81.

Prove that the equation cos(2x) + asin(x) = 2a - 7 possesses a solution if $2 \le \mathrm{a} \le 6$.

Find the values of x. $\left(- \mathrm{\pi } < \mathrm{x} < \mathrm{\pi }, \mathrm{x} \ne 0\right)$ satisfying the equation,

${\mathrm{g}}^{1 + \left|\cos \left(\mathrm{x}\right)\right| + \left|{\cos }^2\left(\mathrm{x}\right)\right| + ... \infty = 4^3}$

Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is $\left(- \frac{1}{2}, \frac{1}{2}\right)$

Prove by induction that for n $\in$ N, n² + n is an even integer (n $\ge$ 1)

If N = n!(n $\left(\mathrm{n} \in \mathrm{N}, \mathrm{n} > 2\right)$, then find

$\underset{\mathrm{N}\to \infty }{\lim }\left[{\left({\log }_2\left(\mathrm{N}\right)\right)}^{- 1} + \left({\log }_3{\left(\mathrm{N}\right)}^{- 1}\right) + ... + {\log }_{\mathrm{n}}{\left(\mathrm{N}\right)}^{- 1}\right]$

Use the formula $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{a}}^{\mathrm{x}} - 1}{\mathrm{x}} = {\log }_{\mathrm{e}}\left(\mathrm{a}\right) \mathrm{to} \mathrm{compute} \underset{\mathrm{x}\to 0}{\lim }\frac{2^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}}{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle 1}}$

If A, B are two square matrices such that AB = A and BA = B, then prove that B² = B

If $\frac{d\mathrm{y}}{d\mathrm{x}} + \sqrt{\frac{1 - {\mathrm{y}}^2}{1 - {\mathrm{x}}^2}} = 0$ prove that, $\mathrm{x}\sqrt{1 - {\mathrm{y}}^2} + \mathrm{y}\sqrt{1 - {\mathrm{x}}^2} = \mathrm{A}$ where A is constant.

Evaluate the following integral

${\int }_{- 1}^2\left|\mathrm{xsin}\left(\mathrm{\pi x}\right)\right|d\mathrm{x}$

If f(a) = 2, f'(a) = 1, g(a) = - 1 and g'(a) = 2, find the value of $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{\mathrm{g}\left(\mathrm{a}\right)\mathrm{f}\left(\mathrm{a}\right) - \mathrm{g}\left(\mathrm{a}\right)\mathrm{f}\left(\mathrm{x}\right)}{\left(\mathrm{x} - \mathrm{a}\right)}$