Number and Algebra — Free IB Maths AI SL Practice Questions

1FoundationSAQ-SExponential functions and their graphs5 marksPaper 1~8 min

The graph of an exponential function

f(x) = a \times b^x

passes through the points

(0, 8)

and

(2, 32)

.

(a)

State the value of

a

. [1 mark]

(b)

Calculate the value of

b

. [2 marks]

(c)

Determine the value of

x

for which

f(x) = 8192

. [2 marks]

Solutions

2MasterySAQ-SExponential functions and their graphs5 marksPaper 1~8 min

A colony of penguins on a remote island has a population modelled by

P(t) = 1200 \times (1.03)^t

where

P

is the population and

t

is the number of years after 1 January 2010.

(a)

State the population of the colony on 1 January 2010. [1 mark]

(b)

Calculate the population 1 January 2020, giving your answer to the nearest whole number. [2 marks]

(c)

Determine the year in which the population first exceeds

2000

. [2 marks]

Solutions

3ChallengeSAQ-LExponential functions and their graphs8 marksPaper 1~12 min

The graph of an exponential function

f(x) = a \times b^x + c

passes through the points

(0,\,7)

,

(2,\,15)

, and

(4,\,39)

.

(a)

Determine the values of

a

,

b

, and

c

. Give

b

correct to 3 significant figures. [4 marks]

(b)

Write down the equation of the horizontal asymptote of

f(x)

. [1 mark]

(c)

The parameter

c

is increased by 5 to give a new function

g(x) = a \times b^x + (c+5)

. Determine the coordinates of the

y

-intercept of

g(x)

. [1 mark]

(d)

A student claims that increasing

c

by 5 does not change the rate at which

f(x)

grows for large values of

x

. Justify whether this claim is correct. [2 marks]

Solutions

4FoundationSAQ-SExponential functions and their graphs5 marksPaper 1~8 min

The graph of

f(x) = 3 \times 2^x

is shown for

-2 \leq x \leq 5

.

(a)

State the

y

-intercept of the graph. [1 mark]

(b)

Calculate the value of

f(2)

. [2 marks]

(c)

Find the value of

x

for which

f(x) = 96

. [2 marks]

Solutions

5MasterySAQ-SExponential functions and their graphs5 marksPaper 1~8 min

The graph of an exponential function

f(x) = a \times b^x

passes through the points

(0, 5)

and

(3, 40)

.

(a)

Write down the value of

a

. [1 mark]

(b)

Hence, find the value of

b

. [2 marks]

(c)

On the following grid, sketch the graphs of

y = f(x)

and

y = f(x) - 10

for

-2 \leq x \leq 4

. On each graph, clearly label the

y

-intercept and the horizontal asymptote. *[A blank coordinate grid with axes labelled

x

and

y

, ranging from

x = -2

to

4

and

y = -15

to

50

, with grid lines.]* [2 marks]

Solutions

6FoundationSAQ-SBinomial expansion and coefficients7 marksPaper 1~11 min

A college basketball team plays

8

games in a season. The probability of winning any particular game is

0.25

, independently of other games.

(a)

State the value of the binomial coefficient

\dbinom{8}{3}

. [1 mark]

(b)

Find the probability that the team wins exactly

3

of the

8

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

(c)

Given that the team wins at least

1

game, find the probability that the team wins exactly

3

games. Give your answer correct to 3 significant figures. Binomial probability:

P(X = r) = \dbinom{n}{r} p^{r}(1-p)^{n-r}

[3 marks]

Solutions

7MasterySAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

The coefficient of

x^5

in the expansion of

(1 + ax)^8

is

1680

, where

a > 0

.

(a)

Show that

a^5 = 30

. [3 marks]

(b)

Hence find the value of

a

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

Solutions

8ChallengeSAQ-LBinomial expansion and coefficients8 marksPaper 1~12 min

A triangular number pattern is formed by writing the binomial coefficients in Pascal's triangle. The

2n

th row of Pascal's triangle contains the coefficients

\binom{2n}{0}, \binom{2n}{1}, \ldots, \binom{2n}{2n}

. Consider the expansion of

(1 + x)^{2n}

.

