Problems

Prove that \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{1}{v^2}\left(v\frac{du}{dx} - u\frac{dv}{dx}\right). \] Differentiate \[ \frac{x\sin^{-1}x}{(1-x^2)^{\frac{1}{2}}} + \frac{1}{2}\log(1-x^2), \] and find the value of \[ \frac{d^8}{dx^8}\left(\frac{\sin^3x \cos x}{256}\right) \] when \(x=\frac{\pi}{12}\).

Shew that the altitude of the right circular cone of maximum volume which can be inscribed in a sphere of radius \(a\) is \(4a/3\). What is the value of the altitude (in terms of \(a\)) when the area of the curved surface is a maximum?

A curve is given by the parametric equations \[ x = 3\cos\theta - \cos3\theta, \quad y = 3\sin\theta - \sin3\theta. \] Shew that the angle which the tangent at any point makes with the \(x\) axis is \(2\theta\). If \(s\) is the length of the arc of the curve measured from the point for which \(\theta=0\), prove that \[ s = 12 \sin^2\frac{\theta}{2}. \]

Sketch the curve whose equation is \[ y^2=c^2\frac{(x-a)}{(b-x)} \quad (b>a) \] and shew that the area enclosed by the curve and its asymptote is \(\pi c(b-a)\).

(i) Evaluate \[ \int \frac{(x-3)dx}{4x^2+5x+1}. \] (ii) Given \(\log_{10}e = 0.4343\), prove that \[ \int_1^3 x\log_x\left(1+\frac{1}{x}\right)dx = 1.601. \] (iii) Find a reduction formula for \(\int_0^{\pi/2}\sin^n x\,dx\) where \(n\) is a positive integer, and evaluate \[ \int_0^{\pi/2} \sin^6 x\,dx. \]

\(ABC\) is a triangle in which the angles \(ABC, ACB\) are each equal to twice the angle \(BAC\). Prove that \(AB^2 = BC^2 + AB \cdot BC\). Hence show how to inscribe geometrically a regular pentagon in a given circle.

Prove that the orthocentre \(H\), the centroid \(G\) and the centre \(O\) of the circumcircle of a triangle are collinear, and that \(HG=2GO\). The ends of any diameter \(BC\) of a given circle are joined to a fixed point \(A\) in the plane of the circle; find the locus of the orthocentre of the triangle \(ABC\) and deduce the locus of the circumcentre.

Explain what is meant by a centre of similitude. Prove that two circles have two centres of similitude, and that the circle on the line joining them as diameter is coaxial with the given circles. Given three circles \(S_1, S_2, S_3\), prove that the join of a centre of similitude of \(S_1, S_2\) to a centre of similitude of \(S_1, S_3\) passes through a centre of similitude of \(S_2, S_3\).

Show that in general two spheres can be inscribed in a right circular cone to touch a given plane not passing through the vertex, and that the plane cuts the cone in a conic whose foci are the points of contact of the spheres with the plane. When is the conic a parabola? Show that the foci of parabolic sections of a right circular cone lie upon another right circular cone.

Explain what is meant by a projective correspondence (or homography) between the points on a straight line, and show that there is a unique projective correspondence in which to given points \(A, B, C\) there correspond given points \(A', B', C'\). \(M\) is a self-corresponding point of a projective correspondence on a line and to \(A, B\) there correspond respectively \(A', B'\). On any line through \(M\) two points \(S, S'\) are taken; \(SA, S'A'\) meet in \(P\) and \(SB, S'B'\) in \(Q\). Prove that the point \(N\) in which \(PQ\) meets the given line is the other self-corresponding point. Deduce, or prove otherwise, that if in a projective correspondence the two self-corresponding points coincide in \(M\) (a parabolic correspondence), and to \(A\) corresponds \(A'\) and to \(A'\) corresponds \(A''\), then \(M, A'\) are harmonically separated by \(A, A''\). (Assume the harmonic property of the quadrilateral.)