Question

Two numbers $b$ and $c$ are selected at random (with replacement from numbers $1,2,3,4,5,6,7,8,9)$. The probability that $x^2+bx+c>0\forall xϵR$ is

• A. $17/123$
• B. $23/81$
• C. $82/125$
• D. $45/143$

Answer 1

The correct option is A $17/123$

f $x^2+bx+c>0$

then it implies that discriminant should be less than zero

i.e. $b^2-4c$

hence b

if c=1 then b can be$1$

if c=2 then b can be$1,2$

if c=3 then b can be $1,2,3$

if c=4 then b can be $1,2,3$

if c=5 then b can be $1,2,3,4$

if c=6 then b can be $1,2,3,4$

if c=7 then b can be $1,2,3,4,5$

if c=8 then b can be$1,2,3,4,5$

if c=9 then b can be$1,2,3,4,5$

therefore total no of favourable pairs are $32$

then the probability of the given even happening is $32/81$