A thin rod of mass \(M\), not necessarily uniform, is suspended from one end \(O\) and can turn freely about \(O\) in a vertical plane. The rod is set in motion from its equilibrium position by a horizontal blow \(J\) delivered at some point \(X\) below \(O\). The impulsive reaction at \(O\) must not exceed a value \(H\) owing to the weakness of the hinge. Find the maximum value of the kinetic energy of the instantaneous motion of the rod, and prove that the depth below \(O\) of the point \(X\) in this case is given by $$OX = \left(1 + \frac{H}{J}\right)\frac{k^2}{h},$$ where \(k\) is the radius of gyration of the rod about \(O\), and \(h\) the depth below \(O\) of its centre of mass.
The lines joining a point \(P\) to the vertices of a triangle \(ABC\) meet the opposite sides at the points \(D\), \(E\), \(F\). The lines \(EF\), \(FD\), \(DE\) meet \(BC\), \(CA\), \(AB\) in \(L\), \(M\), \(N\) respectively. Prove that \(L\), \(M\), \(N\) are collinear. \(LMN\) is called the polar of \(P\) with respect to the triangle \(ABC\). If \(Q\) lies on the polar of \(P\), does \(P\) necessarily lie on the polar of \(Q\)? Justify your answer.
\(A_1\), \(A_2\), \(A_3\), \(B_1\), \(B_2\), \(B_3\) are six points on a conic. \(P_1\) is the meet of \(A_2A_4\) and \(B_2B_3\); \(Q_1\) is the meet of \(A_2B_2\) and \(A_3B_3\); \(P_2\), \(Q_2\), \(P_3\), \(Q_3\) are defined similarly. Prove that the triangles \(P_1P_2P_3\), \(Q_1Q_2Q_3\) are in perspective.
\(A\), \(B\), \(C\) are three distinct points on the complex projective line. Let \(A'\) be the harmonic conjugate of \(A\) with respect to \(B\) and \(C\), and let \(B'\) and \(C'\) be similarly defined. Prove that \(A\) and \(A'\), \(B\) and \(B'\), \(C\) and \(C'\) are pairs in involution. Let \(D\), \(E\) be the double points of this involution. Prove that it is impossible to choose coordinates so that \(A\), \(B\), \(C\), \(D\), \(E\) all have real coordinates.
Let \(f(x) = (x-a)(x-b)(x-c)(x-d)\) where \(a\), \(b\), \(c\), \(d\) are distinct. Resolve \(e^x f(x)\) into partial fractions, for \(n = 0\), \(1\), \(2\), \(3\). Let $$K_n = \sum \frac{a^n}{(a-b)(a-c)(a-d)},$$ the sum of four cyclic terms. Prove that \(K_n = 0\) for \(n = 0\), \(1\), \(2\), and find \(K_3\).
Factorize the determinant $$\begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix}.$$ Given that \(a\), \(b\), \(c\) are real and not all equal and that \(a+b+c \neq 0\), solve \begin{align} ax + by + cz &= 1,\\ cx + ay + bz &= 0,\\ bx + cy + az &= 0. \end{align} What happens when \(a+b+c = 0\)?
Prove that if \(|x| < 1\) then \(\sum_{n=1}^{\infty} x^n\) is convergent. Prove that, if \(0 < \theta < 1\), \(\sum_{n=1}^{\infty} \sin(\theta^n)\) is convergent to sum \(S\), where $$\frac{\sin\theta}{1-\theta} < S < \frac{\theta}{1-\theta}.$$ Is \(\sum \cos(\theta^n)\) also convergent?
Let $$f(y) = \int_{-1}^{1} \frac{dx}{2\sqrt{(1-2xy+y^2)}},$$ where the positive value of the square root is taken. Prove that \(f(y) = 1\) if \(|y| \leq 1\). Find the value of \(f(y)\) when \(|y| > 1\). Hence or otherwise prove that if \(|y| < 1\), then $$\int_{y}^{1} \frac{(x-y)dx}{(1-2xy+y^2)^{3/2}} = \int_{-y}^{1} \frac{(x+y)dx}{(1+2xy+y^2)^{3/2}}.$$
A particle \(Q\) of mass \(2m\) is attached to one end of a light elastic string \(PQ\) of length \(2a\) and modulus of elasticity \(\lambda\); a particle \(R\) of mass \(3m\) is attached to the mid-point of the string. The system is then hung in equilibrium from a fixed point \(P\). The particle \(Q\) is given a small downward impulse \(\epsilon\sqrt{\frac{m\lambda}{a}}\). After time \(t\) the ensuing displacements of \(Q\), \(R\) from the equilibrium position are \(x\), \(y\), respectively. Prove that \(\ddot{x} = -3\omega^2(x-y), \quad \ddot{y} = 2\omega^2(x-2y), \quad \text{where } \omega = \sqrt{\frac{\lambda}{6am}}.\) Verify that \(x = \epsilon\left(\frac{3\sqrt{6}}{10}\sin\omega t + \frac{1}{5}\sin\sqrt{6}\omega t\right)\) satisfies the initial conditions. Deduce that this is the correct solution for \(x\), by finding a similar formula for \(y\), which, together with that for \(x\), satisfies the equations of motion and the initial conditions. Is the motion periodic?
A smooth ball of unit mass collides with a second equal ball, which is initially at rest. The coefficient of restitution \(e\) is less than 1. The first ball is aimed so as to suffer the maximum change of direction in the collision. Find this change of direction, and also the proportion of energy which is lost in the collision.