T1-5 T.pdf

T1-5 1.

If z1

[376 marks]

– = a + a √3i and z2 = 1 − i, where a is a real constant, express z1 and z2 in the form r cis θ, and hence find [7 marks]

an expression for

2.

( z12 ) z

6

in terms of a and i.

Given that z is the complex number x + iy and that |z | + z = 6 − 2i , find the value of x and the value of y .

[6 marks]

Given that (4 − 5i)m + 4n = 16 + 15i , where i2 = −1, find m and n if 3a. m and n are real numbers;

[3 marks]

3b. m and n are conjugate complex numbers.

[4 marks]

4a. Given that 2

[2 marks]

4b. Hence find the two square roots of

[5 marks]

any complex number z , show that 4c. For ∗ 2 2 ∗

[3 marks]

4d. Hence write down the two square roots of

[2 marks]

(x + iy) = −5 + 12i, x, y ∈ R . Show that (i) x2 − y2 = −5 ; (ii) xy = 6 .

−5 + 12i .

(z ) = (z ) .

−5 − 12i .

The graph of a polynomial function f of degree 4 is shown below.

4e. Explain why, of the four roots of the equation

[2 marks]

4f. The curve passes through the point

[5 marks]

4g. Find the two complex roots of the equation

[2 marks]

f(x) = 0 , two are real and two are complex.

(−1, −18) . Find f(x) in the form f(x) = (x − a)(x − b)(x2 + cx + d), where a, b, c, d ∈ Z .

f(x) = 0 in Cartesian form.

4h.

Draw the four roots on the complex plane (the Argand diagram).

4i. Express each of the four roots of the equation in the form

[2 marks]

[6 marks]

reiθ .

Consider the complex numbers z1 = 2√–3cis 3π2 and z2 = −1 + √–3i . 5a. (i)

Write down z1 in Cartesian form. (ii) Hence determine (z1 + z2 )∗ in Cartesian form.

[3 marks]

5b. (i)

[6 marks]

Write z2 in modulus-argument form. (ii) Hence solve the equation z 3 = z2 .

5c. Let z

= r cisθ , where r ∈ R+

(i) if z 2 = (1 + z2 )2 ; (ii) if z = − z12 .

and 0

⩽ θ < 2π . Find all possible values of r and θ ,

[6 marks]

5d.

6.

Find the smallest positive value of n for which n ( zz12 ) ∈ R+ .

[4 marks]

Let ω = cos θ + isin θ . Find, in terms of θ , the modulus and argument of (1 − ω2 )∗ .

[7 marks]

The complex numbers z1 = 2 − 2i and z2 = 1 − √–3i are represented by the points A and B respectively on an Argand diagram. Given that O is the origin, 7a.

[3 marks]

Find AB, giving your answer in the form −−−−−− a√b − √–3 , where a , b ∈ Z+ .

7b.

Calculate

[3 marks]

^ B in terms of AO π.

8a. Factorize

z3

[2 marks]

+ 1 into a linear and quadratic factor.

[9 marks]

8b. Let

γ= (i)

1+i√3 . 2

Show that

γ is one of the cube roots of −1. (ii)

Show that

γ 2 = γ − 1. (iii)

Hence find the value of

(1 − γ)6 .

9.

[6 marks]

Consider the complex numbers z = 1 + 2i and w = 2 + ai , where a∈R. Find a when (a) |w| = 2 |z|; ; (b) Re(zw) = 2 Im(zw) .

[9 marks]

10a. If z is a non-zero complex number, we define

L(z) by the equation L(z) = ln|z| + iarg(z), 0 ⩽ arg(z) < 2π. (a)

Show that when z is a positive real number,

L(z) = ln z . (b)

Use the equation to calculate

(i) L(−1) ; (ii) L(1 − i) ; (iii) L(−1 + i) . (c)

Hence show that the property

L(z1 z2 ) = L(z1 ) + L(z2 ) does not hold for all values of z1 and z2 .

10b. Let f be a function with domain

[14 marks]

R that satisfies the conditions, f(x + y) = f(x)f(y) , for all x and y and f(0) ≠ 0 . (a)

Show that

f(0) = 1. (b)

Prove that

f(x) ≠ 0 , for all x∈R. (c)

Assuming that

f ′ (x) exists for all x ∈ R , use the definition of derivative to show that f(x) satisfies the differential equation f ′ (x) = k f(x) , where k = f ′ (0) . (d)

Solve the differential equation to find an expression for

f(x) .

