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JEE Main202621 January 2026Evening ShiftMathematicsVector AlgebraActual

For a triangle ABC, let p = BC , q = CA and r = BA . If | p |=2 3 ,| q |=2 and = 1 3 , where is the angle between p and q , then | p ( q -3 r )|²+3| r |² is equal to :

Options

  1. A410
  2. B340
  3. C200
  4. D220

Correct answer

C. 200

Step-by-step solution

Given | p | = 2 3 , | q | = 2 , and = 1 3 where is the angle between p and q . Since p = BC , q = CA , and r = BA , from triangle closure r = p + q . p q = | p || q | = 2 3 2 1 3 = 4 | r |^2 = | p |^2 + | q |^2 + 2 p q = 12 + 4 + 8 = 24 q - 3 r = q - 3( p + q ) = -3 p - 2 q p ( q - 3 r ) = -2( p q ) ^2 = 1 - 1 3 = 2 3 | p q |^2 = 12 4 2 3 = 32 | p ( q - 3 r )|^2 = 4 32 = 128 Result: 128 + 3(24) = 128 + 72 = 200

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