If \(y=a+x\sin y\), prove that when \(x=0\), \[ \frac{dy}{dx}=\sin a, \quad \text{and} \quad \frac{d^2y}{dx^2}=\sin 2a. \] Show by expanding \(e^{\frac{1}{x}}\) in descending powers of \(x\), that \[ \frac{d^n}{dx^n}(x^{n-1}e^{\frac{1}{x}}) = (-1)^n x^{-n-1}e^{\frac{1}{x}}. \] Is this procedure strictly legitimate?
Prove that the cone of greatest volume which can be inscribed in a given sphere has an altitude equal to \(\frac{2}{3}\) the diameter of the sphere.
Prove the formula \[ \rho = \frac{\{1+\left(\frac{dy}{dx}\right)^2\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} \] for the radius of curvature of a curve. For the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta), \] show that at a point defined by the angle \(\theta\), \(\frac{dy}{dx}=\tan\frac{1}{2}\theta\), and \(\rho=4a\cos\frac{1}{2}\theta\).
Prove that \(\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)\). Solve the equation \(\cot^{-1}x - \cot^{-1}(x+2) = \frac{\pi}{12}\).
Prove that \begin{align*} 1 - \cos^2\theta - \cos^2\phi &- \cos^2\psi + 2\cos\theta\cos\phi\cos\psi \\ &= 4\sin\sigma\sin(\sigma-\theta)\sin(\sigma-\phi)\sin(\sigma-\psi), \end{align*} where \(2\sigma = \theta+\phi+\psi\). If \(x\) and \(y\) satisfy the equation \[ \sin x + \sin y = \sqrt{3}(\cos y - \cos x), \] prove that \(\sin 3x + \sin 3y = 0\).
In any triangle, prove that
Prove that \begin{align*} \sin n\theta/\sin\theta = 2^{n-1}\cos^{n-1}\theta &- \frac{n-2}{1}2^{n-3}\cos^{n-3}\theta \\ &+ \frac{(n-3)(n-4)}{2!}2^{n-5}\cos^{n-5}\theta - \dots \end{align*} and find the general term. Prove that \(\prod_{r=1}^{r=5} \cos \frac{r\pi}{11} = \frac{1}{2^5}\).
Determine the conditions that a system of coplanar forces acting at a point should be in equilibrium. Four forces act along the sides of a quadrilateral \(ABCD\) and are represented in direction and magnitude by \(BA, BC, AD, CD\). Prove that their resultant is parallel to one diagonal and bisects the other.
State the laws of friction. A uniform rod lying on a rough inclined plane can rotate about a point on it at distances \(a\) and \(b\) from its ends; shew that if \(\mu\) be the coefficient of friction, and \(\alpha\) the angle of the plane, the inclination of the rod to the line of greatest slope cannot be greater than \[ \sin^{-1}\left[\mu\cot\alpha\sqrt{\frac{a^2+b^2}{a^2\sim b^2}}\right]. \]
State the Principle of Virtual Work and shew how it can be applied to find the stress in a rod of a jointed framework. A triangle \(ABC\) of any shape is formed of light rods smoothly jointed to each other at their ends. It is placed in a vertical plane with \(A\) downwards and the rods \(AB, AC\) resting on two smooth pegs in a horizontal line. A weight \(W\) is suspended from \(A\); prove that the stress in the rod \(BC\) is \[ \frac{1}{2}\frac{Wl}{p}\text{cosec}^2 A, \] where \(2l\) is the distance between the pegs and \(p\) is the perpendicular from \(A\) on \(BC\).