Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Binomial Theorem

If the coefficients of x^4, x^5 and x^6 in the expansion of (1+x)^n...

If the coefficients of x4,x5x^4, x^5 and x6x^6 in the expansion of (1+x)n(1+x)^n are in the arithmetic progression, then the maximum value of nn is:

Options

  1. A7
  2. B21
  3. C28
  4. D14

Correct answer

D. 14

Step-by-step solution

 Coeff. of  x4=nC4 Coeff. of x5=nC5 Coeff. of x6=nC6nC4,nC5,nC6….AP2C5=nC4+nC62=nC4nC5+nC6nC5{nCrnCr=n−r+1r}2=5n−4+n−5612(n−4)=30+n2−9n+20n2−21n+98=0(n−14)(n−7)=0nmax⁡=14nmin⁡=7\begin{aligned} & \text{ Coeff.\text{ of } } x^4={ }^n C_4 \\ & \text{ Coeff. of } x^5={ }^n C_5 \\ & \text{ Coeff. of } x^6={ }^n C_6 \\ & { }^n C_4,{ }^n C_5,{ }^n C_6 \ldots . A P \\ & 2 C_5={ }^n C_4+{ }^n C_6 \\ & 2=\frac{{ }^n C_4}{{ }^n C_5}+\frac{{ }^n C_6}{{ }^n C_5} \quad\left\{\frac{{ }^n C_r}{{ }^n C_r}=\frac{n-r+1}{r}\right\} \\ & 2=\frac{5}{n-4}+\frac{n-5}{6} \\ & 12(n-4)=30+n^2-9 n+20 \\ & n^2-21 n+98=0 \\ & (n-14)(n-7)=0 \\ & n_{\max }=14 \quad n_{\min }=7\end{aligned}
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