Algebraic Expressions and Identities — Free IB MYP5 Mathematics (Extended) Questions

1QuestionAlgebraic Fraction Word ProblemsConcept Practice

4 marks~6 minCriterion A

A water treatment plant monitors chemical concentration using the rational function

C(t) = \frac{2t + 4}{t + 1}

where

C

is the concentration in milligrams per litre and

t

is the time in hours after treatment begins. The safe upper limit for the chemical is

2.6 \text{ mg/L}

.

a

Calculate

C(3)

, simplifying your answer fully. [2]

b

Advise the plant operator whether the concentration at

t = 3

hours requires immediate action, justifying your answer with reference to the safe limit. [2]

Solutions

2QuestionNegative and Zero ExponentsConcept Practice

2 marks~3 minCriterion A

A nanoscale sensor chip contains a square circuit element with side length

3^{-2}

m.

a

Calculate the area of the circuit element, expressing your answer as a simplified fraction. [1]

b

A second square element has an area of

\dfrac{1}{6561}

m

^2

. Deduce whether its side length is greater than, equal to, or less than that of the first element, justifying your answer using index notation. [1]

Solutions

3QuestionPerfect Square IdentitiesConcept Practice

2 marks~3 minCriterion A

A square tile has side length

(x + 4)

cm. The tile is divided into four rectangular regions with areas

x^2

,

4x

,

4x

, and

16

cm² respectively.

Show that the total area of the tile can be written as

(x + 4)^2 = x^2 + 8x + 16

. [2]

Solutions

4QuestionCombining Like Terms After ExpansionConcept Practice

4 marks~6 minCriterion C

A mobile data company advertises two pricing plans:

f(x) = 2(x + 3) \quad \text{and} \quad g(x) = 2x + 6

where

x

is the number of gigabytes used and the output is the cost in dollars.

a

Show that

f(x) = 2x + 6

by expanding using the distributive property. [1]

b

Explain why

f(x)

and

g(x)

represent the same function for all values of

x

. [1]

c

A customer claims that Plan

f

is cheaper than Plan

g

. Advise the customer whether this claim is correct, and what it means for their choice between the two plans. [2]

Solutions

5QuestionExpanding Double Brackets BinomialsConcept Practice

4 marks~6 minCriterion A

A farmer models the area of a rectangular field (in square metres) using the expression

(x+5)(x+3)

, where

x

is a positive side-length in metres.

a

Expand and simplify

(x+5)(x+3)

. [2]

b

The farmer needs a field with an area of at least

77 \text{ m}^2

. Given that

x = 7

, justify whether this field meets the requirement, using your expression from part (a). [1]

c

Discuss one strength and one limitation of using this algebraic expression as a model for planning crop planting. [1]

Solutions

6QuestionMultiple Substitutions in Complex ExpressionsConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a triangular flower bed. The base of the triangle is

b = (2x + 3)

cm and the height is

h = 4y

cm, where

x

and

y

are positive integers determined by the available space.

The area of a triangle is

A = \dfrac{1}{2}bh

.

When

x = 2

and

y = 1

, calculate the area of the flower bed, in cm², showing all substitution steps. [2]

Solutions

7QuestionChoosing Appropriate Factorization MethodsConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a rectangular garden bed. The area of the bed is given by

x^2 + 8x + 15

square metres, where

x > 0

. One side of the bed measures

(x + 3)

metres.

a

Factorise

x^2 + 8x + 15

. [1]

b

The architect needs the missing side to be longer than 7 metres when

x = 3

. Justify whether this condition is met. [1]

Solutions

8QuestionAdding and Subtracting with Common DenominatorsAssessment Practice

6 marks~9 minCriterion B

A chemical engineer models the net concentration of a reagent in a reactor using the expression

\frac{2x^2+3}{x-5} + \frac{5x-1}{x-5} - \frac{x^2+2}{x-5}

where

x

is temperature in °C and

x \neq 5

.

