If I is incentre of $∆$ABC, then I is

• $\frac{\mathrm{a}\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{b}\overrightarrow{\mathrm{b}} +\hspace{0.17em}\mathrm{c}\overrightarrow{\mathrm{c}}}{\mathrm{a} +\hspace{0.17em}\mathrm{b} +\hspace{0.17em}\mathrm{c}}$

• $\frac{\mathrm{a}\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{b}\overrightarrow{\mathrm{b}} +\hspace{0.17em}\mathrm{c}\overrightarrow{\mathrm{c}}}{\sqrt{{\mathrm{a}}^2 +\hspace{0.17em}{\mathrm{b}}^2 +\hspace{0.17em}{\mathrm{c}}^2}}$

• $\frac{1}{3}\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right]$

• $\frac{\overrightarrow{\mathrm{a}} +\hspace{0.17em}\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{\mathrm{a} +\hspace{0.17em}\mathrm{b} +\hspace{0.17em}\mathrm{c}}$

What are the DR's of vector parallel to (2, - 1, 1) and (3 4, - 1) ?

• (1, 5, - 2)

• (- 2, - 5, 2)

• (- 1, 5, 2)

• (- 1, - 5, - 2)

Let ABCD be a parallelogram whose diagonals intersect at P and O be the origin, then $\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OB}} + \overrightarrow{\mathrm{OC}} + \overrightarrow{\mathrm{OD}}$

• $\overrightarrow{\mathrm{OP}}$

• $2\overrightarrow{\mathrm{OP}}$

• $3\overrightarrow{\mathrm{OP}}$

• $4\overrightarrow{\mathrm{OP}}$

A unit vector in the plane of $\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$ and $\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$ and perpendicular to $2\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$, is

• $\hat{\mathrm{j}} -\hspace{0.17em}\hat{\mathrm{k}}$

• $\frac{\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}}{\sqrt{2}}$

• $\frac{\hat{\mathrm{j}} -\hspace{0.17em}\hat{\mathrm{k}}}{\sqrt{2}}$

The line joining the points $6\overrightarrow{\mathrm{a}} - 4\overrightarrow{\mathrm{b}} + 4\overrightarrow{\mathrm{c}}, - 4\overrightarrow{\mathrm{c}}$ and the line joining the points $- \overrightarrow{\mathrm{a}} - 2\overrightarrow{\mathrm{b}} - 3\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{b}} - 5\overrightarrow{\mathrm{c}}$ intersect at

• $- 4\overrightarrow{\mathrm{a}}$

• $4\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}$

• $4\overrightarrow{\mathrm{c}}$

• None of the above

The symmetric equation of lines 3x + 2y + z - 5 = 0 and x + y - 2z - 3 = 0, is

• $\frac{\mathrm{x} - 1}{5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} + 1}{5} = \frac{\mathrm{y} + 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} + 1}{- 5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

• $\frac{\mathrm{x} - 1}{- 5} = \frac{\mathrm{y} - 4}{7} = \frac{\mathrm{z} - 0}{1}$

   

The equation of the plane containing the line $\frac{\mathrm{x} + 1}{- 3} = \frac{\mathrm{y} - 3}{2} = \frac{\mathrm{z} + 2}{1}$ and the point (0, 7, - 7) is

• x + y + z = 1

• x + y + z = 2

• x + y + z = 0

• None of these

Angle of intersection of the curve r = $\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)$ and r = 2$\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• None of these

239.

A point on XOZ - plane divides the join of (5, - 3, - 2) and (1, 2, - 2) at

• $\left(\frac{13}{5}, 0, - 2\right)$

• $\left(\frac{13}{5}, 0, 2\right)$

• (5, 0, 2)

• (5, 0, - 2)

If the line $\overrightarrow{\mathrm{OR}}$ makes angles ${\mathrm{\theta }}_1, {\mathrm{\theta }}_2, {\mathrm{\theta }}_3$, with the planes XOY, YOZ, ZOX respectively, then ${\cos }^2\left({\mathrm{\theta }}_1\right) + {\cos }^2\left({\mathrm{\theta }}_2\right) + {\cos }^2\left({\mathrm{\theta }}_3\right)$, is equal to

• 1

• 2

• 3

• 4