IB Mathematics AHL 1.12 Complex numbers AI HL Paper 1- Exam Style Questions- New Syllabus

(a)

Expand \(z^2\):

\(z^2=(2+ai)^2\)

\(z^2=4+4ai+a^2i^2\).

Since \(i^2=-1\),

\(z^2=4-a^2+4ai\).

The real part is therefore

\(\operatorname{Re}(z^2)=4-a^2\).

✅ Answer: \(4-a^2\)

(b)

Using \(z^2=4-a^2+4ai\) and \(w=b+4i\),

\(z^2+2w=(4-a^2+4ai)+2(b+4i)\).

\(z^2+2w=(4-a^2+2b)+(4a+8)i\).

This is equal to \(3+6i\). Therefore, the corresponding real and imaginary parts must be equal.

Comparing imaginary parts:

\(4a+8=6\)

\(4a=-2\)

\(a=-\dfrac{1}{2}\).

Comparing real parts:

\(4-a^2+2b=3\).

Substituting \(a=-\dfrac{1}{2}\),

\(4-\left(-\dfrac{1}{2}\right)^2+2b=3\)

\(4-\dfrac{1}{4}+2b=3\)

\(\dfrac{15}{4}+2b=\dfrac{12}{4}\)

\(2b=-\dfrac{3}{4}\)

\(b=-\dfrac{3}{8}\).

✅ Answer: \(a=-\dfrac{1}{2}\), \(b=-\dfrac{3}{8}\)