Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20203 SeptemberEvening Shift
Exam
JEE Main
Year
2020
Date
3 September
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2020 · 3 September · Evening Shift · Mathematics · Binomial Theorem

If the term independent of x in the expansion of ((3)/(2) x ^ 2 -...

If the term independent of xx in the expansion of (32x2−13x)9is{\left(\dfrac{3}{2}{x}^{2} - \dfrac{1}{3x}\right)}^{9} isk,then, then18k is equal to:

Options

  1. A1111
  2. B55
  3. C99
  4. D77

Correct answer

D. 77

Step-by-step solution

Tr+1=9Cr(3x22)9−r(−13x)r{T}_{r + 1}{=}^{9}{C}_{r}{\left(\dfrac{3{x}^{2}}{2}\right)}^{9 - r}{\left(- \dfrac{1}{3x}\right)}^{r}

=9Cr(32)9−r(−13)rx18−3r{=}^{9}{C}_{r}{\left(\dfrac{3}{2}\right)}^{9 - r}{\left(- \dfrac{1}{3}\right)}^{r}{x}^{18 - 3r}

For the term independent of xx, put r=6r = 6

⇒T7=9C6(32)3(−13)6=9C3(16)3=9×8×73×2×1(16)3=(718)⇒ {T}_{7}{=}^{9}{C}_{6}{\left(\dfrac{3}{2}\right)}^{3}{\left(- \dfrac{1}{3}\right)}^{6}{=}^{9}{C}_{3}{\left(\dfrac{1}{6}\right)}^{3} = \dfrac{9 \times 8 \times 7}{3 \times 2 \times 1}{\left(\dfrac{1}{6}\right)}^{3} = \left(\dfrac{7}{18}\right)

⇒k=T7⇒k=718⇒18k=7⇒ k = {T}_{7} ⇒ k = \dfrac{7}{18} ⇒ 18k = 7

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs