A particle is moving along the -x-axis in simple harmonic motion - HSC - SSCE Mathematics Extension 1 - Question 13 - 2015 - Paper 1

The maximum speed of the particle corresponds to the maximum value of v². This can be determined from the vertex of the parabola shown in the graph. The y-coordinate of the vertex gives the maximum value for v², thus the maximum speed v can be calculated as ( v_{max} = \sqrt{v^2_{max}} ).

To determine the constants a, c, and n in the equation ( v^2 = n^2(a^2 - (x - c)^2) ), we can compare coefficients from the derived equation and the provided equation. It can be shown that a represents the amplitude, c the equilibrium position of the motion, and n relates to the angular frequency of the harmonic motion, derived from the context of the problem.

To find an expression for a₂ in the binomial expansion, we use the formula: ( a_k = \binom{n}{k} a^{n-k} b^{k} ). Applying this for k=2, we obtain ( a_2 = \binom{18}{2} (2x)^{16} (1/3x)^2 ). Simplifying this gives the required expression.

The term independent of x in the expansion is the constant term, achieved when k equals the power of x in the binomial expansion not appearing in the product. This corresponds to finding the middle term of the binomial expansion, which can be represented as ( a_9 ).

To prove this by induction, begin with the base case when n=1. Then, assume it holds for n=k, and show it holds for n=k+1. Upon simplification, both sides should match, completing the induction step.

To show that f(x) is constant, calculate f'(x). If f'(x) results in zero across the permissible domain, then f(x) must be constant. This involves using the chain rule for derivatives applied to inverse trigonometric functions.

From the established constant nature of f(x), we know that for any x in the domain, the relationship must hold true. Therefore, we can derive that cos⁻¹(−x) reflects the properties of cosine across quadrants, yielding the equation cos⁻¹(−x) = π - cos⁻¹(x).