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

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

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

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

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

If $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2& 1 + {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& 1 + {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& 1 + {\mathrm{c}}^3\end{array}\right|$ = 0 and the vectors $\overrightarrow{\mathrm{A}} = \left(1, \mathrm{a}, {\mathrm{a}}^2\right), \overrightarrow{\mathrm{B}} = \left(1, \mathrm{b}, {\mathrm{b}}^2\right) \mathrm{and} \overrightarrow{\mathrm{C}} = \left(1, \mathrm{c}, {\mathrm{c}}^2\right)$ are non- coplana, then abc is equal to

• 0

• - 1

• 1

• None of these

The magnitude of the two forces forming a couple is 36 N each and the arm of the couple is 4m. The magnitude of each force of an equivalent couple whose arm is 9m, is (in Newtons)

• 18

• 26

• 16

• 15

If A, B and C are the vertices of a triangle whose position vectors are $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ respectively and G is the centroid of the $∆$AABC , then $\overrightarrow{\mathrm{GA}} + \overrightarrow{\mathrm{GB}} + \overrightarrow{\mathrm{GC}}$ is

• $\overrightarrow{0}$

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

• $\frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{3}$

• $\frac{\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}}{3}$

Two bodies of masses m and 4m are moving with equal momentum. The ratio of their KE is

• 1 : 4

• 4 : 1

• 1 : 1

• 1 : 2

A particle is projected from the top oftower 5 m high and at the same moment another particle is projected upward from the bottom of the tower with a speed of 10 m/s, meet at distance h' from the top oftower, then h is equal to

• 1.20 m

• 2.5 m

• 1.25 m

• None of these