Number and Algebra — Free IB Maths AI HL Practice Questions

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

The graph of an exponential function

f(x) = a \cdot 2^x

passes through the point

(3, 40)

.

(a)

Find the value of

a

. [2 marks]

(b)

State the equation of the horizontal asymptote of

f(x)

. [1 mark]

(c)

A second function is defined as

g(x) = f(x+2) - 20

. Determine the equation of the horizontal asymptote of

g(x)

and find the value of

x

for which

g(x) = 0

. [2 marks]

Solutions

2MasterySAQ-SExponential functions and their graphs6 marksPaper 1~9 min

The value of a rare painting is modelled by the function

V(t) = 8000 \times 10^{kt}

, where

V

is the value in USD,

t

is the number of years since purchase, and

k

is a constant. The painting was purchased for USD 8000. After

5

years, its value had increased to USD 11000.

(a)

Show that

k = \dfrac{1}{5}\log_{10}\!\left(\dfrac{11}{8}\right)

. [2 marks]

(b)

Calculate the value of

k

. Give your answer correct to

3

significant figures. [1 mark]

(c)

Find the number of years it takes for the painting to reach a value of USD 20000. Give your answer correct to the nearest year. [3 marks]

Solutions

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

A new smartphone app's number of downloads,

D

(in thousands), is modelled by

D(t) = 50 \times 2^{kt}

, where

t

is the time in weeks since the app was released. After 3 weeks, there were 200 thousand downloads.

(a)

Determine the value of

k

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

(b)

Determine the number of weeks it takes for the number of downloads to reach 1000 thousand. Give your answer correct to 3 significant figures. [2 marks]

(c)

A competing app has downloads modelled by

E(t) = 100 \times 1.5^{t}

. State which app has a faster initial rate of growth. Justify your answer by comparing the initial rates of change of

D

and

E

. [2 marks]

(d)

Determine the value of

t

at which both apps have the same number of downloads. Give your answer correct to 3 significant figures. [2 marks]

Solutions

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

The graph of

f(x) = 4 \times 2^{-x} - 1

for

x \in \mathbb{R}

is shown below.

(a)

State the equation of the horizontal asymptote of

f

. [1 mark]

(b)

A second function is defined as

g(x) = f(x) + 2

. State the equation of the horizontal asymptote of

g

. [1 mark]

(c)

Find the value of

x

for which

f(x) = \dfrac{3}{2}

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

Solutions

5FoundationSAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

Consider the expansion of

(2x + 3)^5

.

(a)

Write down the general term of the expansion in terms of

r

, where

r = 0, 1, \ldots, 5

. [2 marks]

(b)

Hence find the coefficient of

x^2

in the expansion. [3 marks]

Solutions

6MasterySAQ-SBinomial expansion and coefficients5 marksPaper 1~8 min

Consider the expansion of

\left(2x + \dfrac{1}{x^2}\right)^n

, where

n \in \mathbb{Z}^+

and

n

is a multiple of 3.

(a)

Show that the coefficient of the term independent of

x

is

\dbinom{n}{\frac{n}{3}} 2^{\frac{2n}{3}}

. [3 marks]

(b)

Hence find the smallest value of

n

such that this coefficient exceeds 5000. [2 marks]

Solutions

7ChallengeSAQ-LApplications of binomial expansions8 marksPaper 1~12 min

A pharmaceutical company is testing a new drug. Clinical trials show that the probability of a patient experiencing a minor side effect is

0.08

. The drug is tested on

100

patients. Let

X

be the number of patients who experience the side effect.

(a)

Calculate the expected value and standard deviation of

X

. [2 marks]

(b)

Determine

P(X = 10)

, giving your answer correct to 4 significant figures. [2 marks]

(c)

Evaluate whether

P(8 \leq X \leq 12) > 0.5

. Justify your answer by calculating each relevant probability and summing them. [4 marks]

Solutions

8FoundationSAQ-SBinomial expansion and coefficients10 marksPaper 1~15 min

The 6th row of Pascal's triangle (where row 0 is

1

) is:

1,\quad 6,\quad 15,\quad 20,\quad 15,\quad 6,\quad 1

(a)

Using the 6th row of Pascal's triangle, write down the full expansion of

(x + 1)^6

in descending powers of

x

. [2 marks]

(b)

Hence determine the coefficient of

x^3

in the expansion of

(2x + 1)^6

. [2 marks]

(c)

