If f(x) is continuous on $\left[- \mathrm{\pi }, \mathrm{\pi }\right]$, where

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{- 2\sin \left(\mathrm{x}\right), \mathrm{for} - \mathrm{\pi } \le \mathrm{x} \le - \frac{\mathrm{\pi }}{2} \\ \mathrm{\alpha sin}\left(\mathrm{x}\right) + \mathrm{\beta }, \mathrm{for} - \frac{\mathrm{\pi }}{2} < \mathrm{x} < \frac{\mathrm{\pi }}{2} \\ \cos \left(\mathrm{x}\right), \mathrm{for} \frac{\mathrm{\pi }}{2} \le \mathrm{x} \le \mathrm{\pi }\right\} \end{gathered}$$

then $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are

• - 1, - 1

• 1, - 1

• 1, 1

• - 1, 1

If f(x) = $$\begin{gathered} \left\{{\log }_{\left(1 - 3\mathrm{x}\right)}\left(1 +\hspace{0.17em}3\mathrm{x}\right), \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$ continuous at x = 0, then k is equal to

• - 2

• 2

• 1

• - 1

If $\mathrm{A} = \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, then A² - 5A is equal to

• 2I

• - 2I

• 3I

• null matrix

If A = $\left[\begin{array}{cc}2& 1\\ - 1& 2\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& - 2\\ 2& 1\end{array}\right], \mathrm{C} = \left[\begin{array}{cc}1& - 3\\ 2& 1\end{array}\right]$, then

• A + B = B + A and A + (B + C) = (A + B) + C

• A + B = B + A and AC = BC

• A + B = B + A and AB = BC

• AC = BC and A = BC

$\mathrm{A} = \left[\begin{array}{cc}- 2& 4\\ - 1& 2\end{array}\right]$, then A² is equal to

• null matrix

• unit matrix

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

If $\mathrm{x} = \log \left(1 + {\mathrm{t}}^2\right) \mathrm{and} \mathrm{y} = \mathrm{t} - {\tan }^{-1}\left(\mathrm{t}\right)$. Then, $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• e^x - 1

• t² - 1

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

• e^x - y

7.

If A = [x y z], B = $\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$. Then, ABC = O, if

• [ax² + by² + cz² + 2gxy + 2fyz + 2czx] = 0

• [ax² + cy² + bz² + xy + yz + zx] = 0

• [ax² + by² + cz² + 2hxy + 2by + 2cz] = 0

• [ax² + by² + cz² + 2zx + 2hxy + 2fyz] = 0

A = $\left[\begin{array}{ccc}0& 3& 3\\ - 3& 0& - 4\\ - 3& 4& 0\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$, then B'(AB) is

• null matrix

• singular matrix

• unit matrix

• symmetric matrix

A square matrix is an orthogonal matrix, if

• AA' = 0

• A + A' = I

• AA' = I

• None of these

If I is incentre of $∆$ABC, then I is

• $\frac{\mathrm{a}\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{b}\overrightarrow{\mathrm{b}} +\hspace{0.17em}\mathrm{c}\overrightarrow{\mathrm{c}}}{\mathrm{a} +\hspace{0.17em}\mathrm{b} +\hspace{0.17em}\mathrm{c}}$

• $\frac{\mathrm{a}\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{b}\overrightarrow{\mathrm{b}} +\hspace{0.17em}\mathrm{c}\overrightarrow{\mathrm{c}}}{\sqrt{{\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2 +\hspace{0.17em}{\mathrm{c}}^2}}$

• $\frac{1}{3}\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right]$

• $\frac{\overrightarrow{\mathrm{a}} +\hspace{0.17em}\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{\mathrm{a} +\hspace{0.17em}\mathrm{b} +\hspace{0.17em}\mathrm{c}}$