If $\left|\begin{array}{ccc}1 + {\sin }^2\left(\mathrm{\theta }\right)& {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right)\\ {\sin }^2\left(\mathrm{\theta }\right)& 1 + {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right)\\ {\sin }^2\left(\mathrm{\theta }\right)& {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right) - 1\end{array}\right| = 0$ and $0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}$, then $\cos \left(4\mathrm{\theta }\right)$ is equal to

• $\frac{\sqrt{3}}{2}$

• 0

• $\frac{- 1}{2}$

• $\frac{1}{2}$

If $\left|\begin{array}{ccc}\mathrm{x} +\hspace{0.17em}1& \mathrm{x} + 2& \mathrm{x} + \mathrm{a}\\ \mathrm{x} + 2& \mathrm{x} + 3& \mathrm{x} + \mathrm{b}\\ \mathrm{x} + 3& \mathrm{x} + 4& \mathrm{x} +\hspace{0.17em}\mathrm{c}\end{array}\right|$ = 0, then a, b, c are

• in GP

• in HP

• equal

• 9

213.

The value of $\left|\begin{array}{ccc}1& {\log }_{\mathrm{x}}\left(\mathrm{y}\right)& {\log }_{\mathrm{x}}\left(\mathrm{z}\right)\\ {\log }_{\mathrm{y}}\left(\mathrm{x}\right)& 1& {\log }_{\mathrm{y}}\left(\mathrm{z}\right)\\ {\log }_{\mathrm{z}}\left(\mathrm{x}\right)& {\log }_{\mathrm{z}}\left(\mathrm{y}\right)& 1\end{array}\right|$ is equal to

• 0

• 1

• xyz

• log(xyz)

If A and B are square matrices ofthe same order such that (A + B)(A - B) = A² - B², then (ABA^-1)² is equal to

• B²

• I

• A²B²

• A²

If A = $\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$, then A² + xA + yI = 0 for (x, y) is

• (- 4, 1)

• (- 1, 3)

• (4, - 1)

• (1, 3)

The constant term of the polynomial $\left|\begin{array}{ccc}\mathrm{x} + 3& \mathrm{x}& \mathrm{x} + 2\\ \mathrm{x}& \mathrm{x} + 1& \mathrm{x} - 1\\ \mathrm{x} + 2& 2\mathrm{x}& 3\mathrm{x} + 1\end{array}\right|$ is

• 0

• 2

• - 1

• 1

If A is a 3 x 3 non-singular matrix and if $\left|\mathrm{A}\right| = 3, \mathrm{then} \left|{\left(2\mathrm{A}\right)}^{- 1}\right|$ is

• 24

• 3

• $\frac{1}{3}$

• $\frac{1}{24}$

If lines represented by x + 3y - 6 = 0, 2x + y - 4 = 0 and kx - 3y + 1 = 0 are concurrent, then the value of k is

• $\frac{6}{19}$

• $\frac{19}{6}$

• $- \frac{19}{6}$

• $- \frac{6}{19}$

If A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then A . A' is

• I

• A

• - A

• A²

If $\left[\begin{array}{ccc}1& 2& - 1\\ 1& \mathrm{x} - 2& 1\\ \mathrm{x}& 1& 1\end{array}\right]$ is singular, then the value of x is

• 2

• 3

• 1

• 0