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JEE Main202312 Apr 2023Morning ShiftMathematicsVector AlgebraActual

Let a , b , c be three distinct real numbers, none equal to one. If the vectors a i ^ + j ^ + k ^ , i ^ + b j ^ + k ^ and i ^ + j ^ + c k ^ are coplanar, then 1 1 - a + 1 1 - b + 1 1 - c is equal to

Options

  1. A2
  2. B- 1
  3. C- 2
  4. D1

Correct answer

D. 1

Step-by-step solution

Given that the vectors are coplanar. That means the determinant is equal to zero. ⇒ a 1 1 1 b 1 1 1 c = 0 ⇒ R 1 → R 1 – R 2 and R 2 → R 2 – R 3 ⇒ a - 1 1 - b 0 0 b - 1 1 - c 1 1 c = 0 Expand the determinant along C 1 . ⇒ ( a - 1 )   [ c ( b - 1 ) - ( 1 - c ) ] + 1 [ ( 1 - b )   ( 1 - c ) ] = 0 ⇒ c ( a - 1 )   ( b - 1 ) - ( a - 1 )   ( 1 - c ) + ( 1 - b )   ( 1 - c ) = 0 ⇒ c ( 1 - a )   ( 1 - b ) + ( 1 - a )   ( 1 - c ) +

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