If the direction cosines of two lines are connected by the equations l + m + n = 0, l² + m² - n² = 0, then the angle between the lines is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

The equation of the plane which contains the origin and the line of intersection of the planes r · a = d₁ and r · b = d₂, is

• r . (d₁a + d₂b) = 0

• r . (d₂a - d₁b) = 0

• r . (d₂a + d₁b) = 0

• r . (d₁a - d₂b) = 0

If from a point P(a, b, c) perpendiculars PA and PB are drawn to YZ and ZX - planes, then the equation of the plane OAB is

• bcx + cay + abz = 0

• bcx + cay - abz = 0

• - bcx + cay + abz = 0

• bcx - cay + abz = 0

If (2, 7, 3) is one end of a diameter of the sphere x² + y² + z² - 6x - 12y - 2z + 20 = 0, then the coordinates of the other end of the diameter are

• (- 2, 5, - 1)

• (4, 5, 1)

• (2, - 5, 1)

• (4, 5. - 1)

If a line segment OP makes angles of $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 3}}$ with X-axis and Y-axis, respectively. Then, the direction cosines are

• $\frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}}$

• $1, \sqrt{3}, 1$

• $1, \frac{1}{\sqrt{3}}, 1$

If a plane passing through the point (2, 2, 1) and is perpendicular to the planes 3x + 2y + 4z + 1 = 0 and 2x + y + 3z + 2 = 0. Then, the equation of the plane is

• 2x - y - z - 1 = 0

• 2x + 3y + z - 1 = 0

• 2x + y + z + 3 = 0

• x - y + z - 1 = 0

If the points (1, 2, 3) and (2, - 1, 0) lie on the opposite sides of the plane 2x + 3y - 2z = k, then

• k < 1

• k > 2

• k < 1 or k > 2

• 1 < k < 2

The triangle formed by the tangent to the curve f (x) = x² + bx - b at the point (1, 1) and the coordinate axes lies in the first quadrant. If its area is 2, then the value of b is

• - 1

• 3

• - 3

• 1

409.

If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is

• x + 2y + 4z = 12

• 4x + 2y + z = 12

• x + 2y + 4z = 3

• 4x + 2y + z = 3

The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is

• 60

• 600

• 720

• 400