Given (a x b) x (c x d) = 5c x 6d, then the value of (a . b) x (a + c + 2d) is

• 7

• 16

• - 1

• 4

Let a, b and c be non-zero vectors such that $\left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c} = \frac{- 1}{4}\left|\mathrm{b}\right|\left|\mathrm{c}\right|\mathrm{a}$.  If $\mathrm{\theta }$ the acute angle between the vectors b and c, then the angle between a and c is equal to

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

A vector of magnitude 12 unit perpendicular to the plane containing the vectors 4i + 6j - k and 3i + 8j + k is

• - 8i + 4j + 8k

• 8i + 4j + 8k

• 8i - 4j + 8k

• 8i - 4j - 8k

Forces of magnitudes 3 and 4 unit acting alon 6i + 2j + 3k and 3i - 2j + 6k, respectively act on a particle and displace it from (2, 2,- 1) to (4, 3, 1). The work done is

• $\frac{124}{7}$

• $\frac{120}{7}$

• $\frac{125}{7}$

• $\frac{121}{7}$

415.

If ABCD be a parallelogram and M be the point of intersection of the diagonals. If O is any point, then OA + OB + OC + OD is

• 3OM

• 4OM

• OM

• 2OM

If D, E and F are the mid points of the sides BC, CA and AB, respectively of the $∆$ABC and G is the centroid of the triangle then GD + GE + GF is

• 0

• 2AB

• 2GA

• 2GC

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are position vectors of the vertices of the triangle ABC , then $\frac{\left|\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}\right) \times \left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{a}}\right)\right|}{\left(\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right) . \left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{a}}\right)}$ is equal to

• cot(A)

• cot(C)

• - tan(C)

• tan(A)

If $\overrightarrow{\mathrm{a}}$ is a vector of magnitude 50, collinear with the vector $\overrightarrow{\mathrm{b}} = 6\hat{\mathrm{i}} - 8\hat{\mathrm{j}} - \frac{15}{2}\hat{\mathrm{k}}$ and makes an acute angle with the positive direction of z-axis, then $\overrightarrow{\mathrm{a}}$ is equal to

• $- 24\hat{\mathrm{i}} + 32\hat{\mathrm{j}} + 30\hat{\mathrm{k}}$

• $24\hat{\mathrm{i}} - 32\hat{\mathrm{j}} - 30\hat{\mathrm{k}}$

• $12\hat{\mathrm{i}} - 16\hat{\mathrm{j}} - 15\hat{\mathrm{k}}$

• $- 12\hat{\mathrm{i}} + 16\hat{\mathrm{j}} - 15\hat{\mathrm{k}}$

If the volume of a parallelopiped with $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$ as coterminus edges is 9 cu units, then the volume of the parallelopiped with $\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right), \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right) \times \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right), \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right) \times \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ as coterminus edges is

• 9 cu unit

• 729 cu unit

• 81 cu unit

• 27 cu unit

If the constant forces $2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$ and $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ act on a particle due to which it is displaced from a point A ( 4, - 3, - 2) to a point B (6, 1, - 3) then the work done by the forces is

• 10 unit

• - 10 unit

• 9 unit

• None of the above