Let (1+x+x^2)^ 10 =a_0+a_1 x+a_2 x^2+ .+a_ 20 x^ 20 . If (a_1+a_3+a_5+ .+a_ 19 )-11 a_2=121 k , then k is equal to .

Correct answer: 0 (1+x+x^2)^ 10 =a_0+a_1 x+a_2 x^2+ .+a_ 20 x^ 20 3^ 10 =a_0+a_1+a_2+ .+a_ 20 ...(i) 1=a_0-a_1+a_2 . .+a_ 20 ...(ii) (i) - (ii) a_1+a_3+ .+a_ 19 = 3^ 10 -1 2 =29524 aligned Also \ & 1+x(1+x)\ ^ 10 =1 & + ^ 10 C_1 x(1+x)+ ^ 10 C_2 x^2(1+x)^2+ . aligned a_2= ^ 10 C_1+ ^ 10 C_2=55 (a_1+a_3+ +a_ 19 )-11 a_2 121 =239

JEE Main 20253 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 3 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2025 · 3 April · Evening Shift · Mathematics · Binomial Theorem

Let (1+x+x^2)^ 10 =a_0+a_1 x+a_2 x^2+ .+a_ 20 x^ 20 . If...

Let

\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots .+a_{20} x^{20}

. If \left(a_1+a_3+a_5+\ldots .+a_{19}\right)-11 \mathrm{a}_2=121 \mathrm{k}

, then k is equal to

\qquad$ .

Correct answer

Step-by-step solution

\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots .+a_{20} x^{20} \therefore 3^{10}=\mathrm{a}_0+\mathrm{a}_1+\mathrm{a}_2+\ldots .+\mathrm{a}_{20}

...(i)

1=a_0-a_1+a_2 \ldots . .+a_{20}

...(ii)

\text{ (i) - (ii) } \Rightarrow a_1+a_3+\ldots .+a_{19}=\frac{3^{10}-1}{2}=29524

\begin{aligned} \text{ Also }\{ & 1+\mathrm{x}(1+\mathrm{x})\}^{10}=1 \\ & +{ }^{10} \mathrm{C}_1 \mathrm{x}(1+\mathrm{x})+{ }^{10} \mathrm{C}_2 \mathrm{x}^2(1+\mathrm{x})^2+\ldots . \end{aligned}

\therefore \mathrm{a}_2={ }^{10} \mathrm{C}_1+{ }^{10} \mathrm{C}_2=55 \therefore \frac{\left(\mathrm{a}_1+\mathrm{a}_3+\ldots+\mathrm{a}_{19}\right)-11 \mathrm{a}_2}{121}=239

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