11. Find the values of n such that

[5 marks]

– n (1 + √3i) is a real number.

12a.

[8 marks]

Let

z = x + iy be any non-zero complex number. (i) 1 z

Express

in the form

u + iv . (ii)

If

z + 1z = k , k ∈ R , show that either y = 0 or x2 + y2 = 1. (iii)

Show that if

x2 + y2 = 1 then |k| ⩽ 2 .

[14 marks]

12b. Let

w = cos θ + isin θ . (i)

Show that

wn + w−n = 2 cos nθ , n∈Z. (ii)

Solve the equation

3w2 − w + 2 − w−1 + 3w−2 = 0, giving the roots in the form x + iy .

13a. If w = 2 + 2i , find the modulus and argument of w.

[2 marks]

[4 marks]

13b. Given

z = cos( 5π6 ) + isin( 5π6 ), find in its simplest form w4 z 6 .

14. (a)

[7 marks]

Solve the equation

z 3 = −2 + 2i, giving your answers in modulus-argument form. (b)

15.

Hence show that one of the solutions is 1 + i when written in Cartesian form.

[16 marks]

Consider ω = cos( 2π3 ) + isin( 2π3 ). (a)

Show that (i) ω3 = 1; (ii) 1 + ω + ω2 = 0

(b)

(i)

Deduce that

i(θ+

eiθ + e

(ii) θ= (c)

(i)

2π ) 3

i(θ+ 4π )

+e

3

= 0.

Illustrate this result for π 2

on an Argand diagram. Expand and simplify

F(z) = (z − 1)(z − ω)(z − ω2 ) where z is a complex number. (ii)

Solve

F(z) = 7, giving your answers in terms of ω.

[7 marks]

16. Given that

z = cos θ + isin θ show that (a) Im(z n + z1n ) = 0, n ∈ Z+ ; (b) Re( z−1 ) = 0, z ≠ −1. z+1

17. Consider the complex number

ω=

z+i , z+2

[19 marks]

where

z = x + iy and −− − i = √−1 . (a)

If

ω = i, determine z in the form z = r cis θ . (b) ω= (c)

Prove that (x2 +2x+ y 2 +y)+i(x+2y+2) (x+2) 2+ y 2

.

Hence show that when

Re(ω) = 1 the points (x, y) lie on a straight line, l1 , and write down its gradient. (d)

Given

arg(z) = arg(ω) = π4 , find |z|.

Given the complex numbers z1 = 1 + 3i and z2 = −1 − i. 18a.

Write down the exact values of

[2 marks]

|z1 | and arg(z2 ).

18b. Find the minimum value of

|z1 + αz2 |, where α ∈ R.

[5 marks]

[7 marks]

19. The complex numbers

z1 and z2 have arguments between 0 and π radians. Given that – z1 z2 = −√3 + i and z1 z2

= 2i, find the modulus and argument of

z1 and of z2 .

[7 marks]

20. Given that

z=

2−i 1+i

− 6+8i , find the values of u, u u+i

∈ R , such that Re z = Im z.

21a.

[1 mark]

Show that

∣eiθ ∣ = 1.

[2 marks]

21b. Consider the geometric series

1 + 13 eiθ + 19 e2iθ + … . Write down the common ratio, z, of the series, and show that |z| = 13 .

21c. Find an expression for the sum to infinity of this series.

21d.

22.

[8 marks]

Hence, show that

sin θ + 13 sin 2θ + 19 sin 3θ + … =

[2 marks]

9 sin θ 10−6 cos θ

.

Given that z = 2−i , z+2 z ∈ C , find z in the form a + ib .

[4 marks]

Write down the expansion of (cos θ + isin θ) 3 in the form a + ib , where a and b are in terms of sin θ and cos θ .

[2 marks]

23a.

23b.

Hence show that cos 3θ = 4cos3 θ − 3 cos θ .

[3 marks]

23c. Similarly show that

[3 marks]

23d. Hence solve the equation

[6 marks]

cos 5θ = 16cos5 θ − 20cos3 θ + 5 cos θ .

cos 5θ + cos 3θ + cos θ = 0 , where θ ∈ [− π2 , π2 ] .

