Functions — Free IB Maths AA SL Practice Questions

1FoundationSAQ-SReal-world applications of functions (e.g. growth models)s7 marksPaper 1~11 min

A small business purchases a delivery van for

28\,000

dollars. The value of the van depreciates linearly each year. After

5

years, the van is valued at

13\,000

dollars.

(a)

Calculate the annual depreciation of the van in dollars per year. [2 marks]

(b)

Find the value of the van after

8

years. [2 marks]

(c)

The business plans to sell the van when its value first reaches

1\,000

dollars. Determine the year in which this occurs and evaluate whether the linear model is appropriate for this prediction. [3 marks]

Solutions

2MasterySAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min

The population of a small town is modelled by

P(t) = 50\,000 + 1200t

, where

P

is the population and

t

is the number of years after 1 January 2020.

(a)

Show that

P(t+1) - P(t) = 1200

for all values of

t

. [2 marks]

(b)

Find the value of

t

for which the population first exceeds

65\,000

. [2 marks]

(c)

A government grant is available only towns whose population remains below

80\,000

throughout the entire period

0 \leq t \leq 20

. Determine, with justification, whether this town is eligible for the grant. [1 mark]

Solutions

3ChallengeSAQ-LReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min

A company produces solar panels. The total revenue

R

(in thousands of dollars) from selling

x

thousand panels is given by

R(x) = 120x - 3x^2

, for

0 \leq x \leq 40

. The total cost

C

(in thousands of dollars) is given by

C(x) = 20x + 400

.

(a)

Show that the profit function is

P(x) = -3x^2 + 100x - 400

. [2 marks]

(b)

Calculate the value of

x

that maximises

P(x)

, and the corresponding maximum profit. [3 marks]

(c)

For

x > 40

, the revenue function changes to

R_2(x) = 80x - x^2

, while

C(x)

remains the same. Evaluate whether the company can achieve a profit greater than the maximum found in part (b) for any

x > 40

. Justify your answer. [3 marks]

Solutions

4FoundationSAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min

The height of a plant is recorded weekly. At week

1

, the plant is

4\,\text{cm}

tall. At week

4

, the plant is

13\,\text{cm}

tall. The growth is assumed to be linear.

(a)

Find the linear function

H(w) = mw + c

that gives the height

H

in cm after

w

weeks. [3 marks]

(b)

A second plant grows according to

G(w) = 5w - 3

. Determine the week at which both plants are the same height, and state which plant is taller at week

10

. [2 marks]

Solutions

5MasterySAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min

A biologist models the number of bacteria in a culture as

N(t) = 200 \times 2^{kt}

, where

N

is the number of bacteria and

t

is the time in hours after the start of the experiment. After 4 hours, there are 800 bacteria.

(a)

Show that

k = \dfrac{1}{2}

. [2 marks]

(b)

Write down the number of bacteria after 6 hours. [1 mark]

(c)

Determine the value of

t

at which the number of bacteria first exceeds 50 000. Give your answer correct to the nearest minute. [2 marks]

Solutions

6FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 1~12 min

A function

f

is defined by

f(x) = \dfrac{2x + 3}{x - 1}

, for

x \in \mathbb{R},\ x \neq 1

.

(a)

State the equation of the horizontal asymptote of

f

. [1 mark]

(b)

Show that

f(a) = f(b) \Rightarrow a = b

, and hence state whether

f

is a one-to-one function. [3 marks]

(c)

Write down the range of

f

. [1 mark]

(d)

Find

f^{-1}(x)

, stating its domain. [3 marks]

Solutions

7MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

A function

g

is defined by

g(x) = x^2 - 4x + 5

, for

x \geq 2

,

x \in \mathbb{R}

.

(a)

Show that

g

is strictly increasing on its domain, and hence that

g

is a one-to-one function. [3 marks]

(b)

Hence find the range of

g

. [2 marks]

Solutions

8ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 1~12 min

Consider the function

f: \mathbb{R} \setminus \{1\} \to \mathbb{R}

defined by

f(x) = \dfrac{2x + 3}{x - 1}

. - A function

f: A \to B

is one-to-one if $f

(a)

= f

(b)

\Rightarrow a = b

for all

a, b \in A$. - A function $f: A \to B

is onto if for every

y \in B$, there exists $x \in A$ such that $f(x) = y$. (a) Justify algebraically that $f$ is a one-to-one function. [4] (i) Show that the range of $f

is

\mathbb{R} \setminus \{2\}$. [3]
(ii) Hence determine, with justification, whether $f$ is an onto function. [1 mark]

Solutions

9FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

Consider the function

g: \mathbb{R} \to \mathbb{R}

defined by

g(x) = x^3 - 4x

.

