<p><strong>Question nos. 690 to 692</strong><br>Column-1 represents a quadratic equation with some given conditions. Column-2 represents number of non-positive integral values of 'k' and column-3 represents number of prime values of 'k'. Then match the following.</p><table border='1'><tr><th>Column-1</th><th>Column-2</th><th>Column-3</th></tr><tr><td>(I) Let α and β are real roots of \(x^2 - 8x + k^2 - 6k = 0\) such that \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2\).</td><td>(i) 0</td><td>(P) 0</td></tr><tr><td>(II) If one root of the equation \((k-2)x^2 - (8-2k)x + (3k+8) = 0\) is negative and other is positive.</td><td>(ii) 1</td><td>(Q) 1</td></tr><tr><td>(III) If difference between the real roots of equation \(4x^2 - 2kx + 1 = 0\) is less than \(\sqrt{3}\).</td><td>(iii) 2</td><td>(R) 2</td></tr><tr><td>(IV) If quadratic expression \(2kx^2 - (4k-5)x - 10\) is negative for exactly three distinct integral values of x.</td><td>(iv) 3</td><td>(S) 3</td></tr></table><br><strong>692.</strong> Which of the following options is the only <strong>correct</strong> combination?</p>
Step-by-Step Solution
Key Concept: For each condition, find the range of k values, then count non-positive integers and primes separately. The critical insight is that 'non-positive' means k ≤ 0, and we must verify each k satisfies the original condition.
<p><strong>Problem I:</strong> α and β real roots with α/β + β/α = 2</p><p>α/β + β/α = (α² + β²)/(αβ) = [(α+β)² - 2αβ]/(αβ) = 2</p><p>(α+β)² = 4αβ. With α+β = 8, αβ = k² - 6k: 64 = 4(k² - 6k)</p><p>k² - 6k - 16 = 0 → k = 8 or k = -2</p><p><strong>Non-positive k:</strong> only k = -2 → count = 1; <strong>Primes:</strong> none → count = 0</p><p><strong>Problem II:</strong> One root negative, other positive → product of roots < 0</p><p>(3k+8)/(k-2) < 0 and k ≠ 2</p><p>Critical points: k = -8/3, k = 2. Solution: -8/3 < k < 2</p><p><strong>Non-positive k:</strong> k ∈ {-2, -1, 0} → count = 3; <strong>Primes:</strong> none → count = 0</p><p><strong>Problem III:</strong> Difference of roots < √3</p><p>|α - β| = √(Δ)/|a| = √(k² - 4)/2 < √3</p><p>k² - 4 < 12 → k² < 16 → -4 < k < 4</p><p><strong>Non-positive k:</strong> k ∈ {-3, -2, -1, 0} → count = 4; <strong>Primes:</strong> k = 2, 3 outside range → count = 0</p><p><strong>Problem IV:</strong> 2kx² - (4k-5)x - 10 < 0 for exactly three distinct integral x</p><p>For k > 0: parabola opens upward. Roots: x = [-5/(2k), 2]. Between roots gives negative values.</p><p>Testing k = 2: roots at x = -5/4, 2. Integers in interval: {-1, 0, 1} → exactly 3 ✓</p><p><strong>Non-positive k:</strong> k = 0 makes it linear, k < 0 opens downward (infinite negatives) → count = 0; <strong>Primes:</strong> k = 2 is prime → count = 1</p><p>∴ Answer: <strong>D</strong></p>
Correct Answer: D