Using your answer to part (b), or otherwise, find the exact value of

\left(2 \cdot \dfrac{1}{2} + 1\right)^6

by evaluating every term of the expansion of

(2x+1)^6

at

x = \dfrac{1}{2}

, and verify that the sum equals

2^6

. [1] Revised cleaner version (removing the forced verify): The 6th row of Pascal's triangle (where row 0 is

1

) is:

1,\quad 6,\quad 15,\quad 20,\quad 15,\quad 6,\quad 1

(a) Using the 6th row, write down the full expansion of

(x + 1)^6

in descending powers of

x

. [2] (b) Hence determine the coefficient of

x^3

in the expansion of

(2x + 1)^6

. [2] (c) By substituting

x = \dfrac{1}{2}

into the expansion of

(2x + 1)^6

, show that

\sum_{k=0}^{6} \binom{6}{k} = 64.

[1 mark]

Solutions

9FoundationSAQ-SDefinition and general term of arithmetic sequences5 marksPaper 1~8 min

Data

The total distance from the first tree to the fifth tree is the sum of the four consecutive gaps.

A farmer is planting trees in a straight line. The distance from the first tree to the second tree is

5\,\text{m}

. The distance between consecutive trees increases by a constant amount, forming an arithmetic sequence. The total distance from the first tree to the fifth tree is

38\,\text{m}

.

(a)

State the value of

u_1

, the first gap, and the value of

S_4

, the total of the first four gaps. [2 marks]

(b)

Calculate the common difference,

d

, of the arithmetic sequence of gaps. [2 marks]

(c)

Hence, find the total distance from the first tree to the tenth tree. [1 mark]

Solutions

10MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 1~9 min

A biologist models a growing bacterial colony using an arithmetic sequence. At 12:00, she counts

1200

bacteria. The population increases by a constant amount each hour. At 15:00, there are

2700

bacteria.

(a)

State the values of

u_1

and

u_4

in the arithmetic sequence that models the number of bacteria, where

n = 1

corresponds to 12:00. [2 marks]

(b)

Show that the common difference

d = 500

. [2 marks]

(c)

Hence, find the number of bacteria at 18:00. [2 marks]

Solutions

11ChallengeSAQ-LDefinition and general term of arithmetic sequences8 marksPaper 1~12 min

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

18\,\text{cm}

. Each subsequent step is

4\,\text{cm}

taller than the previous step. The staircase must have a total height of exactly

480\,\text{cm}

from the ground to the top of the final step.

(a)

Show that the height of the

n

th step above the ground is

u_n = 14 + 4n\,\text{cm}

, and state the domain of

n

, justifying any restriction its upper bound. [3 marks]

(b)

Determine the number of steps in the staircase. [2 marks]

(c)

The architect redesigns the staircase so that the step heights form an arithmetic sequence with first term

a\,\text{cm}

and common difference

d\,\text{cm}

. The total height remains

480\,\text{cm}

and the number of steps is

12

. - Show that

2a + 11d = 80

. - Hence determine all possible integer values of

d

such that

a

is also a positive integer and

d > 0

. [3 marks]

Solutions

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

A company is stacking boxes in a warehouse. The bottom row has

30

boxes. Each row above has

4

fewer boxes than the row below, forming an arithmetic sequence. The top row has

6

boxes.

(a)

State the value of

u_1

and the common difference

d

. [1 mark]

(b)

Calculate the number of rows of boxes. [2 marks]

(c)

Find the total number of boxes in the stack. [2 marks]

Solutions

13FoundationSAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min

A new smartphone model is released. In its first month of sales,

5000

units are sold. The company predicts that each subsequent month, sales will be

92\%

of the sales from the previous month.

(a)

State why this situation can be modelled by a geometric sequence. [1 mark]

(b)

Find the number of units sold in the

4

th month. Give your answer to the nearest whole unit. [2 marks]

(c)

The company considers the model financially viable as long as monthly sales remain above

2000

units. Determine the month in which sales first fall below

2000

units. [2 marks]

Solutions

14MasterySAQ-SDefinition and general term of geometric sequences5 marksPaper 1~8 min

The first three terms of a geometric sequence are

x-3

,

x+3

, and

4x-3

, where

x \in \mathbb{R}

and

x \neq 0

.

(a)

Show that

x = 7

. [3 marks]

(b)

Hence find the general term

u_n

of the sequence, giving your answer in the form

u_n = a \cdot r^{n-1}

. [2 marks]

Solutions

15ChallengeSAQ-LDefinition and general term of geometric sequences8 marksPaper 1~12 min

A company is designing a logo consisting of concentric circles whose radii form a geometric sequence. The smallest circle has a radius of

2\,\text{cm}

. The radius of the fourth circle is

54\,\text{cm}

.

