The value of k, if the line $\frac{\mathrm{x} - 4}{1} = \frac{\mathrm{y} - 2}{1} = \frac{\mathrm{z} - \mathrm{k}}{1}$ lies on the plane 2x - 4y + z = 7, will be

• 5

• 7

• 9

• 11

If the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2 = 2 makes angle $\mathrm{\alpha }$ with positive direction of x  axis, then $\cos \left(\mathrm{\alpha }\right)$ will be equal to

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{5}}$

• $\frac{1}{\sqrt{7}}$

• $\frac{1}{\sqrt{3}}$

Value of a for which the vectors (2, - 1, 1) (1, 2, - 3) and (3, a, 5) become coplanar will be

• 4

• - 4

• no such exists

• None of these

If l , m, n are the DC's of a line, then

• l² + m² + n² = 0

• l² + m² + n² = 1

• l + m + n = 1

• l = m = n = 1

The length of the perpendicular from the point (1 2, 3) on the line $\frac{\mathrm{x} - 6}{3} = \frac{\mathrm{y} - 7}{2} = \frac{\mathrm{z} - 7}{- 2}$ is

• 3 units

• 4 units

• 5 units

• 7 units

The equation of the plane passing through the intersection ofthe planes 2x - 3y + z - 4 = 0 and x - y + z + 1 = 0 and perpendicular to the plane x + 2y - 3z + 6 = 0 is

• x - 5y + 3z - 23 = 0

• x - 5y - 3z - 23 = 0

• x + 5y - 3z + 23 = 0

• x - 5y + 3z + 23 = 0

The angle between the lines $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} + 1}{- 2} = \mathrm{z} = 2$ and $\frac{{\displaystyle \mathrm{x} - 1}}{{\displaystyle 1}} = \frac{{\displaystyle \mathrm{y} + 3}}{{\displaystyle 3}} = \frac{{\displaystyle \mathrm{z} + 5}}{2}$ is

• ${\cos }^{-1}\left(\frac{- 3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{5}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{- 5}{\sqrt{182}}\right)$

If a line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{and} \mathrm{\delta }$ with the four diagonals of a cube, then${\cos }^2\left(\mathrm{\alpha }\right) + {\cos }^2\left(\mathrm{\beta }\right) + {\cos }^2\left(\mathrm{\gamma }\right) +\hspace{0.17em}{\cos }^2\left(\mathrm{\delta }\right)$ is

• 1

• 2

• 2/3

• 4/3

The angle between the straight lines $\frac{\mathrm{x} + 1}{2} = \frac{\mathrm{y} - 2}{5} = \frac{\mathrm{z} + 3}{4}$ and $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y} + 2}{2} = \frac{\mathrm{z} - 3}{- 3}$ is

• $45°$

• $30°$

• $60°$

• $90°$

The ratio in which the join of (1, - 2,3) and (4, 2, - 1) is divided by the XOY plane is

• 1 : 3

• 3 : 1

• - 1 : 3

• None of these