 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/11/c/solution

A counterclockwise rotation through

α

about the origin in the complex plane sends

z_{j}

e^{i α} z_{j}

. By part (a), and because the complex conjugate of

e^{i α}

e^{- i α}

(e^{i α} z_{1}) (e^{i α} z_{2})^{*} = e^{i α} z_{1} e^{- i α} z_{2}^{*} = z_{1} z_{2}^{*} .

Thus

Q

is invariant under the common rotation.

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