If A denotes the sum of all the coefficients in the expansion of (1-3 x+10 x 2 ) n and B denotes the sum of all the coefficients in the expansion of (1+x 2 ) n , then :

Mathematics · Binomial Theorem

JEE Main 2024 — 27 January, Shift 1 — Question 5

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^{2}\right)^{n}$ and $B$ denotes the sum of all the coefficients in the expansion of $\left(1+x^{2}\right)^{n}$ , then :

Option A:
$A=B^{3}$
Correct
Option B:
$3 A=B$
Option C:
$B=A^{3}$
Option D:
$\mathrm{A}=3 \mathrm{~B}$

Answer: A

Step-by-step solution

Sum of coefficients in the expansion of $\left(1-3 \mathrm{x}+10 \mathrm{x}^{2}\right)^{\mathrm{n}}=\mathrm{A}$

then $\mathrm{A}=(1-3+10)^{\mathrm{n}}=8^{\mathrm{n}}($ put $\mathrm{x}=1)$

and sum of coefficients in the expansion of $\left(1+x^{2}\right)^{n}=B$

Then $B=(1+1)^{\mathrm{n}}=2^{\mathrm{n}}$

$A=B^{3}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 27 January, Shift 1
Subject: Mathematics
Chapter: Binomial Theorem
Topic: Multinomial Theorem