CBSE Class 12 Mathematics Three Dimensional Geometry Important Questions Set 02

Question. The foot of perpendicular from \( (\alpha, \beta, \gamma) \) on y-axis is
(a) \( (\alpha, 0, 0) \)
(b) \( (0, \beta, 0) \)
(c) \( (0, 0, \gamma) \)
(d) \( (0, \beta, \gamma) \)
Answer: (b)

Question. Find the equation of the plane through the points (2, 1, –1), (–1, 3, 4) and perpendicular to the plane \( x - 2y + 4z = 10 \)
(a) \( 18x + 17y + 4z = 49 \)
(b) \( 20x - 12y + 3z = 11 \)
(c) \( 3x - 2y - 4z = 17 \)
(d) \( 7x - 2y - 3z = 0 \)
Answer: (a)

Question. The coordinates of the point where the line through (3, – 4, – 5) and (2, – 3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, – 1, 0) are
(a) \( (0, -2, 7) \)
(b) \( (3, -2, 5) \)
(c) \( (1, -2, -7) \)
(d) \( (1, -2, 7) \)
Answer: (d)

Question. The distance between the parallel planes \( x + 2y - 3z = 2 \) and \( 2x + 4y - 6z + 7 = 0 \) is
(a) \( \frac{2}{\sqrt{14}} \) unit
(b) \( \frac{11}{\sqrt{56}} \) unit
(c) \( \frac{7}{\sqrt{56}} \) unit
(d) none of these
Answer: (b)

Question. The distance of the plane \( 2x - 3y + 6z + 14 = 0 \) from origin is _____________ units.
Answer: 2

Question. The Cartesian equation of the line joining the points (–2, 1, 3) and (3, 1, –2) is _____________ .
Answer: \( \frac{x + 2}{5} = \frac{y - 1}{0} = \frac{z - 3}{-5} \)

Question. Find the angle between the line : \( \frac{x - 2}{3} = \frac{y + 1}{-1} = \frac{z - 3}{2} \) and the plane \( 3x + 4y + z + 5 = 0 \).
Answer: \( \sin^{-1} \left( \frac{7}{2\sqrt{91}} \right) \)

Question. Find the angle between the line: \( \vec{r} = (5\hat{i} - \hat{j} - 4\hat{k}) + \lambda(2\hat{i} - \hat{j} + 3\hat{k}) \) and the plane \( \vec{r} \cdot (3\hat{i} + 4\hat{j} + \hat{k}) + 5 = 0 \).
Answer: \( \sin^{-1} \left( \frac{5}{2\sqrt{91}} \right) \)

Question. Find the co-ordinates of the point where the line \( \frac{x - 1}{3} = \frac{y + 4}{7} = \frac{z + 4}{2} \) cuts the XY-plane.
Answer: (7, 10, 0)

Question. If the line drawn from the point (–2, –1, –3) meets a plane at right angle at the point (1, –3, 3), find the equation of the plane.
Answer: \( 3x - 2y + 6z - 27 = 0 \)

Question. Find the distance of the point whose position vector is \( (2\hat{i} + \hat{j} - \hat{k}) \) from the plane \( \vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9 \).
Answer: \( \frac{13}{\sqrt{21}} \)

Question. If a plane meets the coordinate axes in A, B, C such that the centroid of the \( \Delta ABC \) is the point \( (\alpha, \beta, \gamma) \), then find the equation of the plane.
Answer: \( \frac{x}{\alpha} + \frac{y}{\beta} + \frac{z}{\gamma} = 3 \)

Question. Find the distance of the point (2, 3, 4) from the plane \( 3x + 2y + 2z + 5 = 0 \) measured parallel to the line \( \frac{x + 3}{3} = \frac{y - 2}{6} = \frac{z}{2} \).
Answer: 7 units

Question. Find the distance of the point, whose position vector is \( (2\hat{i} + \hat{j} - \hat{k}) \) from the plane \( \vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9 \).
Answer: \( \frac{13}{\sqrt{21}} \)

Question. Let \( P(3, 2, 6) \) be a point in the space and Q be a point on the line \( \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(-3\hat{i} + \hat{j} + 5\hat{k}) \), then find the value of \( \mu \) for which the vector \( \vec{PQ} \) is parallel to the plane \( x - 4y + 3z = 1 \).
Answer: \( \mu = \frac{1}{4} \)

Question. Find the equation of perpendicular from the point (3, –1, 11) to the line \( \frac{x}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \). Also find the foot of the perpendicular and length of the perpendicular.
Answer: Foot of perpendicular: (2, 5, 7); Length of perpendicular: \( \sqrt{53} \) units; Equation of perpendicular: \( \frac{x - 3}{1} = \frac{y + 1}{-6} = \frac{z - 11}{4} \)