Import Question JSON

Current Question (ID: 19119)

Question:

$\text{The motion of a particle in the } xy\text{-plane is described by the following equations:}$ $x = 4 \sin\left(\frac{\pi}{2} - \omega t\right) \text{ m}, \quad y = 4 \sin(\omega t) \text{ m.}$ $\text{Which of the following best describes the path traced by the particle?}$

Options:

1. $\text{circular}$
2. $\text{helical}$
3. $\text{parabolic}$
4. $\text{elliptical}$

Solution:

$\text{Hint: Eliminate } t \text{ to find the relation between } x \text{ and } y.$ $\text{Step 1: Write the given equation in the form of } \sin(\omega t) \text{ and } \cos(\omega t).$ $x = 4 \sin\left(\frac{\pi}{2} - \omega t\right); \quad y = 4 \sin(\omega t)$ $\Rightarrow x = 4 \cos(\omega t); \quad y = 4 \sin(\omega t)$ $\text{Step 2: Find the path of the particle.}$ $x^2 + y^2 = 4^2 \cos^2 \omega t + 4^2 \sin^2 \omega t = 4^2$ $\Rightarrow x^2 + y^2 = 4^2$ $\text{Therefore, the path of the particle is circular.}$ $\text{Hence, option (1) is the correct answer.}$

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
}