Question

The equation \(x^3-24x^2+\alpha x-\beta=0\) has positive integral roots \(p,q,r\). If geometric mean of the three roots is 8, then \(\alpha=\)

• A. 192
• B. 224
• C. 128
• D. 256

Hint:

Whenever AM = GM for positive numbers, all the numbers must be equal.

Answer 1

The Correct Option is A

Concept: For a cubic equation \[ x^3-24x^2+\alpha x-\beta=0, \] \[ p+q+r=24 \] and \[ \alpha=pq+qr+rp. \]

Step 1: Use the geometric mean condition.
Given, \[ \sqrt[3]{pqr}=8 \] Therefore, \[ pqr=8^3=512 \] Also, \[ p+q+r=24 \]

Step 2: Apply AM-GM inequality.
\[ \frac{p+q+r}{3}\geq \sqrt[3]{pqr} \] Substituting values, \[ \frac{24}{3}\geq 8 \] \[ 8\geq 8 \] Equality holds. Hence, \[ p=q=r=8 \]

Step 3: Find \(\alpha\).
\[ \alpha=pq+qr+rp \] \[ =8\times8+8\times8+8\times8 \] \[ =64+64+64 \] \[ =192 \] \centerline{{192}}