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Current Question (ID: 19269)

Question:

$\text{A block of mass } 100 \text{ grams is placed on a smooth surface and moves with an acceleration of } a = 2x. \text{ The change in kinetic energy can be given as } \left( \frac{x^n}{10} \right). \text{ The value of } n \text{ is:}$

Options:

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

Solution:

$\text{Hint: Use the work-energy theorem.}$ $\text{Step: Find the value of } n.$ $\text{The acceleration of the block is given by:}$ $a = v \frac{dv}{dx} = 2x$ $\Rightarrow \int_{v_i}^{v_f} v \, dv = \int 2x \, dx$ $\Rightarrow \frac{1}{2} \left( v_f^2 - v_i^2 \right) = x^2$ $\Rightarrow \frac{1}{2} m \left( v_f^2 - v_i^2 \right) = mx^2$ $\Rightarrow \Delta K.E = (0.1x^2) = \left( \frac{x^2}{10} \right)$ $\Rightarrow \Delta K.E = \left( \frac{x^2}{10} \right) = \left( \frac{x^n}{10} \right)$ $\text{Therefore, the value of } n \text{ is } 2.$ $\text{Hence, option (4) is the correct answer.}$

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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
}