Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

(a)

225

(b)

275

(c)

325

(d)

300

If the points whose position vectors \(10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\) and \(a\hat{i}+11\hat{j}\) are collinear then a is equal to

(a)

6

(b)

3

(c)

5

(d)

8

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

(a)

5

(b)

7

(c)

26

(d)

10

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

(a)

\(7\over4\)

(b)

\(15\over4\)

(c)

\(3\over4\)

(d)

\(17\over4\)

The value of m for which the vectors \(3\hat { i } -6\hat { j } +\hat { k } \) and \(2\hat { i } -4\hat { j } +\lambda \hat { k } \) are parallel is __________ .

(a)

\(\frac { 2 }{ 3 } \)

(b)

\(\frac { 3 }{ 2 } \)

(c)

\(\frac { 5 }{ 2 } \)

(d)

\(\frac { 2 }{ 5 } \)

The value of the expression \({ \left| \overrightarrow { a } \times \overrightarrow { b } \right| }^{ 2 }+{ (\overrightarrow { a } .\overrightarrow { b } ) }^{ 2 }\) is ___________ .

(a)

cos² \(\theta\)

(b)

sin² \(\theta\)

(c)

\({ |\overrightarrow { a } | }^{ 2 }|{ \overrightarrow { b } | }^{ 2 }\)

(d)

\({ (|\overrightarrow { a } |+|\overrightarrow { b } |) }^{ 2 }\)

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

(a)

\(\overrightarrow { a } =\overrightarrow { b } \)

(b)

\(\overrightarrow { a } =\overrightarrow { -b } \)

(c)

\(\overrightarrow { a } =\pm \overrightarrow { b } \)

(d)

both are null vectors

The vector having initial and terminal points as (2, 5, 0) and (-3, 7, 4) respectively is _________ .

(a)

\(-\hat{i}+12\hat{j}+4\vec{k}\)

(b)

\(-5\hat{i}+2\hat{j}-4\vec{k}\)

(c)

\(-5\hat{i}+2\hat{j}+4\vec{k}\)

(d)

\(\hat{i}+\hat{j}+\vec{k}\)

The direction cosines of the vector \(2\hat{i}+2\hat{j}-\hat{k}\) are ___________ .

(a)

\(\frac { 2 }{ 3 } ,\frac { 2 }{ 3 } ,-\frac { 1 }{ 3 } \)

(b)

\(\frac { 2 }{ 3 } ,\frac { 2 }{ 3 } ,\frac { 1 }{ 3 } \)

(c)

\(-\frac { 2 }{ 3 } ,-\frac { 2 }{ 3 } ,\frac { 1 }{ 3 } \)

(d)

\(-\frac { 2 }{ 3 } ,-\frac { 2 }{ 3 } ,-\frac { 1 }{ 3 } \)

The angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes\(\sqrt{3}\) and 4 respectively and \(\vec{a}.\vec{b}=2\sqrt{3}\) is ________ .

(a)

\(\frac{\pi}{6}\)

(b)

\(\frac { \pi }{ 3 } \)

(c)

\(\frac { \pi }{ 2 } \)

(d)

\(\frac { 5\pi }{ 2 } \)