An MX2 AT1 on complex numbers (1 Viewer)

Luca26 · Dec 13, 2025

Below are all the questions in an actual MX2 AT1 on complex numbers, which was worth 40 marks.

I'm interested in what you think is a fair amount of time for this test (because I thought that the given time was unfair).

Question 1 (6 marks)

Consider the quadratic equation $(4 - 24i)z^2 - (16 + 12i)z - 9 = 0$

(a) Show that the discriminant of this equation is $256 - 480i$
(b) Hence, show that $\sqrt{16-30i} = 5 - 3i$
(c) Hence, show that the solutions of the equation are

$z=-\frac12 \quad \text{and} \quad z =\frac{9}{74}+\frac{27}{37}i$

Question 2 (7 marks)

(a) Using De Moivre's Theorem, show that $\cos{4\theta} = 8\cos^4\theta - 8\cos^2\theta + 1$
(b) Using part (a) and the substitution $x = \cos\theta$ , show that the equation $16x^4 - 16x^2 + 1 = 0$ has roots $\pm\cos{\frac{\pi}{12}}$ and $\pm\cos{\frac{5\pi}{12}}$
(c) Hence, or otherwise, prove that $\cos{\frac{5\pi}{12}} = \frac{\sqrt{2-\sqrt3}}{2}$

Question 3 (4 marks)

(a) Express $2 + 2\sqrt3 i$ in modulus-argument form
(b) Hence, or otherwise, find the four fourth roots of $2 + 2\sqrt3 i$

Question 4 (8 marks)

(a) If $z = x + iy$ and $|z - 1 + i| = |z + 4 - 4i|$ , show that $y = x + 3$
(b) Hence, or otherwise, shade the region that satisfies both $|z - 1 + i| > |z + 4 - 4i|$ and $|z - 4 - 2i| < 5$
(c) If $\theta$ represents Arg( $z$ ), find all possible values of $\theta$ correct to 2 decimal places and represent your answer as an inequality

Question 5 (5 marks)
Let $w$ be the non-real fifth root of unity where $w = e^{\frac{2\pi i}{5}}$

(a) Show that $w^4 + w^3 + w^2 + w + 1 = 0$
(b) Explain why $w^{5-k} = \overline{w}^k$ where $k = 1,\, 2,\, 3,$ and $4$
(c) Using (a) and (b), show that $x = \text{Re}\left(2e^{\frac{2\pi i}{5}}\right)$ is a root of $x^2 + x - 1 = 0$

Question 6 (5 marks)
Consider the polynomial $P(x) = x^7 + 4x^4 - 3x^3 + 8x^2 - 2x + 4$ over the complex numbers

(a) Show that $x = i$ is a root of multiplicity 2, that is, is a double root
(b) Hence, or otherwise, express $P(x)$ as a product of linear factors

Question 7 (5 marks)
Let points $A$ and $B$ represent the complex numbers $z$ and $zw$ , respectively}

(a) Draw an Argand diagram that shows points $A$ and $B$ if $w=e^{\frac{i\pi}{3}}$ and $z = re^{i\theta}$ where $0 < \theta < \pi$ and $r > 0$
(b) If $O$ is the origin, explain why $OAB$ is an equilateral triangle
(c) Write the complex number $z - zw$ in exponential form in terms of $r$ and $\theta$

I note that question 1(b) shouldn't be a "hence" and would be better expressed as something like

Show that $z = \pm(5 - 3i)$ are the solutions of $z^2 = 16 - 30i$

and that question 1(c) was probably meant to be "Hence, using (a) and (b), show that ..."

Also, any thoughts on the most time-efficient approach to these questions?

Joshua Presley · Dec 13, 2025

an hour is good but i think this should probably be 70 mins

f7eeting · Dec 13, 2025

maybe around an hour? maybe ill attempt this and see how long it takes me to get through it

idkkdi · Dec 13, 2025

Luca26 said:
Below are all the questions in an actual MX2 AT1 on complex numbers, which was worth 40 marks.

