1.

If $\left|\begin{array}{ccc}6\mathrm{i}& 3\mathrm{i}& - 1\\ 4& 3\mathrm{i}& - 1\\ 20& 3& \mathrm{i}\end{array}\right|$ = x + iy, then

• x = 3, y = 1

• x = 1, y = 3

• x = 0, y = 3

• x = 0, y = 0

$\left|\begin{array}{cc}1& {\log }_{\mathrm{b}}\left(\mathrm{a}\right)\\ {\log }_{\mathrm{a}}\left(\mathrm{b}\right)& 1\end{array}\right|$ is equal to

• 1

• 0

• ${\log }_{\mathrm{a}}\left(\mathrm{b}\right)$

• ${\log }_{\mathrm{b}}\left(\mathrm{a}\right)$

If A = $\left[\begin{array}{cc}1& 2\\ - 3& 0\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{cc}- 1& 0\\ 2& 3\end{array}\right]$, then

• A² = A

• B² = B

• $\mathrm{AB} \ne \mathrm{BA}$

• AB = BA

If A = $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{b}& \mathrm{a}\end{array}\right] \mathrm{and} {\mathrm{A}}^2 = \left[\begin{array}{cc}\mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\beta }& \mathrm{\alpha }\end{array}\right]$, then

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = \mathrm{ab}$

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = 2\mathrm{ab}$

• $\mathrm{\alpha } = {\mathrm{a}}^2 + {\mathrm{b}}^2, \mathrm{\beta } = {\mathrm{a}}^2 - {\mathrm{b}}^2$

• $\mathrm{\alpha } = 2\mathrm{ab}, \mathrm{\beta }= {\mathrm{a}}^2 + {\mathrm{b}}^2$

If matrix A = $\left[\begin{array}{cc}1& - 1\\ 1& 1\end{array}\right]$, then

• $\mathrm{A}\text{'} = \left[\begin{array}{cc}1& 1\\ 1& - 1\end{array}\right]$

• ${\mathrm{A}}^{- 1} = \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right]$

• $\mathrm{A} . \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right] = 2\mathrm{I}$

• $\mathrm{\lambda A} = \left[\begin{array}{cc}\mathrm{\lambda }& - \mathrm{\lambda }\\ - 1& 1\end{array}\right], \mathrm{where} \mathrm{\lambda } \mathrm{is} \mathrm{a} \mathrm{non}- \mathrm{zero} \mathrm{scalar}$

If for AX = B, B = $\left[\begin{array}{c}9\\ 52\\ 0\end{array}\right] \mathrm{and} {\mathrm{A}}^{- 1} = \left[\begin{array}{ccc}3& - 1/2& - 1/2\\ - 4& 3/4& 5/4\\ 2& - 1/4& - 3/4\end{array}\right]$, then X is equal to

• $\left[\begin{array}{c}1\\ 3\\ 5\end{array}\right]$

• $\left[\begin{array}{c}- 1/2\\ - 1/2\\ 2\end{array}\right]$

• $\left[\begin{array}{c}- 4\\ 2\\ 3\end{array}\right]$

• $\left[\begin{array}{c}3\\ 3/4\\ - 3/4\end{array}\right]$

If ${\cos }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right) + {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = \mathrm{\alpha }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{2\mathrm{xy}}{\mathrm{ab}}\cos \left(\mathrm{\alpha }\right) + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}$, then $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{2\mathrm{xy}}{\mathrm{ab}}\cos \left(\mathrm{\alpha }\right) + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}$ is equal to

• ${\sin }^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{acos}}^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{atan}}^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{acot}}^2\left(\mathrm{\alpha }\right)$

The two curves x³ - 3xy² + 2 = 0 and 3x²y - y³ - 2 = 0

• cut at right angle

• touch each other

• cut at an angle $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• cut at an angle $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

The domain of the function ${\sin }^{-1}\left({\log }_2\left(\frac{{\mathrm{x}}^2}{2}\right)\right)$ is

• $\left[- 2, 2\right] ~ \left(- 1, 1\right)$

• $\left[- 1, 2\right] ~ \left\{0\right\}$

• [1,2]

• $\left[- 2, 2\right] ~ \left\{0\right\}$

If the function f(x) = $$\begin{gathered} \left\{1 + \sin \left(\frac{\mathrm{\pi x}}{2}\right), \mathrm{for} - \infty < \mathrm{x} \le 1 \\ \mathrm{ax} +\hspace{0.17em}\mathrm{b}, \mathrm{for} 1 < \mathrm{x} <\hspace{0.17em}3 \\ 6\tan \left(\frac{{\displaystyle \mathrm{\pi x}}}{{\displaystyle 12}}\right), \mathrm{for} 3 \le \mathrm{x} < 6\right\} \end{gathered}$$ is continuous in the interval (- $\infty$, 6), then the values of a and b are respectively

• 0, 2

• 1, 1

• 2, 0

• 2, 1