Express \(1-\cos^2\theta-\cos^2\phi-\cos^2\psi+2\cos\theta\cos\phi\cos\psi\) as the product of four sines. \par Solve the equation \(\cot^{-1}\frac{x}{a}+\cot^{-1}\frac{x}{b}+\cot^{-1}\frac{x}{c}+\cot^{-1}\frac{x}{d} = \frac{\pi}{2}\).
If \(S\) be the area of a quadrilateral whose sides are \(a,b,c,d\), prove that \[ S^2 = (s-a)(s-b)(s-c)(s-d)-abcd\cos^2\alpha, \] where \(2s=a+b+c+d\) and \(2\alpha\) is the sum of two opposite angles. \par If the quadrilateral can be inscribed in a circle, prove that the radius of the circle is \[ \frac{1}{4}\left\{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}\right\}^{\frac{1}{2}}. \]
If \[ \cos^{-1}(\alpha+i\beta) = A+iB, \] prove that \[ \alpha^2\sec^2A-\beta^2\operatorname{cosec}^2A=1 \] and \[ \alpha^2\operatorname{sech}^2B+\beta^2\operatorname{cosech}^2B=1. \] Prove that \(\tan^{-1}\frac{\tan 2\theta+\tanh 2\phi}{\tan 2\theta-\tanh 2\phi} + \tan^{-1}\frac{\tan\theta-\tanh\phi}{\tan\theta+\tanh\phi} = \tan^{-1}(\cot\theta\coth\phi)\).
In what sense is a couple a vector? Give reasons for your answer. \par If forces completely represented by the sides of a triangle taken in order are in equilibrium with three equal forces acting at the corners of the triangle along the tangents to the circumcircle the same way round, prove that the triangle must be equilateral.
A rod of length \(2l\) with one end on a horizontal plane leans against a circular cylinder of radius \(a\) whose axis lies in the plane, \(\lambda\) and \(\lambda'\) being the angles of friction at the two contacts. If \(\theta\) be the greatest angle the rod can make with the horizontal in such a position, shew that \[ l\sin\theta\sin(\theta+\lambda-\lambda') = a\sin\lambda\cos\lambda' \] provided this value of \(\theta\) is greater than \(\tan^{-1}(a/2l)\).
Explain how the principle of virtual work may be used to determine the unknown reactions of a system in equilibrium. \par A regular octahedron formed of twelve equal rods of weight \(w\) freely jointed is suspended from one corner. Prove that the thrust in each horizontal rod is \(3w/\sqrt{2}\).
Prove that the orbit of a projectile in vacuo is a parabola. \par If any number of particles are projected simultaneously from the same point with equal velocities in the same vertical plane, prove that at any instant the particles all lie on a circle, also that their relative directions as seen from one another remain unchanged.
Enunciate and prove the principle of conservation of linear momentum. \par Two equal particles connected by a light string rest with the string taut on a smooth horizontal plane. If one of the particles is struck a blow at right angles to the string, determine the paths of the particles.
State the principle of conservation of energy and prove it for the motion of a particle under gravity. \par Two small rings of equal mass can slide on a smooth parabolic wire with axis vertical and vertex upwards. The rings are connected by an elastic string equal in length to the latus rectum, the modulus of elasticity being equal to four times the weight of either ring. If the rings slide down the two sides of the parabola starting from rest at the vertex, shew that they will come to rest at a depth equal to the latus rectum.
Find the linear factors of \[ a(b-c)^3+b(c-a)^3+c(a-b)^3. \] If \[ x^3+y^3+z^3=mxyz \quad \text{and} \quad \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0, \quad a^2x+b^2y+c^2z=0, \] shew that \[ a^3+b^3+c^3=mabc. \]