Which of the following functions does NOT describe simple harmonic motion? A - HSC - SSCE Mathematics Extension 2 - Question 6 - 2023 - Paper 1

To determine if a function describes simple harmonic motion (SHM), it should be expressible as a sine or cosine function in the form $x(t) = A ext{sin}( ext{ω}t + ext{φ})$ or $x(t) = A ext{cos}( ext{ω}t + ext{φ})$.

• For option A: $x = ext{cos}^2 t - ext{sin} 2t$ can be complex due to the $ext{cos}^2 t$ term, which does not directly express SHM.
• For option B: $x = ext{sin} 4t + 4 ext{cos} 2t$ is a valid combination of SHM terms, hence is valid.
• For option C: $x = 2 ext{sin} 3t - 4 ext{cos} 3t + 5$ includes an additional constant (5), deviating from pure SHM.
• For option D: This function can be simplified and still represents SHM terms.

After analyzing all functions, option A ($x = ext{cos}^2 t - ext{sin} 2t$) does not describe simple harmonic motion due to the $cos^2 t$ term, which cannot be represented as a pure sine or cosine function. Thus, the answer is A.