Binomial Theorem
Constant Term and Coefficient Relation in Binomial
nta_pyq_2023_apr
Grade 11
Question:
Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\dfrac{6}{x^{3/2}}\right)^n$, $n\leq 15$. If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of $x^{-n}$ is $\lambda\alpha$, then $\lambda$ is equal to ________.
Step-by-Step Solution
Key Concept: General term: $T_{r+1}=\binom{n}{r}(-6)^r x^{(n-4r)/2}$. Constant term requires $n=4r$. Sum of all coefficients (put $x=1$): $(1-6)^n=(-5)^n$. Remaining sum $=649+\alpha$... actually: $(-5)^n-\alpha=649$.
$n=4,\ \alpha=-24$. Coeff of $x^{-4}$: $\binom{4}{3}(-6)^3=-864=36\times(-24)$. $\lambda=36$.
Correct Answer: 36