Argand diagrams Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Argand diagrams questions. See exactly how to solve problems on argand, modulus, argument, modulus-argument-form.

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

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
Find the exact modulus of the complex number z=5+12iz=5 + 12 i.

Worked solution

  1. Write down the real and imaginary parts

    x=5, y=12x=5,\ y=12

    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∣=(5)2+(12)2=169\left|z\right|=\sqrt{\left(5\right)^2+\left(12\right)^2}=\sqrt{169}

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

  3. State the modulus

    ∣z∣=13\left|z\right|=13

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

Answer
1313
Question 3
2 markseasy
Find the exact modulus of the complex number z=1+iz=1 + i.

Worked solution

  1. Write down the real and imaginary parts

    x=1, y=1x=1,\ y=1

    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∣=(1)2+(1)2=2\left|z\right|=\sqrt{\left(1\right)^2+\left(1\right)^2}=\sqrt{2}

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

  3. State the modulus

    ∣z∣=2\left|z\right|=\sqrt{2}

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

Answer
2\sqrt{2}
Question 4
2 markseasy
Find the exact modulus of the complex number z=−8+15iz=-8 + 15 i.

Worked solution

  1. Write down the real and imaginary parts

    x=−8, y=15x=-8,\ y=15

    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∣=(−8)2+(15)2=289\left|z\right|=\sqrt{\left(-8\right)^2+\left(15\right)^2}=\sqrt{289}

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

  3. State the modulus

    ∣z∣=17\left|z\right|=17

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

Answer
1717
Question 5
2 markseasy
Find the argument of the complex number z=1+iz=1 + i, giving your answer in radians in the interval −π<arg⁡z≤π-\pi<\arg z\le\pi.

Worked solution

  1. Plot the point and identify its quadrant or axis

    z ⟶ (1, 1)z\ \longrightarrow\ (1,\,1)

    The position of the point determines how the angle is measured.

  2. Find the acute angle to the real axis

    α=arctan⁡∣11∣=π4\alpha=\arctan\left|\frac{1}{1}\right|=\frac{\pi}{4}

    This is the acute angle the vector makes with the real axis.

  3. State the argument

    arg⁡z=π4\arg z=\frac{\pi}{4}

    This is the principal argument in radians.

Answer
π4\frac{\pi}{4}

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