Import Question JSON

Current Question (ID: 19080)

Question:

$\text{A shell is fired from a fixed artillery gun with an initial speed } u \text{ such that it hits the target on the ground at a distance } R \text{ from it. If } t_1 \text{ and } t_2 \text{ are the values of the time taken by it to hit the target in two possible ways, the product } t_1 t_2 \text{ is:}$

Options:

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

Solution:

$\text{Hint: The range of projectiles is the same for complementary angles.}$ $\text{For } \theta \text{ and } 90 - \theta \text{ angle of projection, range will be same}$ $\text{Time of flight for } \theta : t_1 = \frac{2u \sin \theta}{g}$ $\text{Time of flight for } 90 - \theta : t_2 = \frac{2u \sin(90 - \theta)}{g} = \frac{2u \cos \theta}{g}$ $t_1 t_2 = \frac{4u^2 \sin \theta \cos \theta}{g^2} = \frac{2u^2}{g} \left( \frac{\sin 20}{g} \right) = \frac{2}{g} \left( \frac{u^2 \sin 20}{g} \right)$

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
}