If A = $\left[\begin{array}{cc}\mathrm{\alpha }& 2\\ 2& \mathrm{\alpha }\end{array}\right]$ and $\left|{\mathrm{a}}^3\right|$ = 27, then $\mathrm{\alpha }$ is equal to

• $\pm \sqrt{7}$

• $\pm 1$

• $\pm \sqrt{5}$

• $\pm 2$

If P = $\left|\begin{array}{cc}\mathrm{x}& 1\\ 1& \mathrm{x}\end{array}\right|$ and Q = $\left|\begin{array}{ccc}\mathrm{x}& 1& 1\\ 1& \mathrm{x}& 1\\ 1& 1& \mathrm{x}\end{array}\right|$, then $\frac{\mathrm{dQ}}{\mathrm{dX}}$ is equal to

• - 3P

• 3P + 1

• 3P

• 1 - 3P

$\left|\begin{array}{ccc}2\mathrm{a}& {\mathrm{x}}_1& {\mathrm{y}}_1\\ 2\mathrm{b}& {\mathrm{x}}_2& {\mathrm{y}}_2\\ 2\mathrm{c}& {\mathrm{x}}_3& {\mathrm{y}}_3\end{array}\right| = \frac{\mathrm{abc}}{2} \ne 0$, then the area of triangle whose vertices are $\left(\frac{{\mathrm{x}}_1}{\mathrm{a}}, \frac{{\mathrm{y}}_1}{\mathrm{a}}\right), \left(\frac{{\mathrm{x}}_2}{\mathrm{b}}, \frac{{\mathrm{y}}_2}{\mathrm{b}}\right), \left(\frac{{\mathrm{x}}_3}{\mathrm{c}}, \frac{{\mathrm{y}}_3}{\mathrm{c}}\right)$, is

• $\frac{1}{4}$

• $\frac{1}{4}\mathrm{abc}$

• $\frac{1}{8}$

• $\frac{1}{8}\mathrm{abc}$

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

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

Evaluate $\left|\begin{array}{cc}\cos \left(15\right)& \sin \left(15\right)\\ \sin \left(75\right)& \cos \left(75\right)\end{array}\right|$

• 2

• 1

• 3

• 0

If f(x) = $$\begin{gathered} \left\{3\mathrm{x} - 8, \mathrm{if} \mathrm{x} \le 5 \\ 2\mathrm{k}, \mathrm{if} \mathrm{x} > 5\right\} \end{gathered}$$ is continuous, then find k.

• 4/7

• 2/7

• 7/2

• 3/7

A balloon which always remains spherical is being inflated by pumping in 10 cu cm ofgas persecond. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

• $\frac{1}{30\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{90\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{9\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

Solve for $\mathrm{x} {\tan }^{-1}\left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right) = \frac{1}{2}{\tan }^{-1}\left(\mathrm{x}\right) . \mathrm{x} > 0$

• - 1

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• 1

The system of linear equations x + y + z = 6,
x + 2y + 3z = 10 and x + 2y + az = b has no solution when

• b = 2, a = 3

• $\mathrm{a} = 2, \mathrm{b} \ne 3$

• $\mathrm{b} = 3, \mathrm{a} \ne 10$

• $\mathrm{a} = 3, \mathrm{b} \ne 10$

If $\frac{{\left(\mathrm{x} + 1\right)}^2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{\mathrm{A}}{\mathrm{x}} + \frac{\mathrm{Bx} + \mathrm{C}}{{\mathrm{x}}^2 + 1}$, then ${\csc }^{-1}\left(\frac{1}{\mathrm{A}}\right) + {\cot }^{-1}\left(\frac{1}{\mathrm{B}}\right) + {\sec }^{-1}\left(\mathrm{C}\right)$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• 0