Complex numbers

Do not use a calculator in answering this question. Showing your working, find the complex numbers $z$ and $w$ which satisfy the simultaneous equations

The complex numbers $z$ and $w$ satisfy $iw^2+2wz=2i$ and $z+iz=2+iw$. Find $z$ and $w$, giving your answers in the form $a+ib$, where $a$ and $b$ are real numbers. ****

(a) Given $\arg(a+ib)=\theta$, where $a,b$ are real and positive, find in terms of $\theta$ and/or $\pi$:

A complex number is given by $z=a+(a-3)i$, where $a$ is a positive real constant. It is given that $\arg z=\theta$, where $-\pi/2<\theta<0$. Find, leaving answers in terms of $\theta$:

(i) The roots of $w^2=4i$ are $w_1,w_2$. Find $w_1,w_2$ in Cartesian form $x+iy$, showing your working. ****

The complex numbers $z,w$ satisfy $2z+1=|w|$ and $2w-z=4+24i$. Find $z,w$, giving answers in the form $a+ib$, where $a,b$ are real numbers. ****

The complex number $z$ satisfies $z^2+(2-4i)z=c$, where $c$ is a complex constant.

$f(x)$ is a cubic polynomial with real coefficients. The diagram shows $y=f(x)$.

(i) One root of $x^3-3x+c=0$, where $c$ is real, is $2-3i$. Without using a calculator, find $c$ and the other roots. ****

A graphic calculator is not to be used in answering this question.

(a) $w^3+pw^2+qw+30=0$, with $p,q$ real constants, has root $w=2-i$. Find $p,q$, showing working. ****

(i) Solve $z^2-(1+2i)z+1+7i=0$. ****

Do not use a calculator in answering this question.

(a) $f(x)$ is a degree-4 polynomial with real coefficients. Explain whether $f(x)=0$ can have 3 non-real roots and 1 real root. ****

A, B and C represent complex numbers $a=0,b=3,c=-2+i$. These numbers are roots of $f(z)=0$, where $f(z)$ is a quartic polynomial with real coefficients and $z$ is complex.

$z$ has modulus $r$ and argument $\theta$, with $0<r<1$ and $\pi/2<\theta<\pi$. $w$ satisfies $|w|=|z|$ and $\arg w+\arg z=\pi$.

$w=-1-i\sqrt3$ and $z^*+z\ne0$. P and Q represent $w$ and $z^*+z$ respectively. Find $\arg w$ exactly. Hence state possible values of $\angle POQ$, where O is the origin. ****

A, B, C and D represent $-2+5i,z_1,4+i,z_2$ respectively. ABCD is a square labelled anticlockwise. Show $z_1=-1$ and find $z_2$. ****

$\arg z=-\pi/3$, $|w|=2$, $\arg w=\pi/4$ and $\arg(w-z)=\pi/3$. Let W and Z represent $w,z$ respectively.

$z=\sqrt3+i$, $w=-1+i\sqrt3$. Sketch an Argand diagram with origin O showing Z, W and P representing $z,w,z+w$ respectively. Describe the geometrical shape OWPZ. ****

A graphic calculator is not to be used in answering this question. $p=1-i$, $q=-1+\sqrt3i$, $z=p/q$.