Variation 3 · answer guide

Coordinate geometry — Endpoint check for option C form

Question

For , , , which point is the bottom of the arc?

This practice question was inspired by Question 8 in the 2021 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: convert between parametric and Cartesian forms carefully, noting the direction of the parameter.At : . Endpoints: give and .
Finish the calculation, then check that the result meets the question’s conditions.The bottom of this lower semicircle (centre , radius ) is .
Match the result to the choices and state the correct option.Therefore option A is correct.

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

Verification: Same structure as the exam option C, translated.

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