Import Question JSON

Current Question (ID: 17340)

Question:

$\text{When an AC source of emf } e = E_0 \sin(100t) \text{ is connected across a circuit, the phase difference between the emf } e \text{ and the current } i \text{ in the circuit is observed to be } \frac{\pi}{4} \text{ as shown in the diagram. If the circuit consists only of } RC \text{ or } LC \text{ in series, then what is the relationship between the two elements?}$

Options:

1. $R = 1 \text{ k}\Omega, C = 10 \mu\text{F}$
2. $R = 1 \text{ k}\Omega, C = 1 \mu\text{F}$
3. $R = 1 \text{ k}\Omega, L = 10 \text{ H}$
4. $R = 1 \text{ k}\Omega, L = 1 \text{ H}$

Solution:

$\text{As the current } i \text{ leads the voltage by } \frac{\pi}{4}, \text{ it is an } RC \text{ circuit,}$ $\text{Hence, } \tan \phi = \frac{X_c}{R} \Rightarrow \tan \frac{\pi}{4} = \frac{1}{\omega CR}$ $\Rightarrow \omega CR = 1 \text{ as } \omega = 100 \text{ rad/sec}$ $\Rightarrow CR = \frac{1}{100} \text{ sec}^{-1}$ $\text{From all the given options, only option (1) is correct.}$

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
}