Differentiate ab initio \(\log x\), \(\tan^{-1} x\). Differentiate \(e^{\sin(\log x)}\).
Describe the cycle on which (a) a four-stroke gas engine, (b) a Diesel engine, works. Indicate on a diagram in each case the points at which the chief events of the cycle occur, and state briefly the relative advantages of each cycle.
A particle describes backwards and forwards an arc of a circle subtending an angle of 2 radians at the centre, so that its acceleration resolved along the tangent is equal to \(\mu\) times its distance measured along the arc from the middle point; show that the least resultant acceleration is \(\sqrt{3}\mu a\), where \(a\) is the radius of the circle.
Find the equation of the tangent from the origin to the curve \[ y=e^{\lambda x} \quad (\lambda > 0). \] Deduce, or prove otherwise, that the equation \(x=e^{\lambda x}\) has 0, 1, or 2 real roots according as \(e\lambda > 1, e\lambda=1, e\lambda < 1\).
Prove that if \[ y = xe^{-x}\cos x, \] then \[ x^2 \frac{d^2y}{dx^2} + 2x(x-1)\frac{dy}{dx} + 2(x^2-x+1)y=0. \]
Establish a formula of reduction for the integral \[ u_n = \int x^n (1+x^2)^{-\frac{1}{2}}dx. \]
Evaluate the integrals \[ \int \frac{x^2+2x+2}{(x+1)^2}dx, \quad \int x \sin x dx, \quad \int_{-1}^1 (1-x^2)^{\frac{3}{2}}dx. \]
A circular cylinder has its volume fixed: find its shape when the sum of the length and the girth is a minimum.
Calculate the volume common to two spheres, each of radius \(a\), which are so placed that the centre of each lies on the surface of the other.
Show that if four forces in equilibrium act along the sides of a quadrilateral inscribed in a circle, the forces are proportional each to the side opposite to that in which it acts.