Math Answers

1. Use mathematical induction to show that 8 │ (5^2n + 7).
Hint: 5^2(k+1) + 7 = 5^2(5^2k + 7) +(7 - 5^2·7)

2. Use the Division Algorithm to establish that 3a^2 – 1 is never a perfect square.

3. Use the Euclidean Algorithm to obtain integers x and y satisfying
gcd(1769,2378) = 1769x + 2378y.

4. Determine all solutions in the positive integers of the following Diophantine equation:
123x + 360y = 99.

5. Find the prime factorization of integer 1234, 10140, and 36000.

6. Give an example of a^2 ≡ b^2 (mod n) need not imply a ≡ b (mod n).

7. Show the following statements are true:
a. For any integer a, the unit digit of a^2 is 0, 1, 4, 5, 6 or 9.
b. Any one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occur as the units digit of a^3.
c. For any inter a, the units digit of a^4 is 0, 1, 5 or 6.

Evaluate integration of (t+1)^3e^t ,x=x to x=-1 is decreasing

Find the area enclosed by the curve r = a(1-cos theta)

If xsiny=sin(p+y), p belongs to R ,show that sinp.dy/dx + sin^2y=0

1) g(t) = t/2t +6

i) g(0)
ii) g(-3)
iii) g(10)
iv) g(x²)
v) g(t + h)
vi) g(t² - 3t + 1)

2) R(x) = √3+x- 4/x+1

i) R(0)
ii) R(6)
iii) R(-9)
iv) R(x+1)
v) R(x•4 - 3)
vi) R [ 1/x - 1]

1) f(x) = 3 - 5x - 2x²

i) f(4)
ii) f(0)
iii) f(-3)
iv) f(6-t)
v) f(7-4x)
vi) f(x+h)

Scientists are studying the temperature on a distant planet. They find that the surface temperature at one location is 20 Celsius. They also find that the temperature decreases by 3 Celsius for each kilometer you go up from the surface.
Let represent the temperature (in Celsius), and let be the height above the surface (in kilometers). Write an equation relating to , and then graph your equation using the axes below.

A jogger runs from her home to a point A, which is 6 km away. For there 6 km, shebegins by running at a constant speed till she reaches a hilly portion 2 km from herhome. Here her speed slows down while she runs up the hill, which is a 1-km run.Then she speeds up while running down the hill.The last 2 km of the run are again atconstant speed. Draw a graph to show the jogger’s speed as a function of the distancefrom her home. Also find the range of this function.

Find the area enclosed by the curve r = a(1- cos theta)