If A = $\left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right]$ and $\left|{\mathrm{a}}^3\right|$ = 27, then $\mathrm{\alpha }$ is equal to

• $\pm \sqrt{7}$

• $\pm 1$

• $\pm \sqrt{5}$

• $\pm 2$

$\left|\begin{array}{ccc}2\mathrm{a}& {\mathrm{x}}_1& {\mathrm{y}}_1\\ 2\mathrm{b}& {\mathrm{x}}_2& {\mathrm{y}}_2\\ 2\mathrm{c}& {\mathrm{x}}_3& {\mathrm{y}}_3\end{array}\right| = \frac{\mathrm{abc}}{2} \ne 0$, then the area of triangle whose vertices are $\left(\frac{{\mathrm{x}}_1}{\mathrm{a}}, \frac{{\mathrm{y}}_1}{\mathrm{a}}\right), \left(\frac{{\mathrm{x}}_2}{\mathrm{b}}, \frac{{\mathrm{y}}_2}{\mathrm{b}}\right), \left(\frac{{\mathrm{x}}_3}{\mathrm{c}}, \frac{{\mathrm{y}}_3}{\mathrm{c}}\right)$, is

• $\frac{1}{4}$

• $\frac{1}{4}\mathrm{abc}$

• $\frac{1}{8}$

• $\frac{1}{8}\mathrm{abc}$

Evaluate $\left|\begin{array}{cc}\cos \left(15\right)& \sin \left(15\right)\\ \sin \left(75\right)& \cos \left(75\right)\end{array}\right|$

• 2

• 1

• 3

• 0

The system of linear equations x + y + z = 6,
x + 2y + 3z = 10 and x + 2y + az = b has no solution when

• b = 2, a = 3

• $\mathrm{a} = 2, \mathrm{b} \ne 3$

• $\mathrm{b} = 3, \mathrm{a} \ne 10$

• $\mathrm{a} = 3, \mathrm{b} \ne 10$

If A = $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then A² is equal to

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

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

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

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

If x, y,z are all different and not equal to zero and $\left|\begin{array}{ccc}1+ \mathrm{x}& 1& 1\\ 1& 1 + \mathrm{y}& 1\\ 1& 1& 1 + \mathrm{z}\end{array}\right|$ = 0, then the value of x^-1 + y^-1 + z^-1 is equal to

• xyz

• x^-1y^-1z^-1

• - x - y - z

• - 1

If A is any square matrix of order 3 x 3, then $\left|3\mathrm{A}\right|$ is equal to

• $3\left|\mathrm{A}\right|$

• $\frac{1}{3}\left|\mathrm{A}\right|$

• $27\left|\mathrm{A}\right|$

• $9\left|\mathrm{A}\right|$

If A = $\frac{1}{\mathrm{\pi }}\left[\begin{array}{cc}{\sin }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{{\displaystyle \mathrm{x}}}{{\displaystyle \mathrm{\pi }}}\right)& {\cot }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right]$, B = $\frac{1}{\mathrm{\pi }}\left[\begin{array}{cc}- {\cos }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{{\displaystyle \mathrm{x}}}{{\displaystyle \mathrm{\pi }}}\right)& - {\tan }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right]$, then A - B is equal to

• I

• 0

• 2I

• $\frac{1}{2}\mathrm{I}$

If A is a matrix of order m x n and B is  a matrix, such that AB' and B'A are both defined, then the order of the matrix B is

• m x m

• n x n

• n x m

• m x n

450.

If x, y, z are not equal and $\ne 0, \ne 1$, then the value of $\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \log \left(\mathrm{y}\right)& \log \left(\mathrm{z}\right)\\ \log \left(2\mathrm{x}\right)& \log \left(2\mathrm{y}\right)& \log \left(2\mathrm{z}\right)\\ \log \left(3\mathrm{x}\right)& \log \left(3\mathrm{y}\right)& \log \left(3\mathrm{z}\right)\end{array}\right|$ is equal to

• $\log \left(\mathrm{xyz}\right)$

• $\log \left(6\mathrm{xyz}\right)$

• 0

• $\log \left(\mathrm{x} + \mathrm{y} + \mathrm{z}\right)$