Import Question JSON

Current Question (ID: 9635)

Question:

$\text{A particle is projected from a horizontal plane (}x\text{-}z\text{ plane) such that its velocity vector at time }t\text{ is given by }\vec{v} = a\hat{i} + (b - ct)\hat{j}\text{. Its range on this horizontal plane is given by:}$

Options:

1. $\frac{ba}{c}$
2. $\frac{2ba}{c}$
3. $\frac{3ba}{c}$
4. $\text{None}$

Solution:

$\text{Hint: } R = \frac{2u_x u_y}{g}$ $\text{Step 1: Identify the axes}$ $\text{[A coordinate system diagram is shown with Y-axis vertical, X-axis horizontal, and acceleration a pointing downward]}$ $\text{Step 2: Find }u_x\text{, }u_y\text{ and acceleration of the particle.}$ $\vec{v} = u_x \hat{i} + (u_y - gt)\hat{j}$ $u_x = a, u_y = b, g = c$ $\text{Step 3: Find the range of the particle.}$ $R = \frac{2u_x u_y}{g} = \frac{2ab}{c}$ $\text{Hence, option (2) is the correct answer.}$

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
}