Two uniform ladders \(AB, AC\), of the same length and of the same weight, \(W\), are smoothly jointed at \(A\) and stand with \(B\) and \(C\) in contact with a rough horizontal plane. If a man of weight \(W\) can stand on any rung of the ladders, prove that the coefficient of friction must not be less than \(\frac{3}{2}\tan\frac{1}{2}(BAC)\). Also find the minimum coefficient of friction necessary if in all cases two men (of the same weight \(W\)) can stand one on each ladder.
Shew by partial integration that \[ f(a+b) = f(a) + bf'(a) + \dots + \frac{b^n}{n!}f^{(n)}(a) + \int_0^b \frac{(b-t)^n}{n!} f^{n+1}(a+t)dt. \] Expand \((\sin^{-1}x)^2\) in ascending powers of \(x\).
Find for what values of \(a\) and \(b\) the roots of the equation \[ x^4 - 4x^3 + ax^2 + bx - 1 = 0 \] are in arithmetical progression.
Tangents are drawn to the parabola \(y^2=4ax\) at the points whose ordinates are \(2am_1, 2am_2, 2am_3\). Find the equation of the circumcircle of the resulting triangle, and from the equation show that it passes through the focus. Show also that, whatever be the value of \(k\), the same circle circumscribes the triangle formed by the tangents at three points whose ordinates are \(2a\mu_1, 2a\mu_2, 2a\mu_3\), where \(\mu_1, \mu_2, \mu_3\) are the roots of \[ (\mu-m_1)(\mu-m_2)(\mu-m_3)+k(\mu^2+1)=0. \]
Shew that a uniform flexible chain hangs under gravity in a catenary whose Cartesian equation can be written in the form \[ y = c \cosh \frac{x}{c}. \] If the chain is of length \(l\) and hangs between two points whose horizontal and vertical distances apart are \(a, b\), respectively, prove that the parameter \(c\) of the catenary is given by \[ (l^2-b^2)^{\frac{1}{2}} = 2c\sinh\frac{a}{2c}. \]
Reduce the equation \(x^3+3px^2+3qx+r=0\) to the form \(y^3+3y+m=0\) by assuming \(x=\lambda y + \mu\); and solve this equation by assuming \(y=z-\frac{1}{z}\). Hence prove the condition for equal roots to be \[ 4(p^2-q)^3 = (2p^3-3pq+r)^2. \]
Prove that \[ \frac{1}{3\left(1 - \frac{1}{2^2}\right)} - \frac{1}{4\left(1 + \frac{1}{3^2}\right)} + \frac{1}{5\left(1 - \frac{1}{4^2}\right)} - \dots = \frac{1}{2}. \]
Show that the locus of the point of intersection of the normals at the pairs of points in which a given straight line meets the conics of a confocal system is a straight line.
A uniform beam \(AB\) of length \(l\) and weight \(w\) per unit length is smoothly hinged at \(A\), and is kept at an inclination of 45\(^\circ\) to the upward vertical through \(A\) by a light horizontal rope which joins a fixed point to the point of the beam distant \(l/3\) from \(A\). Find expressions for the thrust, shearing force and bending moment at any point of the beam.
Trace the curve \(4(x^2+2y^2-2ay)^2=x^2(x^2+2y^2)\) and find the radii of curvature of the two branches at the origin.