Step-by-step Solution:

Step 1: Given Equations The three quadratic equations are: 1. \[ x^2 + px + qr = 0 \] 2. \[ x^2 + qx + rp = 0 \] 3. \[ x^2 + rx + pq = 0 \] It is given that every pair of these equations has a common root. Step 2: Let the Common Roots Be \( \alpha, \beta, \gamma \) Let: - \( \alpha \) be the common root of equations 1 and 2, - \( \beta \) be the common root of equations 2 and 3, - \( \gamma \) be the common root of equations 3 and 1. Thus: - \( \alpha \) satisfies equations 1 and 2, - \( \beta \) satisfies equations 2 and 3, - \( \gamma \) satisfies equations 3 and 1. Step 3: Write the Equations for the Common Roots 1. For \( \alpha \): \[ \alpha^2 + p\alpha + qr = 0 \quad \text{(1)} \] \[ \alpha^2 + q\alpha + rp = 0 \quad \text{(2)} \] 2. For \( \beta \): \[ \beta^2 + q\beta + rp = 0 \quad \text{(2)} \] \[ \beta^2 + r\beta + pq = 0 \quad \text{(3)} \] 3. For \( \gamma \): \[ \gamma^2 + r\gamma + pq = 0 \quad \text{(3)} \] \[ \gamma^2 + p\gamma + qr = 0 \quad \text{(1)} \] Step 4: Subtract the Equations to Eliminate the Quadratic Term 1. Subtract equation (2) from equation (1) for \( \alpha \): \[ (p - q)\alpha + qr - rp = 0 \] Simplify: \[ (p - q)\alpha = r(p - q) \] If \( p \neq q \), then: \[ \alpha = r \] 2. Subtract equation (3) from equation (2) for \( \beta \): \[ (q - r)\beta + rp - pq = 0 \] Simplify: \[ (q - r)\beta = p(q - r) \] If \( q \neq r \), then: \[ \beta = p \] 3. Subtract equation (1) from equation (3) for \( \gamma \): \[ (r - p)\gamma + pq - qr = 0 \] Simplify: \[ (r - p)\gamma = q(r - p) \] If \( r \neq p \), then: \[ \gamma = q \] Step 5: Verify the Common Roots Substitute \( \alpha = r \), \( \beta = p \), and \( \gamma = q \) into the original equations to verify: 1. For \( \alpha = r \): \[ r^2 + pr + qr = 0 \] This holds true if \( r + p + q = 0 \). 2. For \( \beta = p \): \[ p^2 + qp + rp = 0 \] This holds true if \( p + q + r = 0 \). 3. For \( \gamma = q \): \[ q^2 + rq + pq = 0 \] This holds true if \( p + q + r = 0 \). Thus, the condition \( p + q + r = 0 \) must be satisfied. Step 6: Find the Product of the Common Roots The product of the common roots \( \alpha, \beta, \gamma \) is: \[ \alpha \beta \gamma = r \cdot p \cdot q = pqr \] Final Answer: The product of the three common roots is: \[ \boxed{A} \]