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

1. The distance of point (2, 5, 7) from the x-axis is
(a) 2
(b) √74
(c) √29
(d) √53

Explaination:
(b), as distance of point (2, 5, 7) from the x-axis is

2. P is a point on the line segment joining the points (3, 5, -1) and (6, 3, -2). If y-coordinate of point P is 2, then its x-coordinate will be
(a) 2
(b) $$\frac{17}{3}$$
(c) $$\frac{15}{2}$$
(d) -5

Explaination:
(c), as let P divides the join of (3, 5, -1) and (6, 3, -2) in the ratio k : 1

3. Direction ratios of a line are 2, 3, -6. Then direction cosines of a line making obtuse angle with the y-axis are

Explaination:
(c), as direction cosines of a line whose direction ratio are 2,3, -6 are $$\frac{2}{7}, \frac{3}{7}, \frac{-6}{7}$$.
As angle with the y-axis is obtuse,
∴ cos β < 0,
Therefore direction ratios are $$\frac{-2}{7}, \frac{-3}{7}, \frac{6}{7}$$.

4. A line makes angle α, β, γ with x-axis, y-axis and z-axis respectively then cos 2α + cos 2β + cos 2γ is equal to
(a) 2
(b) 1
(c) -2
(d) -1

Explaination:

5. The equations of y-axis in space are
(a) x = 0, y = 0
(b) x = 0, z = 0
(c) y = 0, z = 0
(d) y = 0

Explaination: (b), as on the y-axis, x-coordinate and z-coordinate are zeroes.

6. If the direction cosines of a line are $$\frac{k}{3}, \frac{k}{3}, \frac{k}{3}$$, then value of k is
(a) k > 0
(b) 0 < k < 1.
(c) k = $$\frac{1}{3}$$
(d) k = ± 73

Explaination:

7. Distance of plane $$\vec{r} \cdot(2 \hat{i}+3 \hat{i}-6 \hat{k})+2=0$$, from origin is
(a) 2
(b) 14
(c) $$\frac{2}{7}$$
(d) –$$\frac{2}{7}$$

Explaination:

8. Distance between planes

Explaination:

9. The line joining the points (0, 5, 4) and (1, 3, 6) meets XY-plane at the point ________ .

Explaination:
(-2, 9, 0), as line is $$\frac{x-1}{1}=\frac{y-3}{-2}=\frac{z-6}{2}=\lambda$$
General point on line is (λ + 1, -2λ + 3, 2λ + 6)
If it meets AT-plane, then 2λ + 6 = 0
⇒ λ = – 3
∴ Point is (-2, 9, 0)

10. A line makes angles α, β, γ with z-axis, x-axis and y-axis respectively. Then direction cosines of line are cos β, cos γ, cos α. State true or false.

Explaination: True, as direction cosines of a line are cosines of the angles which a line makes with x, y and z-axes respectively.

11. A line makes angles $$\frac{\pi}{4}, \frac{3 \pi}{4}$$ with x-axis and y-axis respectively. Then the angle which it makes with z-axis can be ________ .

Explaination:

12. The vector equation of the line

State true or false.

Explaination:

13. The Cartesian equation of a line AB is

Find the direction cosines of a line parallel to AB.

Explaination:

14. Find the direction cosines of the line passing through the following points: (-2, 4, -5), (1, 2, 3) [NCERT]

Explaination:

15. Find the Cartesian equation of the line which passes through the point (-2,4, -5) and is parallel to the line $$\frac { x+3 }{ 3 } =\frac { 4-y }{ 5 } =\frac { z+8 }{ 6 }$$ [Delhi 2013]

Explaination:

16. Write the vector equation of the following line: $$\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { 6-z }{ 2 }$$

Explaination:
The line passes through the point (5, -4, 6) and dr’s of the line are 3, 7, – 2.
∴ vector equation is

17. Write the Cartesian equation of the following line given in vector form:

Explaination:
Point through which line passes is (2, 1, -4) and dr’s: 1, – 1, – 1.
∴ Cartesian equation of line

