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Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax2 + bx + 1 = 0 has real roots, is equal to
- a)1/2
- b)1/4
- c)1/8
- d)1/16
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Each of a and b can take values 1 or 2 with equal probability. The pro...
Solution:
The given equation is ax² + bx + 1 = 0.
For the equation to have real roots, the discriminant b² - 4ac should be greater than or equal to zero.
Substituting the values of a and b, we get
b² - 4ac = b² - 4a
Since a and b can take values 1 or 2 with equal probability, the probability of b² - 4a being greater than or equal to zero is the same as the probability of b² - 4a being less than zero.
Therefore, P(b² - 4a ≥ 0) = P(b² - 4a < 0)="" />
Now, we need to find the probability that the discriminant is greater than or equal to zero, i.e., P(b² - 4ac ≥ 0).
Case 1: a = 1
If a = 1, then the discriminant becomes b² - 4c.
For the discriminant to be greater than or equal to zero, we need b² ≥ 4c.
Since b can take values 1 or 2 with equal probability, the probability of b² ≥ 4c is the same as the probability of b² < />
Therefore, P(b² - 4c ≥ 0) = P(b² - 4c < 0)="" />
Case 2: a = 2
If a = 2, then the discriminant becomes b² - 8c.
For the discriminant to be greater than or equal to zero, we need b² ≥ 8c.
Since b can take values 1 or 2 with equal probability, the probability of b² ≥ 8c is the same as the probability of b² < />
Therefore, P(b² - 8c ≥ 0) = P(b² - 8c < 0)="" />
The total probability of the equation having real roots is the sum of the probabilities from both cases:
P(ax² + bx + 1 = 0 has real roots) = P(a = 1) × P(b² - 4c ≥ 0) + P(a = 2) × P(b² - 8c ≥ 0)
= (1/2) × (1/2) + (1/2) × (1/2) = 1/4
Therefore, the correct option is (B) 1/4.
The given equation is ax² + bx + 1 = 0.
For the equation to have real roots, the discriminant b² - 4ac should be greater than or equal to zero.
Substituting the values of a and b, we get
b² - 4ac = b² - 4a
Since a and b can take values 1 or 2 with equal probability, the probability of b² - 4a being greater than or equal to zero is the same as the probability of b² - 4a being less than zero.
Therefore, P(b² - 4a ≥ 0) = P(b² - 4a < 0)="" />
Now, we need to find the probability that the discriminant is greater than or equal to zero, i.e., P(b² - 4ac ≥ 0).
Case 1: a = 1
If a = 1, then the discriminant becomes b² - 4c.
For the discriminant to be greater than or equal to zero, we need b² ≥ 4c.
Since b can take values 1 or 2 with equal probability, the probability of b² ≥ 4c is the same as the probability of b² < />
Therefore, P(b² - 4c ≥ 0) = P(b² - 4c < 0)="" />
Case 2: a = 2
If a = 2, then the discriminant becomes b² - 8c.
For the discriminant to be greater than or equal to zero, we need b² ≥ 8c.
Since b can take values 1 or 2 with equal probability, the probability of b² ≥ 8c is the same as the probability of b² < />
Therefore, P(b² - 8c ≥ 0) = P(b² - 8c < 0)="" />
The total probability of the equation having real roots is the sum of the probabilities from both cases:
P(ax² + bx + 1 = 0 has real roots) = P(a = 1) × P(b² - 4c ≥ 0) + P(a = 2) × P(b² - 8c ≥ 0)
= (1/2) × (1/2) + (1/2) × (1/2) = 1/4
Therefore, the correct option is (B) 1/4.
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Community Answer
Each of a and b can take values 1 or 2 with equal probability. The pro...
As it has real roota so b^2-4ac must be greater than or equal to zero so b can be selected by 1\2 also a can also be selected by 1\2
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