Binomial Theorem
Equal Coefficients in Two Binomial Expansions — Finding Product
nta_pyq_2023_apr
Grade 11
Question:
If the coefficient of $x^7$ in $\left(ax-\dfrac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\dfrac{1}{bx^2}\right)^{13}$ are equal, then $a^4b^4$ is equal to:
Step-by-Step Solution
Key Concept: For $x^7$ in first: $13-3r=7\Rightarrow r=2$; coeff $=\binom{13}{2}a^{11}b^{-2}$. For $x^{-5}$ in second: $13-3r=-5\Rightarrow r=6$; coeff $=\binom{13}{6}a^7b^{-6}$.
$a^4b^4=\frac{\binom{13}{6}}{\binom{13}{2}}=22$.
Correct Answer: 3