If the coefficient of x^7 in (ax^2+(1)/(2bx))^11 and x^-7 in (ax-13bx^2)^11 are equal, the…

Question

If the coefficient of ${x}^{7}$ in ${(a{x}^{2}+\frac{1}{2bx})}^{11}$ and ${x}^{-7}$ in ${(ax-\frac{1}{3b{x}^{2}})}^{11}$ are equal, then :

Accepted Answer

General term of $\left(a x^2+\frac{1}{2 b x}\right)^{11}$ is

$$
T_{r+1}={ }^{11} C_r\left(a x^2\right)^{11-r}\left(\frac{1}{2 b x}\right)^r={ }^{11} C_r(a)^{11-r}\left(\frac{1}{2 b}\right)^r x^{22-3 r}
$$

$$
\[\begin{gathered} \text { Now, } 22-3 r  =7 \ \Rightarrow  15  =3 r \ \Rightarrow  r  =5 \end{gathered}$$
\]

and general term of $\left(a x-\frac{1}{3 b x^2}\right)^{11}$ is

$$
\begin{aligned}
T_{r+1} & ={ }^{11} C_r(a x)^{11-r}\left(-\frac{1}{3 b x^2}\right)^r \\
& ={ }^{11} C_r a^{11-r}\left(-\frac{1}{3 b}\right)^r x^{11-3 r}
\end{aligned}
$$

Now, $11-3 r=-7$

$$
\Rightarrow 18=3 r \Rightarrow r=6
$$

$\begin{aligned} & \text { Since, coefficient of } x^7 \text { in }\left(a x^2+\frac{1}{2 b x}\right)^{11} \\ & =\text { Coefficient of } x^{-7} \text { in }\left(a x-\frac{1}{3 b x^2}\right)^{11} \\ & \Rightarrow{ }^{11} C_5(a)^6\left(\frac{1}{2 b}\right)^5={ }^{11} C_6(a)^5\left(-\frac{1}{3 b}\right)^6 \\ & \Rightarrow \frac{a}{32 b^5}=\frac{1}{729 b^6} \Rightarrow 729 a b=32\end{aligned}$

Solutions · 1