Find the equation of the angle bisector of the angles

Question

1. find the equation of the angle bisector of the angles between the lines 2x + y = 5 and x − 2y = 1.

2. An awning that covers a sliding glass door that is 88 inches tall forms an angle of 50 degrees with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the Sun is more than 65 degrees. Find the length of the awning.

3. In how many ways can 5 boys and 3 girls be seated in a round table so that no girls are together?

4. Prove that all medians of a triangle are concurrent.

5. Show that n² − 2 < 2ⁿ for all integers n > 2.

6. Give an example of that x|yz implies x|y ∧ x|z is not valid.

7. If y = f(x) is defined on a ≤ x ≤ b, suppose we rotate the graph of this function around X axis. Calculate the surface are of solid generated.

8. Let α > 1 and M ≥ 0. Suppose f : R −→ R satisfies |f(x) − f(y)| ≤ M|x − y| α for all x, y ∈ R. Prove that f is a constant function?

9. 0 < α < 1,prove x−α ∈ L([0, 1]),and calculate

10. We prove that

11. Is f(z) = zⁿ + nwz one-to-one?