Problems

Given that the equation \[x^6 - 5x^5 + 5x^4 + 9x^3 - 14x^2 - 4x + 8 = 0\] has three coincident roots, find the value of this multiple root, and hence, or otherwise, solve the equation completely.

Find the limits of the following expressions \[\frac{x - \sin x}{x^3} \quad \text{and} \quad \frac{1 - \frac{1}{2}x^2 - \cos x}{x^4}\] as \(x \to 0\). Find also the limits of \[\frac{\cos \frac{1}{2}x}{x^2 - \pi^2}\] as \(x \to \pi\), and as \(x \to \infty\).

Establish Leibnitz' theorem for the \(n\)th derivative of a product of two functions. If \[f(x) = \frac{px + q}{ax^2 + 2bx + c},\] and \(f_n\) denotes the \(n\)th derivative of \(f(x)\), prove that \[(ax^2 + 2bx + c) f_{n+2} + 2(ax + b)(n + 2) f_{n+1} + a(n + 1)(n + 2) f_n = 0.\]

Evaluate the integrals: \[\int_0^{\pi/2} (a^2 \cos^2 \theta + b^2 \sin^2 \theta)^{-1} d\theta,\] \[\int_0^{\pi/4} (\cos \theta + \sin \theta)(9 + 32 \cos \theta \sin \theta)^{-1} d\theta,\] \[\int_1^{\infty} \frac{dx}{x^2 + x}.\]

A rigid parabola rolls without slipping on a fixed straight line. Find the locus described by its focus.

The sides \(AB\), \(BC\), \(CD\), \(DA\) of a deformable but plane quadrilateral are of fixed lengths \(a\), \(b\), \(c\), \(d\) respectively. Show that its area is greatest when the shape is such that \(A\), \(B\), \(C\), \(D\) are concyclic.

In Fig. 1, \(A\) and \(B\) are fixed points at the same level 6 in. apart, to which are hinged the stiff rods \(AD\), 5 in. long, and \(BF\), 3 in. long. \(C\) is the middle point of \(AD\) and the hinged rods \(DE\), \(EF\) and \(CF\) are each \(2\frac{1}{2}\) in. long. A weight of 10 lb. is hung from the point \(F\) and the system is maintained in equilibrium with \(BF\) horizontal by a vertical force \(P\) acting at \(E\). Determine the magnitude of \(P\) and the force in the rod \(FB\), neglecting the weights of the rods and any friction at the hinges.

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A four-wheeled trolley of weight \(w\) has wheels of radius \(r\) which can turn freely on their axles. The distance between the axles is \(2l\) and the centre of gravity is equidistant from them. The trolley is standing on level ground with the front wheels in contact with a vertical step of height \(h\) (\(< r\)). A horizontal force \(P\) is applied to the trolley at a height \(x\) above the ground, in a direction at right angles to the step, and increased until motion occurs. Show that the front or rear wheels will leave the ground first according as \(x\) is less or greater than $$r + \frac{l(r-h)}{\sqrt{2rh-h^2}}.$$ Determine the value of \(P\) which will just cause motion if \(x = r\).

A uniform circular disc of radius \(r\) has a particle, of mass \(m\), attached to it at a distance \(a\) from its axis. It is caused to roll without slipping in a vertical plane on a rough horizontal surface at a constant angular velocity \(\omega\) by means of a varying horizontal force applied through its axis. Obtain an expression for this force at the instant when the particle is at the above the horizontal.

A ship is observed from a lighthouse in a direction \(30^\circ\) east of north, and at the instant of observation this angle is found to be increasing at the rate of \(6^\circ\) a minute. Ten minutes later the ship is due east of the lighthouse. Calculate the course of the ship, assuming it to be travelling in a straight line at a uniform speed.