Import Question JSON

Current Question (ID: 9562)

Question:

$\text{The coordinates of a moving particle at any time } t \text{ are given by } x = \alpha t^3 \text{ and } y = \beta t^3. \text{ The speed of the particle at a time } t \text{ is given by:}$

Options:

1. $\sqrt{\alpha^2 + \beta^2}$
2. $3t\sqrt{\alpha^2 + \beta^2}$
3. $3t^2\sqrt{\alpha^2 + \beta^2}$
4. $t^2\sqrt{\alpha^2 + \beta^2}$

Solution:

\text{Step: Find the speed of the particle.} x = \alpha t^3 \text{ and } y = \beta t^3 \text{ (given)} v_x = \frac{dx}{dt} = 3\alpha t^2 \text{ and } v_y = \frac{dy}{dt} = 3\beta t^2 \text{Resultant speed } = \sqrt{v_x^2 + v_y^2} = \sqrt{(3\alpha t^2)^2 + (3\beta t^2)^2} = \sqrt{9\alpha^2 t^4 + 9\beta^2 t^4} = \sqrt{9t^4(\alpha^2 + \beta^2)} = 3t^2\sqrt{\alpha^2 + \beta^2} \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
}