Import Question JSON

Current Question (ID: 9577)

Question:

$\text{A particle projected from origin moves in the } x\text{-}y \text{ plane with a velocity } \vec{v} = 3\hat{i} + 6x\hat{j}, \text{ where } \hat{i} \text{ and } \hat{j} \text{ are the unit vectors along the } x \text{ and } y\text{-axis. The equation of path followed by the particle is:}$

Options:

1. $y = x^2$
2. $y = \frac{1}{x^2}$
3. $y = 2x^2$
4. $y = \frac{1}{x}$

Solution:

$\text{Hint: } \tan \theta = \frac{dy}{dx}$ $\text{Step: Find the equation of the path followed by the particle.}$ $\text{Given:}$ $\vec{v} = 3\hat{i} + 6x\hat{j}$ $\text{Here, } v_x = 3, v_y = 6x$ $\text{Now,}$ $\tan \theta = \frac{dy}{dx} = \frac{v_y}{v_x}$ $\Rightarrow \frac{dy}{dx} = \frac{6x}{3} = 2x$ $\text{Integrating both sides:}$ $\int dy = \int 2x dx$ $y = \frac{2x^2}{2}$ $\Rightarrow y = x^2$ $\text{Hence, option (1) is the correct answer.}$

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
}