(a)

State what it means for a function

f: A \to B

to be onto (surjective). [1 mark]

(b)

Show that

g

is onto. [2 marks]

(c)

The codomain of

g

is now restricted to

[0, +\infty)

, giving

h: D \to [0, +\infty)

where

h(x) = x^3 - 4x

and

D \subseteq \mathbb{R}

. Find the smallest domain

D

such that

h

is onto

[0, +\infty)

. [2 marks]

Solutions

10MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

The function

h

is defined by

h(x) = \dfrac{1}{x^2+1}

, for

x \in \mathbb{R}

.

(a)

Show that

h

is not a one-to-one function. [2 marks]

(b)

The domain of

h

is restricted to

x \geq 0

. Show that

h

is now one-to-one, and find an expression for

h^{-1}(x)

, stating its domain. [3 marks]

Solutions

11FoundationSAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The function

f

is defined by

f(x) = x^2 - 4x + 3

.

(a)

Write down the coordinates of the vertex of the graph of

f

. [1 mark]

(b)

The graph of

f

is stretched vertically by a scale factor of

2

about the

x

-axis. Write down the vertex form of the resulting function. [2 marks]

(c)

The graph from part (b) is then translated

1

unit to the right. Find the equation of the resulting function

g

, giving your answer in the form

g(x) = a(x-h)^2 + k

. [2 marks]

Solutions

12MasterySAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The function

f

is defined by

f(x) = x^2 + 4x + 5

, for

x \in \mathbb{R}

.

(a)

The graph of

f

is translated by the vector

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

to give the graph of a function

g

. Show that

g(x) = x^2 - 2x - 1

. [2 marks]

(b)

The graph of

g

is then reflected in the

y

-axis to give the graph of a function

h

. Find an expression for

h(x)

in the form

h(x) = ax^2 + bx + c

. [3 marks]

Solutions

13ChallengeSAQ-LTranslation, reflection, stretching, and compressions8 marksPaper 1~12 min

The function

f

is defined by

f(x) = x^2

, for

x \in \mathbb{R}

. The graph of

f

undergoes the following transformations in sequence: 1. A horizontal stretch by scale factor

\frac{1}{2}

2. A translation by the vector

\begin{pmatrix} 1 \\ -3 \end{pmatrix}

3. A reflection in the

x

-axis Let

g

be the resulting function.

(a)

Determine an expression for

g(x)

in the form

a(x-h)^2 + k

, where

a, h, k \in \mathbb{R}

. [4 marks]

(b)

The graphs of

g

and

f

intersect at two points. Determine the

x

-coordinates of these points and hence find the exact area of the region enclosed between the two graphs. [4 marks]

Solutions

14FoundationSAQ-STranslation, reflection, stretching, and compressions4 marksPaper 1~6 min

The graph of

y = f(x)

consists of two straight line segments: from

(-4,\,0)

to

(0,\,4)

, and from

(0,\,4)

to

(4,\,0)

. The function

g

is obtained by translating

f

by

2

units to the left and

1

unit down.

(a)

Write down the coordinates of the vertex of

g

. [1 mark]

(b)

Find the equation of the right-hand line segment of

g

in the form

y = mx + c

. [3 marks]

Solutions

15MasterySAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The following shows the graph of

y = f(x)

, where

f(x) = -x^2 + 4x - 3

, for

x \in \mathbb{R}

. The graph of

f

is reflected in the

x

-axis to give the graph of

g

.

(a)

Write down

g(x)

. [1 mark]

(b)

Express

g(x)

in the form

(x - h)^2 + k

, where

h, k \in \mathbb{Z}

. [2 marks]

(c)

The graph of

g

is then stretched horizontally by a scale factor of

2

about the

y

-axis to give the graph of

p

. Write down the coordinates of the vertex of

p

. [2 marks]

Solutions

16FoundationSAQ-SSine, cosine, and tangent functionss5 marksPaper 1~8 min

A water sprinkler rotates back and forth. The horizontal distance

d

(in metres) from the sprinkler to the water jet's landing point is modelled by

d(\theta) = 15\sin(2\theta)

where

\theta

is the angle of the sprinkler head from the vertical, in radians.

