If the term without x in the expansion of (x(2)/(3) + α x ^ 3 ) ^ 22 is 7315 , then |α| is equal to _____ .

Correct answer: 0 General term in the binomial expansion of (x(2)/(3) + α x ^ 3 ) ^ 22 is T _ r + 1 = Cr 22 (x(2)/(3)) ^ 22 - r ( α x ^ 3 ) ^ r ⇒ T _ r + 1 = Cr 22 (x)(44 - 11r)/(3) α ^ r For term independent of x , (44 - 11r)/(3) = 0 ⇒ r = 4 So, T _ 5 = C4 22 α ^ 4 = 7315 ⇒ ((22 !)/(4 ! × 18 !)) α ^ 4 = 7315 ⇒ 7315 α ^ 4 = 7315 ⇒ α ^ 4 = 1 ⇒ |α| = 1

JEE Main 20231 FebruaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2023 · 1 February · Evening Shift · Mathematics · Binomial Theorem

If the term without x in the expansion of (x(2)/(3) + α x ^ 3 ) ^...

If the term without

x

in the expansion of

{\left(x\dfrac{2}{3} + \dfrac{\alpha}{{x}^{3}}\right)}^{22} is

7315

, then \left|\alpha\right|

is equal to _____ .

Correct answer

Step-by-step solution

General term in the binomial expansion of ${\left(x\dfrac{2}{3} + \dfrac{\alpha}{{x}^{3}}\right)}^{22} is$ {T}_{r + 1} = Cr 22 {\left(x\dfrac{2}{3}\right)}^{22 - r}{\left(\dfrac{\alpha}{{x}^{3}}\right)}^{r} $$ ⇒ {T}_{r + 1} = Cr 22 \left(x\right)\dfrac{44 - 11r}{3}{\alpha}^{r} $For term independent of$ x $, $ \dfrac{44 - 11r}{3} = 0 $$ ⇒ r = 4 $So,$ {T}_{5} = C4 22 {\alpha}^{4} = 7315 $$ ⇒ \left(\dfrac{22 !}{4 ! \times 18 !}\right){\alpha}^{4} = 7315 $$ ⇒ 7315{\alpha}^{4} = 7315 $$ ⇒ {\alpha}^{4} = 1 $$ ⇒ \left|\alpha\right| = 1

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