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MHT CET202522 Apr 2025Morning ShiftMathematicsVector AlgebraActual

If the vectors m i +m j +n k , i + k , n i +n j +p k lie in a plane then...

Options

  1. Am+n+p=0
  2. Bm , n , p are in A.P.
  3. Cm , n , p are in G.P.
  4. Dn , m , p are in G.P.

Correct answer

C. m , n , p are in G.P.

Step-by-step solution

Given vectors a = m i + m j + n k , b = i + k , and c = n i + n j + p k , coplanarity requires the scalar triple product to vanish. The determinant condition yields vmatrix m & m & n 1 & 0 & 1 n & n & p vmatrix = 0 Expanding along the first row: m(0 p - 1 n) - m(1 p - 1 n) + n(1 n - 0 n) = -mn - m(p - n) + n^2 = -mp + n^2 = 0 This simplifies to n^2 = mp , indicating m , n , and p form a geometric progression. Among the options, only C satisfies this condition.

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