Central Tendency & DispersionPYQ Jun 24Question 3145 of 473
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If Arithmetic mean and Coefficient of variations of x\displaystyle x are 5 and 20 respectively, the variance of 12−3x\displaystyle 12-3x is

Options

A9\displaystyle 9
B81\displaystyle 81
C3\displaystyle 3
D100\displaystyle 100
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Correct Answer

✅ Option a — 9\displaystyle 9

All Options:

  • A9\displaystyle 9
  • B81\displaystyle 81
  • C3\displaystyle 3
  • D100\displaystyle 100

Detailed Solution & Explanation

**Given:** xˉ=5\displaystyle \bar{x} = 5, CV of x\displaystyle x = 20 **Step 1: Find SD of x\displaystyle x.** CV=σxxˉ×100CV = \frac{\sigma_x}{\bar{x}} \times 100 20=σx5×10020 = \frac{\sigma_x}{5} \times 100 σx=20×5100=1\sigma_x = \frac{20 \times 5}{100} = 1 **Step 2: Find Var(x\displaystyle x).** Var(x)=σx2=12=1\text{Var}(x) = \sigma_x^2 = 1^2 = 1 **Step 3: Find Var(12−3x\displaystyle 12 - 3x).** Var(a+bx)=b2⋅Var(x)\text{Var}(a + bx) = b^2 \cdot \text{Var}(x) Var(12−3x)=(−3)2⋅Var(x)=9×1=9\text{Var}(12 - 3x) = (-3)^2 \cdot \text{Var}(x) = 9 \times 1 = 9 Wait — the answer is 9, but the given `correct_option` is 'b' (81). Let me check if I made an error. σx=1\displaystyle \sigma_x = 1, Var(x)=1\displaystyle \text{Var}(x) = 1. Var(12−3x)=9×1=9\displaystyle \text{Var}(12-3x) = 9 \times 1 = 9. So Option A (9) is the mathematically correct answer. Option B (81) would require Var(x)=9\displaystyle \text{Var}(x) = 9, i.e., σx=3\displaystyle \sigma_x = 3. That would happen if CV = 60, not 20. Hence, **Option A** is the correct answer.

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