Chapter 11: Three Dimensional Geometry — Online MCQ Test
MATHS · CLASS 12 SECOND PUC · Karnataka State Board
Practice Chapter 11: Three Dimensional Geometry with a free chapter-wise online MCQ test.
This chapter covers: Direction Cosines Relation: The direction cosines (\(l, m, n\)) of a line in 3D space are unique and always satisfy the identity \(l^2 + m^2 + n^2 = 1\).Direction Ratios: Direction....
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Chapter 11: Three Dimensional Geometry — Important Questions & Answers
What is the fundamental identity that direction cosines (l, m, n) of a line must always satisfy?
- A. l + m + n = 1
- B. l² + m² + n² = 1
- C. lm + mn + nl = 0
- D. l/m + m/n + n/l = 1
Answer: B. l² + m² + n² = 1
Direction cosines are the cosines of angles made by a line with coordinate axes and always satisfy l² + m² + n² = 1 by definition.
Direction cosines are the cosines of angles made by a line with coordinate axes and always satisfy l² + m² + n² = 1 by definition.
Which of the following sets represents valid direction cosines?
- A. (1, 0, 0)
- B. (1/2, 1/2, 1/2)
- C. (1/√2, 1/√2, 0)
- D. (2, 3, 4)
Answer: C. (1/√2, 1/√2, 0)
For (1/√2)² + (1/√2)² + 0² = 1/2 + 1/2 + 0 = 1, which satisfies the direction cosine identity.
For (1/√2)² + (1/√2)² + 0² = 1/2 + 1/2 + 0 = 1, which satisfies the direction cosine identity.
The direction ratios of line L₁ are (1, 2, 3) and of line L₂ are (2, 4, 6). What can you conclude?
- A. The lines are skew
- B. The lines are parallel
- C. The lines intersect at right angles
- D. The lines are perpendicular
Answer: B. The lines are parallel
Since (2, 4, 6) = 2(1, 2, 3), the direction ratios are proportional, meaning the lines are parallel.
Since (2, 4, 6) = 2(1, 2, 3), the direction ratios are proportional, meaning the lines are parallel.
Consider two lines with direction vectors b⃗₁ = (1, 2, 1) and b⃗₂ = (2, 1, -1). What is b⃗₁ × b⃗₂?
- A. (-3, 3, -3)
- B. (-3, -3, -3)
- C. (3, 3, 3)
- D. (1, 2, 1)
Answer: A. (-3, 3, -3)
b⃗₁ × b⃗₂ = |i j k | = i(2·(-1) - 1·1) - j(1·(-1) - 1·2) + k(1·1 - 2·2) = (-3, 3, -3).
b⃗₁ × b⃗₂ = |i j k | = i(2·(-1) - 1·1) - j(1·(-1) - 1·2) + k(1·1 - 2·2) = (-3, 3, -3).
For two skew lines, the shortest distance between them is zero. What does this necessarily imply?
- A. The lines are parallel
- B. The lines have the same direction ratios
- C. The lines intersect at some point
- D. The lines are perpendicular
Answer: C. The lines intersect at some point
By definition, if shortest distance is zero, the lines must intersect; this is a fundamental property used to determine intersection in 3D.
By definition, if shortest distance is zero, the lines must intersect; this is a fundamental property used to determine intersection in 3D.