The coefficients a , b and c of the quadratic equation, a x2+b x+c=0 are obtained by throwing a dice three times. The probability that this equation has equal roots is

Question

The coefficients a , b and c of the quadratic equation, a x2+b x+c=0 are obtained by throwing a dice three times. The probability that this equation has equal roots is (A) (1/72) (B) (5/216) (C) (1/36) (D) (1/54)

Tardigrade Admin · Accepted Answer

a x2+b x+c=0 For equal roots D =0 ⇒ b 2=4 ac Case I: ac =1 ( a , b , c )=(1,2,1) Case II: ac = 4 (a, b, c)=(1,4,4) or (4,4,1) or (2,4,2) Case III: ac =9 (a, b, c)=(3,6,3) Required probability =(5/216)

Q. 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

$\frac{1}{72}$

$\frac{5}{216}$

$\frac{1}{36}$

$\frac{1}{54}$

$a x^{2} + b x + c = 0$
For equal roots $D = 0$
$\Rightarrow b^{2} = 4 a c$
$C a se I : a c = 1$
$(a, b, c) = (1, 2, 1)$
$C a se II : a c = 4$
$(a, b, c) = (1, 4, 4)$
or $(4, 4, 1)$
or $(2, 4, 2)$
$C a se III : a c = 9$
$(a, b, c) = (3, 6, 3)$
Required probability $= \frac{5}{216}$

Q. 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

721​

2165​

361​

541​