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Complex numbers practice

QCE Specialist Maths · Original practice questions with worked solutions

All complex numbers practice questions

42 original questions · Page 2 of 2

Q176 · Original practice · 5 marks
The polynomial $p(z)=z^3-4z^2+9z-10$ has zero $1+2i$ .
Further complex numbers
Q177 · Original practice · 1 mark
Let $a=4\operatorname{cis}(3\pi/4)$ and $b=2\operatorname{cis}(-3\pi/4)$ . Which is a/b?
Further complex numbers
Q183 · Original practice · 5 marks
Four corners of a square game arena have complex coordinates satisfying $z^4=-81$ . The dashed circle is centred at the origin.
Further complex numbers
Q184 · Original practice · 10 marks
Let $\omega=\operatorname{cis}(2\pi/3)$ , so the cube roots of unity are $1,\omega,\omega^2$ .
Further complex numbers
Q185 · Original practice · 7 marks
A game map has triangular zone with complex coordinates $0,2,2i$ . A portal transforms coordinates by $w=(1+i)z+(2-i)$ .
Further complex numbers
Q186 · Original practice · 5 marks
The polynomial $p(z)=z^3+az^2+bz+10$ has real coefficients a,b and zero $1+i$ .
Further complex numbers
Q187 · Original practice · 1 mark
Which set contains all solutions of $z^2=(1+i)/(1-i)$ ?
Further complex numbers
Q248 · Original practice · 12 marks
A scanner spot has complex position $z(t)=1+i+2\frac{1+it}{1-it}$ , where $t\ge0$ is time in seconds and the real and imaginary parts give coordinates in metres. A straight detection gate lies on $x+2y=7$ .
Further complex numbers
Q258 · Original practice · 4 marks
Let $z=a+bi$ and $w=c+di$ , where $a,b,c,d\in\mathbb R$ .
Further complex numbers
Q259 · Original practice · 5 marks
Let $z,w\in\mathbb C$ , with $w\ne0$ . You may use $u\overline u=|u|^2$ and the modulus product identity.
Further complex numbers
Q260 · Original practice · 5 marks
The complex coordinates of two adjacent sides of a parallelogram are $z$ and $w$ . Its diagonals are represented by $z+w$ and $z-w$ .
Further complex numbers
Q261 · Original practice · 5 marks
A graphics tool maps a complex number $z$ to $w=(z+2)/(2z+1)$ . Assume $|z|=1$ .
Further complex numbers
Q262 · Original practice · 5 marks
For this question, the principal argument is $\operatorname{Arg}z\in(-\pi,\pi]$ . Let $z,w\ne0$ .
Further complex numbers
Q263 · Original practice · 6 marks
Three lights have complex coordinates satisfying $z^3=8i$ . Blank Argand axes are provided.
Further complex numbers
Q264 · Original practice · 6 marks
Consider $S=\{z:|z-1|=2\}$ and $T=\{z:\operatorname{Im}z=1\}$ .
Further complex numbers
Q265 · Original practice · 6 marks
For $z\ne0$ , use $\operatorname{Arg}z\in(-\pi,\pi]$ . Define $R=\{z:1\le|z|\le2,\ 0<\operatorname{Arg}z\le\pi/2\}$ .
Further complex numbers
Q271 · Original practice · 6 marks
The polynomial $P(z)=z^3+az^2+bz+2$ has $a,b\in\mathbb C$ . Its remainder on division by $z-i$ is $2-2i$ , and $P(1)=0$ .
Further complex numbers
Q295 · Original practice · 1 mark
Which example shows that $|z+w|=|z|+|w|$ is not true for every pair of complex numbers?
Further complex numbers