Question

The coefficients \( a, b, c \) of the quadratic equation \( ax^2 + bx + c = 0 \) are obtained by throwing a die three times. The probability that this equation has equal roots is:

• A. \( \dfrac{1}{72} \)
• B. \( \dfrac{5}{216} \)
• C. \( \dfrac{1}{36} \)
• D. \( \dfrac{1}{54} \)

Hint:

N/A

Answer 1

The Correct Option is B

Equal roots condition: Discriminant \( D = b^2 - 4ac = 0 \Rightarrow b^2 = 4ac \) Total outcomes when throwing a die 3 times: \[ 6 \times 6 \times 6 = 216 \] Count favorable outcomes where \( b^2 = 4ac \) for integer values \( a, b, c \in \{1, 2, 3, 4, 5, 6\} \):
Check possible values:
\[ (1, 2, 1) \Rightarrow b^2 = 4, 4ac = 4
(1, 4, 4) \Rightarrow 16 = 4(1)(4)
(2, 4, 2) \Rightarrow 16 = 4(2)(2)
(4, 4, 1) \Rightarrow 16 = 4(4)(1)
(1, 6, 9) \Rightarrow 36 = 4(1)(9) \] Valid tuples: \[ (1,2,1), (1,4,2), (2,4,1), (4,4,1), (3,6,3) \Rightarrow 5 favorable cases \] \[ \text{Required Probability} = \frac{5}{216} \]