81.

A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is  $\frac{1}{2}\left(\mathrm{u} + \mathrm{v}\right)\mathrm{t}$.

Show that

$\frac{\sin \left(\mathrm{\theta }\right)}{\cos \left(3\mathrm{\theta }\right)} + \frac{{\displaystyle \sin \left(3\mathrm{\theta }\right)}}{{\displaystyle \cos \left(9\mathrm{\theta }\right)}} + \frac{{\displaystyle \sin \left(9\mathrm{\theta }\right)}}{{\displaystyle \cos \left(27\mathrm{\theta }\right)}} = \frac{1}{2}\left(\left(\tan 27\mathrm{\theta }\right) - \tan \left(\mathrm{\theta }\right)\right)$

Show that, for a positive integer n, the coefficient of x^k, ($0 \le \mathrm{k} \le \mathrm{n}$) in the expansion of 1 + (1 + x) + (1 + x) + ... + (1 + x)ⁿ is ${}^{\mathrm{n} + 1}\mathrm{C}_{\mathrm{n} - \mathrm{k}}$

Find the values of 'a' for which the expression x² - (3a - 1)x + 2a² + 2a - 11 is always positive

Find the sum of the first n terms of the series 0.2 + 0.22 + 0.222 + ...

If $\mathrm{x} = \sin \left(\mathrm{t}\right), \mathrm{y} = \sin \left(2\mathrm{t}\right)$, prove that

$\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} + 4\mathrm{y}$ = 0

If f is differentiable at x = a, find the value of

$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\mathrm{x}}^2\mathrm{f}\left(\mathrm{a}\right) - {\mathrm{a}}^2\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - \mathrm{a}}$

Find the angle subtended by the double ordinate of length 2a of the parabola y² = ax at its vertex.

The equation to the pairs of opposite sides of a parallelogram are x² - 5x + 6 = 0 and y² - 6y + 5. Find the equations its diagonals.

If m, n be integers, then find the value of

${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}{\left(\cos \left(\mathrm{mx}\right) - \sin \left(\mathrm{nx}\right)\right)}^{2}d\mathrm{x}$