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JEE Main202315 Apr 2023Morning ShiftMathematicsBinomial TheoremActual

Let a + b x + c x 2 10 = ∑ i = 10 20 p i x i , a , b , c ∈ ℕ . If p 1 = 20 and p 2 = 210 , then 2 a + b + c is equal to

Options

  1. A6
  2. B15
  3. C12
  4. D8

Correct answer

C. 12

Step-by-step solution

Given that a + b x + c x 2 10 = ∑ i = 10 20 p i x i ,   a ,   b ,   c ∈ ℕ General term : 10 ! r 1 !   r 2 !   r 3 ! a r 1 b x r 2 c x 2 r 3 For coefficient. of x : r 2 + 2 r 3 = 1 Let r 1 r 2 r 3 9 1 0 ∴ coefficient of x = 10 ! 9 ! a 9 b 1 = 20 ⇒ a 9 · b = 2   . . . . . . i Coefficient of ⇒ x 2 : 10 ! 8 !   2 !   0 ! a 8 · b 2 + 10 ! 9 !   0 !   1 ! · a 9 · c = 210 ⇒ 45 a 8 · b 2 + 10 · a 9 &#18

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