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

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

396.

If a = $- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} + \hat{\mathrm{k}}$, then the vector satisfyin the following conditions

(i) it is coplanar witha and b,

(ii) it is perpendicular to b and

(iii) a · c = 7, is

• $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $- \frac{3}{2}\hat{\mathrm{i}} + \frac{5}{2}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• $- 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $- 6\hat{\mathrm{i}} + \hat{\mathrm{k}}$

If the vectors $\mathrm{b} = \left(\tan \left(\mathrm{\alpha }\right), - 1, 2\sqrt{\sin \left(\frac{\mathrm{\alpha }}{2}\right)}\right)$ and $\mathrm{c} = \left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\alpha }\right), - \frac{3}{\sqrt{\sin \left({\displaystyle \raisebox{1ex}{$\mathrm{\alpha }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}}\right)$ are orthagonal and  a vector a = $\left(1, 3, \sin \left(2\mathrm{\alpha }\right)\right)$ makes an obtuse angle with the Z-axis, then the value of $\mathrm{\alpha }$ is

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

${\left(\mathrm{r} . \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{k}}\right)}^2$ is equal to

• 0

• 1

• r²

• 3r²

The component of $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ along $\hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{k}}$ will be

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{2}$

• $\frac{\hat{\mathrm{j}} + \hat{\mathrm{k}}}{2}$

• $\frac{\hat{\mathrm{k}} + \hat{\mathrm{i}}}{2}$

• None of these

If $\mathrm{a} = 2\hat{\mathrm{i}} + 5\hat{\mathrm{j}}$ and $\mathrm{b} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}}$, then the unit vector along a + b will be

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

• $\sqrt{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$