Import Question JSON

Current Question (ID: 9576)

Question:

$\text{The coordinates of a moving particle at a time } t \text{ are given by } x = 5 \sin 10t, y = 5 \cos 10t. \text{ It can be deduced that the speed of the particle will be:}$

Options:

1. $25 \text{ units}$
2. $50 \text{ units}$
3. $10 \text{ units}$
4. $30 \text{ units}$

Solution:

$\text{Hint: } v_x = \frac{dx}{dt}$ $\text{Step: Find the speed of the particle.}$ $x = 5 \sin 10t, \quad y = 5 \cos 10t$ $v_x = \frac{dx}{dt} = 5 \times 10 \cos 10t = 50 \cos 10t$ $v_y = \frac{dy}{dt} = -50 \sin 10t$ $\text{Velocity } \vec{v} = v_x \hat{i} + v_y \hat{j}$ $= 50 \cos 10t \hat{i} - 50 \sin 10t \hat{j}$ $\text{Speed = Magnitude of velocity}$ $= \sqrt{v_x^2 + v_y^2}$ $= \sqrt{(50)^2 \cos^2 10t + (-50)^2 \sin^2 10t}$ $= 50\sqrt{\cos^2 10t + \sin^2 10t} \text{ units}$ $= 50 \text{ units}$ $\text{Hence, option (2) is the correct answer.}$

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
}