Addis Ababa University
Department Of Mathematics
Final Examination on Freshman Mathematics
[For Political Science Students in the Evening Division]
Date: September 30/2021
Time allowed:
2:15
Total marks:
50%
Name: -------------------------------------------------------------------Id. No: ----------------General instruction:
This examination booklet contains three parts [10 multiple
choice, 10 short answer and 5 workout questions] in 5 pages. Read the specific instruction
for each part and write your answers accordingly.
Part 1 [multiple choice questions]: For the next 10 questions encircle the letter that
corresponds to the correct choice. [1.5 marks each]
1. Let 𝑝 and 𝑞 be propositions. If the truth value of p is true and that of q is false, then
which compound proposition is necessarily true?
A) 𝑝 ∨ 𝑞
B) 𝑝 ∧ 𝑞
C) 𝑝 ⟺ 𝑞
D) 𝑞 ⟹ 𝑝.
2. Let 𝑝 and 𝑞 be propositions. Which compound proposition is neither a contradiction
nor a tautology?
A) 𝑝 ∨ 𝑝 B) 𝑝 ∧ 𝑝 C) [𝑝 ⟹ 𝑞] ⟺ [𝑞 ⟹ 𝑝] D) [ q ⟹ 𝑝] ⟺ [𝑝 ⟹ 𝑞].
3. Which pairs of compound propositions are not equivalent?
A) 𝑝 ∨ 𝑞 , (𝑝 ∧ 𝑞)
B) p ⟹ (q ⟹ r), (p ∧ q) ⟹ r
C) p ⟹ (q ∨ r), (p ∧ r) ⟹ q
D)p ⟹ q, 𝑝 ∨ 𝑞.
4. Which term describes the relationship between the statement 𝑞 ⟹ 𝑝 and the
statementq ⟹ 𝑝?
A) Converse
B) inverse
C) contrapositive
D) equivalence
5.
Let A= {1, 2, 3} and B= {4, 5, 6}. Consider the following relations.
I. {(1, 4), (2, 5), (3, 6)}
III. {(1, 4), (2, 4), (3, 4)}
II. {(1, 4), (2, 4)}
IV. {(1, 4), (1, 2), (1, 3)}
Which of the above relations represents a function from A into B?
A) I and III only
C) III and IV only
B) I and IV only
D) I , II and III only
6. For any real numbers a, b and c, we have(𝑎 + 𝑏)𝑐 = 𝑎𝑐 + 𝑏𝑐. This property is called
A) Closure property B). invertibility C) commutative property D) distributive property
1
7. One solution of 3 𝑥 2 + 7 𝑥 − 6 = 0 is
2
2
A) − 3
B) 3
C) 3
D) 2
8. 15, 000 Birr is invested in an account that yields 5% interest per year. After how
many years will the account be worth 91, 221.04 Birr if the interest is compounded
yearly?
A) 1
B) 2
C) 3
D) none
9. Which one is false about the matrix below?
A) Its rank is 3
C) It has no inverse
B) Its determinant is 0
D) none
2
1 −1
𝑎 1
2
10. If 𝐴 = [
], B= [
] and ( 𝐴 + 𝐵 ) = 𝐴 + 𝐵2 , then the values
2 −1
𝑏 −1
of a and b are
A) 𝑎 = 4, 𝑏 = 1
B)𝑎 = 1, 𝑏 = 4
C)𝑎 = 0, 𝑏 = 4
D)𝑎 = 2, 𝑏 = 4
Part 2 [short answer questions]: For the next 10 questions write your most simplified
answers only. [1.5 marks each]
1. If ¬r ⇒ ¬(𝑝 ∧ 𝑞) is true, then the truth value of (p ⇔ r) ∨q⇔ (¬𝑝 ∧ 𝑟) is ………..
2. The truth value of (∀𝑥 ∈ ℝ)(𝑥 2 ≤ 0 ⟹ 𝑥 = 0) is …………………………
3. The truth value of (∀𝑥 ∈ ℝ)(∃𝑦 ∈ ℝ)(𝑥 + 𝑦 = 5) is………………………………….
4. The truth value of (∃𝑥 ∈ ℝ)(∀𝑦 ∈ ℝ)(𝑥 + 𝑦 = 5) is …………………………
5. If a set has 63 proper subsets, how many elements are there in the set? ………………
6.
Write the set {2, 4, 6, 8, 10} using set builder method.
Answer: ……………………………………………………………………………………………………..
7. If 𝐺𝐶𝐹(𝑎, 𝑏) =5 and 𝑎𝑏 = 250, then 𝐿𝐶𝑀(𝑎, 𝑏) = ………………………………..
̅̅̅̅ as a fraction in simplified form.
8. Write 2.234
Answer:………………………………..
9. Find the least value of x so that the number 91876x2 is divisible by 8.
answer: …………………………………….
10. What is the remainder R when the polynomial p(x) = x3 - 5x2 + 2x – 8 is divided
by (x-5)?
Answer:………………………………..
2
Part 3 [workout questions]: Solve each of the following five problems. Show your
steps neatly and briefly. [4 marks each]
1. Test the validity of the following argument by truth table method.
If you invest in agro industry, then you get rich. You didn't invest in agro industry.
Therefore, you didn't get rich.
2. Of the 240 passengers of a ship 135 like swimming, 150 like dancing, 65 like soccer
games ,80 like swimming and dancing, 40 like swimming and soccer games, 25 like
dancing and soccer games and 15 like all three activities. How many passengers:
a) Like exactly two of the three activities:
b) Like none of these activities
3
3. Solve the following equations.
a) 6 − 3(4𝑥 − 1) = 4𝑥 − 7
b) 𝑥 2 − 5𝑥 = 24
4. Solve the following inequalities.
a) −2(𝑥 + 8) + 6 ≤ 20
b) 𝑥 2 − 5𝑥 ≤ 24
4
5. Solve the following system of equations by Cramer’s rule.
{
9x + 2y = 8
7x + 2y = 4
5