Import Question JSON

Current Question (ID: 19472)

Question:

$\text{At depth } d \text{ from the surface of the earth acceleration due to gravity is}$ $\text{same as its value at height } d \text{ above the surface of the earth. If the}$ $\text{earth is a sphere of radius } 6400 \text{ km, then the value of } d \text{ is equal to:}$

Options:

1. $2975 \text{ km}$
2. $3955 \text{ km}$
3. $2525 \text{ km}$
4. $4915 \text{ km}$

Solution:

$\text{Given } g_h = g_d$ $\text{We know that:}$ $g_0 \left(1 - \frac{d}{R}\right) = \frac{g_0}{\left(1 + \frac{d}{R}\right)^2}$ $\left(1 + \frac{d}{R}\right)^2 \left(1 - \frac{d}{R}\right) = 1$ $\text{On solving:}$ $\frac{d}{R} = 0, -\frac{\sqrt{5} + 1}{2}, \frac{\sqrt{5} - 1}{2}$ $\text{So,}$ $d = \left(\frac{\sqrt{5} - 1}{2}\right) R$ $d = 3955 \text{ km}$

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
}