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NDA 1 2024 Maths, question 54

Topic: Analytical geometry

Q54

If ⟨l,m,n⟩⟨l,m,n⟩ are the direction cosines of a normal to the plane 2x−3y+6z+4=02x-3y+6z+4=0, then what is the value of 49(7l2+m2−n2)49(7l^{2}+m^{2}-n^{2}) ?

  1. (a) 00
  2. (b) 11Correct answer
  3. (c) 33
  4. (d) 7171

Solution.

Answer: (b) 11

  1. A normal vector is (2, −3, 6), of magnitude 7.
  2. The direction cosines have squares 4/49, 9/49 and 36/49.
  3. Substitution gives 49(7 × 4 + 9 − 36)/49 = 1.