21.

The radius of the circle x² + y² + z² - 2y - 4z - 11 = 0 and x + 2y + 2z - 15 = 0 is :

• $\sqrt{7}$

• 3

$\int {\mathrm{x}}^2{\left(\mathrm{ax} + \mathrm{b}\right)}^{- 2}\mathrm{dx}$ is equal to :

• $\frac{2}{{\mathrm{a}}^2}\left(\mathrm{x} - \frac{\mathrm{b}}{\mathrm{a}}\log \left(\mathrm{ax} +\hspace{0.17em}\mathrm{b}\right)\right)$ + c

• $\frac{2}{{\mathrm{a}}^2}\left(\mathrm{x} - \frac{\mathrm{b}}{\mathrm{a}}\log \left(\mathrm{ax} +\hspace{0.17em}\mathrm{b}\right)\right) - \frac{{\mathrm{x}}^2}{\mathrm{a}\left(\mathrm{ax} + \mathrm{b}\right)}$

• $\frac{2}{{\mathrm{a}}^2}\left(\mathrm{x} + \frac{\mathrm{b}}{\mathrm{a}}\log \left(\mathrm{ax} +\hspace{0.17em}\mathrm{b}\right)\right) - \frac{{\mathrm{x}}^2}{\mathrm{a}\left(\mathrm{ax} + \mathrm{b}\right)} + \mathrm{c}$

• $\frac{2}{{\mathrm{a}}^2}\left(\mathrm{x} + \frac{\mathrm{b}}{\mathrm{a}}\log \left(\mathrm{ax} +\hspace{0.17em}\mathrm{b}\right)\right) - \frac{{\mathrm{x}}^2}{\mathrm{a}\left(\mathrm{ax} + \mathrm{b}\right)} + \mathrm{c}$

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}, \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5 \mathrm{and} \left|\overrightarrow{\mathrm{c}}\right| = 7$, then angle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is :

• 0

• 30°

• 45°

• 60°

If f(t) is an odd function, then ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is :

• an odd function

• an even function

• neither even nor odd

• 0

The projection of $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ is :

• 1/7

• - 1/7

• 7

• - 7

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = 0$ and $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$ = 0, then :

• $\overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} \parallel \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{and} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} = \overrightarrow{0} \mathrm{or} \overrightarrow{\mathrm{b}} = \overrightarrow{0}$

If the area bounded by the parabola y = 2 - x² and the line x + y = 0 is A sq unit, then A equals:

• 1/2

• 1/3

• 2/9

• 9/2

The points A (4, 5, 1), B (0, - 1, - 1), C(3, 9, 4) and D(- 4, 4, 4) are

• collinear

• coplanar

• non-coplanar

• non-collinear

${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2$ is equal to :

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2 - \left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)$

• ${\overrightarrow{\mathrm{a}}}^2 {\overrightarrow{\mathrm{b}}}^2 - {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2$

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2 - 2\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2 - 2 . \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

Let F denotes the family of ellipses whose centre is at the origin and major axis is the y-axis. Then, equation of the family F is :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\mathrm{xy}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\mathrm{xy}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$