(a)

Determine the common ratio of the geometric sequence. [3 marks]

(b)

Determine the radius of the seventh circle. [2 marks]

(c)

Show that the ratio

\dfrac{A_{n+1}}{A_n}

equals

9

, where

A_n = \pi r_n^2

is the area of the

n

th circle. [1 mark]

(d)

Determine the value of

n

for which the area of the

n

th circle first exceeds

10000\,\text{cm}^2

. [2 marks]

Solutions

16FoundationSAQ-SDefinition and general term of geometric sequences7 marksPaper 1~11 min

A biologist is studying a population of bacteria. At the start of an experiment (

t = 0

), there are

200

bacteria. The population triples every hour.

(a)

State the common ratio,

r

, of this geometric sequence. [1 mark]

(b)

Write an expression for

P(n)

, the number of bacteria present after

n

hours. [2 marks]

(c)

Calculate the number of bacteria present after

5

hours. [1 mark]

(d)

The laboratory container can support a maximum of

5 \times 10^6

bacteria. Determine the number of complete hours after which this limit is first exceeded. [3 marks]

Solutions

17FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

The volume,

V

cubic metres, of a rectangular box is given by

V(x) = x^3 + 6x^2 + 11x + 6, \quad x > 0

where

x

is a positive real number representing a scaled length.

(a)

Show that

(x+1)

is a factor of

V(x)

. [1 mark]

(b)

Hence write

V(x)

as a product of three linear factors. [2 marks]

(c)

The three linear factors of

V(x)

represent the three edge lengths of the box. Determine the value of

x

for which the box is a cube, and find the corresponding volume. [2 marks]

Solutions

18MasterySAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A polynomial function is defined as

f(x) = 2x^3 + kx^2 - 8x - 12

, where

k

is a constant. It is given that

(x - 2)

is a factor of

f(x)

.

(a)

Show that

k = 3

. [2 marks]

(b)

Hence, write

f(x)

as a product of three linear factors. [2 marks]

(c)

Sketch the graph of

y = f(x)

for

-3 \leq x \leq 3

, labelling all

x

-intercepts and the

y

-intercept. [1 mark]

Solutions

19ChallengeSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A polynomial function

f

of degree 4 has a positive leading coefficient. The derivative of

f

is

f'(x) = 4(x+2)(x-1)(x-3).

(a)

Write down the

x

-coordinates of all local extrema of

f

, and state, with a reason, which is a local maximum and which is a local minimum. [2 marks]

(b)

Given that

f(0) = -36

, find

f(x)

in the form

f(x) = x^4 + bx^3 + cx^2 + dx + e

. [2 marks]

(c)

The tangent to the graph of

f

at

x = 3

passes through the point

(p,\, 0)

. Determine the value of

p

. [1 mark]

Solutions

20FoundationSAQ-SPolynomial functions and their graphs5 marksPaper 1~8 min

A company's profit,

P(x)

thousand dollars, from selling

x

hundred units of a product is modelled by

P(x) = -2x^3 + 12x^2 - 10x, \quad x \geq 0.

(a)

Find the values of

x

for which

P(x) = 0

. [2 marks]

(b)

Find

P'(x)

and hence determine the value of

x

that gives maximum profit. [2 marks]

(c)

The company's fixed costs increase, shifting the model to

Q(x) = P(x) - k

, where

k > 0

. Determine the range of values of

k

for which the company can still achieve a positive profit for some

x \geq 0

. [1 mark]

Solutions

21MasterySAQ-SExponential functions and their graphs6 marksPaper 2~9 min

The population of a certain species of bird on an island is modelled by

P(t) = 1200 \times e^{-0.15t}

, where

t

is the time in years since the population was first recorded.

(a)

State the initial population of the birds. [1 mark]

(b)

Calculate the population after 5 years, giving your answer correct to the nearest whole number. [2 marks]

(c)

Calculate the time, in years, for the population to decrease to 300 birds. Give your answer correct to 2 decimal places. [3 marks]

Solutions

22MasterySAQ-SExponential functions and their graphs8 marksPaper 2~12 min

The graph of

y = 2 \times a^x

passes through the point

(2, 18)

.

