Hard Further Maths Argand diagrams Questions

Challenging, exam-style Further Maths Argand diagrams questions with worked solutions. Stretch yourself on the hardest argand, modulus, products, quotients problems.

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Further Maths34 questionsStep-by-step solutions
Question 1
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
Question 2
8 markschallenging
Which of the following best describes the locus of points zz on an Argand diagram satisfying arg⁡(z−(2−i))=2π3\arg\left(z-\left(2 - i\right)\right)=\frac{2 \pi}{3}?
Show worked solution

Worked solution

  1. Interpret the argument condition

    arg⁡(z−(2−i))=2π3\arg\left(z-\left(2 - i\right)\right)=\frac{2 \pi}{3}

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

  2. Recall the half-line locus

    half-line from (2, −1) at angle 2π3\text{half-line from }\left(2,\,-1\right)\text{ at angle }\frac{2 \pi}{3}

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

  3. Note the endpoint is excluded

    z≠2−iz\ne 2 - i

    At the endpoint the argument is undefined.

  4. Reject the opposite direction

    arg⁡(z−(2−i))≠−π3\arg\left(z-\left(2 - i\right)\right)\ne - \frac{\pi}{3}

    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. 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 correct description

    half-line from (2, −1) at angle 2π3 to the real axis\text{half-line from }\left(2,\,-1\right)\text{ at angle }\frac{2 \pi}{3}\text{ to the real axis}

    This matches the locus exactly.

Answer
A half-line starting at (2, −1)(2,\,-1) making an angle of 2π3\frac{2 \pi}{3} with the positive real axis
Question 3
8 markschallenging
Which of the following best describes the locus of points zz on an Argand diagram satisfying ∣z−(−1+4i)∣=∣z−(5−2i)∣\left|z-\left(-1 + 4 i\right)\right|=\left|z-\left(5 - 2 i\right)\right|?
Show worked solution

Worked solution

  1. Interpret each modulus as a distance

    ∣z−(−1+4i)∣=∣z−(5−2i)∣\left|z-\left(-1 + 4 i\right)\right|=\left|z-\left(5 - 2 i\right)\right|

    zz is equidistant from the two fixed points.

  2. Recall the equidistant locus

    perpendicular bisector of AB\text{perpendicular bisector of }AB

    The set of points equidistant from AA and BB is the perpendicular bisector of ABAB.

  3. Name the two fixed points

    A(−1, 4), B(5, −2)A\left(-1,\,4\right),\ B\left(5,\,-2\right)

    These are the points represented by the two complex numbers.

  4. Give the Cartesian equation as a check

    y=x−1y = x - 1

    The locus is a straight line, not a circle.

  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 correct description

    perpendicular bisector of the segment AB\text{perpendicular bisector of the segment }AB

    This matches the locus exactly.

Answer
The perpendicular bisector of the line segment joining (−1, 4)(-1,\,4) and (5, −2)(5,\,-2)
Question 4
8 markschallenging
Which of the following is the argument of z=−5−5iz=-5 - 5 i, in the interval −π<θ≤π-\pi<\theta\le\pi?
Show worked solution

Worked solution

  1. Locate the point on the Argand diagram

    z ⟶ (−5, −5)z\ \longrightarrow\ \left(-5,\,-5\right)

    The quadrant determines the sign and size of the argument.

  2. Measure the angle from the positive real axis

    arg⁡z=−3π4\arg z=- \frac{3 \pi}{4}

    Anticlockwise is positive, clockwise is negative.

  3. Reject the reflected angle

    −arg⁡z=3π4-\arg z=\frac{3 \pi}{4}

    This would be the argument of the conjugate.

  4. Confirm the principal range

    −π<−3π4≤π-\pi<- \frac{3 \pi}{4}\le\pi

    The principal argument lies in this interval.

  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 matching option

    arg⁡z=−3π4\arg z=- \frac{3 \pi}{4}

    This option equals the computed argument.

Answer
−3π4- \frac{3 \pi}{4}
Question 5
8 markschallenging
Given z=−1+iz=-1 + i, find the exact value of ∣z6∣\left|z^{6}\right|.
Show worked solution

Worked solution

  1. Find the modulus of zz

    ∣z∣=2\left|z\right|=\sqrt{2}

    Apply x2+y2\sqrt{x^2+y^2} to zz.

  2. Use ∣zn∣=∣z∣n\left|z^n\right|=\left|z\right|^n

    ∣z6∣=(2)6=8\left|z^{6}\right|=\left(\sqrt{2}\right)^{6}=8

    The modulus of a power is the power of the modulus.

  3. Justify the rule from the product rule

    ∣z6∣=∣z∣⋯∣z∣⏟6\left|z^{6}\right|=\underbrace{\left|z\right|\cdots\left|z\right|}_{6}

    Repeated use of ∣z1z2∣=∣z1∣∣z2∣\left|z_1z_2\right|=\left|z_1\right|\left|z_2\right|.

  4. Raise the modulus to the power

    (2)6=8\left(\sqrt{2}\right)^{6}=8

    Evaluate the power exactly.

  5. Note the argument plays no part

    ∣z6∣ depends only on ∣z∣\left|z^{6}\right|\ \text{depends only on }\left|z\right|

    Only the modulus is required here.

  6. Simplify to exact form

    ∣z6∣=8\left|z^{6}\right|=8

    Leave surds in exact form.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. State the exact value

    ∣z6∣=8\left|z^{6}\right|=8

    This is the modulus of the required power.

Answer
88

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