Prove that \[ \left( \sum_{n=1}^N a_n b_n \right)^2 \le \sum_{n=1}^N a_n^2 \sum_{n=1}^N b_n^2, \] where \(a_1, \dots, a_N, b_1, \dots, b_N\) are positive. \par Hence or otherwise, prove that, for all real values of \(\rho\) and \(\sigma\), \[ \left( \sum_{n=1}^N c_n^\rho \right)^2 \le \sum_{n=1}^N c_n^{\rho-\sigma} \sum_{n=1}^N c_n^{\rho+\sigma}, \] where \(c_1, \dots, c_N\) are positive. \par Hence or otherwise, prove that \[ \sum_{n=1}^N a_n^2 \sum_{n=1}^N b_n^2 \le \sum_{n=1}^N \frac{a_n^3}{b_n} \sum_{n=1}^N a_n b_n. \]
Prove that \[ (x^2-1) \prod_{\nu=1}^{n-1} \left( x^2 - 2x \cos\frac{\pi\nu}{n} + 1 \right) = x^{2n}-1. \] Hence or otherwise, prove that \[ \prod_{\nu=1}^{\frac{1}{2}(n-1)} \cos\frac{\pi\nu}{n} = 2^{\frac{1}{2}(1-n)} \quad \text{for } n=3,5,7,\dots, \] and \[ \prod_{\nu=1}^{\frac{1}{2}n-1} \cos\frac{\pi\nu}{n} = \sqrt{n} \, 2^{1-n} \quad \text{for } n=4,6,8,\dots. \]
The functions \(\phi(x)\) and \(\psi(x)\) are differentiable in the interval \(a < x < b\); and \(\psi'(x ) > 0\) for \(a < x < b\). Prove that there is at least one number \(\xi\) between \(a\) and \(b\) such that \[ \frac{\phi(\xi)-\phi(a)}{\psi(b)-\psi(\xi)} = \frac{\phi'(\xi)}{\psi'(\xi)}. \] If \(\phi(x)=x^2\) and \(\psi(x)=x\), find a value of \(\xi\) in terms of \(a\) and \(b\).
Find the coefficients in the polynomial \(f_n(x)\) defined by \(f_n(x) = e^{-x} \frac{d^n}{dx^n} (x^n e^x)\). \par Prove the following identities, where \(n\) is a positive integer: \begin{align*} f_{n+1}(x) - (2n+1+x)f_n(x) + n^2 f_{n-1}(x) &= 0; \\ x f_n''(x) + (1+x)f_n'(x) - n f_n(x) &= 0. \end{align*}
Through each point \(P\) of a parabola with focus \(S\) a line \(PQ\) is drawn parallel to a fixed direction (in a definite sense) and of length equal to \(PS\). Show that \(Q\) describes a second parabola whose axis passes through a fixed point independent of the direction chosen.
The coordinates of two points \(P(p',0,p)\) and \(Q(q',q,0)\) on the sides \(y=0\) and \(z=0\) of the triangle of reference are connected by the homographic relation \[ apq + bpq' + cp'q + dp'q' = 0, \] where \(a, b, c, d\) are constants and \(ad \neq bc\). Show that the line \(PQ\) in general envelops a conic \(\Sigma\) touching \(y=0\) and \(z=0\), and find the tangential equation of \(\Sigma\). \par What are the points of contact of \(\Sigma\) with \(y=0\), \(z=0\), and what is the condition that \(\Sigma\) should degenerate into a pair of points? \par From a point \(P\) on a given straight line \(l\) in general position, the line \(\lambda\) is drawn perpendicular to the polar of \(P\) with respect to a conic \(S\). What is the envelope of \(\lambda\) as \(P\) moves on \(l\)? \par Discuss the particular case where \(l\) is parallel to a principal axis of \(S\).
A uniform rod, of length \(2l\), passes through a small smooth ring, and its lower end is attached by a light inextensible string, of length \(a\), to a point at a depth \(h\) vertically below the ring. Show that the position of equilibrium in which the rod is vertical is stable or unstable according as \(l\) is less than or greater than \((h+a)^2/a\). \par Investigate the critical case \(l=(h+a)^2/a\).
Two uniform circular cylinders, each of weight \(W\) and radius \(a\), rest in contact, with their axes horizontal, on a plane inclined at an angle \(\theta\) (less than \(\pi/4\)) to the horizontal. All the surfaces have the same coefficient of friction \(\mu\) (less than 1). A couple, of moment \(2Wa\sin\theta\), is applied to the lower cylinder, tending to roll the cylinders up the plane. Show that equilibrium is possible if \(\tan\theta < 4\mu\). \par If the moment of the couple is gradually increased, show that equilibrium will be broken by the cylinders rolling up the plane, provided that \[ \tan\theta < \frac{\mu(1-\mu)}{2-\mu+\mu^2}. \]
A particle of mass \(m\) is attached to one end \(B\) of a light elastic string \(AB\), the other end \(A\) being fixed. When the particle hangs in equilibrium the length of the string exceeds the unstretched length by \(a\). Prove that if the particle executes small vertical oscillations about the position of equilibrium, the period is the same as that of a simple pendulum of length \(a\). \par A similar string \(BC\), carrying a particle of mass \(\frac{1}{2}m\) at \(C\), is now attached to the first particle at \(B\), and the system hangs in equilibrium from \(A\). If now the particles oscillate vertically, the downward displacements of \(B\) and \(C\) from their equilibrium positions being \(x\) and \(y\), establish the equations of motion \[ \ddot{x} = n^2(y-2x), \quad \ddot{y} = n^2(x-y), \] where \(n^2=g/a\). It is assumed that the strings remain taut. \par Prove that a motion is possible in which \(x/2 = y/3\) throughout.
A particle moves in a plane under an attraction \(n^2 r\) per unit mass towards a fixed point \(O\), where \(r\) denotes distance from \(O\). The particle is projected from a point \(P\), distant \(c\) from \(O\), with velocity \(nb\). Show that the path of the particle is an ellipse. \par Let \(Q\) be a point of this ellipse at which the tangent is perpendicular to the tangent at \(P\). Show that at \(Q\) the ellipse touches a second ellipse, which has \(O\) as centre, \(P\) as one focus, and major axis \(2a\), where \(a^2 = b^2+c^2\). Hence determine what part of the plane is accessible by projection from \(P\) in different directions with the given velocity \(nb\).