(a)

Calculate the value of

a

. [2 marks]

(b)

The graph of

y = 2 \times a^x

is transformed to give the graph of

y = 2 \times a^x - 8

. Write down the equation of the horizontal asymptote of this transformed graph and explain what feature of the function determines it. [2 marks]

(c)

The graph of

y = 2 \times a^x

is reflected in the

y

-axis to give the graph of

g(x)

. Determine the

y

-intercept of

g(x)

and state whether it is the same as or different from the

y

-intercept of

y = 2 \times a^x

. Justify your answer. [2 marks]

(d)

Using your value of

a

, calculate the value of

x

when

y = 162

. [2 marks]

Solutions

23MasterySAQ-SLogarithmic functions and their properties9 marksPaper 2~14 min

A biologist is studying the population of a rare species of bird on a remote island. The population size

P

, after

t

years from the start of the study, is modelled by

P = 250\ln(3t + 5), \quad t \geq 0.

(a)

Calculate the initial population of the birds. [2 marks]

(b)

Determine the number of years it will take for the population to reach

500

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

(c)

Find the rate of change of the population at the time found in part (b), and hence comment on whether the model predicts the population is growing faster or slower at that moment compared to the start of the study. [4 marks]

Solutions

24MasterySAQ-SLogarithmic functions and their properties6 marksPaper 2~9 min

The loudness of a sound, measured in decibels (dB), is modelled by

L = 10\log_{10}\!\left(\frac{I}{I_0}\right)

where

I

is the sound intensity in

\text{W\,m}^{-2}

and

I_0 = 1.0 \times 10^{-12}\,\text{W\,m}^{-2}

is the reference intensity. A concert speaker produces a sound with intensity

I_1 = 5.0 \times 10^{-3}\,\text{W\,m}^{-2}

.

(a)

Calculate the loudness

L_1

of this sound. Give your answer correct to the nearest whole number. [2 marks]

(b)

A second speaker produces a sound that is

15\,\text{dB}

louder than the first. Using your unrounded value of

L_1

, find the intensity

I_2

of the second speaker's sound. Give your answer in the form

a \times 10^{k}\,\text{W\,m}^{-2}

, where

1 \leq a < 10

and

k \in \mathbb{Z}

. [4 marks]

Solutions

25MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

A triangular pattern of numbers is shown below, where each row corresponds to the coefficients of

(a+b)^n

.

\begin{array}{c} \text{Row 0: } 1 \\ \text{Row 1: } 1 \quad 1 \\ \text{Row 2: } 1 \quad 2 \quad 1 \\ \text{Row 3: } 1 \quad 3 \quad 1 \\ \text{Row 4: } 1 \quad 4 \quad 6 \quad 4 \quad 1 \end{array}

(a)

State the numbers in Row 5 of Pascal's triangle. [1 mark]

(b)

Hence, expand

(2x - 3y)^5

in descending powers of

x

. Give all coefficients in simplified form. [3 marks]

(c)

Using your expansion from part (b), find the value of

k

such that the coefficient of

x^3 y^2

in the expansion of

(1 + k)(2x - 3y)^5

is equal to

2880

. [2 marks]

Solutions

26MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

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

0.02

of being defective. The probability of exactly

k

defective components in a batch of

n

components is modelled by

P(X = k) = \binom{n}{k}(0.98)^{n-k}(0.02)^k

A quality control engineer selects a batch of

8

components. Let

X

be the number of defective components.

(a)

Write down the term

P(X = k)

for this batch, substituting

n = 8

. [1 mark]

(b)

Calculate the probability of exactly

3

defective components in the batch. Give your answer correct to

3

significant figures. [2 marks]

(c)

Calculate the probability of at least one defective component in the batch. Give your answer correct to

3

significant figures. [3 marks]

Solutions

27MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

The expansion of

(1 + ax)^n

, where

n

is a positive integer, is given by:

1 + 18x + 144x^2 + \cdots

(a)

Show that

n = 9

and

a = 2

. [3 marks]

(b)

Hence, find the coefficient of

x^3

in the expansion. [3 marks]

Solutions

28MasterySAQ-SBinomial expansion and coefficients6 marksPaper 2~9 min

A farmer uses a triangular plot of land for planting crops. The three sides of the triangle have lengths

a

,

b

, and

c

metres, where

a = 3x - 1

,

b = 2x + 3

, and

c = x + 5

.

