Prove that \[ 2^{n-1} \sin\frac{\pi}{n} \sin\frac{2\pi}{n} \dots \sin\frac{(n-1)\pi}{n} = n. \] Hence or otherwise prove that \(\displaystyle\int_0^\pi \log \sin x dx = -\pi \log 2\).
An engine of weight \(W\) tons can exert a maximum tractive effort of \(P\) tons weight and develop at most \(H\) horse-power. The resistances to motion are constant and equal to \(R\) tons weight. Show that starting from rest the engine will first develop its full horse-power when its velocity is \(\dfrac{55H}{224P}\) f.s. after at least \(\dfrac{55WH}{224Pg(P-R)}\) seconds. What is the greatest velocity which the engine can attain?
\(A\) is a variable point \((X,0)\), \(P\) and \(Q\) are points \((h, k)\), \((h', k')\), respectively. Show that \(AP+AQ\) and \(AP-AQ\) have stationary values if \(X\) has one or other of the values \[ \frac{hk'-h'k}{k'-k}, \quad \frac{hk'+h'k}{k'+k}. \] Interpret these results according as \(P\) and \(Q\) are on the same or opposite sides of the axis of \(x\).
Find the asymptotes of the curve \[ (x+y-1)^3 = x^3+y^3, \] and show that they meet the curve only at infinity. Show also that there is no point on the curve for which \(x\) lies between \(a\) and \(1\), where \(a\) is the real root of the equation \[ 3a^3+3a^2-3a+1=0. \] Give a sketch of the curve showing its general form.
A smooth straight tube rotates in a horizontal plane about a point in itself with uniform angular velocity \(\omega\). At time \(t=0\) a particle is inside the tube, at rest relatively to the tube, and a distance \(a\) from the point of rotation. Show that at time \(t\) the distance of the particle from the point of rotation is \[ a \cosh(\omega t). \] Find the force the tube is then exerting on the particle.
Evaluate \(\displaystyle\int \frac{xdx}{\sqrt{(5+2x+x^2)}}\), \(\displaystyle\int_0^1 x^2 \tan^{-1} x dx\), \(\displaystyle\int_0^{\frac{1}{2}\pi} \sin^6 x dx\).
If \(A\) is the area bounded by the curve \(r=f(\theta)\) and the straight lines \(\theta = \theta_1\), \(\theta = \theta_2\), show that the cartesian coordinates of the centroid of the area are \(\xi, \eta\), where \[ A\xi = \frac{1}{3} \int_{\theta_1}^{\theta_2} r^3 \cos \theta d\theta, \quad A\eta = \frac{1}{3} \int_{\theta_1}^{\theta_2} r^3 \sin \theta d\theta. \]
A number of rods are freely-jointed together at the ends to form a convex polygon, and each corner is joined to an internal point \(O\) by means of a rod. Show that if the rods are in a state of stress, the figure obtained by reciprocating the polygon with regard to a circle whose centre is at \(O\), and joining each vertex of the reciprocal polygon to \(O\), is a stress diagram for the frame. If the frame is as shown, \(ABCDEF\) being a regular hexagon, show that the tension in \(OD\) is \(\frac{2}{3}\) that in \(OA\).
Two equal uniform ladders of weight \(w\) are rigidly fastened together at one end to form a step ladder, which stands on a plank. The angle between the ladders is \(2\alpha\). A man of weight \(W\) stands on the top. The plank is slowly tilted about one end. Show that if \(\beta\) be the inclination of the plank to the horizontal, the system will overturn if \(\tan \beta\) reaches \(\dfrac{W+2w}{W+w}\tan\alpha\), and will slip if \(\tan \beta\) reaches \(\mu\), the coefficient of friction.
State the principle of virtual work. The ends of a uniform rod \(AB\) of length \(2l\) and weight \(w\) rest on two smooth planes inclined at \(45^\circ\) to the horizontal and with their line of intersection horizontal. A light rod \(BC\) of length \(h\) is fixed at \(B\) so that the angle \(ABC\) is \(\beta\), \(BC\) being above \(AB\). A weight \(W\) is applied at \(C\), and the frame \(ABC\) remains always in a vertical plane normal to both of the smooth planes. If the inclination of \(AB\) to the horizon is \(\theta\), show that in the position of equilibrium \[ \tan\theta = \frac{W(h \cos\beta - l)}{wl + Wl + Wh \sin\beta}. \]