Find the resultant acceleration of a point which moves in any manner round a circle. \par The wheel axles of a motor car are 4 feet long and the height of the C.G. is 2 feet. Find the speed of the car if in going round a level track of 400 feet radius the inner wheels just leave the ground.
A particle moves with an acceleration towards a point equal to \(\mu \times\) distance from the point. Find the amplitude of the motion having given the velocity in any position, and find also the time of a complete oscillation. \par A weightless rod of length 3 feet, with equal heavy rings at the ends, one of which can slide on a smooth horizontal wire, is describing small oscillations in the vertical plane containing the wire. Shew that the period of oscillation is about 1.36 seconds.
Solve the equation \[ (x-1)(x+2)(x+3)(x+6)=160. \] Eliminate \(x,y,z\) from \[ x+y-z=a, \quad x^2+y^2-z^2=b^2, \quad x^3+y^3-z^3=c^3, \quad xyz=d^3. \]
Find the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). \par Prove that the function \(\frac{(x-b)(x-c)}{x-a}\) can take all values for real values of \(x\) if \(a\) lies between \(b\) and \(c\); but if this condition does not hold it can take all values except certain values which lie in an interval \(4\sqrt{(a-b)(a-c)}\).
If \[ (1+x)^n = c_0+c_1x+c_2x^2+\dots+c_nx^n, \] where \(n\) is a positive integer, find \(c_0^2+c_1^2+\dots+c_n^2\). \par If \begin{align*} s_0 &= c_0+c_3+c_6+\dots, \\ s_1 &= c_1+c_4+c_7+\dots, \\ s_2 &= c_2+c_5+c_8+\dots, \end{align*} prove that \[ s_0^2+s_1^2+s_2^2=1+s_1s_2+s_2s_0+s_0s_1. \]
Shew that every mixed periodic continued fraction, which has more than one non-periodic element, is a root of a quadratic equation with rational coefficients whose roots are both of the same sign. \par Find the value of the \(2n\)th convergent to the continued fraction \[ \frac{1}{2+}\frac{1}{4+}\frac{1}{2+}\frac{1}{4+}\dots. \]
Prove that in an equation with real coefficients imaginary roots occur in pairs of the type \(\lambda \pm i\mu\). \par If \(\alpha, \beta, \gamma\) are the roots of \[ x^3-px^2+qx-r=0, \] express \((\alpha^2-\beta\gamma)(\beta^2-\gamma\alpha)(\gamma^2-\alpha\beta)\) in terms of the coefficients.
Differentiate \[ x^x, \quad \sin^{-1}\frac{x}{\sqrt{a^2-x^2}}, \quad \log\frac{x^2+x\sqrt{2}+1}{x^2-x\sqrt{2}+1}. \] Prove that, if \(y=\frac{\sin x}{\sqrt{x}}\), \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\tfrac{1}{4})y=0. \]
Shew how to evaluate the indeterminate forms \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). \par Find the limit when \(\theta=\frac{\pi}{2}\) of \((\log(\sec\theta+\tan\theta)).(\log(\operatorname{cosec}\theta+\cot\theta))\).
Find the equations of the tangent and normal at any point of the curve \[ x=3\sin t-2\sin^3t, \quad y=3\cos t-2\cos^3t. \] Prove that the evolute is \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{2}{3}}\).