Prove that the locus of the poles of normal chords of the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is the curve \[ \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2-b^2)^2. \]
Two adjacent corners \(A, B\) of a rigid rectangular lamina \(ABCD\) slide on the rectangular axes \(XOX'\), \(YOY'\), all the motion being in one plane. Prove that the locus of \(C\) is an ellipse of area \(\pi AD^2\).
Prove that the three lines, each of which forms a harmonic pencil with the three lines \[ y=0, \quad ax^2+2hxy+by^2=0, \] are \[ ax+hy=0, \quad a(ax^2+2hxy+by^2)+8(ab-h^2)y^2=0. \]
If the equations \begin{align*} axy+bx+cy+d&=0, \\ ayz+by+cz+d&=0, \\ azw+bz+cw+d&=0, \\ awx+bw+cx+d&=0, \end{align*} are satisfied by values of \(x, y, z, w\) which are all different, show that \[ b^2+c^2=2ad. \]
Prove that the number of primes is infinite. Find \(n\) consecutive numbers, none of which are primes.
Prove that \[ \frac{(1-\sin\theta)(1+\sin 15\theta)}{(1+\sin 3\theta)(1-\sin 5\theta)} = (16\sin^4\theta - 8\sin^3\theta - 16\sin^2\theta+8\sin\theta+1)^2, \] and find the values of \(\theta\) for which the expression on the right vanishes. (Note: The numbers in the trigonometric functions on the LHS are very difficult to read from the scan and may be inaccurate.)
The centres of the circumcircle and the inscribed circle of a triangle are \(O\) and \(I\), the radii are \(R\) and \(r\). Prove that \[ OI^2 = R^2 - 2Rr. \] Triangles are inscribed in a circle, centre \(O\), and circumscribed to a circle, centre \(I\). Show that the centres of their escribed circles lie on a circle of radius \(2R\), whose centre \(I'\) is such that \(O\) bisects \(II'\).
Find the asymptotes of the curve \[ x^2(x+y)=x+4y, \] and trace the curve.
If \(w\) is a function of \(x\) and \(y\), and if \[ x=u^3-3uv^2, \quad y=3u^2v-v^3, \] prove that \begin{align*} 3(u^2+v^2)\left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}\right) &= u\frac{\partial w}{\partial u} - v\frac{\partial w}{\partial v}, \\ 9(u^2+v^2)^2\left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}\right) &= \frac{\partial^2 w}{\partial u^2} + \frac{\partial^2 w}{\partial v^2}. \end{align*}
If \[ u_n = \int_0^{\pi/2} \sin n\theta \cos^{n+1}\theta \operatorname{cosec}\theta \, d\theta, \] find the relation between \(u_n\) and \(u_{n-1}\), and hence evaluate \(u_n\).