An infinite ladder network of resistances is constructed with $1\Omega$ and $2\Omega$ resistances as shown

$x=\frac{\left(x+1\right)2}{2+x+1}\Rightarrow x^2+3x-2x-2=0\Rightarrow x^2+x-2=0$

$x=1\Omega ;i=\frac{V}{R}=\frac{6}{1}=6A$

Option (b) is correct. Option (a) is wrong.
$6\text{A}$ current flows through battery. Because $x=1\Omega$ , the current divides equally at the first junction of the upper branch. Current through $2\Omega$ resistor nearest to battery is 3A. Option (d) is correct, option (c) is wrong.