Given that $\left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 4, \left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right| = 10, \mathrm{then} {\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|}^2$  equals :

• 88

• 44

• 22

• None of these

If two like parallel to forces of $\frac{\mathrm{P}}{\mathrm{Q}} \mathrm{N} \mathrm{and} \frac{\mathrm{O}}{\mathrm{P}} \mathrm{N}$ have a resultant 2 N, then :

• P = Q

• 2P = Q

• P² = Q

• P = 2Q

The value of $\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right]$ is :

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• 1

• None of these

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, then $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ equals :

• $5\hat{\mathrm{i}} - 7\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} + 7\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $5\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• zero

If $\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}} = 2\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$, then the area of $∆ \mathrm{ABC}$ is :

• 3 sq unit

• 4 sq unit

• 16 sq unit

• 9 sq unit

If $\overrightarrow{\mathrm{a}} = \left(1, \mathrm{p}, 1\right), \overrightarrow{\mathrm{b}} = \left(\mathrm{q}, 2, 2\right), \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} = \mathrm{r} \mathrm{and} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ = (0, - 3, - 3), then p, q, r are in that order :

• 1, 5, 9

• 9, 5, 1

• 5, 1, 9

• None of these

287.

If the circle passes through the point (a, b) and cuts the circle x² + y² = k orthogonally, then the locus of its centre is given by :

• 2ax + 2by - (a² + b² + k²) = 0

• 2ax + 2by - (a² + b² - k²) = 0

• 2ax + 2by + (a² + b² + k²) = 0

• None of these

Let a, b, c be distinct non-negative numbers. If the vectors $\mathrm{a}\hat{\mathrm{i}} + \mathrm{a}\hat{\mathrm{j}} + \mathrm{c}\hat{\mathrm{k}}$, $\hat{\mathrm{i}} + \hat{\mathrm{k}}$ and $\mathrm{c}\hat{\mathrm{i}} + \mathrm{c}\hat{\mathrm{j}} + \mathrm{b}\hat{\mathrm{k}}$ lie in a plane, then :

• c² = ab

• a² = bc

• b² = ac

• None of these

$\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are two non-zero vectors, then $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) . \left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right)$ is equal to :

• a + b

• (a - b)²

• (a + b)²

• (a² - b²)

Angle between the vectors $\sqrt{3}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ and $\overrightarrow{\mathrm{b}} - \left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{a}}$ is :

• $\frac{\mathrm{\pi }}{2}$

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$