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JEE MainMathematicsBinomial Theorem

Let A be the sum of the coefficients in the expansion of (2x^2 - x + 5)¹⁰ and B be the sum of the coefficients in the expansion of (x + 2)¹⁰ . If A = B^2 , then the positive value of is :

Options

  1. A4
  2. B2
  3. C16
  4. D10

Correct answer

C. 16

Step-by-step solution

The sum of the coefficients in the expansion of a polynomial is obtained by substituting x = 1 . For the expansion of (x + 2)¹⁰ , substituting x = 1 gives: B = (1 + 2)¹⁰ = 3¹⁰ For the expansion of (2x^2 - x + 5)¹⁰ , substituting x = 1 gives: A = (2 - + 5)¹⁰ = (7 - )¹⁰ It is given that A = B^2 . Substituting the values of A and B : (7 - )¹⁰ = (3¹⁰)^2 (7 - )¹⁰ = 3²⁰ = 9¹⁰ Taking the 10^ th root on both sides gives: 7 - = 9 or 7 - = -9 Solving for : = -2 or = 16 The positive value of is 16 . Answer: 16

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