Explanation
To solve this, we need to find how many values of x (from 1 to 100) satisfy the inequality x + \frac{100}{x} > 29.
1. Formulate the Quadratic Inequality
Given:
x + \frac{100}{x} > 29
Since x is a natural number (x∈{1,2,…,100}), we know x > 0. Multiplying by x:
2. Find the Critical Points
We solve the equation x2−29x+100=0 using factorization:
The roots are x=4 and x=25.
3. Analyze the Inequality
For the expression (x - 4)(x - 25) > 0 to be true:
4. Count the Favorable Natural Numbers
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Case 1 (x < 4): The natural numbers are {1,2,3}.
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Case 2 (x > 25): The natural numbers from 26 to 100.
Total favorable values of x=3+75=74.
5. Calculate the Probability
The total number of possible outcomes is 100.
P(E)=Total OutcomesFavorable Outcomes
Now, simplifying the fraction by dividing both numerator and denominator by 2:
Correct Answer: (A) 5037
