Functions, MET2 2023 VCAA 9 MC
The function \(\large f\) is given by
\(f(x) = \begin {cases}
\tan\Bigg(\dfrac{x}{2}\Bigg) &\ \ 4 \leq x \leq 2\pi \\
\sin(ax) &\ \ \ 2\pi\leq x\leq 8
\end{cases}\)
The value of \(\large a\) for which \(\large f\) is continuous and smooth at \(\large x\) = \(2\pi\) is
- \(-2\)
- \(-\dfrac{\pi}{2}\)
- \(-\dfrac{1}{2}\)
- \(\dfrac{1}{2}\)
- \(2\)
Graphs, MET2 2023 VCAA 1 MC
The amplitude, \(A\), and the period, \(P\), of the function \(f(x)=-\dfrac{1}{2}\sin(3x+2\pi)\) are
- \(A=-\dfrac{1}{2},\ P=\dfrac{\pi}{3}\)
- \(A=-\dfrac{1}{2},\ P=\dfrac{2\pi}{3}\)
- \(A=-\dfrac{1}{2},\ P=\dfrac{3\pi}{2}\)
- \(A=\dfrac{1}{2},\ P=\dfrac{\pi}{3}\)
- \(A=\dfrac{1}{2},\ P=\dfrac{2\pi}{3}\)
Calculus, MET1 2022 VCAA 7
A tilemaker wants to make square tiles of size 20 cm × 20 cm.
The front surface of the tiles is to be painted with two different colours that meet the following conditions:
- Condition 1 - Each colour covers half the front surface of a tile.
- Condition 2 - The tiles can be lined up in a single horizontal row so that the colours form a continuous pattern.
An example is shown below.
There are two types of tiles: Type A and Type B.
For Type A, the colours on the tiles are divided using the rule `f(x)=4 \sin \left(\frac{\pi x}{10}\right)+a`, where `a \in R`.
The corners of each tile have the coordinates (0,0), (20,0), (20,20) and (0,20), as shown below.
- i. Find the area of the front surface of each tile. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
ii. Find the value of `a` so that a Type A tile meets Condition 1. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Type B tiles, an example of which is shown below, are divided using the rule `g(x)=-\frac{1}{100} x^3+\frac{3}{10} x^2-2 x+10`.
- Show that a Type B tile meets Condition 1. (3 marks)
--- 8 WORK AREA LINES (style=lined) ---
- Determine the endpoints of `f(x)` and `g(x)` on each tile. Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2. (2 marks)
--- 8 WORK AREA LINES (style=lined) ---
Graphs, MET1 2022 VCAA 6
The graph of `y=f(x)`, where `f:[0,2 \pi] \rightarrow R, f(x)=2 \sin(2x)-1`, is shown below.
- On the axes above, draw the graph of `y=g(x)`, where `g(x)` is the reflection of `f(x)` in the horizontal axis. (2 marks)
--- 0 WORK AREA LINES (style=lined) ---
- Find all values of `k` such that `f(k)=0` and `k \in[0,2 \pi]`. (3 marks)
--- 5 WORK AREA LINES (style=lined) ---
- Let `h: D \rightarrow R, h(x)=2 \sin(2x)-1`, where `h(x)` has the same rule as `f(x)` with a different domain.
- The graph of `y=h(x)` is translated `a` units in the positive horizontal direction and `b` units in the positive vertical direction so that it is mapped onto the graph of `y=g(x)`, where `a, b \in(0, \infty)`.
-
- Find the value for `b`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Find the smallest positive value for `a`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Hence, or otherwise, state the domain, `D`, of `h(x)`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Find the value for `b`. (1 mark)
Functions, MET1 2021 VCAA 3
Consider the function `g: R -> R, \ g(x) = 2sin(2x).`
- State the range of `g`. (1 mark)
- State the period of `g`. (1 mark)
- Solve `2 sin(2x) = sqrt3` for `x ∈ R`. (3 marks)
Calculus, MET1-NHT 2018 VCAA 7
Let `f : [ 0, (pi)/(2)] → R, \ f(x) = 4 cos(x)` and `g : [0, (pi)/(2)] → R, \ g(x) = 3 sin(x)`.
- Sketch the graph of `f` and the graph of `g` on the axes provided below. (2 marks)
`qquad qquad `
- Let `c` be such that `f(c) = g(c)`, where `c∈[0, (pi)/(2)]`
Find the value of `sin(c)` and the value of `cos(c)`. (3 marks) - Let `A` be the region enclosed by the horizontal axis, the graph of `f` and the graph of `g`.
-
- Shade the region `A` on the axes provided in part a. and also label the position of `c` on the horizontal axis. (1 mark)
- Calculate the area of the region `A`. (3 marks)
Trigonometry, MET2-NHT 2019 VCAA 2
The wind speed at a weather monitoring station varies according to the function
`v(t) = 20 + 16sin ((pi t)/(14))`
where `v` is the speed of the wind, in kilometres per hour (km/h), and `t` is the time, in minutes, after 9 am.
- What is the amplitude and the period of `v(t)`? (2 marks)
- What are the maximum and minimum wind speeds at the weather monitoring station? (1 mark)
- Find `v(60)`, correct to four decimal places. (1 mark)
- Find the average value of `v(t)` for the first 60 minutes, correct to two decimal places. (2 marks)
A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function
`v_1(t) = 28 + 18 sin((pi(t - k))/(7))`
where `v_1` is the speed of the wind, in kilometres per hour, `t` is the time, in minutes, after 9 am and `k ∈ R^+`. The wind speed after 9 am is shown in the diagram below.
