\(P\) is a variable point that moves so that the sum of its distances from fixed points \(S, S'\) is constant. By finding the equation of the locus of \(P\), or otherwise, show that the tangent to this locus at \(P\) bisects the angle \(SPS'\) externally.
Let \(I_n(z) = \int_0^1 (1-y)^n(e^{yz}-1)dy\), for all \(n \geq 0\). Prove that for all \(n \geq 1\), \(I_{n-1}(z) = \frac{z}{n}I_n(z) + \frac{z}{n(n+1)}\). Deduce that for all \(n \geq 1\), \(e^z = \sum_{r=0}^n \frac{z^r}{r!} + \frac{z^n}{(n-1)!}I_{n-1}(z)\).
The graph of \(y = f(x)\) for \(x \geq 0\) is a continuous smooth curve passing through the origin and lying in the first quadrant. It is such that, to each value of \(y\) there corresponds exactly one value of \(x\). Solving for \(x\), we obtain the equation \(x = g(y)\). Let \(F(X) = \int_0^X f(x)dx\), \(G(Y) = \int_0^Y g(y)dy\). Give a geometric argument to show that \(F(a) + G(b) \geq ab\) for any positive \(a, b\). When is equality obtained? Prove that \(u\log u - u + e^v - uv \geq 0\) if \(u \geq 1\) and \(v \geq 0\).
Sketch the graph of the function \(f(x) = -x\textrm{cosec} x\) in the range \(0 < x < 2\pi\). Prove that there is a unique value \(\bar{x}\) of \(x\) which minimises \(f(x)\) in the range \(\pi < x < 2\pi\), and show that the corresponding minimum value of \(f\) is \(\sqrt{(1+\bar{x}^2)}\).
Sketch the plane curve \(C\) whose polar equation is \(r = a\textrm{cosec}^2\frac{1}{2}\theta\), where \(0 < \theta < 2\pi\). Calculate: (i) the length of the arc \(C_1\) consisting of those points of \(C\) such that \(\frac{1}{2}\pi \leq \theta \leq \pi\); (ii) the area enclosed by the arc \(C_1\) and the radii \(\theta = \frac{1}{2}\pi\) and \(\theta = \pi\).
(i) Find, for every real non-negative integer \(k\), all the solutions of the differential equation \[\left(\frac{dy}{dx}\right)^2 = x^{2k}\] that pass through the origin. (ii) Solve, for every real non-negative integer \(k\), the equation \[\frac{1}{y}\frac{dy}{dx} = x^{-1}(\log x)^k\] with the condition \(y = 1\) at \(x = e\).