\\\\(\ extbf{Step 1: Initial situation}\\\\)Let mass of man in \\\\(M\\ kg\\\\) and speed is \\\\(V\\\\) , mass of boy is \\\\(M/2\\\\) and speed of boy is \\\\(V_{1}\\\\)\\\\(K.E_{man} = \\dfrac {1}{2} (K.E_{boy})\\\\)\\\\(\\dfrac {1}{2} MV^{2} = \\dfrac {1}{2} \\left (\\dfrac {1}{2} \\dfrac {M}{2} V_{1}^{2}\\right )\\\\) \\\\(.... (1)\\\\)\\\\(\ extbf{Step 2: Final situation}\\\\)Man speeds up by \\\\(1m/s\\\\) so new speed of man \\\\(=V+1\\\\)\\\\(K.E_{man}=K.E_{boy}\\\\)\\\\(\\dfrac {1}{2} M(V + 1)^{2} = \\dfrac {1}{2} \\left (\\dfrac {M}{2}\\right ) V_{1}^{2}\\\\) \\\\(.... (2)\\\\)\\\\(\ extbf{Step 3: Divide equation (1) by equation (2)}\\\\)\\\\(\\dfrac {V^{2}}{(V + 1)^{2}} = \\dfrac {1}{2}\\\\)\\\\(\\Rightarrow V = \\dfrac {1}{\\sqrt {2} - 1}\\\\)Hence option (C) is correct.