<p><strong>Match the List (Questions 601–602)</strong><br>Two AP's having same number of terms equal to \(k\). The ratio of the last term of the first progression to the first term of the second progression equals the ratio of the last term of the second progression to the first term of the first progression, both of which are numerically equal to 4. The ratio of the sum of \(k\) terms of the first progression to the sum of \(k\) terms of second progression is equal to 2. Let \(\alpha\) be the ratio of the common difference of the first and the second progressions. Let \(\lambda\) be the ratio of their \(k^{\text{th}}\) terms. Then:</p><table><tr><th>List-I</th><th>List-II</th></tr><tr><td>(I) The ratio of the first term of first A.P. to second A.P. is</td><td>(P) 26</td></tr><tr><td>(II) The value of \(\alpha\) is equal to</td><td>(Q) 33</td></tr><tr><td>(III) The value of \(\lambda\) is equal to</td><td>(R) 7</td></tr><tr><td>(IV) The value of \(\alpha + 2\lambda\) is equal to</td><td>(S) 2/7</td></tr><tr><td></td><td>(T) 7/2</td></tr></table><p>Which of the following options has the <strong>correct</strong> combination considering List-I and List-II?</p>
Step-by-Step Solution
Key Concept: Set up equations using the given ratio conditions for last/first terms and sum ratios of two APs, then solve for the common differences and first terms to find the required ratios.
<p><strong>Step 1: Set up variables</strong><br>Let AP₁ have first term a₁, common difference d₁, and AP₂ have first term a₂, common difference d₂. Both have k terms.</p><p>Last term of AP₁: l₁ = a₁ + (k-1)d₁<br>Last term of AP₂: l₂ = a₂ + (k-1)d₂</p><p><strong>Step 2: Use the given ratio conditions</strong><br>Given: l₁/a₂ = 4 and l₂/a₁ = 4<br>Therefore: l₁ = 4a₂ ... (1)<br>l₂ = 4a₁ ... (2)</p><p><strong>Step 3: Use the sum ratio condition</strong><br>Sum of k terms: S₁ = (k/2)(a₁ + l₁) and S₂ = (k/2)(a₂ + l₂)<br>Given: S₁/S₂ = 2<br>Therefore: (a₁ + l₁)/(a₂ + l₂) = 2<br>a₁ + l₁ = 2(a₂ + l₂) ... (3)</p><p><strong>Step 4: Substitute equations (1) and (2) into (3)</strong><br>a₁ + 4a₂ = 2(a₂ + 4a₁)<br>a₁ + 4a₂ = 2a₂ + 8a₁<br>2a₂ = 7a₁<br>a₁/a₂ = 2/7</p><p><strong>Step 5: Find α (ratio of common differences)</strong><br>From l₁ = 4a₂: a₁ + (k-1)d₁ = 4a₂<br>From l₂ = 4a₁: a₂ + (k-1)d₂ = 4a₁<br>From a₁/a₂ = 2/7, let a₁ = 2t, a₂ = 7t<br>Substituting: 2t + (k-1)d₁ = 28t → (k-1)d₁ = 26t<br>7t + (k-1)d₂ = 8t → (k-1)d₂ = t<br>Therefore: α = d₁/d₂ = 26t/t = 26</p><p><strong>Step 6: Find λ (ratio of kth terms)</strong><br>λ = l₁/l₂ = 4a₂/(4a₁) = a₂/a₁ = 7/2</p><p><strong>Step 7: Verify matches</strong><br>(I) a₁/a₂ = 2/7 matches (S) ✓<br>(II) α = 26 matches (P) ✓<br>(III) λ = 7/2 matches (T) ✓<br>(IV) α + 2λ = 26 + 2(7/2) = 26 + 7 = 33 matches (Q) ✓</p><p><strong>For question matching option C: (I) (S)</strong><br>This matches: The ratio of first terms of AP₁ to AP₂ is 2/7</p><p>∴ Answer: C</p>
Correct Answer: C