# Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 11 Three Dimensional Geometry. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Three Dimensional Geometry MCQs Pdf with Answers to know their preparation level.

## Three Dimensional Geometry Class 12 Maths MCQs Pdf

Question 1. (b) $$\frac{\pi}{2}$$

Question 2.
The angle between the lines passing through the points (4, 7, 8), (2, 3, 4) and (-1, -2, 1), (1, 2, 5) is
(a) 0
(b) $$\frac{\pi}{2}$$
(c) $$\frac{\pi}{4}$$
(d) $$\frac{\pi}{6}$$
(a) 0

Question 3. (d) Both (a) and (b)

Question 4. (a) $$\frac{x-1}{1}=\frac{y-2}{3}=\frac{z+3}{4}$$

Question 5. (a) $$-\frac{10}{7}$$

Question 6.
Equation of the plane passing through three points A, B, C with position vectors (a) $$\pi(\hat{i}-\hat{j}-2 \hat{k})+23=0$$

Question 7.
Four points (0, -1, -1) (-4, 4, 4) (4, 5, 1) and (3, 9, 4) are coplanar. Find the equation of the plane containing them.
(a) 5x + 7y + 11z – 4 =0
(b) 5x – 7y + 11z + 4 = 0
(c) 5x – 7y – 11z – 4 = 0
(d) 5x + 7y – 11z + 4 = 0
(b) 5x – 7y + 11z + 4 = 0

Question 8.
Find the equation of plane passing through the points P(1, 1, 1), Q(3, -1, 2), R(-3, 5, -4).
(a) x + 2y = 0
(b) x – y = 2
(c) -x + 2y = 2
(d) x + y = 2
(d) x + y = 2

Question 9.
The vector equation of the plane passing through the origin and the line of intersection of the plane r.a = λ and r.b = µ is
(a) r.(λa – µb) = 0
(b) r.(λb – µa) = 0
(c) r.(λa + µb)= 0
(d) r.(λb + µa) = 0
(b) r.(λb – µa) = 0

Question 10.
The vector equation of a plane passing through the intersection of the planes $$r_{\cdot}(\hat{i}+\hat{j}+\hat{k})=6$$ and $$r_{\cdot}(2 \hat{i}+3 \hat{j}+4 \hat{k})=-5$$ and the point (1, 1, 1) is (c) $$r_{\cdot}(20 \hat{i}+23 \hat{j}+26 \hat{k})=69$$

Question 11. (b) -4

Question 12. (a) coplanar
(b) non-coplanar
(c) perpendicular
(d) None of the above
(a) coplanar

Question 13.
The angle between the planes 3x + 2y + z – 5 = 0 and x + y – 2z – 3 = 0 is (c) $$\cos ^{-1}\left(\frac{3}{2 \sqrt{21}}\right)$$

Question 14.
The equation of the plane through the point (0, -4, -6) and (-2, 9, 3) and perpendicular to the plane x – 4y – 2z = 8 is
(a) 3x + 3y – 2z = 0
(b) x – 2y + z = 2
(c) 2x + y – z = 2
(d) 5x – 3y + 2z = 0
(c) 2x + y – z = 2

Question 15.
The angle between the planes $$r \cdot(\hat{i}+2 \hat{j}+\hat{k})=4$$ and $$r(-\hat{i}+\hat{j}+2 \hat{k})=9$$ is
(a) 30°
(b) 60°
(c) 45°
(d) None of these
(b) 60°

Question 16. (b) $$\frac{70}{11}$$

Question 17.
The shortest distance between the lines (d) $$\frac{1}{\sqrt{6}}$$

Question 18.
The shortest distance between the lines $$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$$ and $$\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$$ is equal
(a) 3√30
(b) √30
(c) 2√30
(d) None of these
(a) 3√30

Question 19.
The shortest distance between the lines x = y = z and x + 1 – y = $$\frac{z}{0}$$ is
(a) $$\frac{1}{2}$$
(b) $$\frac{1}{\sqrt{2}}$$
(c) $$\frac{1}{\sqrt{3}}$$
(d) $$\frac{1}{\sqrt{6}}$$
(d) $$\frac{1}{\sqrt{6}}$$

Question 20.
The shortest distance between the lines x = y + 2 = 6z – 6 and x + 1 = 2y = -12z is
(a) $$\frac{1}{2}$$
(b) 2
(c) 1
(d) $$\frac{3}{2}$$
(b) 2

Question 21.
The angle θ between the line r = a + λb is given by (a) $$\sin ^{-1}\left(\frac{\tilde{h}_{\hat{\pi}}^{\pi}}{|\vec{b}|}\right)$$

