# CIE A level -Pure Mathematics 3 :Topic : 3.9 Complex numbers :complex numbers expressed in polar form: Exam Style Questions Paper 3

### Question

(a) Showing all necessary working, express the complex number$$\frac{2+3i}{1-2i}$$in the form $$re^{i\Theta }$$where r > 0 and −0 < 1 ≤ 0. Give the values of r and 1 correct to 3 significant figures.                                    [5]

(b) On an Argand diagram sketch the locus of points representing complex numbers z satisfying the
equation| z − 3 + 2i |= 1. Find the least value of z for points on this locus, giving your answer
in an exact form. [4]

EITHER: Multiply numerator and denominator by 1 + 2i, or equivalent,or equate to x + iy, obtain two equations in x and y and solve for x or for y

Obtain quotient $$\frac{4}{5}+\frac{7}{5}$$i

Use correct method to find either r or θ

Obtain r = 1.61

Obtain θ = 2.09

OR: Find modulus or argument of 2 + 3i or of 1 –2i Use correct method to find r

Obtain r = 1.61

Use correct method to find θ

Obtain θ = 2.09

(ii) Show a circle with centre 3 – 2i

Show a circle with radius 1

Carry out a correct method for finding the least value of |z|

Obtain answer$$\sqrt{13}-1$$

### Question

(a) The complex number u is given by \ (u=-3(-2\sqrt{10})i\) Showing all necessary working and without
using a calculator, find the square roots of u. Give your answers in the form a + ib, where the numbers a and b are real and exact.

(b) On a sketch of an Argand diagram shade the region whose points represent complex numbers
z satisfying the inequalities z − 3 − i ≤ 3, arg z ≥$$\frac{1}{4}\pi$$and Im z ≥ 2, where Im z denotes the imaginary part of the complex number z.

(a) Square a + ib and equate real and imaginary parts to – 3 and$$-\sqrt[2]{10}$$ respectively

Obtain$$a^{2}-b^{2}=-3 and 2ab=-\sqrt[2]{10}$$ Eliminate one unknown and find an equation in the other

Obtain$$a^{4}+3a^{2}-10=0 ,or b^{4}-3b^{2}-10 =0$$,or horizontal 3-term equivalent

Obtain answers $$\pm (\sqrt{2-\sqrt{5i}})$$ , or exact equivalent

(b) Show point representing 3 + i in relatively correct position Show a circle with radius 3 and centre not at the origin

Show correct half line from the origin at$$\frac{1}{4}\pi$$ to the real axis

Show horizontal line y = 2

### Question

Throughout this question the use of a calculator is  not permitted.

The complex number $$\sqrt{3}$$ + i is denoted by u.

(i) Express u in the form $$re^{i\Theta }$$ , where r > 0 and −0 < 1 ≤ 0, giving the exact values of r and 1. Hence or otherwise state the exact values of the modulus and argument of  $$u^{4}$$

(ii) Verify that u is a root of the equation$$z^{3}$$− 8z +\sqrt[8]{3}\) = 0 and state the other complex root of this equation.

(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities| z − u |≤ 2 and Imz ≥ 2, where Im z denotes the imaginary part of z.

State or imply r=2

State or imply$$\Theta =\frac{1}{6}\pi$$

Use a correct method for finding the modulus or the arrangement of $$u^{4}$$

Obtain modulus 16

Obtain argument $$\frac{2}{3}\pi$$

(ii) Substitute u and carry out a correct method for finding  $$u^{3}$$

Verify u is a root of the given equation

State that the other root is $$\sqrt{3}-i$$

Alternative Method

State that the other root is $$\sqrt{3}-i$$

Verify that remainder is zero and hence that u is a root of the given equation

(iii)

Show the point representing u in a relatively correct position.

Show a circle with center u and radius 2

Show the line y=2Shade the correct region

Show that the line and circle4 intersect on x=0

Question

(a) Showing all working and without using a calculator, solve the equation

$$(1+i)z^{2}-(4+3i)z+5+i=0.$$

Give your answers in the form x + iy, where x and y are real.

