(i) If \(A\) and \(B\) are constants, obtain a differential equation, not involving \(A\) and \(B\), which is satisfied by \[ y = (A \sin x + B \cos x)/x. \] (ii) Evaluate: \[ \int_a^b \frac{dx}{\sqrt{(b-x)(x-a)}}, \quad \text{where } b > a; \qquad \int_0^1 x \sin^{-1} x \, dx. \]
Prove that if a circle and an ellipse cut in four points then the sum of their eccentric angles is an integral multiple of \(2\pi\). A variable chord \(PQ\) of an ellipse is such that the circle on \(PQ\) as diameter touches the ellipse at a point other than \(P\) or \(Q\). Prove that the chords \(PQ\) all touch a similar ellipse.
An elastic ring of mass \(M\), natural length \(2\pi a\), and modulus of elasticity \(\lambda\) is placed upon a rough, horizontal, steadily rotating turntable; the coefficient of friction is \(\mu\). Find the range of values of \(\omega\), the angular velocity of the turntable, for which the ring can remain on the turntable, without slipping, in the form of a circle of radius \(2a\) with its centre at the centre of rotation.
Find the \(n\)th derivative of the function \[ y = \frac{1}{x^2+c}, \] where \(c\) is a real constant, distinguishing the various cases that can arise. If \(c > 0\), prove that \[ \left| \frac{1}{n!} \frac{d^n y}{dx^n} \right| \le \frac{1}{c^{\frac{1}{2}n+1}} \] for all real values of \(x\).
If, as \(n\) tends to infinity, \(a(n)\) and \(b(n)\) tend to finite limits \(a\) and \(b\), respectively, prove that \(a(n)+b(n)\) and \(a(n)b(n)\) tend to \(a+b\) and \(ab\), respectively. If \[ f(n,m) = \frac{n+k}{n} \cdot \frac{n+1+k}{n+1} \cdot \frac{n+2+k}{n+2} \dots \frac{n+m+k}{n+m}, \] where \(k\) is a fixed positive integer, investigate the limits
Show that the feet of the normals from the point \((f, g)\) to the rectangular hyperbola \(xy=c^2\) are the intersections of the hyperbola with the rectangular hyperbola \[ x^2 - y^2 - fx + gy = 0. \] Show that, if \(fg=4c^2\), the same four points are the feet of the normals to the second hyperbola from the point \((-\frac{1}{3}f, -\frac{1}{3}g)\).
A rope of length \(\pi a\) and mass \(m\) per unit length is laid symmetrically over the upper half of the rim of a light pulley of radius \(a\), and set in motion so that the pulley rotates and the rope descends vertically as it leaves the pulley. Prove that when the pulley has turned through an angle \(\theta\) (\(<\pi\)) the tension in the rope at the upper end of the straight part is \(mag\,\theta(\pi - \theta - \sin\theta)/\pi\). [The centroid of a circular arc of radius \(a\) and length \(2a\phi\) is at a distance \((a\sin\phi)/\phi\) from the centre.]
Prove that a set of necessary and sufficient conditions for the equilibrium of a system of coplanar forces is that the sums of their resolved parts in two directions in the plane should vanish, and that the sum of their moments about one point in the plane should also vanish. Two coplanar forces \(F_1, F_2\) act at the points \((x_1,y_1), (x_2,y_2)\) referred to axes \(Ox, Oy\) in the plane of the forces; they make angles \(\theta_1, \theta_2\) with \(Ox\). Shew that the line of action of their resultant is \[ F_1 x_1 \sin\theta_1 + F_2 x_2 \sin\theta_2 - F_1 y_1 \cos\theta_1 - F_2 y_2 \cos\theta_2 \] \[ = x(F_1 \sin\theta_1 + F_2\sin\theta_2) - y(F_1\cos\theta_1 + F_2 \cos\theta_2). \] Deduce that, if \((x_1, y_1), (x_2, y_2)\) are fixed points, and \(F_1, F_2\) are fixed in magnitude but varied in direction so that \(\theta_2 - \theta_1\) is equal to a constant \(\alpha\), then the resultant passes through a fixed point. Find the coordinates of this point.
Sketch, in the same figure, the curves whose equations in polar coordinates are:
\(ABC\) is a triangle inscribed in a conic. The tangents at \(B\) and \(C\) meet at \(D\), the tangents at \(C\) and \(A\) meet at \(E\), and the tangents at \(A\) and \(B\) meet at \(F\). Prove that \(AD, BE, CF\) meet at a point. If a straight line drawn through this point meets \(BC, CA, AB\) at \(L, M, N\) respectively, prove that the triangle whose sides are \(DL, EM, FN\) is self conjugate with respect to the conic.