If `alpha , beta ` are the roots of the quadratic equation ` (p^2 + p + 1) x^2 + (p - 1) x + p^2 = 0 ` such that unity lies between the roots then the set of values of ` p ` is (i) `O/` (ii) `p in (-infty,-1) cup (0,infty)` (iii) `p in (-1,infty)` (iv) `(-1,1)`

Answer

To solve the problem, we need to analyze the quadratic equation given and the conditions under which the roots (α and β) allow unity (1) to lie between them. The quadratic equation is:

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

### Step 1: Identify the coefficients
In the quadratic equation of the form \( ax^2 + bx + c = 0 \), we have:
- \( a = p^2 + p + 1 \)
- \( b = p - 1 \)
- \( c = p^2 \)

### Step 2: Condition for unity to lie between the roots
For unity (1) to lie between the roots α and β, we need two conditions:
1. The leading coefficient \( a \) must be positive (i.e., \( p^2 + p + 1 > 0 \)).
2. The value of the quadratic equation at \( x = 1 \) must be negative (i.e., \( f(1) < 0 \)).

### Step 3: Analyze the first condition
The expression \( p^2 + p + 1 \) is a quadratic function. To determine if it is always positive, we can check its discriminant:

\[ D = b^2 - 4ac = 1^2 - 4(1)(1) = 1 - 4 = -3 \]

Since the discriminant is negative, \( p^2 + p + 1 \) has no real roots and is always positive for all real values of \( p \).

### Step 4: Analyze the second condition
Now we calculate \( f(1) \):

\[ f(1) = (p^2 + p + 1)(1^2) + (p - 1)(1) + p^2 \]
\[ = p^2 + p + 1 + p - 1 + p^2 \]
\[ = 2p^2 + 2p \]

We need \( f(1) < 0 \):

\[ 2p^2 + 2p < 0 \]
\[ p^2 + p < 0 \]

### Step 5: Factor the inequality
Factoring gives:

\[ p(p + 1) < 0 \]

### Step 6: Determine the intervals
To solve \( p(p + 1) < 0 \), we find the roots, which are \( p = 0 \) and \( p = -1 \). The sign of the product \( p(p + 1) \) changes at these points. Testing intervals:
- For \( p < -1 \): both factors are negative, so the product is positive.
- For \( -1 < p < 0 \): \( p \) is negative and \( p + 1 \) is positive, so the product is negative.
- For \( p > 0 \): both factors are positive, so the product is positive.

Thus, the solution to \( p(p + 1) < 0 \) is:

\[ p \in (-1, 0) \]

### Conclusion
The set of values of \( p \) such that unity lies between the roots is:

\[ \boxed{(-1, 0)} \]