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(C) The general term $T_{r+1}$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is given by:$T_{r+1} = {}^{10}C_r (\sqrt{x})^{10-r} \lef

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$\pm 3$

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(C) The general term $T_{r+1}$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is given by:$T_{r+1} = {}^{10}C_r (\sqrt{x})^{10-r} \left(-\frac{k}{x^2}\right)^r$$T_{r+1} = {}^{10}C_r (-k)^r x^{\frac{10-r}{2}} x^{-2r}$$T_{r+1} = {}^{10}C_r (-k)^r x^{\frac{10-5r}{2}}$For the term to be independent of $x$,the exponent of $x$ must be $0$:$\frac{10-5r}{2} = 0 \implies 10-5r = 0 \implies r = 2$Substituting $r=2$ into the expression:$T_3 = {}^{10}C_2 (-k)^2 = 405$$45 \cdot k^2 = 405$$k^2 = \frac{405}{45} = 9$$k = \pm 3$

If the term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is $405$,then $k=$

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