(a)

Write down the coefficient of

x^n

in the expansion of

(1 + x)^{2n}

in terms of

n

. [1 mark]

(b)

Show that the coefficient of

x^n

in

(1 + x)^{2n}

equals the middle term of the

2n

th row of Pascal's triangle. [1 mark]

(c)

Given that the coefficient of

x^n

in

(1 + x)^{2n}

is

252

, determine the value of

n

and hence find the sum of all coefficients in the expansion. [3 marks]

(d)

A student claims that for all positive integers

n

, the middle term of the

2n

th row of Pascal's triangle is always even. Determine whether this claim is correct, justifying your answer. [3 marks]

Solutions

9FoundationSAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

Consider the expansion of

(1+x)^6

.

(a)

Calculate

\dbinom{6}{2}

. [1 mark]

(b)

Show that the sum of all binomial coefficients in the row

n = 6

of Pascal's triangle equals

64

. [2 marks]

(c)

Find the value of

x

for which the term in

x^2

equals the term in

x^3

in the expansion of

(1+x)^6

, given

x \neq 0

. [2 marks]

Solutions

10MasterySAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

The coefficient of

x^3

in the expansion of

(2 + kx)^5

is

1080

, where

k > 0

.

(a)

Find the value of

k

. [3 marks]

(b)

Hence find the coefficient of

x^4

in the expansion of

(2 + kx)^5

. [2 marks]

Solutions

11FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min

A construction company is building a new housing estate. The first house is built

20\,\text{m}

from the main road. Each subsequent house is built

12\,\text{m}

further from the road than the previous house.

(a)

State the value of

u_1

and the value of

d

for this arithmetic sequence. [2 marks]

(b)

Find the distance from the main road to the

15

th house. [2 marks]

(c)

The estate has

n

houses in total. The furthest house must be built no more than

500\,\text{m}

from the main road. Find the maximum value of

n

. [3 marks]

Solutions

12MasterySAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min

A company is designing a trapezoidal solar panel. The panel is an isosceles trapezium. The three distinct lengths that appear in the panel — the two parallel sides and one non-parallel (slant) side — form an arithmetic sequence. The shortest parallel side has length

1.2\,\text{m}

and the longest parallel side has length

2.4\,\text{m}

.

(a)

Show that the common difference of the arithmetic sequence is

0.6\,\text{m}

. [2 marks]

(b)

Find the perpendicular height of the trapezium. [2 marks]

(c)

Hence find the area of the solar panel. [1 mark]

Solutions

13ChallengeSAQ-LDefinition and general term of arithmetic sequences8 marksPaper 1~12 min

A construction company is building a tower of concrete blocks for an art installation. The blocks are arranged in horizontal rows. The top row contains

4

blocks. Each subsequent row contains

3

more blocks than the row above it. The tower has

15

rows in total.

(a)

Determine the number of blocks in the bottom row. [2 marks]

(b)

Calculate the total number of blocks in the tower. [2 marks]

(c)

The company claims that by adding extra rows below the existing

15

rows, keeping the same common difference, the total number of blocks in the extended tower will reach exactly

1000

. Determine the total number of rows the extended tower would need, and evaluate whether the company's claim is valid. Booklet formulae provided:

u_n = u_1 + (n-1)d

S_n = \frac{n}{2}(2u_1 + (n-1)d)

S_n = \frac{n}{2}(u_1 + u_n)

[4 marks]

Solutions

14FoundationSAQ-SDefinition and general term of arithmetic sequences7 marksPaper 1~11 min

A scientist monitors the growth of a bacterial colony. On day 1 there are

500

bacteria. The number of bacteria increases by the same amount each day. On day 8 there are

1480

bacteria.

(a)

Write down the general term

u_n

for the number of bacteria on day

n

, identifying the values of

u_1

and

d

. [1 mark]

(b)

Show that the common difference

d = 140

. [2 marks]

(c)

Find the number of bacteria on day 15. [2 marks]

(d)

The scientist claims the colony will exceed

5000

bacteria before day 35. Determine whether this claim is correct. [2 marks]

Solutions

15MasterySAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min

A liquid cools in a container. Its temperature, in degrees Celsius, is recorded every

10

minutes. The temperatures form an arithmetic sequence. The temperature at

t = 10

minutes is

48^\circ\text{C}

, and the temperature at

t = 40

minutes is

30^\circ\text{C}

.

(a)

Show that the common difference of the sequence is

-6^\circ\text{C}

per

10

-minute interval. [2 marks]

(b)

Find the temperature at

t = 0

minutes. [1 mark]

(c)

Find the value of

t

at which the model predicts the temperature first reaches

0^\circ\text{C}

. [2 marks]

Solutions

16FoundationSAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min

The first three terms of a geometric sequence are

6

,

18

, and

54

.

