If $\mathrm{a} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} - \mathrm{m}\hat{\mathrm{k}}$ and $\mathrm{b} = \frac{4}{7}\hat{\mathrm{i}} - \frac{2}{7}\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$  are collinear, then the value of m is equal to

• - 7

• - 1

• 2

• 7

Let $\mathrm{a} = 2\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 7\hat{\mathrm{k}}$ and $\mathrm{b} = \hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$.  Then, (3a - 5b) . (4a $\times$ 5b) is equal to

• - 7

• 0

• - 13

• 1

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 = 0 and a + b makes an angle of 60° with b, then $\left|\mathrm{a}\right|$ is equal to

• 0

• $\frac{1}{\sqrt{3}}\left|\mathrm{b}\right|$

• $\frac{1}{\left|\mathrm{b}\right|}$

• $\sqrt{3}\left|\mathrm{b}\right|$

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

• $\sqrt{41}$

• $\sqrt{39}$

• $\sqrt{19}$

• $\sqrt{29}$

The straight line r = $(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}) + \mathrm{a}(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}})$ meets the XY - plane at the point

• (2, - 1, 0)

• (3, 4, 0)

• $\left(\frac{1}{2}, \frac{3}{4}, 0\right)$

• $\left(\frac{1}{2}, \frac{5}{4}, 0\right)$

47.

The equation of the plane passing through (- 1, 5, - 7) and parallel to the plane 2x - 5y + 7z + 11 = 0, is

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} - 7\hat{\mathrm{k}}\right) + 76 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}}\right) + 76 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} -+ 7\hat{\mathrm{k}}\right) + 75 = 0$

• $\mathrm{r}. \left(2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 7\hat{\mathrm{k}}\right) + 65 = 0$

The angle subtended at the point (1, 2, 3) by the points P(2, 4, 5) and Q(3, 3, 1) is

• 90°

• 60°

• 30°

• 0°

If the two lines $\frac{\mathrm{x} - 1}{2} = \frac{1 - \mathrm{y}}{- \mathrm{a}} = \frac{\mathrm{z}}{4}$ and $\frac{\mathrm{x} - 3}{1} = \frac{2\mathrm{y} - 3}{4} = \frac{\mathrm{z} - 2}{2}$ are perpendicular, then the value of a is equal to

• - 4

• 5

• - 5

• 4

If the line $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} + 1}{3} = \frac{\mathrm{z} + 1}{4}$ meets the plane x + 2y + 3z = 14 at P, then the distance between P and the origin is

• $\sqrt{14}$

• $\sqrt{15}$

• $\sqrt{13}$

• $\sqrt{12}$