Import Question JSON

Current Question (ID: 19120)

Question:

$\text{A ball is projected from the ground with a speed } 15 \text{ ms}^{-1} \text{ at an angle } \theta \text{ with horizontal so that its range and maximum height are equal, then } \tan \theta \text{ will be equal to:}$

Options:

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $2$
4. $4$

Solution:

$\text{Given that range } R \text{ and maximum height } H \text{ are equal:}$ $R = \frac{u^2 \sin 2\theta}{g}$ $H = \frac{u^2 \sin^2 \theta}{2g}$ $\text{Equating } R \text{ and } H:$ $\frac{u^2 \sin 2\theta}{g} = \frac{u^2 \sin^2 \theta}{2g}$ $\Rightarrow \sin 2\theta = \frac{1}{2} \sin^2 \theta$ $\Rightarrow 2 \sin \theta \cos \theta = \frac{1}{2} \sin^2 \theta$ $\Rightarrow 4 \cos \theta = \sin \theta$ $\Rightarrow \tan \theta = 4$

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
}