List of top Mathematics Questions on Three Dimensional Geometry asked in CUET (PG)

The plane \(x + y + z = \sqrt{3}\lambda\) touches the sphere \(x^2 + y^2 + z^2 - 2x - 2y - 2z - 6 = 0\) if:

The equation of cone with vertex at (0, 0, 0) and passing through the circle given by
\(x^2 + y^2 + z^2 + x - 2z + 3y - 4 = 0, x - y + z = 2\), is

Which of the following statements are true?
(A) The equations of the plane passing through the point (1, -1, 2) having 2, 3, 2 as direction ratios of normal to the plane is 2x + 3y + 2z = 3
(B) Angle between the normal to the plane 2x - y + z = 6 and x + y + 2z = 7 is \(\frac{\pi}{3}\)
(C) The angle at which the normal vectors to the plane 4x + 8y + z = 5 is inclined to the z-axis is \( \sin^{-1}(\frac{1}{9}) \)
(D) The equation of the plane passing through the point (3, -3, 1) and normal to the line joining the points (3, 4, -1) and (2, -1, 5) is x + 5y + 6z = -18
(E) A normal vector to the plane 2x - y + 2z = 5 is \( \frac{1}{3}(2\vec{i} - \vec{j} + 2\vec{k}) \)
Choose the correct answer from the options given below: