21.

If $\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ are unit vectors along the positive direction of x, y and z - axes, then a false statement in the following is

• $\sum \hat{\mathrm{i}} \times \left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} \times \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} . \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) = \overrightarrow{0}$

• $\sum \hat{\mathrm{i}} . \left(\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = \overrightarrow{0}$

If $\int \mathrm{f}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right) . \cos \left(\mathrm{x}\right)\mathrm{dx} = \frac{1}{2\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)}\log \left(\mathrm{f}\left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$, where c is the constant of integration, then f(x) is

• $\frac{2}{\mathrm{abcos}\left(2\mathrm{x}\right)}$

• $\frac{2}{\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)\cos \left(2\mathrm{x}\right)}$

• $\frac{2}{\mathrm{ab}\sin \left(2\mathrm{x}\right)}$

• $\frac{2}{\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)\sin \left(2\mathrm{x}\right)}$

If $\int \frac{\sqrt{\mathrm{x}}}{\mathrm{x}\left(\mathrm{x} + 1\right)}\mathrm{dx} = {\mathrm{ktan}}^{-1}\left(\mathrm{m}\right)$, then (k, m) is

• (2, x)

• (1, x)

• $\left(1, \sqrt{\mathrm{x}}\right)$

• $\left(2, \sqrt{\mathrm{x}}\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{3 + \sin \left(2\mathrm{x}\right)}\mathrm{dx}$ is

• $\frac{1}{4}\log \left(3\right)$

• $\log \left(3\right)$

• $\frac{1}{2\log \left(3\right)}$

• $2\log \left(3\right)$

${\int }_0^1\mathrm{x}{\left(1 - \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ is

• $- \frac{2}{35}$

• $\frac{4}{35}$

• $\frac{24}{35}$

• $- \frac{8}{35}$

The area bounded by the curve y = $$\begin{gathered} \left\{{\mathrm{x}}^2, \mathrm{x} < 0 \\ \mathrm{x}, \mathrm{x} \ge 0\right\} \end{gathered}$$ and the line y = 4, is

• $\frac{32}{3}$

• $\frac{8}{3}$

• $\frac{40}{3}$

• $\frac{16}{3}$

The order and degree of the differential equation y = $\frac{\mathrm{dp}}{\mathrm{dx}}\mathrm{x} = \sqrt{{\mathrm{a}}^2{\mathrm{p}}^2 +\hspace{0.17em}{\mathrm{b}}^2}$, where p = $\frac{\mathrm{dy}}{\mathrm{dx}}$ (here a and b are arbitrary constants) respectively are

• 2, 2

• 1, 1

• 1, 2

• 2, 1

The general solution of the differential equation $2\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 3$ is a family of

• hyperbolas

• parabolas

• straight lines

• circles