Given the complex z = 2m-1 + (M + 1) I (1) if the point corresponding to the complex Z is in the first quadrant, find the value range of the real number m; (2) if the complex | Z | ≤ 3, find the value range of the real number M

Given the complex z = 2m-1 + (M + 1) I (1) if the point corresponding to the complex Z is in the first quadrant, find the value range of the real number m; (2) if the complex | Z | ≤ 3, find the value range of the real number M

(1) Complex z = 2m-1 + (M + 1) I if the point corresponding to complex Z is in the first quadrant, then 2m − 1 ＞ 0m + 1 ＞ 0, the solution is m ＞ 12, so the value range of the real number m corresponding to the point in the first quadrant is {m | M ＞ 12}. (2) because | Z | ≤ 3, so (2m − 1) 2 + (M + 1) 2 ≤ 3, the solution is 1 − 65 ≤ m ≤ 1 + 65