Further complex numbers Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Further complex numbers questions. See exactly how to solve problems on further-complex, transformations, argand-plane, image-of-a-point.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The transformation TT from the zz-plane to the ww-plane is given by w=3z+2−iw=3 z+2 - i. Find the image of the point z=1+2iz=1 + 2 i under TT, giving your answer in the form a+bia+bi.

Worked solution

  1. Substitute the given value of zz into the transformation

    w=3(1+2i)+2−iw=3 \left(1 + 2 i\right)+2 - i

    The image of a point is found by evaluating f(z)f\left(z\right) at that point.

  2. State the modulus of the image

    ∣w∣=52\left|w\right|=5 \sqrt{2}

    A quick modulus check confirms the arithmetic.

  3. State the image of the point

    w=5+5iw=5 + 5 i

    This is the image of z=1+2iz=1 + 2 i under TT.

Answer
5+5i5 + 5 i
Question 2
2 markseasy
The transformation TT from the zz-plane to the ww-plane is given by w=1zw=\frac{1}{z}. Find the image of the point z=1+iz=1 + i under TT, giving your answer in the form a+bia+bi.

Worked solution

  1. Substitute the given value of zz into the transformation

    w=1(1+i)w=\frac{1}{\left(1 + i\right)}

    The image of a point is found by evaluating f(z)f\left(z\right) at that point.

  2. Multiply the numerator and the denominator by the conjugate of the denominator

    w=(12−i2)(1)(1)(1)w=\frac{\left(\frac{1}{2} - \frac{i}{2}\right)\left(1\right)}{\left(1\right)\left(1\right)}

    This makes the denominator the real number ∣1∣2=1\left|1\right|^{2}=1.

  3. Simplify the real denominator

    w=12−i21w=\frac{\frac{1}{2} - \frac{i}{2}}{1}

    The denominator is now real, so the quotient can be split into real and imaginary parts.

  4. State the image of the point

    w=12−i2w=\frac{1}{2} - \frac{i}{2}

    This is the image of z=1+iz=1 + i under TT.

Answer
12−i2\frac{1}{2} - \frac{i}{2}
Question 3
2 markseasy
The transformation TT from the zz-plane to the ww-plane is given by w=z2w=z^{2}. Find the image of the point z=2−3iz=2 - 3 i under TT, giving your answer in the form a+bia+bi.

Worked solution

  1. Substitute the given value of zz into the transformation

    w=(2−3i)2w=\left(2 - 3 i\right)^{2}

    The image of a point is found by evaluating f(z)f\left(z\right) at that point.

  2. State the modulus of the image

    ∣w∣=13\left|w\right|=13

    A quick modulus check confirms the arithmetic.

  3. State the image of the point

    w=−5−12iw=-5 - 12 i

    This is the image of z=2−3iz=2 - 3 i under TT.

Answer
−5−12i-5 - 12 i
Question 4
2 markseasy
The transformation TT from the zz-plane to the ww-plane is given by w=z+1z−2w=\frac{z + 1}{z - 2}. Find the image of the point z=3iz=3 i under TT, giving your answer in the form a+bia+bi.

Worked solution

  1. Substitute the given value of zz into the transformation

    w=(3i)+1(3i)−2w=\frac{\left(3 i\right) + 1}{\left(3 i\right) - 2}

    The image of a point is found by evaluating f(z)f\left(z\right) at that point.

  2. Multiply the numerator and the denominator by the conjugate of the denominator

    w=(7−9i)(13)(13)(13)w=\frac{\left(7 - 9 i\right)\left(13\right)}{\left(13\right)\left(13\right)}

    This makes the denominator the real number ∣13∣2=169\left|13\right|^{2}=169.

  3. Simplify the real denominator

    w=91−117i169w=\frac{91 - 117 i}{169}

    The denominator is now real, so the quotient can be split into real and imaginary parts.

  4. State the image of the point

    w=713−9i13w=\frac{7}{13} - \frac{9 i}{13}

    This is the image of z=3iz=3 i under TT.

Answer
713−9i13\frac{7}{13} - \frac{9 i}{13}
Question 5
2 markseasy
The transformation TT from the zz-plane to the ww-plane is given by w=2z−3iw=2 z-3 i. Find the complex number zz whose image under TT is w=4+iw=4 + i, giving your answer in the form a+bia+bi.

Worked solution

  1. Make zz the subject of the transformation

    z=w2+3i2z=\frac{w}{2} + \frac{3 i}{2}

    Rearranging gives the inverse map f−1f^{-1}.

  2. Substitute the given value of ww

    z=(4+i)2+3i2z=\frac{(4 + i)}{2} + \frac{3 i}{2}

    The pre-image is the value of f−1f^{-1} at the given point.

  3. State the complex number zz

    z=2+2iz=2 + 2 i

    This is the only point of the zz-plane mapped to w=4+iw=4 + i.

Answer
2+2i2 + 2 i

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