Opsimath

2021/08/31 阅读：63 主题：默认主题

# 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.

Opsimath

2021/08/31 阅读：63 主题：默认主题

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