Observe the following statements

A. Three vectors are coplanar if one of them is expressible as a linear combination of the other two.

R. Any three coplanar vectors are linearly dependent.Then, which of the following is true ?

• Both A and R are true and R is the correct explainaton of A

• Both A and R are true but R is not the correct explainaton of A

• A is true, but R is false

• A is false, but R is true

Observe the following lists

List I	List II
(A) [a b c]	1. $\left|\mathrm{a}\right|\left|\mathrm{b}\right|\cos \left(\mathrm{ab}\right)$
(B) $\left(\mathrm{c} \times \mathrm{a}\right) \times \mathrm{b}$	2 .(a . c)b - (a . b) c
(C) $\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right)$	3. $\mathrm{a} . \mathrm{b} \times \mathrm{c}$
(D) a . b	4. $\left|\mathrm{a}\right|\left|\mathrm{b}\right|$
	5. (b . c)a - (a . b)c

Then the correct match for List I from List II is

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0} \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 4 \mathrm{and} \\ \left|\overrightarrow{\mathrm{c}}\right| = \sqrt{37}, \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{is} : \end{gathered}$$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{If} \hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \mathrm{\lambda }\hat{\mathrm{i}} + 3\hat{\mathrm{j}} \mathrm{are} \mathrm{coplaner}, \mathrm{then} \mathrm{\lambda } = ?$

• - 1

• $\frac{1}{2}$

• $\mathit{-}\mathit{ }\frac{\mathit{3}}{\mathit{2}}$

• 2

425.

The position vector of a point lying on the line joining the points whose positions vectors are $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{is} :$

• $\hat{\mathrm{j}}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{k}}$

• $\overrightarrow{0}$

If the volume of parallelopiped with conterminus edges $4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}, - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ is 34 cubic units, then p is equal

• 4

• - 13

• 13

• 6

$\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = \overrightarrow{\mathrm{a}}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \overrightarrow{\mathrm{a}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) = 1, \mathrm{then} \overrightarrow{\mathrm{a}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\hat{\mathrm{i}} - \hat{\mathrm{k}}$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{points} \mathrm{whose} \mathrm{position} \mathrm{vectors} \mathrm{are} \\ 2\hat{\mathrm{i}} + \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}, 6\hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}} \mathrm{and} 14\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}} \\ \mathrm{are} \mathrm{collinear}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{p} \mathrm{is} \end{gathered}$$

• 2

• 4

• 6

• 8

$$\begin{gathered} \mathrm{The} \mathrm{ratio} \mathrm{in} \mathrm{which} \hat{\mathrm{i}} + 2\hat{\mathrm{j}}\hspace{0.17em}+ 3\hat{\mathrm{k}} \mathrm{divides} \mathrm{the} \mathrm{join} \mathrm{of} - 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} +\hspace{0.17em}5\hat{\mathrm{k}} \mathrm{and} \\ 7\hat{\mathrm{i}} - \hat{\mathrm{k}} \mathrm{is} \end{gathered}$$

• 2 : 1

• 2 : 3

• 3 : 4

• 1 : 4

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = + \mathrm{\lambda }\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \mathrm{the} \mathrm{orthogonal} \mathrm{projection} \mathrm{of} \\ \overrightarrow{\mathrm{b}} \mathrm{on} \overrightarrow{\mathrm{a}} \mathrm{is} \frac{4}{3}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}\right), \mathrm{then} \mathrm{\lambda } = ? \end{gathered}$$

• 0

• 2

• 12

• - 1