(a)

State the common ratio,

r

, of the sequence. [1 mark]

(b)

Find the value of the 8th term,

u_8

. [2 marks]

(c)

Find the smallest value of

n

such that the sum of the first

n

terms exceeds

59\,000

. [2 marks]

Solutions

17MasterySAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min

A new savings account is opened with an initial deposit of USD 2000. At the end of each year, interest is added so that the amount in the account at the start of each year forms a geometric sequence. The amount in the account at the start of the third year is USD 2247.20.

(a)

Show that the common ratio of the geometric sequence is

1.06

. [2 marks]

(b)

Hence find the amount in the account at the start of the sixth year. Give your answer correct to the nearest dollar. [2 marks]

(c)

The account holder states: "By the start of the tenth year, the total interest earned will have exceeded USD 1000." Determine whether this statement is correct. [1 mark]

Solutions

18ChallengeSAQ-LDefinition and general term of geometric sequences8 marksPaper 1~12 min

Data

u_n = u_1 r^{n-1}

S_n = \frac{u_1(r^n - 1)}{r - 1}, \quad r \neq 1

S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1

A fractal pattern is created by drawing squares. The largest square has side length

64\,\text{cm}

. Each subsequent square is drawn inside the previous one, with its vertices at the midpoints of the sides of the previous square, so the side length of each new square is

\dfrac{1}{\sqrt{2}}

times the side length of the previous square.

(a)

Determine the side length of the

6

th square. The areas of the squares form a geometric sequence with first term

u_1 = 4096\,\text{cm}^2

and common ratio

r = \dfrac{1}{2}

. [3 marks]

(b)

Determine the total area of the first

10

squares. The pattern continues indefinitely. The artist claims that the total area of all the squares is less than

9000\,\text{cm}^2

. [3 marks]

(c)

Determine whether the artist's claim is correct. [2 marks]

Solutions

19FoundationSAQ-SDefinition and general term of geometric sequences7 marksPaper 1~11 min

A population of bacteria grows according to a geometric sequence. On day 1, there are

200

bacteria. On day 3, there are

800

bacteria.

(a)

Find the common ratio,

r

, of the sequence. [2 marks]

(b)

Find the total number of bacteria counted over the first 6 days. [2 marks]

(c)

A laboratory container can support a maximum of

500\,000

bacteria. Determine the first day on which the number of bacteria on that day exceeds the container's capacity. [3 marks]

Solutions

20MasterySAQ-SDefinition and general term of geometric sequences7 marksPaper 1~11 min

The graph below shows the first five terms of a geometric sequence plotted against

n

, where

u_n

is the

n

th term.

(a)

Write down the values of

u_1

and

u_2

from the graph. Hence show that the common ratio of the sequence is

r = \dfrac{1}{2}

. [2 marks]

(b)

Find the exact value of

u_8

. [2 marks]

(c)

Find the smallest value of

n

for which

u_n < 0.1

. [3 marks]

Solutions

21FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

The graph of

f(x) = a(x - p)(x - q)(x - r)

is shown below.

(a)

State the number of real zeros of

f(x)

. [1 mark]

(b)

State the degree of

f(x)

. [1 mark]

(c)

Determine the value of

a

, the leading coefficient of

f(x)

. [3 marks]

Solutions

22MasterySAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A ball is thrown vertically upwards from a platform. Its height,

h

metres, above the ground after

t

seconds is given by

h(t) = -4.9t^2 + 19.6t + 2.5, \quad t \geq 0.

(a)

Write down the height of the platform above the ground. [1 mark]

(b)

Find the time at which the ball reaches its maximum height. [2 marks]

(c)

The ball is considered "in safe range" if it remains above

10

metres. Determine the total time, in seconds, that the ball is in safe range. Give your answer correct to 3 significant figures. [2 marks]

Solutions

23ChallengeSAQ-LPolynomial functions and their graphs8 marksPaper 1~12 min

Data

$$

A manufacturing company produces cylindrical containers. Each container has radius

r\,\text{cm}

, height

h\,\text{cm}

, and includes a top and a bottom. The total surface area of each container is fixed at

600\pi\,\text{cm}^2

.

