Import Question JSON

Current Question (ID: 9578)

Question:

$\text{In a two-dimensional motion, instantaneous speed } v_0 \text{ is a positive constant. Then which of the following is necessarily true?}$

Options:

1. $\text{The acceleration of the particle is zero.}$
2. $\text{The acceleration of the particle is increasing.}$
3. $\text{The acceleration of the particle is necessarily in the plane of motion.}$
4. $\text{The particle must be undergoing a uniform circular motion.}$

Solution:

\text{Hint: } v = \frac{dr}{dt} \text{Step 1: Try to prove each option wrong.} \text{For uniform circular motion: } a = \frac{v^2}{R} \text{Let velocity, } \vec{v} = v_x \hat{i} + v_y \hat{j} \text{Now, acceleration, } \vec{a} = \frac{d\vec{v}}{dt} = a_x \hat{i} + a_y \hat{j} \text{As velocity is in the } (x\text{-}y) \text{ plane so the acceleration of the particle is necessarily in the plane of motion.} \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
}