I'm interested in what you think is a fair amount of time for this test (because I thought that the given time was unfair).

Question 1 (6 marks)

Consider the quadratic equation $(4 - 24i)z^2 - (16 + 12i)z - 9 = 0$

(a) Show that the discriminant of this equation is $256 - 480i$

(b) Hence, show that $\sqrt{16-30i} = 5 - 3i$

(c) Hence, show that the solutions of the equation are

$z=-\frac12 \quad \text{and} \quad z =\frac{9}{74}+\frac{27}{37}i$

Question 2 (7 marks)

(a) Using De Moivre's Theorem, show that $\cos{4\theta} = 8\cos^4\theta - 8\cos^2\theta + 1$

(b) Using part (a) and the substitution $x = \cos\theta$ , show that the equation $16x^4 - 16x^2 + 1 = 0$ has roots $\pm\cos{\frac{\pi}{12}}$ and $\pm\cos{\frac{5\pi}{12}}$

(c) Hence, or otherwise, prove that $\cos{\frac{5\pi}{12}} = \frac{\sqrt{2-\sqrt3}}{2}$

Question 3 (4 marks)

(a) Express $2 + 2\sqrt3 i$ in modulus-argument form

(b) Hence, or otherwise, find the four fourth roots of $2 + 2\sqrt3 i$

Question 4 (8 marks)

(a) If $z = x + iy$ and $|z - 1 + i| = |z + 4 - 4i|$ , show that $y = x + 3$

(b) Hence, or otherwise, shade the region that satisfies both $|z - 1 + i| > |z + 4 - 4i|$ and $|z - 4 - 2i| < 5$

(c) If $\theta$ represents Arg( $z$ ), find all possible values of $\theta$ correct to 2 decimal places and represent your answer as an inequality

Question 5 (5 marks)
Let $w$ be the non-real fifth root of unity where $w = e^{\frac{2\pi i}{5}}$

(a) Show that $w^4 + w^3 + w^2 + w + 1 = 0$

(b) Explain why $w^{5-k} = \overline{w}^k$ where $k = 1,\, 2,\, 3,$ and $4$

(c) Using (a) and (b), show that $x = \text{Re}\left(2e^{\frac{2\pi i}{5}}\right)$ is a root of $x^2 + x - 1 = 0$

Question 6 (5 marks)
Consider the polynomial $P(x) = x^7 + 4x^4 - 3x^3 + 8x^2 - 2x + 4$ over the complex numbers

(a) Show that $x = i$ is a root of multiplicity 2, that is, is a double root

(b) Hence, or otherwise, express $P(x)$ as a product of linear factors

Question 7 (5 marks)
Let points $A$ and $B$ represent the complex numbers $z$ and $zw$ , respectively}

(a) Draw an Argand diagram that shows points $A$ and $B$ if $w=e^{\frac{i\pi}{3}}$ and $z = re^{i\theta}$ where $0 < \theta < \pi$ and $r > 0$

(b) If $O$ is the origin, explain why $OAB$ is an equilateral triangle

(c) Write the complex number $z - zw$ in exponential form in terms of $r$ and $\theta$

I note that question 1(b) shouldn't be a "hence" and would be better expressed as something like

Show that $z = \pm(5 - 3i)$ are the solutions of $z^2 = 16 - 30i$

and that question 1(c) was probably meant to be "Hence, using (a) and (b), show that ..."

Also, any thoughts on the most time-efficient approach to these questions?

these qs look like theyre straight out of cambridge

WeiWeiMan · Dec 13, 2025

Luca26 said:
Below are all the questions in an actual MX2 AT1 on complex numbers, which was worth 40 marks.

I'm interested in what you think is a fair amount of time for this test (because I thought that the given time was unfair).

