Import Question JSON

Current Question (ID: 9623)

Question:

$\text{The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is:}$

Options:

1. $\theta = \tan^{-1}\left(\frac{1}{4}\right)$
2. $\theta = \tan^{-1}(4)$
3. $\theta = \tan^{-1}(2)$
4. $\theta = 45°$

Solution:

\text{Hint: } R = \frac{u^2\sin 2\theta}{g} \text{Step: Find the angle of the projectile.} \text{Given,} \text{For a projectile, (range) } R = \text{(maximum height) } H \text{Horizontal Range, } R = \frac{u^2(2\sin\theta\cos\theta)}{g} \text{Vertical Height, } H = \frac{u^2\sin^2\theta}{2g} \text{Hence,} \frac{u^2(2\sin\theta\cos\theta)}{g} = \frac{u^2\sin^2\theta}{2g} 2\cos\theta = \frac{\sin\theta}{2} \tan\theta = 4 \theta = \tan^{-1}(4) \text{Therefore, the angle of the projectile is } \tan^{-1}(4). \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
}