Measures of Central Tendency and DispersionPYQ May 25Question 4370 of 473
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If Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of two numbers are 6.50 and 6 respectively, then the two numbers are

Options

A6 and 7
B9 and 4
C10 and 3
D8 and 5
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Correct Answer

✅ Option b — 9 and 4

All Options:

  • A6 and 7
  • B9 and 4
  • C10 and 3
  • D8 and 5

Detailed Solution & Explanation

Let the two numbers be a\displaystyle a and b\displaystyle b.
We are given:
- The Arithmetic Mean (A.M.) of a\displaystyle a and b\displaystyle b is 6.50\displaystyle 6.50:
A.M.=a+b2=6.5  ⟹  a+b=13\text{A.M.} = \frac{a + b}{2} = 6.5 \implies a + b = 13 - The Geometric Mean (G.M.) of a\displaystyle a and b\displaystyle b is 6\displaystyle 6:
G.M.=ab=6  ⟹  ab=36\text{G.M.} = \sqrt{a b} = 6 \implies a b = 36
Now we can set up a quadratic equation with roots a\displaystyle a and b\displaystyle b:
x2−(a+b)x+ab=0x^2 - (a + b)x + a b = 0 Substituting the sum and product of the roots:
x2−13x+36=0x^2 - 13x + 36 = 0 Solving this quadratic equation using factorization:
x2−9x−4x+36=0x^2 - 9x - 4x + 36 = 0 x(x−9)−4(x−9)=0x(x - 9) - 4(x - 9) = 0 (x−9)(x−4)=0(x - 9)(x - 4) = 0 Thus, the roots are x=9\displaystyle x = 9 or x=4\displaystyle x = 4.
So, the two numbers are 9\displaystyle 9 and 4\displaystyle 4.

Hence, **Option B** is the correct answer.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

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