If a² + b² + c² = - 2 and f(x) = $\left|\begin{array}{ccc}1 +\hspace{0.17em}{\mathrm{a}}^2\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{b}}^2\right)\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{c}}^2\right)\mathrm{x}\\ \left(1 +\hspace{0.17em}{\mathrm{a}}^2\right)\mathrm{x}& 1 +\hspace{0.17em}{\mathrm{b}}^2\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{c}}^2\right)\mathrm{x}\\ \left(1 +\hspace{0.17em}{\mathrm{a}}^2\right)\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{b}}^2\right)\mathrm{x}& 1 +\hspace{0.17em}{\mathrm{c}}^2\mathrm{x}\end{array}\right|$ then f(x) is a polynomial of degree

• 3

• 2

• 1

• 0

If A = $\left|\begin{array}{ccc}1& - 1& 1\\ 0& 2& - 3\\ 2& 1& 0\end{array}\right|$ and B = (adj A), and C = 5A, then $\frac{\left|\mathrm{adj} \mathrm{B}\right|}{\left|\mathrm{C}\right|}$ is

• 5

• 25

• - 1

• 1

If matrix A = $\left|\begin{array}{ccc}1& 0& - 1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right|$ and its inverse is denoted by A^-1 = $\left|\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|$, then the value of a₂₃ is

• $\frac{21}{20}$

• $\frac{1}{5}$

• $- \frac{2}{5}$

• $\frac{2}{5}$

4.

The number of solutions of the system of equations 2x + y - z = 7, x - 3y + 2z = 1 and x + 4y - 3z = 5 is

• 3

• 2

• 1

• 0

If f(x) = $$\begin{gathered} \left\{\frac{1 - \cos \left(4\mathrm{x}\right)}{{\mathrm{x}}^2}, \mathrm{when} \mathrm{x} < 0 \\ \mathrm{a}, \mathrm{when} \mathrm{x} = 0 \\ \frac{\sqrt{\mathrm{x}}}{\sqrt{\left(16 + \sqrt{\mathrm{x}}\right)}{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle 4}}, \mathrm{when} \mathrm{x} > 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of a will be

• 8

• - 8

• 4

• None of these

If f is a real-valued differentiable function satisfying $\left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{y}\right)\right| \le$ (x - y)² , x, y $\in$ R and f(0) = 0, then f(1) is equal to

• 2

• 1

• - 1

• 0

$\tan \left[\frac{\mathrm{\pi }}{4} + \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right] + \tan \left[\frac{\mathrm{\pi }}{4} - \frac{1}{2}{\cos }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]$ is equal to

• $\frac{2\mathrm{a}}{\mathrm{b}}$

• $\frac{2\mathrm{b}}{\mathrm{a}}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{\mathrm{b}}{\mathrm{a}}$

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \sqrt{2}}$ is equal to

• $- \frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right) + \mathrm{C}$

$\int {\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{4}\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\sqrt{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\sqrt{2}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

The part of straight line y = x + 1 between x = 2 and x = 3 is revolved about x-axis, then the curved surface of the solid thus generated is

• $\frac{37\mathrm{\pi }}{3}$

• $7\mathrm{\pi }\sqrt{2}$

• $37\mathrm{\pi }$

• $\frac{7\mathrm{\pi }}{\sqrt{2}}$