Binomial Theorem
Binomial Theorem
nta_abhyas_2025
Grade 11
Question:
If numerically greatest term in the expansion of $(3 - 5x)^{11}$, where $x = \frac{1}{5}$, 729A, then the value of $\frac{a}{b}$ is
Step-by-Step Solution
Key Concept: The greatest term in a binomial expansion occurs where the ratio of consecutive terms transitions from being greater than 1 to less than 1.
We have $(3 - 5x)^{11} = 3^{11}(1 - \frac{5x}{3})^{11}$. The general term is $T_{r+1} = 3^{11} \binom{11}{r}(-\frac{5x}{3})^r = 3^{11} \binom{11}{r}(-1)^r \frac{5^r x^r}{3^r} = (-1)^r 3^{11-r} 5^r \binom{11}{r} x^r$. For the greatest terms, we compute the ratio $\frac{T_{r+1}}{T_r} = \frac{55 - 5r}{3(r+1)}$. Setting this equal to 1: $55 - 5r = 3r + 3$, so $52 = 8r$, giving $r = 6.5$. The greatest term occurs at $r = 6$ and $r = 7$ (they are equal). Computing: $T_7 = T_6$ with ratio giving $\lambda = 65 \times 27 = 1755$ and $\frac{\lambda}{55} = 9.9$.
Correct Answer: 9