The points, whose position vectors are 60i + 3j, 40i - 8j and ai - 52j collinear, if

• a = 40

• a = - 40

• a = 20

• a = - 20

In a $∆$ABC, AB = ri + j, AC = si - j if the area of triangle is of unit magnitude, then

• $\left|\mathrm{r} - \mathrm{s}\right| = 2$

• $\left|\mathrm{r} + \mathrm{s}\right| = 1$

• $\left|\mathrm{r} + \mathrm{s}\right| = 2$

• $\left|\mathrm{r} - \mathrm{s}\right| = 1$

If a = i - j + k, a b = 0, a x b = c, where c = - 2i - j + k, then b 1s equal to

• (1, 0, - 1)

• (0, 1, 1)

• (- 1, - 1, 0)

• (- 1, 0, 1)

The resultant of P and Q is R. If Q is doubled, R is also doubled and if Q is reversed, R is again doubled. Then, P² : Q² : R² gven by

• 2 2 3

• 3 2 2

• 2 3 2

• 2 3 1

Forces ofmagnitudes 3, P, 5, 10 and Q are respectively acting along the sides AB, BC, CD, AD and the diagonal CA of a rectangle ABCD, where AB = 4m and BC = 3m. Ifthe resultant is a single force along the other diagonal BD, then P, Q and the resultant are

• $4, 10\frac{5}{12}, 12\frac{11}{12}$

• 5, 6, 7

• $3\frac{1}{2}, 8, 9\frac{1}{2}$

• None of the above

576.

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

• $\sqrt{25 + 6\sqrt{3}}$

• $\sqrt{25 + 12\sqrt{3}}$

• $\sqrt{37}$

• $\sqrt{13}$