If $a b
eq 0$ and the sum of the coefficients of $x^7$ and $x^4$ in the expansion of $\left(\frac{x^2}{a}-\frac{b}{x} ight)^{11}$ is 0 , then

Question

If $a b 
eq 0$ and the sum of the coefficients of $x^7$ and $x^4$ in the expansion of $\left(\frac{x^2}{a}-\frac{b}{x}ight)^{11}$ is 0 , then

Aicharya · Accepted Answer

Given expansion is $\left(\frac{x^2}{a}-\frac{b}{x}ight)^{11}$. $	herefore$ The general term is $\begin{aligned} T_{r+1} & { }^{11} C_r\left(\frac{x^2}{a}ight)^{11-r}\left(-\frac{b}{x}ight)^r \ & ={ }^{11} C_r(x)^{22-3 r}(-b)^r\left(\frac{1}{a}ight)^{11-r}\end{aligned}$ $	herefore$ For coefficient $x^7$, put $22-3 r=7$ $\begin{aligned} \Rightarrow & & 3 r & =15 \ \Rightarrow & & r & =5\end{aligned}$ and for coefficient of $x^4$, put $22-3 r=4$ $\begin{aligned} & \Rightarrow \quad 3 r=18 \Rightarrow r=6 \ & 	herefore \quad T_6={ }^{11} C_5\left(\frac{1}{a}ight)^6(-b)^5 \ & 	ext { and } T_7={ }^{11} C_6\left(\frac{1}{a}ight)^6(-b)^6\end{aligned}$ According to the given condition, $\begin{array}{rlrl} & { }^{T_6}+T_7=0 \ & { }^{11} C_5\left(\frac{1}{a}ight)^6(-b)^5+{ }^{11} C_6\left(\frac{1}{a}ight)^5(-b)^6=0 \ \Rightarrow & { }^{11} C_5\left(\frac{1}{a}ight)^5(-b)^5\left(\frac{1}{a}-bight)=0 \ \Rightarrow & & \frac{1}{a}-b=0 \ \Rightarrow & a b=1\end{array}$

If $a b \neq 0$ and the sum of the coefficients of $x^7$ and $x^4$ in the expansion of…

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