State which of the following are true or false with a simple T or F?

Question

1. Two integers that are coprime are also prime.
2. If the sum of two complex numbers is real, the summands are complex conjugate of each other.
3. If the derivative of a function is increasing, the function is increasing.
4. If an integer is the product of two primes, then it has four distinct divisors.

Answer 1

Every single one of them is false:

1 is false, because being coprime only means that they have no common factors other than 1 -- they may have multiple factors that they do not share in common.

2 is false, because 1+2 = 3∈R, but 1 is certainly not the complex conjugate of 2.

3 is false, because the slope of the derivative determines the concavity of the function, not its slope. For instance, x² on (-∞, 0) has an increasing derivative (namely, 2x), but is itself decreasing.

4 is false, because 4 is the product of two primes, but has only 3 distinct divisors (namely, 1, 2, and 4). Similarly for p² in general, whenever p is prime.

Answer 2

3 is true in the vector sense. It may not be true of a scalar.

until you've clarified this, 3 cannot be answered with T/F

4 is definitely true.