Chapter Exam

Chapter 11: Three Dimensional Geometry — Online MCQ Test

Q: What is the fundamental identity that direction cosines (l, m, n) of a line must always satisfy?

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.

Q: Which of the following sets represents valid direction cosines?

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.

Q: The direction ratios of line L₁ are (1, 2, 3) and of line L₂ are (2, 4, 6). What can you conclude?

Answer: B. The lines are parallel — Since (2, 4, 6) = 2(1, 2, 3), the direction ratios are proportional, meaning the lines are parallel.

Q: Consider two lines with direction vectors b⃗₁ = (1, 2, 1) and b⃗₂ = (2, 1, -1). What is b⃗₁ × b⃗₂?

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).

Q: For two skew lines, the shortest distance between them is zero. What does this necessarily imply?

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.

MATHS · CLASS 12 SECOND PUC · Karnataka State Board

Practice Chapter 11: Three Dimensional Geometry with a free chapter-wise online MCQ test for Karnataka State Board CLASS 12 SECOND PUC MATHS. 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.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 11: Three Dimensional Geometry — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 12 SECOND PUC MATHS — Chapter 11: Three Dimensional Geometry, with answers and explanations. These are sample questions; the exam has its own separate question set.

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.

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.

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.

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).

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.