Mathematics : Vector Algebra

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

If $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$, $\left|\overrightarrow{\mathrm{a}}\right| = 2, \left|\overrightarrow{\mathrm{b}}\right| = 3, \left|\overrightarrow{\mathrm{c}}\right| = 4$ and the angle between $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ is $\frac{2\mathrm{\pi }}{3}$, then $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$ is equal to

• $4\sqrt{3}$

• $6\sqrt{3}$

• $12\sqrt{3}$

• $18\sqrt{3}$

The vector $\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude, then it becomes $4\hat{\mathrm{i}} + \left(4\mathrm{x} - 2\right)\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. The value of x is

• $\left\{- \frac{2}{3}, 2\right\}$

• $\left\{\frac{1}{3}, 2\right\}$

• $\left\{\frac{2}{3}, 0\right\}$

• {2, 7}

A ball of mass 3 kg moving with a velocity of 3 m/s collides with another ball of mass 1 kg moving with velocity u in the opposite direction. If the first ball comes to rest after the impact and e  = $\frac{2}{7}$, then u is m/s, is

• $\frac{13}{3}$

• $\frac{17}{3}$

• $\frac{19}{3}$

• $\frac{23}{3}$

Three forces ofmagnitude 30, 60 and P acting at a point are in equilibrium. If the angle between the first two is 60°, the value of P is

• $30\sqrt{7}$

• $30\sqrt{3}$

• $20\sqrt{6}$

• $25\sqrt{2}$

Let a and b be two equal vectors inclined at an angle $\mathrm{\theta }$, then $\mathrm{a} \sin \left(\frac{\mathrm{\theta }}{2}\right)$ is equal to

• $\frac{\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|}{2}$

• $\frac{\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|}{2}$

• $\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right|$

• $\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right|$