The motion of particles in the solar system, under the influence of the sun's gravity, is described by the equations (in appropriate units) \begin{align*} r - r\dot{\theta}^2 &= -1/r^2\\ r^2\dot{\theta} &= h = \text{const.} \end{align*} Using the second of these equations to give \(\theta\) as a function of \(r\), or otherwise, show that the first equation has the solution \begin{align*} \frac{1}{r} = \frac{1 + e\cos\theta}{h^2} \end{align*} for any constant \(e\). In the case \(0 \leq e < 1\), find the speed when the particle is nearest to the sun, and when it is furthest from it. A spaceship is in a circular orbit around the sun. Its velocity is increased instantaneously, parallel to itself, by a factor \(5/4\). Show that it will reach out to a distance \(25/7\) times its initial distance.
The real-valued functions \(x(t)\) and \(y(t)\) satisfy the pair of coupled differential equations \begin{align*} \ddot{x} + M\dot{y} - \omega^2 x &= 0\\ \ddot{y} - 2\omega\dot{x} - \omega^2 y &= 0 \end{align*} where \(\omega\) is a real constant, and dot denotes differentiation with respect to \(t\). Obtain a differential equation satisfied by \(x + \lambda y\) for a suitable choice of \(\lambda\), not necessarily real, and hence find the general solution of \((*)\). Describe briefly the shape of the path of the point \((x,y)\) when \(t\) is very large and positive.
Let \(k\), \(n\) be integers, \(k \geq 1\), \(n \geq 1\). Show that if \(n^2\) divides \((n+1)^k - 1\) then \(n\) divides \(k\), and deduce that if \((n+1)^k - 1 = n!\) and \(n \geq 6\), then \(n\) divides \(k\). [Hint: is \(n\) odd or even?] Hence find all pairs \((n, k)\) of positive integers such that \((n+1)^k - 1 = n!\)
Let \(S_n(a, b)\) be the sum of the \(n\)th powers of the roots of the cubic equation \begin{align*} x^3 + ax^2 + bx + 1 = 0. \end{align*} Evaluate \(S_0(a, b)\), \(S_1(a, b)\), \(S_2(a, b)\). Prove (i) \(S_n(a, b) = S_{-n}(b, a)\) (ii) \(S_n(a, b) = -aS_{n-1}(a, b) - bS_{n-2}(a, b) - S_{n-3}(a, b)\). Find by direct calculation the least \(m > 1\) such that \(S_m(0, 1) = 0\), and deduce that the \(m\)-th powers of the roots of \(x^3 + x + 1 = 0\) satisfy an equation of the form \(y^3 + ky + 1 = 0\). Deduce that \(\theta^{22} + \theta^{-11}\) is an integer, where \(\theta\) is a root of \(x^3 + x + 1 = 0\), and calculate its value.
Let \(x_1,\ldots,x_n\) be distinct real numbers. Write down an expression for a polynomial \(e_k\), of degree \(n-1\), such that \begin{align*} e_k(x_l) = \begin{cases} 1 & (l = k),\\ 0 & (l \neq k). \end{cases} \end{align*} Given real numbers \(\alpha_1,\ldots,\alpha_n\), find a polynomial \(p\), of degree at most \(n-1\), for which \(p(x_k) = \alpha_k\) \((k = 1,\ldots,n)\). Show further, given numbers \(\beta_1,\ldots,\beta_n\), that there is a polynomial \(q\), of degree at most \(2n-1\), such that both \(q(x_k) = \alpha_k\) and \(q'(x_k) = \beta_k\) \((k = 1,\ldots,n)\). [It is sufficient to prove the existence of \(q\); you are not expected to find its coefficients in an explicit form. In the last part of the question, you may find it helpful firstly to find a polynomial \(\eta_k\) such that \begin{align*} \eta_k(x_l) = 0 \text{ } (l = 1,\ldots,n), \text{ } \eta'_k(x_k) = 1, \text{ } \eta'_k(x_l) = 0 \text{ } (l \neq k, l = 1,\ldots,n).] \end{align*}
\(C\) is a circle with centre \(O\) and radius \(R\), \(C'\) a circle with centre \(O'\) and radius \(r\) (\(< \frac{1}{2}R\)), and \(C'\) passes through \(O\). From a point \(T\) of \(C'\) a tangent is drawn meeting \(C\) at \(E\), \(F\). If the angle \(TO'O\) equals \(\alpha\), show that \begin{align*} TE^2 + TF^2 = 2[R^2 + 2r^2\cos\alpha(1-\cos\alpha)], \end{align*} and deduce that \begin{align*} 2R^2 - 8r^2 \leq TE^2 + TF^2 \leq 2R^2 + r^2. \end{align*}
\(P\) is a fixed point of a parabola, and \(l_1\), \(l_2\) are lines at right angles to each other passing through \(P\); \(l_1\), \(l_2\) meet the parabola again at \(P_1\) and \(P_2\) respectively. Show that \(P_1P_2\) passes through a fixed point \(P'\). For another point \(Q\) of the parabola, \(Q'\) is similarly defined. If \(P'Q'\) cuts the parabola at \(R\), \(S\), show that \(P\), \(Q\), \(R\), \(S\) lie on the circle with \(RS\) as diameter.
(i) Sketch the graph of \([e^x]\) for \(x \geq 0\); here \([y]\) means the integer part of \(y\). Evaluate \begin{align*} I = \int_0^{\log_e (n+1)} [e^x]dx \end{align*} and show that \(e^I = (n+1)^n/n!\). (ii) If \(f(x) = xg(\sin x)\), show that \begin{align*} f(x) + f(\pi-x) = \pi g(\sin x), \end{align*} and hence (or otherwise) that \begin{align*} \int_0^{\pi} \frac{x\sin x}{2-\sin^2 x}dx = \frac{\pi}{2}\int_{-1}^{1}\frac{du}{u^2+1} = \frac{\pi^2}{4}. \end{align*}
Let \begin{align*} I_n = \int_0^{\pi/4} \tan^n x dx. \end{align*} (i) Show that for \(n \geq 2\) \begin{align*} I_n = \frac{1}{n-1}I_{n-2} \end{align*} (ii) Show that \(\tan x \leq \frac{4x}{\pi}\) for \(0 \leq x \leq \pi/4\), and hence show that \(I_n \to 0\) as \(n\to\infty\). (iii) Hence show that \begin{align*} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n-1} = \frac{\pi}{4} \text{ (Gregory's series)} \end{align*} and \begin{align*} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2n} = \frac{1}{2}\log_e 2. \end{align*}
For a curve defined parametrically by functions \(x(t)\), \(y(t)\), the radius of curvature is given by \begin{align*} \rho = \frac{(\dot{x}^2+\dot{y}^2)^{\frac{3}{2}}}{(\dot{x}\ddot{y}-\ddot{x}\dot{y})}. \end{align*} An ellipse is given by \begin{align*} x = a\cos t, \text{ } y = b\sin t. \end{align*} Find the parametric equations of the centre of curvature of the ellipse, and sketch its locus. Describe the shape carefully near the points corresponding to \(t = 0\), \(\pi/2\), \(\pi\), \(3\pi/2\).