The HM, H of two numbers is 4 and their AM, A and the GM, G satisfy the equation 2A+G2=27 , the numbers are:

Option c: (6,3). Solution: Step 1: Use the relationship GM² = AM × HM. G2 = A × H = 4A Step 2: Substitute into the equation 2A + G2 = 27 . 2A + 4A = 27 ⟹ 6A = 27 ⟹ A = 4.5 Step 3: Find G2 . G2 = 4 × 4.5 = 18 Step 4: Find the sum and product of the two numbers. a + b = 2A = 9, ab = G2 = 18 Step 5: Solve the quadratic t2 - 9t + 18 = 0 . (t-3)(t-6) = 0 ⟹ t = 3 or t = 6 Verification with HM = 4: H = 2ab/a+b = 36/9 = 4 Hence, Option C is the correct answer.

Measures of Central Tendency and DispersionMCQMTP Sep 24 Series IQuestion 3073 of 473

All Questions

The HM, $\displaystyle H$ of two numbers is $\displaystyle 4$ and their AM, $\displaystyle A$ and the GM, $\displaystyle G$ satisfy the equation $\displaystyle 2A+G^2=27$ , the numbers are:

Options

\displaystyle (1,3)

\displaystyle (9,5)

\displaystyle (6,3)

\displaystyle (12,7)

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Correct Answer

✅ Option c — $\displaystyle (6,3)$

All Options:

A $\displaystyle (1,3)$
B $\displaystyle (9,5)$
C $\displaystyle (6,3)$
D $\displaystyle (12,7)$

Detailed Solution & Explanation

**Step 1: Use the relationship GM² = AM × HM.**

G^2 = A \times H = 4A

**Step 2: Substitute into the equation

\displaystyle 2A + G^2 = 27

.**

2A + 4A = 27 \Rightarrow 6A = 27 \Rightarrow A = 4.5

**Step 3: Find

\displaystyle G^2

.**

G^2 = 4 \times 4.5 = 18

**Step 4: Find the sum and product of the two numbers.**

a + b = 2A = 9, \quad ab = G^2 = 18

**Step 5: Solve the quadratic

\displaystyle t^2 - 9t + 18 = 0

.**

(t-3)(t-6) = 0 \Rightarrow t = 3 \text{ or } t = 6

**Verification with HM = 4:**

H = \frac{2ab}{a+b} = \frac{36}{9} = 4 \checkmark

Hence, **Option C** 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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The HM, H\displaystyle HH of two numbers is 4\displaystyle 44 and their AM, A\displaystyle AA and the GM, G\displaystyle GG satisfy the equation 2A+G2=27\displaystyle 2A+G^2=272A+G2=27, the numbers are: