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Current Question (ID: 8749)

Question:

$\text{The displacement } (Y) \text{ as a function of position } (x) \text{ and time } (t) \text{ is given as } Y = Ae^{(bx+Ct)}. \text{ Which of the following expressions has dimensions different from others?}$

Options:

1. $YC$
2. $AC$
3. $\frac{C}{b}$
4. $bC$

Solution:

$\text{Hint: The argument of the exponential function is dimensionless.}$ $\text{Step 1: Find the dimensions of } b \text{ and } C.$ $\text{The given function is } Y = Ae^{(bx+Ct)}$ $\Rightarrow [Y] = [A] = [L]$ $\text{As the argument of the exponential function is dimensionless, so,}$ $[bx + Ct] = [M^0L^0T^0]$ $\Rightarrow [bx] = [M^0L^0T^0] \Rightarrow [b] = [M^0L^{-1}T^0]$ $\Rightarrow [Ct] = [M^0L^0T^0] \Rightarrow [C] = [M^0L^0T^{-1}]$ $\text{Step 2: Check the dimensions for each option.}$ $\Rightarrow [YC] = [LT^{-1}]$ $\Rightarrow [AC] = [LT^{-1}]$ $\Rightarrow \left[\frac{C}{b}\right] = [LT^{-1}]$ $\Rightarrow [bC] = [L^{-1}T^{-1}]$ $\text{Therefore, option (4) has dimensions different from others.}$ $\text{Hence, option (4) is the correct answer.}$

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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
}