- Find the smallest value of `k`, correct to four decimal places, such that `v(t)` and `v_1(t)` are equal and are both increasing at 10 am. (2 marks)
- Another possible value of `k` was found to be 31.4358
Using this value of `k`, the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.
i. Find the value of `t` at which a signal is first sent, correct to two decimal places. (1 mark)
ii. Find the proportion of one cycle, to the nearest whole percent, for which `v_1 > 38`. (2 marks)
- Let `f(x) = 20 + 16 sin ((pi x)/(14))` and `g(x) = 28 + 18 sin ((pi(x - k))/(7))`.
The transformation `T([(x),(y)]) = [(a \ \ \ \ 0),(0 \ \ \ \ b)][(x),(y)] + [(c),(d)]` maps the graph of `f` onto the graph of `g`.State the values of `a`, `b`, `c` and `d`, in terms of `k` where appropriate. (3 marks)
Calculus, MET2 2019 VCAA 3
During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.
The strength, `f`, of a simple dual-tone frequency signal is given by the function `f(t) = sin((pi t)/3) + sin ((pi t)/6)`, where `t` is a measure of time and `t >= 0`.
Part of the graph of `y = f(t)` is shown below
- State the period of the function. (1 mark)
- Find the values of `t` where `f(t) = 0` for the interval `t in [0, 6]`. (1 mark)
- Find the maximum strength of the dual-tone frequency signal, correct to two decimal places. (1 mark)
- Find the area between the graph of `f` and the horizontal axis for `t in [0, 6]`. (2 marks)
Let `g` be the function obtained by applying the transformation `T` to the function `f`, where
`T([(x), (y)]) = [(a, 0), (0, b)] [(x), (y)] + [(c), (d)]`
and `a, b, c` and `d` are real numbers.
- Find the values of `a, b, c` and `d` given that `int_2^0 g(t)\ dt + int_2^6 g(t)\ dt` has the same area calculated in part d. (2 marks)
- The rectangle bounded by the line `y = k, \ k in R^+`, the horizontal axis, and the lines `x = 0` and `x = 12` has the same area as the area between the graph of `f` and the horizontal axis for one period of the dual-tone frequency signal.
Find the value of `k`. (2 marks)
Graphs, MET2 2019 VCAA 10 MC
Which one of the following statements is true for `f: R -> R, \ f(x) = x + sin(x)`?
- The graph of `f` has a horizontal asymptote
- There are infinitely many solutions to `f(x) = 4`
- `f` has a period of `2 pi`
- `f prime (x) >= 0` for `x in R`
- `f prime (x) = cos(x)`
Graphs, MET2 2019 VCAA 1 MC
Let `f: R -> R,\ \ f(x) = 3 sin ((2x)/5) - 2`.
The period and range of `f` are respectively
- `5 pi` and `[-3, 3]`
- `5 pi` and `[-5, 1]`
- `5 pi` and `[-1, 5]`
- `(5 pi)/2` and `[-5, 1]`
- `(5 pi)/2` and `[-3, 3]`
Graphs, MET2 2017 VCAA 1 MC
Let `f : R → R, \ f (x) = 5sin(2x) - 1`.
The period and range of this function are respectively
- `π\ text(and)\ [−1, 4]`
- `2π\ text(and)\ [−1, 5]`
- `π\ text(and)\ [−6, 4]`
- `2π\ text(and)\ [−6, 4]`
- `4π\ text(and)\ [−6, 4]`
Graphs, MET1 SM-Bank 27
Graphs, MET2 2016 VCAA 6 MC
Consider the graph of the function defined by `f: [0, 2 pi] -> R,\ f(x) = sin (2x).`
The square of the length of the line segment joining the points on the graph for which `x = pi/4 and x = (3 pi)/4` is
- `(pi^2 + 16)/4`
- `pi + 4`
- `4`
- `(3 pi^2 + 16 pi)/4`
- `(10 pi^2)/16`
Algebra, MET2 2012 VCAA 1 MC
The function with rule `f(x) = −3sin((pix)/5)` has period
- `3`
- `5`
- `10`
- `pi/5`
- `pi/10`
Functions, MET1 2010 VCAA 4a
Write down the amplitude and period of the function
`qquad f: R -> R,\ \ f(x) = 4 sin ((x + pi)/3)`. (2 marks)
Functions, MET1 2012 VCAA 6
The graphs of `y = cos (x) and y = a sin (x)`, where `a` is a real constant, have a point of intersection at `x = pi/3.`
- Find the value of `a`. (2 marks)
- If `x in [0, 2 pi]`, find the `x`-coordinate of the other point of intersection of the two graphs. (1 mark)
Graphs, MET2 2015 VCAA 1 MC
Let `f: R -> R,\ f(x) = 2sin(3x) - 3.`
The period and range of this function are respectively
- `text(period) = (2 pi)/3 and text(range) = text{[−5, −1]}`
- `text(period) = (2 pi)/3 and text(range) = text{[−2, 2]}`
- `text(period) = pi/3 and text(range) = text{[−1, 5]}`
- `text(period) = 3 pi and text(range) = text{[−1, 5]}`
- `text(period) = 3 pi and text(range) = text{[−2, 2]}`
Graphs, MET2 2008 VCAA 10 MC
The range of the function `f: [pi/8, pi/3) -> R,\ f(x) = 2 sin (2x)` is
- `(sqrt 2, sqrt 3]`
- `[sqrt 2, 2)`
- `[sqrt 2, 2]`
- `(sqrt 2, sqrt 3)`
- `[sqrt 2, sqrt 3)`