451.

If p and q are non-collinear unit vectors and $\left|\mathrm{p} + \mathrm{q}\right| = \sqrt{3}$, then (2p - 3q) · (3p + q) is equal to

• 0

• $\frac{1}{3}$

• $- \frac{1}{3}$

• $- \frac{1}{2}$

The triangle formed by the three points whose position vectors are 2i + 4j - k, 4i + 5j + k and 3i + 6j - 3k, is

• an equilateral triangle

• a right angled triangle but not isosceles

• an isosceles triangle but not right angled triangle

• a right angled isosceles triangle

If (1, 2, 4) and (2, - 3$\mathrm{\lambda }$, - 3) are the initial and terminal points of the vector i + 5j - 7k, then the value of $\mathrm{\lambda }$ is

• $\frac{7}{3}$

• $\frac{- 7}{3}$

• $\frac{- 5}{3}$

• $\frac{5}{3}$

Let u = 5a + 6b + 7c, v = 7a - 8b + 9c and w = 3 a + 20b + 5c, where a, b and c are non-zero vectors. If u = lv + mw, then the values of l and m respectively are

• $\frac{1}{2}, \frac{1}{2}$

• $\frac{1}{2}, - \frac{1}{2}$

• $- \frac{1}{2}, \frac{1}{2}$

• $\frac{1}{3}, \frac{1}{3}$

If $\overrightarrow{\mathrm{\alpha }} = 3\mathrm{i} - \mathrm{k}$ and $\left|\overrightarrow{\mathrm{\beta }}\right| = \sqrt{5}$ and $\overrightarrow{\mathrm{\alpha }} . \overrightarrow{\mathrm{\beta }}$ = 3 then the area of the parallelogram for which $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are adjacent sides, is

• $\frac{\sqrt{17}}{2}$

• $\frac{\sqrt{14}}{2}$

• $\frac{\sqrt{7}}{2}$

• $\sqrt{41}$

If the vectors PQ = - 3i + 4j + 4k and PR = 5i - 2j + 4k are the sides of a $∆$PQR, then the length of the median through P is

• $\sqrt{14}$

• $\sqrt{15}$

• $\sqrt{17}$

• $\sqrt{18}$

If $\mathrm{a} = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \left|\mathrm{b}\right| = 5$ and the angle between a and b is $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$, then the area of the triangle formed by these two vectors as two sides is

• $\frac{15}{4}$

• $\frac{15}{2}$

• 15

• $\frac{15\sqrt{3}}{2}$

If a - b = 0 and a + b makes an angle of 60° with a, then

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

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

• $\left|\mathrm{a}\right| = \sqrt{3}\left|\mathrm{b}\right|$

• $\sqrt{3}\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$

If $\hat{\mathrm{i}} + \hat{\mathrm{j}}, \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} + \hat{\mathrm{k}}$ are the position vectors of the vertices of a $∆$ABC taken in order, then $\angle$A is equal to

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

Let a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If b is a vector such that a · b = ${\left|\mathrm{b}\right|}^2$ and  $\left|\mathrm{a} - \mathrm{b}\right|$ = $\sqrt{7}$, then $\left|\mathrm{b}\right|$ is equal to

• $\sqrt{7}$

• $\sqrt{3}$

• 7

• 3