Two Numbers b and c are Chosen At Random With Replacement From The Numbers 1, 2, 3, 4, 5, 6, 7, 8 And 9. The Probability That x2 + bx + c > 0 For All x Ïµ R is

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Question

Two numbers b and c are chosen at random with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. The probability that x2 + bx + c > 0 for all x Ïµ R is

Solution

Correct option is

32/81

Since b and c both can assume values from 1 to 9. So, total number of ways of choosing b and c is 9 × 9 = 81.

Now, x2 + bx + c = 0 for all x Ïµ R ⇒ discriminate < 0 i.e., b2 – 4c < 0

The following table shows the possible values of b and c which b2 – 4c < 0

c

b

Total

1

1

1

2

1, 2

2

3

1, 2, 3

3

4

1, 2, 3

3

5

1, 2, 3, 4

4

6

1, 2, 3, 4

4

7

1, 2, 3, 4, 5

5

8

1, 2, 3, 4, 5

5

9

1, 2, 3, 4, 5

5

 

 

32

So, favorable number of ways = 32

Hence, required probability = 32/81.

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