JEE PYQ: Quadratic Equation Question 9
Question 9 - 2021 (25 Feb Shift 1)
The coefficients $a$, $b$ and $c$ of the quadratic equation, $ax^2 + bx + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
(1) $\frac{1}{54}$
(2) $\frac{1}{72}$
(3) $\frac{1}{36}$
(4) $\frac{5}{216}$
Show Answer
Answer: (4)
Solution
$ax^2 + bx + c = 0$
$a, b, c \in {1, 2, 3, 4, 5, 6}$
$n(s) = 6 \times 6 \times 6 = 216$
$D = 0 \Rightarrow b^2 = 4ac$
$ac = \frac{b^2}{4}$. If $b = 2, ac = 1 \Rightarrow a = 1, c = 1$
If $b = 4, ac = 4 \Rightarrow a = 1, c = 4$ or $a = 4, c = 1$
If $b = 6, ac = 9 \Rightarrow 2, c = 2$ (wait: no, we need $a=3, c=3$ but also check: no other combo from dice)
$\therefore$ probability $= \frac{5}{216}$
BETA
