Import Question JSON

Current Question (ID: 17336)

Question:

$\text{A bulb is rated at } 100 \text{ V, } 100 \text{ W, and is treated as a resistor. The inductance of an inductor (choke coil) that should be connected in series with the bulb to operate the bulb at its rated power with the help of an AC source of } 200 \text{ V and } 50 \text{ Hz is:}$

Options:

1. $\frac{\pi}{\sqrt{3}} \text{ H}$
2. $100 \text{ H}$
3. $\frac{\sqrt{2}}{\pi} \text{ H}$
4. $\frac{\sqrt{3}}{\pi} \text{ H}$

Solution:

$R = \frac{(100)^2}{100} = 100 \, \Omega$ $\text{now, } V = 200 \sqrt{2} \sin(2 \pi 50 t)$ $\text{Here, } \omega = 100 \pi \, \text{rad/sec}$ $I_{\text{rms}} = \frac{200}{\sqrt{R^2 + (L \omega)^2}}$ $\text{and } I_{\text{rms}}^2 R = 100$ $\Rightarrow \frac{(200)^2}{R^2 + (\omega L)^2} \cdot R = 100$ $\Rightarrow \frac{R^2 + L^2 (100 \pi)^2}{200 \times 200} \times 100 = 100$ $\Rightarrow 1 + L^2 \pi^2 = 4$ $\Rightarrow L = \frac{\sqrt{3}}{\pi} \, \text{H}$

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
}