Chapter 11: Three Dimensional Geometry — Online MCQ Test
MATHS · Grade 12 · CBSE(NCERT)
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.
If direction ratios of a line are (2, -3, 6), what are the corresponding direction cosines?
- A. (2/7, -3/7, 6/7)
- B. (2/6, -3/6, 6/6)
- C. (1/2, -1/3, 1/6)
- D. (2/√49, -3/√49, 6/√49)
Answer: A. (2/7, -3/7, 6/7)
Direction cosines = direction ratios / √(sum of squares) = (2, -3, 6) / √(4+9+36) = (2, -3, 6) / 7.
Direction cosines = direction ratios / √(sum of squares) = (2, -3, 6) / √(4+9+36) = (2, -3, 6) / 7.
What is the relationship between direction ratios (a, b, c) and direction cosines (l, m, n)?
- A. They are always equal
- B. Direction ratios are proportional to direction cosines
- C. They are reciprocals of each other
- D. Direction ratios are always integers while cosines are fractions
Answer: B. Direction ratios are proportional to direction cosines
Direction ratios are any numbers proportional to direction cosines; l = a/√(a²+b²+c²), similarly for m and n.
Direction ratios are any numbers proportional to direction cosines; l = a/√(a²+b²+c²), similarly for m and n.
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.
If line L passes through (1, 0, -1) with direction vector b⃗ = (2, 3, 4), what is the position vector of a point on this line when λ = 2?
- A. (5, 6, 7)
- B. (3, 3, 3)
- C. (5, 6, 8)
- D. (4, 6, 7)
Answer: A. (5, 6, 7)
r⃗ = (1, 0, -1) + 2(2, 3, 4) = (1, 0, -1) + (4, 6, 8) = (5, 6, 7).
r⃗ = (1, 0, -1) + 2(2, 3, 4) = (1, 0, -1) + (4, 6, 8) = (5, 6, 7).
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).
The parametric equations of a line are x = 1 + t, y = 2 + 2t, z = 3 - t. Find the Cartesian form.
- A. (x-1)/1 = (y-2)/2 = (z-3)/-1
- B. (x-1)/1 = (y-2)/2 = (z-3)/1
- C. (x-1)/2 = (y-2)/2 = (z-3)/-1
- D. (x-1)/1 = (y-3)/2 = (z-2)/-1
Answer: A. (x-1)/1 = (y-2)/2 = (z-3)/-1
From parametric form, point is (1, 2, 3) and direction ratios are coefficients of t: (1, 2, -1). Cartesian form is (x-1)/1 = (y-2)/2 = (z-3)/-1.
From parametric form, point is (1, 2, 3) and direction ratios are coefficients of t: (1, 2, -1). Cartesian form is (x-1)/1 = (y-2)/2 = (z-3)/-1.
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.
Consider lines L₁: (x-1)/2 = (y-2)/3 = (z-3)/4 and L₂: (x-3)/2 = (y-5)/3 = (z-7)/4. What is their relationship and why?
- A. Parallel, because direction ratios are proportional
- B. Coincident (same line), because both pass through (1,2,3) and (3,5,7)
- C. Intersecting, because direction ratios are same
- D. Skew, because points don't match
Answer: A. Parallel, because direction ratios are proportional
Both have identical direction ratios (2, 3, 4), so they're parallel. They're not coincident since (1,2,3) on L₁ gives (1-1)/2 = 0 = (2-2)/3 = (3-3)/4 ✓, but (1,2,3) on L₂ gives (1-3)/2 = -1 ≠ 0.
Both have identical direction ratios (2, 3, 4), so they're parallel. They're not coincident since (1,2,3) on L₁ gives (1-1)/2 = 0 = (2-2)/3 = (3-3)/4 ✓, but (1,2,3) on L₂ gives (1-3)/2 = -1 ≠ 0.