Correct option is C. $$\dfrac{24}{7}$$
$${\textbf{Step -1: Solving the given equation by applying relevant Trigonometric identity}}{\textbf{.}}$$
$${\text{sin x + cos x = }}\dfrac{1}{5}$$
$$\Rightarrow {\left( {{\text{sin x + cos x}}} \right)^2} = \dfrac{1}{{25}}$$ $$\textbf{(Squaring both sides)}$$
$$\Rightarrow {\text{1 + 2 sin x cos x = }}\dfrac{1}{{25}}$$
$$\Rightarrow {\text{1 + sin 2x = }}\dfrac{1}{{25}}$$
$$\therefore {\text{sin 2x = }}\dfrac{{ - 24}}{{25}}$$
$$\therefore{\text{co}}{{\text{s}}^2}{\text{ 2x = 1}} - \dfrac{{576}}{{625}}$$ $$\left[ {{\textbf{co}}{{\textbf{s}}^2}{\textbf{ 2x = 1}} - {\textbf{ si}}{{\textbf{n}}^2}{\textbf{ 2x}}} \right]$$
$$\Rightarrow {\cos ^2}{\text{2x = }}\dfrac{{49}}{{625}}$$
$$\Rightarrow\cos {\text{ }}2{\text{x = }} \pm \dfrac{7}{{25}}$$
$${\textbf{Step -2: Placing the value of sin 2x and cos 2x to find the value of tan 2x}}{\textbf{.}}$$
$${\text{tan 2x = }}\dfrac{{\sin {\text{ 2x}}}}{{{\text{cos 2x}}}}$$
$$ = \dfrac{{ - 24}}{{25}} \div \dfrac{{ \pm 7}}{{25}}$$
$$ = \dfrac{{ \pm 24}}{{7}}$$
$${\textbf{ Hence, the value of tan 2x is }} {\mathbf{\dfrac{{ \ 24}}{{7}}.}}$$