$$\begin{gathered} \mathrm{The} \mathrm{volume} (\mathrm{in} \mathrm{cubic} \mathrm{units}) \mathrm{of} \mathrm{the} \mathrm{tetrahedron} \mathrm{with} \mathrm{edges} \hat{\mathrm{i}} + \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}, \\ \hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hspace{0.17em}\hat{\mathrm{k}} \mathrm{is} \end{gathered}$$

• 4

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

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

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

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} = {\mathrm{a}}_1\hat{\mathrm{i}} +\hspace{0.17em}{\mathrm{a}}_2\hat{\mathrm{j}} + {\mathrm{a}}_3\hat{\mathrm{k}} \\ \mathrm{Assertion} \left(\mathrm{A}\right) : \mathrm{The} \mathrm{identity} \\ {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right|}^2 +\hspace{0.17em}{\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right|}^2 = 2{\left|\overrightarrow{\mathrm{a}}\right|}^2 \mathrm{holds} \mathrm{for} \\ \overrightarrow{\mathrm{a}} \\ \mathrm{Reason} \left(\mathrm{R}\right) : \overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}} = {\mathrm{a}}_3\hat{\mathrm{j}} - {\mathrm{a}}_2\hat{\mathrm{k}}, \\ \overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}} = {\mathrm{a}}_1\hat{\mathrm{k}} - {\mathrm{a}}_3\hat{\mathrm{i}}, \overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}} = {\mathrm{a}}_2\hat{\mathrm{i}} - {\mathrm{a}}_1\hat{\mathrm{j}} \\ \mathrm{Wich} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both (A) and (R) are true and (R) is the correct reason for (A)

• Both (A) and (R) are true but (R) is not the correct reason for (A)

• (A) is true, (R) is false

• A) is false, (R) is true

The position vectors of P and Q are $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ respectively. If R is a point on PQ such that $\overrightarrow{\mathrm{PR}} = 5\overrightarrow{\mathrm{PQ}}$, then the position vector of R is

• $5\overrightarrow{\mathrm{b}} - 4\overrightarrow{\mathrm{a}}$

• $5\overrightarrow{\mathrm{b}} + 4\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{b}} + 5\overrightarrow{\mathrm{a}}$

If the points with position vectors $60\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, 40\hat{\mathrm{i}} - 8\hat{\mathrm{j}} \mathrm{and} \mathrm{a}\hat{\mathrm{i}} - 52\hat{\mathrm{j}} \mathrm{are} \mathrm{collinear}, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to}$  are collinear,then a is equal to

• - 40

• - 20

• 20

• 40

If the position vectors of A, B and C are respectively $2\hat{\mathrm{I}} - \hat{\mathrm{J}} + \hat{\mathrm{K}}, \hat{\mathrm{I}} - 3\hat{\mathrm{J}} - 5\hat{\mathrm{K}} \mathrm{and} 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{then} {\cos }^2\left(\mathrm{A}\right) = ?$ 

• 0

• $\frac{6}{41}$

• $\frac{35}{41}$

• 1

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}} \mathrm{be} \mathrm{a} \mathrm{unit} \mathrm{vector}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} =\hat{\mathrm{i}} + 3\hat{\mathrm{k}}. \\ \mathrm{Then}, \mathrm{maximum} \mathrm{value} \mathrm{of}[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b} }\overrightarrow{\mathrm{c}}] \mathrm{is} \end{gathered}$$

• - 1

• $\sqrt{10} + \sqrt{6}$

• $\sqrt{10} - \sqrt{6}$

• $\sqrt{59}$

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}} =\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}$, then observe the following lists

	List-I		List-II
(i)	$\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$	(A)	$\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{d}}$
(ii)	$\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}}$	(B)	3
(iii)	$\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$	(C)	$\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{d}}$
(iv)	$\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$	(D)	$2\hat{\mathrm{i}} - \hat{\mathrm{k}}$
		(E)	$2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$
		(F)	4

Then, correct match of List_I to List-II is

$$\begin{gathered} \mathrm{If} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{are} \mathrm{respectively} \mathrm{the} \mathrm{magnitudes} \mathrm{of} \mathrm{the} \mathrm{vectors} \\ {\overrightarrow{\mathrm{a}}}_1 = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, {\overrightarrow{\mathrm{a}}}_2 = 3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \\ {\overrightarrow{\mathrm{a}}}_3 = \hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} {\overrightarrow{\mathrm{a}}}_4 = - \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}, \mathrm{then} \mathrm{the} \mathrm{correct} \mathrm{order} \mathrm{of} {\mathrm{m}}_1, {\mathrm{m}}_2, {\mathrm{m}}_3 \mathrm{and} {\mathrm{m}}_4 \mathrm{is} \end{gathered}$$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_4 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_1 < {\mathrm{m}}_2 < {\mathrm{m}}_4$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_1 < {\mathrm{m}}_2$

• ${\mathrm{m}}_3 < {\mathrm{m}}_4 < {\mathrm{m}}_2 < {\mathrm{m}}_1$

$\mathrm{Let} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is parallel to the plane containing $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$, then $\mathrm{\lambda }$ is equal to

• 0

• 1

• - 1

• 2

440.

If three unit vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ satisfy $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$, then the angle between a and b is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$