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

## 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

**Answer/Explanation**

Answer: b

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

**Answer/Explanation**

Answer: c

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

**Answer/Explanation**

Answer: c

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

**Answer/Explanation**

Answer: d

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

**Answer/Explanation**

Answer: b

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

**Answer/Explanation**

Answer: d

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) –\(\frac{2}{7}\)

**Answer/Explanation**

Answer: c

Explaination:

8. Distance between planes

**Answer/Explanation**

Answer: c

Explaination:

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

**Answer/Explanation**

Answer:

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.

**Answer/Explanation**

Answer:

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 ________ .

**Answer/Explanation**

Answer:

Explaination:

12. The vector equation of the line

State true or false.

**Answer/Explanation**

Answer:

Explaination:

13. The Cartesian equation of a line AB is

Find the direction cosines of a line parallel to AB.

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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:

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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.

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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 \(\overrightarrow{P Q}\).

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

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 ________ .

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

Explaination:

29. Cartesian equation of the plane

State true or false.

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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]

**Answer/Explanation**

Answer:

Explaination:

33. Find the angle between the planes

**Answer/Explanation**

Answer:

Explaination:

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

**Answer/Explanation**

Answer:

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

**Answer/Explanation**

Answer:

Explaination: