If $a0$ and the coefficient of $x^2$ in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is 60, then $a c^2=$

Question

If $a>0$ and the coefficient of $x^2$ in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is 60, then $a c^2=$

Aicharya · Accepted Answer

General term, $ \begin{aligned} & T_{r+1}={ }^6 C_r\left(a x^3ight)^{6-r}\left(\frac{c}{x}ight)^r \ & ={ }^6 C_r a^{6-r} c^r x^{18-3 r-r} \ & ={ }^6 C_r a^{6-r} c^r x^{18-4 r} \end{aligned} $ According to the question, $ \begin{aligned} 18-4 r & =2 \Rightarrow 4 r=16 \ r & =4 . \end{aligned} $ So, ${ }^6 C_4 a^{6-4} c^4=60 \Rightarrow 15 a^2 c^4=60$ $ \begin{array}{rlrl} & \Rightarrow & a^2 c^4=4 \Rightarrow a c^2= \pm 2 \ \because & a>0 & \ 	herefore & a c^2=2 \end{array} $

If $a>0$ and the coefficient of $x^2$ in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is 60, then $a…

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