Question

If the coefficient of $x^7$ in ${[a{x}^2+\left(\frac{1}{bx}\right)]}^{11}$ equals the coefficient of $x^{-7}$ in ${[a{x}^2-\left(\frac{1}{bx}\right)]}^{11}$, then $a$ and $b$ satisfy the relation

• A. $a-b=1$
• B. $a+b=1$
• C. $\frac{a}{b}=1$
• D. $ab=1$

Answer 1

The correct option is D $ab=1$

Coefficient of $x^7$ in ${(a{x}^2+\frac{1}{bx})}^{11}$ is ${}^{11}{C}_5{a}^6\frac{1}{b^5}$
Coefficient of $x^{-7}$ in ${(ax-\frac{1}{b{x}^2})}^1$ is ${}^{11}{C}_6{a}^5\frac{1}{b^6}$
${}^{11}{C}_6{a}^5\frac{1}{b^6}={}^{11}{C}_5{a}^6\frac{1}{b^5}$
$\Rightarrow ab=\frac{{}^{11}{C}_6}{{}^{11}{C}_5}=\frac{6}{6}=1$