The multiplicative inverse of A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$ is

• $\left[\begin{array}{cc}- \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}- \cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$

The value of 'a' for which the system of equations

a³x + (a + 1)³y + (a + 2)³z = 0

     ax + (a - 1)y + (a + 2)z = 0

                          x + y + z = 0

• 1

• 0

• - 1

• None of these

If f(x) = $$\begin{gathered} \left\{{\left(1 + \left|\sin \left(\mathrm{x}\right)\right|\right)}^{\frac{\mathrm{a}}{\left|\sin \left(\mathrm{x}\right)\right|}}, - \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right. < \mathrm{x} < 0 \\ {\mathrm{e}}^{\frac{\tan \left(2\mathrm{x}\right)}{\tan \left(3\mathrm{x}\right)}}, 0 < \mathrm{x} < \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.\right\} \end{gathered}$$ is continuous at x = 0, find the values of a and b.

• 3/2, e^3/2

• - 2/3, e^{- 3/2}

• 2/3, e^2/3

• None of these

4.

If f(x) = e^xg(x), g(0) = 2, g'(0) = 1 then f'(0) is

• 1

• 3

• 2

• 0

At the point x = 1, the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^3 - 1, 1 < \mathrm{x} < \infty \\ \mathrm{x} - 1, - \infty < \mathrm{x} \le 1\right\} \end{gathered}$$

• continuous and differentiable

• continuous and not differentiable

• discontinuous and differentiable

• discontinuous and not differentiable

If x = 2cos(t) - cos(2t), y = 2sin(t) - sin(2t), then the value of ${\overline{)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}}}_{\mathrm{t} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• 3/2

• 5/2

• 5/3

• - 3/2

y = $\log \left(\tan \left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right) + {\sin }^{-1}\left(\cos \left(\mathrm{x}\right)\right)$, then dy/dx is

• csc(x) - 1

• cs(x)

• csc(x) + 1

• x

For all real x, the minimum value of $\frac{1 - \mathrm{x} + {\mathrm{x}}^2}{1 + \mathrm{x} + {\mathrm{x}}^2}$ is

• 0

• 1/3

• 1

• 3

If x + y = k is normal to y² = 12x, then k is

• 3

• 9

• - 9

• - 3

A particle moves along a straight line according to the law s = 16 - 2t + 3t³, where s metres is the distance of the particle from a fixed point at the end of t second. The acceleration of the particle at the end of 2s is

• 3.6 m/s²

• 36 m/s²

• 36 km/s²

• 360 m/s²