Complex Past Year

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Contents

Preparation

  • Make sure you have have mastered all the materials in the previous parts.
  • If you are using the material here to learn this topic (as opposed to just a revision to supplement your school lessons), its' best you take a day (or more) off after learning the previous parts.
  • In fact, try take a week off, and then revise back all the materials before trying the following questions. This will be a good practice for how you are going to revise for the actual STPM.
  • Find a comfortable place & comfortable time.
  • DO NOT do this while you are half-awake, or directly after a long day in school. Else, you will frustrate yourself. Trust me.
  • Saving trees is a good thing, but DO NOT do this (in fact, any of the exercises) on rough paper/recycled paper.
  • "ROUGH PAPER = ROUGH WORK = CARELESS MISTAKES = LOSS OF MARKS"
  • Know that you will face questions that you have NEVER SEEN which will require you to adapt on the spot.
  • Keep a clock or watch handy and give yourself the suggested time to complete it. Check your answers too during that time limit.
  • After finishing, check the answers given. If you made mistakes or couldn't find the solution, you can refer to the answers/answers with guidance.

Questions

Estimated time :

1) If z=1+2i\,, express the following complex numbers in the form a+bi\, : \left(iz-1\right)^{2} and \frac{z}{4-z^{2}} .


2) Determine the values of x\, and y\, that satisfy the equation \frac{x}{1-i}+\frac{y}{1+i}=1+3i.


3) If z_{1}=1-2i\, and z_{2}=2-3i\,, find the complex numbers \left(z_{1}-z_{2}\right)^{2} and z_{1}z_{2}\, in the form a+bi\,.

If x+yi=\frac{z_{1}}{\left(z_{1}-z_{2}\right)^{2}}-\frac{1}{z_{1}z_{2}}, find the values of x\, and y\,.


4) If z_{1}=1+2i\,, find the real part and imaginary part of z+\frac{1}{z}.


5) If z_{1}=1+2i\,, find the modulus of z+\frac{1}{z^{\ast}}, if z^{\ast} is the complex conjugate of z\,.


6) The complex number z_{1}\, and z_{2}\, satisfy the equation z^{2}=2-2\sqrt{3}i\,

  • a) Express z_{1}\, and z_{2}\, in the form a+bi\,, where a\, and b\, are real numbers.
  • b) Represent z_{1}\, and z_{2}\, in an Argand diagram.
  • c) For each of z_{1}\, and z_{2}\,, find the modulus and the argument in radians.


7) If \left(x+yi\right)^{2}=i, find the real values of x\, and y\,


8) Determine the value of a\, if \frac{\sqrt{2}+ai}{1+\sqrt{2}i} is a real number and find this real number.


9) If \frac{\sqrt{3}-i}{1+i}=x+yi, where x\, and y\, are real numbers, find the values of x\, and y\,.

Answers

1) If z=1+2i\,, express the following complex numbers in the form a+bi\, : \left(iz-1\right)^{2} and \frac{z}{4-z^{2}}.

  • 8-6i ; -\frac{1}{65}+\frac{18}{65}i \,
  • \begin{align} \left(iz-1\right)^{2} &= \left[i\left(1+2i\right)-1\right]^{2}\\ &= \left(i+2i^{2}-1\right)^{2}\\ &= \left(-3+i\right)^{2}\\ &= 9-6i+i^{2}\\ &= 8-6i \\ \frac{z}{4-z^{2}} &=\frac{1+2i}{4-\left(1+2i\right)^{2}}\\ &=\frac{1+2i}{4-\left(1+4i+4i^{2}\right)}\\ &=\frac{1+2i}{7-4i}\times\frac{7+4i}{7+4i}\\ &=\frac{7+18i+8i^{2}}{49+16}\\ &=\frac{-1+18i}{65}\\ &=-\frac{1}{65}+\frac{18}{65}i\\ \end{align}

2) Determine the values of x\, and y\, that satisfy the equation \frac{x}{1-i}+\frac{y}{1+i}=1+3i.

