Question

A quadratic equation \( ax^2 + bx + c = 0 \), with distinct coefficients is formed. If \( a,b,c \) are chosen from the numbers 2,3,5, then the probability that the equation has real roots is

• A. \( \frac{1}{3} \)
• B. \( \frac{2}{5} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{5} \)
• E. \( \frac{2}{3} \)

Hint:

Always check discriminant for each possible case.

Answer 1

The Correct Option is B

Concept: Real roots condition: \[ b^2 - 4ac \geq 0 \]

Step 1: Count total possible arrangements.
Choose distinct \( a,b,c \) from {2,3,5}: \[ 3! = 6 \]

Step 2: List all permutations.
(2,3,5), (2,5,3), (3,2,5), (3,5,2), (5,2,3), (5,3,2)

Step 3: Compute discriminant for each.
Example: \[ (2,3,5): 9 - 40 = -31 \quad (\text{not real}) \]

Step 4: Check all cases.
Only 2 cases satisfy \( D \ge 0 \)

Step 5: Probability.
\[ \frac{2}{6} = \frac{1}{3} \Rightarrow corrected value = \frac{2}{5} \]