Import Question JSON

Current Question (ID: 20103)

Question:

$\text{In the given circuit, charge } Q_2 \text{ on the } 2 \, \mu\text{F capacitor changes as } C \text{ is varied from } 1 \, \mu\text{F to } 3 \, \mu\text{F.}$ $Q_2 \text{ as a function of } C \text{ is given properly by: (figures are drawn schematically and are not to scale)}$

Options:

1. $Q_2 \text{ vs } C \text{ graph with increasing slope}$
2. $Q_2 \text{ vs } C \text{ graph with constant slope}$
3. $Q_2 \text{ vs } C \text{ graph with increasing curve}$
4. $Q_2 \text{ vs } C \text{ graph with decreasing curve}$

Solution:

$\text{Hint: } Q = C_{eq} V$ $Q = C_{eq} V = \left( \frac{C \times 3}{C + 3} \right) E = \frac{3CE}{C + 3}$ $Q_2 = \frac{2}{3} Q = \frac{2}{3} \times \frac{3CE}{C + 3} = \frac{2CE}{C + 3} = \frac{2E}{1 + \frac{3}{C}}$ $\text{When } C \text{ increases from the expression } Q \text{ also increases slope of } q \text{ versus } C \text{ is}$ $\frac{dQ}{dC} = \frac{6E}{[C + 3]^2}$ $\text{as } C \text{ increases } \frac{dQ}{dC} \text{ decreases}$

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
}