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Current Question (ID: 18915)

Question:

$\text{The expression for time, in terms of the universal gravitational constant } G, \text{ Planck's constant } h, \text{ and the speed of light } c, \text{ is proportional to:}$

Options:

1. $\sqrt{\frac{hc^5}{G}}$
2. $\sqrt{\frac{c^3}{Gh}}$
3. $\sqrt{\frac{Gh}{c^5}}$
4. $\sqrt{\frac{Gh}{c^3}}$

Solution:

$\text{Hint: } F = \frac{Gm^2}{r^2}$ $\text{Step 1: Find the value of } a, b \text{ and } p.$ $t \propto G^a h^b c^p$ $\Rightarrow [T] \propto [M^{-1}L^3T^{-2}]^a[M L^2T^{-1}]^b[LT^{-1}]^p$ $\Rightarrow [M^0L^0T] \propto [M]^{-a+b}[L]^{3a+2b+p}[T]^{-2a-b-p}$ $\text{Compare the powers:}$ $\Rightarrow -a + b = 0, 3a + 2b + p = 0, -2a - b - p = 1$ $\Rightarrow a = b = \frac{1}{2} \text{ and } p = -\frac{5}{2}$ $\text{Step 2: Find the expression for time in terms of given constants.}$ $t \propto G^a h^b c^p$ $\Rightarrow t \propto G^{1/2}h^{1/2}c^{-5/2}$ $\Rightarrow t \propto G^{1/2}h^{1/2}c^{-5/2}$ $\Rightarrow t \propto \sqrt{\frac{Gh}{c^5}}$ $\text{Hence, option (3) is the correct answer.}$

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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
}