Question

The number of negative integral solution of $x^2.2^{x+1}+2^{|x-3|+2}=x^2.2^{(|x-3|+4)}+2^{x-1}$ is

• A. $4$
• B. $2$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

Comparing the terms on the RHS to that on the LHS, we get
$x+1=|x-3|+4$
$x=|x-3|+3$
If $x>3$
$x=x-3+3$ .. no solution.
And if $x\le 3$
$x=3-x+3$
$x=3$ ...(i)
And
$|x-3|+2=x-1$
$|x-3|=x-3$
If $x>3$ then
$x-3=x-3$ no solution.
And if $x\le 3$
$3-x=x-3$
$x=3$ ...(ii)
From i and ii, we get $x=3$ as one solution.
Substituting in the equation, we get
LHS $=\left(3{)}^2\right(2^4)+2^2$
$=9\left(16\right)+4$
$=148$
RHS $=3^2\left(2^4\right)+2^2$
$=148$
Hence $x=3$ is one and only solution.
Hence there are no negative integral solution.