18. What are the direction cosines of a line, which makes equal angles with the coordinate axes? [NCERT; Foreign 2011]

Explaination:

19. If the direction cosines of a given line are $$\frac{1}{k}, \frac{1}{k}, \frac{1}{k}$$ then, find the value of k.

Explaination:

20. If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction ofz-axis. [Delhi 2017]

Explaination:
Let angle with z-axis be γ.
cos²90° + cos²60° + cos² γ = 1
⇒ 0 + $$\frac{1}{4}$$ + cos² γ = 1
⇒ cos² γ = $$\frac{3}{4}$$
cos γ = $$\pm \frac{\sqrt{3}}{2}$$
γ = 30°, 150°

21. Find the vector equation of the line passing through the point A(1, 2,-1) and parallel to the line 5x – 25 = 14 – 7y = 35z. [Delhi 2017]

Explaination:
Given line is 5x – 25 = 14 – 7y = 35z
⇒ 5(x – 5) = – 7(y – 2) = 35z

DR’s of line are 7, – 5 and 1
dr’s of line parallel to the given line are 7,-5, 1.
vector equation of line through the point (1, 2, – 1) and having dr’s 7,-5 and 1 is

22. Write the distance of the point (3, – 5, 12) from the x-axis. [Foreign 2017]

Explaination:
Distance of the point (3, – 5, 12) from the x-axis

23. Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular. [Delhi 2011]

Explaination:
DR’s of lines are 2, 7, – 3 and – 1, 2, 4
As 2 × (- 1) + 7 × 2 – 3 × 4 = 0, so lines are perpendicular. Angle = 90°

24. The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its z-coordinate. [AI2017]

Explaination:
Let point R(4, y, z) lies on the line joining P(2, 2, 1) and Q(5, 1, -2). Let R divides PQ in ratio k: 1

25. If P(1, 5, 4) and Q(4, 1, – 2), find the direction ratios of $$\overrightarrow{P Q}$$.

Explaination:
Direction ratios of $$\overrightarrow{P Q}$$ =4 – 1, 1 – 5 and -2 -4, i.e. 3, -4 and – 6.

26. The equations of a line are 5x – 3 = 15y + 7 = 3 – 10z. Write the direction cosines of the line. [All India]

Explaination:
The equation of a line are 5x – 3 = 15y + 7 = -10z + 3

27. Equation of the perpendicular drawn from the point with position vector $$2 \hat{i}- \hat{j}+ \hat{k}$$ to the plane $$\vec{r} \cdot(\hat{i}-3 \hat{k})=5$$ is ________ .

Explaination:

28. General equation of a plane passing through the intersection of two given

Explaination:

29. Cartesian equation of the plane

State true or false.

Explaination:

30. Find the distance of the point (2,3,4) from the plane

Explaination:

31. Write the intercept cut off by the plane 2x + y – z = 5 on the x-axis. [Delhi 2011]

Explaination:
For intercept on the x-axis, put y = 0 and z = 0
⇒ 2x = 5
⇒ x = $$\frac{5}{2}$$
∴ x-intercept = $$\frac{5}{2}$$

32. Find the distance of the plane 3x – 4y + 12z = 3 from the origin. [AI 2012]

Explaination:

33. Find the angle between the planes

Explaination:

34. Find the distance between the planes 2x – y + 2z – 5 and 5x – 2.5y + 5z = 20. [AI 2017]

Explaination:
Planes are 2x – y + 2z = 5
⇒ 2x – y + 2z – 5 = 0
and 5x – 2.5y + 5z = 20
⇒ 2x – y + 2z – 8 = 0

35. A line passes through the point with position vector $$2 \hat{i}-3 \hat{j}+4 \hat{k}$$ and is perpendicular to the plane $$\vec{r} \cdot(3 \hat{i}+4 \hat{j}-5 \hat{k})=7$$. Find the