Import Question JSON

Current Question (ID: 9564)

Question:

\text{A particle moves along the positive branch of the curve } y = \frac{x^2}{2} \text{where } x = \frac{t^2}{2} \text{x and y are measured in metres and t in seconds respectively.} \text{At } t = 2 \text{ s, the velocity of the particle will be:}

Options:

1. $(2\hat{i} - 4\hat{j}) \text{ m/s}$
2. $(4\hat{i} + 2\hat{j}) \text{ m/s}$
3. $(2\hat{i} + 4\hat{j}) \text{ m/s}$
4. $(4\hat{i} - 2\hat{j}) \text{ m/s}$

Solution:

\text{Hint: } \vec{v}(t) = \frac{dx}{dt}(t)\hat{i} + \frac{dy}{dt}(t)\hat{j} \text{Step 1: Express y-coordinate as a function of time} \text{Given } x = \frac{t^2}{2} y = \frac{x^2}{2} = \frac{(\frac{t^2}{2})^2}{2} = \frac{t^4}{8} \text{Step 2: Find the speed in x and y-direction} v_x = \frac{dx}{dt} = \frac{d}{dt}(\frac{t^2}{2}) = t v_y = \frac{dy}{dt} = \frac{d}{dt}(\frac{t^4}{8}) = \frac{t^3}{2} \text{At } t = 2 \text{ sec:} v_x = 2 \text{ m/s} v_y = \frac{2^3}{2} = \frac{8}{2} = 4 \text{ m/s} \text{Therefore: } \vec{v} = v_x\hat{i} + v_y\hat{j} = 2\hat{i} + 4\hat{j} \text{ m/s} \text{Hence, option (3) 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
}