UPSC NDA I 2022 Mathematics Solved Paper

Concept:Use the identity $(a+b)(a-b)=a^2-b^2$ to combine the two factors into a single binomial expression. The number of terms in the expansion of $(1+t)^n$ is $n+1$.Explanation:Step 1: Rewrite the product as $\left[\left(1+\frac{2}{x}\right)\left(1-\frac{2}{x}\right)\right]^9$.Step 2: Apply the identity: $\left(1+\frac{2}{x}\right)\left(1-\frac{2}{x}\right)=1-\left(\frac{2}{x}\right)^2=1-\frac{4}{x^2}$.Step 3: The given expression becomes $\left(1-\frac{4}{x^2}\right)^9$, which is a binomial of the form $(1+y)^9$ with $y=-\frac{4}{x^2}$.Step 4: In the binomial expansion of $(a+b)^n$, the total number of terms is $n+1$. Here $n=9$, so the number of terms is $9+1=10$.Answer:There are $10$ terms in the expansion (Option B).