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35 Maths MCQ for Class 12th (Three Dimensional Geometry)

Three Dimensional Geometry Class 12 Maths 

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


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 13


3. Direction ratios of a line are 2, 3, -6. Then direction cosines of a line making obtuse angle with the y-axis are
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 1

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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 14


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 17


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 31
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 32


8. Distance between planes
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 8
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 9

Answer/Explanation

Answer: c
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 35


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 15


12. The vector equation of the line
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 2
State true or false.

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 16


13. The Cartesian equation of a line AB is
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 3
Find the direction cosines of a line parallel to AB.

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 18


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

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 19


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 20


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 21


17. Write the Cartesian equation of the following line given in vector form:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 4

Answer/Explanation

Answer:
Explaination:
Point through which line passes is (2, 1, -4) and dr’s: 1, – 1, – 1.
∴ Cartesian equation of line
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 22


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

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 23


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 24


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 25
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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 26


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 27


23. Find the angle between the following pair of lines:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 5
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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 28


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 29


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 30


28. General equation of a plane passing through the intersection of two given
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 6

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 33


29. Cartesian equation of the plane
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 7
State true or false.

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 34


30. Find the distance of the point (2,3,4) from the plane
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 10

Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 36


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 37


33. Find the angle between the planes


Answer/Explanation

Answer:
Explaination:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 38


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
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 39


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:
Maths MCQs for Class 12 with Answers Chapter 11 Three Dimensional Geometry 40

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