If the four points with position vectors $- 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \mathrm{\lambda }\hat{\mathrm{j}} + \hat{\mathrm{k}}$ are coplanar, then $\mathrm{\lambda }$ is equal to

• 1

• 2

• - 1

• 0

42.

The equation $\mathrm{z}\overline{\mathrm{z}} + \mathrm{a}\overline{\mathrm{z}} + \overline{\mathrm{a}}\mathrm{z} + \mathrm{b} = 0$, represents a circle, if

• ${\left|\mathrm{a}\right|}^2 = \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 > \mathrm{b}$

• ${\left|\mathrm{a}\right|}^2 <\hspace{0.17em}\mathrm{b}$

•  None of the above

If a, b, c are non-coplanar unit vectors such that

$\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = \frac{\mathrm{b} + \mathrm{c}}{\sqrt{2}}$, then the angle between a and b is

• $\frac{3\mathrm{\pi }}{4}$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\mathrm{\pi }$

If a, b, c and a', b',c' form a reciprocal system of vectors, then a . a'+ b . b' + c · c' is equal to

• 0

• 1

• 2

• 3

If a, b, c are three non - zero vectors which are pairwise non-collinear. Also, a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is

• c

• 0

• a + c

• a

If d = a x b + b x c + c x a is a non-zero vector and $\left|(\mathrm{d} · \mathrm{c}) (\mathrm{a} \mathrm{x} \mathrm{b}) + (\mathrm{d} · \mathrm{a}) (\mathrm{b} \mathrm{x} \mathrm{c}) + (\mathrm{d} . \mathrm{b}) (\mathrm{c} \mathrm{x} \mathrm{a})\right|$ = 0, then

• $\left|\mathrm{a}\right| + \left|\mathrm{b}\right| + \left|\mathrm{c}\right| = \left|\mathrm{d}\right|$

• $\left|\mathrm{a}\right| = \left|\mathrm{b}\right| = \left|\mathrm{c}\right|$

• a, b and c are coplanar

• None of the above

Let u, v and w be such that $\left|\mathrm{u}\right|$ = 1, $\left|\mathrm{v}\right|$ = 2 and $\left|\mathrm{w}\right|$ = 3. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then $\left|\mathrm{u} - \mathrm{v} + \mathrm{w}\right|$ equals

• 2

• $\sqrt{7}$

• $\sqrt{14}$

• 14

If a and b are two vectors, such that a . b < 0 and $\left|\mathrm{a} . \mathrm{b}\right| = \left|\mathrm{a} \times \mathrm{b}\right|$ then the angle between vectors a and b is

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

• $\frac{7\mathrm{\pi }}{4}$

A tetrahedron has vertices at 0(0, 0, 0), A(1, - 2, 1), B (-2, 1, 1) and C (1, - 1, 2). Then, the angle between the faces OAB and ABC will be

• ${\cos }^{-1}\left(\frac{1}{2}\right)$

• ${\cos }^{-1}\left(\frac{- 1}{6}\right)$

• ${\cos }^{-1}\left(\frac{- 1}{3}\right)$

• ${\cos }^{-1}\left(\frac{1}{4}\right)$

If a, b and c are three non-coplanar vectors, then (a + b - c) . [(a - b) x (b - c)] equals

• 0

• a . b $\times$ c

• a . c $\times$ b

• 3a . b $\times$ c