If e$\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{x}}& {\mathrm{e}}^{\mathrm{y}}\\ {\mathrm{e}}^{\mathrm{y}}& {\mathrm{e}}^{\mathrm{x}}\end{array}\right] = \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, [he the values of x and y are respectively

• - 1, - 1

• 1, 1

• 0, 1

• 1, 0

If A = $\left[\begin{array}{cc}\log \left(\mathrm{x}\right)& - 1\\ - \log \left(\mathrm{x}\right)& 2\end{array}\right]$ and if det(A) = 2, then the value of x is equal to

• 2

• e²

• - 2

• e

If $\left|\begin{array}{cc}{\sin }^2\left(\mathrm{\alpha }\right)& {\cos }^2\left(\mathrm{\alpha }\right)\\ {\cos }^2\left(\mathrm{\alpha }\right)& {\sin }^2\left(\mathrm{\alpha }\right)\end{array}\right|$ = 0, $\mathrm{\alpha } \in \left(0, \mathrm{\pi }\right)$, then the value of $\mathrm{\alpha }$ are

• $\frac{\mathrm{\pi }}{2} \mathrm{and} \frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2} \mathrm{and} \frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6} \mathrm{and} \frac{\mathrm{\pi }}{3}$

If A = $\left[\begin{array}{cc}\mathrm{a}& \mathrm{x}\\ \mathrm{y}& \mathrm{a}\end{array}\right]$ and if xy = 1, then det (AA^T) is equal to

• a² - 1

• (a² + 1)²

• 1 - a²

• (a² - 1)²

5.

If f(x) = $\left|\begin{array}{cc}\mathrm{x}& \mathrm{\lambda }\\ 2\mathrm{\lambda }& \mathrm{x}\end{array}\right|$, then $\mathrm{f}\left(\mathrm{\lambda x}\right) - \mathrm{f}\left(\mathrm{x}\right)$ is equal to

• $\mathrm{x}\left({\mathrm{\lambda }}^2 - 1\right)$

• $2\mathrm{\lambda }\left({\mathrm{x}}^2 - 1\right)$

• ${\mathrm{\lambda }}^2\left({\mathrm{x}}^2 - 1\right)$

• ${\mathrm{x}}^2\left({\mathrm{\lambda }}^2 - 1\right)$

If $\mathrm{A} = {\left({\mathrm{a}}_{\mathrm{ij}}\right)}_{\mathrm{m} \times \mathrm{n}}, \mathrm{B} = {\left({\mathrm{b}}_{\mathrm{ij}}\right)}_{\mathrm{n} \times \mathrm{p}}$, and C = ${\left({\mathrm{C}}_{\mathrm{ij}}\right)}_{\mathrm{p} \times \mathrm{q}}$, then the product (BC) A is possible only when

• m = q

• n = q

• p = q

• m = p

The value of ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right) + {\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{5}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

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

• ${\tan }^{-1}\left(\frac{1}{3\sqrt{3}}\right)$

The value of $\cos \left({\sin }^{-1}\left(\frac{2}{3}\right)\right)$ is equal to

• $\frac{\sqrt{3}}{5}$

• $\frac{5}{3}$

• $\frac{5}{\sqrt{3}}$

• $\frac{\sqrt{5}}{3}$

If ${\tan }^{-1}\left(2\mathrm{x}\right) + {\tan }^{-1}\left(3\mathrm{x}\right) = \frac{\mathrm{\pi }}{2}$, then the value of x is equal to

• $\frac{1}{\sqrt{6}}$

• $\frac{1}{6}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

If the function f(x) = $$\begin{gathered} \left\{\mathrm{x}, \mathrm{if} \mathrm{x} \le 1 \\ \mathrm{cx} + \mathrm{k}, \mathrm{if} 1 < \mathrm{x} < 4 \\ - 2\mathrm{x}, \mathrm{if} \mathrm{x} \ge 4\right\} \end{gathered}$$ is continuous everywhere, then the values of c and k are respectively.

• - 3, - 5

• - 3, 5

• - 3, - 4

• - 3, 4