Jimma University
College of Natural Science
Department of Mathematics
Math1011 (Mathematics for Natural Science) Worksheet II
Chapter 2: The Real and Complex Number Systems
1. Prove the following by using the principle of mathematical induction for all 𝑛 ∈ 𝑁:
a. 1 + 3 + 32 + ⋯ + 3𝑛−1 =
1
3𝑛 −1
2
.
1
1
2𝑛
b. 1 + (1+2) + (1+2+3) + ⋯ + (1+2+3+⋯+𝑛) = 𝑛+1.
c. 1 + 2 + 3 + ⋯ + 𝑛 < 8(2𝑛 + 1)2 .
d. 102𝑛−1 + 1 is divisible by 11.
e. 𝑥 2𝑛 − 𝑦 2𝑛 is divisible by 𝑥 + 𝑦.
f. 32𝑛+2 − 8𝑛 − 9 is divisible by 8.
2. Prove by mathematical induction that 6𝑛+2 + 72𝑛+1 is divisible by 43 for each positive
integer 𝑛.
𝑛(𝑛+1) 2
3. Prove by mathematical induction that the sum of 𝑛 cubes is (
2
) .
4. The LCM and GCF of two numbers are 600 and 20 respectively. If one of the numbers is
120, then find the other number.
5. Find:
a. GCF(35, 49, 84).
b. GCF of 2 × 33 × 52 and 23 × 3 × 52 .
c. LCM of 2 × 33 × 52 and 23 × 3 × 52 .
d. an odd natural number 𝑥 if LCM(𝑥, 40) = 1400.
6. When will LCM of two numbers be equal to their product?
7. Mr X has three classes. Each class has 36, 48 and 84 students respectively. Mr X wants to
divide each class into groups so that every group in every class has the same number of
students and there are no students left over. What is the maximum number of students he
can put into each group?
8. In each of the following, ∀𝑛 ∈ ℕ, determine upper bounds, lub(sup), lower bounds and
glb(inf)
a. {1 +
(−1)𝑛
𝑛
}
d. {−𝑛2 − 2𝑛 + 1}
b. {𝑛2 − 2𝑛 + 1}
1
e. {(−1)𝑛 (1 − 𝑛)}
c. [0, ∞)
𝑛
d. {𝑛+1}
9. Which of the following statements are true and which of them are false? For false
statements, give a counter example.
a) The sum of any two rational numbers is rational
b) The sum of any two irrational numbers is irrational
c) The product of any two rational numbers is rational
d) The product of any two irrational numbers is irrational
10. Find the complex number z that satisfies the equation: 4𝑧 − 3𝑧 =
1−18𝑖
2−𝑖
.
11. Find the two solutions of (𝑧 − 𝑖)(𝑧 − 𝑖) = 6𝑧 − 22𝑖.
12. If 𝑧1 = −𝑖 is the solution of a quadratic equation 𝑧 2 − 2𝑧 + 1 − 𝑖, then find the other
solution of the equation.
13. Given that, 𝑤 ∈ ℂ. Show that|𝑧 + 𝑤|2 − |𝑧 − 𝑤|2 = 4Re(z)Re(w).
14. Show that there is no complex number 𝑧 that satisfies the equation |𝑧| − 𝑧 = 𝑖.
15. Find an argument and principal argument of the following complex numbers:
a) 𝑧 =
−1+𝑖√3
b. 𝑧 = (
2
−1+𝑖√3
2
5
c. 𝑧 = (1 − 𝑖)1/3
)
d. √𝑖
16. If 𝑧1 = 1 + 𝑖 and 𝑧2 = 2 + 2𝑖, then find the principal argument of the following:
a) 𝑧1 + 𝑧2
b) 𝑧1 − 𝑧2
d) (𝑧1 /𝑧2 )
c) 𝑧1 . 𝑧2
17. Write the following complex numbers in polar form:
6−2𝑖
a) (4 + 4𝑖)(5 − 3𝑖)
c) (−1 − 6𝑖)3
b) (2+4𝑖)
18. Express the following in rectangular form( i.e., z = x + iy form):
4𝜋
4𝜋
−𝜋
a) 2 [cos ( 3 ) + isin( 3 )]
b)
−𝜋
c) √2[cos ( 3 ) + isin ( 3 )]
7𝜋
c) cos(5𝜋) + isin(5𝜋)
7𝜋
d) √2[cos ( 3 ) + isin ( 3 )]
19. Express the following complex numbers using Euler’s Formula:
a) −1 + 𝑖
b) (−√3 + 𝑖)
5
c) −3𝑖
20. Compute:
a) cube roots of 𝑧 = −1 − 𝑖.
b) fifth roots of 𝑧 = −8
21. Solve the following equations:
a) 𝑧 3 − 27 = 0
b) 𝑧 2 − (1 − 𝑖)𝑧 + 1 = 0
d) −2 − 4𝑖
e)
5𝜋
5𝜋
)+isin( )]
6
6
𝜋
𝜋
12[cos( )+isin( )]
6
6
3 [cos(