Import Question JSON

Current Question (ID: 19096)

Question:

$\text{Two projectiles are thrown at an angle of projection } \alpha \text{ and } \beta \text{ with the horizontal. If } \alpha + \beta = 90^\circ, \text{ then the ratio of the range of two projectiles on the horizontal plane is equal to:}$ $1.\ 1 : 1$ $2.\ 2 : 1$ $3.\ 1 : 2$ $4.\ 1 : 3$

Options:

1. $1 : 1$
2. $2 : 1$
3. $1 : 2$
4. $1 : 3$

Solution:

$\text{Hint: When projectiles are thrown at complementary angles then, } R_1 = R_2.$ $\text{Step 1: Find the range of the first projectile.}$ $R_\alpha = \frac{u^2 \sin 2\alpha}{g}$ $\text{Step 2: Find the range of the second projectile.}$ $R_\beta = R_{(90-\alpha)} = \frac{u^2 \sin 2\beta}{g} = \frac{u^2 \sin 2(90-\alpha)}{g} = \frac{u^2 \sin 2\alpha}{g}$ $\Rightarrow R_\alpha = R_{(90-\alpha)} \Rightarrow R_\alpha : R_{(90-\alpha)} = 1 : 1$ $\text{Therefore, the ratio of the range of two projectiles on the horizontal plane is } 1 : 1.$ $\text{Hence, option (1) is the correct answer.}$

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
}