Nanyang Technological UniversitySPMS/Division of Mathematical Sciences

2013/14 Semester 1 MH1810 Mathematics I Tutorial 1

1. Evaluate the expression and write your answer in the form a + bi, a, b ∈ R.

(a) i179 + i2013

(b)1 + 2i

3− 4i

2. For each of the following, represent the complex number on the Argand diagram. Find the modulusand the principal argument of the complex number. Hence express the complex number in its polarrepresentation.

(a) 1 +√

3i

(b) −1 +√

3i

(c) 1−√

3i

(d) −1−√

3i

3. Find the complex conjugate of each of the following complex numbers.

(a) 2i

(b) 2(c) 1 + 3i

(d) −3− 4i

(e) a complex number with modulus 2 and argument θ = π3 .

4. Sketch the regions defined by

(a) Re(z) ≥ 0(b) Im(z) < 2(c) |z| ≥ 2

(d)π

6≤ arg(z) ≤ π

3(e) |z − i| = |z − 1|(f) |z − (1 + i)| ≤ 2

5. Solve the following equation for the real numbers, x and y.(1 + i

1− i

)2

+1

x + iy= 1 + i

6. Suppose a complex number z = x + iy satisfies

|z − 1| = 12|z − i| .

Show that x and y satisfy the following equation(x− 4

3

)2

+(

y +13

)2

=89,

which represents a circle of radius√

83

centred at(

43,−13

).

7. Express each of the following complex numbers in the form reiθ, with r ≥ 0, and −π < θ ≤ π.

(a)(1 +

√−3

)2(b)

1 + i

1− i

8. Let z = a + ib, where a and b are some real numbers.

Show that

(a) z + z̄ = 2Re(z).

(b) z = z̄ ⇐⇒ z is a real number.

(c) If z is a root of ax3 + bx2 + cx+ d = 0, where a, b, c and d are real constants, then z̄ is also a rootof ax3 + bx2 + cx + d = 0.Remark More generally, we haveSuppose that p(x) = a0 + a1x + ... + anxn is a polynomial in x with real coefficients ak’s. Ifa complex number z is a solution of p(x) = 0, then the conjugate z of z is also a solution ofp(x) = 0.

9. Express each of following complex numbers in the form (i) cos α + i sinα and (ii) x + iy.

(a)(cos π

3 + i sin π3

)7

(b)1(

cos π6 − i sin π

6

)2

(c)cos π

6 + i sin π6(

− cos π3 + i sin π

3

)4

10. Without using any series expansions, prove that

(√

3 + i)n + (√

3− i)n, is real.

Find the value of this expression when n = 12.

11. Find all four distinct fourth roots of −16i.

12. Solve the following equations.

(a) z4 + 4z2 + 16 = 0

(b) z4 + 1 = 0

(c) z3 + z2 + z + 1 = 0

13. Solve the equation(

z − 4i

2i

)3

= i and represent the roots of the equation in an Argand diagram.

14. (a) Find the cube roots of −1 and show that they can be denoted by −1, λ , −λ2.

(b) Consider the factorization(x3 + 1) = (x + 1)(x− λ)(x + λ2).

By comparing the coefficients of x2, prove that λ2−λ+1 = 0. What is the value of (2−λ)(2+λ2)?(Answer: 3)

15. If α is a complex 5th root of unity with the smallest positive principal argument, determine the valueof

(1 + α4)(1 + α3)(1 + α2)(1 + α).

16. Suppose sinθ

26= 0. Prove that

12

+n∑

k=1

cos kθ =sin

[(n + 1

2

)θ]

2 sin θ2

.

(Hint: Let z = cos θ + i sin θ = eiθ. Geometric sum:n∑

k=1

zk = z + z2 + · · ·+ zn =z(zn − 1)

z − 1.)

17. (a) Use De Moivre’s Theorem and binomial expansion to show that

cos 5θ = cos5 θ − 10 cos3 θ sin2 θ + 5 cos θ sin4 θ.

(b) Express sin 5θ in terms of powers of cos θ and sin θ.

(c) Hence, obtain an expression for tan 5θ in terms of powers of tan θ.

(Answer: (b) 5c4s− 10c2s3 + s5 (c)5t− 10t3 + t5

1− 10t2 + 5t4)

18. (a) Let z = cos θ + i sin θ. Show that

(i)1z

= cos θ − i sin θ, and hence cos θ =12

(z +

1z

)and sin θ =

12i

(z − 1

z

).

(ii) zk +1zk

= 2 cos kθ and zk − 1zk

= 2i sin kθ, for k ∈ Z+.

(b) Use the results in part (a) to prove that

cos4 θ =18

(cos 4θ + 4 cos 2θ + 3) .

Hence find the integral∫

8 cos4 θ dθ.