a

Deduce the general rule for adding and subtracting algebraic fractions that share a common denominator

Q

. [2]

b

Apply your rule to simplify the expression above, showing all working. [3]

c

The reactor operates safely only when the net concentration expression yields a value greater than 30 for integer temperatures in the range

6 \leq x \leq 8

. Justify whether the reactor operates safely across this entire range. [1]

Solutions

9QuestionRestrictions and Undefined Values in ExpressionsAssessment Practice

4 marks~6 minCriterion C

A water treatment plant monitors flow rate using the model

y = \dfrac{1}{x - 3}

, where

x

is time in hours after midnight and

y

is flow rate in megalitres per hour. The graph of this function is shown.

a

State the value of

x

for which

\dfrac{1}{x-3}

is undefined. Explain why the denominator determines this restriction. [2]

b

Identify the feature on the graph that corresponds to this restriction, and explain what it indicates about the behaviour of

y

near that value of

x

. [1]

c

Advise the plant operators whether this model can be used to determine flow rate at 3:00 am, justifying your answer with reference to the model's behaviour at

x = 3

. [1]

Solutions

10QuestionMultiplying and Dividing Algebraic FractionsAssessment Practice

6 marks~9 minCriterion D

A biologist models the population of a bacterial colony (in thousands) using

P(t) = \frac{5t^2 + 3t}{t + 1}

where

t

is time in hours. Recorded populations are:

t = 1

: actual

4.0

thousand

t = 2

: actual

8.2

thousand

t = 3

: actual

13.5

thousand

a

Calculate

P(1)

,

P(2)

, and

P(3)

. [3]

b

Deduce, for each value of

t

, whether the model overestimates, underestimates, or matches the actual population, and by how much. [2]

c

Assess whether this model is suitable for predicting the colony's population over time, using the pattern of errors as evidence. [1]

Solutions

11QuestionNegative and Zero ExponentsAssessment Practice

4 marks~6 minCriterion B

A population of bacteria doubles every hour. A scientist models the population using

y = 2^x

, where

x

is the number of hours elapsed (negative values represent time before observation began) and

y

is the population in millions.

Selected values from the model:

x

:

-2

,

-1

,

0

,

1

,

2

y

(millions):

\dfrac{1}{4}

,

\dfrac{1}{2}

,

?

,

2

,

4

a

Explain how the pattern in the table demonstrates that

2^0 = 1

. [2]

b

A second bacterial strain follows

y = 5^x

and a third follows

y = (-3)^x

. Evaluate

5^0

and

(-3)^0

, justifying your answers using the zero exponent rule. [1]

c

The scientist claims the population at

x = 0

is "effectively zero" because no growth has yet occurred. Assess whether the mathematical model supports or contradicts this claim, and interpret what

2^0 = 1

means in this context. [1]

Solutions

12QuestionNegative and Zero ExponentsAssessment Practice

2 marks~3 minCriterion C

A car currently has a value of 8000 dollars. Its value depreciates at a constant rate of 10% per year, modelled by

V = P(1 - r)^{t}

, where

V

is the value after

t

years,

P

is the initial value, and

r

is the annual depreciation rate.

a

Calculate the value of the car 2 years ago. [1]

b

A buyer argues that this model overestimates how much the car was worth 2 years ago. Critique this claim, with reference to both the mathematical result and the real-world depreciation pattern. [1]

Solutions

13QuestionNegative and Zero ExponentsAssessment Practice

6 marks~9 minCriterion D

A patient receives a 500 mg dose of a drug. The amount of drug remaining in the body after

t

hours is modelled by

A = 500 \times 2^{-t},

where

A

is measured in mg. A second dose may be given once

A

falls below 100 mg.

a

Show that the amount of drug remaining after 3 hours is 62.5 mg. [2]

b

Deduce, using your result from part (a), whether a second dose can be given at

t = 3

hours. Justify your answer in the context of the model. [1]

c

Advise a clinician whether the model

A = 500 \times 2^{-t}

alone is sufficient to determine the safe timing of a second dose. In your response, discuss at least two real-world limitations and explain how neglecting these limitations could affect patient safety. [3]