(b) The complex number u is given by u = −1 − i. On a sketch of an  Argand diagram show the point representing u. Shade the region whose points represent complex numbers satisfying the inequalities $$|z|<|z-2i| and \frac{1}{4}\Pi<arg(z-u)<\frac{1}{2}$$

(a) Use quadratic formula to solve for z

Use $$i^{2}1 = −1$$ throughout

Obtain correct answer in any form Multiply numerator and denominator by 1 – i, or equivalent

Obtain final answer, e.g. 1 – i

Obtain second final answer, e.g.$$\frac{5}{2}+\frac{1}{2}i$$

(b) Show the point representing u in relatively correct position

Show the horizontal line through z = i

Show correct half-lines from u, one of gradient 1 and the other vertical

### Question

The complex number Ï is defined by $$Z\frac{9\sqrt{3}+9i}{\sqrt{3-i}}$$

(i) an expression for Ï in the form \)re^{i\Theta } \), where r > 0 and −0 < 1 ≤ 0,

(ii) the two square roots of Ï, giving your answers in the form re^{i\Theta } , where r > 0 and −0 < 1 ≤ 0.

(i)Either Multiply numerator and denominator by

Obtain correct numerator 18 +18

Obtain

Obtain modulus or argument

Obtain

OR Obtain modulus and argument of numerator or denominator, or both
moduli or both arguments

Obtain moduli and argument 18 and

or moduli 18 and 2 or arguments

Obtain $$18e^{\frac{1}{6}\pi i}\div 2e^{-\frac{1}{6}\pi i}$$ or equivalent
Divide moduli and subtract arguments

Obtain $$9e^{\frac{1}{3}\pi i}$$

(ii) State $$3e^{\frac{1}{6}\pi i}$$, following through their answer to part (i)

State $$3e^{\frac{1}{6}\pi i\pm \frac{1}{2}\pi i}$$, following through their answer to part (i)

obtain

### Question

Throughout this question the use of a calculator is not permitted.

(a) The complex numbers u and v satisfy the equations

u + 2v = 2i    and   iu + v = 3.

Solve the equations for u and v, giving both answers in the form x + iy, where x and y are real. [5]

(b) On an Argand diagram, sketch the locus representing complex numbers z satisfying $$\left | z+i \right |=1$$ and the locus representing complex numbers w satisfying $$arg\left ( w-2 \right )=\frac{3}{4}\pi$$ . Find the least value of $$\left | z-w \right |$$ for points on these loci.[5]

Ans:

8 (a) EITHER: Solve for u or for v
Obtain $$u=\frac{2i-6}{1-2i}$$ or $$v=\frac{5}{1-2i}$$, or equivalent
Either: Multiply a numerator and denominator by conjugate of denominator, or equivalent
Or: Set u or v equal to x + iy, obtain two equations by equating real and imaginary parts and solve for x or for y
OR:            Using a + ib and c +id for u and v, equate real and imaginary parts and obtain four equations in a, b, c and d
Obtain b + 2d = 2, a + 2c = 0, a + d = 0 and –b + c = 3, or equivalent
Solve for one unknown M1
Obtain final answer u = –2 –2i, or equivalent
Obtain final answer v = l + 2i, or equivalent

(b) Show a circle with centre –i
Show a circle with radius l
Show correct half line from 2 at an angle of $$\frac{3}{4}π$$ to the real axis
Use a correct method for finding the least value of the modulus
Obtain final answer $$\frac{3}{\sqrt{2}}-1$$, or equivalent, e.g. 1.12 (allow 1.1)

### Question

(a) The complex number u is given by u = 8 − 15i. Showing all necessary working, find the two
square roots of u. Give answers in the form a + ib, where the numbers a and b are real and exact.

(b) On an Argand diagram, shade the region whose points represent complex numbers satisfying
both the inequalities $$|z-2-i|\leq 2 and 0\leqslant arg(z-i)\leqslant \frac{1}{4}\pi$$ .

