Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 201912 AprilEvening Shift
Exam
JEE Main
Year
2019
Date
12 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2019 · 12 April · Evening Shift · Mathematics · Binomial Theorem

If ^ 20 C _ 1 + ( 2 ^ 2 ) ^ 20 C _ 2 + ( 3 ^ 2 ) ^ 20 C _ 3 + . . ....

If 20C1+(22)20C2+(32)20C3+.....+{}^{20}{C}_{1} + ( {2}^{2} ) {}^{20}{C}_{2} + ( {3}^{2} ) {}^{20}{C}_{3} + . . . . . +\left({20}^{2}\right){}^{20}{C}_{20} = A(2β),\left({2}^{\beta}\right), then the ordered pair (A,β)( A, \beta ) is equal to

Options

  1. A(380,19)\left(380, 19\right)
  2. B(420,18)( 420, 18 )
  3. C(420,19)( 420, 19 )
  4. D(380,18)( 380, 18 )

Correct answer

B. (420,18)( 420, 18 )

Step-by-step solution

(1+x)n=nC0+nC1x+nC2x2+......+nCnxn{( 1 + x )}^{n} = {}^{n}{C}_{0} + {}^{n}{C}_{1}x + {}^{n}{C}_{2}{x}^{2} + . . . . . . + {}^{n}{C}_{n}{x}^{n}
Diff. w. r. t. xx
⇒n(1+x)n−1=nC1+nC2(2x)+.....+nCnn(x)n−1⇒ n{( 1 + x )}^{n - 1} = {}^{n}{C}_{1} + {}^{n}{C}_{2} ( 2x ) + . . . . . + {}^{n}{C}_{n}n{( x )}^{n - 1}
Multiply by xx both side
⇒nx(1+x)n−1=nC1x+nC2⇒ nx{( 1 + x )}^{n - 1} = {}^{n}{C}_{1}x + {}^{n}{C}_{2}\left(2{x}^{2}\right)+.....+nCn(nxn)+ . . . . . + {}^{n}{C}_{n}\left(n{x}^{n}\right)
Diff. w. r. t. xx
⇒n[(1+x)n−1+(n−1)x(1+x)n−2]=nC1+nC222X+......+nCn(n2)xn−1⇒ n [ {( 1 + x )}^{n - 1} + ( n - 1 ) x{( 1 + x )}^{n - 2} ] = {}^{n}{C}_{1} + {}^{n}{C}_{2}{2}^{2}X + . . . . . . + {}^{n}{C}_{n} ( {n}^{2} ) {x}^{n - 1}
Put x=1andn=20x = 1 and n = 20
⇒20C1+2220C2+3220C3+......+(20)220C20⇒ {}^{20}{C}_{1} + {2}^{2}{}^{20}{C}_{2} + {3}^{2}{}^{20}{C}_{3} + . . . . . . + {( 20 )}^{2}{}^{20}{C}_{20}
=20×218= 20 \times {2}^{18}\left[2 + 19\right] = 420 \left({2}^{18}\right)=A= A\left({2}^{\beta}\right)<br/>Hence,(A,β)<br />Hence, \left(A, \beta\right) = ( 420, 18 )
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