Solutions

14QuestionDeriving Identities from Expansion and FactorizationAssessment Practice

3 marks~5 minCriterion B

A garden designer models rectangular planting beds using binomial products. She observes the following pattern when expanding

(x + a)(x + b)

:

(x + 2)(x + 3) = x^2 + 5x + 6

(x - 1)(x + 4) = x^2 + 3x - 4

(x - 3)(x - 2) = x^2 - 5x + 6

The designer plans a bed modelled by

(x + 5)(x - 3)

, where

x

represents a length in metres.

a

Deduce the coefficient of

x

and the constant term in the expansion of

(x + 5)(x - 3)

. [1]

b

Construct the fully expanded form of

(x + 5)(x - 3)

. [1]

c

The designer states: "Because the constant term is negative, no positive value of

x

can make this expression equal to zero." Critique this statement, using the roots of the expression to support your reasoning. [1]

Solutions

15QuestionDifference of Squares IdentityAssessment Practice

2 marks~3 minCriterion C

An architect is designing a square patio of side length

a

metres. A square planter of side length

b

metres is removed from one corner, creating an L-shaped paved area (see diagram).

a

Write down an expression for the area of the L-shaped paved region, then show that it factorizes to

(a-b)(a+b)

. [1]

b

The architect states: "The paved area can always be calculated by multiplying the sum and difference of the two side lengths." Justify whether this statement is mathematically valid for all values

a > b > 0

. [1]

Solutions

16QuestionDeriving Identities from Expansion and FactorizationAssessment Practice

6 marks~9 minCriterion D

A packaging company designs square-based open-top boxes with a fixed total surface area of 600 cm². The side length of the square base is

x

cm and the height is

h

cm, giving:

SA = x^2 + 4xh = 600.

a

Using the identity

(a + b)^2 = a^2 + 2ab + b^2

, show that

x = \sqrt{300 + h^2} - h.

[2]

b

For

h = 10

cm, deduce the exact side length

x

and the exact volume

V = x^2 h.

[2]

c

The company can only manufacture to the nearest millimetre. Advise the production manager whether

h = 10

cm is suitable for manufacture at this precision, and what action to take if

h

is changed to 8 cm. [2]

Solutions

17QuestionCombining Like Terms After ExpansionAssessment Practice

8 marks~12 minCriterion B

Consider the following expansions:

(x+1)(x+2) = x^2 + 3x + 2

(x+2)(x+3) = x^2 + 5x + 6

(x+3)(x+4) = x^2 + 7x + 12

(x+4)(x+5) = x^2 + 9x + 20

(a) Using the pattern above, predict:

(i) the constant term in the expansion of

(x+5)(x+6)

. [1]

(ii) the coefficient of

x

in the expansion of

(x+5)(x+6)

. [1]

(b) Deduce a general formula for

(x+a)(x+b)

, showing full algebraic working. [2]

(c) Apply your formula to expand

(x+10)(x+15)

. [1]

(d) A student claims: "The coefficient of $x$ is always equal to the product of the two constants." Evaluate this claim using your general formula and one example from the table above. [2]

(e) A rectangular garden has side lengths

(x + 10)

metres and

(x + 15)

metres. The designer states the garden can only be built if its area exceeds

500\ \text{m}^2

when

x = 10

. Justify whether the designer's condition is met. [1]

Solutions

18QuestionCombining Like Terms After ExpansionAssessment Practice

4 marks~6 minCriterion D

A small business owner models monthly profit (in dollars) as

P(x) = (x + 50)(200 - 2x)

, where

x

is the number of one-dollar price increases applied to the product. The model assumes demand decreases linearly as price rises.

a

Expand

P(x) = (x + 50)(200 - 2x)

and write it in the form

ax^2 + bx + c

, stating the values of

a

,

b

, and

c

. [2]

b

Advise the business owner whether this model is a sufficient basis for determining an optimal pricing strategy. Refer to both the mathematical structure of

P(x)

and real-world factors the model does not capture. [2]

Solutions

19QuestionEvaluating with Negative Numbers and FractionsAssessment Practice

6 marks~9 minCriterion B

A signal-processing engineer models the attenuation (reduction in strength) of a radio signal at stage

n

of a transmission network using the expression

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

.