  • x=4, y=-2\,
  • \begin{align} & \frac{x}{1-i}+\frac{y}{1+i}=1+3i \\ & \frac{x\left(1+i\right)+y\left(1-i\right)}{1+1}=1+3i \\ & x+xi+y-yi=2+6i \\ & x+y+\left(x-y\right)i=2+6i\\ & \therefore x+y=2\frac{\qquad}{}(1)\qquad x-y=6\frac{\qquad}{}(2)\\ & (1)-(2):\\ & 2y=-4\\ & \therefore y=-2, x=4 \end{align}

3) If z_{1}=1-2i\, and z_{2}=2-3i\,, find the complex numbers \left(z_{1}-z_{2}\right)^{2} and z_{1}z_{2}\, in the form a+bi\,.


If x+yi=\frac{z_{1}}{\left(z_{1}-z_{2}\right)^{2}}-\frac{1}{z_{1}z_{2}}, find the values of x\, and y\,.

  • -2i ; -4-7i ; x= \frac{69}{65}, y=\frac{51}{130}\,
  • \begin{align} \left(z_{1}-z_{2}\right)^{2} & = \left[\left(1-2i\right)-\left(2-3i\right)\right]^{2}\\ & = \left(-1+i\right)^{2}\\ & = 1-2i+i^{2} \\ & = -2i \\ z_{1}z_{2} & = \left(1-2i\right)\left(2-3i\right)\\ & = 2-7i+6i^{2} \\ & = -4-7i\\ x+yi& =\frac{z_{1}}{\left(z_{1}-z_{2}\right)^{2}}-\frac{1}{z_{1}z_{2}} \\ & = \frac{1-2i}{-2i}-\frac{1}{-4-7i}\\ & = \frac{1-2i}{-2i}\times\frac{i}{i}+\frac{1}{4+7i}\\ & = \frac{i-2i^{2}}{2}+\frac{1}{4+7i}\times\frac{4-7i}{4-7i}\\ & = \frac{2+i}{2}+\frac{4-7i}{16+49}\\ & = 1+\frac{1}{2}i+\frac{4}{65}-\frac{7}{65}i\\ & = \frac{69}{65}+\frac{51}{130}i\\ \therefore & x= \frac{69}{65}, y=\frac{51}{130} \end{align}

4) If z_{1}=1+2i\,, find the real part and imaginary part of z+\frac{1}{z}.

  • \mbox{Real part } = \frac{6}{5}, \mbox{Imaginary part } = \frac{8}{5}i
  • \begin{align} z+\frac{1}{z} & = 1+2i+\frac{1}{1+2i} \\ & = 1+2i+\frac{1}{1+2i}\times\frac{1-2i}{1-2i} \\ & = 1+2i+\frac{1-2i}{1+4} \\ & = 1+2i+\frac{1}{5}-\frac{2}{5}i \\ & = \frac{6}{5}+\frac{8}{5}i \\ \therefore & \mbox{Real part } = \frac{6}{5}, \mbox{Imaginary part } = \frac{8}{5}i \end{align}

5) If z_{1}=1+2i\,, find the modulus of z+\frac{1}{z^{\ast}}, if z^{\ast} is the complex conjugate of z\,.

  • \frac{6\sqrt{5}}{5}
  • \begin{align} z+\frac{1}{z^{\ast}} & = 1+2i+\frac{1}{1-2i} \\ & = 1+2i+\frac{1}{1-2i}\times\frac{1+2i}{1+2i} \\ & = 1+2i+\frac{1+2i}{1+4} \\ & = 1+2i+\frac{1}{5}+\frac{2}{5}i \\ & = \frac{6}{5}+\frac{12}{5}i \\ \therefore \left |z+\frac{1}{z^{\ast}}\right|& =\sqrt{\left(\frac{6}{5}\right)^{2}+\left(\frac{12}{5}\right)^{2}}\\ & = \frac{\sqrt{180}}{5}\\ & = \frac{6\sqrt{5}}{5}\\ \end{align}

6) The complex number z_{1}\, and z_{2}\, satisfy the equation z^{2}=2-2\sqrt{3}i\,

  • a) Express z_{1}\, and z_{2}\, in the form a+bi\,, where a\, and b\, are real numbers.
  • b) Represent z_{1}\, and z_{2}\, in an Argand diagram.
  • c) For each of z_{1}\, and z_{2}\,, find the modulus and the argument in radians.

