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.
| Column-1 | Column-2 | Column-3 |
|---|---|---|
| (I) Let $\alpha$ and $\beta$ are real roots of $x^2 - 8x + k^2 - 6k = 0$ such that $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2$. | (i) 0 | (P) 0 |
| (II) If one root of the equation $(k-2)x^2 - (8-2k)x + (3k+8) = 0$ is negative and other is positive. | (ii) 1 | (Q) 1 |
| (III) If difference between the real roots of equation $4x^2 - 2kx + 1 = 0$ is less than $\sqrt{3}$. | (iii) 2 | (R) 2 |
| (IV) If quadratic expression $2kx^2 - (4k-5)x - 10$ is negative for exactly three distinct integral values of $x$. | (iv) 3 | (S) 3 |
Which of the following options is the only **correct** combination?
Step-by-Step Solution
Key Concept: The condition $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=2$ forces $(\alpha-\beta)^2=0$, i.e., equal roots, which combined with Vieta's formulas pins down $k$ to exactly two values.
Step 1:
To solve this problem, 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 '$k$' as represented in Column-2 and Column-3, respectively. We will start by examining the condition given in Row (I), which involves a quadratic equation with real roots $\alpha$ and $\beta$ such that $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2$. The equation given is $x^2 - 8x + k^2 - 6k = 0$.
Step 2:
For Row (I), we need to find the values of $k$ that satisfy the given condition. The sum and product of the roots of the quadratic equation $x^2 - 8x + k^2 - 6k = 0$ can be found using the coefficients of the quadratic equation. The sum of the roots, $\alpha + \beta$, equals the negation of the coefficient of $x$ divided by the coefficient of $x^2$, which gives us $8$. The product of the roots, $\alpha \beta$, equals the constant term divided by the coefficient of $x^2$, giving us $k^2 - 6k$. The condition $\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} = 2$ can be rewritten as $\dfrac{\alpha^2 + \beta^2}{\alpha \beta} = 2$. Since $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$, we can substitute the values of $\alpha + \beta$ and $\alpha \beta$ into this equation to solve for $k$.
Step 3:
Substituting the values into the equation, we get $\dfrac{(\alpha + \beta)^2 - 2\alpha \beta}{\alpha \beta} = 2$. This simplifies to $\dfrac{8^2 - 2(k^2 - 6k)}{k^2 - 6k} = 2$. Solving this equation will give us the possible values of $k$. We have $64 - 2k^2 + 12k = 2k^2 - 12k$, which simplifies to $4k^2 - 24k - 64 = 0$. Dividing the entire equation by $4$, we get $k^2 - 6k - 16 = 0$. We can solve this quadratic equation for $k$ using the quadratic formula: $k = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -6$, and $c = -16$. Substituting these values into the quadratic formula gives us $k = \dfrac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-16)}}{2(1)}$. This simplifies to $k = \dfrac{6 \pm \sqrt{36 + 64}}{2}$, which further simplifies to $k = \dfrac{6 \pm \sqrt{100}}{2}$. Therefore, $k = \dfrac{6 \pm 10}{2}$. This gives us two possible values for $k$: $k = \dfrac{6 + 10}{2} = 8$ and $k = \dfrac{6 - 10}{2} = -2$.
Step 4:
Now, let's analyze the number of non-positive integral values of $k$ for Row (I). From our calculation, we found $k = 8$ or $k = -2$. The non-positive integral value of $k$ is $\{-2\}$, so the count is $1$, which corresponds to option (ii). Next, we look at the number of prime values of $k$. Since $k = 8$ is not prime and $k = -2$ is also not prime, the count of prime values is $0$, which corresponds to option (P). Therefore, Row (I) matches with (ii)(P), which is part of option (a).
Step 5:
For Row (II), the given condition involves the quadratic equation $(k-2)x^2 - (8-2k)x + (3k+8) = 0$, where one root is negative and the other is positive. To find the values of $k$ that satisfy this condition, we need to consider the properties of the roots of a quadratic equation. However, based on the provided analysis, $k \in (-8/3, 2)$. The non-positive integers in this range are $\{-2, -1, 0\}$, giving us a count of $3$, which corresponds to option (iv). Since there are no prime values of $k$ in this range, the count of prime values is $0$, corresponding to option (P). Thus, Row (II) does not match with option (b) as it corresponds to (iv)(P), not (i)(Q).
Step 6:
Moving on to Row (III), the condition involves the quadratic equation $4x^2 - 2kx + 1 = 0$, where the difference between the real roots is less than $\sqrt{3}$. The analysis provided indicates that $k \in (-4,-2] \cup [2,4)$. The non-positive integers in this range are $\{-3, -2\}$, resulting in a count of $2$, which corresponds to option (iii). The prime values in this range are $\{2, 3\}$, giving us a count of $2$, which corresponds to option (R). Therefore, Row (III) matches with (iii)(R), not (iv)(S), making option (c) incorrect.
Step 7:
Lastly, for Row (IV), the condition involves the quadratic expression $2kx^2 - (4k-5)x - 10$ being negative for exactly three distinct integral values of $x$. The analysis shows that $k = 2$ (and possibly other values). The non-positive integral values of $k$ are none, resulting in a count of $0$, which corresponds to option (i). The prime value of $k$ is $\{2\}$, giving us a count of $1$, which corresponds to option (Q). Hence, Row (IV) matches with (i)(Q), not (ii)(P), making option (d) incorrect.
Step 8:
After analyzing all the rows and their corresponding conditions, we can conclude that the only correct combination is the one that matches Row (I) with (ii)(P), which is part of option (a). Therefore, the final answer is $\boxed{1}$.
Correct Answer: 1