If a0 and the coefficient of x^2 in the expa | vedclass

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(A) The general term in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is given by: $T_{r+1} = { }^6 C_r (a x^3)^{6-r} \left(\frac{c}{x}\right)^r

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$2$

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(A) The general term in the expansion of $\left(a x^3+\frac{c}{x}\right)^6$ is given by: $T_{r+1} = { }^6 C_r (a x^3)^{6-r} \left(\frac{c}{x}\right)^r$ $= { }^6 C_r a^{6-r} c^r x^{18-3r-r} = { }^6 C_r a^{6-r} c^r x^{18-4r}$ For the coefficient of $x^2$,we set the exponent of $x$ to $2$: $18-4r = 2$ $\Rightarrow 4r = 16$ $\Rightarrow r = 4$ Substituting $r=4$ into the expression for the coefficient: ${ }^6 C_4 a^{6-4} c^4 = 60$ $15 a^2 c^4 = 60$ $a^2 c^4 = 4$ Taking the square root on both sides: $a c^2 = \pm 2$ Since $a > 0$,we have $a c^2 = 2$.

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 $a c^2=$

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