By considering the solutions of the equation cos 5θ = 0 , show that − −−− 5+ √5 π cos 10 = √ 8 and state the value of

23e.

[8 marks]

cos 7π . 10

24. A geometric sequence

[17 marks]

{un }, with complex terms, is defined by un+1 = (1 + i)un and u1 = 3. (a)

Find the fourth term of the sequence, giving your answer in the form

x + yi, x, y ∈ R. (b)

Find the sum of the first 20 terms of

{un }, giving your answer in the form a × (1 + 2m ) where a ∈ C and m ∈ Z are to be determined. A second sequence {vn } is defined by vn = un un+k , k ∈ N . (c)

(i)

Show that

{vn } is a geometric sequence. (ii)

State the first term.

(iii)

Show that the common ratio is independent of k.

A third sequence {wn } is defined by wn = |un − un+1 |. (d)

(i)

Show that

{wn } is a geometric sequence. (ii)

State the geometrical significance of this result with reference to points on the complex plane.

25. Consider the complex numbers

u = 2 + 3i and v = 3 + 2i. (a) 1 u

Given that 1 v

+ =

10 , w

express w in the form

a + bi, a, b ∈ R . (b)

Find

w* and express it in the form reiθ .

[7 marks]

[6 marks]

26. A complex number z is given by

z= (a)

(b)

a+i , a−i

a ∈ R.

Determine the set of values of a such that (i)

z is real;

(ii)

z is purely imaginary.

Show that

|z| is constant for all values of a.

27a. (i) (ii)

Use the binomial theorem to expand (cos θ + i sin θ)5 .

[6 marks]

Hence use De Moivre’s theorem to prove

sin 5θ = 5cos4 θ sin θ − 10cos2 θsin3 θ + sin5 θ. (iii)

State a similar expression for

cos 5θ in terms of cos θ and sin θ.

Let z = r(cos α + i sin α), where positive argument.

27b.

α is measured in degrees, be the solution of z 5 − 1 = 0 which has the smallest

Find the value of r and the value of α .

[4 marks]

4

2

4

27c. Using (a) (ii) and your answer from (b) show that 16sin

27d.

28.

α − 20sin2 α + 5 = 0.

Hence express

[5 marks]

√a+b √c sin 72∘ in the form where d

a, b, c, d ∈ Z.

[5 marks]

Express 1 (1−i√3 )

3

[4 marks]

in the form ab where a, b ∈ Z .

29. Find, in its simplest form, the argument of

(sin θ + i(1 − cos θ))

2

[7 marks]

where

θ is an acute angle.

30. z = (1 + i√– 3)m and z2 = (1 − i)n . 1

(a)

Find the modulus and argument of

z1 and z2 in terms of m and n, respectively. (b)

Hence, find the smallest positive integers m and n such that

z1 = z2 .

[14 marks]

[7 marks]

31. Consider

w=

z z 2 +1

where z = x + iy , y ≠ 0 and z 2 + 1 ≠ 0 .

Given that Im w = 0, show that |z| = 1.

32a. (a)

z4

[12 marks]

Use de Moivre’s theorem to find the roots of the equation = 1−i .

(b) Draw these roots on an Argand diagram. (c) If z1 is the root in the first quadrant and z2 is the root in the second quadrant, find z2 in the form a + ib . z 1

32b.

(a)

[13 marks]

Expand and simplify

(x − 1)(x4 + x3 + x2 + x + 1) . (b)

Given that b is a root of the equation

z 5 − 1 = 0 which does not lie on the real axis in the Argand diagram, show that 1 + b + b2 + b3 + b4 = 0 . (c)

If

u = b + b4 and v = b2 + b3 show that (i)

u + v = uv = −1;

(ii) – u − v = √5 , given that u−v > 0 .

The complex number z = −√–3 + i . 33a. Find the modulus and argument of z , giving the argument in degrees.

[2 marks]

33b. Find the cube root of z which lies in the first quadrant of the Argand diagram, giving your answer in Cartesian form.

[2 marks]

33c. Find the smallest positive integer n for which

[2 marks]

z n is a positive real number.

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