Number and Algebra — Free IB Maths AA SL Practice Questions

1FoundationSAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

A company is designing a rectangular solar panel. The lengths of the sides of the panel, in metres, form an arithmetic sequence with first term

u_1 = 2\,\text{m}

and fourth term

u_4 = 8\,\text{m}

.

(a)

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

(b)

Find the area of the solar panel. [1 mark]

(c)

A second rectangular panel has the same perimeter as the first panel but has side lengths that are equal. Determine whether the second panel has a greater area than the first panel, justifying your answer. [2 marks]

Solutions

2MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 1~9 min

A water tank contains

1200

litres of water. Due to a leak, the tank loses water each day. The volume of water lost on each successive day forms an arithmetic sequence. On the first day,

15

litres are lost, and on the fourth day,

30

litres are lost.

(a)

Show that the common difference of the sequence is

5

litres per day. [2 marks]

(b)

Hence find the volume of water lost on the tenth day. [2 marks]

(c)

Calculate the number of litres of water remaining in the tank at the end of the tenth day. [2 marks]

Solutions

3ChallengeSAQ-LDefinition and general term of arithmetic sequencess8 marksPaper 1~12 min

A farmer is designing a terraced garden on a hillside. The first terrace (closest to the bottom) is a rectangle of length

12\,\text{m}

and width

5\,\text{m}

. Each subsequent terrace moving up the hill has its length decreased by a constant

d\,\text{m}

and its width increased by a constant

e\,\text{m}

compared to the terrace immediately below it, forming arithmetic sequences for both dimensions.

(a)

Given that the length of the 4th terrace is

6\,\text{m}

, determine the value of

d

. [2 marks]

(b)

The width of the 6th terrace is

11\,\text{m}

. Determine the value of

e

. [2 marks]

(c)

The area of the

n

th terrace is

A_n\,\text{m}^2

. (i) Show that

A_n = -2.4n^2 + 19.6n - 7.2

. [2]
(ii) Hence determine the terrace number for which the area is greatest, and state that area. [2 marks]

Solutions

4FoundationSAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

A gardener is building a tiered flower bed. The heights of the tiers, measured in centimetres, form an arithmetic sequence. The first tier has height

30\,\text{cm}

and the fifth tier has height

54\,\text{cm}

.

(a)

Find the common difference of the sequence. [2 marks]

(b)

Find the height of the third tier. [2 marks]

(c)

The gardener has enough soil to build tiers up to a total combined height of

500\,\text{cm}

. Determine whether the gardener can complete a

9

-tier flower bed. [1 mark]

Solutions

5MasterySAQ-SDefinition and general term of arithmetic sequencess5 marksPaper 1~8 min

The first term of an arithmetic sequence is

u_1 = 5

and the common difference is

d = 3

.

(a)

Write down the values of

u_2

and

u_3

. [1 mark]

(b)

Show that the general term of this sequence is

u_n = 3n + 2

. [2 marks]

(c)

The sum of the first

n

terms of the sequence is denoted

S_n

. Given that

S_n = 1365

, find the value of

n

. [2 marks]

Solutions

6FoundationSAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

A rectangular garden has length

(x+3)

metres and width

(x-1)

metres. The area of the garden is

45

square metres.

(a)

Write down an expression for the area of the garden in terms of

x

. [1 mark]

(b)

Show that

x^2 + 2x - 48 = 0

. [2 marks]

(c)

Hence find the value of

x

, justifying why only one value is valid. [2 marks]

Solutions

7MasterySAQ-SPolynomial expressions and their factorizationss6 marksPaper 1~9 min

The volume,

V

cubic metres, of a rectangular box with height

x

metres is given by

V(x) = x(6-x)(8-x)

, where

0 < x < 6

.

(a)

Show that

V(x) = x^3 - 14x^2 + 48x

. [2 marks]

(b)

Hence, find all values of

x

for which the volume is

45

cubic metres. Give your answers in exact form. [4 marks]

Solutions

8ChallengeSAQ-LPolynomial expressions and their factorizationss8 marksPaper 1~12 min

Consider the polynomial function

f(x) = 2x^4 - 5x^3 - 10x^2 + 15x + 18

.

