Problems

Explain the method of finding positive integral values of \(x\) and \(y\) which satisfy the equation \(ax+by=c\), where \(a, b, c\) are positive integers. Illustrate your answer by solving in positive integers the equation \(7x+11y=514\). Find the number of positive integral or zero values of \(x, y, z, w\) satisfying the equation \(x+y+2z+2w=2n\), where \(n\) is a positive integer.

Find the sum to \(n\) terms of the series:

1. [(i)] \(\sin^2\alpha+\sin^2(\alpha+\beta)+\sin^2(\alpha+2\beta)+\dots+\sin^2(\alpha+\overline{n-1}\beta)\);
2. [(ii)] \(\tan\theta+\tan(\theta+\frac{\pi}{n})+\tan(\theta+\frac{2\pi}{n})+\dots+\tan(\theta+\frac{\overline{n-1}\pi}{n})\);
3. [(iii)] \(\cot\theta+\cot(\theta+\frac{\pi}{n})+\cot(\theta+\frac{2\pi}{n})+\dots+\cot(\theta+\frac{\overline{n-1}\pi}{n})\);
4. [(iv)] \(\sec^2\theta+\sec^2(\theta+\frac{\pi}{n})+\sec^2(\theta+\frac{2\pi}{n})+\dots+\sec^2(\theta+\frac{\overline{n-1}\pi}{n})\);

distinguishing where necessary between the cases of \(n\) odd and \(n\) even.

Define a coaxal system of circles and its limiting points. Given a coaxal system of circles \(S\), prove that there is a second coaxal system \(S'\) such that the circles of one system intersect those of the other system orthogonally. Any circle of the system \(S\) is considered with an equal circle of the system \(S'\) and the radical axis of these two circles found. Prove that the envelope of the radical axes of such pairs of circles is a rectangular hyperbola.

Find the equation of the diameters of the conic \(ax^2+by^2=1\) which pass through the points of intersection of the conic and the line \(lx+my+n=0\). Find conditions such that

1. [(a)] the given line shall touch the conic,
2. [(b)] the two diameters shall be perpendicular,
3. [(c)] the two diameters shall be conjugate.

Neglecting \(x^5\) and higher powers of \(x\), obtain by the use of Maclaurin's theorem or otherwise the expansions in series of powers of \(x\) of \(\log_e(1+\sqrt{1+x^2})\) and \(\sin(x\sqrt{1+x^2})\). Prove also that to the same degree of accuracy \[ \log_e\frac{x\sin x}{1-\cos x} = \log_e 2 - \frac{x^2}{12} - \frac{7x^4}{1440}. \]

Each generator of a cylinder touches a sphere of radius \(a\). Two planes are taken perpendicular to the generators of the cylinder and intersecting the sphere. Prove that the area of the zone of the sphere between the two planes is equal to the area of the portion of the cylinder between the planes. \(s\) is the distance of any point from a given point \(P\) distant \(f\) from the centre of the sphere. Prove that the mean value of the reciprocal of \(s\) over the surface of the sphere is \(\frac{1}{f}\) or \(\frac{1}{a}\) according as to whether \(f\) is greater or is less than \(a\).

A uniform lamina of mass \(m\) is bounded by an arc of a parabola of latus rectum \(4a\) and by a chord which is perpendicular to the axis of the parabola and at a distance \(na\) from the vertex. The plane of the lamina is vertical and the lamina can roll with its curved edge in contact with a horizontal plane. A couple \(mg a \lambda\) acts on the lamina in its own plane when its axis makes an angle \(\theta\) with the vertical, the sense of the couple being such as to tend to diminish \(\theta\). Find the potential energy of the system when the axis makes an angle \(\theta\) with the vertical. Show that a necessary condition for oblique positions of equilibrium to be possible is that \(n\) shall be greater than 5.

Hanging over a smooth pulley are two scale pans \(A\) and \(B\). \(A\) is of mass \(m\), and in it is an insect of mass \((n-1)m\). \(B\) is of mass \(2m\), and in it is an insect of mass \((n-2)m\). When the system is in equilibrium the insect in \(B\) jumps vertically with velocity \(c\) relative to the pan. Find the initial velocity of the insect and also the time that elapses before the insect returns to the pan, and prove that the system is then in its initial position.

Establish the principal properties of a compound pendulum. A thin uniform rod of length \(2a\) and mass \(m\) is suspended from one end and a mass \(M\) is rigidly fastened to the other end. If the length of the simple equivalent pendulum is \(\frac{5a}{3}\), find \(M\) in terms of \(m\).

Find the necessary and sufficient conditions that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] represent (i) two straight lines, (ii) two coincident straight lines. Prove that in general three of the conics represented by the equation \[ ax^2+2hxy+by^2+2gx+2fy+c+\lambda(a'x^2+2h'xy+b'y^2+2g'x+2f'y+c') = 0, \] for varying values of \(\lambda\), are line-pairs. State geometrically the conditions that more than three of the conics be line-pairs.