Practice section 7 of 9

Let $z=\dfrac{2+i}{3-i}$. Without using a calculator, find the real and imaginary parts of

(a) $\dfrac{1+z}{1-z}$,

(b) $z+\dfrac1z$.

Hence, find the modulus and argument of $\dfrac{1+z}{1-z}$ and $z+\dfrac1z$.

Find the possible values of $z$ in the form $x+iy$, where $x,y\in\mathbb R$, satisfying both $\dfrac{z}{z^*}=\dfrac35+\dfrac45i$ and $zz^*=5$. [4]

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

Two complex numbers are $p=1-i$ and $q=-1+\sqrt3\,i$, and $z=p/q$. Express $z$ in the form $x+iy$, where $x,y$ are exact real values to be determined. [2]

The complex numbers $w,z$ satisfy $\dfrac wz=\dfrac32i$ and $z^*-2w=4+4i$. Find $w$ and $z$. [4]

The complex numbers $z_1=2$ and $z_2=i$ are roots of $z^3+Az^2+Bz+C=0$, where $A,B,C$ are real.

(i) State the third root and find $A,B,C$. [3]

(ii) Hence, with a suitable substitution, find the roots of $2w^3-w^2+2w-1=0$. [3]

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

One root of $x^4+ax^3+5x^2-x-10=0$, where $a$ is real, is $x=1+2i$. Find $a$ and the other roots. [5]

Hence find the $x$-coordinates of the intersection point(s) of $y=5(x^2-2x^4)$ and $y=x^3+3x-1$. [3]