$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right] \mathrm{and} \det \left({\mathrm{A}}^3\right) = 125, \mathrm{Then} \mathrm{\alpha } = ?$

• $\pm 1$

• $\pm 2$

• $\pm 3$

• $\pm 5$

12.

The equations

x + 2y + 3z = 12

x + y + 3z  = 25

x + 5y + 9z = 4

• have the unique solution

• Have infinitely many solutions

• are inconsistent

• None of the above

What is the value of the determinant

$\left|\begin{array}{ccc}1& 1& 1\\ 1& 1 + \mathrm{xyz}& 1\\ 1& 1& 1 +\hspace{0.17em}\mathrm{xyz}\end{array}\right|$

• 1 + x + y + z

• 2xyz

• x²y²z²

• 2x²y²z²

$\mathrm{If} \left|\begin{array}{ccc}\mathrm{x}& \mathrm{y}& 0\\ 0& \mathrm{x}& \mathrm{y}\\ \mathrm{y}& 0& \mathrm{x}\end{array}\right| = 0, \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\frac{\mathrm{x}}{\mathrm{y}} \mathrm{is} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{cube} \mathrm{roots} \mathrm{of} \mathrm{unity}$

• x is one of the cube roots of unity

• y is one of the cube roots of unity

• $\frac{\mathrm{x}}{\mathrm{y}}$ is one of the cube roots - 1

Consider the set A of all matrices of order 3 x 3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is - 1. Then which one of the following is correct ?

• C is empty

• B as as many elements as C

• $\mathrm{A} = \mathrm{B} \cup \mathrm{C}$

• B has thrice as many elements as C

$\mathrm{If} \left|\mathrm{a}\right|$ denotes the absolute value of an integer, then which of the following are correct ?

$$\begin{gathered} 1. \left|\mathrm{ab}\right| = \left|\mathrm{a}\right|\left|\mathrm{b}\right| \\ 2. \left|\mathrm{a} + \mathrm{b}\right| \le \left|\mathrm{a}\right| + \left|\mathrm{b}\right| \\ 3. \left|\mathrm{a} - \mathrm{b}\right| \ge \left|\left|\mathrm{a}\right| - \left|\mathrm{b}\right|\right| \end{gathered}$$

Select the correct answer using the code given below :

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

The value of the determinant $\left|\begin{array}{cc}{\cos }^2\left(\frac{\mathrm{\theta }}{2}\right)& {\sin }^2\left(\frac{\mathrm{\theta }}{2}\right)\\ {\sin }^2\left(\frac{\mathrm{\theta }}{2}\right)& {\cos }^2\left(\frac{\mathrm{\theta }}{2}\right)\end{array}\right|$

for all values of $\mathrm{\theta }$, is

• 1

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

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

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

The system of equations kx + y + z = 1, x + ky + z = kand x + y + kz = k has no solution if k equal

• 0

• 1

•  - 1

•  - 2

If $\left|\mathrm{z} - \frac{4}{\mathrm{z}}\right| = 2,$ he maximum value of Izl is equal to :

• $1 + \sqrt{3}$

• $1 + \sqrt{5}$

• $1 - \sqrt{5}$

• $\sqrt{5} - 1$

The value of the determinant $\left|\begin{array}{ccc}1 - \mathrm{\alpha }& \mathrm{\alpha } - {\mathrm{\alpha }}^2& {\mathrm{\alpha }}^2\\ 1 - \mathrm{\beta }& \mathrm{\beta } - {\mathrm{\beta }}^2& {\mathrm{\beta }}^2\\ 1 - \mathrm{\gamma }& \mathrm{\gamma } - {\mathrm{\gamma }}^2& {\mathrm{\gamma }}^2\end{array}\right| = ?$

• $\left(\mathrm{\alpha } - \mathrm{\beta }\right)\left(\mathrm{\beta } - \mathrm{\gamma }\right)\left(\mathrm{\alpha } - \mathrm{\gamma }\right)$

• $\left(\mathrm{\alpha } - \mathrm{\beta }\right)\left(\mathrm{\beta } - \mathrm{\gamma }\right)\left(\mathrm{\gamma } - \mathrm{\alpha }\right)$

• $\left(\mathrm{\alpha } - \mathrm{\beta }\right)\left(\mathrm{\beta } - \mathrm{\gamma }\right)\left(\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma }\right)$

• 0