The complex number $w$ has modulus $\sqrt{2}$ and argument $\tfrac{1}{4}\pi$ and the complex number $z$ has modulus $\sqrt{2}$ and argument $\tfrac{5}{4}\pi$.

1. (i)

   By first expressing $w$ and $z$ in the form $x+iy$, find the exact real and imaginary parts of $w+z$.

2. (ii)

   On the same Argand diagram, sketch the points $P$, $Q$, $R$ representing the complex numbers $z$, $w$ and $z+w$ respectively. State the geometrical shape of the quadrilateral $OPRQ$.

3. (iii)

   Referring to the Argand diagram in part (ii), find $\arg(w+z)$ and show that $\tan\left(\frac{11}{24}\pi\right) = \frac{a+\sqrt{2}}{\sqrt{6}+b}$ where $a$ and $b$ are constants to be determined.