Problems

Classify the values of \(a, b\) such that the three equations \begin{align*} 5x + ay - 5z &= 3, \\ 4x + 4y - 7z &= b, \\ -3x + y + 4z &= -2, \end{align*} shall have (i) a unique solution, (ii) no solution, (iii) an infinite number of solutions.

Explain what is meant by ``\(a_n \to a\) as \(n \to \infty\).'' Prove that, if \(a_n \to a\), \(b_n \to b\) as \(n \to \infty\), then

1. [(i)] \(a_n b_n \to ab\),
2. [(ii)] \(\frac{a_n b_{2n} + a_{n+1} b_{2n-1} + \dots + a_{2n} b_n}{n} \to ab\).

The function \(f(x)\) is differentiable and satisfies the functional equation \[ f(x)+f(y) = f\left(\frac{x+y}{1-xy}\right) \text{ for } xy < 1, \] and \(f'(0)=1\). Without assuming any properties of the trigonometric functions, shew that:

1. [(i)] \(f(0)=0\),
2. [(ii)] \(f(-x)=-f(x)\),
3. [(iii)] \(f'(x) = \frac{1}{1+x^2}\),
4. [(iv)] \(f(x)+f(\frac{1}{x})\) is constant, if \(x>0\),
5. [(v)] \(f(x)\) tends to a limit as \(x \to \infty\).

Prove that the geometric mean of \(n\) positive numbers is less than or equal to their arithmetic mean. Shew that, if the equation \[ x^n - nb_1 x^{n-1} + \dots + \frac{n(n-1)\dots(n-r+1)}{r!}(-b_r)^r x^{n-r} + \dots + (-b_n)^n = 0, \] where \(b_1, b_2, \dots, b_n\) are all real and greater than zero, has \(n\) real roots, then \(b_r \le b_{r-1}\) for \(r=1,2,\dots,n-1\).

A conic S may be defined as the locus of intersections of corresponding rays of two coplanar related (homographic) pencils of lines; find the relation between the two pencils, when S is (a) a circle, (b) a rectangular hyperbola, (c) a pair of straight lines.

1. [(i)] Two fixed tangents to a conic meet at O and a variable tangent to the conic meets them at P, Q; prove that the locus of the circumcentre of the triangle OPQ is a hyperbola and find the directions of the asymptotes of this hyperbola.
2. [(ii)] T, T' are the points of contact of the tangents from a point O to a conic, whose foci are H, H'; prove that the conic through O, T, T', H, H' passes through the point of intersection of the lines through H, H' perpendicular to OH, OH' respectively.

Prove that by a suitable choice of homogeneous coordinates \((x, y, z)\) the equation of any conic through the quadrangle A, B, C, D can be written in the form \[ ax^2+by^2+cz^2=0, \] where \(a+b+c=0\).

1. [(i)] Find the equation of the locus of the point P, if the pairs of lines PB, PC and PA, PD are harmonically conjugate.
2. [(ii)] If each pair of opposite sides of the quadrangle are conjugate lines with respect to the conic \(px^2+qy^2+rz^2=0\), find \(p:q:r\).
3. [(iii)] Find the locus of points whose polar lines with respect to \(ax^2+by^2+cz^2=0\) and \((b-c)x^2+(c-a)y^2+(a-b)z^2=0\) are conjugate lines with respect to \[ a(b-c)x^2+b(c-a)y^2+c(a-b)z^2=0. \]

Prove that, if three forces are in equilibrium, their lines of action are in one plane and either meet in a point or are parallel. Three light rods are freely jointed together, not necessarily at their ends, so as to form a triangle ABC. Forces are applied to the rods BC, CA and AB at points P, Q and R respectively, their lines of action meeting in the point O, and the system is in equilibrium. Indicate how to determine the stresses at the joints. Prove that, if P, Q, R are collinear, the lines of action of these stresses are along OA, OB, OC.

A uniform cube of weight \(W\) and edge \(a\) is placed upon a rough plane, and a uniform sphere of weight \(w\) and diameter \(a\) rests upon the plane, touching the cube at the centre of one of its faces. The plane is gradually tilted from a horizontal position about a line lying in the plane, and parallel to the face of contact of the cube with the sphere so that the sphere is above the cube. Shew that, if \(\mu\) is the coefficient of friction at all contacts and \(\mu<1\), equilibrium will be broken by the cube slipping and the sphere rolling down the plane. Find the angle of inclination of the plane to the horizontal when this occurs.

Investigate the oblique impact of two smooth elastic spheres, of masses \(m, m'\), proving that the impulsive pressure between the spheres is \[ (1+e)\frac{mm'}{m+m'}(U-U'), \] where \(U, U'\) are the resolved parts of the velocities before impact along the line of centres. Find the kinetic energy lost in the collision. Two equal circular discs of radius \(a\) lie in contact on a smooth table and are struck simultaneously by a circular disc of radius \(c\) moving perpendicularly to the line of centres of the first two discs. Prove that, if all the discs are uniform solids of the same material and the same thickness, the impinging disc will be reduced to rest, if the coefficient of restitution is \[ \frac{1}{2} \frac{c(a+c)^2}{a^2(2a+c)}. \]

The lower end of a uniform rod of length \(a\) slides on a light smooth inextensible string of length \(2a\) whose ends are fixed to two points distant \(2\sqrt{(a^2-b^2)}\) apart in a horizontal line, and the upper end of the rod slides on a fixed smooth vertical rod which bisects the line joining the two fixed points. If the rod makes an angle \(\theta\) with the vertical at any time, find expressions for the kinetic and potential energy in terms of \(\theta, \dot{\theta}\). Prove that, if \(2b>a\), the time of a small oscillation about the vertical position of equilibrium is \(2\pi/\mu\), where \[ \mu^2 = \frac{3g(2b-a)}{2a^2}. \]