(a)

Write down an expression for the perimeter

P = a + b + c

in terms of

x

, giving your answer in simplest form. [1 mark]

(b)

The perimeter is exactly

49

metres. Calculate the value of

x

. [2 marks]

(c)

The area of the triangle is given by Heron's formula

A = \sqrt{s(s-a)(s-b)(s-c)}

, where

s = \dfrac{P}{2}

. Using your value of

x

from part (b), calculate the area of the triangle. Give your answer correct to 3 significant figures. [3 marks]

Solutions

29MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

The first term of an arithmetic sequence is

u_1 = 12

. The fifth term is

u_5 = 28

.

(a)

State the common difference,

d

. [1 mark]

(b)

Determine the general term

u_n

in the form

u_n = an + b

, where

a, b \in \mathbb{Z}

. [2 marks]

(c)

Find the smallest value of

n

for which the sum of the first

n

terms exceeds

1000

. [3 marks]

Solutions

30MasterySAQ-SDefinition and general term of arithmetic sequences6 marksPaper 2~9 min

A company produces wooden pallets. In week 1, they produce 500 pallets. The production increases by a fixed number each week, forming an arithmetic sequence. In week 10, they produce 680 pallets.

(a)

State the weekly increase in production. [1 mark]

(b)

Determine an expression for the number of pallets produced in week

n

, denoted

P_n

. [2 marks]

(c)

The company receives a contract requiring a cumulative total of at least 40 000 pallets. Determine the minimum number of complete weeks needed to fulfil this contract. [3 marks]

Solutions

31MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

A construction company is building a terraced garden. The first terrace requires

120

bricks. Each subsequent terrace requires

8

fewer bricks than the previous one. Terraces are built in sequence for as long as the number of bricks required is positive. Formulae:

u_n = u_1 + (n-1)d

;

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

(a)

Calculate the number of bricks required for the

10

th terrace. [2 marks]

(b)

Determine the total number of terraces that can be built, and hence calculate the total number of bricks used. [4 marks]

Solutions

32MasterySAQ-SSum of an arithmetic sequence6 marksPaper 2~9 min

Data

u_n = u_1 + (n-1)d

;

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

A farmer plants trees in rows. The first row has

25

trees. Each subsequent row has a constant number of additional trees. The

12

th row has

69

trees.

(a)

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

(b)

Calculate the total number of trees planted in the first

20

rows. [3 marks]

(c)

The farmer has space for at most

30

rows and a total budget allowing a maximum of

3000

trees. Determine which constraint is reached first, justifying your answer. [1 mark]

Solutions

33MasterySAQ-SDefinition and general term of geometric sequences6 marksPaper 2~9 min

A new social media app has

5000

users when it launches. The number of users increases by

8\%

each month. The number of users after

n

months is modelled by a geometric sequence with first term

u_1 = 5000

.

(a)

Write down the common ratio,

r

, of this geometric sequence. [1 mark]

(b)

Calculate the number of users after

6

months. Give your answer correct to the nearest whole number. [2 marks]

(c)

Determine the month in which the number of users first exceeds

12\,000

. [3 marks]

Solutions

34MasterySAQ-SSum of a geometric sequence6 marksPaper 2~9 min

Data

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

,

\quad S_\infty = \dfrac{u_1}{1-r}

for

|r| < 1

A ball is dropped from a height of

10

metres. After each bounce, it reaches

85\%

of the height of the previous bounce. The total vertical distance travelled is the sum of all distances fallen and risen.

(a)

Calculate the total vertical distance travelled by the ball when it hits the ground for the 8th time. Give your answer correct to 3 significant figures. [3 marks]

(b)

Determine the minimum number of bounces required for the total vertical distance travelled to exceed

60

metres. [3 marks]

Solutions

35MasterySAQ-SSum of a geometric sequence6 marksPaper 2~9 min

A population of bacteria in a laboratory culture grows geometrically. At 12:00 there are

200

bacteria. At 14:00 there are

320

bacteria. The population continues to grow with the same common ratio every hour.

(a)

Show that the common ratio per hour is

r = \sqrt{1.6}

. [1 mark]

(b)

Calculate the total number of bacteria present from 12:00 up to and including 20:00. Give your answer correct to 3 significant figures. [3 marks]

(c)

The culture dish can support a maximum of

100\,000

bacteria in total before the population collapses. Determine the time at which the cumulative total number of bacteria first exceeds

100\,000

. [2 marks]

Solutions

36MasterySAQ-SApplications of geometric sequences in finance and growth models6 marksPaper 2~9 min

A company invests

12\,000

dollars in a new technology fund. The value of the investment increases by

4.2\%

each year, compounded annually.