7(a) Square x + iy and equate real and imaginary parts to 8 and –15

Obtain $$x^{2}-y^{2}$$ and 2xy = -15
Eliminate one unknown and find a horizontal equation in the other

Obtain$$4x^{4}-32x^{2}-225=0 or 4y^{4}+32y^{2}-225=0$$,or three term equivalent

Obtain answers $$\pm \frac{1}{\sqrt{2}}(5-3i)$$or equivalent

7(b) Show a circle with centre 2 i + in a relatively correct position B1
Show a circle with radius 2 and centre not at the origin B1
Show line through i at an angle of $$\frac{1}{4} π$$ to the real axis

### Question

(a) Without using a calculator, solve the equation

3w + 2iw* = 17 + 8i,

where w* denotes the complex conjugate of w. Give your answer in the form a + bi. [4]

(b) In an Argand diagram, the loci

$$arg\left ( z-2i \right )=\frac{1}{6}\pi$$ and $$\left | z-3 \right |=\left | z-3i \right |$$

intersect at the point P. Express the complex number represented by P in the form re, giving the exact value of θ and the value of r correct to 3 significant figures.[5]

Ans:

(a) State or imply 3a +  3bi + 2i(a – bi) = 17 + 8i
Consider real and imaginary parts to obtain two linear equations in a and b
Solve two simultaneous linear equations for a or b
Obtain 7 – 2i

(b) Either    Show or imply a triangle with side 2
State at least two of the angles $$\frac{1}{4}\pi ,\frac{2}{3}\pi and \frac{1}{12}\pi$$
State or imply argument is $$\frac{1}{4}\pi$$
Use sine rule or equivalent to find r
Obtain $$6.69e^{\frac{1}{4}\pi i}$$
Or            State y = x.
State $$y=\frac{1}{\sqrt{3}}x+2$$ or $$\frac{\sqrt{3}}{2}=\frac{x}{\sqrt{x^{2}+\left ( y-2 \right )^{2}}}$$ or $$\frac{1}{2}=\frac{y-2}{\sqrt{x^{2}+\left ( y-2 \right )^{2}}}$$
State or imply argument is $$\frac{\pi }{4}$$
Solve for x or y.
Obtain $$6.69e^{\frac{1}{4}\pi i}$$

### Question

Throughout this question the use of a calculator is not permitted. The complex numbers u and w are defined by u = −1 + 7i and w = 3 + 4i.
(i) Showing all your working, find in the form x + iy, where x and y are real, the complex numbers u − 2w and $$\frac{ u}{w}$$ In an Argand diagram with origin O, the points A, B and C represent the complex numbers u, w and u − 2w respectively.
(ii) Prove that angle AOB = $$\frac{1}{4}\pi$$
(iii) State fully the geometrical relation between the line segments OB and CA.

7(i) State that u – 2w = – 7 – i
EITHER:
Multiply numerator and denominator of $$\frac{u }{w}$$
by 3 – 4i, or equivalent
Simplify the numerator to 25 + 25i or denominator to 25
Obtain final answer 1 + i )
OR:
Obtain two equations in x and y and solve for x or for y
Obtain x = 1 or y = 1
Obtain final answer 1 + i )
7(ii) Find the argument of $$\frac{u}{w}$$

7(iii) State that OB and CA are parallel
State that CA = 2OB, or equivalent

### Question

Throughout this question the use of a calculator is not permitted.

The polynomial $$z^{4}+3z^{2} 6z + 10$$ is denoted by p(z). The complex number −1 + i is denoted by u.

(i) Showing all your working, verify that u is a root of the equation p(z) = 0.

(ii) Find the other three roots of the equation p(z) = 0.

(i) Substitute z = −1 +i  and attempt expansions of the $$z^{2} and z^{4}$$

Use$$i^{2}$$ = − 1 at least once Complete the verification correctly

(ii) State second root z =  −1 -i

Carry out a complete method for finding a quadratic factor with zeros -1+i and -1-i

Obtain$$z^{2}+2z+2$$ , or equivalent

Attempt division of p(z) by $$z^{2}+2z+2$$ and reach a partial quotient $$z^{2}+kz$$

Obtain quadratic factor $$z^{2}-2z+5$$

Solve 3-term quadratic and use$$i^{2}=-1$$

Obtain roots 1 + 2i and 1 – 2i

### Question

(a) Verify that $$-1+\sqrt{5}i$$ is a root of the equation $$2x^{3}+x^{2}+6x-18=0$$              [3]

(b) Find the other roots of this equation.                                                                                              [4]

Ans

(a) Substitute $$-1+\sqrt{5}i$$  in the equation and attempt expansions of
x2 and x3

Use i2 = –1 correctly at least once M1 1 – 5 or 4 + 10 see
Complete the verification correctly

