Trace the curve \[ y = x \pm \sqrt{\{x(x-1)(2-x)\}}. \]
Circles are drawn with their centres on the circle \(x^2+y^2=1\) and touching the axis of \(y\). Shew that the axis of \(y\) and the curve \[ 4(x^2+y^2-1)^2 = 27x^2 \] form the envelope of the system of circles; and trace the curve.
A heavy particle slides down a smooth vertical circle of radius \(R\) from rest at the highest point. Shew that on leaving the circle it moves in a parabola whose latus rectum is \(\frac{16}{27}R\).
Differentiate \(\cos x\) from first principles. Differentiate \[ \sin^{-1}\left[\frac{2\sqrt{\{(\alpha-x)(x-\beta)\}}}{\alpha-\beta}\right]. \]
Under the action of constant tractive effort \(P\) by the engine, a train of total mass \(m\) starting from rest at \(A\) attains its maximum speed \(V\); the pull of the engine is then reduced so that for a time the speed is maintained at its value \(V\), after which the steam is shut off and the brakes applied, bringing the train to rest at the point \(B\). The distance \(AB\) is \(l\), the time of run between \(A\) and \(B\) is \(\frac{4l}{3V}\), the rail resistance is \(\frac{3}{5}\frac{mV^2}{l}\), and the brake resistance is \(\frac{12}{5}\frac{mV^2}{l}\), both these being independent of the speed. Prove that \[ P = \frac{18}{5}\frac{mV^3}{l}. \]
Show that the function \[ y = ax^2 + 2bx + c + A \cos mx + B \sin mx \] satisfies an equation of the form \[ \frac{d^4y}{dx^4} + k^2 \frac{d^2y}{dx^2} = 0, \] determining the value of \(k\).
Calculate \[ \int (x \cos x)^2 dx, \quad \int x \log x dx, \quad \int_0^\pi \frac{dx}{13+5 \cos x}. \]
Show that the surface generated by the revolution of the cardioid \[ r = a(1-\cos\theta) \] about the line \(\theta=0\) is \(\displaystyle\frac{32}{5}\pi a^2\).
Three smooth heavy cylinders \(A, B, C\) lie on a table, with \(B\) between \(A\) and \(C\) and touching each of them. \(A\) and \(C\) have equal radii \(a\) and \(B\) has weight \(W\) and radius \(b(
Prove that couples in one plane and of equal and opposite moment are in equilibrium. The ends of a rod are constrained by smooth rings to slide on two horizontal fixed rods which meet at \(A\). To the centre of the rod is attached a string which passes over a smooth pulley at \(A\) and is fastened to a weight which hangs freely. A couple is applied to the rod in a horizontal plane. Show that in any position of equilibrium of the rod the couple applied is proportional to the cosine of the angle which the string makes with the rod.