Import Question JSON

Current Question (ID: 8889)

Question:

$\text{The acceleration of a particle starting from rest varies with time according to the relation } A = -a\omega^2 \sin \omega t. \text{ The displacement of this particle at a time } t \text{ will be:}$

Options:

1. $\frac{1}{2}(a\omega^2 \sin \omega t) t^2$
2. $a\omega \sin \omega t$
3. $a\omega \cos \omega t$
4. $a \sin \omega t$

Solution:

\text{Hint: } v = \int a \, dt \text{Step: Find the displacement of the particle at a time } t. \text{The acceleration of the particle is given as: } a = -A\omega^2 \sin(\omega t) \text{The velocity of the particle is given by:} a = \frac{dv}{dt} \Rightarrow v = \int a \, dt \Rightarrow v = \int (-A\omega^2 \sin(\omega t)) \, dt \Rightarrow v = A\omega \cos(\omega t) \text{The displacement of the particle is given by:} x = \int v \, dt \Rightarrow x = \int A\omega \cos(\omega t) \, dt \Rightarrow x = A \sin(\omega t) \text{Therefore, the displacement of the particle is } A \sin(\omega t). \text{Hence, option (4) is the correct answer.}

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
}