(b) State second root $$-1-\sqrt{5}i$$

Carry out a complete method for finding a quadratic factor with
zeros $$-1+\sqrt{5}i \ and \ -1-\sqrt{5}i$$

Obtain x2 + 2x + 6
Obtain root $$x=\frac{3}{2}$$

Alternative method for question(b)

State second root $$-1-\sqrt{5}i$$

$$(x+1-\sqrt{5}i)(x+1+\sqrt{5}i)(2x+a)=2x^{3}+x^{2}+6x-18$$

$$(1-\sqrt{5}i)(1+\sqrt{5}i)a=-18$$

$$6a=-18\ a=-3 \ leading \ to \ x=\frac{3}{2}$$

(b) Alternative method for question (b)

State second root $$-1-\sqrt{5}i$$

POR = 6 SOR = – 2
Obtain x2 + 2x + 6

Obtain root $$x=\frac{3}{2}$$

Alternative method for question (b)

State second root $$-1-\sqrt{5}i$$

$$POR(-1-\sqrt{5}i)(-1+\sqrt{5}i)a=9$$

Obtain root $$x=\frac{3}{2}$$

Alternative method for question (b)

State second root $$-1-\sqrt{5}i$$

$$SOR\left ( -1-\sqrt{5}i \right )+\left ( -1+\sqrt{5}i \right )+a=-\frac{1}{2}$$

Obtain root $$x=\frac{3}{2}$$

### Question

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities | z | ≥ 2 and | z − 1 + i | ≤ 1.                                                                                  [4]

Show a circle with centre the origin and radius 2
Show the point representing 1 – i
Show a circle with centre 1 – i and radius 1

### Question

(a) The complex number u is defined by $$u=\frac{3i}{a+2i}$$, where a is real.
(i) Express u in the Cartesian form x + iy, where x and y are in terms of a.

(ii) Find the exact value of a for which arg u* $$=\frac{1}{3}\pi$$.

(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex
numbers z satisfying the inequalities $$|z-2i|\leqslant |z-1-i|\ and |z-2-i|\leqslant 2$$

(ii) Calculate the least value of arg z for points in this region.

Ans

(a) (i) Multiply numerator and denominator by a – 2i, or equivalent
Use i2 = –1 at least once A1
Obtain answer $$\frac{6}{a^{2}+4}+\frac{3ai}{a^{2}+4}$$

(a) (ii) Either state that arg $$u=-\frac{1}{3}\pi$$ or express u* in terms of a (FT on u)

Use correct method to form an equation in a
Obtain answer $$a=-2\sqrt{3}$$

(b) (i) Show the perpendicular bisector of points representing 2i and 1 + i
Show the point representing 2 + i
Show a circle with radius 2 and centre 2 + i
(FT on the position of the point for 2 + i)

(b) (ii) State or imply the critical point 2+ 3i

### Question

(a) The complex numbers v and w satisfy the equations
v + iw = 5     and       (1+2i)v-w = 3i.
Solve the equations for v and w, giving your answers in the form x+iy, where x and y are real.
(b) (i) On an Argand diagram, sketch the locus of points representing complex numbers z satisfying |z-2-3i|=1.
(ii) Calculate the least value of arg z for points on this locus.

Ans:

(a) Solve for v or w
Use $$i^2=-1$$
Obtain $$v=-\frac{2i}{1+i}$$ or $$w=\frac{5+7i}{-1+i}$$
Multiply numerator and denominator by the conjugate of the denominator
Obtain v = -1 – i
Obtain w = 1 – 6i
(b) (i) Show a circle with centre 2 + 3i
Show a circle with radius 1 and centre not at the origin
(ii) Carry out a complete method for finding the least value of arg z
Obtain answer $$40.2^o$$ or 0.702 radians

The complex number 1 + 2i is denoted by u. The polynomial 2x3 + ax2 + 4x + b, where a and b are real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.

### (a) Question

Find the values of a and b.

### (b) Question

State a second complex root of this equation.

### (c) Question

Find the real factors of p(x).

### (d) (i) Question

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − u| ≤ $$\sqrt{5}$$ and arg z ≤ $$\frac{1}{4}$$π.

### (ii) Question

Find the least value of Im z for points in the shaded region. Give your answer in an exact form.

