Import Question JSON

Current Question (ID: 19527)

Question:

$\text{An astronaut takes a ball of mass } m \text{ from earth to space. He throws the ball into a circular orbit about earth at an altitude of } 318.5 \text{ km.}$ $\text{From earth's surface to the orbit, the change in total mechanical energy of the ball is } x \frac{G M_e m}{21 R_e}. \text{ The value of } x \text{ is (take } R_e = 6370 \text{ km)}$

Options:

1. $11$
2. $10$
3. $9$
4. $12$

Solution:

$\text{The change in total mechanical energy is given by } \Delta E = \frac{G M_e m}{2} \left( \frac{1}{R_e} - \frac{1}{R_e + h} \right)$ $\text{Substitute } h = 318.5 \text{ km and } R_e = 6370 \text{ km}$ $\Delta E = \frac{G M_e m}{2} \left( \frac{1}{6370} - \frac{1}{6688.5} \right)$ $\Delta E = \frac{G M_e m}{2} \times \frac{318.5}{6370 \times 6688.5}$ $\Delta E = x \frac{G M_e m}{21 R_e}$ $x = 11$

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
}