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.
Two particles whose masses are in the ratio 4:3 are connected by a light string of length \(\pi a\) and rest in equilibrium over a smooth horizontal cylinder of radius \(a\). If equilibrium is disturbed so that the heavier particle begins to descend, find at what point it will leave the surface, and shew that at that instant the pressure on the other particle is slightly greater than two-thirds of its weight.
An elastic string is stretched between two fixed points \(A\) and \(B\) in the same vertical line, \(B\) being below \(A\). Prove that if a particle is fixed to a point \(P\) of the string and released from rest in that position it will oscillate with simple harmonic motion of period \(t\sqrt{\mu}\) and of amplitude \(\mu a\), where \(t\) is the period and \(a\) the amplitude when \(P\) coincides with the mid-point of \(AB\), and \(\mu = 4AP.PB/AB^2\). The string may be assumed taut throughout.
Solve the equation \((x+b+c)(x+c+a)(x+a+b)+abc=0\). Eliminate \(x, y\) from \(x+y=a, x^3+y^3=b^3, x^5+y^5=c^5\).