\(ABC\) is a triangle. Shew that the increases in area resulting from small increases \(\delta a, \delta b, \delta c\) in the sides \(a, b, c\) respectively, the other two sides in each case remaining unchanged, are proportional to \(\cos A \delta a, \cos B \delta b, \cos C \delta c\).
Shew that \[ \frac{d^n}{dx^n} (\tan^{-1} x) = (-1)^{n-1} (n-1)! r^{-n} \sin n\phi, \] where \[ r \cos \phi = x, \quad r \sin \phi = 1. \]
A fine smooth wire of mass \(M\) forms an equilateral triangle \(ABC\). The triangle can move horizontally in a vertical plane, the uppermost side \(BC\) passing through smooth fixed rings in a horizontal line. Beads of masses \(m\) and \(m'\) are free to slide on the wires \(BA, CA\). The system begins to move with the beads at \(B\) and \(C\) respectively. Prove that the velocity of the wire at any instant, while both the beads are moving on it, is equal to the difference of the speeds of the beads relative to the wire, and that the acceleration of the wire is \[ \sqrt{3}(m' \sim m) g / (4M+3m+3m'). \]
Trace the curve given by \(ax^2y = x^2 + y + 1\).
A point \(P\) has polar co-ordinates connected by the relation \[ \theta = \int \frac{\sqrt{a(1-e^2)} dr/r}{ \sqrt{ a(1-e) \sqrt{-a(1-e^2) + 2r - r^2/a} }}, \] where \(a > 0, e^2 < 1\). Shew that \(P\) lies on an ellipse having a focus at the origin and \(a, e\) for its semi-major axis and eccentricity respectively. Shew further that \[ \int \frac{rdr}{\sqrt{a(1-e)\sqrt{-a^2(1-e^2) + 2a^3 r - a^2 r^2}}} = \phi - e \sin \phi, \] where \(\phi\) is the eccentric angle of the point \((r, \theta)\) of the ellipse.
State the principle of the conservation of angular momentum of a system about a fixed axis. A flywheel of moment of inertia \(I\) is rotating with angular velocity \(\Omega\) about a vertical axis. The flywheel contains a pocket at a distance \(a\) from the axis into which is dropped a sphere of mass \(M\), moment of inertia \(i\) and spin \(\omega\) about a vertical axis, without horizontal motion. Find the angular velocity of the system after the sphere has come to relative rest in the pocket.
Evaluate \(\displaystyle \int \frac{x^3 dx}{(x+1)(x^2+1)}\), \(\displaystyle \int_0^{\pi} \sin^n \theta d\theta\), \(\displaystyle \int (x+a) \log (x+b) dx\).
Obtain a reduction formula for \[ \int \frac{x^n dx}{\sqrt{(ax^2+2bx+c)}}. \] Shew that \[ \int_0^1 \frac{x^4+x^2+1}{\sqrt{(x^2+1)}} dx = 3\sqrt{2}/8 + (7/8) \log_e(\sqrt{2}+1). \]
Find the area of a loop of the curve \[ r^2 = a^2 (\sin 2\theta + 2 \sin \theta). \]
Shew that a system of forces acting in one plane on a rigid body can be reduced to a force through a given point and a force in a given line not passing through the point. Shew further that the reduction is unique. Forces 1, 2, 3, 4 act in the sides \(AB, BC, CD, DA\) of a square \(ABCD\). Reduce the system to a force through \(A\) and a force in \(BC\).