Further Maths Argand diagrams Practice Questions

Free Further Maths Argand diagrams practice questions with full step-by-step worked solutions. Covers argand, modulus, argument, modulus-argument-form. Practise exam-style problems and check your method.

argandmodulusargumentmodulus-argument-formdistanceloci
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Find the exact modulus of the complex number z=3+4iz=3 + 4 i.
Show worked solution

Worked solution

  1. Write down the real and imaginary parts

    x=3, y=4x=3,\ y=4

    The point (x,y)(x,y) represents zz on the Argand diagram.

  2. Substitute into ∣z∣=x2+y2\left|z\right|=\sqrt{x^2+y^2}

    ∣z∣=(3)2+(4)2=25\left|z\right|=\sqrt{\left(3\right)^2+\left(4\right)^2}=\sqrt{25}

    Square each part, add, then take the positive square root.

  3. State the modulus

    ∣z∣=5\left|z\right|=5

    This is the exact distance of the point from the origin.

Answer
55
Question 2
2 markseasy
In which quadrant of the Argand diagram does the point representing z=−3−4iz=-3 - 4 i lie?
Show worked solution

Worked solution

  1. Read off the coordinates

    z ⟶ (−3, −4)z\ \longrightarrow\ \left(-3,\,-4\right)

    The real part is the xx-coordinate and the imaginary part the yy-coordinate.

  2. Use the signs of the coordinates

    x<0, y<0x<0,\ y<0

    The pair of signs identifies the quadrant.

  3. Select the quadrant

    quadrant III\text{quadrant }III

    This is where the point lies.

Answer
The third quadrant
Question 3
3 marksintermediate
Which of the following best describes the locus of points zz on an Argand diagram satisfying arg⁡(z−(1+2i))=π4\arg\left(z-\left(1 + 2 i\right)\right)=\frac{\pi}{4}?
Show worked solution

Worked solution

  1. Interpret the argument condition

    arg⁡(z−(1+2i))=π4\arg\left(z-\left(1 + 2 i\right)\right)=\frac{\pi}{4}

    The vector from the fixed point to zz has a fixed direction.

  2. Recall the half-line locus

    half-line from (1, 2) at angle π4\text{half-line from }\left(1,\,2\right)\text{ at angle }\frac{\pi}{4}

    Only one direction is allowed, so the locus is a half-line, not a full line.

  3. Note the endpoint is excluded

    z≠1+2iz\ne 1 + 2 i

    At the endpoint the argument is undefined.

  4. Reject the opposite direction

    arg⁡(z−(1+2i))≠−3π4\arg\left(z-\left(1 + 2 i\right)\right)\ne - \frac{3 \pi}{4}

    The opposite ray corresponds to a different argument.

  5. Recall the modulus formula

    ∣z∣=x2+y2\left|z\right|=\sqrt{x^2+y^2}

    The modulus is the distance of the point from the origin on the Argand diagram.

  6. Select the correct description

    half-line from (1, 2) at angle π4 to the real axis\text{half-line from }\left(1,\,2\right)\text{ at angle }\frac{\pi}{4}\text{ to the real axis}

    This matches the locus exactly.

Answer
A half-line starting at (1, 2)(1,\,2) making an angle of π4\frac{\pi}{4} with the positive real axis
Question 4
5 markshard
Which of the following is the exact modulus of z=−7+24iz=-7 + 24 i?
Show worked solution

Worked solution

  1. Apply the modulus formula

    ∣z∣=(−7)2+(24)2\left|z\right|=\sqrt{\left(-7\right)^2+\left(24\right)^2}

    The modulus is x2+y2\sqrt{x^2+y^2}.

  2. Evaluate the surd

    ∣z∣=25\left|z\right|=25

    Simplify to exact form.

  3. Reject the option x2+y2x^2+y^2

    x2+y2=625x^2+y^2=625

    This is the square of the modulus, not the modulus.

  4. Reject the option ∣x∣+∣y∣\left|x\right|+\left|y\right|

    ∣x∣+∣y∣=31\left|x\right|+\left|y\right|=31

    Adding the parts is not the same as Pythagoras.

  5. Recall the modulus formula

    ∣z∣=x2+y2\left|z\right|=\sqrt{x^2+y^2}

    The modulus is the distance of the point from the origin on the Argand diagram.

  6. Recall how the argument is measured

    arg⁡z=angle from the positive real axis, measured anticlockwise\arg z=\text{angle from the positive real axis, measured anticlockwise}

    The argument is the angle the vector from the origin makes with the positive real axis.

