New

Subjects

# Complex Numbers & Sine Waves

Paper

Paper 1

Difficulty

Easy
Medium
Hard

View

##### Question 1

calculator

hard

[Maximum mark: 5]

Two voltage sources are connected to a circuit. At time $t$ milliseconds (ms), the voltage from the first source is $V_1(t) = 12\cos(20t)$ and the voltage from the second source is $V_2(t) = 18\cos(20t+5)$, where both $V_1(t)$ and $V_2(t)$ are measured in volts.

1. Write, in the form $V(t) = A\cos\hspace{0.05em}(\omega t+\varphi)$, an expression for the total voltage in the circuit at time $t$ ms. 

2. Hence write down the highest voltage in the circuit. 

hard

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 2

calculator

hard

[Maximum mark: 6]

The revenues of a four seasons hotel can be modelled by the function

$\mathrm{R}(t) = 58.2\sin\hspace{0.05em}(0.0172t - 1.25) + 204$,

where $t$ is the number of days after midnight on $31$ December.

In a similar way, the operating costs of the hotel can be modelled by the function

$\mathrm{C}(t) = 31.4\sin\hspace{0.05em}(0.0172t + 1.14) + 85.0$.

Both $\mathrm{R}(t)$ and $\mathrm{C}(t)$ are measured in thousand dollars.

1. Show that the profits of the hotel can be modelled by the function $\mathrm{P}(t) = 83.9\sin\hspace{0.05em}(0.0172t-1.51) + 119$. 

2. According to the model, find:

1. the highest profit the hotel will make;

2. the date on which the highest profit will occur. 

hard

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 3

calculator

hard

[Maximum mark: 6]

Ali is swimming in a public pool with some of his friends. At time $t$ seconds, he $\text{encounters}$ some waves with height $h_1(t) = 0.15\sin\hspace{0.05em}(3t)$ from big Bobby jumping into the pool, and waves of height $h_2(t) = 0.08\sin\hspace{0.05em}(3t+1.25)$ from small Suzie jumping into the pool. Both $h_1(t)$ and $h_2(t)$ are measured in metres.

1. Write, in the form $h(t) = A\sin\hspace{0.05em}(\omega t+\varphi)$, an expression for the total height of the waves Ali encounters at time $t$ seconds. 

2. Find the times in the first $5$ seconds when Ali isn't affected by any waves. 

3. Find the first time when the waves reaching Ali has maximum height. 

hard

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Video (c)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Thank you Revision Village Members

#### #1 IB Math Resource

Revision Village was ranked the #1 IB Math Resources by IB Students & Teachers in 2021 & 2022.