Mathematics : Vector Algebra

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are two non-zero non-collinear vectors, then $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{i}}\right]\hat{\mathrm{i}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{j}}\right]\hat{\mathrm{j}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{k}}\right]\hat{\mathrm{k}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}}\right]$ is equal to

• $2\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$

• $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$

• None of these

If ${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2 + {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2 = 676 \mathrm{and} \left|\overrightarrow{\mathrm{b}}\right|$ = 2, then $\left|\overrightarrow{\mathrm{a}}\right|$ is equal to

• 13

• 26

• 39

• None of these

If $\overrightarrow{\mathrm{a}}$ is any vector, then $\hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right) +\hspace{0.17em}\hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ is equal to

• $\overrightarrow{\mathrm{a}}$

• $2\overrightarrow{\mathrm{a}}$

• $3\overrightarrow{\mathrm{a}}$

• $\overrightarrow{0}$

The product of two times of flight from a point P to another point Q with a given velocity of projection is

• $\frac{\mathrm{PQ}}{\mathrm{g}}$

• $\frac{2\mathrm{PQ}}{\mathrm{g}}$

• $\frac{\mathrm{PQ}}{2\mathrm{g}}$

• $\frac{3\mathrm{PQ}}{\mathrm{g}}$

A stone of mass m is thrown vertically upwards with a velocity of 9.8 ms^-1. The height of the point where KE = PE is (g = 9.8 ms^-2)

• 9.8 m

• 4.9 m

• 2.45 m

• 2 m

Three sides of a square are formed by bending a rod of length 36 cm. The distance of its CG from an open end is

• 12 cm

• $6\sqrt{2} \mathrm{cm}$

• $8\sqrt{2} \mathrm{cm}$

• None of these

If ABCDEF is a regular hexagon with $\overrightarrow{\mathrm{AB}} = \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{BC}} = \overrightarrow{\mathrm{b}}$, then $\overrightarrow{\mathrm{CE}}$ equals

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

• $- \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{b}} - 2\overrightarrow{\mathrm{a}}$

• None of these

Magnitudes of vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are 3, 4, 5 respectively. If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}},$ $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$  are mutually perpendicular, then magnitude of $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$ is

• $4\sqrt{2}$

• $3\sqrt{2}$

• $5\sqrt{2}$

• $3\sqrt{3}$

If forces $\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}, \overrightarrow{\mathrm{R}}$ acting at a point can be represented by the sides of a triangle taken in order, then

• $\overrightarrow{\mathrm{P}} + \overrightarrow{\mathrm{Q}} + \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} - \overrightarrow{\mathrm{Q}} + \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} + \overrightarrow{\mathrm{Q}} - \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} - \overrightarrow{\mathrm{Q}} - \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

A sphere S₁ impings directly on an equal sphere S₂ at rest. If the coefficient of restitution is e, then the velocities of S₁ and S₂ are in the ratio

• $\frac{1 + \mathrm{e}}{1 - \mathrm{e}}$

• $\frac{1 - \mathrm{e}}{1 + \mathrm{e}}$

• $\frac{\mathrm{e} - 1}{\mathrm{e} + 1}$

• $\frac{\mathrm{e} + 1}{\mathrm{e} - 1}$