Import Question JSON

Current Question (ID: 9621)

Question:

$\text{When a particle is projected at some angle to the horizontal, it has a range } R \text{ and time of flight } t_1. \text{ If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is } t_2, \text{ then:}$

Options:

1. $t_1 + t_2 = \frac{2R}{g}$
2. $t_1 - t_2 = \frac{R}{g}$
3. $t_1 t_2 = \frac{2R}{g}$
4. $t_1 t_2 = \frac{R}{g}$

Solution:

\text{Hint: } t = (2u \sin\theta)/g \text{Step: Find the relation between time } t_1 \text{ and } t_2. \text{Let the angles of the projectile are } \theta \text{ and } (90° - \theta) \text{Now, } t_1 = (2u \sin\theta)/g t_2 = (2u \sin(90° - \theta))/g = (2u \cos\theta)/g \text{Now, } t_1 \times t_2 = [(2u \sin\theta)/g] \times [(2u \cos\theta)/g] \text{Therefore } t_1 \times t_2 = (4u^2 \sin\theta \cos\theta)/g^2 = (2u^2 \sin(2\theta))/g^2 = 2R/g \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
}