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CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q222 Marksvector_algebra

CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 22 (Vector Algebra)

Vectors a = 3î − 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

✅ Step-by-Step Solution

Final Answer: Diagonals: 4î−2ĵ+4k̂ and 2î−2ĵ; lengths 6 and 2√2.
  1. Diagonals: a+b = (3+1)î + (−2+0)ĵ + (2+2)k̂ = 4î − 2ĵ + 4k̂.
  2. a−b = (3−1)î + (−2−0)ĵ + (2−2)k̂ = 2î − 2ĵ. Length of first = √(16+4+16)=√36=6. Length of second = √(4+4)=√8=2√2.
  3. Diagonals: 4î−2ĵ+4k̂ and 2î−2ĵ; lengths 6 and 2√2.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Diagonals: a+b = (3+1)î + (−2+0)ĵ + (2+2)k̂ = 4î − 2ĵ + 4k̂.
  • [1 mark] a−b = (3−1)î + (−2−0)ĵ + (2−2)k̂ = 2î − 2ĵ. Length of first = √(16+4+16)=√36=6. Length of second = √(4+4)=√8=2√2.
  • [0.5 marks] Diagonals: 4î−2ĵ+4k̂ and 2î−2ĵ; lengths 6 and 2√2.

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