The value of ${\log }_2\left(20\right){\log }_2\left(80\right) - {\log }_2\left(5\right){\log }_2\left(320\right)$ is equal to

• 5

• 6

• 7

• 8

If the matrix M_r is given by Mr = $\left[\begin{array}{cc}\mathrm{r}& \mathrm{r} - 1\\ \mathrm{r} - 1& \mathrm{r}\end{array}\right]$, r = 1, 2, 3, ..., then the value of det(M₁) + det(M₂) + ... + det(M₂₀₀₈) is

• 2007

• 2008

• (2008)²

• (2007)²

3.

Let A = $\left[\begin{array}{cc}{\mathrm{\alpha }}^2& 5\\ 5& - \mathrm{\alpha }\end{array}\right]$ and $\left|{\mathrm{A}}^{10}\right| = 1024$, then $\mathrm{\alpha }$ is equal to

• 2

• - 2

• 3

• - 3

If w $\ne$ 1 is a cube root of unity, then the value of

$\left|\begin{array}{ccc}1 + 2{\mathrm{w}}^{100} + {\mathrm{w}}^{200}& {\mathrm{w}}^2& 1\\ 1& 1 + {\mathrm{w}}^{100} + 2{\mathrm{w}}^{200}& \mathrm{w}\\ \mathrm{w}& {\mathrm{w}}^2& 2+ {\mathrm{w}}^{100} + {\mathrm{w}}^{200}\end{array}\right|$ is equal to

• 0

• 1

• w

• w²

If $\mathrm{\alpha }$ is a cube root of unity, then the value of

$\left|\begin{array}{ccc}\mathrm{\alpha }& {\mathrm{\alpha }}^3& {\mathrm{\alpha }}^5\\ {\mathrm{\alpha }}^3& {\mathrm{\alpha }}^5& \mathrm{\alpha }\\ {\mathrm{\alpha }}^3& \mathrm{\alpha }& {\mathrm{\alpha }}^3\end{array}\right|$ is equal to

• $3{\mathrm{\alpha }}^3$

• $3\left({\mathrm{\alpha }}^3 + {\mathrm{\alpha }}^6 + {\mathrm{\alpha }}^9\right)$

• $3\left(\mathrm{\alpha } + {\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^3\right)$

• 0

If $\mathrm{f}\left(\mathrm{\alpha }\right) = \left|\begin{array}{ccc}1& \mathrm{\alpha }& {\mathrm{\alpha }}^2\\ \mathrm{\alpha }& {\mathrm{\alpha }}^2& 1\\ {\mathrm{\alpha }}^2& 1& \mathrm{\alpha }\end{array}\right|$, then $\mathrm{f}\left(\sqrt[3]{3}\right)$ is equal to

• 1

• - 4

• 4

• 2

If A = $\left[\begin{array}{ccc}3& 3& 3\\ 3& 3& 3\\ 3& 3& 3\end{array}\right]$, then A⁴ is equal to

• 27 A

• 81 A

• 243 A

• 729 A

Let B be a boolean algebra. If a, b $\in$ B, then (x - y)' is equal to

• a . b

• x . y'

• x' . y'

• x' + y'

The output of the circuit is

• x₃ . (x'₁ + x₂)

• (x'₃ + x₂) . x₁

• x'₃ . (x₁ + x₂)

• (x₁ + x₂) . x₃

If ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^2 + {\left({\cot }^{-1}\left(\mathrm{x}\right)\right)}^2 = \frac{5{\mathrm{\pi }}^2}{8}$, then x is equal to

• 0

• 2

• 1

• - 1