11.

Let f : $\mathrm{R} \to \mathrm{R}$ be defined by f(x) = x⁴, then

• f may be one-one and onto

• f is neither one-one nor onto

• f is one-one and onto

• f is one-one but not onto

If y = log(log(x)), then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• $\frac{- \left(1 + \log \left(\mathrm{x}\right)\right)}{{\mathrm{x}}^2\log \left(\mathrm{x}\right)}$

• $\frac{- \left(1 + \log \left(\mathrm{x}\right)\right)}{{\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^2}$

• $\frac{\left(1 + \log \left(\mathrm{x}\right)\right)}{{\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^2}$

• $\frac{\left(1 + \log \left(\mathrm{x}\right)\right)}{{\mathrm{x}}^2\log \left(\mathrm{x}\right)}$

Binary operation * on R - {- 1} defined by $\mathrm{a} * \mathrm{b} = \frac{\mathrm{a}}{\mathrm{b} + 1}$ is

• * is neither associative not commutative

• * is associative but not commutative

• * is commutative but not commutative

• * is associative and commutative

If y = ${\tan }^{-1}\left(\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 1/2

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• 1

The derivative of ${\cos }^{-1}\left(2{\mathrm{x}}^2 - 1\right) \mathrm{w}.\mathrm{r}.\mathrm{t}. {\cos }^{-1}\left(\mathrm{x}\right)$ is

• 1 - x²

• $\frac{2}{\mathrm{x}}$

• $\frac{1}{2\sqrt{1 - {\mathrm{x}}^2}}$

• 2

The rate of change ofvolume of a sphere with respect to its surface area when the radius is 4 cm is

• 4 cm³/cm²

• 6 cm³/cm²

• 2 cm³/cm²

• 8 cm³/cm²

The value of c in mean value theorem for the function f(x) = x² in [2, 4] is

• 3

• 7/2

• 4

• 2

If ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) = \frac{4\mathrm{\pi }}{5}$, then ${\cot }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{y}\right)$ is equal to

• $\frac{2\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{3\mathrm{\pi }}{5}$

• $\mathrm{\pi }$

If a matrix A is both symmetric and skew symmetric, then

• A is diagonal matrix

• A is a zero matrix

• A is scalar matrix

• A is square matrix

If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

• the LPP under consideration is not solvable

• the LPP under consideration must be reconstructed

• the required optimal solution is at the mid-point of the line joining two points

• the optimal solution occurs at every point on the line joining these two points