(a)

Calculate the value of the investment after

8

years. [3 marks]

(b)

Determine the number of complete years it will take for the investment to first exceed

20\,000

dollars. [3 marks]

Solutions

37MasterySAQ-SPolynomial functions and their graphs6 marksPaper 2~9 min

A company designs a water storage tank. The cross-section of the tank is modelled by the polynomial function

f(x) = -0.02x^3 + 0.3x^2 + 0.6x + 2

for

0 \leq x \leq 12

, where

f(x)

is the height of the tank wall (in metres) at a horizontal distance

x

metres from the left edge.

(a)

State the degree of the polynomial and the value of the leading coefficient. [2 marks]

(b)

Calculate the height of the tank wall at the left edge (

x = 0

) and at the right edge (

x = 12

). [2 marks]

(c)

Determine the value of

x

at which the maximum height occurs and state the maximum height, giving your answer correct to 3 significant figures. Hence comment on whether the maximum occurs at an endpoint or in the interior of the domain. [2 marks]

Solutions

38MasterySAQ-SPolynomial functions and their graphs10 marksPaper 2~15 min

A ball is thrown from ground level on a sports field. Its height above the ground,

h

metres, after

t

seconds is modelled by

h(t) = -5t^3 + 20t^2 + 5t, \quad 0 \leq t \leq 4.5.

(a)

Write down the degree and the leading coefficient of

h(t)

. [2 marks]

(b)

Calculate the height of the ball at

t = 2

seconds. [2 marks]

(c)

Determine the value of

t

when the ball returns to ground level. Give your answer correct to 3 significant figures. [2 marks]

(d)

Find the maximum height reached by the ball and the time at which it occurs. Hence determine whether the ball ever exceeds a height of

55\,\text{m}

, justifying your answer. [4 marks]

Solutions

39MasterySAQ-SPolynomial functions and their graphs8 marksPaper 2~12 min

A rectangular box has a square base of side length

x\,\text{cm}

and height

h\,\text{cm}

. The volume of the box is

500\,\text{cm}^3

. The total surface area (including the lid) is given by

A(x) = 2x^2 + \frac{2000}{x}, \quad x > 0.

(a)

Show that

h = \dfrac{500}{x^2}

, and hence verify the expression for

A(x)

above. [2 marks]

(b)

Calculate the value of

A

when

x = 10\,\text{cm}

. [2 marks]

(c)

Find the value of

x

that minimises

A(x)

. Give your answer correct to 3 significant figures. [1 mark]

(d)

Hence find the corresponding height

h

of the box at this minimum, and determine whether this stationary point is a global minimum for

x > 0

. Justify your answer. [3 marks]

Solutions

40MasterySAQ-SPolynomial functions and their graphs8 marksPaper 2~12 min

A small business sells handmade candles. The weekly profit

P

(in dollars) from selling

x

candles is modelled by

P(x) = -0.5x^3 + 15x^2 + 20x - 200, \quad 0 \leq x \leq 30.

(a)

State the degree and leading coefficient of

P(x)

. [1 mark]

(b)

Calculate the profit when

x = 10

candles are sold. [2 marks]

(c)

Using your GDC, find all values of

x

in the domain

[0, 30]

at which the business breaks even (i.e.

P(x) = 0

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

(d)

Hence determine the range of values of

x

, within the domain, for which the business makes a profit, and find the maximum weekly profit correct to the nearest dollar. [3 marks]

Solutions

41ChallengeLAQExponential functions and their graphs10 marksPaper 3~15 min

Data

\ln 2 \approx 0.693

A researcher is studying the population decay of a rare bacterial colony in a controlled laboratory environment. The population

P

(in thousands of cells) at time

t

(in days) is modelled by

P(t) = Ae^{-kt}

, where

A

and

k

are positive constants. The initial population is

500

thousand cells. After

10

days, the population is

250

thousand cells.

(a)

Show that

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

. [3 marks]

(b)

Determine the number of days for the population to drop below

1

thousand cells. [3 marks]

(c)

The researcher claims the exponential model remains valid indefinitely. Evaluate this claim in the context of the bacterial colony, identifying one mathematical and one biological limitation of the model. [4 marks]

Solutions

42ChallengeLAQExponential functions and their graphs10 marksPaper 3~15 min

Two investment funds offer different growth models. Fund A grows continuously:

A(t) = 1000e^{0.05t}

, where

t

is time in years and

A(t)

is the value in USD. Fund B grows with annual compounding:

B(t) = 1000(1.0512)^t

.

