**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.

**(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.

**▶️Answer/Explanation**

(a)Multiplying by the conjugate:

\( \frac{2+3i}{1-2i} \)

\( \frac{(2+3i)(1+2i)}{(1-2i)(1+2i)} \)

\( \frac{4+7i}{5} \)

So, $r = \frac{4}{5}$ and $7 = 5\sin\theta$, from which we can solve for $\theta$.

\( r = \sqrt{\left(\frac{4}{5}\right)^2 + \left(\frac{7}{5}\right)^2} \)

\( = \sqrt{\frac{16}{25} + \frac{49}{25}} \)

\( = \sqrt{\frac{65}{25}} \)

\( = \frac{\sqrt{65}}{5} \)

\( \approx 1.61 \)

\( \sin\theta = \frac{7}{5} \)

\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\),

\( \theta = \arcsin\left(\frac{7}{5}\right) \)

Since both the real and imaginary parts are positive, $\theta$ lies in the first quadrant.

\( \theta \approx 2.09 \)

(b) Given the equation of the circle with center $3-2i$ and radius $1$:

\( |z-3+2i| = 1 \)

\( |x+iy – (3-2i)| = 1 \)

\( |x+iy – 3 + 2i| = 1 \)

\( |(x-3) + (y+2)i| = 1 \)

\( \sqrt{(x-3)^2 + (y+2)^2} = 1 \)

The center of the circle is $3-2i$, so its distance from the origin is:

\( \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \)

Therefore, the least value of $|z|$ is this distance minus the radius of the circle, which is $1$:

\( \sqrt{13} – 1 \)

Hence, the least value of $|z|$ for points on the locus represented by the circle is $\sqrt{13} – 1$.

**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.

**▶️Answer/Explanation**

(a)Square roots of the complex number $u = -3 – \sqrt{10}i$,

$a+ib$

\( (a+ib)^2 = (a^2 – b^2) + i(2ab) \)

Equating the real and imaginary parts to $-3$ and $-\sqrt{10}$ respectively, we have:

\( a^2 – b^2 = -3 \)

\( 2ab = -\sqrt{10} \)

We can eliminate one unknown by solving one of the equations for $a$ or $b$ and substituting it into the other equation. Let’s solve the second equation for $a$:

\( a = \frac{-\sqrt{10}}{2b} \)

Now, substitute this expression for $a$ into the first equation:

\( \left(\frac{-\sqrt{10}}{2b}\right)^2 – b^2 = -3 \)

\( \frac{10}{4b^2} – b^2 = -3 \)

\( \frac{10}{4b^2} = b^2 – 3 \)

\( 10 = 4b^4 – 12b^2 \)

\( 4b^4 – 12b^2 – 10 = 0 \)

\( a^4 + 3a^2 – 10 = 0 \)

\( a = \pm\sqrt{\frac{-3 \pm \sqrt{49}}{2}} \)

\( b = \pm\sqrt{\frac{3 \pm \sqrt{49}}{2}} \)

Thus, the square roots of $u$ are:

\( \pm\left(\sqrt{\frac{-3 \pm \sqrt{49}}{2}} + i\sqrt{\frac{3 \pm \sqrt{49}}{2}}\right) \)

This simplifies to:

\( \pm(\sqrt{2-\sqrt{5}} + i\sqrt{2+\sqrt{5}}) \)

(b)

1. Point representing $3+i$.

2. Circle with radius $3$ and center at $3+i$.

3. Half-line from the origin at $\frac{1}{4}\pi$ to the real axis.

4. Horizontal line $y=2$.

Shade the region where all these conditions are satisfied.

**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.**

**▶️Answer/Explanation**

(i) \( |u| = \sqrt{3^2 + 1^2} = 2 \), we have \( r = 2 \).

Now, \( \Theta = \tan^{-1}\left(\frac{\text{Im}(u)}{\text{Re}(u)}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \).

Since \( -\frac{\pi}{2} < \Theta \leq \frac{\pi}{2} \) (because \( u \) lies in the first quadrant), we have \( \Theta = \frac{\pi}{6} \).

Thus, \( u = 2 e^{i\frac{\pi}{6}} \).

