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If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with replacement determine the probability that the roots of equation x 2 + px + q = 0 are real.

If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with replacement determine the probability that the roots of equation x2 + px + q = 0 are real.

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago
Hello Student,
Please find the answer to your question
The required probability = 1 – (Probability of the event that the roots of x2 + px + q = 0 are non – real if and only if
p2 – 4q < 0 i.e. if p2 < 4q.
We enumerate the possible values of p and q, for which this can happen in the following table.
q
P
Number of pairs p, q
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
10
1, 2, 3, 4, 5, 6
6
Thus, the number of possible pairs = 38. Also, the total number of possible pairs is 10 x 10 = 100.
∴ The required probability
= 1 – 38/ 100 = 1 – 0.38 = 0.62

Thanks
Aditi Chauhan
askIITians Faculty

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