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AHL 1.12—Complex numbers – Cartesian form and Argand diag - IB Questionbank | RevisionDojo

Practice AHL 1.12—Complex numbers – Cartesian form and Argand diag with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The equation $z^3 - 3z^2 + 7z - 5 = 0$ has one real root and two complex roots.

Verify that $1-2i$ is one of the complex roots.

[3]

Write down the other complex root of the equation.

[1]

Describe how to sketch an Argand diagram showing the point representing the complex number $1-2i$ and the set of points representing the complex numbers $z$ which satisfy $|z-(3+3i)| = |z-i|$ . Your description should include the coordinates of the points and properties of the locus.

[4]

Question 2

HLPaper 1

The polynomial $x^4 + ax^3 + bx^2 - 4x + 6$ , where $a$ and $b$ are constants, is denoted by $p(x)$ . It is given that $p(x)$ is divisible by $x^2 + x - 2$ .

Find the values of $a$ and $b$ .

[5]

When $a$ and $b$ have these values, find the real roots of the equation $p(x) = 0$ .

[2]

Question 3

HLPaper 1

The complex number $w$ is defined by $w = \frac{3+6i}{2-i}$ .

Showing all your working, find the modulus of $w$ and show that the argument of $w$ is $\frac{1}{2}\pi$ .

[4]

For complex numbers $z$ satisfying $\arg(z-w) = \frac{1}{4}\pi$ , find the least possible value of $|z|$ .

[3]

For complex numbers $z$ satisfying $|z-(1+i)w| = 1$ , find the greatest possible value of $|z|$ .

[3]

Question 4

HLPaper 1

Given complex numbers are $z_1 = 3+2i$ and $z_2 = 1-i$ .

Find $z_1 z_2$ in the form $x+iy$ , where $x$ and $y$ are real.

[2]

Question 5

HLPaper 1

Find the roots of the equation $x^2 - 2x + 5 = 0$ , giving your answers in the form $x + iy$ , where $x$ and $y$ are real.

[2]

Question 6

HLPaper 1

The complex number $z$ is given by $z = \frac{5+i}{1-i}$ . Express $z$ in the form $x+iy$ , where $x$ and $y$ are real.

[2]

Question 7

HLPaper 1

The complex number $3+i$ is denoted by $w$ . Its complex conjugate is denoted by $w^*$ .

Show, on a sketch of an Argand diagram with origin $O$ , the points $P$ , $Q$ and $R$ representing the complex numbers $w$ , $w^*$ and $w + w^*$ respectively. Describe in geometrical terms the relationship between the four points $O, P, Q$ and $R$ .

[4]

Express $\frac{w}{w^*}$ in the form $x+iy$ , where $x$ and $y$ are real.

[3]

By considering the argument of $\frac{w}{w^*}$ or otherwise, prove that $\tan^{-1}\left(\frac{3}{4}\right) = 2 \tan^{-1}\left(\frac{1}{3}\right)$ .

[2]

Question 8

HLPaper 1

The complex number $w$ is defined by $w = \frac{3\sqrt{3} + 3i}{1+i\sqrt{3}}$ .

Express $w$ in the form $re^{i\theta}$ , where $r>0$ and $0 \le \theta < 2\pi$ .

[5]

Find the square roots of $w$ , giving your answers in the form $re^{i\phi}$ , where $r>0$ and $0 \le \phi < 2\pi$ .

[4]

Without using a calculator, show that $w^6$ is a real number.

[3]

Question 9

HLPaper 1

The complex number $w$ is defined by $w = 2e^{i\frac{\pi}{4}}$ . In an Argand diagram, the points $A$ , $B$ and $C$ represent the complex numbers $w$ , $w^*$ and $w^2$ respectively (where $w^*$ denotes the complex conjugate of $w$ ).

Draw the Argand diagram showing the points $A$ , B and C, and calculate the area of triangle $ABC$ .

[5]

Question 10

HLPaper 1

Find the roots of the equation $z^2 - 2iz - 2 = 0$ , giving your answers in the form $x+iy$ , where $x$ and $y$ are real.

[2]

State the modulus and argument of each root.

[3]

Showing all your working, verify that each root also satisfies the equation $z^4 = -4$ .

[3]

AHL 1.12—Complex numbers – Cartesian form and Argand diag Questionbank