If l, m and n are real numbers such that l² + m² + n² = 0, then

$\left|\begin{array}{ccc}1 + {\mathrm{l}}^2& \mathrm{lm}& \ln \\ \mathrm{lm}& 1 + {\mathrm{m}}^2& \mathrm{mn}\\ \ln & \mathrm{mn}& 1 + {\mathrm{n}}^2\end{array}\right|$ is equal to

• 0

• 1

• l + m + n + 2

• 2(l + m + n) + 3

If f (x) = x² + 4x - 5 and A = $\left[\begin{array}{cc}1& 2\\ 4& - 3\end{array}\right]$, then f(A) is equal to

• $\left[\begin{array}{cc}0& - 4\\ 8& 8\end{array}\right]$

• $\left[\begin{array}{cc}2& 1\\ 2& 0\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}8& 4\\ 8& 0\end{array}\right]$

$\left|\begin{array}{ccc}\mathrm{\alpha }& - \mathrm{\beta }& 0\\ 0& \mathrm{\alpha }& \mathrm{\beta }\\ \mathrm{\beta }& 0& \mathrm{\alpha }\end{array}\right| = 0$, then

• $\frac{\mathrm{\alpha }}{\mathrm{\beta }}$ is one of the cube roots of unity

• $\mathrm{\alpha }$ is one of the cube roots of unity

• $\mathrm{\beta }$ is one of the cube roots of unity

• $\mathrm{\alpha \beta }$ is one of the cube roots of unity 

4.

The coefficient of x in

$\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}\mathrm{x}& 1 + \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ 1& \log \left(1 + \mathrm{x}\right)& 2\\ {\mathrm{x}}^2& 1 + {\mathrm{x}}^2& 0\end{array}\right|, - 1 < \mathrm{x} \le 1$, is

• 1

• - 2

• - 1

• 0

If $\left|\begin{array}{ccc}\mathrm{x}& 3& 6\\ 3& 6& \mathrm{x}\\ 6& \mathrm{x}& 3\end{array}\right| = \left|\begin{array}{ccc}2& \mathrm{x}& 7\\ \mathrm{x}& 7& 2\\ 7& 2& \mathrm{x}\end{array}\right| = \left|\begin{array}{ccc}4& 5& \mathrm{x}\\ 5& \mathrm{x}& 4\\ \mathrm{x}& 4& 5\end{array}\right| = 0$, then x is equal to

• 9

• - 9

• 0

• - 1

If w be the complex cube root of unity and matrix H = $\left[\begin{array}{cc}\mathrm{w}& 0\\ 0& \mathrm{w}\end{array}\right]$, then H⁷⁰ is equal to

• 0

• - H

• H

• H²

The output of the circuit is

• (x₂ + x₃) . [(x₁ · x₂) . x₃']

• (x₂ + x₃') . [(x₁ · x₂) . x₃']

• (x₂ + x₃) + [(x₁ · x₂) . x₃']

• (x₂ . x₃) . [(x₁ · x₂) . x₃']

If ${\tan }^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right) - {\tan }^{-1}\left(\frac{\mathrm{m} - \mathrm{n}}{\mathrm{m} +\hspace{0.17em}\mathrm{n}}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{\mathrm{n}}{\mathrm{m}}\right)$

• ${\tan }^{-1}\left(\frac{\mathrm{m} +\hspace{0.17em}\mathrm{n}}{\mathrm{m} - \mathrm{n}}\right)$

• $\frac{\mathrm{\pi }}{4}$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

If ${\sec }^{-1}\left(\sqrt{1 + {\mathrm{x}}^2}\right) + {\csc }^{-1}\left(\frac{\sqrt{1 +\hspace{0.17em}{\mathrm{y}}^2}}{\mathrm{y}}\right) + {\cot }^{-1}\left(\frac{1}{\mathrm{z}}\right) = \mathrm{\pi }$, then x + y + z is equal to

• xyz

• 2xyz

• xyz²

• x²yz

If $\mathrm{x} = \frac{2\mathrm{at}}{1 + {\mathrm{t}}^3} \mathrm{and} \mathrm{y} = \frac{2{\mathrm{at}}^2}{{\left(1 + {\mathrm{t}}^3\right)}^2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

• ax

• a²x²

• $\frac{\mathrm{x}}{\mathrm{a}}$

• $\frac{\mathrm{x}}{2\mathrm{a}}$