(a)

Calculate the value of

d

when

\theta = \dfrac{\pi}{6}

. [2 marks]

(b)

Find all values of

\theta

in the interval

0 \leq \theta \leq \dfrac{\pi}{2}

for which

d(\theta) = 7.5

. [3 marks]

Solutions

17MasterySAQ-SSine, cosine, and tangent functionss7 marksPaper 1~11 min

The graph of

y = a\cos(bx) + c

for

0 \leq x \leq 2\pi

has a maximum point at

(0,\, 5)

and a minimum point at

(\pi,\, -1)

.

(a)

Show that

a = 3

,

b = 1

, and

c = 2

. Your working must include: - use of the maximum and minimum values to find

a

and

c

- use of the period to find

b

[3 marks]

(b)

Hence, find all values of

x

in

0 \leq x \leq 2\pi

that satisfy

3\cos x + 2 = \dfrac{1}{2}

, giving your answers in exact form. [2 marks]

(c)

The line

y = k

intersects the graph of

y = 3\cos x + 2

at exactly one point in the open interval

0 < x < 2\pi

. Determine the value of

k

and the corresponding

x

-value. [2 marks]

Solutions

18ChallengeSAQ-LSine, cosine, and tangent functionss10 marksPaper 1~15 min

A water wheel of radius

5

metres rotates anticlockwise at a constant angular speed. The point

P

is located at the lowest point on the wheel at time

t = 0

seconds. The height

h(t)

metres of

P

above the water level is modelled by

h(t) = a\sin(bt) + c, \quad t \geq 0

After

2

seconds,

P

reaches its maximum height of

8

metres for the first time.

(a)

Write down the values of

a

and

c

. [2 marks]

(b)

Show that

b = \dfrac{\pi}{4}

. [2 marks]

(c)

Find the first time

t > 0

at which

h(t) = 2

, giving your answer correct to three significant figures. [3 marks]

(d)

The wheel is submerged whenever

h(t) < 0

. Determine the total length of time, per complete revolution, for which

P

is below the water level. Give your answer in exact form. [3 marks]

Solutions

19FoundationSAQ-SSine, cosine, and tangent functionss5 marksPaper 1~8 min

A ladder of length

5

metres leans against a vertical wall. The angle between the ladder and the ground is

\theta

, where

0 < \theta < \dfrac{\pi}{2}

. The horizontal distance from the base of the ladder to the wall is

x

metres and the vertical height reached by the ladder is

h

metres.

(a)

Show that

x = 5\cos\theta

. [1 mark]

(b)

Given that

\sin\theta = \dfrac{3}{5}

, find the exact value of

x

. [2 marks]

(c)

Find the exact area of the triangle formed by the ladder, the wall, and the ground. [2 marks]

Solutions

20MasterySAQ-SSine, cosine, and tangent functionss5 marksPaper 1~8 min

Consider the function

f(x) = a\cos x + b

, where

a

and

b

are constants. The graph of

y = f(x)

passes through the points

(0,\,5)

and

(\pi,\,-1)

.

(a)

Show that

a = 3

and

b = 2

. [3 marks]

(b)

Hence, find the exact value of

f\!\left(\dfrac{\pi}{3}\right)

. [2 marks]

Solutions

21MasterySAQ-SReal-world applications of functions (e.g. growth models)s8 marksPaper 2~12 min

The population of a small town is growing linearly. In 2015, the population was

12\,400

people. In 2023, the population was

14\,200

people. Let

P

represent the population

t

years after 2015.

(a)

Find the linear function

P(t)

that models the population growth. [3 marks]

(b)

Using your model, determine the year in which the population will reach

16\,000

people. [2 marks]

(c)

The town has a maximum sustainable population of

18\,000

people. Determine the year in which the model predicts this limit will be reached, and evaluate one reason why the linear model may not be reliable for this prediction. [3 marks]

Solutions

22MasterySAQ-SReal-world applications of functions (e.g. growth models)s6 marksPaper 2~9 min

A biologist is studying the growth of a bacterial culture. The number of bacteria,

N

, after

t

hours is modelled by

N(t) = 500\,e^{kt}

, where

k

is a positive constant. After

3

hours there are

1200

bacteria.

