For given equation `x^(2) -ax+b=0` , match conditions in column I with possible values in column II
`{:("column -I " , "Colum-II"),("(A) If roots differ by unity then " a^(2)" is equal to" , "(p) b(ab+2)"),("(B) If roots differ by unity then " 1 + a^(2) " is equal to " , "(q) 1 + 4b"),("(C) If one of the root be twice the other then " 2a^(2) " is equal to " , "(r) 2(1+2b)"),("(D) If the sum of roots of the equation equal to the sum of squares of their reciprocal then " a^(2) " is equal to " , "(s) 9b"):}`

Answer

To solve the problem, we need to match the conditions in column I with the corresponding values in column II based on the given quadratic equation \(x^2 - ax + b = 0\).

### Step-by-Step Solution:

**Step 1: Analyze Condition (A)**
Condition (A) states: "If roots differ by unity then \(a^2\) is equal to".
Let the roots be \(\alpha\) and \(\beta\). If they differ by unity, we can express this as:
\[ \alpha - \beta = 1 \quad \text{or} \quad \beta - \alpha = 1 \]
Assuming \(\alpha - \beta = 1\), we can square both sides:
\[ (\alpha - \beta)^2 = 1^2 \implies \alpha^2 - 2\alpha\beta + \beta^2 = 1 \]
Using the relationships:
\[ \alpha + \beta = a \quad \text{and} \quad \alpha \beta = b \]
We can rewrite \(\alpha^2 + \beta^2\) as:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = a^2 - 2b \]
Substituting into our equation:
\[ a^2 - 2b - 1 = 0 \implies a^2 = 2b + 1 \]
Thus, we can express this as:
\[ a^2 = 1 + 4b \quad \text{(after rearranging)} \]
**Match:** (A) → (q)

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**Step 2: Analyze Condition (B)**
Condition (B) states: "If roots differ by unity then \(1 + a^2\) is equal to".
From the previous step, we found that:
\[ a^2 = 1 + 4b \]
Thus:
\[ 1 + a^2 = 1 + (1 + 4b) = 2 + 4b \]
This can be factored as:
\[ 1 + a^2 = 2(1 + 2b) \]
**Match:** (B) → (r)

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**Step 3: Analyze Condition (C)**
Condition (C) states: "If one of the roots is twice the other then \(2a^2\) is equal to".
Let \(\alpha\) be one root and \(\beta = 2\alpha\). Then:
\[ \alpha + \beta = 3\alpha = a \quad \text{and} \quad \alpha \beta = 2\alpha^2 = b \]
From \(3\alpha = a\), we have \(\alpha = \frac{a}{3}\). Substituting this into the product of the roots:
\[ b = 2\left(\frac{a}{3}\right)^2 = \frac{2a^2}{9} \]
Rearranging gives:
\[ a^2 = \frac{9b}{2} \]
Thus:
\[ 2a^2 = 9b \]
**Match:** (C) → (s)

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**Step 4: Analyze Condition (D)**
Condition (D) states: "If the sum of roots equals the sum of squares of their reciprocals then \(a^2\) is equal to".
We know:
\[ \alpha + \beta = a \quad \text{and} \quad \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{a}{b} \]
The sum of squares of the reciprocals is:
\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\beta^2 + \alpha^2}{\alpha^2\beta^2} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2} \]
Thus:
\[ a = \frac{a^2 - 2b}{b^2} \]
Cross-multiplying gives:
\[ a b^2 = a^2 - 2b \implies a^2 = ab^2 + 2b \]
Factoring out gives:
\[ a^2 = b(ab + 2) \]
**Match:** (D) → (p)

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### Final Matches:
- (A) → (q)
- (B) → (r)
- (C) → (s)
- (D) → (p)