A smooth rigid wire in the form of a parabola is held fixed in a vertical plane with its vertex downwards. A bead moves under gravity on the wire. Prove that at the square of the normal reaction of the bead on the wire is inversely proportional to the cube of the height of the point above the directrix of the parabola.
A particle \(P\) describes an orbit under a force per unit mass directed towards a fixed origin \(O\) of magnitude \(\mu r^{-2} + \lambda r^{-3}\), where \(r\) is the length \(OP\) and \(\mu\) and \(\lambda\) are constants. Prove that if \(\lambda\) is small enough the path in general can be expressed by the equation \(l = r(1 + e\cos n\theta)\), where \(l\), \(e\), and \(n\) are constants, and where \(\theta\) is measured from a suitable radius.
A uniform rough solid sphere is projected up a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal. Initially the velocity of the centre of the sphere is \(v\) and there is no angular velocity. Obtain an equation giving the variation of the velocity between the sphere and the plane at the point of contact for the motion immediately after the projection, and prove that the velocity of slip will vanish instantaneously after a time \(2\mu g(7\mu\cos\alpha + 2\sin\alpha)\), where \(\mu\) is the coefficient of friction between the sphere and the plane. Prove also that whatever the value of \(\mu\), the velocity of slip will either be zero or change in sign during the subsequent motion.
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?