Problems

Prove that \[ 2 - 2 \cos \theta + \cos 2\theta - 2 \cos 3\theta + \cos 4\theta \ge 0. \] What are the values of \(\theta\) for which equality occurs?

If A, B, C, D are four concyclic points, shew that the feet of the perpendiculars from D on the sides of the triangle ABC are collinear (on the pedal line of D with respect to the triangle ABC). Shew further that the pedal lines of A, B, C, D with respect to the triangles BCD, CDA, DAB, ABC are concurrent.

A long ladder of negligible weight rests with one end on the ground and the other projecting over the top of a wall of height \(h\), the vertical plane through the ladder being at right angles to the wall. The coefficients of friction at the ground and the wall are \(\mu_1\) and \(\mu_2\), and the ladder makes an angle \(\theta\) with the horizontal. Shew that, if a man tries to ascend the ladder, equilibrium will be broken by the ladder tilting or sliding according as \(\mu_2\) is greater or less than \(\tan\theta\). If equilibrium is broken by sliding, shew that the man will be unable to reach the top of the wall unless \(\mu_1 > \cot\theta\), and find the maximum height he can attain. Shew further that whatever may be the values of the frictions the man cannot ascend a vertical distance greater than \(h \sec^2\theta\).

Prove that if a circle \(S\) cuts each of two given circles \(S_1, S_2\) orthogonally, then the centre of \(S\) lies on the radical axis of \(S_1, S_2\). Shew that in general there are two circles which cut two given circles \(S_1, S_2\) orthogonally and touch a given circle \(S_3\).

(i) Prove that \[ (2 \cos \theta - 1) (2 \cos 2\theta - 1) (2 \cos 2^2\theta - 1) \dots (2 \cos 2^{n-1}\theta - 1) = \frac{2 \cos 2^n \theta + 1}{2 \cos \theta + 1}. \] (ii) Sum to \(n\) terms the series \[ 1 + 2 \cos \theta + 2^2 \cos 2\theta + 2^3 \cos 3\theta + \dots. \]

A variable tangent to a conic S meets two fixed perpendicular tangents a, b at P, Q respectively, and the perpendiculars to a, b at P, Q meet at L. Prove that, if S is a central conic, the locus of L is a rectangular hyperbola whose asymptotes are the tangents of S parallel to a, b; and that this hyperbola passes through the points of contact of a and b with S. What is the locus of L when S is a parabola?

Describe the principle of virtual work, and illustrate your description by an example. \(ABCDE\) is a framework of four equal rods, each of weight \(w\) and length \(l\), freely hinged at \(B, C\) and \(D\). The ends \(A\) and \(E\) are freely hinged at two fixed points on the same horizontal level, and the points \(B\) and \(D\) are connected by a string. The whole framework hangs symmetrically below \(AE\) in the shape of a W, and \(AB, BC\) make acute angles \(\theta\) and \(\phi\) respectively with the vertical. A weight \(W\) is suspended from \(C\). Using the principle of virtual work, or otherwise, prove that the tension in the string is \[ \frac{3w+W}{2} \tan\theta + \frac{w+W}{2} \tan\phi. \]

The tangents to a central conic \(S\) from a point \(T\) touch \(S\) at \(P\) and \(Q\). \(QQ'\) is the diameter through \(Q\), and \(PR\) is the chord through \(P\) conjugate to \(QQ'\) with respect to \(S\). \(PR\) meets \(TQ'\) in \(L\). Prove that \(3PL = LR\).

Express the function \[ f(x) = \frac{x^3 - x}{(x^2 - 4)^2} \] in partial fractions (with numerical numerators). Find the value of the \(n\)th derivative of \(f(x)\) for \(x=0\).

If \(\alpha = 2\pi/7\), prove that \[ \sin\alpha + \sin 2\alpha + \sin 4\alpha = \tfrac{1}{2}\sqrt{7}. \]