Import Question JSON

Current Question (ID: 10404)

Question:

$\text{The bulk modulus of rubber is } 9.8 \times 10^8 \text{ N/m}^2\text{. To what depth a rubber ball be taken in a lake so that its volume is decreased by 0.1\%?}$

Options:

1. $25 \text{ m}$
2. $100 \text{ m}$
3. $200 \text{ m}$
4. $500 \text{ m}$

Solution:

\text{To make the rubber ball's volume decrease by 0.1%, it needs to be submerged to a depth} \text{where the pressure increase is } 9.8 \times 10^5 \text{ N/m}^2\text{.} \text{Using the bulk modulus formula: } B = -\frac{\Delta P}{\Delta V/V} \text{we find this pressure change. Then, using the pressure in fluid formula:} \Delta P = \rho g h \text{and knowing the density of water and gravity, we calculate the depth:} \Delta P = -B \times \left(\frac{\Delta V}{V}\right) = -\left(9.8 \times 10^8 \text{ N/m}^2\right) \times (-0.001) \Delta P = 9.8 \times 10^5 \text{ N/m}^2 h = \frac{\Delta P}{\rho g} = \frac{9.8 \times 10^5 \text{ N/m}^2}{(1000 \text{ kg/m}^3) \times (9.8 \text{ m/s}^2)} h = 100 \text{ m} \text{The required depth is 100 m.}

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
}