UPSC NDA I 2022 Mathematics Solved Paper

Concept:In the binomial expansion of $(x + y)^{n}$, the term with the greatest coefficient is the middle term.When $n$ is even, there is exactly one middle term: the $\left(\frac{n}{2}+1\right)$th term.Its coefficient is $\binom{n}{n/2}$, which is the largest.For any binomial coefficient, $\binom{n}{r} = \binom{n}{n-r}$ holds.Explanation:The given expansion is $(x + y)^{10}$, so $n = 10$, which is even.The middle term is the 6th term because $\frac{10}{2}+1 = 6$.Thus the coefficient of the 6th term, $\binom{10}{5}$, is the highest among all coefficients.Therefore, Statement 1 is correct.For Statement 2, the $r$th term coefficient is $\binom{10}{r-1}$.The coefficient of the 3rd term is $\binom{10}{2}$.The coefficient of the 9th term is $\binom{10}{8}$.Since $\binom{10}{2} = \binom{10}{8}$, both are equal.Hence Statement 2 is also correct.Answer:Both statements 1 and 2 are correct. So, option C is correct.