Question

The coefficients a, b and c of the quadratic equation, ax² + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :

• A. 1/36
• B. 1/54
• C. 1/72
• D. 5/216

Hint:

In probability problems involving dice, list the cases systematically by fixing one variable (like $b$) to avoid missing combinations.

Answer 1

The Correct Option is D

Step 1: Total outcomes = $6^3 = 216$. Equal roots condition: $D = b^2 - 4ac = 0 \Rightarrow b^2 = 4ac$.
Step 2: $b=2 \Rightarrow ac=1 \Rightarrow (1,1)$ [1 case].
Step 3: $b=4 \Rightarrow ac=4 \Rightarrow (1,4), (4,1), (2,2)$ [3 cases].
Step 4: $b=6 \Rightarrow ac=9 \Rightarrow (3,3)$ [1 case].
Step 5: Total = $1 + 3 + 1 = 5$ cases. $P = 5/216$.