Import Question JSON

Current Question (ID: 9611)

Question:

$\text{A ball is projected at a certain angle with initial velocity } u\text{. It covers horizontal range } R\text{. With what initial velocity it should be projected keeping the angle of projection the same so that its horizontal range becomes } 2.25R\text{?}$

Options:

1. $2.5u$
2. $1.5u$
3. $2.25u$
4. $0.25u$

Solution:

\text{Hint: The formula for the range is given as } R = \frac{u^2 \sin 2\theta}{g} \text{Step 1: Calculate the range when the initial velocity is } u\text{.} R = \frac{u^2 \sin 2\theta}{g} \quad \text{...(1)} \text{Step 2: When the range is } 2.25R\text{.} \text{Let } v_1 \text{ be the initial velocity} \text{So,} 2.25R = \frac{v_1^2 \sin 2\theta}{g} \quad \text{...(2)} \text{Step 3: From equation (1) and (2)} 2.25 = \frac{v_1^2}{u^2} v_1 = 1.5u \text{Therefore, the initial velocity of the ball is } 1.5u\text{.} \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
}