JEE Main 2020 · 3 September · evening · Single correct · +4 / −1

If the term independent of x in the expansion of
( 3/2x² - 1/3x )⁹ is k, then 18 k is equal to :

  1. (a) 5
  2. (b) 9
  3. (c) 7
  4. (d) 11
Answer and solution

Answer (c) 7

Solution

Maths sir · 4 steps

Write the general term, collect the powers of x, and pick the r that makes the power zero.

  1. General term. With a = 3/2x², b = -1/3x and n = 9, the general term is Tᵣ₊₁ = C(9, r) a⁹⁻ʳ bʳ.

    Tᵣ₊₁ = C(9, r)(3/2)⁹⁻ʳ(-1/3)ʳ x¹⁸⁻²ʳ x⁻ʳ = C(9, r)(3/2)⁹⁻ʳ(-1/3)ʳ x¹⁸⁻³ʳ

  2. Find r. The term is independent of x when 18 - 3r = 0, so r = 6. That is the 7th term, T₇.
  3. Substitute r. k = C(9, 6)(3/2)³(-1/3)⁶ = 84 × 27/8 × 1/729. Since 27/729 = 1/27, this is 84/216 = 7/18. The sign is positive because the power 6 is even.
  4. So. 18k = 18 × 7/18 = 7, which is option (c).

The idea to keep. For a term independent of x, write the power of x in Tᵣ₊₁ as a function of r and set it to zero. Remember that a negative power, like 1/x, lowers the exponent.

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 2020.