Taking the distance of the Sun to be 93,000,000 miles, compare the gravitational effect of the Sun and Earth at the surface of the latter, without assuming the masses of the Sun and Earth.
Examine whether the function \[ \frac{\sin^3 x}{x^2 \cos x} \] is a maximum or minimum when \(x=0\).
Two circles \(A, B\) cut orthogonally in \(X\) and \(Y\). A diameter of \(A\) cuts \(B\) in \(P\) and \(Q\). Prove that the points \(X, P, Y, Q\) subtend a harmonic pencil at any point on the circle.
A quadrilateral is inscribed in one circle and circumscribed about another circle. Prove that the internal diagonals intersect in a limiting point of the coaxal family to which the two circles belong.
If \(O\) be the middle point of a chord \(EF\) of a conic and \(POQ, P'OQ'\) any two chords of the conic, prove that any conic through \(P, P', Q, Q'\) will intersect \(EF\) in points equidistant from \(O\).
If a conic touch the sides of a triangle at points where the perpendiculars from the angular points meet the opposite sides, shew that the distances of its centre from the sides are proportional to the lengths of the sides.
Having given \(n\) points on the circumference of a circle shew that \(\frac{1}{2}(n-1)!\) polygons of \(n\) sides can be formed by joining the points. Also shew that if from any other point on the circumference perpendiculars are drawn to all the sides of one of the polygons, the product of these perpendiculars will be the same for all the polygons.
Sum to \(n\) terms \[ \frac{a}{a^2-1} + \frac{a^2}{a^4-1} + \frac{a^4}{a^8-1} + \dots \] and deduce the sum to \(n\) terms of \[ \operatorname{cosec}\theta + \operatorname{cosec}2\theta + \operatorname{cosec}2^2\theta + \dots. \]
If \(u_n - n(1+k)u_{n-1} + n(n-1)ku_{n-2}=0\), and \(u_2=2u_1k\), shew that \[ \frac{u_2}{2!}+\frac{u_3}{3!}+\frac{u_4}{4!}+\dots = u_1 e^k. \]
If \[ \sin^2\theta = \sin(A-\theta)\sin(B-\theta)\sin(C-\theta), \] and \[ A+B+C=\pi, \] prove that \[ \cot\theta = \cot A + \cot B + \cot C. \]