7) If \left(x+yi\right)^{2}=i, find the real values of x\, and y\,

  • x=\frac{\sqrt{2}}{2}, y=\frac{\sqrt{2}}{2} \mbox{ or } x=-\frac{\sqrt{2}}{2}, y=-\frac{\sqrt{2}}{2}
  • \begin{align} & \left(x+yi\right)^{2}=i \\ & x^{2}+2xyi+y^{2}i^{2}=i \\ & x^{2}-y^{2}+2xyi=i\\ & \therefore x^{2}-y^{2}=0\frac{\qquad}{}(1) \quad 2xy=1 \\ & \qquad \qquad \qquad \qquad \qquad \quad y=\frac{1}{2x}\frac{\qquad}{}(2)\\ & (2)\to(1) \\ & x^{2}-\left(\frac{1}{2x}\right)^{2}=0 \\ & x^{2}-\frac{1}{4x^{2}}=0 \\ & x^{2}=\frac{1}{4x^{2}} \\ & x^{4}=\frac{1}{4} \\ & \therefore x^{2} =\pm \frac{1}{2} \\ & \therefore x^{2} = \frac{1}{2} \mbox{ or } -\frac{1}{2}\mbox{(rejected)} \\ & \therefore x =\pm \sqrt{\frac{1}{2}} \\ & \mbox{When } x=\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2}\\ & y=\frac{1}{2\sqrt{\frac{1}{2}}}=\frac{\sqrt{2}}{2} \\ & \mbox{When } x=-\sqrt{\frac{1}{2}}=-\frac{\sqrt{2}}{2}\\ & y=\frac{1}{2\left(-\sqrt{\frac{1}{2}}\right)}=-\frac{\sqrt{2}}{2} \\ & \therefore x=\frac{\sqrt{2}}{2}, y=\frac{\sqrt{2}}{2} \mbox{ or } x=-\frac{\sqrt{2}}{2} \\ \end{align}

8) Determine the value of a\, if \frac{\sqrt{2}+ai}{1+\sqrt{2}i} is a real number and find this real number.

  • a=2; \sqrt{2}
  • \begin{align} \frac{\sqrt{2}+ai}{1+\sqrt{2}i} &= \frac{\sqrt{2}+ai}{1+\sqrt{2}i} \times \frac{1-\sqrt{2}i}{1-\sqrt{2}i}\\ *&=\frac{\sqrt{2}-2i+ai-\sqrt{2}ai^{2}}{1+2}\\ *&=\frac{\sqrt{2}+\sqrt{2}a}{3}+\frac{a-2}{3}i\\ *&\mbox{Since it is a real number, imaginary part } =0 \\ \therefore & \frac{a-2}{3}=0 \\ & a = 2 \\ \therefore \frac{\sqrt{2}+ai}{1+\sqrt{2}i} & =\frac{\sqrt{2}-\sqrt{2}a}{3} \\ & =\frac{\sqrt{2}+\sqrt{2}\left(2\right)}{3} \\ & =\sqrt{2} \\ \end{align}

9) If \frac{\sqrt{3}-i}{1+i}=x+yi, where x\, and y\, are real numbers, find the values of x\, and y\,.

  • x=\frac{\sqrt{3}-1}{2}, y=-\left(\sqrt{3}+1\right)\,
  • \begin{align} x+yi &= \frac{\sqrt{3}-i}{1+i}\\ & = \frac{\sqrt{3}-i}{1+i} \times \frac{1-i}{1-i}\\ & = \frac{\sqrt{3}-\sqrt{3}i-i+i^{2}}{1+1}\\ & = \frac{\sqrt{3}-1-\sqrt{3}i-i}{2}\\ & = \frac{\sqrt{3}-1}{2}-\frac{\sqrt{3}+1}{2}i\\ \therefore & x=\frac{\sqrt{3}-1}{2}, y=-\frac{\left(\sqrt{3}+1\right)}{2} \end{align}

Answers(With guidance)

To Be Done

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