Level 1 — RecognitionComplex Numbers

Complex Numbers

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Chapter: 3.5 Complex Numbers Level: 1 — Recognition (MCQ, Matching, True/False with justification) Time limit: 20 minutes Total marks: 30

Section A — Multiple Choice (1 mark each) — 10 marks

Q1. The value of $i^{15}$ is: (a) $1$ (b) $-1$ (c) $i$ (d) $-i$

Q2. The real and imaginary parts of $z = 4 - 7i$ are: (a) $4,\ 7$ (b) $4,\ -7$ (c) $-7,\ 4$ (d) $7,\ 4$

Q3. The modulus of $z = 3 + 4i$ is: (a) $5$ (b) $7$ (c) $\sqrt{7}$ (d) $25$

Q4. The complex conjugate of $z = -2 + 5i$ is: (a) $2 - 5i$ (b) $-2 - 5i$ (c) $2 + 5i$ (d) $-5 + 2i$

Q5. Euler's formula states that $e^{i\theta}$ equals: (a) $\cos\theta - i\sin\theta$ (b) $\sin\theta + i\cos\theta$ (c) $\cos\theta + i\sin\theta$ (d) $r(\cos\theta + i\sin\theta)$

Q6. In the Argand plane, the number $z = 0 - 3i$ lies on: (a) the positive real axis (b) the negative imaginary axis (c) the positive imaginary axis (d) the origin

Q7. De Moivre's theorem gives $(\cos\theta + i\sin\theta)^n$ equal to: (a) $\cos(n\theta) + i\sin(n\theta)$ (b) $n\cos\theta + i\,n\sin\theta$ (c) $\cos\theta^n + i\sin\theta^n$ (d) $\cos(n\theta) - i\sin(n\theta)$

Q8. The number of distinct $n$ th roots of a non-zero complex number is: (a) $1$ (b) $2$ (c) $n-1$ (d) $n$

Q9. The product $(2 + 3i)(2 - 3i)$ equals: (a) $4 - 9i$ (b) $13$ (c) $-5$ (d) $4 + 9i$

Q10. The polar (exponential) form of a complex number with modulus $r$ and argument $\theta$ is: (a) $r + e^{i\theta}$ (b) $r\,e^{i\theta}$ (c) $e^{r\theta}$ (d) $r\,e^{\theta}$

Section B — Matching (1 mark each) — 5 marks

Q11. Match each expression in Column X with its value in Column Y.

Column X		Column Y
(i) $i^2$		(P) $i$
(ii) $i^3$		(Q) $1$
(iii) $i^4$		(R) $-1$
(iv) $i^{21}$		(S) $-i$
(v) $\dfrac{1}{i}$		(T) $-i$

Write the matches, e.g. (i)–(?).

Section C — True/False WITH justification (2 marks each: 1 verdict + 1 reason) — 15 marks

Q12. The sum of a complex number and its conjugate is always real.

Q13. $|z_1 z_2| = |z_1| + |z_2|$ for all complex numbers.

Q14. The three cube roots of unity are equally spaced on the unit circle at angles $120°$ apart.

Q15. The equation $x^2 + 1 = 0$ has no solutions in the complex number system.

Q16. For $z = 1 + i$ , the argument is $\arg(z) = \dfrac{\pi}{4}$ .

Q17. Dividing complex numbers is carried out by multiplying numerator and denominator by the conjugate of the denominator.

Q18. The sum of all $n$ th roots of unity is equal to $1$ .

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (d) $-i$ . Powers of $i$ cycle with period 4. $15 = 4(3)+3$ , so $i^{15}=i^{3}=-i$ . (Reduce exponent mod 4.)

Q2 — (b) $4,\ -7$ . For $a+bi$ , real part $a=4$ , imaginary part $b=-7$ (the coefficient of $i$ , including its sign).

Q3 — (a) $5$ . $|3+4i|=\sqrt{3^2+4^2}=\sqrt{25}=5$ .

Q4 — (b) $-2 - 5i$ . Conjugate flips the sign of the imaginary part only.

Q5 — (c) $\cos\theta + i\sin\theta$ . Direct statement of Euler's formula.

Q6 — (b) negative imaginary axis. Real part $=0$ , imaginary part $=-3<0$ , so it sits on the imaginary axis below the origin.

Q7 — (a) $\cos(n\theta)+i\sin(n\theta)$ . Statement of De Moivre's theorem.

Q8 — (d) $n$ . A non-zero complex number has exactly $n$ distinct $n$ th roots.

Q9 — (b) $13$ . $(2+3i)(2-3i)=2^2+3^2=4+9=13$ (product of conjugates $=|z|^2$ ).

Q10 — (b) $r\,e^{i\theta}$ . Exponential form of a complex number.

Section B

Q11 (1 mark each):

(i) $i^2 = -1$ → (R)
(ii) $i^3 = -i$ → (S) (or T, both give $-i$ ; accept either)
(iii) $i^4 = 1$ → (Q)
(iv) $i^{21}=i^{21\bmod 4}=i^1=i$ → (P)
(v) $\dfrac{1}{i}=\dfrac{1}{i}\cdot\dfrac{-i}{-i}=\dfrac{-i}{1}=-i$ → (T) (or S)

Award full 5 marks if all values correct; accept the S/T ambiguity for $-i$ .

Section C (1 mark verdict + 1 mark justification)

Q12 — TRUE. If $z=a+bi$ , then $z+\bar z = (a+bi)+(a-bi)=2a$ , which is real (imaginary parts cancel).

Q13 — FALSE. The correct modulus rule is $|z_1 z_2| = |z_1|\,|z_2|$ (product, not sum). Counter-example: $z_1=z_2=i$ gives $|z_1z_2|=|-1|=1$ but $|z_1|+|z_2|=2$ .

Q14 — TRUE. Cube roots of unity are $e^{i\cdot 0},\,e^{i2\pi/3},\,e^{i4\pi/3}$ , i.e. at $0°,120°,240°$ — equally spaced $120°$ apart on the unit circle.

Q15 — FALSE. $x^2+1=0\Rightarrow x^2=-1\Rightarrow x=\pm i$ ; complex numbers were introduced precisely to solve this. Two solutions $\pm i$ exist.

Q16 — TRUE. $z=1+i$ lies in the first quadrant; $\arg(z)=\arctan(1/1)=\pi/4$ .

Q17 — TRUE. Multiplying by the conjugate of the denominator makes the denominator real ( $z\bar z=|z|^2$ ), enabling division into $a+bi$ form.

Q18 — FALSE. The sum of all $n$ th roots of unity is $0$ (for $n\ge 2$ ), since they are roots of $x^n-1=0$ whose coefficient of $x^{n-1}$ is $0$ . (Only for $n=1$ is the sum $1$ .)

[
  {"claim":"i^15 = -i","code":"result = (I**15 == -I)"},
  {"claim":"|3+4i| = 5","code":"result = (Abs(3+4*I) == 5)"},
  {"claim":"(2+3i)(2-3i) = 13","code":"result = (expand((2+3*I)*(2-3*I)) == 13)"},
  {"claim":"sum of cube roots of unity is 0","code":"roots=[exp(2*pi*I*k/3) for k in range(3)]; result = (simplify(sum(roots))==0)"},
  {"claim":"1/i = -i","code":"result = (simplify(1/I) == -I)"},
  {"claim":"arg(1+i) = pi/4","code":"result = (arg(1+I) == pi/4)"}
]