Joint equation of pair of lines through (3, - 2) and parallel to x² - 4xy + 3y² = 0 is

• x² + 3y² - 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y - 45 = 0

• x² + 3y² + 4xy - 14x - 24y - 45 = 0

Equation of the plane passing through (- 2, 2, 2) and (2, - 2, - 2) and perpendicular to the plane 9x - 13y - 3z = 0 is

• 5x + 3y + 2z = 0

• 5x - 3y + 2z = 0

• 5x - 3y - 2z = 0

• 5x + 3y - 2z = 0

If 'f' is the angle between the lines ax² + 2hxy + by² = 0, then angle between x² + 2xy sec$\left(\mathrm{\theta }\right)$ + y² = 0 is

• $\mathrm{\theta }$

• $2\mathrm{\theta }$

• $\frac{\mathrm{\theta }}{2}$

• $3\mathrm{\theta }$

The equation of the plane which passes through (2, - 3, 1) and is normal to the line joining the points (3, 4, - 1) and (2, - 1, 5), is given by

• x + 5y - 6z + 19 = 0

• x - 5y + 6z - 19 = 0

• x + 5y + 6z + 19 = 0

• x - 5y - 6z - 19 = 0

The angle between the lines x² - xy - 6y² - 7x + 31y - 18 = 0 is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

The equation of the lines passing through the origin and having slopes 3 and - $\frac{1}{3}$, is

• 3y² + 8xy - 3x² = 0

• 3x² + 8xy + 3y² = 0

• 3y² - 8xy - 3x² = 0

• 3x² + 8xy - 3y² = 0

The point where the line $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} - 2}{- 3} = \frac{\mathrm{z} + 3}{4}$ meets the plane 2x + 4y - z = 1, is

• (3, - 1, 1)

• (3, 1, 1)

• (1, 1, 3)

• (1, 3, 1)

A vector vis equally inclined to the x-axis, y-axis and z-axis respectively, its direction cosines are

• $< \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} >$

• $< - \frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}} >$

• $< \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} > \mathrm{or} < - \frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}} >$

• None of the above

A plane meets the axes in A, B and C such that centroid of the $∆$ABC is (1, 2, 3). The equation of the plane is

• x + y/2 + z/3 = 1

• x/3 + y/6 + z/9 = 1

• x + 2y + 3z = 1

• None of these

250.

If $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are the angles which a half ray makes with the positive direction of the axes, then ${\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• 1

• 2

• 0

• - 1