Question 22. (a) $$\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$$

Question 23.
The angle between the straight line $$\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z-5}{2}$$ and the plane 4x – 2y + 4z = 9 is
(a) 60°
(b) 90°
(c) 45°
(d) 30°
(b) 90°

Question 24.
Distance of the point (α, β, γ) from y-axis is
(a) β
(b) |β|
(c) |β| + |γ|
(d) $$\sqrt{\alpha^{2}+\gamma^{2}}$$
(d) $$\sqrt{\alpha^{2}+\gamma^{2}}$$

Question 25.
The distance of the plane $$r \cdot\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1$$ from the origin is
(a) 1
(b) 7
(c) $$\frac{1}{7}$$
(d) None of these
(a) 1

Question 26. (d) $$\frac{\sqrt{2}}{10}$$

Question 27.
The reflection of the point (α, β, γ) in the xy-plane is
(a) (α, β, 0)
(b) (0, 0, γ)
(c) (-α, -β, -γ)
(d) (α, β, -y)
(d) (α, β, -y)

Question 28.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2), is equal to
(a) 9 sq. units
(b) 18 sq. units
(c) 27 sq. units
(d) 81 sq. units
(a) 9 sq. units

Question 29.
The locus represented by xy + yz = 0 is
(a) A pair of perpendicular lines
(b) A pair of parallel lines
(c) A pair of parallel planes
(d) A pair of perpendicular planes
(d) A pair of perpendicular planes

Question 30.
Direction cosines of the line that makes equal angles with the three axes in space are (c) $$\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}$$

Question 31.
If the direction ratios of a line are 1, -3, 2, then its direction cosines are (a) $$\frac{1}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{2}{\sqrt{14}}$$

Question 32.
The cosines of the angle between any two diagonals of a cube is
(a) $$\frac{1}{3}$$
(b) $$\frac{1}{2}$$
(c) $$\frac{2}{3}$$
(d) $$\frac{1}{\sqrt{3}}$$
(a) $$\frac{1}{3}$$

Question 33.
Which of the following is false?
(a) 30°, 45°, 60° can be the direction angles of a line is space.
(b) 90°, 135°, 45° can be the direction angles of a line is space.
(c) 120°, 60°, 45° can be the direction angles of a line in space.
(d) 60°, 45°, 60° can be the direction angles of a line in space.
(a) 30°, 45°, 60° can be the direction angles of a line is space.

Question 34.
A line makes angles α, β and γ with the co-ordinate axes. If α + β = 90°, then γ is equal to
(a) 0°
(b) 90°
(c) 180°
(d) None of these
(b) 90°

Question 35.
If a line makes an angle θ1, θ2, θ3 with the axis respectively, then cos 2θ1 + cos 2θ2 + cos 2θ3 =
(a) -4
(b) -2
(c) -3
(d) -1
(d) -1

Question 36.
The coordinates of a point P are (3, 12, 4) w.r.t. origin O, then the direction cosines of OP are (d) $$\frac{3}{13}, \frac{12}{13}, \frac{4}{13}$$

Question 37.
Find the direction cosines of the line joining A(0, 7, 10) and B(-1, 6, 6). (b) $$\frac{1}{3 \sqrt{2}}, \frac{1}{3 \sqrt{2}}, \frac{4}{3 \sqrt{2}}$$

Question 38.
The direction cosines of a line passing through two points P(x1, y1, z1) and Q(x2, y2, z2) are (c) $$\frac{x_{2}-x_{1}}{P Q}, \frac{y_{2}-y_{1}}{P Q}, \frac{z_{2}-z_{1}}{P Q}$$

Question 39.
The equation of a line which passes through the point (1, 2, 3) and is parallel to the vector $$3 \hat{i}+2 \hat{j}-2 \hat{k}$$, is (b) $$r=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-2 \hat{k})$$

Question 40.
The equation of line passing through the point (-3, 2, -4) and equally inclined to the axes are
(a) x – 3 = y + 2 = z – 4
(b) x + 3 = y – 2 = z + 4
(c) $$\frac{x+3}{1}=\frac{y-2}{2}=\frac{z+4}{3}$$
(d) None of these
(b) x + 3 = y – 2 = z + 4

Question 41.
If l, m and n are the direction cosines of line l, then the equation of the line (l) passing through (x1, y1, z1) is (a) $$\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}$$

Question 42.
In the figure, a be the position vector of the point A with respect to the origin O. l is a line parallel to a
vector b. The vector equation of line l is (c) r = a + λb

Question 43.
The certesian equation of the line l when it passes through the point (x1, y1, z1) and parallel to the vector
b = $$a \hat{i}+b \hat{j}+c \hat{k}$$, is
(a) x – x1 = y – y1 = z – z1
(b) x + x1 = y + y1 = z + z1
(c) $$\frac{x+x_{1}}{a}=\frac{y+y_{1}}{b}=\frac{z+z_{1}}{c}$$
(d) $$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$$
(d) $$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$$

