A uniform flexible chain of length \(6l\) hangs in equilibrium over two small smooth pegs at the same horizontal level with a length \(2l\) of chain between the pegs. Prove that the distance between the pegs must be \((\frac{4}{3}\log 3)l\), and find the depth of the free ends of the chain below its mid-point.
A particle hangs in equilibrium from the ceiling of a stationary lift, to which it is attached by an elastic string (of natural length \(l\)) extended to length \(l+a\). The lift descends, moving for time \(T\) with constant acceleration \(f\) and subsequently with constant velocity. Prove that, if \(f < g\), the string never becomes slack, and show that the amplitudes of the oscillations before and after time \(T\) are respectively \(af/g\) and \(2af|\sin\frac{1}{2}nT|/g\), where \(n^2=g/a\).
A particle is projected horizontally with speed \(\sqrt{(\lambda ag)}\), where \(0<\lambda<1\), from the highest point of a fixed smooth sphere of radius \(a\). Find the velocity of the particle at the instant it leaves the sphere. After leaving the sphere the particle describes a parabola. Find the depth of the vertex of this parabola below the point of projection.
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.