If A, G and H are the arithmetic mean, geometric mean and harmonic mean between unequal positive integers. Then, the equation `Ax^2 -|G|x- H=0` has
(a) both roots are fractions (b) atleast one root which is negative fraction (c) exactly one positive root (d) atleast one root which is an integer

Options

A. both roots are fractioins
B. atleast one root which is negative fraction
C. exactly one positive root
D. atleast one root which is an integer

Explanation

To solve the problem, we need to analyze the given equation and the relationships between the arithmetic mean (A), geometric mean (G), and harmonic mean (H) of two unequal positive integers.

### Step 1: Understand the Means
Let the two unequal positive integers be \( a \) and \( b \).
- The Arithmetic Mean \( A \) is given by:
\[ A = \frac{a + b}{2} \]
- The Geometric Mean \( G \) is given by:
\[ G = \sqrt{ab} \]
- The Harmonic Mean \( H \) is given by:
\[ H = \frac{2ab}{a + b} \]

From the properties of these means, we know:
\[ A > G > H \]
and since \( a \) and \( b \) are positive integers, \( A \), \( G \), and \( H \) are all positive.

### Step 2: Analyze the Given Equation
The equation provided is:
\[ Ax^2 - |G|x - H = 0 \]
Since \( G \) is positive, we can remove the absolute value:
\[ Ax^2 - Gx - H = 0 \]

### Step 3: Identify the Roots
Let \( \alpha \) and \( \beta \) be the roots of the equation. By Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = \frac{G}{A} \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = -\frac{H}{A} \]

### Step 4: Analyze the Signs of the Roots
1. **Sum of the Roots**: Since \( G > 0 \) and \( A > 0 \), we have:
\[ \alpha + \beta > 0 \]
This indicates that the sum of the roots is positive.

2. **Product of the Roots**: Since \( H > 0 \) and \( A > 0 \), we have:
\[ \alpha \beta < 0 \]
This indicates that the product of the roots is negative.

### Step 5: Conclusion About the Roots
From the above analysis:
- The sum of the roots being positive implies that at least one of the roots is positive.
- The product of the roots being negative implies that one root must be positive and the other must be negative.

Thus, we conclude that:
- There is exactly one positive root and one negative root.

### Final Answer
The correct option is:
(c) exactly one positive root.