The sides of a triangle are \(p\), \(q\), \(r\); the angles opposite them are (in circular measure) \(P\), \(Q\), \(R\). Prove that $$\frac{\pi}{3} \leq \frac{pP + qQ + rR}{p + q + r} \leq \frac{\pi}{2}.$$ When, if at all, can equality occur?
Show that if \(a\), \(b\), \(c\) are integers it is always possible to find integers \(A\), \(B\), \(C\) such that $$(a + b2^i + c2^j)(A + B2^i + C2^j) = a^2 + 2b^2 + 4c^2 - 6abc.$$ Prove that the right side of this can be zero only if the integers \(a\), \(b\), \(c\) are all zero, and deduce that if now \(a\), \(b\), \(c\) are rational numbers such that \(a + b2^i + c2^j = 0\), then \(a = b = c = 0\).
Evaluate $$\int_0^\infty x \sin nx \, e^{-ax} dx \quad \text{and} \quad \int_0^\infty x^n \sin x \, e^{-x} dx,$$ where \(n\) is a positive integer.
The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined by $$a_0 = a_1 = 1; \quad a_n = a_{n-1} + a_{n-2} \quad (n \geq 2).$$ Prove that, for all \(n \geq 1\), $$a_{2+1}^2 - a_{2-1}^2 = a_{2s-1} \quad \text{and} \quad a_s^2 + a_{s-1}^2 = a_{2s}.$$
A sequence of integers \(n_1\), \(n_2\), \(n_3\), \(\ldots\) is obtained as follows. If \(1 < n_r < 3\) then \(n_{r+1} = n_r - 1\) or \(n_r + 1\), with probability \(\frac{1}{2}\) each; if \(n_r = 9\) then \(n_{r+1} = 8\) (with probability 1) and if \(n_r = 0\) then the sequence terminates at this point. Given that \(n_1 = 9\), calculate (i) the probability that \(n_r\) is never equal to \(0\) for \(r \geq 2\), and (ii) the expected length of the sequence. [For (i), let \(p_k\) be the probability that if \(n_r = k\) for some \(r \geq 2\) then \(n_s = 9\) for \(s \geq r\). Show that \(2p_k = p_{k-1} + p_{k+1}\) for \(1 \leq k \leq 8\), and use the obvious values of \(p_0\) and \(p_9\) to obtain the required probability. A similar method may be used for (ii).]
Two distinct complex numbers \(z_1\) and \(z_2\) are given, with \(|z_1| < 1\), \(|z_2| < 1\). Prove that there is a positive real number \(K\), depending on \(z_1\) and \(z_2\), such that $$|1-z| \leq K(1-|z|)$$ for all complex numbers \(z\) whose representative points in the complex plane lie within, or on a side of, the triangle determined by the points representing \(z_1\), \(z_2\) and \(1\). Determine the smallest possible value of \(K\) in the case \(z_1 = \frac{1}{2}(1+i)\), \(z_2 = \frac{1}{2}(1-i)\).
\(G\) is a group; operations \(\wedge\) and \(\vee\) are introduced for subgroups \(H\), \(K\), \(L\), \(\ldots\) of \(G\) as follows. \(H \wedge K\) is defined to be the set of all elements of \(G\) that are in both \(H\) and \(K\), and \(H \vee K\) is the set of all products formed from elements of \(H\) and \(K\) (taking any number of factors, in any order). Prove that \(H \wedge K\) and \(H \vee K\) are subgroups of \(G\), that \(H \wedge K\) is the largest subgroup of \(G\) contained in both \(H\) and \(K\), and that \(H \vee K\) is the smallest subgroup of \(G\) that contains both \(H\) and \(K\). Prove that \((H \wedge K) \vee (H \wedge L)\) is a subgroup of \(H \wedge (K \vee L)\); by considering the group consisting of the eight elements \(\pm 1\), \(\pm i\), \(\pm j\), \(\pm k\), whose multiplication table is given below (or otherwise), show that in general \((H \wedge K) \vee (H \wedge L)\) is not the whole of \(H \wedge (K \vee L)\).
A cube of mass \(M\) rests on a rough slope inclined at an angle \(\alpha\) to the horizontal. To the mid-point \(A\) of its highest edge is attached a light inextensible string \(AB\) which passes over a peg \(C\), arranged so that \(AC\) is parallel to the slope, and \(m < M/\sqrt{2}\) is attached to \(B\) hangs freely below \(C\). The mass \(m\) is slowly reduced, and equilibrium is broken by sliding. Obtain an inequality which the coefficient of friction between the cube and the slope must satisfy.
A spherical shell of radius \(a\) and mass \(m\) per unit area is cut by two parallel planes distant \(d < a\) apart, one of which passes through the centre of the shell. Calculate the moment of inertia of the portion of the shell between the planes, about an axis through the centroid of the portion and perpendicular to its axis of symmetry.
A particle moves under a central attractive force \(f(r)\) per unit mass when its distance from the centre of force is \(r\). Find the form of \(f(r)\) if the particle describes the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and the centre of force is at the origin.