i² is simply −1 and (2 + 3i) + (4 − i) adds up to 6 + 2i, so arithmetic with i is where this IB complex-number sheet starts. The key also lists the conjugate 3 − 4i and a modulus of 5 for a later pair.
Key Takeaways
i² = −1 (Q1).
(2 + 3i) + (4 − i) = 6 + 2i (Q2).
The key records 3 − 4i as a conjugate and 5 as a modulus.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.What is i²?
(a) 1(b) −1(c) i(d) −i
2.What is (2 + 3i) + (4 − i)?
(a) 6 + 2i(b) 6 + 4i(c) 8 + 2i(d) −2 + 4i
3.What is the conjugate of 3 + 4i?
(a) 3 − 4i(b) −3 − 4i(c) 4 + 3i(d) −3 + 4i
4.What is the modulus of 3 + 4i?
(a) 5(b) 25(c) 7(d) 1
5.What is (1 + i)²?
(a) 2i(b) 1 + 2i(c) 2(d) 0
6.What is i⁴?
(a) 1(b) −1(c) i(d) 0
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Simplify i³.
8.Find (2 − i)(3 + i).
9.Write the conjugate of 7 − 2i.
10.Find the modulus of −12i.
11.Solve z² = −1 for complex z.
12.Find the modulus of 1 + i.
13.Simplify (3 + 4i)(3 − 4i).
14.Find the argument of z = 1 + i, giving your answer in degrees.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Two impedances in a circuit are (3 + 4i) and (1 − 2i) ohms. Find the total impedance when they are in series.
16.In a circuit V = 2 + 2i volts and I = 1 + i amps. Find Z = V ÷ I.
17.If z₁ = 2 + 3i and z₂ = 1 − i, find z₁ × z₂.
18.A point on the complex plane is z = 3 + 4i. Find its distance from the origin.
19.Solve z² = −4 for complex z.
20.Write z = 1 + i in polar form with the argument in degrees.
Answer Key
1. b) −1
2. a) 6 + 2i
3. a) 3 − 4i
4. a) 5
5. a) 2i
6. a) 1
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
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14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is i²?
−1, option (b) in this worksheet (Q1).
What is (2 + 3i) + (4 − i)?
6 + 2i, option (a) in this worksheet (Q2).
What is the conjugate of 3 + 4i?
3 − 4i, from the answer key on this page.
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