Probability 1 Question 18

26. If p and q are chosen randomly from the set 1,2,3,4,5$,$6,7,8,9$and10$ with replacement, determine the probability that the roots of the equation x2+px+q=0 are real.

(1997, 5M)

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Answer:

Correct Answer: 26. 110(n+2)n+7C5

Solution:

  1. The required probability =1 (probability of the event that the roots of x2+px+q=0 are non-real).

The roots of x2+px+q=0 will be non-real if and only if p24q<0, i.e. if p2<4q

The possible values of p and q can be possible according to the following table.

Value of q Value of p Number of pairs of p,q
1 1 1
2 1,2 3
3 1,2,3 3
5 1,2,3 4
6 1,2,3,4 4
7 1,2,3,4 5
9 1,2,3,4,5 5
10 1,2,3,4,5 5

Therefore, the number of possible pairs =38

Also, the total number of possible pairs is 10×10=100

The required probability =138100=10.38=0.62



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