A small satellite moves in an orbit under the influence of the earth's attraction. With the assumptions that the earth is spherically symmetrical and frictional drag is negligible, show that $$\frac{d^2u}{d\theta^2} + u = C,$$ where \(r\), \(\theta\) are polar coordinates of the satellite with the centre of the earth as pole, \(u = 1/r\), and \(C\) is independent of \(r\) and \(\theta\). A satellite originally travels with speed \(V\) in a circular orbit of radius \(VT/(2\pi)\). At some stage an internal impulsive mechanism separates the satellite into two parts moving away from each other with relative velocity \(v\) along the direction of the original orbit at the point of separation. Show that, if \(v/V\) is small, the orbital period of one part differs from that of the other by approximately $$3 \frac{v}{V} T.$$ If \(T\) is 90 minutes and \(v\) is 10 m.p.h., give an approximation to the distance between the two parts when one of them has completed its first orbit.
Five schools play a rugby football competition, each school playing each of the others twice, once at home and once away. The final outcome is given in the following table:
The functions \(f_n(x)\) are defined thus: \begin{align} f_0(x) = 1, \quad f_n(x) = (-\frac{1}{2})^n e^{-x} \frac{d^n}{dx^n}(e^{-x}) \quad (n \geq 1). \end{align} Show that \(f_n(x) = xf_{n-1}(x) - \frac{1}{2}f'_{n-1}(x)\) if \(n \geq 1\), and deduce that \(f_n(x)\) is a polynomial of degree \(n\), with leading coefficient 1. By considering the signs of \(f'_n\) and \(f_n\) at the zeros of \(f_{n-1}\), or otherwise, prove that the equation \(f_n(x) = 0\) has \(n\) distinct real roots, which are separated by the \(n-1\) distinct real roots of \(f_{n-1}(x) = 0\).
By use of the identity \(\cos n\theta + \cos(n-2)\theta - 2\cos\theta\cos(n-1)\theta\), or otherwise, prove that \begin{align} \cos n\theta = \sum_{k=0}^{m} a_{n,k} \cos^{n-2k}\theta, \end{align} where \(m\) is the largest integer \(k\) such that \(2k \leq n\), and the coefficients \(a_{n,k}\) are integers satisfying \begin{align} a_{n,0} = 2a_{n-1,0}, \quad a_{n,k} = 2a_{n-1,k} - a_{n-2,k-1} \quad (1 \leq k \leq \frac{1}{2}n). \end{align} Deduce that \(a_{n,0} = 2^{n-1}\), \(a_{n,1} = -2^{n-3}n\) and \begin{align} \sum_{0 \leq r \leq n} \cos(r + \frac{1}{2})\frac{\pi}{n} \cos(s + \frac{1}{2})\frac{\pi}{n} = -\frac{1}{4}n. \end{align}
A right-angled triangle has integral sides and the lengths of the two shorter sides differ by 1. If the sides are \(m\), \(m+1\), \(n\), show that there exist right-angled triangles having the same property with hypotenuse \(3n \pm (4m + 2)\). Hence or otherwise, show that there are just four such triangles with hypotenuse less than 1000.
A regular octahedron is oriented by assigning a direction along each edge, in such a way that the boundary of each face can be described by a point moving along the edges in the assigned directions. Find the number of ways in which a point, starting from a given vertex of the octahedron, can describe a closed path, following the edges in the given directions and describing each edge of the octahedron once.
Prove that by suitable choice of homogeneous coordinates the general point of a non-singular conic \(S\) can be expressed in the parametric form \begin{align} x:y:z = t^2:t:1. \end{align} In a homography on \(S\) in which \(P_i \leftrightarrow Q_i\), the parameters \(p_i\) and \(q_i\) of the corresponding points are connected by the relation \begin{align} ap_i q_i + bp_i + cq_i + d = 0 \quad (ad \neq bc). \end{align} \(X_{ij}\) \((i \neq j)\) is the intersection of the lines \(P_i Q_j\), \(P_j Q_i\). Show that all the points \(X_{ij}\) lie on a line which meets \(S\) in the (distinct or coincident) self-corresponding points of the homography. \(A_1, A_2, B_1, B_2\) are four distinct points in the given order on a line \(l\) in the Euclidean plane. In a homography \(T\) on \(l\) the self-corresponding points coincide in \(U\) and \(A_1 \leftrightarrow B_1\), \(A_2 \leftrightarrow B_2\). Prove that there are two possible positions for the point \(U\), and outline a geometrical construction for determining them.
The equation of a non-singular conic in homogeneous point coordinates \((x, y, z)\) is \begin{align} ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy = 0, \end{align} Prove that its equation in tangential coordinates \((l, m, n)\) is \begin{align} Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm = 0, \end{align} where \(A = bc - f^2\), \(B = ca - g^2\), \(C = ab - h^2\), \(F = gh - af\), \(G = hf - bg\), \(H = fg - ch\). A non-singular conic \(S\) is in general position with respect to a triangle. \(XYZ\) such that tangents to it from the vertices \(X\), \(Y\), \(Z\) meet the opposite sides in the pairs \((L_1, L_2)\), \((M_1, M_2)\), \((N_1, N_2)\). Prove that the six points \(L_1, L_2, M_1, M_2, N_1, N_2\) lie on a conic \(S'\) passing through the common points of \(S\), \(S'\) and the sides of the triangle.
\(AE_1 E_2\) is a triangle and \(L\) is a point of the line \(E_1 E_2\). Two conics \(S_1, S_2\) touch \(AE_1, AE_2\) and touch \(E_1 E_2\) at \(E_1, E_2\) respectively. \(S_1, S_2\) meet again in \(H\) and \(K\). Prove that \(HK\) meets \(E_1 E_2\) in the harmonic conjugate of \(L\) with respect to \(E_1, E_2\).
A rope hangs over a pulley of radius \(a\) and moment of inertia \(I\), which is smooth on its bearings but perfectly rough to the rope. Two monkeys of equal mass \(m\) sit hanging at the same level, one on each end of the rope. Starting at the same instant, the monkeys climb with constant speeds \(u_1\) and \(u_2\) relative to the rope \((u_1 > u_2)\). Show that when the monkey of speed \(u_1\) is at a height \(h\) above his initial position, he is at a height \begin{align} \frac{hl(u_1 - u_2)}{(I + ma^2)u_1 + ma^2u_2} \end{align} above the other monkey.