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JEE Main202329 Jan 2023Morning ShiftMathematicsVector AlgebraActual

If the vectors a → = λ i ^ + μ j ^ + 4 k ^ , b → = - 2 i ^ + 4 j ^ - 2 k ^ and c → = 2 i ^ + 3 j ^ + k ^ are coplanar and the projection of a → on the vector b → is 54 units, then the sum of all possible values of λ + μ is equal to

Options

  1. A0
  2. B6
  3. C24
  4. D18

Correct answer

C. 24

Step-by-step solution

Given, a → = λ i ^ + μ j ^ + 4 k ^ , b → = - 2 i ^ + 4 j ^ - 2 k ^ and c → = 2 i ^ + 3 j ^ + k ^ are coplanar So, λ μ 4 - 2 4 - 2 2 3 1 = 0 ⇒ λ ( 10 ) - μ ( 2 ) + 4 ( - 14 ) = 0 ⇒ 10 λ - 2 μ = 56 ⇒ 5 λ - μ = 28   . . . . . 1 And given projection of a → on b → is 54 ⇒ a → · b → | b → | = 54 ⇒ - 2 λ + 4 μ - 8 24 = 54 ⇒ - 2 λ + 4 μ - 8 = 54 × 24  

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