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?
\(OABC, OA'B'C'\) are two straight lines; \(AB', BA'\) meet at \(P\); \(BC', CB'\) meet at \(Q\), and \(CA', AC'\) meet at \(R\). Shew that \(P, Q, R\) lie on a straight line. Prove that, if the cross-ratios \(\{OB, AC\}\) and \(\{OB', A'C'\}\) are equal, the line \(PQR\) passes through \(O\).
\(F_1, F_2, F_3 \dots F_n\) are fixed coplanar forces. A new force \(F_{n+1}\) is added, whose point of application \(A\) and line of action are fixed, but whose magnitude can be varied. Shew that the resultant of the forces \(F_1, F_2, F_3 \dots F_{n+1}\) always passes through another fixed point \(B\), and by a suitable choice of \(F_{n+1}\) may be made to pass through any arbitrary point \(C\), which does not lie on the line \(AB\). Discuss any exceptional cases. \(PQRS\) is a square of side \(a\), and forces 1, 2, 3 act along \(PQ, QR, RS\) respectively. A variable force \(F\) acts along \(SP\). Find the fixed point through which the resultant always acts, and the value of \(F\) if it is to pass through the centre of the square.
A, B, C are three points in a straight line. Three semicircles are constructed on AB, BC and AC as diameters, all on the same side of the line ABC, and a circle is drawn touching the three semicircles. Prove that its diameter is equal to the perpendicular distance of its centre from ABC.
(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 \[ 1 + 2 \cos \theta + 2^2 \cos 2\theta + 2^3 \cos 3\theta + \dots. \]
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 \(S_3\).
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 \(h \sec^2\theta\).
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.
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 \(x=0\).
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 \(3PL = LR\).