Question

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

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

Answer 1

Answer: (B)

Let, $t=2^{11 x}$
$\Rightarrow \frac{\left(2^{11 x}\right)^{3}}{2^{2}}+2^{11 x}\left(. 2\right)^{2}=\left(2^{11 x}\right)^{2}.2+1$
$\Rightarrow \frac{t^{3}}{4}+4t=2t^{2}+1$
$\Rightarrow t^{3}-8t^{2}+16t-4=0$
Cubic in $t$ has roots $t_{1},t_{2},t_{3}$
i.e. $t_{1}t_{2}t_{3}=4\Rightarrow 2^{11 x_{1}}. 2^{11 x_{2}}. 2^{11 x_{3}}=4$
$\Rightarrow 2^{11 \left(x_{1} + x_{2} + x_{3}\right)}=2^{2}$
$\Rightarrow 11\left(x_{1} + x_{2} + x_{3}\right)=2\Rightarrow x_{1}+x_{2}+x_{3}=\frac{2}{11}$