Problems

Prove that the acceleration towards the centre of a particle moving in a circle is \(v^2/r\). Two particles describe two circles in a plane uniformly in the same time. Prove that the acceleration of one relative to the other is constant in magnitude and changes its direction uniformly.

A particle is moving in a straight line under a force to a fixed point in the line proportional to the distance from the point. Prove that the motion is simple harmonic and find the period. Two light elastic strings of natural lengths \(l, l'\) and moduli \(E, E'\) respectively are knotted together to form one string, one end of which is fixed while the other is attached to a particle of mass \(m\) which oscillates freely in a vertical line under the action of gravity and the tension of the string. Prove that the period of an oscillation is the same as that of a simple pendulum of length \(mg(l/E+l'/E')\).

Solve the equations

1. [(i)] \(x(y+z) = y(z+x) = z(x+y) = a^2\),
2. [(ii)] \(x+y+z=2, \quad x^2+y^2+z^2=26, \quad x^3+y^3+z^3=38\).

Find the conditions that

1. [(i)] \(ax^2+2bxy+cy^2\).
2. [(ii)] \(ax^2+2hxy+by^2+2gx+2fy+c\)

should be positive for all real values of \(x\) and \(y\). Find the condition that \((x-a)/(x-b)(x-c)\) may be capable of taking all real values.

Prove that the arithmetic mean of \(n\) positive quantities is not less than their geometric mean. Show that the sum of a series of \(n\) positive terms forming a harmonical progression is not less than \(n\) times the harmonic mean between the first and last terms.

Obtain the expansion of \(\log_e(1+x)\) from the exponential theorem. Prove that the sum to infinity of the series \[ \frac{1}{1(p+1)} + \frac{1}{2(p+2)} + \frac{1}{3(p+3)} + \dots \] is \[ \frac{1}{p}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{p}\right). \]

Prove the law of formation of the successive convergents of the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+} \dots. \] Prove that \[ \frac{1}{2+} \frac{2}{3+} \frac{3}{4+} \dots \text{ to infinity} \] is equal to \(\frac{3-e}{e-2}\).

Find from first principles the differential coefficient of \(\cos^{-1}x\). Find the \(n\)th differential coefficients of \(1/(1+x^2)\) and \(\sin^3 x\).

The radius \(R\) of the circumcircle of the triangle \(ABC\) is expressed in terms of \(a,b\) and \(C\); find \(\frac{\partial R}{\partial a}\) and prove that \[ \frac{\partial R}{\partial C} = R \frac{\cos A \cos B}{\sin C}. \]

Prove that if \(\phi\) is the angle between the radius vector and the tangent at any point of a curve \(\tan\phi = r \frac{d\theta}{dr}\). Find \(\phi\) at the point \((r, \theta)\) on the curve \(r^2=a^2\cos 2\theta\) and prove that if \(p\) is the perpendicular from the origin on the tangent at the point \(pa^2=r^3\). Find also the equation of the first positive pedal.