Mathematics Interview Questions (X)
46. Integrate .
Solution. Let , then . As a result, .
47. Draw , then draw , then draw a graph of the number of solutions of against for , and then find the value of where there is only 1 solution.
Solution. The sketch is trivial.
From the graph, we can see that there are infinite solutions when . Suppose then . When they are tangent to each other at some point, say , and . As a result, and . We can then conclude that there is only 1 solution when or using the sketch. Otherwise, there are 2 solutions. The "graph of the number of solutions" is trivial and will be left to the reader.
48. I got a Rubik's cube, held it by two diagonally opposite vertices, and rotated it till it reached the same position. By how many degrees did it take a turn?
Solution. using intuition.
49. Solve for all natural numbers and where .
Solution. Obviously . We can prove that only 2 sets of solutions exist for all : and .
WLOG, suppose . Taking natural logarithms on both sides gives , i.e., since . Suppose . Using quotient rule, which is 0 when only. Therefore, iff . can therefore only be 1 or 2; if , then must also be 1, contradicting . When , , so . Since is increasing when and decreasing when , must be greater than . Trial and error gives , and by monotonicity, no more solutions exist. The equation is symmetric about , so and are the only solutions.
50. There is a game with 2 players (A&B) who take turns to roll a die and have to roll a six to win. What is the probability of person A winning?
Solution 1. Assume A throws first. Let the probability be . Then because either A wins on the first roll, or it's as if B started first and we want the probability he doesn't win. Solving gives .
Solution 2. Assume A throws first. The probability of A wins on the first roll is . The probability of A wins on the second roll is because both A and B have to fail on the first roll. Repeat the calculation, and we end up getting a geometric series .