81.

If A = $\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$, then by the principle of Mathematical induction, prove that Aⁿ = $\left[\begin{array}{cc}1& 2\mathrm{n}\\ 0& 1\end{array}\right]$

Find the general solution of $\sec \left(\mathrm{\theta }\right) + 1 = \left(2 + \sqrt{3}\right)\tan \left(\mathrm{\theta }\right)$

Prove that for all values of m, except zero the straight line

y =  mx + $\frac{\mathrm{a}}{\mathrm{m}}$ touches the parabola y² = 4ax.

If the tangent to the parabola y = x(2 - x) at the point (1, 1) intersects the parabola at P. Find the coordinate of P.

If an unbiased coin is tossed n times. Find the probability that head appears an odd number of times.

Evaluate $\int \frac{{\mathrm{x}}^2}{\mathrm{x}\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}\mathrm{dx}$