\(P\) 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 \(x = e\).
Sketch the graph of \(z(t) = (\log t)/t\) in \(t > 0\). Find the maximum value of \(z(t)\) in this range. How many positive values of \(t\) correspond to a given value of \(z\)? Hence find how many positive values of \(y\) satisfy \(x^y = y^x\) for a given positive value of \(x\). Sketch the graph of \(x^y = y^x\) in \(x > 0\), \(y > 0\).
Prove that the average (straight-line) distance apart of 2 points \(P, Q\) chosen at random on the surface of a sphere of unit radius is \(\int_0^{\pi} \sin\frac{1}{2}\theta \cdot \sin\theta d\theta\) and evaluate this.
A string is wound around the perimeter of a fixed disc of radius \(a\); one end is then unwound, the string remaining taut throughout, the portion remaining in contact with the disc not slipping and the motion being in the plane of the disc. Show that the equation of the curve described by the end of the string is given, with respect to suitably chosen axes, and a parameter \(t\), by \(x = a(\cos t + t\sin t)\), \(y = a(\sin t - t\cos t)\). Express this relationship in terms of intrinsic coordinates \(s\) and \(\psi\), and hence find the radius of curvature at the point with the parameter \(t\). Find also the area swept out by the string as its end moves from \(t = t_1\) to \(t = t_2\).
Let \(y(x) = \sin^{-1}x\), and write \(y^{(r)}(x)\) for the value of the \(r\)th derivative \(\frac{d^r y}{dx^r}\) at the point \(x\). Prove that \((1-x^2)y^{(2)}(x) - xy^{(1)}(x) = 0\), and deduce that for all \(n \geq 0\) \((1-x^2)y^{(n+2)}(x) - (2n+1)xy^{(n+1)}(x) - n^2y^{(n)}(x) = 0\). Hence show that for all \(r \geq 0\) \(y^{(2r)}(0) = 0\), \(y^{(2r+1)}(0) = \left[\frac{(2r)!}{2^r r!}\right]^2\).