MCD4490 Advanced Mathematics Assignment (Trimester 1, 2024)

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MCD4490 Advanced Mathematics

Assignment (Trimester 1, 2024)

1. Consider the functions f : (1,∞) → R, g : (−∞, 1) ∪ (1, 2) → R and h : A → R that are defined by

where A = {x ∈ dom(f) : f(x) ∈ dom(g)}.

(a) Determine the domain of h, and an algebraic expression for h(x).

(b) Show algebraically that h is one-to-one. That is, suppose that a and b in dom(h) satisfy h(a) = h(b) and deduce that a must be equal to b.

(c) As h is one-to-one, it has an inverse . Determine the domain and range of , and an algebraic expression for (x).

[5 + 1 + 6 = 12 marks]

2. In radio communication, a message M(t) is not usually transmitted directly. Instead, a technique called Amplitude Modulation (AM) may be used, in which a carrier wave C(t) = Ac sin(ωct) has its amplitude modulated by the message. In short, the signal  is transmitted instead of M(t) itself.

(a) Prove for all a and b in R that

2 sin(a) cos(b) = sin(a + b) + sin(a − b).                 (1)

(b) Suppose that you receive an AM radio signal of the form

S(t) = 2 sin(98t) + 12 sin(100t) + 2 sin(102t).

Assuming that the message is of the form M(t) = Am cos(ωmt), determine expressions for the message and carrier wave that were used to create this signal.

[3 + 9 = 12 marks]

3. Consider the complex number

(a) Express w in Cartesian form and in polar form.

(b) Use your answer to part (a) to calculate the exact values of sin() and cos(). Express your answers with rational denominators.

(c) Find all of the solutions of the equation , expressed in polar form.

(d) Plot all of the solutions found in part (c) on an Argand diagram.

[7 + 4 + 9 + 3 = 23 marks]