A heavy, uniform circular cylinder of radius \(r\) lies on a rough horizontal plane with its axis horizontal. A heavy, uniform rod \(AB\) of length \(l\) lies in the vertical plane which bisects the axis of the cylinder at right angles. Its end \(A\) rests on the plane and point \(C\), distinct from \(B\), is in contact with the cylinder. The coefficient of friction is the same value, \(\mu\), at each of the three points of contact between the rod, cylinder and plane. The rod makes an angle \(\alpha\) with the horizontal, and the friction is limiting at Show, by a geometrical method or otherwise, that for fixed values of \(r\), \(l\) and \(\alpha\), there exists a value of \(\mu\) such that this situation is a possible equilibrium state of the system provided \[r\cot(\frac{1}{2}\alpha) \leq l\cos\alpha \leq 2r\cot(\frac{1}{2}\alpha).\]
The earth may be assumed to be a homogeneous sphere and then the gravitational acceleration within it may be shown to be directed towards and to vary directly as the distance from the centre. A straight tunnel connects two points on the surface of the earth which subtend an angle \((\pi - 2\alpha)\) at the centre. A small particle is placed at one end of the tunnel. The limiting coefficient of static friction between the particle and the tunnel is \(\mu_s\), and the coefficient of dynamic friction is \(\mu_d\), where \[\mu_d < \mu_s < 1.\] Describe the subsequent motion of the particle and show that, if the particle moves initially and does not reach the half-way point in the tunnel, then \[\frac{1}{2}\cot\alpha < \mu_d < \mu_s < \cot\alpha.\]
A tumbler which has square cross-section of side \(2a\) and height \(Ka\) is closed at one end and this end rests on a rough horizontal table. The tumbler is filled to a height \(ka\) with liquid of uniform density. Assuming that no sliding takes place and that the weight of the tumbler is negligible compared with that of the liquid, show that when the table is tilted slowly through an angle \(\theta\) about an axis parallel to one face of the tumbler then, provided \[\tan\theta \leq k \leq K - \tan\theta,\] the tumbler will topple when \(\theta\) is given by \[\tan^3\theta + (3k^2 + 2)\tan\theta - 6k = 0.\] If it is required to tilt the table through an angle \(\tan^{-1}\frac{1}{2}\) without spillage, determine what height of tumbler is required and how full it can be.
Two players play a dice game on a board marked with squares numbered 0 to 13. Each player has a counter that is initially on square 0 and they take turns to throw a six-sided die. A player's counter is not moved until he throws a six, when it moves to square 6. Thereafter, if it is on square \(m\) and he throws an \(n\), it advances to square \(m + n\) if \(m + n \leq 13\), and 'rebounds' to square \(26 - (m + n)\) if \(m + n > 13\). The winner is the player whose counter first reaches square 13. Find (i) the probability that the first player to throw is the first to move his counter; (ii) the probability that the loser's counter never leaves square 0.
Show that, if the polynomial \[f(x) = x^3+3ax+b \quad (a \neq 0)\] can be expressed in the form \[A(x-p)^3+B(x-q)^3,\] where \(A\) and \(B\) are constants, then \(p \neq q\), and \(p, q\) are the roots of the equation \[at^2+bt-a^2 = 0.\] Prove, conversely, that if this equation has unequal roots then \(f(x)\) can be written in the form (1). Hence or otherwise find the real root of the equation \[x^3+54x+54 = 0.\]
Show that, if \(a > b > 0\) and \(m\) is a positive integer, then \[a^{m+1}- b^{m+1} \leq (a-b)(a+b)^m.\] Deduce that \[1^m+ 3^m+ ... + (2n - 1)^m \geq n^{m+1}.\] Interpret this result in terms of the position of the centre of mass of equal particles suitably placed on the curve \(y = x^m\). For what values of \(m\) other than positive integers do you think the result (2) is true?
Show that, if \(z_0\) is any non-zero complex number, then there is a complex number \(w_0\) such that \(z_0 w_0 = 1\). Let \(A\) be the set of all complex numbers \(z = x+iy\) such that \(x\) and \(y\) are integers, and let \(B\) be the set of all complex numbers such that \(x\) and \(y\) are rational. Let \(z_3 \neq 0\). Show that, if \(z_1, z_2\) belong to \(A\), then \(z_1/z_2\) need not belong to \(A\); but that if \(z_1, z_2\) belong to \(B\), then \(z_1/z_2\) must belong to \(B\). Find a number \(z_3\) of \(A\) such that \[\left|\frac{2+7i}{3+i}-z_3\right| \leq \frac{1}{\sqrt{2}},\] and show, more generally, that for any \(z_1, z_2\) of \(A\) with \(z_2 \neq 0\) there is a \(z_3'\) of \(A\) such that \[\left|\frac{z_1}{z_2}-z_3'\right| \leq \frac{1}{\sqrt{2}}.\] Find also a member \(z_4\) of \(A\) such that \[\left|\frac{z_4}{3+i}-z\right| \geq \frac{1}{\sqrt{2}}\] for all \(z\) of \(A\).
Let \(g\) be an element of a group \(G\), and let \(\langle g \rangle\) denote the set of elements \(g^i\) for all integers \(i\) (positive, negative or zero); let \(e\) be the identity element of \(G\). Prove the following results.
The rectangular Cartesian coordinates of \(P'\) are \((x', y')\) and a mapping \(\alpha\) of the plane into itself sends \(P\) to \(P' = (x', y')\), where \[\begin{pmatrix} x' \\ y' \end{pmatrix} = A \begin{pmatrix} x \\ y \end{pmatrix},\] \(A\) being a \(2 \times 2\) matrix. The \(2 \times 2\) matrix \(B\) yields a mapping \(\beta\) in the same way. Show how the mapping determined by the matrix \(AB\) is related to \(\alpha\) and \(\beta\). Hence or otherwise show that, if \(l, m\) are two distinct fixed lines through the origin, and if \(P'\) is the projection of \(P\) onto \(l\) parallel to \(m\) (i.e. \(P'\) is on \(l\) and \(PP'\) is parallel to \(m\)), then \(A^2 = A\). It may be assumed without proof that this mapping is of the type (3). Now let \(B\) be any \(2 \times 2\) matrix which is neither the zero matrix nor the identity matrix, and which satisfies \(B^2 = B\). Show that \(B\) must take one or other of the forms \[\text{(i)} \begin{pmatrix} a & \lambda a \\ c & \lambda c \end{pmatrix} \text{ with } a+\lambda c = 1, \quad \text{(ii)} \begin{pmatrix} 0 & b \\ 0 & 1 \end{pmatrix}.\] Deduce that the mapping \(\beta\) is a projection onto one line parallel to another.
\(ABCDE\) is a regular pentagon of side 1. \(BD\) and \(CE\) meet in \(A'\), and \(DA\) and \(BC\) meet in \(C'\). Find the length of \(A'C'\). (Your answer should not contain trigonometric functions, but may contain square roots.)