(a)

(i) Show that

x = 2

is a zero of

f(x)

. [1]
(ii) Hence find a quadratic factor of

f(x)

, showing all division steps. [3 marks]

(b)

Determine the exact values of all real zeros of

f(x)

. [2 marks]

(c)

State the behaviour of

f(x)

as

x \to \pm\infty

. Hence deduce the number of local turning points of

y = f(x)

, and determine the number of those turning points that are local minima. Justify your answer. [2 marks]

Solutions

9FoundationSAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

Consider the polynomial

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

.

(a)

Show that

x = 1

is a root of

P(x)

. [2 marks]

(b)

Hence write

P(x)

in the form

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

, where

a

,

b

,

c \in \mathbb{Z}

. [2 marks]

(c)

Find all real roots of

P(x)

. [1 mark]

Solutions

10MasterySAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

Consider the polynomial

P(x) = 2x^3 + ax^2 + bx - 6

, where

a

and

b

are real constants. When

P(x)

is divided by

(x-1)

, the remainder is

-6

.

(x+2)

is a factor of

P(x)

.

(a)

Show that

a = 3

and

b = -5

. [3 marks]

(b)

Hence, factorise

P(x)

completely. [2 marks]

Solutions

11FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The mass of a sample of a radioactive isotope is modelled by

M(t) = M_0 \cdot 2^{-\frac{t}{8}}

, where

M_0

is the initial mass in grams and

t

is the time in years.

(a)

Show that the half-life of the isotope is 8 years. [1 mark]

(b)

Find the value of

t

for which

M(t) = \dfrac{M_0}{4}

. [3 marks]

(c)

A second isotope has the same initial mass

M_0

and a decay model

N(t) = M_0 \cdot 2^{-\frac{t}{k}}

, where

k > 0

. Given that

N(t) > M(t)

for all

t > 0

, determine the set of possible values of

k

. [1 mark]

Solutions

12MasterySAQ-SExponent laws and propertiess5 marksPaper 1~8 min

A radioactive isotope has initial mass

100\,\text{g}

. The mass

m

(in grams) remaining after

t

years is modelled by

m = 100 \times 2^{-\frac{t}{10}}

(a)

Show that the mass remaining after

30

years is

12.5\,\text{g}

. [2 marks]

(b)

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

t

for which the mass remaining is

3.125\,\text{g}

. [3 marks]

Solutions

13ChallengeSAQ-LExponent laws and propertiess8 marksPaper 1~12 min

The number of bacteria in a laboratory culture,

N(t)

, after

t

hours is modelled by

N(t) = A \cdot 2^{kt}

where

A

and

k

are positive constants. Initially there are

500

bacteria. After

3

hours there are

4000

bacteria.

(a)

Write down the value of

A

. [1 mark]

(b)

Determine the exact value of

k

. [2 marks]

(c)

The culture reaches a critical threshold when

N(t) = 500 \times 2^{10}

. (i) Show that the time

T

to reach the critical threshold satisfies

T = \dfrac{10}{k}

. [2]
(ii) Hence determine the exact value of

T

, and state what this reveals about the effect of doubling

k

on the time to reach the critical threshold. [3 marks]

Solutions

14FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

Consider the expression

\dfrac{2x^3 \times 4x^5}{8x^2}

.

(a)

Simplify the expression, giving your answer in the form

kx^n

where

k \in \mathbb{Z}

and

n \in \mathbb{Z}

. [3 marks]

(b)

The simplified expression equals

\dfrac{1}{x^p}

for some

p \in \mathbb{Z}^+

. Determine the value of

x

for which this possible, and find the corresponding value of

p

. [2 marks]

Solutions

15MasterySAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The intensity

I

of light (in lumens) at a depth

x

metres below the surface of a lake is given by

I = I_0 \times 10^{-0.2x}

, where

I_0

is the intensity at the surface.

(a)

Show that a depth of

5

metres, the intensity is

\dfrac{I_0}{10}

. [2 marks]

(b)

Find the depth at which the intensity is

\dfrac{I_0}{500}

. [3 marks]

Solutions

16FoundationSAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

Consider the binomial expansion of

(2 + x)^5

.

