If $\left|\overrightarrow{\mathrm{a}}\right| = \left|\overrightarrow{\mathrm{b}}\right| = 1 \mathrm{and} \left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| = \sqrt{3}$, then the value of $\left(3\overrightarrow{\mathrm{a}} - 4\overrightarrow{\mathrm{b}}\right) . \left(2\overrightarrow{\mathrm{a}} +\hspace{0.17em}5\overrightarrow{\mathrm{b}}\right)$ is

• - 21

• $- \frac{21}{2}$

• 21

• $\frac{21}{2}$

If $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, $\left|\overrightarrow{\mathrm{a}}\right|$ = 2, $\left|\overrightarrow{\mathrm{b}}\right|$ = 3, $\left|\overrightarrow{\mathrm{c}}\right|$ = 4 and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{2\mathrm{\pi }}{3}$, then [$\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$] is equal to

• $4\sqrt{3}$

• $6\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$

If $\overrightarrow{\mathrm{a}}$, $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are perpendicular to $\overrightarrow{\mathrm{b}} +\hspace{0.17em}\overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{c}} +\hspace{0.17em}\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\overrightarrow{\mathrm{b}}$ respectively and if $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|$ = 6, $\left|\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ = 8 and $\left|\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right|$ = 10 then $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right|$ is equal to

• $5\sqrt{2}$

• 50

• $10\sqrt{2}$

• 10

If the vectors $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{\lambda }\overrightarrow{\mathrm{b}} +\hspace{0.17em}3\overrightarrow{\mathrm{c}}$, $- 2\overrightarrow{\mathrm{a}} +\hspace{0.17em}3\overrightarrow{\mathrm{b}} -\hspace{0.17em}4\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} -\hspace{0.17em}3\overrightarrow{\mathrm{b}} +\hspace{0.17em}5\overrightarrow{\mathrm{c}}$ are coplanar, then the value of $\mathrm{\lambda }$ is

• 2

• - 1

• 1

• - 2

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

If the vectors $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{a}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{a}}^2\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}}= \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{b}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{b}}^2\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{c}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{c}}^2\hat{\mathrm{k}}$ are three non-coplanar vectors and $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2& 1 + {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& 1 + {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& 1 + {\mathrm{c}}^3\end{array}\right| = 0$, then the value of abc is

• 0

• 1

• 2

• - 1

Let $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$  be three vectors. A vector in the plane of $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ whose projection on $\overrightarrow{\mathrm{a}}$ is magnitude $\sqrt{\frac{2}{3}}$, is

• $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 5\hat{\mathrm{k}}$

If the constant forces $\mathrm{}$ 2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$\stackrel{}{}$ and $$$- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$act on a particle due to which it is displaced from a point A (4,- 3, - 2) to a point B (6, 1,- 3), then the work done by the forces is

• 15 unit

• 9 unit

• - 15 unit

• - 9 unit

189.

If $\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = 4, \mathrm{then} \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right) . \left(2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$ is equal to

• 12

• 2

• 0

• - 12

$\overrightarrow{\mathrm{a}} \times \left[\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right]$ is equal to

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

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

• $\left[\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)\right]\overrightarrow{\mathrm{a}}$

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