(a)

State the continuous growth rate of Fund A and determine the effective annual interest rate of Fund B. [2 marks]

(b)

Show that

A(t) < B(t)

for all

t > 0

by analysing the function

D(t) = B(t) - A(t)

. You should demonstrate that

D(t)

is strictly increasing for

t > 0

and state the value of

D(0)

. [4 marks]

(c)

Calculate the difference in value between the two funds at

t = 20

years, giving your answer to the nearest USD. Hence evaluate whether the difference is financially significant as a percentage of the initial investment. [4 marks]

Solutions

43ChallengeLAQBinomial expansion and coefficients10 marksPaper 3~15 min

A mathematics student investigates the Catalan numbers, defined for

n \in \mathbb{N},\ n \geq 2

by:

a_n = \frac{1}{n+1}\binom{2n}{n}

(a)

Show that

a_n = \dbinom{2n}{n} - \dbinom{2n}{n+1}

. [4 marks]

(b)

Show that

\dfrac{a_{n+1}}{a_n} = \dfrac{2(2n+1)}{(n+1)(n+2)} \cdot (n+1) = \dfrac{4n+2}{n+2}

. [3 marks]

(c)

The student claims that

\{a_n\}

is strictly decreasing for

n \geq 2

. Determine whether this claim is correct. Hence, by considering

\dfrac{a_{n+1}}{a_n}

as

n \to \infty

, describe the long-run behaviour of consecutive terms and explain what this implies about the rate of growth of

a_n

. [3 marks]

Solutions

44ChallengeLAQApplications of binomial expansions13 marksPaper 3~20 min

A player rolls a fair six-sided die repeatedly. A "success" is defined as rolling a 6. The random variable

X

represents the number of successes in exactly 8 rolls.

(a)

(i) Show that the probability of obtaining exactly

k

successes in 8 rolls is

P(X = k) = \binom{8}{k}\left(\frac{1}{6}\right)^k\left(\frac{5}{6}\right)^{8-k}, \quad k = 0, 1, 2, \ldots, 8.

[2]
(ii) Hence show that

\displaystyle\sum_{k=0}^{8} P(X = k) = 1

. [2 marks]

(b)

Calculate

P(X \leq 2)

, giving your answer correct to 4 significant figures. [3 marks]

(c)

The game is now modified: the player rolls the die until the first 6 appears. The random variable

Y

denotes the number of rolls required. There is no upper limit on the number of rolls. (i) Show that

P(Y = k) = \left(\frac{5}{6}\right)^{k-1}\!\left(\frac{1}{6}\right)

for

k = 1, 2, 3, \ldots

[2]
(ii) Calculate

P(Y \leq 3)

, expressing your answer as an exact fraction. [1]
(iii) In a further variation, the die is rolled at most 8 times and the player loses if no 6 appears. Determine

P(Y > 8)

and hence explain why the values

P(Y = 1), P(Y = 2), \ldots, P(Y = 8)

do not sum to 1. [3 marks]

Solutions

45ChallengeLAQSum of an arithmetic sequence10 marksPaper 3~15 min

A farmer constructs a series of trapezoidal irrigation channels. The cross-section of each channel is a trapezium with a constant bottom width of

1.0\,\text{m}

and a constant depth of

0.5\,\text{m}

. The first channel has a top width of

2.0\,\text{m}

, and each subsequent channel has a top width

0.4\,\text{m}

wider than the previous one.

(a)

Prove that the cross-sectional area

A_n

of the

n

th channel satisfies

A_n = 0.5\bigl(3 + 0.4(n-1)\bigr)\,\text{m}^2.

[3 marks]

(b)

The cross-sectional areas

A_1, A_2, \ldots

form an arithmetic sequence. (i) Write down the first term

a

and common difference

d

of this arithmetic sequence. [2 marks]
(ii) Find the total cross-sectional area of the first

20

channels. [2 marks]
(iii) The farmer wants to build channels until the total cross-sectional area first exceeds

200\,\text{m}^2

. Find the minimum number of channels needed. [3 marks]

Solutions

46ChallengeLAQApplications of arithmetic sequences in real-world contexts10 marksPaper 3~15 min

A construction company is building a high-rise tower. The first floor has a height of

5.0\,\text{m}

. Each subsequent floor has a height

0.10\,\text{m}

less than the floor below it. The building has

40

floors in total.

