Import Question JSON

Current Question (ID: 9569)

Question:

$\text{A particle starts moving from the origin in the XY plane and its velocity after time t is given by } \vec{v} = 4\hat{i} + 2t\hat{j}\text{. The trajectory of the particle is correctly shown in the figure:}$

Options:

1. $\text{Upward opening parabola (concave up)}$
2. $\text{Straight diagonal line}$
3. $\text{Rightward opening parabola (about y-axis)}$
4. $\text{Horizontal straight line}$

Solution:

\text{Hint: } v_x = \frac{dx}{dt} \text{Step: Find the trajectory of the particle.} \text{Given: } v_x = 4 \Rightarrow \frac{dx}{dt} = 4 \text{Integrating: } \int dx = \int 4 \, dt \Rightarrow x = 4t + C_1 \text{At } t = 0, x = 0 \Rightarrow C_1 = 0 \text{, so } x = 4t \text{ or } t = \frac{x}{4} \text{Also given: } v_y = 2t \Rightarrow \frac{dy}{dt} = 2t \text{Integrating: } \int dy = \int 2t \, dt \Rightarrow y = t^2 + C_2 \text{At } t = 0, y = 0 \Rightarrow C_2 = 0 \text{, so } y = t^2 \text{Substituting } t = \frac{x}{4} \text{ into } y = t^2: y = \left(\frac{x}{4}\right)^2 = \frac{x^2}{16} \text{Therefore: } x^2 = 16y \text{The trajectory of the particle is a parabola about the y-axis.} \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
}