(a)

Show that the volume of the container can be expressed as $$V(r) = 300\pi r - \pi r^3. \quad [3 marks]

(b)

State the values of

r

for which

V(r) > 0

, giving a reason based on the physical constraints of the container. [2 marks]

(c)

Calculate the value of

r

that maximises

V(r)

, correct to 3 significant figures. [2 marks]

(d)

A competitor claims their container holds more than

6400\,\text{cm}^3

while satisfying the same surface area constraint. Determine whether this claim is possible, justifying your answer. [1 mark]

Solutions

24FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

The graph of a polynomial function

f(x)

is shown below.

(a)

State the number of distinct real zeros of

f(x)

. [1 mark]

(b)

State the degree of

f(x)

. [1 mark]

(c)

Given that

f(x) = a(x+1)(x-2)^{2}(x-4)

, find the value of the leading coefficient

a

. [3 marks]

Solutions

25MasterySAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

Consider the cubic polynomial function

f(x) = 2x^3 - 5x^2 - 4x + 3

.

(a)

Show that

f(1) = -4

. [2 marks]

(b)

Hence, write down a linear factor of

f(x) + 4

. [1 mark]

(c)

Hence fully factorise

f(x) + 4

, giving your answer in the form

(x-1)(ax^2 + bx + c)

where

a

,

b

,

c \in \mathbb{Z}

. [2 marks]

Solutions

26MasterySAQ-SExponential functions and their graphs6 marksPaper 2~9 min

The population of a colony of bees,

P

(in thousands), is modelled by the function

P(t) = 12 \times 2^{kt}

, where

t

is the time in years after the colony is established. After 3 years, the population is 24 thousand bees.

(a)

Calculate the value of

k

. [2 marks]

(b)

Determine the population after 6 years. [2 marks]

(c)

A second colony is modelled by

Q(t) = 20 \times 1.2^{t}

. Determine the value of

t

at which both colonies have the same population. Give your answer correct to one decimal place. [2 marks]

Solutions

27MasterySAQ-SExponential functions and their graphs6 marksPaper 2~9 min

The graph below shows the function

f(x) = a \times b^x

, where

a

and

b

are constants,

b > 0

,

b \neq 1

. The graph passes through the points

(0, 5)

and

(2, 20)

.

(a)

Determine the values of

a

and

b

. [2 marks]

(b)

Calculate the value of

f(4)

. [2 marks]

(c)

Find the value of

x

for which

f(x) = 100

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

Solutions

28MasterySAQ-SExponential functions and their graphs7 marksPaper 2~11 min

The population of a small island nation is modelled by

P(t) = 25000 \times e^{0.024t}

, where

P

is the population and

t

is the number of years after 1 January 2010.

(a)

State the population of the island on 1 January 2010. [1 mark]

(b)

Calculate the year in which the population will first reach

40\,000

. [3 marks]

(c)

The government states that the population will have doubled from its 2010 value by 1 January 2040. Determine whether this claim is correct, justifying your answer with a calculation. [3 marks]

Solutions

29MasterySAQ-SExponential functions and their graphs6 marksPaper 2~9 min

The function

f(x) = 3 \times 2^x + 1

is defined for

-2 \leq x \leq 3

.

(a)

Write down the

y

-intercept of the graph of

f

. [1 mark]

(b)

The graph of

g(x) = 5x + 4

intersects the graph of

f

at

x = 0

. Determine the other value of

x

at which

f

and

g

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

(c)

Write down the equation of the horizontal asymptote of

f

. [1 mark]

(d)

The function

h(x) = 3 \times 2^x + k

has a horizontal asymptote at

y = -2

. Given that

h

passes through the point

(2,\, p)

, find the value of

p

. Revised mark total: 6 [1 mark]

Solutions

30MasterySAQ-SExponential functions and their graphs6 marksPaper 2~9 min

The graph below shows the function

f(x) = k \cdot e^{-0.5x} + 2

, for

x \geq 0

. The graph passes through the point

(0, 10)

.

(a)

Calculate the value of

k

. [2 marks]

(b)

Calculate the value of

f(4)

, correct to 3 significant figures. [2 marks]

(c)

The function

g(x) = 2x + 1

intersects

f(x)

at a point

P

. Determine the

x

-coordinate of

P

, correct to 3 significant figures, and hence state which function has the greater rate of change at

P

. [2 marks]

Solutions

31MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

A company produces electronic chips. The probability that a randomly selected chip is defective is

0.15

. The company packs chips in boxes of

8

. Let

X

be the number of defective chips in a randomly selected box.

