Explanation

(c)
$As\ m \in \left( \frac{1}{2},3 \right)$
$\therefore\$Line $y = mx + 4$ lies between
$y = 3x + 1$ and $2y = x + 3$
$Slope\ of\ given\ lines\ are\ m_{2} = 3,\ m = m\ and\ m_{1} = \frac{1}{2}$
$\therefore\tan\theta = \frac{3 - m}{1 + 3m}$
$and\tan\theta = \frac{m - \frac{1}{2}}{1 + \frac{m}{2}}$
$\Rightarrow \ \frac{3 - m}{1 + 3m} = \frac{2m - 1}{2 + m}$
$\Rightarrow \ \ {7m}^{2} - 2m - 7 = 0$
$\therefore\ m = \frac{2 \pm \sqrt{4 + 196}}{2 \times 7} = \frac{1}{7}(1 \pm 5\sqrt{2})$