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JEE Main 202427 JanuaryMorning Shift
Exam
JEE Main
Year
2024
Date
27 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2024 · 27 January · Morning Shift · Mathematics · Binomial Theorem

If A denotes the sum of all the coefficients in the expansion of (1...

If AA denotes the sum of all the coefficients in the expansion of (1−3x+10x2)nand{\left(1 - 3x + 10{x}^{2}\right)}^{n} andBdenotesthesumofallthecoefficientsintheexpansionofdenotes the sum of all the coefficients in the expansion of{\left(1 + {x}^{2}\right)}^{n}$, then :

Options

  1. AA=B3A = {B}^{3}
  2. B3A=B3A = B
  3. CB=A3B = {A}^{3}
  4. DA=3BA = 3B

Correct answer

A. A=B3A = {B}^{3}

Step-by-step solution

We have,

A=A =Sum of the coefficients in the expansion of

⇒(1−3x+10x2)n=(1−3+10)n=(8)n=(2)3n⇒ {\left(1 - 3x + 10{x}^{2}\right)}^{n} = ( 1 - 3 + 10{)}^{n} = ( 8{)}^{n} = ( 2{)}^{3n}

(putting x=1x = 1)

Now, B=B =Sum of the coefficients in the expansion of

⇒(1+x2)n=(1+1)n=2n⇒ {\left(1 + {x}^{2}\right)}^{n} = ( 1 + 1{)}^{n} = {2}^{n}

Clearly, A=B3A = {B}^{3}.

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