Let `x^2 - 2(a - 1)x + a - 1 = 0 (a in R)` be a quadratic equation, then find the values of "a" for which
(i) Exactly one root lies in (0,1)
(ii) Both roots lies in (0, 1).
(iii) Atleast one root lies in (0,1).
(iv) One root is greater than 1 and other root is smaller than 0.

Answer

To solve the problem step by step, we will analyze the quadratic equation given by:

\[ x^2 - 2(a - 1)x + (a - 1) = 0 \]

This can be rewritten in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = 1 \)
- \( b = -2(a - 1) \)
- \( c = a - 1 \)

### Part (i): Exactly one root lies in (0, 1)

1. **Condition for real and distinct roots**: The discriminant \( D \) must be greater than 0.
\[ D = b^2 - 4ac = [-2(a - 1)]^2 - 4(1)(a - 1) = 4(a - 1)^2 - 4(a - 1) \]
Simplifying gives:
\[ D = 4[(a - 1)^2 - (a - 1)] = 4(a - 1)(a - 2) \]
We require:
\[ 4(a - 1)(a - 2) > 0 \]
This implies:
\[ (a - 1)(a - 2) > 0 \]
The solution to this inequality is:
\[ a < 1 \quad \text{or} \quad a > 2 \]

2. **Condition for one root in (0, 1)**: We need to check the function values at \( x = 0 \) and \( x = 1 \):
- At \( x = 0 \):
\[ f(0) = a - 1 \]
We require \( f(0) < 0 \):
\[ a - 1 < 0 \implies a < 1 \]

- At \( x = 1 \):
\[ f(1) = 2 - a \]
We require \( f(1) > 0 \):
\[ 2 - a > 0 \implies a < 2 \]

3. **Combining conditions**: From the above, we have:
- \( a < 1 \) (from \( f(0) < 0 \))
- \( a < 2 \) (from \( f(1) > 0 \))
- \( a < 1 \) is the stricter condition.

Thus, the values of \( a \) for which exactly one root lies in (0, 1) are:
\[ \boxed{(-\infty, 1)} \]

### Part (ii): Both roots lie in (0, 1)

1. **Sum of roots**: The sum of the roots \( \alpha + \beta = -\frac{b}{a} = 2(a - 1) \) must be less than 2:
\[ 2(a - 1) < 2 \implies a - 1 < 1 \implies a < 2 \]

2. **Product of roots**: The product of the roots \( \alpha \beta = \frac{c}{a} = a - 1 \) must be positive:
\[ a - 1 > 0 \implies a > 1 \]

3. **Combining conditions**: We need:
- \( a < 2 \)
- \( a > 1 \)

Thus, the values of \( a \) for which both roots lie in (0, 1) are:
\[ \boxed{(1, 2)} \]

### Part (iii): At least one root lies in (0, 1)

1. **Condition for real roots**: The discriminant must be non-negative:
\[ 4(a - 1)(a - 2) \geq 0 \]
This gives:
\[ a \leq 1 \quad \text{or} \quad a \geq 2 \]

2. **Checking function values**:
- For \( f(0) < 0 \): \( a < 1 \)
- For \( f(1) > 0 \): \( a < 2 \)

3. **Combining conditions**: The values of \( a \) for which at least one root lies in (0, 1) are:
\[ \boxed{(-\infty, 1) \cup (2, \infty)} \]

### Part (iv): One root is greater than 1 and the other root is smaller than 0

1. **Condition for real roots**: The discriminant must be positive:
\[ 4(a - 1)(a - 2) > 0 \implies a < 1 \quad \text{or} \quad a > 2 \]

2. **Function values**:
- At \( x = 0 \): \( f(0) = a - 1 < 0 \implies a < 1 \)
- At \( x = 1 \): \( f(1) = 2 - a < 0 \implies a > 2 \)

3. **Combining conditions**: The values of \( a \) for which one root is greater than 1 and the other is smaller than 0 are:
\[ \boxed{(-\infty, 1) \cup (2, \infty)} \]

### Summary of Results:
- (i) \( a \in (-\infty, 1) \)
- (ii) \( a \in (1, 2) \)
- (iii) \( a \in (-\infty, 1) \cup (2, \infty) \)
- (iv) \( a \in (-\infty, 1) \cup (2, \infty) \)