1. Identify the basicity of the acids

   For oxyacids of phosphorus, only the hydrogen atoms attached to oxygen are ionisable.

   • $H_3PO_3$ has structure $HPO(OH)_2$. So it has 2 ionisable H atoms $\Rightarrow$ it is dibasic.

   • $H_3PO_2$ has structure $H_2PO(OH)$. So it has 1 ionisable H atom $\Rightarrow$ it is monobasic.

2. For $50\,\text{mL}$ of $1\,M\; H_3PO_3$

   Volume $= 50\,\text{mL} = 0.050\,L$

   Moles of $H_3PO_3$: $n = M \times V = 1 \times 0.050 = 0.050\,\text{mol}$

   Since $H_3PO_3$ is dibasic, moles of $NaOH$ required: $0.050 \times 2 = 0.100\,\text{mol}$

   For $1\,M\; NaOH$, volume required is: $V = \frac{n}{M} = \frac{0.100}{1} = 0.100\,L = 100\,\text{mL}$

3. For $100\,\text{mL}$ of $2\,M\; H_3PO_2$

   Volume $= 100\,\text{mL} = 0.100\,L$

   Moles of $H_3PO_2$: $n = M \times V = 2 \times 0.100 = 0.200\,\text{mol}$

   Since $H_3PO_2$ is monobasic, moles of $NaOH$ required: $0.200 \times 1 = 0.200\,\text{mol}$

   For $1\,M\; NaOH$, volume required is: $V = \frac{n}{M} = \frac{0.200}{1} = 0.200\,L = 200\,\text{mL}$

4. Final result

   Required volumes of $1\,M\; NaOH$ are: $100\,\text{mL} \text{ and } 200\,\text{mL}$

5. Option check

   • A: $100\,\text{mL}$ and $50\,\text{mL}$ $\rightarrow$ incorrect
   • B: $100\,\text{mL}$ and $200\,\text{mL}$ $\rightarrow$ correct
   • C: $100\,\text{mL}$ and $100\,\text{mL}$ $\rightarrow$ incorrect
   • D: $50\,\text{mL}$ and $50\,\text{mL}$ $\rightarrow$ incorrect

Therefore, the correct answer is B.