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Final Preparation Problems Section E: Farid Aliniaeifard
The following are practice problems are for Math 1200 Final Exam-Section D. Some
of these may appear on the exam version for your section. To use them well, solve
the problems, then discuss and compare your methods and answers. See if you can
identify what particular knowledge and skills were required.
1. Prove that if 5x + 12 is even then x must be even.
2. Prove that for any positive integer m, m2 is even if and only if m is even.
3. Find a complex number z such that z 3 = 2i − 2.
1+i
4. Express √
in the form x + iy, where x, y ∈ R. By writing each of 1 + i
3+i
√
and 3 + i in polar form, deduce that
√
√
π
π
3+1
3−1
cos
= √
and sin
= √
.
12
12
2 2
2 2
5. (a) Find a complex number in the form x + yi whose square is 5 + 12i.
(b) Extend your method in (a) to prove that every complex number has a
square root.
6. Prove or disprove the following statement: If m and n are even integers then
m + n is even.
7. Prove or disprove the following statement: If st is even and s is odd, then t
must be even.
8. Prove that if x + y is irrational and x is rational then y is irrational.
9. Prove or disprove: If x + y is irrational and x is rational then y is irrational.
10. Prove by direct method:
If 5s + 3t is even then both s and t are even or both s and t are odd.
11. Prove that for all integers n > 0, 3|22n+1 + 1.
12. Students are asked to consider the following statement for an interger t: If t2
is odd then t is odd.
A student provides the following:
t = 2t is even where t is an integer.
t2 = (2t)2 = 2(2t2 ) which is even since t is even, by proof by contradiction the statment is true.
(a) Evaluate the student’s work and offer feedback.
(b) Indicate what you would have given the student’s work out of five points.
13. Prove that for any integer positive integer n, if n2 is a multiple of 3 then n is
a multiple of 3.
14. Prove or disprove: If 5x − 7 is odd then x is even.
15. Prove that if x and y are both odd then x2 + y 2 is even but not divisible by 4.
16. Prove that if x and y are both rational then xy is rational.
17. Prove or disprove the following: If st is even and s is odd then t must be even.
18. Prove or disprove the following: If x and y are positive irrational numbers then
xy is an irrational number.
19. Prove that for any postive integer m, if m2 is a multiple of 2 then m is a
multiple of 2.
20. Prove using contradiction that if 5s + 3t is even then both s and t are even or
s and t are both odd.
21. Prove that the sum and product of two rational numbers are rational.
22. Suppose k is a rational number and d is an irrational number.
(a) Prove or disprove that k − d is irrational.
(b) Prove or disprove that if d 6= 0 then
k
d
is irrational.
23. Prove that if ab is odd and a is even then b must be odd.
24. Prove that for every non-negative integer n, 2n > n.
25. Prove that for all n ≥ 1, 1 + 4 + . . . + (3n − 2) =
n(3n−1)
.
2
26. Prove that for all positive integers n, 7n − 2n is divisible by 5.
27. Prove that for any positive integer n, 2(2) + 3(22 ) + . . . + (n + 1)(2n ) = n(2n+1 ).
28. Show that for any positive integer n, 2n3 + n is divisible by 3.
29. Show that for any positive integer n, 6n − 1 is divisible by 5.
30. Prove that
Pn
1
i=1 i(i+1)
=
n
.
n+1
31. Prove that for any real number X > −1, and any positive integer n, (1+X)n ≥
1 + nX.
32. Show that the sum of first n positive integers is
n(n+1)
.
2
33. Prove that for any positive integer n, 13 + 23 + . . . + n3 =
n2 (n+1)2
.
4
34. (a) Prove by induction that if b is an odd number and n a positive integer,
then bn is also odd.
(b) Using the conclusion of the first part of this question, show that the equation x19 + x + 1 = 0 has no solutions that are integers.
35. Show that n! < nn for any positive integer n ≥ 2.
36. Show that 3n < n! for any positive integer n ≥ 7.
37. Show that there are non-negative integers a and b such that 3a + 2b = n for
any integer n ≥ 5.
38. Prove that every positive integer n > 1 is product of some prime numbers.
39. Prove that for every non-negative integer n, 12|(n4 − n2 ).
40. Let F1 = 1, F2 = 1 and Fn = Fn−1 + Fn−2 (this sequence is the well-known
Fibonacci sequence). Show that for every positive integer n, Fn < 2n .
41. Argand diagram, a complex number z = a + bi can be represented graphically
as the vector from the origin to the point (a, b).
(a) Find the polar form of z = −1 + i. Graph z = −1 + i.
(b) Compute (−1 + i)2 . Find the polar form of (−1 + i)2 . Graph (−1 + i)2 .
(c) Compute (−1 + i)3 . Find the polar form of (−1 + i)3 . Graph (−1 + i)3 .
(d) Is there a pattern? What do you notice? Write down a formula for the
pattern you noticed. Remember for questions 5 and 6 to explain your
thinking :)
(e) Compute (−1 + i)50 .
(f) Find all complex numbers z such that z 3 = 8i.