(a)

Write down the general term of the expansion in the form

\binom{5}{r} 2^{5-r} x^r

, and state the values of

r

that give valid terms. [1 mark]

(b)

Find the term in

x^3

in the expansion. [2 marks]

(c)

Given that the coefficient of

x^3

in the expansion of

(k + x)^5

is

270

, find the value of

k

. [2 marks]

Solutions

17MasterySAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

Consider the binomial expansion of

\left(2x^2 - \dfrac{3}{x}\right)^6

.

(a)

Show that the coefficient of

x^3

in the expansion is

-4320

. [3 marks]

(b)

Hence, find the coefficient of

x^3

in the expansion of

\left(2x^2 - \dfrac{3}{x}\right)^6(x + 2)

. [2 marks]

Solutions

18ChallengeSAQ-LBinomial expansion and coefficientss9 marksPaper 1~14 min

The expansion of

(1 + ax)^n

, where

n \in \mathbb{Z}^+

and

a \in \mathbb{R}

, has coefficients of

x^2

and

x^3

equal to

112

and

448

respectively.

(a)

Determine the values of

a

and

n

. [5 marks]

(b)

Find the sum of all coefficients in the expansion of

(1 + ax)^n

for the values found in part (a). [1 mark]

(c)

The expansion of

(1 + ax)^n

is multiplied by

(1 + x)

. Determine the coefficient of

x^5

in the resulting expansion, and hence deduce whether this coefficient is greater than, equal to, or less than the coefficient of

x^5

in the expansion of

(1 + ax)^n

alone. [3 marks]

Solutions

19FoundationSAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

Consider the expansion of

(1 + 2x)^5

.

(a)

Write down the six numbers in Row 5 of Pascal's triangle. [2 marks]

(b)

Using your answer to part (a), expand

(1 + 2x)^5

, giving each coefficient in simplified form. [3 marks]

Solutions

20MasterySAQ-SBinomial expansion and coefficientss5 marksPaper 1~8 min

The coefficient of

x^2

in the expansion of

(1 + kx)^5

is

90

.

(a)

Show that

k = 3

or

k = -3

. [3 marks]

(b)

Hence, for the case where

k > 0

, find the coefficient of

x^4

in the expansion of

(1 + kx)^5

. [2 marks]

Solutions

21MasterySAQ-SDefinition and general term of arithmetic sequencess8 marksPaper 2~12 min

A company manufactures cylindrical containers. The heights of the containers, measured in centimetres, form an arithmetic sequence. The third container has a height of

16\,\text{cm}

and the seventh container has a height of

28\,\text{cm}

.

(a)

Find the common difference and the height of the first container. [3 marks]

(b)

Find the height of the twelfth container. [2 marks]

(c)

The company needs the total height of the first

n

containers to reach at least

500\,\text{cm}

. Find the minimum value of

n

. [3 marks]

Solutions

22MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min

A farmer is planting trees in rows. The number of trees in each row forms an arithmetic sequence. The first row has

15

trees. The fifth row has

27

trees.

(a)

Show that the common difference of the sequence is

3

. [2 marks]

(b)

Calculate the number of trees in the twelfth row. [2 marks]

(c)

The total number of trees planted in the first

n

rows is

540

. Calculate the value of

n

. [2 marks]

Solutions

23MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min

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

12\,\text{cm}

. Each subsequent step has a height

2\,\text{cm}

greater than the previous step.

(a)

Calculate the height of the 25th step. [2 marks]

(b)

Show that the total vertical height of a staircase with 25 steps is exactly

900\,\text{cm}

. [2 marks]

(c)

The company needs the total vertical height to exceed

900\,\text{cm}

. Determine the minimum number of steps required. [2 marks]

Solutions

24MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min

A biologist monitors the growth of a bamboo plant. On the first day of observation, the plant has a height of

40\,\text{cm}

. The plant grows by a constant amount each day. On the

10

th day of observation, the plant has a height of

103\,\text{cm}

.

(a)

Calculate the daily growth rate of the bamboo plant. [2 marks]

(b)

Determine on which day the height of the plant first exceeds

200\,\text{cm}

. [2 marks]

(c)

The biologist claims that the plant will reach a height of exactly

250\,\text{cm}

on a particular day of observation. Determine whether this claim is correct, justifying your answer. [2 marks]

Solutions

25MasterySAQ-SDefinition and general term of arithmetic sequencess6 marksPaper 2~9 min

A gardener plants trees in a straight line along a path. The first tree is planted

5\,\text{m}

from the start of the path. Each subsequent tree is planted

3\,\text{m}

further from the start than the previous tree.