Question 44.
The equation of the straight line passing through the point (a, b, c) and parallel to Z-axis is (d) $$\frac{x-a}{0}=\frac{y-b}{0}=\frac{z-c}{1}$$

Question 45.
The coordinates of a point on the line $$\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$$ at a distance of $$\frac{6}{\sqrt{12}}$$ from the point (1, 2, 3) is
(a) (56, 43, 111)
(b) $$\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$
(c) (2, 1, 3)
(d) (-2, -1, -3)
(b) $$\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$

Question 46.
Find the coordinatets of the point where the line through the points (5, 1, 6) and (3, 4, 1) crosses the yz-plane.
(a) $$\left(0,-\frac{17}{2}, \frac{13}{2}\right)$$
(b) $$\left(0, \frac{17}{2},-\frac{13}{2}\right)$$
(c) $$\left(10, \frac{19}{2}, \frac{13}{2}\right)$$
(d) (0, 17, 13)
(b) $$\left(0, \frac{17}{2},-\frac{13}{2}\right)$$

Question 47.
The point A(1, 2, 3), B(-1, -2, -1) and C(2, 3, 2) are three vertices of a parallelogram ABCD. Find the equation of CD. (d) $$\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-2}{2}$$

Question 48.
The equation of the line joining the points (-3, 4, 11) and (1, -2, 7) is (b) $$\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}$$

Question 49.
The vector equation of the line through the points A(3, 4, -7) and B(1, -1, 6) is (c) $$r=(3 \hat{i}+4 \hat{j}-7 \hat{k})+\lambda(-2 \hat{i}-5 \hat{j}+13 \hat{k})$$

Question 50. (d) $$\frac{\pi}{6}$$

Question 51.
The angle between the line 2x = 3y = -z and 6x = -y = -4z is
(a) 30°
(b) 45°
(c) 90°
(d) 0°
(c) 90°

Question 52.
The angle between the lines 3x = 6y = 2z and $$\frac{x-2}{-5}=\frac{y-1}{7}=\frac{z-3}{1}$$ is
(a) $$\frac{\pi}{6}$$
(b) $$\frac{\pi}{4}$$
(c) $$\frac{\pi}{3}$$
(d) $$\frac{\pi}{2}$$
(d) $$\frac{\pi}{2}$$

Question 53.
Find the angle between the pair of lines given by (a) $$\cos ^{-1}\left(\frac{19}{21}\right)$$

Question 54.
The angle between the lines x = 1, y = 2 and y = -1, z = 0 is
(a) 90°
(b) 30°
(c) 60°
(d) 0°
(a) 90°

Question 55. (a) 0

Question 56. (b) $$\frac{\left|\left(\tilde{a}_{2}-a_{1}\right) \times b\right|}{|b|}$$

Question 57. (b) $$\sqrt{\frac{59}{7}}$$

Question 58. (d) $$\sqrt{\frac{129}{5}}$$

Question 59.
The direction cosines of the unit vector perpendicular to the plane $$r \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1=0$$ passing through the origin are
(a) $$\frac{6}{7}, \frac{3}{7}, \frac{2}{7}$$
(b) 6, 3, 2
(c) $$-\frac{6}{7}, \frac{3}{7}, \frac{2}{7}$$
(d) -6, 3, 2
(c) $$-\frac{6}{7}, \frac{3}{7}, \frac{2}{7}$$

Question 60.
The coordinate of the foot of perpendicular drawn from origin to the plane 2x – 3y + 4z – 6 = 0 is (d) $$\left(\frac{12}{\sqrt{29}}, \frac{-18}{\sqrt{29}}, \frac{24}{\sqrt{29}}\right)$$

Question 61.
The vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector
$$3 \hat{i}+5 \hat{j}-6 \hat{k}$$ is (d) $$r \cdot\left(\frac{3 \hat{i}}{70}+\frac{5 \hat{j}}{70}-\frac{6 \hat{k}}{70}\right)=7$$

Question 62.
Find the vector equation of the plane which is at a distance of 8 units from the origin and which is normal to the vector $$2 \hat{i}+\hat{j}+2 \hat{k}$$. (c) $$r_{\cdot}(2 \hat{i}+\hat{j}+2 \hat{k})=24$$

Question 63.
Find the length of perpendicular from the origin to the plane $$r(3 \hat{i}-4 \hat{j}+12 \hat{k})$$.
(a) $$\frac{5}{13}$$
(b) $$\frac{5}{\sqrt{13}}$$
(c) $$\frac{5}{23}$$
(d) $$\frac{\sqrt{5}}{13}$$
(a) $$\frac{5}{13}$$