(a)

State the probability distribution of

X

, including the values of all parameters. [1 mark]

(b)

Calculate the probability that a box contains exactly

3

defective chips. [2 marks]

(c)

The company considers a box "acceptable" if it contains at most

2

defective chips. Calculate the probability that a box is acceptable, and determine whether it is more likely than not that two independently selected boxes are both acceptable. [3 marks]

Solutions

32MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

The expansion of

(2x + 3)^5

is written in the form

a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0

.

(a)

Calculate the coefficient

a_3

of the

x^3

term. [2 marks]

(b)

Calculate the coefficient

a_1

of the

x

term. [2 marks]

(c)

It is given that

a_5 = 32

,

a_4 = 240

,

a_2 = 1080

, and

a_0 = 243

. Using your answers to parts (a) and (b), find the sum

a_0 + a_1 + a_2 + a_3 + a_4 + a_5

, and verify this result by evaluating

(2(1)+3)^5

. [2 marks]

Solutions

33MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

A company produces electronic chips. The probability that a chip is defective is

0.15

. A quality control inspector examines a batch of

8

chips. Let

X

be the number of defective chips found.

(a)

State the probability distribution of

X

, including the name of the distribution and the values of its parameters. [1 mark]

(b)

Calculate the probability that exactly

3

chips are defective. [2 marks]

(c)

Determine the probability that most

2

chips are defective. [3 marks]

Solutions

34MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

Pascal's triangle has entries

\binom{n}{r}

in row

n

, position

r

, where

r = 0, 1, \ldots, n

. The first three rows are shown below.

\text{Row 0: } 1

\text{Row 1: } 1 \quad 1

\text{Row 2: } 1 \quad 2 \quad 1

(a)

Write down row 5 of Pascal's triangle. [2 marks]

(b)

Calculate the sum of all entries in row 10. [1 mark]

(c)

The

k

-th entry of row 12 (where

k = 0, 1, \ldots, 12

) equals 220. Determine the value(s) of

k

. [3 marks]

Solutions

35MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

A factory produces electronic components. Each component independently has a probability of

0.02

of being defective. The number of defective components in a batch of

n

components is modelled by a binomial distribution.

(a)

State the first three terms of the expansion of

(0.98 + 0.02)^{8}

in ascending powers of

0.02

. [2 marks]

(b)

Hence calculate the probability that in a batch of

8

components, at most

2

are defective. Give your answer correct to

4

decimal places. [2 marks]

(c)

Find the minimum value of

n

such that the probability of having at most

2

defective components in a batch of

n

components is less than

0.99

. [2 marks]

Solutions

36MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A new smartphone app records the number of push-ups a user completes each day. On the first day, the user completes

12

push-ups. Each subsequent day, the user completes

5

more push-ups than the previous day. The number of push-ups completed on day

n

is denoted by

u_n

.

(a)

Write down an expression for

u_n

in terms of

n

. [1 mark]

(b)

Calculate the number of push-ups the user completes on day

20

. [2 marks]

(c)

Determine the first day on which the user completes at least

100

push-ups. [3 marks]

Solutions

37MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A theme park has a queue for a new ride. On the first day,

120

people queue. The park managers predict that each subsequent day, the number of people queuing will increase by a constant amount. On the 5th day,

200

people queue.

(a)

Calculate the common difference in the number of people queuing each day. [2 marks]

(b)

Calculate the number of people predicted to queue on the 12th day. [2 marks]

(c)

The ride can accommodate a maximum of

500

people per day. Determine the first day on which the predicted number of people queuing exceeds this capacity. [2 marks]

Solutions

38MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A biologist is studying the growth of a bamboo plant. On the first day of measurement, the plant is

25\,\text{cm}

tall. The biologist models the height as an arithmetic sequence, where the plant grows by a constant amount each day.

(a)

On the 8th day, the plant is

74\,\text{cm}

tall. Calculate the daily growth rate. [2 marks]

(b)

Calculate the height of the plant on the 15th day. [2 marks]

(c)

A second bamboo plant also starts at

25\,\text{cm}

on day 1 and grows according to the model

h(n) = 25 \times (1.09)^{n-1}

, where

n

is the day number. Determine the first day on which the second plant is taller than the first plant, and evaluate whether the arithmetic model or the exponential model predicts a greater height on day 30. [2 marks]

Solutions

39MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A construction company is building a staircase. The first step has a height of

12\,\text{cm}

. Each subsequent step is

2\,\text{cm}

higher than the previous step.

