4. Write an analysis of proof for each proposition below by using construction method. a) If a, b, and c are integers for which a|b, and blc, then alc. b) Ifn is an integer and n is an odd integer, then n² + 3 is divisible by 2. c) If p and q are rational and q + 0, then 2 is rational. d) If the function f(x) = Vx + 1 is continuous in the interval [0,3), then there exists a value c that satisfy the Mean Value Theorem for integral. e) Prove that if m < n are consecutive integers and m is even, then 4 divides m? + n2 – 1.

A rare genetic disease is known to be carried by DD people among 10000 people in the society. a) In a random sample of 1000 people, calculate the probability that less than GG carry the disease? (Please solve this section with Binomial distribution) a) In a random sample of 2000 people, calculate the probability that more than HH carry the disease? (Please solve this section with Poisson distribution) c) If people are randomly tested for the disease, what is the probability that the first person who is found to carry the disease is the LL person?

V-10 oV - 35

6. 7 V6 31 2V3

Evaluate

Calculate C(12, 4). 40 320 990 11 880 495

Let AABC be a triangle. Fix the points X, Y, and Z on AB, AC, and BC, respectively (as in the figure) such that BX = - = CZ AY = 4. Assume that the area of AXYZ is one and calculate the area of AABC. ХА YC ZB

Evaluate. 15C1

b. a = Vx-36, b = 6