(a)

Find the value of

k

, correct to three significant figures. [3 marks]

(b)

Find the time at which the number of bacteria first exceeds

10\,000

, giving your answer correct to the nearest minute. [3 marks]

Solutions

23MasterySAQ-SReal-world applications of functions (e.g. growth models)s8 marksPaper 2~12 min

A biologist is studying the growth of a bacterial colony. At 09:00, the colony has an area of

8.0\,\text{cm}^2

. The area of the colony increases by a constant amount each hour. At 14:00, the area is

21.5\,\text{cm}^2

.

(a)

Calculate the constant hourly increase in area. [2 marks]

(b)

Find the area of the colony at 18:00. [2 marks]

(c)

A second model proposes that the area grows exponentially, with the same values at 09:00 and 14:00. Determine the ratio of the area predicted by the exponential model to the area predicted by the linear model at 18:00, and hence evaluate which model predicts faster growth over the interval 09:00 to 18:00. [4 marks]

Solutions

24MasterySAQ-SReal-world applications of functions (e.g. growth models)s6 marksPaper 2~9 min

The amount

A

(in grams) of a radioactive substance remaining after

t

hours is modelled by

A(t) = A_0 e^{-\lambda t}

. The graph of

A

against

t

passes through the points

(0,\,240)

and

(10,\,120)

.

(a)

Show that

\lambda = \dfrac{\ln 2}{10}

. [2 marks]

(b)

Calculate the value of

\lambda

correct to three significant figures. [1 mark]

(c)

The substance becomes safe to handle when its amount falls below

15\,\text{g}

. Using your value of

\lambda

from part (b), determine the time at which the substance first becomes safe to handle. Give your answer correct to the nearest hour. [3 marks]

Solutions

25MasterySAQ-SReal-world applications of functions (e.g. growth models)s6 marksPaper 2~9 min

A city's population is growing linearly. In the year 2000, the population was

45\,000

. In the year 2010, the population was

54\,000

.

(a)

State the annual rate of population growth. [1 mark]

(b)

Calculate the population in the year 2025. [2 marks]

(c)

A second city has population model

Q(t) = 30\,000 + 1\,400t

, where

t

is the number of years after 2000. Determine the year in which the two cities will have the same population, and state which city has the larger population in the year 2040. [3 marks]

Solutions

26MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

A company designs a water tank. The volume of water in the tank,

V

litres, at time

t

hours after filling begins is modelled by

V(t) = 2000 - 1000e^{-0.2t}, \quad t \geq 0.

(a)

Determine the range of

V

. [2 marks]

(b)

The tank is considered full when the volume reaches

1900

litres. Find the exact time, in hours, for the tank to become full, giving your answer in the form

a\ln b

where

a, b \in \mathbb{Q}^+

. [2 marks]

(c)

Hence state the time for the tank to become full, correct to the nearest minute. [2 marks]

Solutions

27MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s7 marksPaper 2~11 min

Consider the function

g(x) = x^2 - 4x + 5

, for

x \in \mathbb{R}

.

(a)

Write

g(x)

in the form

a(x-h)^2 + k

, where

a

,

h

,

k \in \mathbb{Z}

. [1 mark]

(b)

Hence explain why

g

is not a one-to-one function. [2 marks]

(c)

The domain of

g

is restricted to

x \geq p

so that

g

becomes one-to-one. State the minimum value of

p

. [1 mark]

(d)

For the restricted domain part (c), find

g^{-1}(x)

, stating its domain. [3 marks]

Solutions

28MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

A function

h

is defined by

h(x) = \dfrac{2x-1}{x+3}

, for

x \in \mathbb{R},\ x \neq -3

.

(a)

Justify that

h

is a one-to-one function. [2 marks]

(b)

The codomain of

h

is

\mathbb{R} \setminus \{c\}

. Find the value of

c

. [2 marks]

(c)

Hence, calculate the value of

h^{-1}(5)

. [2 marks]

Solutions

29MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 2~12 min

The function

f

is defined by

f(x) = \ln(x^2 - 4x + 5)

, for

x \in \mathbb{R}

.

(a)

Determine the domain of

f

. [2 marks]

(b)

Show that

f

is not a one-to-one function. [2 marks]

(c)

The domain of

f

is restricted to

x \geq 2

so that

f

becomes one-to-one. Find an expression for

f^{-1}(x)

, stating its domain. [4 marks]

Solutions

30MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s6 marksPaper 2~9 min

A company designs a logo consisting of a curve and a line. The curve is modelled by

f(x) = -x^2 + 4x + 5

for

0 \leq x \leq 5

. The line is modelled by

g(x) = 2x + 1

for

0 \leq x \leq 4

.

