A curve in the Cartesian plane goes through the origin, touching the \(x\)-axis there; at any point the product of its radius of curvature \(R\) and its arc-length \(s\) (measured from \(O\)) is a constant, \(a^2\). Obtain the intrinsic equation of the curve and deduce that it may be parametrized thus: \[\begin{cases} dx = a(2\psi)^{-\frac{1}{2}}\cos\psi d\psi,\\ dy = a(2\psi)^{-\frac{1}{2}}\sin\psi d\psi. \end{cases}\] Draw a rough sketch of the curve. [You may assume if you wish that \[\int_0^\infty \frac{\cos\psi}{\sqrt{\psi}}d\psi = \int_0^\infty \frac{\sin\psi}{\sqrt{\psi}}d\psi = \sqrt{\frac{\pi}{2}}.\]]
\(ABC\) is an acute angled triangle and \(P\) is the foot of the perpendicular from \(A\) to \(BC\). \(X\) is a variable point on the line \(BC\) and is equally likely to be anywhere between \(B\) and \(C\). Let \(u\), \(v\) and \(h\) denote the lengths of \(BP\), \(CP\) and \(AP\) respectively. Find expressions for the expected value and the variance of the area of the triangle \(APX\) in terms of the parameters \(u\), \(v\) and \(h\).
A box contains \(b\) black and \(r\) red balls. Balls are drawn from it at random one at a time. After each draw the drawn ball is replaced and \(c\) balls of its colour are added to the box. Prove by induction or otherwise that the probability \(p(n)\) that a black ball is drawn on the \(n\)th occasion is \(b/(r+b)\). What is the expected number of black balls in the box immediately before the \((n+1)\)th draw?
Two independent random variables \(X\) and \(Y\) are each uniformly distributed between 0 and 2. Find the probability that \(X^m Y^n \leq 1\) in the cases (i) \(m = n = 1\), (ii) \(m = 2\), \(n = -1\).
In a sample of 50 male undergraduates at Cambridge in 1900 the mean height was found to be 68.93 in. In a sample of 25 male undergraduates at Cambridge in 1974 the mean height was 70.66 in. It may be assumed that the heights of male undergraduates are always normally distributed with a standard deviation of 2.5 in. Is it reasonable to suppose that there has been an increase in the average height of male undergraduates at Cambridge over the past 74 years? Explain carefully the reasoning you use.
A simple pendulum has length \(l\) and is deflected through an angle \(\theta(t)\) from the vertical. Without making any approximations, write down the equation of motion and deduce the equation of energy if \(\alpha\) is the greatest value of \(\theta\) reached. Show that the period is given by \[2\left(\frac{l}{g}\right)^\frac{1}{2} \int_0^\alpha (\sin^2 \frac{1}{2}\alpha - \sin^2 \frac{1}{2}\theta)^{-\frac{1}{2}} d\theta.\] By making the substitution \(\sin \frac{1}{2}\theta = \sin \frac{1}{2}\alpha \sin \psi\) and expanding the integrand appropriately, show that, for small values of \(\alpha\), the period is approximately \[2\pi\left(\frac{l}{g}\right)^\frac{1}{2} \left(1 + \frac{1}{16}\alpha^2\right).\]
Two identical small smooth spheres \(S_1\) and \(S_2\) of radius \(b\) are free to slide inside a long smooth hollow tube whose inner circular cross-section is just large enough to contain the spheres. The tube has length \(2\pi a\) where \(a\) is much greater than \(b\), and it is bent in the form of a large circle of radius \(a\) and closed on itself. Suppose that the tube is held fixed in a horizontal plane, that \(S_1\) and \(S_2\) are initially touching each other, that \(S_1\) is at rest and that \(S_2\) is projected away from \(S_1\) with speed \(U\). Find, in terms of \(U\), \(a\), and the coefficient of restitution \(e\) for collision between the spheres,
For the purpose of this question it may be assumed that, when any car travelling at speed \(v\) on a straight road makes an emergency stop, it stops in a distance \(t_0 v + bv^n\), where \(t_0\) is the reaction time of the driver (the same for all drivers) and \(b\) and \(n\) are positive constants (the same for all cars) determined by the efficiency of the brakes. A car \(C_1\) travelling at speed \(v_1\) is following a car \(C_2\) travelling at speed \(v_2\) (\(< v_1\)). When the cars are separated by a distance \(d\), the driver of \(C_2\) detects a hazard ahead and makes an emergency stop. When the driver of \(C_1\) sees the brake-lights of \(C_2\) (which light up after the first reaction time \(t_0\)) he also makes an emergency stop. Show that a collision is inevitable if \(v_1 > \lambda v_2\), where \[\lambda^n + \epsilon\lambda = A\] and \(\epsilon\) and \(A\) are to be found, as functions of \(t_0\), \(b\), \(n\), \(d\) and \(v_2\). When \(\epsilon\) is small the solution of this equation may be found as a power series \[\lambda = \lambda_0 + \epsilon\lambda_1 + \epsilon^2\lambda_2 + \ldots\] By substituting, and equating coefficients of different powers of \(\epsilon\), find \(\lambda_0\), \(\lambda_1\) and \(\lambda_2\).
One end \(A\) of a uniform rod \(AB\) of length \(2a\) and weight \(W\) can turn freely about a fixed smooth hinge; the other end \(B\) is attached by a light elastic string of unstretched length \(a\) to a fixed support at the point vertically above \(A\) and distant \(4a\) from \(A\). If the equilibrium of the vertical position of the rod with \(B\) above \(A\) is stable, find the minimum modulus of elasticity of the string.
Serving a ball in the game of lawn tennis can be modelled by the following problem. A projectile is emitted with horizontal and vertical components of velocity \(u\) and \(v\) (\(u > 0\)) from a point at a height \(h\) above horizontal ground. The player can adjust \(u\) and \(v\) but not \(h\). At a horizontal distance \(a\) from the point of emission, there is a net of height \(c\) which the ball must clear; the ball must also strike the ground at a horizontal distance not greater than \(a+b\) from the point of emission. Establish two inequalities involving the quantities \(\xi = u^2\) and \(\eta = uv\) linearly, and show diagrammatically, using the \((\xi, \eta)\) plane, what is required for a valid serve, distinguishing between the cases where \(h\) is less than or greater than \(c(1 + a/b)\).