Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #822
KNOWLEDGE BASED
APPLY
1 Marks 2023 MCQ SINGLE
The angle between the lines 2x=3y=-z and 6x=-y=-4z is
(A) 0°
(B) 30°
(C) 45°
(D) 90°
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AI Tutor Explanation

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Step-by-Step Solution

  1. Rewrite the equations of the lines in the standard form: Line 1: 2x = 3y = -z => x/(1/2) = y/(1/3) = z/(-1) Line 2: 6x = -y = -4z => x/(1/6) = y/(-1) = z/(-1/4)
  2. Find the direction ratios of the lines: Line 1: a1 = 1/2, b1 = 1/3, c1 = -1 Line 2: a2 = 1/6, b2 = -1, c2 = -1/4
  3. Use the formula to find the angle θ between the lines: cos θ = |(a1a2 + b1b2 + c1c2) / (√(a1² + b1² + c1²) * √(a2² + b2² + c2²))|
  4. Substitute the values: cos θ = |((1/2)(1/6) + (1/3)(-1) + (-1)(-1/4)) / (√((1/2)² + (1/3)² + (-1)²) * √((1/6)² + (-1)² + (-1/4)²))| cos θ = |(1/12 - 1/3 + 1/4) / (√(1/4 + 1/9 + 1) * √(1/36 + 1 + 1/16))| cos θ = |(1/12 - 4/12 + 3/12) / (√(9/36 + 4/36 + 36/36) * √(4/144 + 144/144 + 9/144))| cos θ = |(0) / (√(49/36) * √(157/144))| cos θ = 0
  5. Find the angle θ: θ = arccos(0) = 90°

Correct Answer: 90°

AI Suggestion: Option D

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their knowledge of direction ratios/vectors of lines and the formula for finding the angle between two lines.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to find the angle between the lines, including converting the given equations to standard form, identifying direction ratios, and applying the angle formula.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of concepts related to 3D geometry as covered in the textbook.

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