Three forces of magnitudes 1, 2 and 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is

• 114 dynes

• 6 dynes

• 5 dynes

• None of the above

The vectors AB = 3i + 5j + 4k and AC = 5i - 5j + 2k are side of a $∆$ABC. The length ofthe median through A is

• $\sqrt{13} \mathrm{units}$

• $2\sqrt{5} \mathrm{units}$

• 5 units

• 10 units

Let a = 2i + j + k, b = i + 2j - 1, and a unit vector c be caplanar. If c is perpendicular to a, then c is

• $\frac{1}{\sqrt{2}}\left(- \mathrm{j} + \mathrm{k}\right)$

• $\frac{1}{\sqrt{3}}\left(- \mathrm{i} - \mathrm{j} - \mathrm{k}\right)$

• $\frac{1}{\sqrt{5}}\left(\mathrm{i} - 2\mathrm{j}\right)$

• $\frac{1}{\sqrt{3}}\left(\mathrm{i} - \mathrm{j} - \mathrm{k}\right)$

The area of a parallelogram whose adjacent sides are represented by the vectors $\mathrm{a} = - \hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $\mathrm{b} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ is

• $\sqrt{14}$

• $\sqrt{6}$

• $\frac{49}{36}$

• $4\sqrt{10}$

The resultant of two forces of 3 kg-wt and 4 kg-wt which are inclined at 30° angles to each other is

• $\sqrt{25 + 6\sqrt{3}}$

• $\sqrt{25 + 12\sqrt{3}}$

• $\sqrt{37}$

• $\sqrt{13}$

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

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

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

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

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