10273 problems found
A regular pentagon ABCDE is formed of five uniform heavy rods each of weight \(w\) smoothly jointed at their ends. It is suspended from A and its form is maintained by two light struts BD and CE. Prove that the thrust in each strut is \(\frac{1}{2}w \cot 18^\circ \operatorname{cosec} 18^\circ\).
Three equal particles attract one another so that the potential energy between two of the particles at a distance \(r\) apart is \(-Ar^{-m}+Br^{-n}\), where A, B, m and n are positive constants, and \(m
A particle is projected vertically upwards in air, which produces a resistance \(g\mu^2 v^2\) per unit mass, where \(v\) is the velocity of the particle. Find the maximum height attained, and show that the particle returns to its initial position after a time \[ \frac{1}{\mu g}[\tan^{-1}\mu u + \log\{\mu u + \sqrt{(1+\mu^2 u^2)}\}], \] where \(u\) is the initial velocity.
A particle rests in equilibrium on the outer surface of a rough uniform cylindrical shell of radius \(a\), which is free to turn about its axis, which is horizontal. The particle and the shell have equal masses, and the coefficient of friction is \(\mu\). The equilibrium is disturbed by giving the cylinder a small angular velocity. Show that, if the particle slips, it does so when the cylinder has turned through an angle \(\theta\) given by \(4\mu\cos\theta - \sin\theta = 2\mu\). \par Investigate whether the particle can leave the surface before slipping occurs.
Express \[ \frac{3x^2+1}{(x-1)^3(x^2+2)(x-3)} \] in terms of partial fractions, and expand as a series of increasing powers of \(x\), stating the coefficients of \(x^{2n}\) and \(x^{2n+1}\).
Find the limitations on the value of \(a\), in order that \(\dfrac{x^2+4x-5}{x^2+2x+a}\) may take every real value for real values of \(x\). \par Determine the restriction on the values of the function for all real values of \(x\), when \(a\) does not satisfy the limitations.
Shew how to find the equation whose roots are the squares of those of a given algebraic equation. \par Hence or otherwise find the equation whose roots are \[ (b+c+d-a)^2, (c+d+a-b)^2, (d+a+b-c)^2, (a+b+c-d)^2, \] where \(a,b,c,d\) are the roots of the equation \(x^4+px^3+qx+r=0\). \par Verify for the case \(p=r=0\), that the equation obtained is \(x^4-64q^2x=0\).
By the method of differences, or otherwise, shew that the series \[ 1+5+15+35+70+126+\dots, \] can be represented as a series whose \(n\)th term is a polynomial in \(n\) with rational coefficients. Find the \(n\)th term and the appropriate scale of relation.
If \(x>0\), prove that \((x-1)^2\) is not less than \(x(\log x)^2\). \par Discuss the general behaviour of the function \((\log x)^{-1}-(x-1)^{-1}\) for positive values of \(x\) and with special reference to \(x=1\). \par Sketch the graph of the function.
Prove that, if \(u\) and \(v\) are functions of \(x\) and \(y\) such that \(\dfrac{\partial u}{\partial x}\dfrac{\partial v}{\partial y}-\dfrac{\partial u}{\partial y}\dfrac{\partial v}{\partial x}=0\), then \(v\) may be expressed in terms of \(u\) alone. \par Verify that this condition is satisfied in the case \(u=\dfrac{x-y}{1+xy}, v=\dfrac{(x-y)(1+xy)}{(1+x^2)(1+y^2)}\), and express \(v\) in terms of \(u\).