31.

A· {(B + C) x (A + B + C)} equals

• [A B C]

• [B A C]

• 0

• 1

If equation y + $\frac{{\mathrm{y}}^3}{3} + \frac{{\mathrm{y}}^5}{5} + ...\infty = 2\left[\mathrm{x} + \frac{{\displaystyle {\mathrm{x}}^3}}{{\displaystyle 3}} + \frac{{\displaystyle {\mathrm{x}}^5}}{{\displaystyle 5}} + ...\infty \right]$, then value of y is

• $\frac{\mathrm{x}}{1 - {\mathrm{x}}^2}$

• $\frac{2\mathrm{x}}{1 - {\mathrm{x}}^2}$

• $\frac{1 - {\mathrm{x}}^2}{2\mathrm{x}}$

• None of these

If $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0$, then

• z = xy

• $\mathrm{z} = \frac{1}{\mathrm{xy}}$

• $\mathrm{z} = - \frac{1}{\mathrm{xy}}$

• None of these

If $\left|\mathrm{A}\right| \ne 0$ and A is of order n, then adj(adj A) is equal to

• ${\left|\mathrm{A}\right|}^{\mathrm{n}}$

• ${\left|\mathrm{A}\right|}^2$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1} . \mathrm{I}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 2} . \mathrm{A}$

Which is incorrect ?

• (AB)' = B'A'

• ${\left(\mathrm{AB}\right)}^{\mathrm{\theta }} = {\mathrm{B}}^{\mathrm{A}}{\mathrm{A}}^{\mathrm{\theta }}$

• $\overline{\mathrm{AB}} = \overline{\mathrm{B}} \overline{\mathrm{A}}$

• ${\left(\mathrm{AB}\right)}^{-1} = {\mathrm{B}}^{-1}{\mathrm{A}}^{-1}$

A square matrix A is called an orthogonal matrix, if

• $\mathrm{A} \overline{\mathrm{A}} = \mathrm{I}$

• AA' = I

• ${\mathrm{AA}}^{\mathrm{\theta }} = \mathrm{I}$

• A² = I

The value of 'c' of Rolle's theorem for f(x) = e^x sin(x) in [0, x] is given by

• $\frac{\mathrm{\pi }}{4}$

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

${\int }_{- \mathrm{a}}^{\mathrm{a}}\mathrm{x}\sqrt{\left({\mathrm{a}}^2 - {\mathrm{x}}^2\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{8}$

• 0

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{xsin}\left(\mathrm{x}\right) . \cos \left(\mathrm{x}\right)}{{\cos }^4\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{\pi }}^2}{8}$

• $\frac{{\mathrm{\pi }}^2}{16}$

• 1

• 0

Use Simpson's $\frac{1}{3}$ rule to find the value of ${\int }_1^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ given,

• 140.88

• 256.66

• 160.26

• None of these