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MHT CET20255 May 2025Evening ShiftMathematicsVector AlgebraActual

If c =5 a +6 b and 3 c = a -4 b then

Options

  1. Aa , b , c are non-collinear
  2. Ba , b , c are in the same direction
  3. Ca , c are in the same direction but a , b are in the opposite direction
  4. Dc , b are in the opposite direction and a , b are in the same direction

Correct answer

C. a , c are in the same direction but a , b are in the opposite direction

Step-by-step solution

Given the vector equations c = 5 a + 6 b and 3 c = a - 4 b , substituting the first into the second yields: 3(5 a + 6 b ) = a - 4 b 15 a + 18 b = a - 4 b Rearranging terms gives 14 a = -22 b , or after simplification 7 a = -11 b . This shows a and b are scalar multiples with opposite directions. Expressing b = - 7 11 a and substituting into the first equation: c = 5 a + 6(- 7 11 a ) = 5 a - 42 11 a = 13 11 a Thus, a and c are in the same direction. Evaluating the options: Option A is false as all vectors are collin

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