If a + 2b - c =  0 and $\mathrm{a} \times \mathrm{b} + \mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} = \mathrm{\lambda }\left(\mathrm{a} + \mathrm{b}\right)$, then the value of $\mathrm{\lambda }$ is equal to

• 5

• 4

• 2

• - 2

If a + b and a - b are perpendicular and b = $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then $\left|\mathrm{a}\right|$ is equal to

• $\sqrt{41}$

• $\sqrt{39}$

• $\sqrt{19}$

• $\sqrt{29}$

If the vectors $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}}, 2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are coplanar, then the value of $\mathrm{\lambda }$ is

• - 10

• 1

• 0

• 2

Let a be a unit vector. If (x - a) . (x + a) = 12, then the magnitude of x is

• $\sqrt{8}$

• $\sqrt{9}$

• $\sqrt{10}$

• $\sqrt{13}$

A force of magnitude 5 unit acting along the vector $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ displaces the point of application from (1, 2, 3) to (5, 3, 7). Then the work done is :

• 50/7 unit

• 50/3 unit

• 25/3

• 25/4

An unit vector perpendicular to both $\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

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

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

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

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

The area of the triangle whose vertices are (1, 2, 3), (2, 5, -1) and (-1, 1, 2) is :

• 150 sq unit

• 145 sq unit

• $\frac{\sqrt{155}}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{155}{2} \mathrm{sq} \mathrm{unit}$

For any three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$, $\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$ + $\overrightarrow{\mathrm{b}} \times \left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right)$ + $\overrightarrow{\mathrm{c}} \times \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)$ is :

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) . \overrightarrow{\mathrm{c}}$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are any three vectors, then $\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right]$ is equal to

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• 0

• $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• ${\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]}^2$

The volume of the parallelopiped whose coterminus edges are $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}, 2\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ is :

• 4 cu unit

• 3 cu unit

• 2 cu unit

• 8 cu unit