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Q.
If the constant term in the expansion of $\left(3 x^{3}-2 x^{2}+\frac{5}{x^{5}}\right)^{10}$ is $2^{k} . l$, where $l$ is an odd integer, then the value of $k$ is equal to :
General term
$ T _{ r _{ r }}=\frac{\lfloor 10}{\lfloor r _{1}\lfloor r _{2} \lfloor r _{3}}(3)^{ r _{1}}(-2)^{ r _{2}}(5)^{ r _{3}}( x )^{3 r _{1}+2 r _{2}-5 r _{3}} $
$3 r _{1}+2 r _{2}-5 r _{3}=0 ....$(1)
$ r _{1}+ r _{2}+ r _{3}=10 \ldots $(2)
from equation (1) and (2)
$r _{1}+2\left(10- r _{3}\right)-5 r _{3}=0$
$ r _{1}+20=7 r _{3}$
$\left( r _{1}, r _{2}, r _{3}\right)=(1,6,3)$
constant term $=\frac{\lfloor 10}{\lfloor 1 \lfloor 6 \lfloor3}(3)^{1}(-2)^{6}(5)^{3}$
$=2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 7^{1} $
$l=9$