If R denotes the set of all real numbers, then the function f : defined f(x) =  is :

• one-one only

• onto only

• both one-one and onto

• neither one-one nor onto

$\left(\mathrm{a} . \hat{\mathrm{i}}\right)\hat{\mathrm{i}} + \left(\mathrm{a} . \hat{\mathrm{j}}\right)\hat{\mathrm{j}} +\hspace{0.17em}\left(\mathrm{a} . \hat{\mathrm{k}}\right)\hat{\mathrm{k}}$ is equal to

• a

• 2a

• 3a

• 0

13.

If $\left|\mathrm{a}\right| = 3, \left|\mathrm{b}\right| = 4$, then a value of $\mathrm{\lambda }$ for which a + $\mathrm{\lambda b}$ is perpendicular to a - $\mathrm{\lambda b}$ is

• $\frac{9}{16}$

• $\frac{3}{4}$

• $\frac{3}{2}$

• $\frac{4}{3}$

The projection of $\mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ on $\mathrm{b} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is

• $\frac{1}{\sqrt{14}}$

• $\frac{2}{\sqrt{14}}$

• $\sqrt{14}$

• $\frac{- 2}{\sqrt{14}}$

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1 + \cos \left(2\mathrm{y}\right)}{1 - \cos \left(2\mathrm{y}\right)} = 0$ is given by

• $\tan \left(\mathrm{y}\right) + \cot \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - \cot \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{x}\right) - \cot \left(\mathrm{y}\right) = \mathrm{c}$

• $\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{y}\right) = \mathrm{c}$

The degree of the differential equation ${\left(1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} = {\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ is

• 2

• 4

• 9

• 1

The area enclosed between the curves y = x³ and y = $\sqrt{\mathrm{x}}$ is, (in square units)

• $\frac{5}{3}$

• $\frac{5}{4}$

• $\frac{5}{12}$

• $\frac{12}{5}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}{\cos }^3\left(4\mathrm{\theta }\right)\mathrm{d\theta }$ is equal to

• $\frac{5}{3}$

• $\frac{5}{4}$

• $\frac{1}{3}$

• $\frac{1}{6}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right) \sin \left(\mathrm{x}\right)\hspace{0.17em}}\mathrm{dx}$ is equal to

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^7 + 1\right)}$ is equal to

• $\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\log \left(\frac{{\mathrm{x}}^7 + 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7 + 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$