Problems

The ends of a uniform rod \(AB\) of length \(2l\) slide without friction, the end \(A\) along the horizontal line \(OX\) and the end \(B\) along the vertical line \(OY\) and \(Y\) is below \(O\). The system \(XOY\) is forced to revolve with angular velocity \(\omega\) about \(OY\). Shew that, if \(3g<4l\omega^2\), there is a position of equilibrium other than the vertical position.

Two equal uniform rods \(AB, BC\) each of length \(2l\) and mass \(m\) are freely hinged together at \(B\). They are falling with \(ABC\) horizontal and with velocity \(v\) and without spin, when the rod \(BC\) hits a fixed inelastic peg at a point which is at a distance \(x\) (\(<2l\)) from \(C\). Find the angular velocities of the two rods immediately after the impact and the impulsive reaction at the joint.

Determine \(\lambda\) so that the equation in \(x\) \[ \frac{2A}{x+a} + \frac{\lambda}{x} - \frac{2B}{x-a} = 0 \] may have equal roots; and if \(\lambda_1, \lambda_2, x_1, x_2\) be the two values of \(\lambda\) and the two corresponding values of \(x\), prove that \[ x_1 x_2 = a^2, \quad \lambda_1 \lambda_2 = (A-B)^2. \]

If \(n\) is a positive integer, prove that \(3 \cdot 5^{2n+1} + 2^{3n+1}\) is divisible by 17 and \(3^{2n+2}-8n-9\) is divisible by 64.

Evaluate the integrals:

1. \(\int \frac{x-1}{x^2}e^x dx\); \quad (ii) \(\int \frac{(2x^3+1)dx}{x(x^3+1)\sqrt{x^6(1+x^3)^2+1}}\).

Draw the graph of the curve \[ (x^2-1)(x^2-4)y^2 - x^2 = 0 \] and find the area bounded by the line \(x=1\) and the two arcs of the curve passing through the origin.

(i) Prove that if \[ H_n(x) = e^{x^2} \frac{d^n e^{-x^2}}{dx^n}, \] then \[ \frac{dH_n(x)}{dx} + 2nH_{n-1}(x) = 0. \] (ii) Find the \(n\)th derivative of the function \(y = \frac{x}{x^2-1}\).

Two planes are inclined at an angle \(\theta\). A straight line makes angles \(\alpha\) and \(\beta\) with the normals to the two planes. Prove that if its projections on the two planes are perpendicular, then \[ 1 - \cos^2\alpha - \cos^2\beta \pm \cos\alpha\cos\beta\cos\theta = 0. \]

Find the limits of

1. \(n\{e-(1+\frac{1}{n})^n\}\); \quad (ii) \(n\left( \frac{\gamma^n-1}{\log n} \right)\)

Find the two nearest points on the curves \(y^2-4x=0\), \(x^2+y^2-6y+8=0\), and evaluate their distance.