Binomial Theorem
Coefficient matching; AM-GM for minimum
MJMT_Full_Test_05
Grade 12
Question:
If the coefficient of $x^7$ in the expansion of $\left(ax^2 + \frac{1}{bx}\right)^{11}$ and coefficient of $x^{-7}$ in the expansion of $\left(ax - \frac{1}{bx^2}\right)^{11}$ are equal, then the minimum value of $a^2 + b^2$ is
Step-by-Step Solution
Key Concept: Write the general term $T_{r+1}$ for each expansion, set the power of $x$ equal to the target value to find $r$, then equate the two coefficient expressions.
$T_{r+1} = {}^{11}C_r \frac{a^{11-r}}{b^r} x^{22-3r}$. For $x^7$: $r=5$. Second expansion: $t_{r+1} = (-1)^r {}^{11}C_r \frac{a^{11-r}}{b^r} x^{11-3r}$. For $x^{-7}$: $r=6$. Equating $T_6 = t_7 \Rightarrow ab=1$. By AM-GM: $a^2+b^2 \geq 2$.
Correct Answer: 3