(a)

Determine whether

f

is a one-to-one function the given domain. Justify your answer. [2 marks]

(b)

(i) Find the range of

f

. [2]
(ii) Find the range of

g

. Hence determine whether

g

is surjective onto the range of

f

, giving a specific counterexample if it is not. > [2 marks]

Solutions

31MasterySAQ-STranslation, reflection, stretching, and compressions13 marksPaper 2~20 min

The below shows the graph of

y = f(x)

for

-4 \leq x \leq 4

. The graph has a local maximum at

A(-2,\,5)

and a local minimum at

B(1,\,-3)

. The graph of

f

is transformed to obtain the graph of

g

, where

g(x) = 2f(x+1) - 3

.

(a)

Calculate the coordinates of the images of

A

and

B

on the graph of

g

. [4 marks]

(b)

The function

f

has a horizontal asymptote at

y = 0

. Write down the equation of the horizontal asymptote of

g

. [1 mark]

(c)

Determine the range of values of

x

for which

g(x) > 0

, given that

g

has exactly two zeros and that the zero with the larger

x

-value lies between the images of

A

and

B

found in part (a). Justify your answer. [1] Revised (c) — replacing trivial reflection with genuine AO3 discriminator: (c) The graph of

g

is further transformed by the mapping

(x,\,y) \mapsto (x,\,-y+k)

for some constant

k

. The image of

A'

under this combined transformation has the same

y

-coordinate as the image of

B'

. Find the value of

k

. [1] Final clean version: (a) Calculate the coordinates of the images of

A

and

B

on the graph of

g

. [4] (b) The function

f

has a horizontal asymptote at

y = 0

. Write down the equation of the horizontal asymptote of

g

. [1] (c) The graph of

g

is reflected in the

x

-axis to give the graph of

h

. The graph of

h

is then translated by

\begin{pmatrix}0\\k\end{pmatrix}

so that the image of

A'

and the image of

B'

have the same

y

-coordinate. Find the value of

k

. [1 mark]

Solutions

32MasterySAQ-STranslation, reflection, stretching, and compressions7 marksPaper 2~11 min

The graph of

y = f(x)

consists of a straight line segment from

A(-4, -2)

to

B(0, 2)

, and another straight line segment from

B(0, 2)

to

C(4, 0)

. The graph of

f

is transformed to the graph of

g

by a horizontal stretch by scale factor

2

, followed by a reflection in the

y

-axis, followed by a translation by the vector

\begin{pmatrix} 1 \\ 0 \end{pmatrix}

.

(a)

Calculate the coordinates of the images of

A

,

B

, and

C

under this composite transformation. [3 marks]

(b)

Calculate the gradient of the image of segment

BC

on the graph of

g

. [2 marks]

(c)

Hence find the equation of the image of segment

AB

on the graph of

g

, giving your answer in the form

y = mx + c

. [2 marks]

Solutions

33MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The graph of

y = f(x)

has a local maximum at

(-2,\,5)

and a local minimum at

(1,\,-3)

. The curve passes through

(0,\,1)

. The function

g

is defined by

g(x) = f(2x) - 4.

(a)

Determine the coordinates of the local maximum and local minimum of

g

. [4 marks]

(b)

The graph of

g

crosses the

y

-axis at the point

P

. Write down the coordinates of

P

, and hence state the number of

x

-intercepts of

g

that lie in the interval

-1 \leq x \leq 1

. [2 marks]

Solutions

34MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The graph of

y = f(x)

for

-4 \leq x \leq 4

is shown below. It has

x

-intercepts at

x = -3

,

x = 0

, and

x = 2

. The function

g

is obtained from

f

by the following sequence of transformations: - a vertical compression with scale factor

\dfrac{1}{2}

, - followed by a reflection in the

x

-axis, - followed by a translation of

3

units in the positive

x

-direction.

(a)

Determine the

x

-intercepts of the graph of

g

. [3 marks]

(b)

The function

f

has a local maximum at

(-1.5,\; 2)

. Calculate the coordinates of the corresponding point on the graph of

g

. [3 marks]

Solutions

35MasterySAQ-STranslation, reflection, stretching, and compressions6 marksPaper 2~9 min

The function

f

is defined by

f(x) = x^2 - 4x + 5

for

x \in \mathbb{R}

.

