# CIE A level -Pure Mathematics 3 :Topic : 3.9 Complex numbers :operations of Complex numbers: Exam Style Questions Paper 3

### 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

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

(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.                                  [3]

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

(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex
numbers z satisfying the inequalities $$|z-2i|\leqslant |z-1-i|\ and |z-2-i|\leqslant 2$$      [4]

(ii) Calculate the least value of arg z for points in this region.                                                           [2]

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

### 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

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

### 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