Question

The sum of the roots of the equation $2^{\left(33x-2\right)}+2^{\left(11x+2\right)}=2^{\left(22x+1\right)}+1$ is

• A. $\frac{1}{11}$
• B. $\frac{2}{11}$
• C. $\frac{3}{11}$
• D. $\frac{4}{11}$

Answer 1

Answer: (B)

Let, $t=2^{11x}$
$\Rightarrow \frac{{\left(2^{11x}\right)}^3}{2^2}+2^{11x}{\left(.2\right)}^2={\left(2^{11x}\right)}^2.2+1$
$\Rightarrow \frac{t^3}{4}+4t=2t^2+1$
$\Rightarrow t^3-8t^2+16t-4=0$
Cubic in $t$ has roots $t_1,t_2,t_3$
i.e. $t_1{t}_2{t}_3=4\Rightarrow 2^{11x_1}.2^{11x_2}.2^{11x_3}=4$
$\Rightarrow 2^{11\left(x_1+x_2+x_3\right)}=2^2$
$\Rightarrow 11\left(x_1+x_2+x_3\right)=2\Rightarrow x_1+x_2+x_3=\frac{2}{11}$