Let two like parallel forces are acting, at points A and B such that the force at A is double than the force at B. If C is the point of action ofthe resultant, then AC/BC is

• 2

• 1/2

• 1/3

• 1/4

Let a, b and c represent vector quantities. Which of the following points are collinear?

• a - 2b + 3c, 2a + 3b - 4c, - 7b + 10c

• - 2a + 3b + 5c, a + 2b + 3c, 7a - c

• a, b, 3a - 2b

• a + b - c, 3a - 4c, 2a + b + 3c

The values of $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ for which the vectors a = $2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} - \hat{\mathrm{k}}$ is perpendicular to the vector b = $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{\mu }\hat{\mathrm{k}}$ with $\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$ are

• $\mathrm{\lambda } = \frac{41}{12}, \mathrm{\mu } = \frac{31}{12}$

• $\mathrm{\lambda } = \frac{41}{12}, \mathrm{\mu } = - \frac{31}{12}$

• $\mathrm{\lambda } = - \frac{41}{12}, \mathrm{\mu } = \frac{31}{12}$

• None of these

At a point the addition of two active force is 18 N. If the magnitude of resultant is 12 N and meet at right angle. Then, magnitude of forces are

• 5 N, 13 N

• 6N, 12 N

• 8 N, 10 N

• None of these

The angle between two active forces P + Q and P  - Q is 2$\mathrm{\alpha }$. If their resultant make angle $\mathrm{\theta }$ with bisector of angle. Then

• $\mathrm{Pcos}\left(\mathrm{\theta }\right) = \mathrm{Qcos}\left(\mathrm{\alpha }\right)$

• $\mathrm{Ptan}\left(\mathrm{\theta }\right) = \mathrm{Qtan}\left(\mathrm{\alpha }\right)$

• $\mathrm{Qcos}\left(\mathrm{\theta }\right) = \mathrm{Pcos}\left(\mathrm{\alpha }\right)$

• $\mathrm{Qtan}\left(\mathrm{\theta }\right) = \mathrm{Ptan}\left(\mathrm{\alpha }\right)$

The vector $\overrightarrow{\mathrm{a}}$ is equal to

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{a}}\right)\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

587.

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are the positton vectors of the vertices f an equilateral triangle whose orthocentre is at the origin, then

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$

• ${\overrightarrow{\mathrm{a}}}^2 = {\overrightarrow{\mathrm{b}}}^2 + {\overrightarrow{\mathrm{c}}}^2$

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

• None of these

$\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}}, \overrightarrow{\mathrm{C}}$ are three non-zero vectors; no two of them are parallel. If $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$ is collinear to $\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is collinear to $\overrightarrow{\mathrm{A}}$, then $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is equal to

• $\overrightarrow{\mathrm{A}}$

• $\overrightarrow{\mathrm{B}}$

• $\overrightarrow{\mathrm{C}}$

• $\overrightarrow{0}$

Let $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. If c is vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}} = \left|\overrightarrow{\mathrm{c}}\right| \left|\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right| = 2\sqrt{2}$ the angle between $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{is} 30°$ is, then $\left|\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \overrightarrow{\mathrm{c}}\right|$ is equal to

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

• 3

If the vectors $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, - 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{\lambda }\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ are linearly dependent, then the value of $\mathrm{\lambda }$ is equal to

• 0

• 1

• 2

• 3