The coefficients $a, b and c$ of the quadratic equation, $a x^{2} + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :

\frac{1}{54}

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\frac{1}{72}

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\frac{1}{36}

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\frac{5}{216}

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Solution

The correct option is D $\frac{5}{216}$
$a x^{2} + b x + c = 0$
$a, b, c \in 1, 2, 3, 4, 5, 6$
$n (S) = 6 \times 6 \times 6 = 216$
$D = 0 \Rightarrow b^{2} = 4 a c$
$a c = \frac{b^{2}}{4} If b = 2, a c = 1 \Rightarrow a = 1, c = 1$
$If b = 4, a c = 4 \Rightarrow a = 1, c = 4$
$a = 4, c = 1$
$a = 2, c = 2$
$If b = 6, a c = 9 \Rightarrow a = 3, c = 3$
$∴$ probability $\frac{5}{216}$

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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 :

The correct option is D 5216ax2+bx+c=0 a,b,c ∈1,2,3,4,5,6 n(S)=6×6×6=216 D=0⇒b2=4ac ac=b24 If b=2,ac=1⇒a=1,c=1 If b=4,ac=4⇒a=1,c=4 a=4,c=1 a=2,c=2 If b=6,ac=9⇒a=3,c=3 ∴ probability 5216

The coefficients $a, b and c$ of the quadratic equation, $a x^{2} + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :