^T g(x) \sin x \, dx \geq 0 \quad \text{for all } T > 0.\]
(i) Show that the general solution of \[(1 + ax) w'(x) + \frac{1}{2}aw(x) = 0\] is \[w(x) = A(1 + ax)^{-\frac{1}{2}}, \quad A \text{ constant}.\] (ii) Find polynomials \(p(x)\), \(q(x)\), \(r(x)\), \(s(x)\) such that \[(1-x^2)y''(x) - xy'(x) + \frac{1}{4}y(x) = \left[p(x)\frac{d}{dx}+q(x)\right]\left[r(x)\frac{d}{dx}+s(x)\right]y(x),\] for all \(y(x)\). (There are two possible answers; only one is required.) Hence, or otherwise, find the general solution of \[(1-x^2)y''(x) - xy'(x) + \frac{1}{4}y(x) = 0.\] [You are reminded that \[\left[f(x)\frac{d}{dx}+g(x)\right]h(x)\] is defined to be \(f(x)h'(x)+g(x)h(x)\) for any functions \(f(x)\), \(g(x)\) and \(h(x)\) of \(x\). You may assume if necessary that \[\int(1-ax)^{-\frac{1}{2}}(1+ax)^{-\frac{1}{2}}dx = a^{-1}(1+ax)^{\frac{1}{2}}(1-ax)^{-\frac{1}{2}}.\]
Let \(C_p\) denote the set of all points \(z\) in the Argand diagram such that \[\left|\frac{z-i}{z+i}\right| = p,\] where \(p\) is a positive real constant. Show that, if \(p \neq 1\), then \(C_p\) is a circle. What is \(C_1\)? Show that \(C_p\) is orthogonal to the circle \(|z| = 1\).
Let \(f(1) = 0\) and \[f(n) = 1 + \frac{1}{2} + \ldots + \frac{1}{n-1} - \log_e(n), \quad (n = 2, 3, \ldots).\] By interpreting \(f(n)\) as an area, or otherwise, show that \(f(n)\) is an increasing sequence, and that \(f(n) < 1\) for all \(n\). Show also that \[\frac{1}{2n(n+1)} < f(n+1) - f(n) < \frac{1}{n(n+1)}.\]
The chance of a batsman at the crease being out to the next ball he faces is \(p\) if he has not yet faced 20 balls, and is \(q < p\) thereafter. Find, for a completed innings,
Under Atypical Tennis Players rules, a game is won when either player has scored two more points than his opponent. If the chance of the first player winning any given point is \(p\), independently of the outcomes of all other points, evaluate the probability \(f(p)\) that he wins the game. Show that, for \(\frac{1}{2} < p < 1\), \(f(p) > p\), and that, for \(p > \frac{1}{2}\), \(f(p) - \frac{1}{2} \geq 2(p - \frac{1}{2})\).
The random variable \(C\) takes integral values in the range \(-5\) to \(5\), with probabilities \[\text{Pr}[C = -5] = \text{Pr}[C = 5] = \frac{1}{20}; \quad \text{Pr}[C = i] = \frac{1}{10} \quad (-4 \leq i \leq 4).\] Calculate the mean and variance of \(C\). A shopper buys 36 items at random in a supermarket, and, instead of adding up his bill exactly, he rounds the cost of each item to the nearest 10p, rounding an odd 5p up or down with equal probability. Should he suspect a mistake if the cashier asks him for 20p less than he had estimated?
An experiment was conducted to investigate the effect of a new fertilizer on the yield of tomato plants. Ten plants were grown using the new fertilizer, and ten using the one previously recommended, giving yields (in kg): New \(1.5\) \(1.9\) \(1.7\) \(1.8\) \(1.5\) \(2.0\) \(2.0\) \(1.8\) \(1.9\) \(1.8\) Old \(1.4\) \(1.3\) \(1.3\) \(1.5\) \(1.8\) \(1.3\) \(1.1\) \(1.3\) \(1.4\) \(1.6\) Assuming that the yields are normally and independently distributed, with means \(\mu_N\) for plants having the new fertilizer and \(\mu_0\) for those having the old one, and with standard deviation 0.3 kg whichever fertilizer was used, test whether or not there is evidence that the new fertilizer is an improvement on the old one. How would you estimate the standard deviation of the yield of a tomato plant if it was not known to be 0.3 kg?
An artificial satellite moves in the earth's upper atmosphere. If air resistance were ignored the orbit would be exactly circular. Write down an expression for the total energy of the satellite. The effect of air resistance can be represented by a force whose magnitude is extremely small and depends only on the velocity, and whose direction directly opposes the motion of the satellite. Show that its effect is to cause the satellite to spiral inwards, the speed increasing at the same rate that would occur in rectilinear motion (i.e. with no gravitational force) with the sign of the resistance reversed.
The behaviour of some radial-ply tyres on icy roads can be approximated as follows. The tyre can withstand a horizontal force in any direction up to a limiting magnitude \(F_0\). If a greater force is applied the tyre starts to slip and the coefficient of sliding friction is negligible. An imprudent motorist using such tyres is travelling fast on a wide almost empty icy road when he encounters a T-junction with a long brick wall opposite him. If a collision can be avoided compare the strategies of braking in a straight line or swerving without braking. If a collision is inevitable how can the driver reduce the kinetic energy as fast as possible?