The two random variables \(U\) and \(V\) are independent and each is uniformly distributed on \((0, 1)\). The random variables \(X\) and \(Y\) are defined by \(X = \log_e(1/U)\), \(Y = \log_e(1/V)\). Prove that the probability that \(X + Y \leq z\) is \[\int_0^z te^{-t}dt \quad (z > 0).\]
An entomologist measures the lengths of 8 specimens of each of two closely related species of bees. His measurements of species \(A\) and of species \(B\) have mean values 15 mm and 17 mm respectively. If he believes that in each species length is normally distributed with standard deviation 2 mm, should he conclude that the mean lengths of the two species differ? What procedure should he use if he does not know the standard deviation (though still believing it to be the same for both species)?
A breakdown truck tows away a car of mass \(m\) by means of an extensible rope whose unstretched length is \(l\) and whose modulus of elasticity is \(\lambda\). Initially the rope is slack and the car stationary; the truck then moves off with speed \(v\) which it maintains constant. The movement of the car is opposed by a constant frictional force \(F\). Determine the motion of the car as a function of time elapsed from the instant the rope becomes taut.
A smooth wedge of mass \(M\) is free to slide on a smooth horizontal plane and has one face inclined at an angle \(\alpha\) to the horizontal. A smooth particle of mass \(m\) is placed on this inclined face of the wedge. The particle and the wedge are initially at rest. Prove that the particle moves in a straight path inclined to the horizontal at an angle \[\tan^{-1}\left[\left(1+\frac{m}{M}\right)\tan\alpha\right].\] Find the velocity of the wedge when the particle has fallen a vertical height \(h\).
A uniform circular disc of mass \(M\) and radius \(a\) is free to rotate about a fixed vertical axis through its centre and perpendicular to it. A shallow groove is cut in the upper surface of the disc along a diameter. Two insects each of mass \(m\) are together on the upper surface of the disc at one end of the groove and the disc is rotating with angular velocity \(\Omega\). At time \(t = 0\) one insect starts to crawl along the groove with uniform velocity \(V\) relative to the disc. Show that when it reaches the other end of the diameter the disc has rotated through an angle \[\frac{2a}{V}\sqrt{\frac{2m}{M+2m}}\left\{\Omega\left(2+\frac{M}{2m}\right)\right\}\tan^{-1}\sqrt{\frac{2m}{M+2m}}.\] If at time \(t = 0\) the other insect starts to crawl round the circumference in the direction of rotation of the disc at such a constant velocity relative to the disc that it arrives at the far end of the diameter at the same instant as its companion, find the angle the disc has turned through when they meet.
Two uniform rough cylinders each with radius \(a\) and mass \(M\) lie touching each other on a rough horizontal table. A third identical cylinder lies on these two. The end faces of all three cylinders are coplanar. The coefficient of friction for all pairs of surfaces in contact has the same value, \(\mu\). Outward horizontal forces \(P\) are applied to the axes of both the lower cylinders. Find the greatest value that \(P\) can have before slipping occurs. Show that when \(\mu = \sqrt{3}\), this value is \(\frac{Mg}{2}\left(1+\frac{1}{\sqrt{3}}\right)\).
A large massive circular cylinder, radius \(a\), rotates about its axis with constant angular velocity \(\Omega\). A projectile is launched from the inside curved surface in a plane perpendicular to the axis of the cylinder with velocity \(V\) and elevation \(\alpha\) relative to the cylinder. Show that the particle hits the cylinder again after a time \begin{equation*} \frac{2aV\sin\alpha}{V^2+a^2\Omega^2+2aV\Omega\cos\alpha}. \end{equation*} Write down the condition that the projectile passes through the axis of the cylinder and find in this case the set of solutions to the condition that the particle strikes the cylinder at its launching point. [You may ignore the effects of gravity.]
A spaceship gathers interstellar gas as it travels at a rate \(\alpha V\) where \(V\) is its velocity. Its motors burn the gas and expel it at the same rate at which they acquire it, with velocity \(V_0\) relative to the ship. The ship, of mass \(M\), experiences a constant force \(Mg\) directly opposing its motion. Show that if the ship is initially travelling at a speed \(\frac{1}{2}V_0\) and \(\alpha = 2Mg/V_0^2\), then it will come to rest after a time \(\pi V_0/4g\). Find the distance travelled in this time.
A vector \(\mathbf{k}\) is of unit length but its direction varies as a function of time. Show that \begin{align*} \mathbf{k}\cdot\dot{\mathbf{k}} &= 0\\ \mathbf{k}\cdot\frac{d}{dt}(\mathbf{k}\wedge\dot{\mathbf{k}}) &= 0, \end{align*} where \(\dot{\mathbf{k}} = d\mathbf{k}/dt\). If \(\mathbf{k}\) also satisfies the equation of motion \begin{equation} \frac{d}{dt}(C_0\mathbf{k}+A\mathbf{k}\wedge\dot{\mathbf{k}}) = G\mathbf{l}\wedge\mathbf{k} \tag{*} \end{equation} where \(A\), \(C\), \(G\) and \(\mathbf{l}\) are constants, and \(|\mathbf{l}| = 1\), show that \(\omega = \text{constant}\). Show also that (*) has a solution where \begin{align*} \dot{\mathbf{k}} &= \Omega\mathbf{l}\wedge\mathbf{k}\\ \mathbf{l}\cdot\mathbf{k} &= \cos\alpha, \end{align*} \(\Omega\), \(\alpha\) being constants related by \begin{equation*} \Omega^2A\cos\alpha-C\omega\Omega+G = 0. \end{equation*}
Let \(x\) be a positive non-zero integer. \(S^1(x)\) will denote the sum of the digits of \(x\) when written in the scale of 10 (e.g. \(S^1(193) = 1+9+3 = 13\)). For \(i = 1, 2, \ldots\) we define \(S^{i+1}(x) = S^1(S^i(x))\). Show that \(x - S^1(x)\) is divisible by 9 for all \(i\). Denote by \(\mathcal{S}(x)\) the set \(\{S^1(x), \ldots, S^n(x)\}\) where \(n\) is the least integer such that \(S^n(x) \leq 9\) (e.g. \(\mathcal{S}(193) = \{13, 4\}\)). Show that if the smallest element of \(\mathcal{S}(x^2)\) is not a square, then it is 7, and the smallest element in \(\mathcal{S}(x)\) is 4 or 5. Deduce that if \begin{equation*} 0 < x^2 < 1000, \end{equation*} and no element of \(\mathcal{S}(x^2)\) is a square, then \(x = 4\) or 5.