Question

The sum of the coefficients in the expansion of $(1-2x+5x^2{)}^n$ is a and the sum of the coefficients in the expansion of $(1+x{)}^{2n}$ is $b$. Then

• A. $a=b$
• B. $(x-a{)}^2-(x-b{)}^2$ $=$ $0$, has real roots
• C. $si{n}^{2}a+co{s}^{2}b$ $=$ $1$
• D. $ab=1$

Answer 1

The correct option is A $a=b$

Let $(1-2x+5x^2{)}^n=a_o+a_{1}x+a_2{x}^2+...\dots ..+a_n{x}^n$

To find sum of coefficient put $x=1$

$\Rightarrow a=a_o+a_1+...\dots .+a_n=(1-2+5{)}^n$

$\Rightarrow a=4^n=2^{2n}$

For $(1+x{)}^{2n}=c_o+c_{1}x+c_2{x}^2+.....+c_{2n}{x}^{2n}$ at $x=1$

sum$=c_o+c_1+c_2+....+c_n=b=2^{2n}$

$\Rightarrow b=2^{2n}$

$\Rightarrow a=b$.