(a)

Show that the height of the

n

th floor is given by

u_n = 5.1 - 0.1n

metres, and state the height of the 40th floor. [2 marks]

(b)

Calculate the total height of the building from ground level to the top of the 40th floor. [3 marks]

(c)

The building must comply with a regulation stating that the average floor height must be at least

3.0\,\text{m}

. Evaluate whether this building meets the regulation. Justify your conclusion. [2 marks]

(d)

A second tower has the same number of floors. Its floor heights also form an arithmetic sequence, with a first floor height of

5.0\,\text{m}

and a 40th floor height of

3.5\,\text{m}

. Show that the common difference of this sequence differs from that of the first tower. [1 mark] (e) Determine which tower has the greater total height, and calculate the difference in total heights between the two towers. [2 marks]

Solutions

47ChallengeLAQSum of a geometric sequence10 marksPaper 3~15 min

A geometric sequence has first term

u_1 = a

and common ratio

r

, where

r \neq 1

. The sum of the first

n

terms is

S_n

. An industrial robot is purchased for

120\,000

dollars. Its value at the end of each year is modelled as a fixed percentage of its value at the start of that year, forming a geometric sequence. After

10

years the robot is sold for

15\,000

dollars.

(a)

Prove that

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

. [4 marks]

(b)

Show that the value of the robot after

n

complete years is

120\,000 \cdot r^n

dollars, and hence calculate the annual percentage rate of depreciation, correct to three significant figures. [3 marks]

(c)

The total depreciation over the 10-year life is defined as the difference between the purchase price and the final sale value. Using your value of

r

from part (b), calculate the total depreciation as a percentage of the purchase price. Evaluate whether this percentage would change if the robot had instead been purchased for

200\,000

dollars but depreciated at the same annual rate for the same number of years. Justify your answer. [3 marks]

Solutions

48ChallengeLAQSum of a geometric sequence13 marksPaper 3~20 min

A geometric sequence has first term

u_1 = p

and common ratio

r

, where

r \neq 1

. The sum of the first

n

terms is

S_n

. A scientist studies a bacterial population that reproduces in discrete generations. The population at the start of generation 1 is

5000

. In each subsequent generation the population multiplies by a constant factor

k

, where

k > 1

. The total number of bacteria that have ever lived up to and including generation

n

is

S_n

.

(a)

Prove that

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

for

r \neq 1

. [4 marks]

(b)

After 12 generations, the total number of bacteria that have ever lived is

81\,915\,000

(to the nearest thousand). Calculate the value of

k

correct to three significant figures. [3 marks]

(c)

Show that the number of bacteria alive in generation 12 only is approximately

433\,000

. [2 marks]

(d)

Using your results from parts (b) and (c), determine the value of

n

for which the population in generation

n

alone first exceeds the total of all bacteria that lived in all previous generations combined. Justify your answer. (HL only — extended) [4 marks]

Solutions

49ChallengeLAQPolynomial functions and their graphs10 marksPaper 3~15 min

A polynomial function

P

of degree

n

is defined by

P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \quad a_n \neq 0,\; n \in \mathbb{N}.

Consider

f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12

.

(a)

Show that

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

. [4 marks]

(b)

The graph of

y = f(x)

has exactly three turning points. A student claims: *"If a polynomial function has degree

n

, then its graph has exactly

n - 1

turning points."* (i) Using the factored form from (a), explain why

f(x)

has exactly three turning points. [2]
(ii) Determine whether the student's claim is valid. Justify your answer by constructing a counterexample and stating the correct general result. [4 marks]

Solutions

50ChallengeLAQPolynomial functions and their graphs10 marksPaper 3~15 min

A manufacturing company produces solar panels. The profit

P(x)

, in thousands of dollars, from selling

x

hundred panels is modelled by

P(x) = -0.5x^3 + 6x^2 - 13.5x + 8, \quad x \geq 0.

(a)

(i) Show that

x = 1

is a root of

P(x) = 0

. [2]
(ii) Hence fully factorise

P(x)

over the real numbers. [3 marks]

(b)

The company claims that profit is non-negative for all production levels between the break-even points. - State three break-even values of

x

, correct to three significant figures. - Determine the local maximum value of

P(x)

, correct to three significant figures. - Evaluate the company's claim, justifying your conclusion with reference to the sign of

P(x)

between consecutive roots and the behaviour of a cubic with a negative leading coefficient. [5 marks]

Solutions

Number and Algebra — Free Maths AI HL Practice Questions