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.
Explain the term `angle of friction.' A cylinder rests inside a fixed hollow cylinder whose axis is horizontal and subtends an angle \(2\alpha\) at this axis. A cylinder equal to the former is placed so as to rest in contact with both without disturbing the former. Show that if the surfaces are equally rough, the angle of friction must be greater than each of \(\frac{\pi}{4}-\frac{\alpha}{2}\) and \(\alpha\).
A regular hexagon ABCDEF formed by equal heavy rods connected by smooth joints is kept in shape by light rods FB, FC, FD and is suspended from A. Draw the force diagram and calculate the ratio of the stress in the rod FC to the weight of one of the rods.