A wedge of mass \(M\), whose faces are inclined at angles \(\alpha, \beta\) to the horizontal, is free to move on a horizontal plane. Particles of masses \(m_1, m_2\) are placed on its two faces respectively. Taking all the surfaces to be smooth, prove that the particle \(m_1\) will remain at rest relative to the wedge provided that \[ (M+m_1)\tan(\alpha+\beta)=(M+m_1+m_2)\tan\beta. \]
A heavy particle is projected from a point with velocity \(V\) so as to pass through another point at a distance \(r\). Prove that there are in general two possible directions of projection, and that the times of flight are roots of the equation \[ g^2t^4-2(V^2+V_1^2)t^2+4r^2=0, \] where \(V_1\) is the velocity at the second point.
Find the time of a small oscillation of a simple pendulum; find also the pressure on the point of suspension in any position when the oscillation is not necessarily small. \(ABC\) is a light framework in the form of an isosceles triangle, having \(B=C=30^\circ\) and \(AB=9\) in. Equal masses are fixed at \(B, C\) and the system is suspended from \(A\). Prove that the time of a small oscillation is nearly 1.36 sec.
Write down the first four terms of the expansion of \((1-x)^{-\frac{1}{2}}\) in ascending powers of \(x\) and also give the coefficient of \(x^n\). Prove that, when \(x\) is very small, \[ \frac{(1+2x)^{\frac{1}{2}}(1-5x)^{\frac{1}{3}}}{(1-11x)^{\frac{1}{4}}} = 1-\frac{1}{2}x. \]
Calculate to four places of decimals \[ (\cdot 0035)^{-\frac{1}{2}} \times (32\cdot 17)^{\frac{1}{5}} \quad \text{and} \quad \operatorname{cosec} 2^\circ 17'. \]
Find the conditions that the equation \(ax^2+2bx+c=0\) should have (i) both its roots positive and (ii) two equal roots.
By drawing the graph of \(y=\sin x\), prove that the equation \(x=10\sin x\) has seven real roots.
Prove that in a triangle \(\tan\frac{A-B}{2} = \frac{a-b}{a+b}\cot\frac{C}{2}\). In a triangle \(a=7\cdot 5, b=5, C=36^\circ 12'\), prove that \(A=103^\circ 22'\).
Find the condition that \(lx+my+n=0\) should touch the circle \(x^2+y^2+2ax=0\).
Differentiate \(x^{x^2}\), \((ax^2+b)^n\), \(x^2 \sin x\) and \(\frac{x+2}{(x+1)(x+3)}\).