  7. Plot the number as a point

    z=x+iy ⟶ (x, y)z=x+iy\ \longrightarrow\ (x,\,y)

    A complex number is represented by the point (x,y)(x,y) on the Argand diagram.

  8. Identify the quadrant from the signs

    signs of x and y fix the quadrant\text{signs of }x\text{ and }y\text{ fix the quadrant}

    The quadrant tells you how to adjust the acute angle to get the argument.

  9. Recall modulus-argument form

    z=r(cos⁡θ+isin⁡θ)z=r\left(\cos\theta+i\sin\theta\right)

    Here r=∣z∣r=\left|z\right| and θ=arg⁡z\theta=\arg z.

  10. Select the matching option

    ∣z∣=25\left|z\right|=25

    This option equals the computed modulus.

Answer
2525
Question 5
8 markschallenging
In which quadrant of the Argand diagram does the point representing z=4−9iz=4 - 9 i lie?
Show worked solution

Worked solution

  1. Read off the coordinates

    z ⟶ (4, −9)z\ \longrightarrow\ \left(4,\,-9\right)

    The real part is the xx-coordinate and the imaginary part the yy-coordinate.

  2. Use the signs of the coordinates

    x>0, y<0x>0,\ y<0

    The pair of signs identifies the quadrant.

  3. Recall the quadrant convention

    I: x>0,y>0; II: x<0,y>0; III: x<0,y<0; IV: x>0,y<0\text{I}:\ x>0,y>0;\ \text{II}:\ x<0,y>0;\ \text{III}:\ x<0,y<0;\ \text{IV}:\ x>0,y<0

    Quadrants are numbered anticlockwise from the positive real axis.

  4. Confirm with the argument

    arg⁡z=−atan⁡(94)\arg z=- \operatorname{atan}{\left(\frac{9}{4} \right)}

    The argument also confirms the quadrant.

  5. Recall the modulus formula

    ∣z∣=x2+y2\left|z\right|=\sqrt{x^2+y^2}

    The modulus is the distance of the point from the origin on the Argand diagram.

  6. Recall how the argument is measured

    arg⁡z=angle from the positive real axis, measured anticlockwise\arg z=\text{angle from the positive real axis, measured anticlockwise}

    The argument is the angle the vector from the origin makes with the positive real axis.

  7. Plot the number as a point

    z=x+iy ⟶ (x, y)z=x+iy\ \longrightarrow\ (x,\,y)

    A complex number is represented by the point (x,y)(x,y) on the Argand diagram.

  8. Identify the quadrant from the signs

    signs of x and y fix the quadrant\text{signs of }x\text{ and }y\text{ fix the quadrant}

    The quadrant tells you how to adjust the acute angle to get the argument.

  9. Recall modulus-argument form

    z=r(cos⁡θ+isin⁡θ)z=r\left(\cos\theta+i\sin\theta\right)

    Here r=∣z∣r=\left|z\right| and θ=arg⁡z\theta=\arg z.

  10. Recall the multiplication rule for moduli

    ∣z1z2∣=∣z1∣∣z2∣\left|z_1z_2\right|=\left|z_1\right|\left|z_2\right|

    Moduli multiply when complex numbers are multiplied.

  11. Recall the addition rule for arguments

    arg⁡(z1z2)=arg⁡z1+arg⁡z2\arg\left(z_1z_2\right)=\arg z_1+\arg z_2

    Arguments add when complex numbers are multiplied.

  12. Recall the division rule for moduli

    ∣z1z2∣=∣z1∣∣z2∣\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}

    Moduli divide when complex numbers are divided.

  13. Interpret ∣z−a∣\left|z-a\right| as a distance

    ∣z−a∣=distance from z to a\left|z-a\right|=\text{distance from }z\text{ to }a

    This geometric reading is the key to every locus question.

  14. Recall the circle locus

    ∣z−a∣=r is a circle, centre a, radius r\left|z-a\right|=r\ \text{is a circle, centre }a\text{, radius }r

    All points a fixed distance from a fixed point form a circle.

  15. Select the quadrant

    quadrant IV\text{quadrant }IV

    This is where the point lies.

Answer
The fourth quadrant

Unlock 65 more Argand diagrams questions

Create a free account to work through every Further Maths Argand diagrams question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Argand diagrams practice

Hard QuestionsWorked Solutions

Related Pure Maths topics

Complex numbers: arithmeticModelling with differential equationsComplex numbers: de MoivreFirst-order differential equationsFurther calculusHyperbolic functions