If $\mathrm{\theta }$ is the angle between the vectors a = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and b = $6\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then

• $\cos \left(\mathrm{\theta }\right) = \frac{4}{21}$

• $\cos \left(\mathrm{\theta }\right) = \frac{3}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{2}{19}$

• $\cos \left(\mathrm{\theta }\right) = \frac{5}{21}$

22.

If a, b and c are three vectors, such that $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right| = 0$, $\left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2, \left|\mathrm{c}\right| = 3$, then a . b + b . c + c . a is equal to

• 0

• - 7

• 7

• 1

If the position vectors of P and Q are $\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 7\hat{\mathrm{k}}\right)$ and $\left(5\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$, then $\left|\mathrm{PQ}\right|$ is

• $\sqrt{158}$

• $\sqrt{160}$

• $\sqrt{161}$

• $\sqrt{162}$

Let u = $\hat{\mathrm{i}} + \hat{\mathrm{j}}$, v = $\hat{\mathrm{i}} - \hat{\mathrm{j}}$ and w = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\hat{\mathrm{n}}$ is a unit vector such that $\mathrm{u} . \hat{\mathrm{n}}$ = 0 and $\mathrm{v} . \hat{\mathrm{n}}$ = 0, then $\left|\mathrm{w} . \hat{\mathrm{n}}\right|$ is equal to

• 0

• 1

• 2

• 3

If A = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$, B = $2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and C = $\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, then $\left(\mathrm{A} \times \mathrm{B}\right) \times \mathrm{C}$ is

• $5\left(- \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $4\left(- \hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

• $5\left(- \hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$

• $4\left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$

A particle is acted upon by constant forces $4\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ which displace it from a point $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ to the point $5\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + \hat{\mathrm{k}}$. The work done in standard unit by the forces is given by

• 40

• 30

• 25

• 15

The volume ofa parallelepiped whose sides are given by a = $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}}$, b = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$ and c = $3\hat{\mathrm{i}} - \hat{\mathrm{k}}$ is

• 6 cu unit

• 5 cu unit

• 4 cu unit

• 3 cu unit

If a, b, c are non-coplanar vectors and 11, is a real number, then the vectors a + 2b + 3c, $\mathrm{\lambda }$b + 4c and (2$\mathrm{\lambda }$ - 1)c are non - coplanar for

• all values of $\mathrm{\lambda }$

• all except one value of $\mathrm{\lambda }$

• all expect two values of $\mathrm{\lambda }$

• no value of $\mathrm{\lambda }$

$\int \frac{\mathrm{dx}}{1 - {\mathrm{x}}^2}$ is equal to

• ${\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{c}$

• ${\sin }^{-1}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{1}{2}\log \left|\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right| + \mathrm{c}$

$\int \frac{\cos \left(2\mathrm{x}\right) - 1}{\cos \left(2\mathrm{x}\right) + 1}\mathrm{dx}$ is equal to

• $\tan \left(\mathrm{x}\right) - \mathrm{x} + \mathrm{c}$

• $\mathrm{x} + \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x} - \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $- \mathrm{x} - \cot \left(\mathrm{x}\right) + \mathrm{c}$