Find the asymptotes of the curve \[ x(x+y)^2 = 2(5x-3y), \] and trace the curve.
A light spiral spring is fixed at its lower end with its axis vertical; a mass, which would compress the spring a distance \(d\) when at rest on it, is dropped on the spring from a height \(h\): show that it will be shot off on the rebound after remaining on the spring for a time \[ \sqrt{\frac{d}{g}}\left\{\pi + 2 \tan^{-1}\sqrt{\frac{d}{2h}}\right\}. \]
Shew that, if the base \(AB\) of a triangle \(ABC\) is fixed and the vertex \(C\) moves along the arc of a circle of which \(AB\) is a chord, then \[ \frac{da}{\cos A} + \frac{db}{\cos B} = 0. \]
If \(n\) is a positive integer, and \[ I_n = \int_0^\infty \frac{dx}{(1+x^2)^n}, \] find a reduction formula for \(I_n\) in terms of \(I_{n-1}\) and elementary functions. Prove that \[ \int_{-\infty}^\infty \frac{dx}{(1+x^2)^{n+1}} = \frac{1 \cdot 3 \dots (2n-1)}{2 \cdot 4 \dots 2n}\pi. \]
A light rigid rod has particles each of mass \(m\) attached at \(A, B\) and \(C\), where \(AB = a, BC=b\). A blow \(P\) perpendicular to the rod is applied at the middle point of \(AC\): show that the angular velocity acquired is \(P \frac{|a-b|}{4m a^2 + ab + b^2}\).
Find \(\int \frac{x^2 dx}{x^2-x-2}\), \(\int e^{ax}(a \sin x + b \cos x)dx\), \(\int x^a \log x dx\).
Shew that the area, contained by the straight lines \(\theta = 0\), \(\theta = \frac{\pi}{3}\) and the part of the curve \(r = a \cos\frac{\theta}{2}\) for which \(\theta\) lies between 0 and \(\frac{\pi}{3}\), is \(\frac{a^2}{4}\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\).
The figure represents the main part of the framework of a folding step-ladder. The bar \(AB\) which carries the steps is 6 ft. long. The top end of the bar \(CDE\) slides along \(AB\) without friction from \(A\) to a stop at \(E\), which is 1 ft. from \(A\). The point \(D\) is connected to \(B\) by a rod 4 ft. long, freely hinged at \(B\) and \(D\) to \(AB\) and \(CE\) respectively. \(CD\) is 4 ft. and \(DE\) is 1 ft. 6 ins. A man of weight 160 lbs. ascends to the top of the steps, his centre of gravity being then vertically above \(A\). Assuming that there is no friction with the ground at \(C\), find (i) the pressure on the ground at \(C\), (ii) the normal pressure on \(AB\) at \(E\), (iii) the pressure on the stop along \(EB\), (iv) the tension in the bar \(BD\).
A uniform solid cube of edge \(2c\) rests on two parallel horizontal bars placed under one face parallel to edges of that face at distances \(b\) from the centre of it. The plane containing the bars makes an angle \(\theta\) with the horizontal. Shew that, if equilibrium exists and \(b > \mu c\), then \(\tan\theta < \frac{(\mu' + \mu)b}{2b+(\mu'-\mu)c}\), where \(\mu, \mu'\) are the coefficients of friction between the cube and the lower and upper rails respectively.
A chimney of brickwork 18 in. thick has an external diameter of 13 ft. at the base, and 9 ft. at the top, which is 100 ft. above the base. Shew that the centre of gravity of the chimney is about 3.5 ft. below the middle point of the axis.