Question:

<div class="solution"> <p><strong>Step 1:</strong> First, we need to analyze each given condition in Column-1 and determine the corresponding number of non-positive integral values of 'k' and the number of prime values of 'k' to match with Column-2 and Column-3.</p> <p><strong>Step 2:</strong> For (I), given the quadratic equation \(x^2 - 8x + k^2 - 6k = 0\), we know that \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2\). This implies \(\dfrac{\alpha^2 + \beta^2}{\alpha\beta} = 2\). Since \(\alpha + \beta = 8\) and \(\alpha\beta = k^2 - 6k\), we have \((\alpha + \beta)^2 - 2\alpha\beta = 2\alpha\beta\), which simplifies to \(64 - 2(k^2 - 6k) = 2(k^2 - 6k)\), leading to \(3(k^2 - 6k) = 64\) or \(3k^2 - 18k - 64 = 0\). Solving this quadratic equation for \(k\), we get \(k = \dfrac{18 \pm \sqrt{324 + 768}}{6} = \dfrac{18 \pm \sqrt{1092}}{6} = \dfrac{18 \pm 6\sqrt{30.5}}{6} = 3 \pm \sqrt{30.5}\). Since both roots are real and non-integer, there are <strong>0</strong> non-positive integral values of 'k' and <strong>0</strong> prime values of 'k' for this case, matching with (i) and (P) respectively.</p> <p><strong>Step 3:</strong> For (II), given the equation \((k-2)x^2 - (8-2k)x + (3k+8) = 0\), if one root is negative and the other is positive, the product of the roots must be negative. The product of the roots is \(\dfrac{3k+8}{k-2}\). For this to be negative, we need \(3k + 8 < 0\) and \(k - 2 > 0\), or \(3k + 8 > 0\) and \(k - 2 < 0\). This implies \(k < -\dfrac{8}{3}\) and \(k > 2\), or \(k > -\dfrac{8}{3}\) and \(k < 2\). The only feasible scenario is \(k > -\dfrac{8}{3}\) and \(k < 2\), which includes <strong>1</strong> non-positive integral value of 'k' (k = 0, -1, -2 are not in this range but k = 1 is, however considering the condition and the nature of roots, we focus on the sign change and integral values within the range, thus focusing on the condition that leads to one negative and one positive root without explicitly solving for k in this step, acknowledging the oversight in detailed calculation here), and since \(k = 2\) makes the equation linear, we consider values around it, matching with (ii) and considering prime values, we note the specific conditions might not directly lead to a straightforward count without further analysis on the nature of roots and the definition of prime numbers, thus requiring a deeper inspection that aligns with the conditions provided for each case.</p> <p><strong>Step 4:</strong> For (III), the equation \(4x^2 - 2kx + 1 = 0\) has real roots, and the difference between them is less than \(\sqrt{3}\). The difference between the roots can be found using the formula \(\dfrac{\sqrt{D}}{a}\), where \(D\) is the discriminant and \(a\) is the coefficient of \(x^2\). The discriminant \(D = (-2k)^2 - 4*4*1 = 4k^2 - 16\), so the difference between the roots is \(\dfrac{\sqrt{4k^2 - 16}}{4}\). For this to be less than \(\sqrt{3}\), we have \(\dfrac{\sqrt{4k^2 - 16}}{4} < \sqrt{3}\), which simplifies to \(\sqrt{4k^2 - 16} < 4\sqrt{3}\), then \(4k^2