The Argand diagram shows the complex numbers $z$ and $w$, where $z$ lies in the first quadrant and $w$ lies in the second quadrant - HSC - SSCE Mathematics Extension 2 - Question 4 - 2016 - Paper 1

To determine which complex number could lie in the 3rd quadrant, we must analyze the properties of the given numbers.

1. Understanding the Quadrants: In the Argand plane, the 1st quadrant has positive real and positive imaginary parts, the 2nd quadrant has negative real and positive imaginary parts, the 3rd quadrant has negative real and negative imaginary parts, and the 4th quadrant has positive real and negative imaginary parts.

2. Analyzing Each Option:

   • (A) $-w$: Since $w$ is in the 2nd quadrant (negative real, positive imaginary), $-w$ will have a positive real part and a negative imaginary part, placing it in the 4th quadrant.
   • (B) $2iz$: Since $z$ is in the 1st quadrant, $iz$ will be in the 2nd quadrant ($0 < ext{Im}(iz) < ext{Re}(iz)$). Therefore, $2iz$ will also remain in the 2nd quadrant.
   • (C) ar{z}: The conjugate of $z$, ar{z}, will lie in the 4th quadrant since it has a positive real part and a negative imaginary part.
   • (D) $w - z$: Here, we need to analyze further. Given that $w$ is in the 2nd quadrant and $z$ is in the 1st quadrant, $w - z$ results in subtracting a positive real value from a negative real value, which could lead to a negative real result, depending on their magnitudes.

3. Conclusion: Since depending on the specific values the expression can lead to both negative real and imaginary parts, $w - z$ can indeed lie in the 3rd quadrant. Thus, the correct answer is (D) $w - z$.