The value of $\left(\mathrm{i} + \mathrm{j}\right) . \left[\left(\mathrm{j} + \mathrm{k}\right) \times \left(\mathrm{k} + \mathrm{i}\right)\right]$ is

• 0

• 1

• - 1

• 2

If $\hat{\mathrm{a}} \mathrm{and} \hat{\mathrm{b}}$ are unit vectors and 0 is the angle between them, then $\sin \left(\raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ is equal to

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{2}$

• $\frac{\hat{\mathrm{a}} - \hat{\mathrm{b}}}{2}$

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{2}$

• $\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|$

If a, b and c are three non-zero, non-coplanar vectors, then the value of a x a' + b x b'+ c x c' is

• 1

• 0

• - 1

• None of the above

Three concurrent edges of a parallelopiped are given by

$$\begin{gathered} \mathrm{a} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \\ \mathrm{b} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \\ \mathrm{and} \mathrm{c} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \end{gathered}$$

The volume of the parallelopiped is

• 14 cu units

• 20 cu units

• 25 cu units

• 60 cu units

The number of vectors of unit length perpendicular to vectors a = $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and b = $\hat{\mathrm{j}} + \hat{\mathrm{k}}$

• infinite

• one

• two

• three

If $\mathrm{a} = \frac{\hat{\mathrm{i}} - 2\hat{\mathrm{j}}}{\sqrt{5}} \mathrm{and} \mathrm{b} = \frac{2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}}{\sqrt{14}}$ are vectors in space, then the value of $\left(2\mathrm{a} +\hspace{0.17em}\mathrm{b}\right) . \left[\left(\mathrm{a} \times \mathrm{b}\right) \times \left(\mathrm{a} - 2\mathrm{b}\right)\right]$ is

• 0

• 1

• 5

• 4

The value of $\hat{\mathrm{i}} . \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) + \hat{\mathrm{j}} . \left(\hat{\mathrm{k}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{k}} . \left(\hat{\mathrm{i}} \times \hat{\mathrm{j}}\right)$ is

• 0

• 1

• 3

• - 3

738.

The values of $\mathrm{\lambda }$, such that (x, y, z) if (0, 0, 0) and $\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$x + $\left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$y + $\left(- 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}}\right)$ are

• 0, 1

• - 1, 1

• - 1, 0

• - 2, 0

If G and G' are respectively centroid of $∆$ABC and $∆$A' B' C', then AA' + BB' + CC' is equal to

• 2GG'

• 3GG'

• $\frac{2}{3}\mathrm{GG}\text{'}$

• $\frac{1}{3}\mathrm{GG}\text{'}$

If a = $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$, b = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $- 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$, and if [·] is the least integer function, then [a + b + c] is equal to

• 1

• 2

• 3

• 0