11.

Let M be the set of all 2 x 2 matrices with entries from the set ofreal number R. Then, the function f : M R defined by f(A) = $\left|\mathrm{A}\right|$ for ever A $\in$ M, is

• one-one and onto

• neither one-one nor onto

• one-one but not onto

• onto but not one-one

For how many values of x in the closed interval [- 4, - 1] the matrix $\left[\begin{array}{ccc}3& \mathrm{x} - 1& 2\\ 3& - 1& \mathrm{x} + 2\\ \mathrm{x} + 3& - 1& 2\end{array}\right]$ is singular

• zero

• 2

• 1

• 3

G = $\left\{\left(\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right) : \mathrm{\theta } \in \mathrm{R}\right\}$is a group under matrix multiplication. Then, which one ofthe following statements in respect of G is ture

• $\left(\begin{array}{cc}- 1& 0\\ 0& - 1\end{array}\right)$ is inverse of itself

• G is finite group

• $\left(\begin{array}{cc}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$- \sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right)$ is not an element of G

• $\left(\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right)$ is an element of G

If two circles (x - 1)² + (y - 3)² = r² and x² + y² - 8x + 2y + 8 = 0 intersect in two distinct points, then

• r < 2

• 8 < r < 10

• r = 2

• 2 < r < 8

If f(x) = $$\begin{gathered} \left\{\mathrm{xsin}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right), \mathrm{if} \mathrm{x} \ne 0 \\ 0, \mathrm{if} \mathrm{x} = 0\text{'}\right\} \end{gathered}$$, then at x = 0 the function f is

• continuous but not differentiable

• differentiable but not continuous

• continuous and differentiable

• not continuous

The perimeter of a sector is P. The area of the sectoris maximum when its radius is

• $\frac{1}{\sqrt{\mathrm{P}}}$

• $\frac{\mathrm{P}}{2}$

• $\frac{\mathrm{P}}{4}$

• $\sqrt{\mathrm{P}}$

The function f(x) = x³ - 3x is

• increasing on $\left(- \infty , - 1\right) \cup [1, \infty ) \mathrm{and} \mathrm{decreasing} \mathrm{on} (- 1, 1)$

• $\mathrm{decreasing} \mathrm{on} \left(- \infty , - 1\right) \cup [1, \infty ) \mathrm{and} \mathrm{increasing} \mathrm{on} (- 1, 1)$

• $\mathrm{increasing} \mathrm{on} (0, \infty ) \mathrm{and} \mathrm{decreasing} \mathrm{on} (- \infty , 0)$

• $\mathrm{decreasing} \mathrm{on} (0, \infty ) \mathrm{and} \mathrm{increasing} \mathrm{on} (- \infty , 0)$

The differential of ${\mathrm{e}}^{{\mathrm{x}}^3}$ with respect to log(x) is

• ${\mathrm{e}}^{{\mathrm{x}}^3}$

• $3{\mathrm{x}}^2{\mathrm{e}}^{{\mathrm{x}}^3} + 3{\mathrm{x}}^2$

• $3{\mathrm{x}}^2{\mathrm{e}}^{{\mathrm{x}}^3}$

• $3{\mathrm{x}}^3{\mathrm{e}}^{{\mathrm{x}}^3}$

If a, b, c are mutually perpendicular unit vectors, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• 3

• $\sqrt{3}$

• 0

• 1

If ABCDEF is a regular hexagon, then AD + EB + FC equals

• 0

• 2 AB

• 4 AB

• 3 AB