Ans:(a)

Substitute 1 + 2i in the polynomial and attempt expansions of x2 and x3

Equate real and/or imaginary parts to zero

Obtain a = – 1

Obtain b = 15

Ans:(b)

State second root 1 – 2i

Ans:(c)

State the quadratic factor x2 – 2x + 5

State the linear factor 2x +3

Ans:d(i)

Show a circle with centre 1 + 2i

Show circle passing through the origin

Show the half line y = x in the first quadrant (accept chord of circle)

Shade the correct region on a correct diagram

Ans:d(ii)

State answer $$2 – \sqrt{5}$$

### Question

(a) Solve the equation z2 − 2piz − q = 0, where p and q are real constants.                                                          [2]

In an Argand diagram with origin O, the roots of this equation are represented by the distinct points
A and B.
(b) Given that A and B lie on the imaginary axis, find a relation between p and q.                                              [2]

(c) Given instead that triangle OAB is equilateral, express q in terms of p.                                                           [3]

Ans

(a) Use quadratic formula and i2 =  -1

Obtain answers $$pi+\sqrt{q-p^{2}}\ and\ pi – \sqrt{q-p^{2}}$$

(b) State or imply that the discriminant must be negative

State condition  q < p2

(c) Carry out a correct method for finding a relation, e.g. use the fact that the
argument of one of the roots is (±) 60°

State a correct relation in any form, e.g  $$\frac{p}{\sqrt{q-p^{2}}}=(\pm )\sqrt{3}$$

Simplify to  $$q=\frac{4}{3}p^{2}$$

Alternative method for Question 5(c)
Carry out a correct method for finding a relation, e.g. use the fact that the
sides have equal length

State a correct relation in any form, e.g  $$4(q-p^{2})=p^{2}+q-p^{2}$$

Simplify to $$q=\frac{4}{3}p^{2}$$

### Question

The complex numbers u and v are defined by u = -4 + 2i and v = 3 + i.
(a) Find $$\frac{u}{v}$$ in the form x + iy, where x and y are real.
(b) Hence express $$\frac{u}{v}$$ in the form $$re^{i\theta}$$, where r and $$\theta$$ are exact.
In an Argand diagram. with origin O, the points A, B and C represents the complex numbers u, v and 2u+v respectively.
(c) State fully the geometrical relationship between OA and BC.
(d) Prove that angle $$AOB=\frac{3}{4}\pi$$.

Ans:

1. Multiply numerator and denominator by 3 – i
Obtain numerator -10 +10i or denominator 10
Obtain final answer -1 + i
2. State or imply $$r=\sqrt{2}$$
State or imply that $$\theta=\frac{3}{4}\pi$$
3. State that OA and BC are parallel
State that BC = 2OA
4. Use angle AOB = arg u- arg v = arg$$\frac{u}{v}$$
Alternative method for question 8(d)
Obtain tan AOB from gradients of OA nad OB and the tan(A±B) formula
Alternative method for question 8(d)
Obtain cos AOB by using the cosine rule or a scalar product

### Question

(a) Showing all your working and without the use of a calculator, find the square roots of the complex  number $$7 -(6\sqrt{2})i$$ Give your answers in the form x + iy, where x and y are real and exact.

(b) (i) On an Argand diagram, sketch the loci of points representing complex numbers w and z such that |w − 1 − 2i |= 1 and arg
(z − 1) =$$\frac{3}{4}\pi$$

(ii) Calculate the least value of| w − z |for points on these loci

.

(a) Square x+ iy and equate real and imaginary parts to 7 and $$-6\sqrt{2}$$ respectively

Obtain equations$$9 x^{2}-y^{2}=7$$ and $$2xy=-6\sqrt{2}$$

Eliminate one variable and find an equation in the other

Obtain$$x^{4}-7x^{2}-18=0 or y^{4}+7y^{2}-18=0$$,or 3-term equivalent

Obtain answers$$± (3-i\sqrt{2})$$

(b) (i) Show point representing 1 + 2i
Show circle with radius 1 and centre 1 + 2i
Show a half line from the point representing 1
Show line making the correct angle with the real axis

(ii) State or imply the relevance of the perpendicular from 1 + 2i to the line

Obtain answer$$\sqrt{2}-1$$ (or 0.414)

### Question

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z + 2 − 3i| ≤ 2 and arg z ≤ $$\frac{3}{4}$$ π.

