The vector $\overrightarrow{\mathrm{a}}$ is equal to

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

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

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

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

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are the positton vectors of the vertices f an equilateral triangle whose orthocentre is at the origin, then

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$

• ${\overrightarrow{\mathrm{a}}}^2 = {\overrightarrow{\mathrm{b}}}^2 + {\overrightarrow{\mathrm{c}}}^2$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}}$

• None of these

$\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}}, \overrightarrow{\mathrm{C}}$ are three non-zero vectors; no two of them are parallel. If $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$ is collinear to $\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is collinear to $\overrightarrow{\mathrm{A}}$, then $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is equal to

• $\overrightarrow{\mathrm{A}}$

• $\overrightarrow{\mathrm{B}}$

• $\overrightarrow{\mathrm{C}}$

• $\overrightarrow{0}$

Let $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. If c is vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}} = \left|\overrightarrow{\mathrm{c}}\right| \left|\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right| = 2\sqrt{2}$ the angle between $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{is} 30°$ is, then $\left|\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \overrightarrow{\mathrm{c}}\right|$ is equal to

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

• 3

If the vectors $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, - 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{\lambda }\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ are linearly dependent, then the value of $\mathrm{\lambda }$ is equal to

• 0

• 1

• 2

• 3

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are two non-zero non-collinear vectors, then $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{i}}\right]\hat{\mathrm{i}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{j}}\right]\hat{\mathrm{j}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{k}}\right]\hat{\mathrm{k}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}}\right]$ is equal to

• $2\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$

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

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

• None of these

247.

If ${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2 + {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2 = 676 \mathrm{and} \left|\overrightarrow{\mathrm{b}}\right|$ = 2, then $\left|\overrightarrow{\mathrm{a}}\right|$ is equal to

• 13

• 26

• 39

• None of these

If $\overrightarrow{\mathrm{a}}$ is any vector, then $\hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right) +\hspace{0.17em}\hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ is equal to

• $\overrightarrow{\mathrm{a}}$

• $2\overrightarrow{\mathrm{a}}$

• $3\overrightarrow{\mathrm{a}}$

• $\overrightarrow{0}$

The product of two times of flight from a point P to another point Q with a given velocity of projection is

• $\frac{\mathrm{PQ}}{\mathrm{g}}$

• $\frac{2\mathrm{PQ}}{\mathrm{g}}$

• $\frac{\mathrm{PQ}}{2\mathrm{g}}$

• $\frac{3\mathrm{PQ}}{\mathrm{g}}$

A stone of mass m is thrown vertically upwards with a velocity of 9.8 ms^-1. The height of the point where KE = PE is (g = 9.8 ms^-2)

• 9.8 m

• 4.9 m

• 2.45 m

• 2 m