For \( u^4 \), we have:

\( u^4 = (2 e^{i\frac{\pi}{6}})^4 = 16 e^{i\frac{4\pi}{6}} = 16 e^{i\frac{2\pi}{3}}. \)

So, the modulus of \( u^4 \) is \( |u^4| = |16| = 16 \), and the argument is \( \arg(u^4) = \frac{2}{3}\pi \).

(ii) To verify that \( u = \sqrt{3} + i \) is a root of the equation \( z^3 – 8z + \sqrt[8]{3} = 0 \),

\( u^3 – 8u + \sqrt[8]{3} = (\sqrt{3} + i)^3 – 8(\sqrt{3} + i) + \sqrt[8]{3} \)

\( = (2i) – 8(\sqrt{3} + i) + \sqrt[8]{3} \)

\( = -8\sqrt{3} + (2 – 8i) + \sqrt[8]{3} = 0. \)

This confirms that u is a root of the equation.

The other root is the conjugate of \( u \), which is \( \sqrt{3} – i \).

(iii)

1. Plot \( u = \sqrt{3} + i \) in the first quadrant.

2. Draw a circle with center \( u \) and radius 2.

3. Draw the line \( y = 2 \).

4. Shade the region where the circle and the line intersect. This shading covers the region where \( |z – u| \leq 2 \) and \( \text{Im}(z) \geq 2 \).

5. The intersection of the circle and line occurs at \( x = 0 \), indicating the correct region.

**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 \(|z|<|z-2i| and \frac{1}{4}\Pi<arg(z-u)<\frac{1}{2}\)**

**Answer/Explanation**

**(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

Shade the correct region

**Question**

**Question**

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

Find, showing all your working,

**(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.

**Answer/Explanation**

**(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]**

**Answer/Explanation**

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 \(|z-2-i|\leq 2 and 0\leqslant arg(z-i)\leqslant \frac{1}{4}\pi\) .**

**Answer/Explanation**

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

Shade the correct region

*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 ^{iθ}, giving the exact value of θ and the value of r correct to 3 significant figures.[5]**

**Answer/Explanation**

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.

**Answer/Explanation**

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 \(\frac{u}{w}\)

Obtain the given answer

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.

**Answer/Explanation**

**(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]

**Answer/Explanation**

Ans

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

x^{2} and x^{3}

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 x^{2} + 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 x^{2} + 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]

**Answer/Explanation**

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

Shade the appropriate region

*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. **

**Answer/Explanation**

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)

Shade the correct region

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

Obtain answer 56.3° or 0.983 radians

**Question**

**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.**

**Answer/Explanation**

**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 2x^{3} + ax^{2} + 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*

*Question*

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

**Answer/Explanation**

**Ans:(a)**

Substitute 1 + 2i in the polynomial and attempt expansions of x^{2} and x^{3}

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 x^{2} – 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]

**Answer/Explanation**

Ans

(a) Use quadratic formula and i^{2} = -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 < p^{2}

(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**

**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 \(AOB=\frac{3}{4}\pi\).

**Answer/Explanation**

**Ans:**

- Multiply numerator and denominator by 3 – i

Obtain numerator -10 +10i or denominator 10

Obtain final answer -1 + i - State or imply \(r=\sqrt{2}\)

State or imply that \(\theta=\frac{3}{4}\pi\) - State that OA and BC are parallel

State that BC = 2OA - Use angle AOB = arg u- arg v = arg\(\frac{u}{v}\)

Obtain the given answer

Alternative method for question 8(d)

Obtain tan AOB from gradients of OA nad OB and the tan(A±B) formula

Obtain the given answer

Alternative method for question 8(d)

Obtain cos AOB by using the cosine rule or a scalar product

Obtain the given answer

**Question**

**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

**Answer/Explanation**

.

**(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.

**Answer/Explanation**

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

Shade the correct region.

*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]**

**Answer/Explanation**

Ans:

** (a) **Substitute and obtain a correct equation in *x* and *y*

Use i^{2} = − 1 and equate real and imaginary parts

Obtain two correct equations, e.g. *x* + 2*y* +1 = 0 and *y* + 2*x* = 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

Shade the correct region

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

Obtain answer 0.588 radians (accept 33.7°)

**Question**

**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.

**Answer/Explanation**

**(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**

**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|.**

**Answer/Explanation**

**(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**

**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.

**Answer/Explanation**

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 \(\sqrt{3}-i\)

Forn the quadratic factor and divide cubic by quadratic

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