If A = $\left[\begin{array}{ccc}1& - 5& 7\\ 0& 7& 9\\ 11& 8& 9\end{array}\right]$, then trace of matrix A is

• 17

• 25

• 3

• 12

If x, y and z are all distinct and $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& \mathrm{z}& 1 + {\mathrm{z}}^3\end{array}\right|$ = 0, then the value of xyz is 

• - 2

• - 1

• - 3

• None of these

If B is a non-singular matrix and A is a square matrix, then det (B^-1AB) is equal to

• det(A^-1)

• det(B^-1)

• det(A)

• det(B)

54.

${\left[\begin{array}{cc}0& \mathrm{a}\\ \mathrm{b}& 0\end{array}\right]}^4 = \mathrm{I}$, then

• a = 1 = 2b

• a = b

• a = b²

• ab = 1

If $\mathrm{A}\left(\mathrm{\theta }\right) = \left[\begin{array}{cc}1& \tan \left(\mathrm{\theta }\right)\\ - \tan \left(\mathrm{\theta }\right)& 1\end{array}\right] \mathrm{and} \mathrm{AB} = \mathrm{I}, \mathrm{then} \left({\sec }^2\left(\mathrm{\theta }\right)\right)\mathrm{B}$ is equal to :

• $\mathrm{A}\left(\mathrm{\theta }\right)$

• $\mathrm{A}\left(\frac{\mathrm{\theta }}{2}\right)$

• $\mathrm{A}\left(- \mathrm{\theta }\right)$

• $\mathrm{A}\left(\frac{- \mathrm{\theta }}{2}\right)$

If the rank of the matrix $\left[\begin{array}{ccc}- 1& 2& 5\\ 2& - 4& \mathrm{a} - 4\\ 1& - 2& \mathrm{a} +\hspace{0.17em}1\end{array}\right]$ is 1, then the value of a is :

• - 1

• 2

• - 6

• 4

If l, m and n are real numbers such that l² + m² + n² = 0, then

$\left|\begin{array}{ccc}1 + {\mathrm{l}}^2& \mathrm{lm}& \ln \\ \mathrm{lm}& 1 + {\mathrm{m}}^2& \mathrm{mn}\\ \ln & \mathrm{mn}& 1 + {\mathrm{n}}^2\end{array}\right|$ is equal to

• 0

• 1

• l + m + n + 2

• 2(l + m + n) + 3

If f (x) = x² + 4x - 5 and A = $\left[\begin{array}{cc}1& 2\\ 4& - 3\end{array}\right]$, then f(A) is equal to

• $\left[\begin{array}{cc}0& - 4\\ 8& 8\end{array}\right]$

• $\left[\begin{array}{cc}2& 1\\ 2& 0\end{array}\right]$

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

• $\left[\begin{array}{cc}8& 4\\ 8& 0\end{array}\right]$

$\left|\begin{array}{ccc}\mathrm{\alpha }& - \mathrm{\beta }& 0\\ 0& \mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\beta }& 0& \mathrm{\alpha }\end{array}\right| = 0$, then

• $\frac{\mathrm{\alpha }}{\mathrm{\beta }}$ is one of the cube roots of unity

• $\mathrm{\alpha }$ is one of the cube roots of unity

• $\mathrm{\beta }$ is one of the cube roots of unity

• $\mathrm{\alpha \beta }$ is one of the cube roots of unity 

The coefficient of x in

$\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}\mathrm{x}& 1 + \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ 1& \log \left(1 + \mathrm{x}\right)& 2\\ {\mathrm{x}}^2& 1 + {\mathrm{x}}^2& 0\end{array}\right|, - 1 < \mathrm{x} \le 1$, is

• 1

• - 2

• - 1

• 0