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Current Question (ID: 8742)

Question:

$\text{Which of the following relations is dimensionally wrong? [The symbols have their usual meanings]}$

Options:

1. $s = ut + \frac{1}{6}at^2$
2. $v^2 = u^2 + \frac{2as^2}{\pi}$
3. $v = u - 2at$
4. $\text{All of these}$

Solution:

\text{Hint: For the dimensionally correct, the dimension of LHS should be equal to that of RHS.} \text{Step: Check the dimensions for the given equations.} \text{Apply dimension analysis for each equation.} \Rightarrow s = ut + \frac{1}{6}at^2 [s] = [L] [ut] = [LT^{-1} \times T] = [L] [at^2] = [LT^{-2} \times T^2] = [L] \text{Therefore, this equation is dimensionally correct.} \Rightarrow v^2 = u^2 + \frac{2as^2}{\pi} [v^2] = [L^2T^{-2}] [u^2] = [L^2T^{-2}] [as^2] = [LT^{-2}L^2] = [L^3T^{-2}] \text{Therefore, this equation is dimensionally wrong.} \Rightarrow v = u - 2at [v] = [LT^{-1}] [u] = [LT^{-1}] [at] = [LT^{-2}T] = [LT^{-1}] \text{Therefore, this equation is dimensionally correct.} \text{Therefore, the dimensionally incorrect equation is } v^2 = u^2 + \frac{2as^2}{\pi}\text{.} \text{Hence, option (2) 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
}