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2021 HSC Mathematics Extension 1

Question 7:
Trigonometric equation

1 markadvanced trig equations

Which curve best represents the graph of the function given that the constants and are both positive?

(Four sketched sinusoidal graphs A–D are supplied in the paper; discriminate using value and gradient at the origin.)

The four graph options supplied for y = -a sin x + b cos x, where a and b are positive.
The four graph options supplied for y = -a sin x + b cos x, where a and b are positive.
  1. Sinusoid with and
  2. Sinusoid with and
  3. Sinusoid with and
  4. Sinusoid with and

How to recognise this question

Question type: advanced trig equations

  • The command and notation point to advanced trig equations.
  • The requested response is a multiple choice.

How to handle it: rewrite the equation into a familiar trig form, solve each branch, then filter by the interval.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. Evaluate at the origin: .
  2. Differentiate: , so .
  3. The correct graph therefore crosses the positive -axis with a negative slope (starting to decrease through a positive intercept).
  4. Only option D matches and .
  5. Answer: D. (NESA multiple-choice key: D.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

For with , which is true?

  1. and
  2. and
  3. and
  4. and
See the step-by-step answer guide →

Question 2

For , which is true?

  1. and
  2. and
  3. and
  4. and
See the step-by-step answer guide →

Question 3

For with , which is true?

  1. and
  2. and
  3. and
  4. and
See the step-by-step answer guide →