Import Question JSON

Current Question (ID: 19390)

Question:

$\text{Earth shrinks to } \frac{1}{64} \text{ times of its initial volume. Time period of earth rotation is found to be } \frac{24}{x} \text{ hrs. The value of } x \text{ is:}$

Options:

1. $16$
2. $20$
3. $4$
4. $10$

Solution:

$\text{Hint: } I \omega = \text{constant}$ $\text{Step 1: Find the new radius of the earth.}$ $V = \left( \frac{V_0}{64} \right)$ $\frac{4}{3} \pi R^3 = \frac{1}{64} \times \frac{4}{3} \pi R_0^3$ $\Rightarrow R = \left( \frac{R_0}{4} \right)$ $\text{Step 2: Find the time period of the earth's rotation.}$ $\text{Since } M \text{ Remains the same.}$ $\Rightarrow I \omega = \text{constant}$ $\Rightarrow \frac{2}{5} M \left( R_0^2 \right) \frac{2 \pi}{(24 \text{hrs})} = \frac{2}{5} \times M \times \frac{R_0^2}{16} \times \frac{2 \pi}{T}$ $\Rightarrow T = \left( \frac{24}{16} \text{ hrs} \right)$ $\text{As it is provided that time period is } \frac{24}{x} \text{ hrs. So, } x = 16$ $\text{Hence, option (1) is the correct answer.}$

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
}