Mathematics Interview Questions (II)
6. Integrate .
Solution. Consider integration by parts:
7. Sketch the curve . What does it look like?
Solution. Letting or , interchanging and , or letting and do not affect the equation of the curve, so it is symmetric about . Hence, we may only consider the proportion where .
When , so it does not intersect with the -axis; using the same logic, it does not intersect with the -axis as well.
Differentiating both sides implicitly gives , so . When , or , where . When , and .
Differentiating again gives . When , it can be easily verified that , so is a maximum.
For the part when for one value, there is only one corresponding value, we must have . That is the point .
The rest of the curve can be sketched using symmetries. You can view here for a sample.
8. 3 girls and 4 boys were standing in a circle. What is the probability that two girls are together but one is not with them?
Solution 1. There are permutations in total. Select two girls that are sitting together, so ways. Two boys must be adjacent to them, so ways. Finally, the other 3 can be arranged in ways. In total, the probability that two girls are together but one is not with them is .
Solution 2. Ignore the boys, and seat any two girls together. The 3rd girl has 3 permissible places out of 5; thus, the probability is .
9. Prove that .
Solution. We know that . The LHS of that equation is greater than , so .
10. Is there such number that 7 divided ?
Solution. Consider . Since , . Therefore, this number is nonexistent.