Binomial Theorem
Equal Coefficients in Two Binomial Expansions
nta_pyq_2023_apr
Grade 11
Question:
If the coefficients of $x^7$ in $\left(ax^2+\dfrac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\dfrac{1}{3bx^2}\right)^{11}$ are equal, then
729ab=32
32ab=729
64ab=243
243ab=64
Step-by-Step Solution
Key Concept: For $x^7$ in first: $22-3r=7\Rightarrow r=5$; coefficient $=\binom{11}{5}a^6(2b)^{-5}$. For $x^{-7}$ in second: $11-3r=-7\Rightarrow r=6$; coefficient $=\binom{11}{6}a^5(-3b)^{-6}$.
Equating coefficients: $729ab=32$.
Correct Answer: 1