(a)

Calculate the distance from the start of the path to the

20

th tree. [2 marks]

(b)

The last tree is planted

152\,\text{m}

from the start of the path. Find the total number of trees planted. [2 marks]

(c)

A water pipe is to run from the start of the path to each tree and back again. Calculate the total length of pipe required to service all trees exactly once each. [2 marks]

Solutions

26MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

A company manufactures cylindrical containers. The volume

V

, in

\text{cm}^3

, of a container with radius

x\,\text{cm}

(

x > 0

) is given by

V = \frac{2\pi x^3 + 4\pi x^2}{x + 2}.

(a)

State the domain of

V

in the context of this problem. [1 mark]

(b)

Show that

V = 2\pi x^2

. [2 marks]

(c)

Hence, determine the radius

x

when

V = 72\pi\,\text{cm}^3

. Give your answer in exact form. [3 marks]

Solutions

27MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

Consider the rational function

f(x) = \dfrac{2x^2 - 5x - 3}{x^2 - 9}

.

(a)

Determine the domain of

f

. [2 marks]

(b)

Show that

f(x)

simplifies to

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

, stating any restriction

x

. [2 marks]

(c)

The graph of

f

has two discontinuities. Determine the coordinates of each discontinuity and, for each one, state whether it is a vertical asymptote or a removable discontinuity (hole). Justify your answer. [2 marks]

Solutions

28MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

A rectangle has an area given by

A(x) = \dfrac{x^2 + 2x - 15}{x - 3}

square units, where

x

is a positive real number.

(a)

Simplify

A(x)

by factorising the numerator and cancelling any common factors. [2 marks]

(b)

State the value of

x

that must be excluded from the domain of

A(x)

, and explain why this value is also not physically meaningful in the context of the rectangle. [2 marks]

(c)

Calculate the values of

x

for which the area of the rectangle equals

9

square units, and determine which value is valid given the domain restriction found in part (b). [2 marks]

Solutions

29MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

The concentration of a chemical in a solution,

C(t)

parts per million, is modelled by

C(t) = \frac{3t^2 - 6t}{t^2 - 4t + 4}, \quad t > 2,

where

t

is the time in hours after the experiment starts.

(a)

Show that

C(t)

can be written in the form

A + \dfrac{B}{t-2}

, where

A

and

B

are integers to be determined. [3 marks]

(b)

State the value of

t

for which

C(t)

is undefined, and explain why this value is excluded from the domain. [2 marks]

(c)

Determine the time at which the concentration is

4

parts per million. [1 mark]

Solutions

30MasterySAQ-SRational expressions and their simplifications6 marksPaper 2~9 min

A rectangular field has length

x

metres and width

y

metres. The area of the field is

180\,\text{m}^2

and the perimeter is

54

metres.

(a)

Write down two equations in

x

and

y

. [1 mark]

(b)

Show that

x

satisfies

x^2 - 27x + 180 = 0

. [2 marks]

(c)

Find the length and width of the field. [2 marks]

(d)

A second rectangular field has its length and width each increased by

3

metres compared to the first field. Determine whether the ratio of the area of the second field to the area of the first field is greater than

1.5

. [1 mark]

Solutions

31MasterySAQ-SExponent laws and propertiess8 marksPaper 2~12 min

The population of a colony of ants,

P

, is modelled by the function

P(t) = 500 \times 3^{kt}

, where

t

is the time in weeks after the colony is first observed. After 4 weeks, the population is 4500 ants.

(a)

Show that

k = \dfrac{1}{2}

. [3 marks]

(b)

Calculate the population after 10 weeks. Give your answer in the form

a \times 3^b

, where

a

and

b

are integers. [1 mark]

(c)

Determine the number of complete weeks it takes for the population to first exceed 1 000 ants. [4 marks]

Solutions

32MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

The intensity of light,

I

(in lux), at a depth

x

metres below the surface of a lake is modelled by

I(x) = I_0 \times 10^{-0.2x}

, where

I_0

is the intensity at the surface. At the surface, the intensity is

1200

lux.

