Mathematics Interview Questions (VIII)
36. Integrate .
Solution. Let , then . As a result, .
37. Integrate between limits of 1 and 0. Draw that graph.
To draw the graph, consider its derivative , which is greater than 0 when or . Also, it passes through the origin. As , using L'Hôpital's rule; as , as well. You can view here for a sample.
38. Integrate between limits of 1 and -1. Draw the graph. Why is it -2 and not infinity, as it appears to be on the graph?
For its graph, please refer to question 35.
The reason why it is -2 is because of the singularity (tends to ) at . That integral is therefore improper and does not refer to the area under the graph.
39. Write down 3 digits, and then write the number again next to itself, e.g., 145145. Why is it divisible by 13?
Solution. Such a number has the form where . Since , the number always contains a factor of 13.
40. You are given a triangle with a fixed perimeter and you want to maximise the area. What shape will it be? Prove it.
Solution 1. Suppose that the lengths of the sides of the triangle are , , and such that the perimeter of the triangle is fixed and it is . Using the Heron's formula, the area .
The AM-GM inequality (with 3 terms) states that for , with equality at .
Let , , and . Since where , . Similarly, . We can then use the AM-GM inequality to obtain that . As a result, and , with equality when , i.e., . The shape is therefore equilateral.
Solution 2. Suppose that the lengths of the sides of the triangle are , , and such that the perimeter of the triangle is fixed and it is . Let , , and , respectively, be the vertices of the triangle opposite the sides of length , , and .
Given , we'll show that it must be the case that , proving that the largest triangle is isosceles. Repeating the argument given or will prove it must be equilateral.
Place and on the x-axis, and consider where can lie. Wherever it is, the sum of the distances from to the two points and is , which is constant, so the locus of is an ellipse with foci and . The area of is , where is the -coordinate of P, so the triangle's area is greatest when is at a point on the ellipse as far from the -axis as possible. Therefore is at an endpoint of the ellipse's vertical axis, and by symmetry, .