JEE MAIN - Mathematics (2021 - 22th July Evening Shift - No. 22)
If the constant term, in binomial expansion of $${\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}$$ is 180, then r is equal to __________________.
Answer
8
Explanation
$${\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}$$
General term $$ = {}^{10}{C_R}{(2{x^2})^{10 - R}}{x^{ - 2R}}$$
$$ \Rightarrow {2^{10 - R}}{}^{10}{C_R} = 180$$ ....... (1)
& (10 $$-$$ R)r $$-$$ 2R = 0
$$r = {{2R} \over {10 - R}}$$
$$r = {{2(R - 10)} \over {10 - R}} + {{20} \over {10 - R}}$$
$$ \Rightarrow r = - 2 + {{20} \over {10 - R}}$$ ....... (2)
R = 8 or 5 reject equation (1) not satisfied
At R = 8
$$ \Rightarrow {2^{10 - R}}\times{}^{10}{C_R} = 180 \Rightarrow r = 8$$
General term $$ = {}^{10}{C_R}{(2{x^2})^{10 - R}}{x^{ - 2R}}$$
$$ \Rightarrow {2^{10 - R}}{}^{10}{C_R} = 180$$ ....... (1)
& (10 $$-$$ R)r $$-$$ 2R = 0
$$r = {{2R} \over {10 - R}}$$
$$r = {{2(R - 10)} \over {10 - R}} + {{20} \over {10 - R}}$$
$$ \Rightarrow r = - 2 + {{20} \over {10 - R}}$$ ....... (2)
R = 8 or 5 reject equation (1) not satisfied
At R = 8
$$ \Rightarrow {2^{10 - R}}\times{}^{10}{C_R} = 180 \Rightarrow r = 8$$
Comments (0)
