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Solution


Q.

The coefficients a,b,c in the quadratic equation ax2+bx+c=0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216p equals:                    [2024]
1 76  
2 38  
3 57  
4 19  

Ans.

(2)

We need to find the probability that the given equation ax2+bx+c=0 has real and distinct roots.

       [∵ One root will be bigger if roots are distinct]

∴    D>0

⇒b2-4ac>0

b≠1,2          ∵ if b=1or 2

then ac<14 or ac<1 which is not possible as

a,b,c∈{1,2,3,4,5,6}.

If b=3

⇒ac<94⇒(a,c)={(1,1),(1,2),(2,1)}

i.e., 3 ways

If b=4

⇒ac<4⇒(a,c)={(1,1),(1,2),(2,1),(3,1),(1,3)}.

i.e., 5 ways

If b=5

⇒ac<254⇒(a,c)={(1,1),(1,2),(2,1),(3,1),(1,3),(2,2),(1,4),(1,5),(2,3),(3,2),(4,1),(5,1),(6,1),(1,6)}.

i.e., 14 ways

If b=6

⇒ac<9⇒(a,c)={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(4,1),(4,2),(5,1),(6,1)}

i.e., 16 ways

∴    Total number of ways = 3 + 5 + 14 + 16 = 38

⇒ Required probability =3863=p

So, 216p=38216×216=38