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JEE Main20253 Apr 2025Morning ShiftMathematicsQuadratic EquationActual

Let and be the roots of x^2+ 3 x -16=0 , and and be the roots of x^2+3 x-1=0 . If P_n= ^n+ ^n and Q_n= ^n+ ^n , then P ₂₅+ 3 P ₂₄ 2 P ₂₃ + Q ₂₅- Q ₂₃ Q ₂₄ is equal to

Options

  1. A3
  2. B4
  3. C5
  4. D7

Correct answer

C. 5

Step-by-step solution

aligned & x^2+ 3 x -16=0 & P _ n + 3 P _ n -1 -16 P _ n -2 =0 & P ₂₅+ 3 P ₂₄-16 P ₂₃=0 & P ₂₅+ 3 P ₂₄ 2 P ₂₃ =8 aligned Similarly x^2+3 x-1=0 _ ^ Q_n= ^n+ ^n aligned & Q ₂₅- Q ₂₃= ²⁵+ ²⁵- ²³- ²³ &= ²³ ( ^2-1 )+ ²³ ( ^2-1 ) &= ²³(-3 )+ ²³(-3 ) &=-3 [ ²⁴+ ²⁴ ] &=-3 Q ₂₄ & Q ₂₅- Q ₂₃ Q ₂₄ =-3 aligned P ₂₅+ 3 P ₂₄ 2 P ₂₃ + Q ₂₅- Q ₂₃ Q ₂₄ =8-3=5

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