If 1 and z are real numbers and the coefficient before I is m 2-3m = 0, then m = 0 or m=3.
2, z is an imaginary number, then the coefficient m 2-3m before I is not equal to 0, so m is not equal to 0 or m is not equal to 3.
3, z is a pure imaginary number, then m 2-5m+6 = 0, and m 2-3m is not equal to 0.
M=2 or m=3, and m is not equal to 0 or 3.
So only m=2 can meet the requirements.
4.z=a+bi, where a stands for real part and b stands for imaginary part.
The coordinates (+,+) are in the first quadrant.
The coordinates (-,+) are in the second quadrant.
The coordinates (-,-) are in the third quadrant.
The coordinates (+,-) are in the fourth quadrant.
Z is in the second quadrant, so the real part m 2-5m+6: 0.
(m-2)(m-3)& lt; 0, and m (m-3) >; 0, so 2
Such m does not exist, so the absence of m makes the complex number z in the second quadrant.
I hope this is the right answer. I wish you progress in your study.