(a)

Show that the heights of the steps form an arithmetic sequence with common difference

d = 2\,\text{cm}

. [1 mark]

(b)

Calculate the height of the

15

th step. [2 marks]

(c)

The total combined height of all steps must equal exactly

\text{620}\,\text{cm}

. Determine the number of steps in the staircase. [3 marks]

Solutions

40MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A biologist is studying a population of bacteria in a petri dish. At 08:00, there are

500

bacteria. The number of bacteria increases by a fixed amount each hour. At 11:00, there are

800

bacteria.

(a)

Calculate the common difference of this arithmetic sequence. [2 marks]

(b)

Calculate the number of bacteria at 15:00. [2 marks]

(c)

The petri dish has a maximum capacity of

2000

bacteria. The biologist claims the dish will reach capacity before 20:00 on the same day. Determine whether the biologist's claim is correct, showing all working. [2 marks]

Solutions

41MasterySAQ-SDefinition and general term of geometric sequences10 marksPaper 2~15 min

A ball is dropped from a height of

2\,\text{m}

. On each bounce, it rebounds to

60\%

of its previous height. The height of the ball on the

n

th bounce is

h_n

metres, modelled by a geometric sequence where

n = 1

is the first bounce.

(a)

Write down the values of

u_1

and

r

for this geometric sequence, and hence write down an expression for

h_n

. [2 marks]

(b)

Calculate the height of the ball on the 5th bounce. [2 marks]

(c)

Determine the smallest value of

n

for which

h_n < 0.1\,\text{m}

. [2 marks]

(d)

The ball is considered to have "effectively stopped bouncing" once the total vertical distance it has travelled (both falling and rising) from the moment is dropped exceeds

9.5\,\text{m}

. Determine whether this occurs before or after the 10th bounce, justifying your answer. [4 marks]

Solutions

42MasterySAQ-SDefinition and general term of geometric sequences8 marksPaper 2~12 min

A company produces a new phone. In month 1, they sell

8000

phones. Sales increase by

12\%

each month. The number of phones sold in month

n

is modelled by a geometric sequence

u_n

.

(a)

State the value of

u_1

and the common ratio

r

. [1 mark]

(b)

Calculate the number of phones sold in month

4

. [2 marks]

(c)

Determine the total number of phones sold in the first

6

months. [2 marks]

(d)

The company's warehouse can dispatch a maximum of

70\,000

phones in total. Determine the month in which cumulative sales first exceed this capacity, and hence evaluate whether the model suggests the company will face a supply problem within the first

8

months. [3 marks]

Solutions

43MasterySAQ-SDefinition and general term of geometric sequences6 marksPaper 2~9 min

Data

u_n = u_1 r^{n-1}

,

\quad S_n = \dfrac{u_1(r^n - 1)}{r - 1}

A new smartphone app is downloaded by

120

users in its first week. The developers model the number of new weekly downloads a geometric sequence, expecting a decrease of

15\%

each week.

(a)

Write down the common ratio and show that the number of new downloads in the third week is

86.7

, correct to 3 significant figures. [2 marks]

(b)

Calculate the total number of new downloads in the first

8

weeks. [2 marks]

(c)

Determine the first week in which the number of new downloads falls below

20

. [2 marks]

Solutions

44MasterySAQ-SDefinition and general term of geometric sequences6 marksPaper 2~9 min

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

N

, in thousands, is recorded at the end of each hour. The first four recorded values are:

5.0,\quad 7.5,\quad 11.25,\quad 16.875

(a)

Show that the sequence is geometric and state the value of the common ratio

r

. [2 marks]

(b)

Write down the general term

u_n

for the number of bacteria (in thousands) at the end of hour

n

. [1 mark]

(c)

Calculate the number of bacteria at the end of hour 10, giving your answer correct to 3 significant figures. [1 mark]

(d)

Determine after how many complete hours the number of bacteria first exceeds one million. Hence state whether the colony has exceeded one million bacteria by the end of hour 14. Justify your answer. [2 marks]

Solutions

45MasterySAQ-SDefinition and general term of geometric sequences9 marksPaper 2~14 min

The population of a small island on 1 January 2010 was

15\,000

. The population decreases geometrically each year. On 1 January 2015, the population was

12\,000

.

