Import Question JSON

Current Question (ID: 19893)

Question:

$x \text{ and } y \text{ displacements of a particle are given as } x(t) = a \sin \omega t \text{ and } y(t) = a \sin 2\omega t. \text{ Its trajectory will look like:}$

Options:

1. $\text{Option 1: Graph with two loops along the } y \text{-axis}$
2. $\text{Option 2: Graph with a single loop along the } y \text{-axis}$
3. $\text{Option 3: Elliptical graph}$
4. $\text{Option 4: Graph with two loops along the } x \text{-axis}$

Solution:

$\text{Hint: Express } x \text{ in terms of } y.$ $x = a \sin \omega t$ $y = a \sin 2\omega t = 2a \sin \omega t \cos \omega t$ $\Rightarrow \frac{y}{2a} = \sin \omega t \cos \omega t$ $\Rightarrow \sin \omega t = \pm \sqrt{\frac{1 \pm \sqrt{1 - \left(\frac{y}{a}\right)^2}}{2}}$ $\Rightarrow x = a \sin \omega t = a \left(\pm \sqrt{\frac{1 \pm \sqrt{1 - \left(\frac{y}{a}\right)^2}}{2}}\right)$

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
}