If the vectors $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}, \mathrm{\lambda }\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are orthogonal to each other, then $\mathrm{\lambda }$ is equal to

• 5

• - 5

• 8

• - 8

The vector c . (b + c) x (a + b + c) is equal to

• c . b x a

• 0

• c . a x b

• a . c x b

If the vector $\mathrm{a} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and b are collinear and $\left|\mathrm{b}\right| = 21$, then b is equal to

• $\pm \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

• $\pm 3\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

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

• $\pm 21\left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 6\hat{\mathrm{k}}\right)$

If a and b are unit vectors, then the vector (a + b) x (a x b) is parallel to the vector

• a - b

• a + b

• 2a - b

• 2a + b

I. Two non-zero, non-collinear vectors arelinearly independent.

II. Any three coplanar vectors are linearlydependent.Which ofthe above statements is/are true?

• Only I

• Only II

• Both I and II

• Neither I nor II

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

768.

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

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

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

• - 1

• $\frac{1}{2}$

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

• 2

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

• $\hat{\mathrm{j}}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{k}}$

• $\overrightarrow{0}$