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 can lie in the 3rd quadrant, we should recall the location of numbers in the Argand diagram:

• A complex number lies in the 3rd quadrant if both its real and imaginary components are negative.

Let's analyze the options:

1. Option A: -w
   Since w lies in the second quadrant, it has a negative real part and a positive imaginary part. Therefore, -w will have a positive real part and a negative imaginary part, placing it in the 4th quadrant.

2. Option B: 2iz
   The number z lies in the first quadrant, meaning both its real and imaginary parts are positive. Thus, 2iz will have a negative real part (due to the multiplication by i) and a positive imaginary part, placing it in the 2nd quadrant.

3. Option C: z
   As stated, z is in the first quadrant. It cannot lie in the 3rd quadrant.

4. Option D: w - z
   Here, we need to evaluate:

   • w has a negative real part and a positive imaginary part.
   • z has a positive real part and a positive imaginary part.

   Therefore, w - z could have:

   • A negative real part (because the negative quantity of w dominates) and a negative imaginary part (if the positive imaginary part of z exceeds the positive imaginary part of w).

   This configuration suggests that w - z can indeed lie in the 3rd quadrant.

In conclusion, the complex number that could lie in the 3rd quadrant is (D) w - z.