(a)

Show that the common ratio,

r

, of the geometric sequence is

0.956

, correct to three significant figures. [3 marks]

(b)

Calculate the population of the island on 1 January 2020. [2 marks]

(c)

The island's infrastructure requires a minimum population of

10\,000

to remain sustainable. Determine the year in which the population first falls below this threshold. [4 marks]

Solutions

46MasterySAQ-SPolynomial functions and their graphs8 marksPaper 2~12 min

A small brewery produces craft beer. The profit, in thousands of dollars, from selling

x

hundred litres of beer is modelled by

P(x) = -0.1x^3 + 1.5x^2 + 3.6x - 10, \quad 0 \leq x \leq 15.

(a)

Write down the value of

P(0)

and state what this value represents for the brewery. [2 marks]

(b)

Calculate the number of litres of beer the brewery must sell to break even. Give your answer correct to the nearest litre. [2 marks]

(c)

Determine the maximum profit and the quantity of beer sold at which it occurs. Give the quantity correct to the nearest hundred litres and the profit correct to the nearest thousand dollars. The brewery's storage capacity limits production to

900

litres. Determine whether operating at maximum profit or at the storage limit gives greater profit, and find the difference in profit in dollars. [4 marks]

Solutions

47MasterySAQ-SPolynomial functions and their graphs8 marksPaper 2~12 min

The height of a ball thrown upwards is modelled by

h(t) = -4.9t^2 + 14.7t + 1.2

, where

h

is the height in metres and

t

is the time in seconds after the ball is thrown.

(a)

Write down the height of the ball at the moment is thrown. [1 mark]

(b)

Calculate the time at which the ball reaches its maximum height. [2 marks]

(c)

Calculate the time when the ball hits the ground. Give your answer correct to 3 significant figures. [2 marks]

(d)

The model predicts

h(t) < 0

for

t > 3.08\,\text{s}

. Explain why the domain of

h(t)

should be restricted to

0 \leq t \leq 3.08

, and state one limitation of using a quadratic model for this context. [3 marks]

Solutions

48MasterySAQ-SPolynomial functions and their graphs6 marksPaper 2~9 min

A rectangular box has a square base of side length

x\,\text{cm}

and a height of

(10 - x)\,\text{cm}

, where

0 < x < 10

. The volume of the box is

V(x) = 10x^2 - x^3\,\text{cm}^3

.

(a)

Find

V'(x)

and state what

V'(x)

represents in this context. [2 marks]

(b)

Calculate the value of

x

that maximises the volume. Give your answer correct to 3 significant figures. [2 marks]

(c)

The manufacturer claims that a box with a square base of side length

5\,\text{cm}

uses the same material but gives a volume within

10\%

of the maximum. Determine whether this claim is correct, justifying your answer with calculations. [2 marks]

Solutions

49MasterySAQ-SPolynomial functions and their graphs6 marksPaper 2~9 min

The revenue

R(x)

, in thousands of dollars, from selling

x

hundred units of a product is modelled by

R(x) = -0.05x^3 + 0.6x^2 + 2x, \quad 0 \leq x \leq 12.

The cost

C(x)

, in thousands of dollars, is modelled by

C(x) = 0.1x^2 + 0.5x + 3

.

(a)

State the degree of

R(x)

and the sign of its leading coefficient. [2 marks]

(b)

Show that the profit function is

P(x) = -0.05x^3 + 0.5x^2 + 1.5x - 3

, and find the values of

x

, within the domain

0 \leq x \leq 12

, at which the company breaks even. [3 marks]

(c)

Calculate the number of hundred units that must be sold to maximise profit, and state this maximum profit in thousands of dollars. [1 mark]

Solutions

50MasterySAQ-SPolynomial functions and their graphs6 marksPaper 2~9 min

The population of a small town (in thousands) is modelled by

P(t) = -0.02t^3 + 0.3t^2 + 2.4t + 5

, where

t

is the number of years since 2010, for

0 \leq t \leq 15

.

(a)

State the degree of the polynomial and the value of the constant term. [2 marks]

(b)

Find the value of

t

at which the population is growing most rapidly, and state the corresponding population at that time. [2 marks]

(c)

Calculate the calendar year in which the population first reaches

10\,000

people. [2 marks]

Solutions

Number and Algebra — Free Maths AI SL Practice Questions