Import Question JSON

Current Question (ID: 9656)

Question:

$\text{A particle starts moving on a circular path from rest, such that its tangential acceleration varies with time as } a_t = kt. \text{ Distance travelled by particle on the circular path in time } t \text{ is:}$

Options:

1. $\frac{kt^3}{3}$
2. $\frac{kt^2}{6}$
3. $\frac{kt^3}{6}$
4. $\frac{kt^2}{2}$

Solution:

$\text{Given, } a_t = Kt$ $\frac{dv}{dt} = Kt$ $dv = Kt dt$ $\text{On integrating -}$ $\int dv = \int Kt dt \Rightarrow v = \frac{Kt^2}{2}$ $v = \frac{dx}{dt} = \frac{Kt^2}{2}$ $dx = \frac{Kt^2}{2} dt$ $\text{Integrating again-}$ $\int dx = \int \frac{Kt^2}{2} dt$ $\Rightarrow x = \frac{Kt^3}{6}$

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
}