If the greatest value of the term independent of x in the expansion of (xsinα + a(cosα)/(x)) ^ 10 is 10 ! (5 !) ^ 2 , then the value of a is equal to:

Correct answer: D. 2 We have, r ^ th term in the expansion of (xsinα + a(cosα)/(x)) ^ 10 is T _ r + 1 = Cr 10 (xsinα) ^ 10 - r ((acosα)/(x)) ^ r r = 0, 1, 2, , 10 T _ r + 1 will be independent of x, when 10 - 2r = 0 ⇒ r = 5 Now, T _ 6 = C5 10 (xsinα) ^ 5 × ((acosα)/(x)) ^ 5 = C5 10 × a ^ 5 × 1 2 ^ 5 ( 2α) ^ 5 T _ 6 will be greatest, when 2α = 1 Therefore, C5 10 a ^ 5 2 ^ 5 = C5 10 ⇒ a = 2

JEE Main 202125 JulyEvening Shift

Exam: JEE Main
Year: 2021
Date: 25 July
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2021 · 25 July · Evening Shift · Mathematics · Binomial Theorem

If the greatest value of the term independent of x in the expansion...

If the greatest value of the term independent of

x

in the expansion of

{\left(xsin\alpha + a\dfrac{cos\alpha}{x}\right)}^{10}

\dfrac{10 !}{{\left(5 !\right)}^{2}},

then the value of

a

is equal to:

Options

A $- 1$
B $1$
C $- 2$
D $2$

Correct answer

D. $2$

Step-by-step solution

We have, ${r}^{th}$ term in the expansion of ${\left(xsin\alpha + a\dfrac{cos\alpha}{x}\right)}^{10} is$ {T}_{r + 1} = Cr 10 {\left(xsin\alpha\right)}^{10 - r}{\left(\dfrac{acos\alpha}{x}\right)}^{r} $$ r = 0, 1, 2, \ldots , 10 $$ {T}_{r + 1} $will be independent of$ x, $when$ 10 - 2r = 0 $$ ⇒ r = 5 $Now,$ {T}_{6} = C5 10 {\left(xsin\alpha\right)}^{5} \times {\left(\dfrac{acos\alpha}{x}\right)}^{5} $$ = C5 10 \times {a}^{5} \times \dfrac{1}{{2}^{5}}{\left(\sin 2\alpha\right)}^{5} $$ {T}_{6} $will be greatest, when$ \sin 2\alpha = 1 $Therefore,$ C5 10 \dfrac{{a}^{5}}{{2}^{5}} = C5 10 $$ ⇒ a = 2

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