If a, b, c are three vectors such that [a b c] = 5, then the value of [a x b, b x c, c x a] is

• 15

• 25

• 20

• 10

Three forces each of magnitude F are applied along the edges of a regular hexagon as shown in the figure. Each side of hexagon is a. What is the resultant moment (torque) of these three forces about centre O ?

• $\frac{3\sqrt{3}}{2}\mathrm{aF}$

• $\frac{1}{2}\mathrm{aF}$

• 3aF

• $\frac{\sqrt{3}}{2}\mathrm{aF}$

The coordinates of a moving point particle in a plane at time t is given by x = a(t + sin(t)), y = a(1 - cos(t)). The magnitude of acceleration of the particle is acceleration ofthe particle is

• 2a

• $\frac{\sqrt{3}}{2}\mathrm{a}$

• a

• $\sqrt{3}\mathrm{a}$

Two vectors A = 3 and B = 4 are perpendicular. Resultant of both these vectors is R. The projection of the vector B on the vector R is

• 5

• 1.25

• 3.2

• 2.4

A vector R is given by R = A x (B x C), which of the following is true ?

• R must be perpendicular to 8

• R is parallel to A

• R must be parallel to B

• None of these

The maximum resultant of two forces is 2P and the minimum resultant is 2Q, the two forces are at right angles, then the resultant is

• P + Q

• P - Q

• $\sqrt{\frac{{\mathrm{P}}^2 + {\mathrm{Q}}^2}{2}}$

• $\sqrt{2\left({\mathrm{P}}^2 + {\mathrm{Q}}^2\right)}$

In a right angle $∆\mathrm{ABC}$, $\angle$A = 90° and sides a, b, c are respectively 10 cm, 8 cm and 6 cm. If a force F has moments 0, 64 and 36 N-cm respectively about vertices A, B and C, then magnitude of F is

• 9

• 4

• 10

• 8

If the vectors a and b are linearly independent satisfying $\left(\sqrt{3}\tan \left(\mathrm{\theta }\right) + 1\right)\mathrm{a} + \left(\sqrt{3}\sec \left(\mathrm{\theta }\right) - 2\right)\mathrm{b} = 0$, then the value of $\mathrm{\theta }$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{11\mathrm{\pi }}{6}$

If a and b are two unit vectors inclined at an angle $\frac{\mathrm{\pi }}{3}$, then $\left\{\mathrm{a} \times \left(\mathrm{b} + \mathrm{a} \times \mathrm{b}\right)\right\} . \mathrm{b}$ is equal to

• $\frac{1}{4}$

• $\frac{- 3}{4}$

• $\frac{3}{4}$

• $\frac{1}{2}$

Let u, v and w be such that $\left|\mathrm{u}\right| = 1, \left|\mathrm{v}\right| = 3 \mathrm{and} \left|\mathrm{w}\right| = 2$. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then $\left|\mathrm{u} - \mathrm{v} + \mathrm{w}\right|$ equals

• 2

• $\sqrt{7}$

• $\sqrt{14}$

• 14