Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202118 MarchEvening Shift
Exam
JEE Main
Year
2021
Date
18 March
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2021 · 18 March · Evening Shift · Mathematics · Binomial Theorem

The term independent of x in the expansion of [ x + 1 x ^ 2 / 3 - x...

The term independent of xx in the expansion of [x+1x2/3−x1/3+1−x−1x−x1/2]10,x≠1{\left[\dfrac{x + 1}{{x}^{2 / 3} - {x}^{1 / 3} + 1} - \dfrac{x - 1}{x - {x}^{1 / 2}}\right]}^{10}, x \ne 1, is equal to ___.

Correct answer

0

Step-by-step solution

Given [x+1x2/3−x1/3+1−x−1x−x1/2]10{\left[\dfrac{x + 1}{{x}^{2 / 3} - {x}^{1 / 3} + 1} - \dfrac{x - 1}{x - {x}^{1 / 2}}\right]}^{10}

⇒((x1/3+1)−(x+1x))10⇒ {\left(\left({x}^{1 / 3} + 1\right) - \left(\dfrac{\sqrt{x} + 1}{\sqrt{x}}\right)\right)}^{10}

⇒(x1/3−x−1/2)10⇒ {\left({x}^{1 / 3} - {x}^{- 1 / 2}\right)}^{10}

General term is Tr+1=10Cr(x1/3)10−r(−x−1/2)r{T}_{r + 1} = 10{C}_{r}{\left({x}^{1 / 3}\right)}^{10 - r}{\left(- {x}^{- 1 / 2}\right)}^{r}

∴10−r3−r2=0⇒20−2r−3r=0∴ \dfrac{10 - r}{3} - \dfrac{r}{2} = 0 ⇒ 20 - 2r - 3r = 0

⇒r=4⇒ r = 4

Then, T5=C410=10×9×8×74×3×2×1=210{T}_{5} = C4 10 = \dfrac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210

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