If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the cube roots of unity, then the value of the determinant $\left|\begin{array}{ccc}{\mathrm{e}}^{\mathrm{\alpha }}& {\mathrm{e}}^{2\mathrm{\alpha }}& \left({\mathrm{e}}^{3\mathrm{\alpha }} - 1\right)\\ {\mathrm{e}}^{\mathrm{\beta }}& {\mathrm{e}}^{2\mathrm{\beta }}& \left({\mathrm{e}}^{3\mathrm{\beta }} - 1\right)\\ {\mathrm{e}}^{\mathrm{\gamma }}& {\mathrm{e}}^{2\mathrm{\gamma }}& \left({\mathrm{e}}^{3\mathrm{\gamma }} - 1\right)\end{array}\right|$ is equal to

• - 2

• - 1

• 0

• 1

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

• det(A^-1)

• det (B^-1)

• det(B)

• det(A)

3.

If 1, w, w² are the cube roots of unity and if

, a² + b² is equal to

• 1 + w²

• w² - 1

• 1 + w

• (1 + w)²

If the three linear equations

x + 4ay + az = 0

x + 3by + bz = 0

x + 2cy + cz = 0

have a non-trivial solution, where a$\mathrm{a} \ne 0, \mathrm{b} \ne 0, \mathrm{c} \ne 0$, then ab + bc is equal to

• 2ac

• - ac

• ac

• - 2ac

If A = $\left|\begin{array}{ccc}1& 0& 0\\ \mathrm{x}& 1& 0\\ \mathrm{x}& \mathrm{x}& 1\end{array}\right| \mathrm{and} \mathrm{I} = \left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$, then A³ - 4A² + 3A + I is equal to

• 3I

• I

• - I

• - 2I

If A = $\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$, then the value of the determinant $\left|{\mathrm{A}}^{2009} - 5{\mathrm{A}}^{2008}\right|$ is

• - 6

• - 5

• - 4

• 4

The boolean expression corresponding to the combinational circuit is

• (x₁ + x₂ · x'₃)x₂

• (x₁ . (x₂ + x₃)) + x₂

• (x₁ . (x₂ + x'₃)) + x₂

• (x₁ + (x₂ + x'₃)) + x₃

In a boolean algebra B with respect to '+' and '.', x' denotes the negation of x $\in$ B. Then

• x - x' = 1 and x · x' = 1

• x + x' = 1 and x . x' = 0

• x + x' = 0 and x . x' = 0

• x + x' = 0 and x . x' = 0

If ${\cos }^{-1}\left(\frac{5}{13}\right) - {\sin }^{-1}\left(\frac{12}{13}\right) = {\cos }^{-1}\left(\mathrm{x}\right)$,  then x is equal to

• 1

• $\frac{1}{\sqrt{2}}$

• 0

• $\frac{\sqrt{3}}{2}$

The value of $\cos \left[{\tan }^{-1}\left\{\sin \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right\}\right]$ is

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 - 1}}$

• $\sqrt{\frac{1 - {\mathrm{x}}^2}{{\mathrm{x}}^2 + 2}}$

• $\sqrt{\frac{1 - {\mathrm{x}}^2}{1 + {\mathrm{x}}^2}}$

• $\sqrt{\frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 + 2}}$