Question 1 (6 marks)

Consider the quadratic equation $(4 - 24i)z^2 - (16 + 12i)z - 9 = 0$

(a) Show that the discriminant of this equation is $256 - 480i$

(b) Hence, show that $\sqrt{16-30i} = 5 - 3i$

(c) Hence, show that the solutions of the equation are

$z=-\frac12 \quad \text{and} \quad z =\frac{9}{74}+\frac{27}{37}i$

Question 2 (7 marks)

(a) Using De Moivre's Theorem, show that $\cos{4\theta} = 8\cos^4\theta - 8\cos^2\theta + 1$

(b) Using part (a) and the substitution $x = \cos\theta$ , show that the equation $16x^4 - 16x^2 + 1 = 0$ has roots $\pm\cos{\frac{\pi}{12}}$ and $\pm\cos{\frac{5\pi}{12}}$

(c) Hence, or otherwise, prove that $\cos{\frac{5\pi}{12}} = \frac{\sqrt{2-\sqrt3}}{2}$

Question 3 (4 marks)

(a) Express $2 + 2\sqrt3 i$ in modulus-argument form

(b) Hence, or otherwise, find the four fourth roots of $2 + 2\sqrt3 i$

Question 4 (8 marks)

(a) If $z = x + iy$ and $|z - 1 + i| = |z + 4 - 4i|$ , show that $y = x + 3$

(b) Hence, or otherwise, shade the region that satisfies both $|z - 1 + i| > |z + 4 - 4i|$ and $|z - 4 - 2i| < 5$

(c) If $\theta$ represents Arg( $z$ ), find all possible values of $\theta$ correct to 2 decimal places and represent your answer as an inequality

Question 5 (5 marks)
Let $w$ be the non-real fifth root of unity where $w = e^{\frac{2\pi i}{5}}$

(a) Show that $w^4 + w^3 + w^2 + w + 1 = 0$

(b) Explain why $w^{5-k} = \overline{w}^k$ where $k = 1,\, 2,\, 3,$ and $4$

(c) Using (a) and (b), show that $x = \text{Re}\left(2e^{\frac{2\pi i}{5}}\right)$ is a root of $x^2 + x - 1 = 0$

Question 6 (5 marks)
Consider the polynomial $P(x) = x^7 + 4x^4 - 3x^3 + 8x^2 - 2x + 4$ over the complex numbers

(a) Show that $x = i$ is a root of multiplicity 2, that is, is a double root

(b) Hence, or otherwise, express $P(x)$ as a product of linear factors

Question 7 (5 marks)
Let points $A$ and $B$ represent the complex numbers $z$ and $zw$ , respectively}

(a) Draw an Argand diagram that shows points $A$ and $B$ if $w=e^{\frac{i\pi}{3}}$ and $z = re^{i\theta}$ where $0 < \theta < \pi$ and $r > 0$

(b) If $O$ is the origin, explain why $OAB$ is an equilateral triangle

(c) Write the complex number $z - zw$ in exponential form in terms of $r$ and $\theta$

I note that question 1(b) shouldn't be a "hence" and would be better expressed as something like

Show that $z = \pm(5 - 3i)$ are the solutions of $z^2 = 16 - 30i$

and that question 1(c) was probably meant to be "Hence, using (a) and (b), show that ..."

Also, any thoughts on the most time-efficient approach to these questions?

I think 50-60 minutes is a reasonable amount of time. Realistically closer to 60 tho

People who have done a decent amount of complex should be able to finish this in like 30-45 minutes (I'm making up that number tbh but none of the questions seem that bad).

Luca26 · Dec 14, 2025

WeiWeiMan said:
I think 50-60 minutes is a reasonable amount of time. Realistically closer to 60 tho

People who have done a decent amount of complex should be able to finish this in like 30-45 minutes (I'm making up that number tbh but none of the questions seem that bad).

The exam allowed 70 minutes. No one finished. So much of it is messy, IMO.

WeiWeiMan · Dec 14, 2025

Luca26 said:
The exam allowed 70 minutes. No one finished. So much of it is messy, IMO.

dang what
I’ll try it if I have time

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An MX2 AT1 on complex numbers (1 Viewer)

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