Obtain the expansion of \(\sin x\) in ascending powers of \(x\). For what values of \(x\) is this series convergent? \par A small arc \(PQ\) of a circle of radius 1 is of length \(x\). The arc \(PQ\) is bisected at \(Q_1\) and the arc \(PQ_1\) is bisected at \(Q_2\). The chords \(PQ\), \(PQ_1\) and \(PQ_2\) are of lengths \(c\), \(c_1\) and \(c_2\), respectively. Prove that \(\frac{1}{45}(c - 20c_1 + 64c_2)\) differs from \(x\) by a quantity of order \(x^7\).
If a variable chord of a parabola subtends a right angle at the focus, prove that the locus of its pole is a rectangular hyperbola. Generalize this result by projection.
Two trains \(A\) and \(B\) are travelling along the same straight line with velocities \(u\) and \(v\) respectively. Prove that, on certain assumptions, the velocity of \(B\) as measured by a man in \(A\) is \(v-u\). What are these assumptions?
As \(x\) tends to \(a\), the functions \(f(x), g(x), f'(x)\) and \(g'(x)\) tend to the limits \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \neq 0\), \(f(x)/g(x)\) tends to \(b/c\). \par Evaluate the following limits:
If \[ I_{m, n} = \int \cos^m x \cos nx dx, \] prove that \[ (m+n) I_{m,n} = \cos^m x \sin nx + m I_{m-1, n-1}. \] Find \[ \int_0^{\pi/2} \cos^m x \cos nx dx, \] when \(m\) and \(n\) are integers such that \(0 < m < n\).
Two conics touch at \(A\) and intersect at \(B\) and \(C\). Prove that the point \(A\), the middle points of \(BC, CA\) and \(AB\), and the centres of the conics lie on a conic.
A smooth wedge of mass \(M\) and angle \(\alpha\) lies on a horizontal table, and a particle of mass \(m\) is allowed to slide from rest down the inclined face of the wedge. If in the ensuing motion the wedge slides without rotating, show that its acceleration is \[ (mg \cos \alpha \sin \alpha)/(M + m \sin^2 \alpha). \] Find also the reaction between the particle and the wedge, and that between the wedge and the plane.
A heavy non-uniform rod, inclined at an angle \(\theta\) to the horizontal, is wedged between two rough planes inclined at angles \(\alpha_1, \alpha_2\) to the vertical, as shown in the figure. The planes intersect in a horizontal line, and the rod lies in a vertical plane at right angles to this line. The angles of friction between the rod and the two planes are \(\lambda_1, \lambda_2\) respectively. Prove geometrically, or otherwise, that equilibrium is possible if and only if \[ \lambda_1 - \alpha_1 > \theta > \alpha_2 - \lambda_2. \] (A diagram shows a rod wedged between two planes. The planes make angles \(\alpha_1\) and \(\alpha_2\) with the vertical. The rod makes an angle of \(90^\circ - \theta\) with the vertical.)
Sketch the graph of the function \[ y = e^{-a(x+b/x^2)}, \] where \(a\) and \(b\) are both positive. Prove that there are always at least two points of inflexion. Find the abscissae of the points of inflexion when \(a=10/27\), \(b=5\).
Find the coordinates of the mirror image of the point \((h,k)\) in the line \[ lx+my+n=0. \] Prove that the rectangular hyperbolas \begin{align*} xy &= c^2, \\ xy - 2c(x+y)+3c^2 &= 0 \end{align*} touch each other and that each is the mirror image of the other in the common tangent.