JEE Main 2021 · 25 July · evening · Single correct · +4 / −1

The sum of all those terms which are rational numbers in the
expansion of (2 1/3 + 3 1/4 )¹² is :

  1. (a) 89
  2. (b) 27
  3. (c) 35
  4. (d) 43
Answer and solution

Answer (d) 43

Solution

Maths sir · 4 steps

Write the general term and ask when both powers become whole numbers at the same time.

  1. General term. In (2^(1/3) + 3^(1/4))¹² the (r+1)th term carries 2 to the power (12-r)/3 and 3 to the power r/4, for r = 0, 1, …, 12.

    Tᵣ₊₁ = C(12, r) 2^((12-r)/3) 3^(r/4)

  2. Condition for rational. The term is rational only when both exponents are integers. So r must be a multiple of 4, and 12 - r must be a multiple of 3, which means r is a multiple of 3. A multiple of both 3 and 4 is a multiple of 12, so r = 0 or r = 12.
  3. Find the two terms. For r = 0: C(12, 0) 2⁴ 3⁰ = 16. For r = 12: C(12, 12) 2⁰ 3³ = 27.
  4. So. The sum of the rational terms is 16 + 27 = 43, which is option (d).

The idea to keep. For rational terms, make every fractional exponent in Tᵣ₊₁ an integer. The allowed r values are the common multiples, here the LCM of 3 and 4 inside 0 ≤ r ≤ n.

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 2021.