The nine numbers \(a_{ij}\) (\(i,j=1, 2, 3\)) satisfy the equations \[ a_{i1} a_{j1} + a_{i2} a_{j2} + a_{i3} a_{j3} = \delta_{ij} \quad (i, j = 1, 2, 3), \] where \(\delta_{ij}=0\) if \(i \neq j\) but \(\delta_{ii}=1\) if \(i=j\). Show that they also satisfy the equations \[ a_{1i} a_{1j} + a_{2i} a_{2j} + a_{3i} a_{3j} = \delta_{ij} \quad (i, j=1, 2, 3). \] Prove also that \(a_{22} a_{33} - a_{23} a_{32} = \pm a_{11}\).
A circle is divided into \(n\) sectors by drawing \(n\) radii. Show that the number of ways of colouring the \(n\) sectors using three given colours so that neighbouring sectors are coloured differently is \[ 2^n + (-1)^n 2. \] (When \(n\) is even, all three colours need not be used.)
Given three collinear points \(A, B, C\) in a plane, explain how to construct the harmonic conjugate of \(C\) with respect to \(A\) and \(B\), using the ruler alone. If \(XYZ\) is the diagonal point triangle of the quadrangle \(ABCD\), prove that \(X\) is the pole of \(YZ\) with respect to any conic \(S\) through \(ABCD\). If a fifth point \(E\) on \(S\) is given, show how to construct with a ruler at least one more point on \(S\).
Given two points \(A, B\) on a conic \(S\), show that there is a unique conic \(S'\) touching \(S\) at \(A\) and \(B\) and such that there exists a triangle inscribed in \(S\) whose sides touch \(S'\). If \(XYZ\) is such a triangle and if \(YZ\) meets \(AB\) in \(U\) and touches \(S'\) in \(V\), show that \(U, V\) separate \(Y, Z\) harmonically.
In the Argand diagram, the points \(P_0\) and \(P_1\) represent the complex numbers \(4+6i\) and \(10+2i\) respectively. Find the complex numbers which correspond to the other five vertices of the regular hexagon with centre \(P_0\) and one vertex at \(P_1\).
(i) Defining \(\log x\) for \(x > 0\) to be \[ \int_1^x \frac{dt}{t}, \] prove \(\log xy = \log x + \log y\). (ii) Prove that \(0 < \log(1+x) - \frac{2x}{2+x} < \frac{1}{12}x^3\), where \(x > 0\).
If \(y_n = \int_0^X \frac{dx}{(x^3+1)^{n+1}}\), prove that \[ 3n y_n - (3n-1) y_{n-1} = \frac{X}{(X^3+1)^n}. \] Show that \(\int_0^\infty \frac{dx}{x^3+1} = \frac{2\pi}{3\sqrt{3}}\) and hence deduce the value of \(\int_0^\infty \frac{dx}{(x^3+1)^{n+1}}\) for positive integer values of \(n\).
A plane lamina bounded by the curve \(C\) moves in a plane so that its edge \(C\) rolls along a fixed straight line \(l\). If the instantaneous point of contact \(P\) of \(C\) with \(l\) travels along \(l\) with speed \(v\), show that the velocity of any point \(S\) fixed in the lamina is \(kvr\), where \(k\) is the curvature of \(C\) at \(P\) and \(r=SP\). \(N\) is the foot of the perpendicular from \(S\) on to \(l\). If the lamina is held fixed and the tangent \(l\) to \(C\) is made to roll along \(C\) with angular velocity \(kv\), show that \(N\) describes its locus (the pedal curve of \(C\) with respect to \(S\)) with velocity \(kvr\). Hence or otherwise prove that, if an ellipse \(C\) is rolled along a straight line in its plane, each focus will describe a curve of length \(2\pi a\) during each complete revolution of \(C\), where \(a\) is the semi-major axis of \(C\).
A particle of unit mass is projected vertically upwards with velocity \(v_0\) in a slightly resisting medium, the resistance being \(kgv^\lambda\), where \(v\) is the velocity and \(k, \lambda\) are constants; \(k\) is so small that \(k^2\) may be neglected. If the time which elapses before the particle returns to the point of projection is \(\frac{2v_0}{g}(1-\alpha)\), show that, to the first order in \(k\), \[ \alpha = k v_0^\lambda / (\lambda+2). \]
A straight rod with centroid at \(G\) and radius of gyration about \(G\) equal to \(k\) moves on a smooth horizontal table. At a moment when the rod is rotating about a point \(A\) of itself distant \(x\) from \(G\) it hits an inelastic peg at a point \(B\) distant \(y\) from \(G\) on the opposite side to \(A\). Prove that a fraction \[ \frac{k^2 (x+y)^2}{(k^2+x^2)(k^2+y^2)} \] of the kinetic energy is lost during the impact.