Import Question JSON

Current Question (ID: 19229)

Question:

$\text{It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is } P_d; \text{ while for its similar collision with carbon nucleus at rest, fractional loss of energy is } P_c. \text{ The values of } P_d \text{ and } P_c \text{ are respectively:}$

Options:

1. $0.89, \ 0.28$
2. $0.28, \ 0.89$
3. $0, \ 0$
4. $0, \ 1$

Solution:

$\text{Hint: Fractional loss of energy} = 1 - \left( \frac{m_1 - m_2}{m_1 + m_2} \right)^2$ $\mathbf{v_1 = \left( \frac{m_1 - m_2}{m_1 + m_2} \right) u_1}$ $\text{fractional loss of energy}$ $= \frac{1}{2} m_1 u_1^2 - \frac{1}{2} m_1 \left( \frac{m_1 - m_2}{m_1 + m_2} \right)^2 u_1^2$ $= \frac{1}{2} m_1 u_1^2 \left( 1 - \left( \frac{m_1 - m_2}{m_1 + m_2} \right)^2 \right)$ $= 1 - \left( \frac{m_1 - m_2}{m_1 + m_2} \right)^2$ $\textbf{For neutron deuteron collision: } m_1 = m \text{ and } m_2 = 2m$ $P_d = 1 - \left( \frac{m - 2m}{m + 2m} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9} = 0.89$ $\textbf{For neutron-carbon collision: } m_1 = m \text{ and } m_2 = 12m$ $P_c = 1 - \left( \frac{m - 12m}{m + 12m} \right)^2 = 1 - \frac{121}{169} = 0.28$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}