Ans:

Show a circle with centre – 2 + 3i

Show a circle of radius 2 and centre not at the origin.

Show correct half line from the origin

### Question

(a) Find the complex number z satisfying the equation z* + 1 = 2iz, where z* denotes the complex conjugate of z. Give your answer in the form x + iy, where x and y are real.[5]

(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities $$\left | z+1-3i \right |\leqslant 1$$ and Im $$z\geqslant 3$$ , where Im z denotes the imaginary part of z.[4]

(ii) Determine the difference between the greatest and least values of arg z for points lying in this region.[2]

Ans:

(a) Substitute and obtain a correct equation in x and y
Use i2 = − 1 and equate real and imaginary parts
Obtain two correct equations, e.g. x + 2y +1 = 0 and y + 2x = 0
Solve for x or for y
Obtain answer $$z=\frac{1}{3}-\frac{2}{3}$$ i

(b) (i) Show a circle with centre -1 + 3 i
Show a circle with radius 1
Show the line Im z = 3

(ii) Carry out a complete method to calculate the relevant angle

### Question

The complex number $$1+(\sqrt{2})i$$ is denoted by u. The polynomial$$x^{4}$$+ $$x^{2}$$+ 2x + 6 is denoted by p(x).

(i) Showing your working, verify that u is a root of the equation p(x) = 0, and write down a second complex root of the equation.
(ii) Find the other two roots of the equation p(x) = 0.

(i) EITHER Substitute x = 1 + √2 i and attempt the expansions of the$$x^{2}$$
and$$x^{4}$$ terms

Use$$i^{2}$$= –1 correctly at least once  Complete the verification
State second root 1 – √2 i
OR 1 State second root 1 –√2i
Carry out a complete method for finding a quadratic factor with zeros 1 ± √2 i
Obtain$$x^{2}$$ – 2x + 3, or equivalent

Show that the division of p(x) by$$x^{2}$$– 2x + 3 gives zero remainder and

complete the verification
OR 2 Substitute x = 1 + √2 i and use correct method to express$$x^{2}$$ and$$x^{4}$$ in polar form

Obtain$$x^{2}$$ and$$x^{4}$$ in any correct polar form (allow decimals here) Complete an exact verification  State second root 1 – √2 i, or its polar equivalent (allow decimals here)

(ii) Carry out a complete method for finding a quadratic factor with zeros 1 ± √2 i
Obtain $$x^{2}$$– 2x + 3, or equivalent

Attempt division of p(x) by $$x^{2}$$ – 2x + 3 reaching a partial quotient$$x^{2}$$+ kx, or equivalent
Obtain quadratic factor $${x^{2}$$– 2x + 2

Find the zeros of the second quadratic factor, using $$i^{2}$$ = –1
Obtain roots –1 + i and –1 –i  [The second  is earned if inspection reaches an unknown factor$$x^{2}$$+ Bx + C and an

equation in B and/or C, or an unknown factor $$Ax^{2}$$ + Bx + (6/3) and an equation in A and/or B]
[If part (i) is attempted by the OR 1 method, then an attempt at part (ii) which uses or quotes relevant working or results obtained in part (i) should be marked using the scheme for part (ii)]

### Question

The complex number u is defined by u $$\frac{(1+2i)^{2}}{2+i}$$

(i) Without using a calculator and showing your working, express u in the form x + iy, where x and
y are real.
(ii) Sketch an Argand diagram showing the locus of the complex number ß such that |ß − u| = |u|.

(i) Either Expand (1 + 2i)^{2} to obtain –3 + 4i or unsimplified equivalent
Multiply numerator and denominator by 2 – i
Obtain correct numerator –2 + 11i or correct denominator 5

Obtain$$-\frac{2}{5}+\frac{11}{5}i$$ or equivalent

Or Expand (1 + 2i)^{2} to obtain –3 + 4i or unsimplified equivalent
Obtain two equations in x and y and solve for x or y

Obtain final answer $$x=-\frac{2}{5}$$

Obtain final answer $$y=\frac{11}{5}$$

(ii) Draw a circle
Show centre at relatively correct position, following their u  Draw circle passing through the origin