Let A = $\left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& - \sin \left(\mathrm{\alpha }\right)\\ \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$, $\mathrm{\alpha } \in \mathrm{R}$ such that A³² = $\left[\begin{array}{cc}0& - 1\\ 1& 0\end{array}\right]$ then a value of $\mathrm{\alpha }$ is

• 0

• $\frac{\mathrm{\pi }}{32}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{64}$

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x⁴ + 12x³ - 36x² - 25, x $\in$ R, then:

• S₁ = { - 1}; S₂ = {0, 2}

• S₁ = { - 2, 1}, S₂ = {0}

• S₁ = { - 2}; S₂ = {0, 1}

• S₁ = { - 2, 0}; S₂ = {1}

3.

The greatest value of c $\in$ R or which the system of linear equations

x – cy – cz = 0

cx - y + cz = 0

cx + cy – z = 0 has non – trivial solution, is:

• - 1

• 0

• $\frac{1}{2}$

• 2

If $\mathrm{\alpha } = {\cos }^{-1}\left(\frac{3}{5}\right), \mathrm{\beta } = {\tan }^{-1}\left(\frac{1}{3}\right)$, where $0 < \mathrm{\alpha }, \mathrm{\beta } < \frac{\mathrm{\pi }}{2}$, then $\mathrm{\alpha } - \mathrm{\beta }$ is equal to :

• ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

• ${\tan }^{-1}\left(\frac{9}{14}\right)$

• ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

• ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

Let f : [0, 2] $\to$ R be a twice differentiable function such that f’’(x) > 0, for all  $\mathrm{x} \in \left(0, 2\right)$. If $\mathrm{\varphi }\left(\mathrm{x}\right)$ = f(x) + f(2 - x), then $\mathrm{\varphi }$ is :

• Increasing on (0, 1) and decreasing on (1, 2)

• Decreasing on (0, 1) and increasing on (1, 2)

• Decreasing on (0, 2)

• Increasing on (0, 2)

If f(x) = $\frac{2 - \mathrm{xcos}\left(\mathrm{x}\right)}{2 + \mathrm{xcos}\left(\mathrm{x}\right)} \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)\hspace{0.17em}= {\log }_{\mathrm{e}}\left(\mathrm{x}\right), (\mathrm{x} > 0)$ then the value of integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{g}\left(\mathrm{f}\right(\mathrm{x}\left)\right)$dx is :

• log_e(3)

• log_e(1)

• log_e(2)

• log_ee

The magnitude of the projection of the vector $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on the vector perpendicular to the plane containing the vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is

• $\frac{\sqrt{3}}{2}$

• $3\sqrt{6}$

• $\sqrt{6}$

• $\sqrt{\frac{3}{2}}$

Let y = y(x) be the solution of the differential equation, $\left({\mathrm{x}}^2 + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{x}\left({\mathrm{x}}^2 + 1\right)\mathrm{y} = 1$ such that y(0) = 0. If $\sqrt{\mathrm{a}}\mathrm{y}\left(1\right) = \frac{\mathrm{\pi }}{32}$, then the value of 'a' is :

• $\frac{1}{4}$

• 1

• $\frac{1}{16}$

• $\frac{1}{2}$

The shortest distance between the line y = x and the curve
y² = x - 2 is

• $\frac{11}{4\sqrt{2}}$

• $\frac{7}{8}$

• 2

• $\frac{7}{4\sqrt{2}}$

The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1 , 1, 0) is :

• x - 3y - 2z = - 2

• x - y - z = 0

• 2x - z = 2

• x + 3y + z = 4