Determinants

Let A = $\left(\begin{array}{cc}\mathrm{x} + 2& 3\mathrm{x}\\ 3& \mathrm{x} + 2\end{array}\right)$, B = $\left(\begin{array}{cc}\mathrm{x}& 0\\ 5& \mathrm{x} + 2\end{array}\right)$, Then all solutions of equation det(AB) = 0 is

• 1, - 1, 0, 2

• 1, 4, 0, - 2

• 1, - 1, 4, 3

• - 1, 4, 0, 3

The value of det A, where

A = $\mathrm{A} = \left(\begin{array}{ccc}1& \cos \left(\mathrm{\theta }\right)& 0\\ - \cos \left(\mathrm{\theta }\right)& 1& \cos \left(\mathrm{\theta }\right)\\ - 1& - \cos \left(\mathrm{\theta }\right)& 1\end{array}\right)$, lies

• in the closed interval [1, 2]

• in the closed interval [0, 1]

• in the open interval (0, 1)

• in the open interval (1, 2)

The points (- a, - b), (a, b), (0, 0) and (a, ab), $\mathrm{a} \ne 0, \mathrm{b} \ne 0$ are always

• collinear

• vertices of a parallelogram

• vertices of a rectangle

• lie on a circle

The number of distinct real roots of

$\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)\end{array}\right| = 0 \mathrm{in} \mathrm{the} \mathrm{interval} - \frac{\mathrm{\pi }}{4} \le \hspace{0.17em}\mathrm{x} \le \frac{\mathrm{\pi }}{4}$ is

• 0

• 2

• 1

• > 2

$$\begin{gathered} \mathrm{If} \mathrm{f} : \left[0, \mathrm{\pi }/2\right] \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{\theta }\right) = \left|\begin{array}{ccc}1& \tan \left(\mathrm{\theta }\right)& 1\\ - \tan \left(\mathrm{\theta }\right)& 1& \tan \left(\mathrm{\theta }\right)\\ - 1& - \tan \left(\mathrm{\theta }\right)& 1\end{array}\right| \mathrm{Then}, \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\left(2, \infty \right)$

• $(-\infty , - 2]$

• $[2, \infty )$

• $(- \infty , 2]$

If a is an imaginary cube root of unity, then the value of the determinant

$\left|\begin{array}{ccc}1 + \mathrm{w}& {\mathrm{w}}^2& - \mathrm{w}\\ 1 + {\mathrm{w}}^2& \mathrm{w}& - {\mathrm{w}}^2\\ \mathrm{w} + {\mathrm{w}}^2& \mathrm{w}& - {\mathrm{w}}^2\end{array}\right|$ is

• - 2w

• - 3w²

• - 1

• 0

Let $\mathrm{n} \ge 2$ be an integer,

$\mathrm{A} = \left[\begin{array}{ccc}\cos \left(2\mathrm{\pi }/3\right)& \sin \left(2\mathrm{\pi }/\mathrm{n}\right)& 0\\ - \sin \left(2\mathrm{\pi }/\mathrm{n}\right)& \cos \left(2\mathrm{\pi }/\mathrm{n}\right)& 0\\ 0& 0& 1\end{array}\right]$ and I is the identity matrix of order 3. Then,

• Aⁿ = I and A^{n - 1} $\ne$ I

• A^m $\ne$ I for any positive integer m

• A is not invertible 

• A^m = 0 for a positive integer m 

The value of determinant

$\left|\begin{array}{ccc}1 + {\mathrm{a}}^2 - {\mathrm{b}}^2& 2\mathrm{ab}& - 2\mathrm{b}\\ 2\mathrm{ab}& 1 - {\mathrm{a}}^2 + {\mathrm{b}}^2& 2\mathrm{a}\\ 2\mathrm{b}& - 2\mathrm{a}& 1 - {\mathrm{a}}^2 - {\mathrm{b}}^2\end{array}\right|$ is

• 0

• (1 + a² + b²)

• (1 + a² + b²)²

• (1 + a² + b²)³

Consider the system of equations x + y + z = 0, $\mathrm{\alpha x} + \mathrm{\beta y} + \mathrm{\gamma z} = 0$ and ${\mathrm{\alpha }}^2\mathrm{x} + {\mathrm{\beta }}^2\mathrm{y} + {\mathrm{\gamma }}^2\mathrm{z} = 0$. Then, the system of equations has

• a unique solution for all values of $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$

• infinite number of solutions, if any two of $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are equal.

• a unique solution, if $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are distinct

• more than one, but finite number of solutions depending on values of $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$

If P = $\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]$, Q = PP^T, then the value of the determinant of Q is

• 2

• - 2

• 1

• 0