Complex Numbers | DP IB Maths: AA HL Topic Questions 2021

1a

3 marks

Consider the quadratic equation $z^{2} - 8 z + 25 = 0, z \in ℂ$ .

The roots of the equation are $z_{1} = a + b i$ and $z_{2} = a - b i$ where $a, b \in ℤ .$

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

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1b

4 marks

Sketch $z_{1}, z_{2}, z_{1} + z_{2}$ and $z_{1} - z_{2}$ on the Argand diagram below, be sure to include an appropriate scale.

q1b-1-8-ib-aa-hl-complex-numbers-hard-maths_dig

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1a

2 marks

Consider the complex numbers $z_{1} = 2 + 2 i$ and $z_{2} = 2 + 2 \sqrt{3} i$ .

Sketch $z_{1}$ and $z_{2}$ on the Argand diagram below, be sure to include an appropriate scale.

q1a_1-8_complex-numbers_medium_ib-maths-aa-hl

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1b

3 marks

Find the modulus of $z_{1}$ and $z_{2}$ .

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1c

3 marks

Find the argument of $z_{1}$ and $z_{2}$ .

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1a

4 marks

Consider the complex numbers $z_{1} = \sqrt{3} + 2 i$ and $z_{2} = i - 3 \sqrt{3}$ .

Find

(i)

u = z_{1} z_{2}

(ii)

v = \frac{z_{1}}{z_{2}}

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1b

3 marks

The complex numbers $u$ and $v$ are represented by the points $A$ and $B$ respectively on an Argand diagram with origin $O$ .

Determine whether the angle made by $OA$ with the positive horizontal axis is greater than or less than the angle made by $OB$ with the positive horizontal axis. Give a reason for your answer.

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DP IB Maths: AA HL

Topic Questions

1.8 Complex Numbers