Question

If the constant term in the expansion of ${\left(3x^3-2x^2+\frac{5}{x^5}\right)}^{10}$ is $2^k.l$, where $l$ is an odd integer, then the value of $k$ is equal to :

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (D)

General term
$T_{r_r}=\frac{\lfloor 10}{\lfloor r_1\lfloor r_2\lfloor r_3}(3{)}^{r_1}(-2{)}^{r_2}(5{)}^{r_3}(x{)}^{3r_1+2r_2-5r_3}$
$3r_1+2r_2-5r_3=\mathrm{0....}$(1)
$r_1+r_2+r_3=10\dots$(2)
from equation (1) and (2)
$r_1+2\left(10-r_3\right)-5r_3=0$
$r_1+20=7r_3$
$\left(r_1,r_2,r_3\right)=(1,6,3)$
constant term $=\frac{\lfloor 10}{\lfloor 1\lfloor 6\lfloor 3}(3{)}^1(-2{)}^6(5{)}^3$
$=2^9\cdot 3^2\cdot 5^4\cdot 7^1$
$l=9$