(a)

Express

f(x)

in the form

(x - h)^2 + k

, where

h, k \in \mathbb{Z}

. [2 marks]

(b)

The graph of

f

is translated by the vector

\begin{pmatrix} 2 \\ -3 \end{pmatrix}

to give the graph of

g

. Find an expression for

g(x)

in the form

(x - a)^2 + b

, where

a, b \in \mathbb{Z}

. [2 marks]

(c)

The graph of

g

is reflected in the

x

-axis to give the graph of

p

, where

p(x) = -(x-4)^2 + 2

. Determine the number of intersections between the graphs of

g

and

p

, justifying your answer. [2 marks]

Solutions

36MasterySAQ-SSine, cosine, and tangent functionss7 marksPaper 2~11 min

A Ferris wheel at a theme park has a diameter of

40

metres. The lowest point of the wheel is

2

metres above the ground. The wheel completes one full revolution in

60

seconds. A rider starts at the lowest point at time

t = 0

seconds. The height

h(t)

metres of the rider above the ground after

t

seconds is modelled by

h(t) = a\cos(bt) + c

where

a

,

b

, and

c

are constants.

(a)

State the value of

a

. [1 mark]

(b)

Calculate the value of

b

. [2 marks]

(c)

State the value of

c

. [1 mark]

(d)

Calculate the height of the rider after

25

seconds. Give your answer correct to 3 significant figures. [3 marks]

Solutions

37MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

The depth of water in a harbour varies sinusoidally over time due to tides. At low tide at 4:00 AM, the depth is

3

metres. At high tide at 10:00 AM, the depth is

11

metres. The depth

d(t)

metres at time

t

hours after midnight is modelled by

d(t) = A\sin(B(t - C)) + D

where

A

,

B

,

C

,

D

are constants.

(a)

State the value of

A

. [1 mark]

(b)

Calculate the value of

B

. [2 marks]

(c)

State the value of

D

. [1 mark]

(d)

Calculate the depth of water at 2:00 PM. Give your answer correct to 3 significant figures. [2 marks]

Solutions

38MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

The voltage

V(t)

in an electrical circuit is modelled by

V(t) = 240\sin(100\pi t)

, where

t

is time in seconds and

V

is in volts.

(a)

State the amplitude of the voltage. [1 mark]

(b)

State the period of the voltage. [1 mark]

(c)

Calculate the voltage when

t = 0.0025

seconds. Give your answer in exact form. [2 marks]

(d)

Determine the smallest positive value of

t

for which

V(t) = 120

volts. Give your answer in exact form. [2 marks]

Solutions

39MasterySAQ-SSine, cosine, and tangent functionss6 marksPaper 2~9 min

A point on a bicycle tyre traces a circular path as the bicycle moves forward. Its height above the ground is modelled by

h(\theta) = 35 + 35\sin\!\left(\theta - \frac{\pi}{2}\right)

where

h

is in centimetres and

\theta

is the total angle, in radians, through which the tyre has rotated from its starting position.

(a)

Show that

h(\theta)

can be written as

h(\theta) = 35 - 35\cos\theta

. [2 marks]

(b)

State the maximum height of the point above the ground and find the smallest positive value of

\theta

at which this maximum occurs. [2 marks]

(c)

Determine the first positive value of

\theta

for which

h = 52.5\,\text{cm}

. Give your answer correct to 3 significant figures. [2 marks]

Solutions

40MasterySAQ-SSine, cosine, and tangent functionss7 marksPaper 2~11 min

The number of visitors to a museum on a particular day is modelled by

N(t) = 500 + 300\sin\!\left(\dfrac{\pi}{12}(t - 9)\right)

, where

t

is the number of hours after midnight. The museum is open from 9:00 AM to 5:00 PM.

(a)

State the maximum number of visitors predicted by the model. [1 mark]

(b)

Calculate the time at which this maximum occurs. Give your answer in hours and minutes. [2 marks]

(c)

Calculate the number of visitors at 1:00 PM, giving your answer correct to the nearest whole number. [2 marks]

(d)

Determine the first time after opening when the number of visitors equals 650. Give your answer in hours and minutes. [2 marks]

Solutions

Functions — Free Maths AA SL Practice Questions