If `pa n dq` 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 the equation `x^2+p x+q=0` are real.

Options

A. are real is 33/50
B. are imaginary is 19/50
C. are real and equal is 3/50
D. are real and distinct is 3/5

Answer

B, C, D

Explanation

Roots of `x^(2) + px + q = 0` will be real if `p^(2) ge` 4q. The possible selection are as follows:

Therefore, number of favourable ways is 62 and total number of ways is `10^(2) = 100`. Hence, the required probaility is
`62//100 = 31//50`.
The probability that the roots are imaginary is
`1 - 31//50 = 19//50`.
Roots are equal when (p, q) = (2, 1), (4, 4), (6, 9). The probability that the roots are real and equal is `3//50`. Hence, probability that the roots are real and distinct is `3//5`.