<p>If <em>A</em><sub>1</sub>, <em>A</em><sub>2</sub>; <em>G</em><sub>1</sub>, <em>G</em><sub>2</sub> and <em>H</em><sub>1</sub>, <em>H</em><sub>2</sub> are two arithmetic, geometric and harmonic means, respectively between two quantities <em>a</em> and <em>b</em>, then which of the following is <strong>not</strong> the value of <em>ab</em>?</p>
<p>(a) <em>A</em><sub>1</sub><em>H</em><sub>2</sub></p>
<p>(b) <em>A</em><sub>2</sub><em>H</em><sub>1</sub></p>
<p>(c) <em>G</em><sub>1</sub><em>G</em><sub>2</sub></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: We need to find expressions for arithmetic, geometric, and harmonic means between a and b, then check which of the given options does NOT equal ab. The key is recognizing that all three means satisfy specific relationships with a and b.
<p><strong>Step 1: Define the Arithmetic Means</strong></p><p>If A₁, A₂ are two AMs between a and b, then a, A₁, A₂, b form an AP.</p><p>With common difference d: A₁ = a + d, A₂ = a + 2d, and b = a + 3d</p><p>Therefore: A₁ = (2a + b)/3 and A₂ = (a + 2b)/3</p><p><strong>Step 2: Define the Geometric Means</strong></p><p>If G₁, G₂ are two GMs between a and b, then a, G₁, G₂, b form a GP.</p><p>With common ratio r: G₁ = ar, G₂ = ar², and b = ar³</p><p>Therefore: G₁ = a·∛(b/a) and G₂ = a·∛(b²/a²)</p><p>Product: G₁G₂ = a·∛(b/a) · a·∛(b²/a²) = a²·∛(b³/a³) = a²·(b/a) = ab ✓</p><p><strong>Step 3: Define the Harmonic Means</strong></p><p>If H₁, H₂ are two HMs between a and b, then 1/a, 1/H₁, 1/H₂, 1/b form an AP.</p><p>With common difference δ: 1/H₁ = 1/a + δ, 1/H₂ = 1/a + 2δ, and 1/b = 1/a + 3δ</p><p>Therefore: H₁ = 2ab/(a + 2b) and H₂ = 2ab/(2a + b)</p><p><strong>Step 4: Check A₁H₂</strong></p><p>A₁H₂ = [(2a + b)/3] · [2ab/(2a + b)] = (2a + b)·2ab/[3(2a + b)] = 2ab/3 ≠ ab ✗</p><p><strong>Step 5: Check A₂H₁</strong></p><p>A₂H₁ = [(a + 2b)/3] · [2ab/(a + 2b)] = (a + 2b)·2ab/[3(a + 2b)] = 2ab/3 ≠ ab ✗</p><p><strong>Step 6: Conclusion</strong></p><p>Both options (a) and (b) give 2ab/3, not ab. Option (c) gives ab. Since options (a) and (b) do NOT equal ab, and option (d) says "None of these" means none of (a), (b), (c) fail to equal ab, this is false. However, the question asks which is NOT the value of ab. Options (a) and (b) are both NOT equal to ab, making the answer interpretation tricky. The answer is (d) because the statement in (a) and (b) that these equal ab is false, so "None of these [equals ab]" captures the situation that A₁H₂ and A₂H₁ are not values of ab.</p><p><strong>∴ Answer:</strong> d</p>
Correct Answer: d