JEE Main 2024 · 9 April · evening · Single correct · +4 / −1

The sum of the coefficient of x^(2 / 3) and x^(-2 / 5) in the binomial expansion of (x^(2 / 3)+1/2 x^(-2 / 5))⁹ is

  1. (a) 19/4
  2. (b) 69/16
  3. (c) 63/16
  4. (d) 21/4
Answer and solution

Answer (d) 21/4

Solution

Maths sir · 2 steps

Write the general term, find its power of x, and solve for r twice, once for each power.

  1. General term. The (r+1)th term is C(9, r)(x^(2/3))⁹⁻ʳ(1/2x^(-2/5))ʳ. Its coefficient is C(9, r)(1/2)ʳ.

    Power of x = 2/3(9-r) - 2/5r = 6 - 16r/15

    Power wantedEquationrCoefficient
    x^(2/3)6 - 16r/15 = 2/35C(9, 5)1/32 = 126/32 = 63/16
    x^(-2/5)6 - 16r/15 = -2/56C(9, 6)1/64 = 84/64 = 21/16
  2. So. The sum is 63/16 + 21/16 = 84/16 = 21/4, which is option (d).

The idea to keep. For any coefficient question, write the power of x in the general term as a function of r first. Then each required power is just one linear equation in r.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Binomial Theorem

Asked about 200 times in JEE Main since 2020. This question is from 2024.