JEE MAIN - Mathematics (2024 - 27th January Morning Shift - No. 19)
If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :
$\mathrm{B}=\mathrm{A}^3$
$3 \mathrm{A}=\mathrm{B}$
$A=3 B$
$\mathrm{A}=\mathrm{B}^3$
Explanation
Sum of coefficients in the expansion of $$\left(1-3 \mathrm{x}+10 \mathrm{x}^2\right)^{\mathrm{n}}=\mathrm{A}$$
then $$A=(1-3+10)^n=8^n$$ (put $$x=1$$)
and sum of coefficients in the expansion of
$$\begin{aligned} & \left(1+x^2\right)^n=B \\ & \text { then } B=(1+1)^n=2^n \\ & A=B^3 \end{aligned}$$
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