Question

Prove or disprove the statements: (a) If x is a real number such that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2.

(b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2 + 2x − 1 ≤ 2.

(c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2 + 2x − 1 ≤ 2.

(d) If x is a real number such that |x + 2| + |x| ≤ 5, then x 2 + 2x − 1 ≤ 2.

(2) Prove or disprove the statements: (a) If z is a complex number such that |z + 1| + |z − 1| ≤ 3, then |z 2 − 1| ≤ 2.

(b) If z is a complex number such that if |z 2 − 1| ≤ 2, then |z + 1| + |z − 1| ≤ 3.

(3) A clock with a face that has the numbers 1 through 12 has three hands that indicate the second, minute and hour of the day.

Assume that the center of the clock is at position (0, 0), and at noon the end points of the hands are (respectively) at (0, 1), (0, 3/4), (0, 1/2).

(a) Give the position of the end points of each of the hands at time t where t represents the number of seconds after noon in both polar and rectangular coordinates (make sure that you label which you are using clearly).

(b) At what times do your equations say that the hands of the clock will all align?

Answer 1