Given the circumcentre, the nine-point circle and the difference of two angles of a triangle, construct the triangle.
The tangents at the points \(P, Q\) of \(x^2/a^2+y^2/b^2=1\) meet on the confocal \[ x^2/(a^2+\lambda) + y^2/(b^2+\lambda)=1. \] \(R\) is the other extremity of the diameter of the first conic through \(Q\). Prove that the tangents at \(P\) and \(R\) meet on the confocal \[ \frac{x^2}{a^2(b^2+\lambda)} + \frac{y^2}{b^2(a^2+\lambda)} = \frac{1}{\lambda}. \]
A conic is inscribed in a triangle. Prove that the straight lines drawn from the vertices of the triangle to the points of contact of the opposite sides meet in a point \(P\) and shew that if the centre of the conic moves along a straight line the locus of \(P\) is a conic through the vertices of the triangle.
If \(1, \alpha, \alpha^2, \alpha^3, \alpha^4\) are the fifth roots of unity, prove that \[ \alpha\tan^{-1}\alpha + \alpha^2\tan^{-1}\alpha^2 + \alpha^3\tan^{-1}\alpha^3 + \alpha^4\tan^{-1}\alpha^4 \] \[ = \pi\cos\frac{3\pi}{5} + \sin\frac{3\pi}{5}\log\left(\tan\frac{\pi}{20}\right) + \sin\frac{\pi}{5}\log\left(\tan\frac{3\pi}{20}\right). \]
\(P(x), Q(x)\) are given polynomials of which the latter can be expressed as the product of real linear factors. Into what Partial Fractions can the function \(P(x)/Q(x)\) be decomposed? Prove your result and shew that such decomposition is possible in one way only.
\(x_1, x_2, y_1, y_2, z_1, z_2\) are given. Shew that the numbers \begin{align*} X &= \lambda x_1 + \mu x_2 \\ Y &= \lambda y_1 + \mu y_2 \\ Z &= \lambda z_1 + \mu z_2 \end{align*} satisfy for all values of \(\lambda, \mu\) a relation of the form \[ aX+bY+cZ = 0, \] where all of \(a, b, c\) are not zero. State and prove the converse proposition.
State without proof conditions that the expression \[ a\lambda^2 + 2h\lambda\mu + b\mu^2 \] should be positive for all real values of \(\lambda\) and \(\mu\). By integrating \((\lambda f(x) + \mu\phi(x))^2\) or otherwise shew that \[ \left(\int_a^b f(x)\phi(x)dx\right)^2 \le \int_a^b (f(x))^2 dx \cdot \int_a^b (\phi(x))^2 dx. \] Shew that \(\int_0^{\pi/2} \sqrt{\sin x} \, dx\) lies between \(\frac{16}{5\pi}\) and \(\frac{1}{2}\sqrt{2\pi}\).
What is meant by the Mean Value of a function \(f(x)\) with respect to a variable \(x\)? A point moves from rest along a straight line in such a way that its average velocity with respect to distance travelled bears a constant ratio \(k\) to that with respect to time elapsed. Shew that \(k > 1\).
\(A\) and \(B\) are points on opposite sides of a stream 10 feet wide which are connected by a bridge formed by two equal uniform planks \(AC, CD\), each of which is \(7\frac{1}{2}\) feet long and of weight \(W_1\). The plank \(CD\) projects \(2\frac{1}{2}\) feet over the stream and is kept in position by a weight \(W_2\) placed at \(D\). The plank \(AC\) is hinged at \(A\) and just overlaps the other plank at \(C\). Prove that for a man to be able to cross the bridge in safety his weight must not exceed \(2W_2\).
A particle of mass \(m\) is placed on the inclined face of a wedge of mass \(M\) which rests on a rough horizontal table. Prove that, if the particle slides down, the wedge will begin to move provided that \[ \frac{m}{M} > \frac{\cos\lambda\sin\lambda'}{\cos\alpha\sin(\alpha-\lambda-\lambda')}, \] where \(\alpha\) is the inclination of the face of the wedge to the horizontal, \(\lambda\) is the angle of friction for the particle and the wedge, and \(\lambda'\) is the angle of friction for the wedge and the table.