A uniform rod of length \(2l\) and mass \(M\) is gently disturbed from its position of equilibrium in a vertical position with one end resting on a smooth horizontal table. Prove that the angular velocity \(\dot{\theta}\) of the rod when it makes an angle \(\theta\) with the vertical is given by \[ \dot{\theta}^2 = \frac{6g(1-\cos\theta)}{l(1+3\sin^2\theta)}. \] Find the reaction of the table on the rod at the same instant.
Three roots of the quartic equation \[ (x^2+1)^2 = ax(1-x^2)+b(1-x^4) \] satisfy the equation \[ x^3+px^2+qx+r=0. \] Prove that \[ p^2-q^2-r^2+1=0. \]
Prove that the arithmetic mean of a set of positive numbers cannot be less than their geometric mean. If \(x,y\) are positive numbers and \(m,n\) are positive integers prove that \[ \frac{x^m y^n}{(x+y)^{m+n}} \le \frac{m^m n^n}{(m+n)^{m+n}}. \]
Prove that the coefficient of \(x^{2n}\) in the expansion of \((1+x^2)^n(1-x)^{-4}\) in ascending powers of \(x\) is \[ \frac{1}{3}(n+2)(n^2+7n+3) \cdot 2^{n-1}. \]
Find the number of different arrangements of \(n\) different articles in \(m\) different pigeon-holes. An event happens irregularly but in the long run occurs once a year on an average. Show that the chance that it will not take place in a particular future year is \(1/e\).
Two functions of \(x\), \(f(x)\) and \(\phi(x)\), have the following properties for all real values of \(x\): \(f(-x)=f(x)\), \(f'(x)=\phi(x)\), \(\phi'(x)=f(x)\). Deduce that \(\phi(-x)=-\phi(x)\) for all real values of \(x\). If it is further given that \(f(x+y)=f(x)f(y)+\phi(x)\phi(y)\) for all real values of \(x\) and \(y\), and that \(f(0)=1, \phi(0)=0\), deduce that \(\phi(x+y)=\phi(x)f(y)+\phi(y)f(x)\) for all real values of \(x\) and \(y\). Find the differential equation satisfied by \(f(x)\) and \(\phi(x)\), and obtain explicit forms for them.
If for \(q>1\), \(I(p,q)\) denote \(\int_0^\pi e^{px}\sin^q x \,dx\), derive the reduction formula \[ (p^2+q^2)I(p,q) = q(q-1)I(p, q-2). \] Hence show that for a positive even integral value of \(q\), \[ I(p,q) = q!(e^{p\pi}-1)/p \cdot (p^2+4)(p^2+16)\dots(p^2+q^2). \] Find the corresponding result when \(q\) is an odd integer greater than unity.
Show how, by graphical means, a general indication of the position of the real roots of the equation \(x\cos x = 1\) may be determined, and obtain an approximate value for that one of them lying nearest to \(3\pi/2\).
Prove that for an algebraic equation \(f(x)=0\), there can at most be only one real root in a range of values of \(x\) not containing any real root of the derived equation \(f'(x)=0\). Consider the equation \(3x^5-25x^3+60x+k=0\) for different real values of \(k\), and prove that it cannot have more than three real roots, and that it will have more than one real root only if \(16 \le |k| \le 38\).
If for the segment of a sphere intercepted by a plane, \(\lambda\) denotes the ratio of the area of the curved part of the surface of the segment to that of the whole sphere, and \(\mu\) denotes the ratio of the volume of the segment to that of the whole sphere, prove that \(\mu=\lambda^2(3-2\lambda)\). If \(\lambda'\) denote the ratio of the area of the total surface of the segment to that of the sphere, show that \((\lambda'-\mu)^2=4(\lambda'-1)(\mu-\lambda')\).