Binomial Theorem
Equal Coefficients — Relationship between a and b
nta_pyq_2023_jan
Grade None
Question:
If the coefficient of $x^{15}$ in the expansion of $\left(ax^3+\dfrac{1}{bx^{1/3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(ax^{1/3}-\dfrac{1}{bx^3}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:
$a=b$
$ab=1$
$a=3b$
$ab=3$
Step-by-Step Solution
Key Concept: Coeff of $x^{15}$ at $r=9$: ${}^{15}C_9\cdot a^6/b^9$. Coeff of $x^{-15}$ at $r=6$: ${}^{15}C_6\cdot a^9/b^6$. ${}^{15}C_9={}^{15}C_6$.
$ab=1$.
Correct Answer: 2