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CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q345 Marksperpendicular_lines

CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Question 34 (Three Dimensional Geometry)

Find the intersection of the line through A=(1,2,3), B=(5,8,11) and r=(4,1,0)+t(5,2,1). Find the line through the intersection perpendicular to both.

✅ Step-by-Step Solution

Final Answer: Intersection (−1,−1,−1). Required line r=(−1,−1,−1)+λ(−5,18,−11).
  1. Parametrize AB by (1+2s,2+3s,3+4s). Equate z and y to get t=3+4s and 2+3s=1+2t.
  2. This yields s=t=−1, and the x-coordinates also agree at −1, giving (−1,−1,−1).
  3. The cross product of (2,3,4) and (5,2,1) is (−5,18,−11). Use this direction through the intersection.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Parametrize AB by (1+2s,2+3s,3+4s). Equate z and y to get t=3+4s and 2+3s=1+2t.
  • [3 marks] This yields s=t=−1, and the x-coordinates also agree at −1, giving (−1,−1,−1).
  • [1 mark] The cross product of (2,3,4) and (5,2,1) is (−5,18,−11). Use this direction through the intersection.

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All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/2/3

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