The direction cosines of the line 4x - 4 = 1 - 3y = 2 - 1 are :

• $\frac{3}{\sqrt{56}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{56}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{56}}}$

• $\frac{3}{\sqrt{29}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{29}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{29}}}$

• $\frac{3}{\sqrt{61}}, \frac{{\displaystyle - 4}}{{\displaystyle \sqrt{61}}}, \frac{{\displaystyle 6}}{{\displaystyle \sqrt{61}}}$

• 4, - 3, 2

The centre of the sphere passing through the origin and through the intersection pomts of the plane$\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 1$ with axes is :

• $\left(\frac{\mathrm{a}}{2}, 0, 0\right)$

• $\left(0, \frac{\mathrm{a}}{2}, 0\right)$

• $\left(0, 0, \frac{\mathrm{a}}{2}\right)$

• $\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{b}}{2}, \frac{\mathrm{c}}{2}\right)$

An integrating factor of the differential equation, (1 + y + x²y)dx + (x + x³)dy = 0 is :

• $\log \left(\mathrm{x}\right)$

• x

• e^x

• $\frac{1}{\mathrm{x}}$

The line 2x - y = 1 bisects angle between two lines. If equation of one line is y = x, then the equation of the other line is :

• 7x - y - 6 = 0

• x - 2y + 1 = 0

• 3x - 2y - 1 = 0

• x - 7y + 6 = 0

If $\overrightarrow{\mathrm{A}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{B}} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{C}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}$, then $\overrightarrow{\mathrm{A}} + \mathrm{t}\overrightarrow{\mathrm{B}}$ is perpendicular to $\overrightarrow{\mathrm{C}}$, if t is equal to :

• - 5

• 4

• 5

• - 4

The angle between the lines $\sqrt{3}\mathrm{x} - \mathrm{y} - 2 = 0$ and x - $\sqrt{3}\mathrm{y} + 1 = 0$ is :

• 90°

• 60°

• 45°

• 30°

Let f(x) = x - [x], for every real x, where [x] is the greatest integer less than or equal to x. Then, ${\int }_{- 1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is :

• 1

• 2

• 3

• 0

If ${\int }_0^{{\mathrm{x}}^2}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt} = \mathrm{xcos}\left(\mathrm{\pi x}\right)$, then the value of f(4) is :

• 1

• $\frac{1}{4}$

• - 1

• $\frac{- 1}{4}$

A force of magnitude $\sqrt{6}$ acting along the line joining the points A(2, - 1, 1) and B (3, 1, 2) displaces a particle from A to B. The work done by the force is :

• 6

• $6\sqrt{6}$

• $\sqrt{6}$

• 12

50.

Equation of the plane parallel to the planes x + 2y + 3z - 5 = 0, x + 2y + 3z - 7 = 0 and equidistant from them is :

• x + 2y + 3z - 6 = 0

• x + 2y + 3z - 1 = 0

• x + 2y + 3z - 8 = 0

• x + 2y + 3z - 3 = 0