Download Sample Questions for Test 2

MATH 1200 (SECTION E): TERM TEST 2 PREPARATION
PROBLEMS
The following practice problems are for Math 1200 Term test 2-Section E. 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. Prove or disprove the following statement: If m and n are even integers then
m + n is even.
4. Prove or disprove the following statement: If st is even and s is odd, then t
must be even.
5. Prove that if x + y is irrational and x is rational then y is irrational.
6. Prove or disprove: If x + y is irrational and x is rational then y is irrational.
7. Prove or disprove: If 5s + 3t is even then both s and t are even or both s
and t are odd.
8. Students are asked to consider the following statement for an integer 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
statement 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.
9. Prove or disprove that for any integer positive integer n, if n2 is a multiple of
3 then n is a multiple of 3.
10. Prove or disprove: If 5x − 7 is odd then x is even.
11. Prove that if x and y are both odd then x2 + y 2 is even but not divisible by
4.
1
2
12. Prove that if x and y are both rational then xy is rational.
13. Prove or disprove the following: If st is even and s is odd then t must be
even.
14. Prove or disprove the following: If x and y are positive irrational numbers then
xy is an irrational number.
15. Prove that for any positive integer m, if m2 is a multiple of 2 then m is a
multiple of 2.
16. Prove that the sum and product of two rational numbers are rational.
17. 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 kd is irrational.
18. Prove that if ab is odd and a is even then b must be odd.
19. Prove or disprove: Suppose x ∈ Z. If x2 + 6x + 5 is even, then x is odd.
20. Prove or disprove:
√
2 is irrational.
√
21. Let x be a real number. You may
√ use the fact that 2 is irrational.
(a) Prove or disprove that, if x + √2 is rational then x is irrational.
(b) Prove or disprove that, if x + 2 is irrational√then x is rational.
(c) Prove or disprove that, if x √
is rational then x 2 is irrational.
(d) Prove or disprove that, if x 2 is irrational then x is rational.
22. Prove or disprove: Suppose n ∈ Z. Then n is even if and only if n − 1 is
odd.
Induction
23. Use Mathematical Induction to prove that for all integers n,
1+
1
1
1 1
+ + ··· + 2 ≤ 2 − .
4 9
n
n
24. How many subsets does {1} have? How many subsets does {1, 2} have? Find a
formula for the number of subsets of {1, 2, . . . , n}, and use mathematical induction
to prove your formula.
3
25. Use Mathematical Induction to prove that
n(n + 1)(2n + 1)
12 + 22 + 32 + · · · + n2 =
.
6
26. Find a formula for the sum
1
1
1
1
+
+
+ ··· +
,
1.2 2.3 3.4
n.(n + 1)
and use Mathematical Induction to prove your formula is correct.
27. Use Mathematical Induction to prove that n3 + 2n is a multiple of 3 for every positive integer n.
28. Use Mathematical Induction to prove that for all positive integers n, 2n 32n − 1
is a multiple of 17.
29. Prove or Disprove: If p > −1, then (1 + p)n ≥ 1 + np.
30. Show that if A1 , · · · , An are sets, then
(A1 ∪ · · · ∪ An )c = Ac1 ∩ · · · ∩ Acn .