In the expansion of [(x3)+3x2]10, the constant term is ___

Given, [x3+3x2]10General term, Tr+1= 10Cr(x3)12(10−r)(3x2)r ⇒Tr+1 =10Cr(13)10−r2(3)tx12(10−r)−2rFor term independent of x put12(10−r)−2r=0 ⇒r=2 ∴ T2+1=T3= 10C2(13)82(3)2⇒=45×1×381=53

In the expansion of [(x3)+3x2]10, the constant term is ___

Given, [x3+3x2]10General term, Tr+1= 10Cr(x3)12(10−r)(3x2)r ⇒Tr+1 =10Cr(13)10−r2(3)tx12(10−r)−2rFor term independent of x put12(10−r)−2r=0 ⇒r=2 ∴ T2+1=T3= 10C2(13)82(3)2⇒=45×1×381=53

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In the expansion of ${[\sqrt{(\frac{x}{3})} + \frac{\sqrt{3}}{x^{2}}]}^{10},$ the constant term is ___

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$\frac{5}{3}$

$\frac{4}{5}$

$\frac{1}{2}$

answer is A.

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Detailed Solution

Given, ${[\sqrt{\frac{x}{3}} + \frac{\sqrt{3}}{x^{2}}]}^{10}$
General term, $T_{r + 1}$
$=^{10} C_{r} {(\frac{x}{3})}^{\frac{1}{2} (10 - r)} {(\frac{\sqrt{3}}{x^{2}})}^{r}$ $\Rightarrow T_{r + 1}$ $=^{10} C_{r} {(\frac{1}{3})}^{\frac{10 - r}{2}} {(\sqrt{3})}^{t} x^{\frac{1}{2} (10 - r) - 2 r}$
For term independent of x put

$\frac{1}{2} (10 - r) - 2 r = 0$ $\Rightarrow r = 2$ $∴ T_{2 + 1} = T_{3} =^{10} C_{2} {(\frac{1}{3})}^{\frac{8}{2}} {(\sqrt{3})}^{2}$