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

76.

If $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}} \mathrm{and} \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}} = \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}$, then

• $\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{d}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{d}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right)$

• $\left(\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{d}}\right)$

• $\left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right) = \mathrm{\lambda }\left(\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{d}}\right)$

A unit vector in xy-plane makes an angle of 45° with the vector $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and an angle of 60° with the vector $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}}$, is

• $\hat{\mathrm{i}}$

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

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

• None of these

Let a, b, c be distinct non-negative numbers. If the vector $\mathrm{a}\hat{\mathrm{i}} + \mathrm{a}\hat{\mathrm{j}}+ \mathrm{c}\hat{\mathrm{k}}, \hat{\mathrm{i}}+ \hat{\mathrm{k}} \mathrm{and} \mathrm{c}\hat{\mathrm{i}} + \mathrm{c}\hat{\mathrm{j}} + \mathrm{b}\hat{\mathrm{k}}$, i+f and ci + cj + bk lies in a plane, then c is

• the GM of a and b

• the GM of a and b

• the HM of a and b

• equal to zero

If $\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}} = \overrightarrow{\mathrm{r}} \mathrm{and} \overrightarrow{\mathrm{q}} \times \overrightarrow{\mathrm{r}} = \overrightarrow{\mathrm{p}}$, then

• r = 1, p = q

• p = 1, q = 1

• r = 2p, q = 2

• q = 1, p = r

Vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are inclined at an angle 8 = 120°. If $\left|\overrightarrow{\mathrm{a}}\right| = 1 \mathrm{and} \left|\overrightarrow{\mathrm{b}}\right| = 2,$ then ${\left[\left(\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}}\right) \times \left(3\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right)\right]}^2$ is equal to 

• 190

• 275

• 300

• 192