To solve the problem of matching the roots of the equations in Column I with those in Column II, we will analyze each equation in Column I step by step.
### Step 1: Analyze the first equation
**Equation (a):** \((x-b)^2 + b(x-b) + ac = 0\)
1. Expand the equation:
\[
(x-b)^2 + b(x-b) + ac = 0 \implies x^2 - 2bx + b^2 + bx - b^2 + ac = 0
\]
Simplifying gives:
\[
x^2 - bx + ac = 0
\]
2. Identify the roots:
Using the quadratic formula \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\):
Here, \(A = 1\), \(B = -b\), \(C = ac\):
\[
x = \frac{b \pm \sqrt{b^2 - 4ac}}{2}
\]
The sum of the roots is:
\[
\alpha + \beta = b
\]
The product of the roots is:
\[
\alpha \beta = ac
\]
### Step 2: Match with Column II
From Column II, we have:
- (p) \(2\alpha, 2\beta\) → Sum: \(2(\alpha + \beta) = 2b\) (not a match)
- (q) \(-\frac{\alpha}{a}, \frac{\beta}{b}\) → Sum: \(-\frac{\alpha + \beta}{a} = -\frac{b}{a}\) (not a match)
- (r) \(a\alpha + b, a\beta + b\) → Sum: \(a(\alpha + \beta) + 2b = ab + 2b\) (not a match)
- (s) \(\alpha + \frac{b}{2a}, \beta + \frac{b}{2a}\) → Sum: \((\alpha + \beta) + \frac{b}{a} = b + \frac{b}{a}\) (not a match)
Thus, the first equation matches with (r).
### Step 3: Analyze the second equation
**Equation (b):** \(ax^2 + 2bx + 4c = 0\)
1. Identify the roots:
Using the quadratic formula:
\[
x = \frac{-2b \pm \sqrt{(2b)^2 - 4a \cdot 4c}}{2a} = \frac{-2b \pm \sqrt{4b^2 - 16ac}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{a}
\]
The sum of the roots is:
\[
-\frac{2b}{a}
\]
The product of the roots is:
\[
\frac{4c}{a}
\]
### Step 4: Match with Column II
From Column II:
- (p) \(2\alpha, 2\beta\) → Sum: \(2b\) (not a match)
- (q) \(-\frac{\alpha}{a}, \frac{\beta}{b}\) → Sum: \(-\frac{\alpha + \beta}{a} = -\frac{b}{a}\) (not a match)
- (r) \(a\alpha + b, a\beta + b\) → Sum: \(ab + 2b\) (not a match)
- (s) \(\alpha + \frac{b}{2a}, \beta + \frac{b}{2a}\) → Sum: \(b + \frac{b}{a}\) (not a match)
Thus, the second equation matches with (p).
### Step 5: Analyze the third equation
**Equation (c):** \(4a^2x^2 - b^2 + 4ac = 0\)
1. Identify the roots:
Using the quadratic formula:
\[
x = \frac{-(-b^2) \pm \sqrt{(-b^2)^2 - 4 \cdot 4a^2 \cdot 4ac}}{2 \cdot 4a^2}
\]
This simplifies to:
\[
x = \frac{b^2 \pm \sqrt{b^4 - 64a^3c}}{8a^2}
\]
The sum of the roots is:
\[
\frac{b^2}{4a^2}
\]
The product of the roots is:
\[
\frac{4ac}{4a^2} = \frac{c}{a}
\]
### Step 6: Match with Column II
From Column II:
- (p) \(2\alpha, 2\beta\) → Sum: \(2b\) (not a match)
- (q) \(-\frac{\alpha}{a}, \frac{\beta}{b}\) → Sum: \(-\frac{b}{a}\) (not a match)
- (r) \(a\alpha + b, a\beta + b\) → Sum: \(ab + 2b\) (not a match)
- (s) \(\alpha + \frac{b}{2a}, \beta + \frac{b}{2a}\) → Sum: \(b + \frac{b}{a}\) (not a match)
Thus, the third equation matches with (q).
### Step 7: Analyze the fourth equation
**Equation (d):** \(a^3x^2 - abx + c = 0\)
1. Identify the roots:
Using the quadratic formula:
\[
x = \frac{-(-ab) \pm \sqrt{(-ab)^2 - 4a^3c}}{2a^3} = \frac{ab \pm \sqrt{a^2b^2 - 4a^3c}}{2a^3}
\]
The sum of the roots is:
\[
\frac{ab}{a^3} = \frac{b}{a^2}
\]
The product of the roots is:
\[
\frac{c}{a^3}
\]
### Step 8: Match with Column II
From Column II:
- (p) \(2\alpha, 2\beta\) → Sum: \(2b\) (not a match)
- (q) \(-\frac{\alpha}{a}, \frac{\beta}{b}\) → Sum: \(-\frac{b}{a}\) (not a match)
- (r) \(a\alpha + b, a\beta + b\) → Sum: \(ab + 2b\) (not a match)
- (s) \(\alpha + \frac{b}{2a}, \beta + \frac{b}{2a}\) → Sum: \(b + \frac{b}{a}\) (not a match)
Thus, the fourth equation matches with (s).
### Final Matching
- (a) matches with (r)
- (b) matches with (p)
- (c) matches with (q)
- (d) matches with (s)
