Vectors — Area formula and numerical check, then maximise

Question

(i) Show that the area of with , is , and verify with , .

(ii) Hence, for arising as the area factor for the double-angle circle configuration of the exam, find all that maximise when (equivalently maximise the triangle area).

This practice question was inspired by Question 14(c) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

Set up the solution: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.(i) Perpendicular vector has equal magnitude; its dot product with yields , so area .
Simplify the previous line carefully, keeping exact values where possible.Check: .
Finish the calculation, then check that the result meets the question’s conditions.(ii) or .
State the final answer clearly in the form the question requested.On intervals where , the maximum occurs at (at , is a minimum of the area factor).

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Both parts: geometric proof plus calculus maximisation of the area factor.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.

← Back to 2023 HSC Extension 1 paper Question 14(c)