A curve is given by the parametric equations \[ x=f(t), \quad y=g(t). \] Explain the significance of the expression \[ \frac{1}{2} \int_{t_0}^{t_1} (fg' - f'g) dt. \] Sketch the curve whose parametric equations are \[ x = \frac{1-t^2}{1+t^2}, \quad y = \frac{t(1-t^2)}{1+t^2}, \] and calculate the area of the loop.
Obtain a recurrence relation between integrals of the type \[ I_n = \int x^n e^{ax} \cosh bx \, dx. \]
\(Q, R, S\) are the points \((\alpha, \beta)\), \((-l, 0)\) and \((l, 0)\), and \(P\) is a variable point \((x, 0)\) on \(RS\). Evaluate \[ \int_{-l}^l \frac{dx}{PQ}, \] and show that it can be expressed as a function of \(RS\) and \(QR+QS\).
Show that the solution of the equation \[ y'' + n^2 y = a \sin pt \] (where \(n\neq 0\) and \(p^2 \neq n^2\)), such that \(y=0\) and \(y'=0\) when \(t=0\), is \[ y = \frac{a}{n^2-p^2}\left(\sin pt - \frac{p}{n}\sin nt\right). \] Show also that, as \(p\) tends to \(n\), \(y\) tends to \[ \frac{a}{2n}\left(\frac{1}{n}\sin nt - t\cos nt\right), \] and verify that this is the solution when \(p\) is equal to \(n\).
The roots of the cubic equation \(x^3-px+q=0\) are \(\alpha, \beta, \gamma\). Evaluate \(\alpha^7+\beta^7+\gamma^7\) in terms of \(p\) and \(q\). Hence, or otherwise, solve the equation \[ 64\sin^7\theta + \sin 7\theta = 0. \]
Solution: Let \(s_i = \alpha^i + \beta^i + \gamma^i\) then \(s_n = ps_{n-2} - qs_{n-3}\) for \(n \geq 3\), so \begin{align*} && s_0 &= 3 \\ && s_1 &= 0 \\ && s_2 &= 0^2 - 2(-p) = 2p \\ && s_3 &= -3q \\ &&s_4 &= p(2p) - q(0) = 2p^2 \\ &&s_5 &= p(-3q) - q(2p) = -5pq \\ && s_6 &= p(2p^2) - q(-3q) = 2p^3 + 3q^2 \\ && s_7 &= p(-5pq) - q(2p^2) = -7p^2q \end{align*} \begin{align*} && \sin 7 \theta + 64 \sin^7 \theta &= \frac{e^{7i\theta} -e^{-7i \theta}}{2i} + \frac12 \left ( 2 \sin \theta\right)^7 \\ &&&= \frac12 \left ( -i(e^{i \theta})^7 + i(e^{-i\theta})^7 + (2 \sin \theta)^7\right) \\ &&&= \frac12\left ( (ie^{i \theta})^7 + (-ie^{-i\theta})^7 + (2 \sin \theta)^7 \right) \\ \end{align*} since \(2 \sin \theta - i(e^{i \theta} - e^{-i\theta}) = 0\) we must have \begin{align*} && \sin 7 \theta + 64 \sin^7 \theta &= \frac12\left ( (ie^{i \theta})^7 + (-ie^{-i\theta})^7 + (2 \sin \theta)^7 \right) \\ &&&= -\frac{7}{2} \left (ie^{i \theta}(-ie^{-i\theta}) + ie^{i \theta}(2 \sin \theta) + (-ie^{-i\theta})(2 \sin \theta) \right)^2 \left (-ie^{i \theta}(-ie^{-i\theta})(2 \sin \theta) \right) \\ &&&= -\frac72 \left (1 - 4\sin^2 \theta \right)^2(-2 \sin \theta) \\ &&&= 7 \sin \theta(1- 4\sin^2 \theta)^2 \end{align*} Therefore for our equation to equal zero we need \(\sin \theta = 0\) or \(\sin \theta = \pm \frac12\), ie \(\theta = n \pi , \frac{\pi}{6} + n \pi, \frac{5 \pi}{6} + n \pi\)
Prove that \(|z_1+z_2| \le |z_1|+|z_2|\) where \(z_1, z_2\) are complex numbers. Show that if \(|a_n|<2\) for \(1 \le n \le N\) then the equation \[ 1+a_1z+\dots+a_Nz^N = 0 \] has no solution such that \(|z|<\frac{1}{3}\). Is the converse true?
The function \(f(x)\) is ``bounded as \(x\to 0\) through positive values'' if and only if there exist positive constants \(K, \delta\) such that \(|f(x)| < K\) for \(0< x< \delta\). Show that if \(f(x), g(x)\) are bounded as \(x\to 0\) through positive values then so are \(f(x)+g(x)\), \(f(x)g(x)\), \(\int_x^\delta f(t)\,dt\). Show that \[ \int_x^1 \frac{e^{-t}}{t}dt + \log x \] is bounded as \(x\to 0\) through positive values.
A set of functions \(J_n(x)\), \(n=0, \pm 1, \pm 2, \dots\), satisfy the following equations: \begin{align*} J_{n-1}(x)+J_{n+1}(x) &= \frac{2n}{x}J_n(x), \\ J_{n-1}(x)-J_{n+1}(x) &= 2\frac{d}{dx}J_n(x). \end{align*} Show that \begin{align*} \left(\frac{1}{x}\frac{d}{dx}\right)^m x^n J_n(x) &= x^{n-m}J_{n-m}(x), \\ \left(\frac{1}{x}\frac{d}{dx}\right)^m x^{-n} J_n(x) &= (-)^m x^{-n-m}J_{n+m}(x). \end{align*} Also prove that \(x^2\dfrac{d^2J_n(x)}{dx^2} + x\dfrac{dJ_n(x)}{dx} + (x^2-n^2)J_n(x)=0\).
Prove that the point whose rectangular cartesian coordinates are \[ x=\frac{2t}{a(1+t^2)}, \quad y=\frac{1-t^2}{b(1+t^2)}, \] where \(t\) is a variable parameter, describes an ellipse. Find the equation of the chord joining the points where \(t\) has the values \(t_1, t_2\), and the equation of the tangent at the point \(t_1\). Prove that, if the four points \(t_1, t_2, t_3, t_4\) are concyclic, then \[ \Sigma t_1 = \Sigma t_1 t_2 t_3, \] and that, if they lie on a rectangular hyperbola through the origin, then \[ a^2(1+\Sigma t_1 t_2 + t_1 t_2 t_3 t_4) + 2b^2(1+t_1t_2t_3t_4)=0. \]
Show that the equation of the pair of tangents to the conic whose equation, in homogeneous coordinates, is \[ S \equiv ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] at the points where it is met by the line \[ L \equiv \lambda x + \mu y + \nu z = 0, \] is \[ S\Sigma - \Delta L^2 = 0, \] where, in the usual notation, \(\Sigma = A\lambda^2+B\mu^2+C\nu^2+2F\mu\nu+2G\nu\lambda+2H\lambda\mu\), \(A=bc-f^2\), \(F=gh-af\), etc., and \(\Delta=abc+2fgh-af^2-bg^2-ch^2\). If \(x, y\) are rectangular cartesian coordinates, prove that, if the line \(\lambda x+\mu y+\nu=0\) is a directrix of the conic \(ax^2+by^2+c+2fy+2gx+2hxy=0\), then \[ \frac{\lambda^2-\mu^2}{a-b} = \frac{\lambda\mu}{h} = \frac{\Sigma}{\Delta}. \] \subsubsection*{SECTION B}