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 \(u_n\).
If \[ \phi(c-x) = \phi(x), \] show that \[ \int_0^c x^3\phi(x)dx = \frac{3c}{2}\int_0^c x^2\phi(x)dx - \frac{c^3}{4}\int_0^c \phi(x)dx. \]
The function \(\log x\) is defined for real positive values of \(x\) by the equation \[ \log x = \int_1^x \frac{dt}{t}. \] Prove that
Define Simple Harmonic Motion and obtain an expression for the periodic time. Consider the case of a bead sliding on a smooth wire in the form of a cycloid which is fixed with its axis vertical and vertex downwards, and shew that the equation of the hodograph of the motion can be written in the form \[ (x^2+y^2)^2 = (C+4ga)x^2+Cy^2, \] where \(C\) is a constant and \(a\) is the radius of the generating circle of the cycloid.