How many terms are there in the expansion of ((a^2)/(b^2)+(b^2)/(a^2)+2)^ 21 where a ≠ 0, b ≠ 0 ?

Correct answer: D. 43 aligned & ((a^2)/(b^2)+(b^2)/(a^2)+2)^ 21 =((a^4+b^4+2 a^2 b^2)/(a^2 b^2))^ 21 & =(((a^2+b^2)/(a b))^2)^ 21 =((a)/(b)+(b)/(a))^ 42 & Number ofterms =42+1=43. aligned

NDA 2021

Exam: NDA
Year: 2021
Subject: Mathematics
Chapter: Binomial Theorem

NDA 2021 · Mathematics · Binomial Theorem

How many terms are there in the expansion of...

How many terms are there in the expansion of

\left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}

where

a \neq 0, b \neq 0

Options

Correct answer

D. 43

Step-by-step solution

\begin{aligned} & \left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}=\left(\frac{a^4+b^4+2 a^2 b^2}{a^2 b^2}\right)^{21} \\ & \quad=\left(\left(\frac{a^2+b^2}{a b}\right)^2\right)^{21}=\left(\frac{a}{b}+\frac{b}{a}\right)^{42} \\ & \therefore \quad \text{ Number ofterms }=42+1=43. \end{aligned}

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