Import Question JSON

Current Question (ID: 19639)

Question:

$\text{The heat passing through the cross-section of a conductor, varies with time } t \text{ as } Q(t) = \alpha t - \beta t^2 + \gamma t^3 \ (\alpha, \beta \text{ and } \gamma \text{ are positive constants}).$ $\text{The minimum heat current through the conductor is:}$

Options:

1. $\alpha - \frac{\beta^2}{2\gamma}$
2. $\alpha - \frac{\beta^2}{3\gamma}$
3. $\alpha - \frac{\beta^2}{\gamma}$
4. $\alpha - \frac{3\beta^2}{\gamma}$

Solution:

$\text{Heat through cross section of rod, } Q = \alpha t - \beta t^2 + \gamma t^3$ $\text{Heat current, } i = \frac{dQ}{dt} = \alpha - 2\beta t + 3\gamma t^2$ $\text{For heat current to be minimum, } \frac{di}{dt} = 0$ $\Rightarrow -2\beta + 6\gamma t = 0$ $\Rightarrow t = \frac{\beta}{3\gamma}$ $\text{So minimum heat current, } i_{\text{min}} = \alpha - 2\beta \times \frac{\beta}{3\gamma} + 3\gamma \left(\frac{\beta}{3\gamma}\right)^2$ $= \alpha - \frac{\beta^2}{3\gamma}$

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
}