Import Question JSON

Current Question (ID: 17286)

Question:

$\text{The output current versus time curve of a rectifier is shown in the figure. The average value of the output current in this case will be:}$

Options:

1. $0$
2. $\frac{I_0}{2}$
3. $\frac{2I_0}{\pi}$
4. $I_0$

Solution:

$\text{Hint: } I_{\text{av}} = \frac{\int_0^{T/2} i \, dt}{\int_0^{T/2} dt}$ $\text{Step: Find the average value of the output current.}$ $I_{\text{av}} = \frac{\int_0^{T/2} I_0 \sin(\omega t) \, dt}{\int_0^{T/2} dt} = \frac{2I_0}{T} \left[ -\frac{\cos \omega t}{\omega} \right]_0^{T/2}$ $= \frac{2I_0}{T} \left[ -\frac{\cos \left( \frac{\omega T}{2} \right)}{\omega} + \frac{\cos 0^\circ}{\omega} \right]$ $= \frac{2I_0}{\omega T} \left[ -\cos \pi + \cos 0^\circ \right]$ $\Rightarrow I_{\text{av}} = \frac{2I_0}{2\pi} [1 + 1] = \frac{2I_0}{\pi}$ $\text{Hence, option (3) is the correct answer.}$

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
}