(a)

Calculate the intensity at a depth of

3

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

(b)

Show that the depth at which the intensity equals

60

lux satisfies

x = \dfrac{\log_{10}(0.05)}{-0.2}

. [2 marks]

(c)

A diver requires a minimum intensity of

15

lux to work safely. Determine the maximum depth, in metres, at which the diver can work. Give your answer correct to 2 significant figures. [2 marks]

Solutions

33MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

The value of a rare painting,

V

dollars, is modelled by the function

V(t) = 2000 \times 3^{kt}

, where

t

is the number of years since its purchase. After 4 years, the painting is valued at USD 54000.

(a)

Calculate the value of

k

. [3 marks]

(b)

Calculate the value of the painting after 10 years, giving your answer correct to the nearest dollar. [2 marks]

(c)

Determine the number of complete years it takes for the painting's value to first exceed USD 1 000. [1 mark]

Solutions

34MasterySAQ-SExponent laws and propertiess6 marksPaper 2~9 min

A biologist models the population of bacteria in a petri dish,

P

cells, using the equation

P = 500 \times 2^{\frac{t}{d}}

where

t

is the time in hours and

d

is a positive constant. After

6

hours, the population is

8000

cells.

(a)

Calculate the value of

d

. [3 marks]

(b)

Calculate the time, in hours, for the population to reach

64000

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

Solutions

35MasterySAQ-SExponent laws and propertiess8 marksPaper 2~12 min

A scientist is studying the decay of a radioactive isotope. The mass

m(t)

of the isotope, in grams, after

t

years is given by

m(t) = 100 \times 2^{-\frac{t}{30}}

(a)

Calculate the mass of the isotope after

60

years. [2 marks]

(b)

Determine the value of

t

for which the mass is

20

grams. [3 marks]

(c)

A second isotope has initial mass

60\,\text{g}

and decays according to

n(t) = 60 \times 2^{-\frac{t}{20}}

. Determine the value of

t

at which both isotopes have the same mass, and state which isotope has greater mass for

t

beyond this value. Total: 8 marks > (Note: total adjusted to 8 to accommodate the added AO3 sub-part; reduce to 6 by omitting (c) if required.) [3 marks]

Solutions

36MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

In the expansion of

(2x + k)^7

, where

k \in \mathbb{Z}^+

, the coefficient of the

x^3

term is

22\,680

.

(a)

Calculate the value of

k

. [4 marks]

(b)

Hence, determine the coefficient of the

x^4

term in the expansion of

(2x + k)^7

. Binomial theorem:

(a+b)^n = \displaystyle\sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r

[2 marks]

Solutions

37MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

Consider the binomial expansion of

\left(2x - \dfrac{1}{x}\right)^8

.

(a)

Show that the general term of the expansion can be written as

\dbinom{8}{r} 2^{8-r}(-1)^r\, x^{8-2r}

. [1 mark]

(b)

Hence calculate the term independent of

x

. [2 marks]

(c)

The constant term of

\left(2x - \dfrac{1}{x}\right)^8 \times (ax^2 + bx)

equals

-1\,120

, where

a, b \in \mathbb{Z}

. Given that

a = 1

, find the value of

b

. [3 marks]

Solutions

38MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

The probability that a randomly selected patient experiences a side effect from a new drug is

0.15

. A clinical trial involves

8

independent patients. Let

X

be the number of patients who experience a side effect.

(a)

State the probability distribution of

X

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

(b)

Calculate

P(X = 3)

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

(c)

Calculate

P(X \leq 2)

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

Solutions

39MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

Consider the binomial expansion of

(2x - 3)^6

.

(a)

State the coefficient of

x^3

in the expansion. [2 marks]

(b)

Find the constant term in the expansion of

\left(2x - \dfrac{3}{x^2}\right)^6

. [4 marks]

Solutions

40MasterySAQ-SBinomial expansion and coefficientss6 marksPaper 2~9 min

Consider the binomial expansion of

\left(2x^2 - \dfrac{1}{x}\right)^6

.

(a)

Show that the general term of the expansion can be written as

\dbinom{6}{r} 2^{6-r}(-1)^r x^{12-3r}

. [2 marks]

(b)

Hence calculate the coefficient of

x^3

in the expansion. [2 marks]

(c)

Determine the constant term in the expansion. [2 marks]

Solutions

Number and Algebra — Free Maths AA SL Practice Questions