The first five values are:

\left(1 + \frac{1}{1}\right)^{-1} = \frac{1}{2}, \quad \left(2 + \frac{1}{2}\right)^{-1} = \frac{2}{5}, \quad \left(3 + \frac{1}{3}\right)^{-1} = \frac{3}{10}, \quad \left(4 + \frac{1}{4}\right)^{-1} = \frac{4}{17}, \quad \left(5 + \frac{1}{5}\right)^{-1} = \frac{5}{26}

a

Deduce a general formula for the attenuation at stage

n

, expressing your answer in the form

\dfrac{n}{f(n)}

. [2]

b

Calculate the attenuation at stage

10

using your formula. [2]

c

The engineer claims that the attenuation will never fall below

0.09

for any stage in this network (

n \leq 12

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

Solutions

20QuestionMultiple Substitutions in Complex ExpressionsAssessment Practice

12 marks~18 minCriterion C

A structural engineer uses the following values to model load-bearing stress in a beam:

a = 2, \quad b = -3, \quad c = \tfrac{1}{2}

a

Evaluate

2a^2 - 3b + 4c

. [2]

b

Evaluate

\dfrac{a^3 + b^2}{c}

. [2]

c

The two expressions give different results. Justify which expression produces the greater stress value and explain how the structure of each expression — including the effect of negative values and division by a fraction — leads to that outcome. [8]

Solutions

21QuestionSolving Word Problems with SubstitutionAssessment Practice

2 marks~3 minCriterion D

A local bakery calculates delivery cost using

C = 2n + 5

, where

C

is the total delivery cost in dollars and

n

is the number of items ordered. The fixed fee is 5 dollars; each item costs an additional 2 dollars.

Advise the bakery owner whether the formula

C = 2n + 5

is fair for customers placing small orders. Support your advice with at least one calculation and justify a specific modification to the formula that would address any identified problem. [2]

Solutions

22QuestionFactoring Difference of SquaresAssessment Practice

6 marks~9 minCriterion B

Consider the following numerical pattern:

1.

4^2 - 3^2 = (4-3)(4+3) = 1 \times 7 = 7

2.

7^2 - 2^2 = (7-2)(7+2) = 5 \times 9 = 45

3.

10^2 - 5^2 = (10-5)(10+5) = 5 \times 15 = 75

a

Calculate

12^2 - 7^2

using the same method. Show all steps. [2]

b

Deduce a general formula for

a^2 - b^2

in terms of

a

and

b

, and apply it to factorise

25x^2 - 16y^2

completely. [2]

c

A student claims that

49x^2 - 9

cannot be factorised because it contains both a variable term and a constant. Justify whether this claim is correct. [2]

Solutions

23QuestionFactoring Difference of SquaresAssessment Practice

4 marks~6 minCriterion C

A landscape designer plans a square garden with side length

a = 10

m. A square pond with side length

b = 2

m will be placed in one corner. The remaining garden area (excluding the pond) is modelled using the identity

a^2 - b^2 = (a-b)(a+b)

.

a

Show that the area of the garden excluding the pond is

96 \text{ m}^2

, using the difference of squares identity. [1]

b

The pond's side length is remeasured as

2.1

m. Calculate the revised area and deduce the percentage change from the original result. [1]

c

The designer considers replacing the square pond with a circular pond of diameter

2

m. Advise the designer whether the difference of squares identity remains a valid model and whether the area calculation must be adjusted. [2]

Solutions

24QuestionChoosing Appropriate Factorization MethodsAssessment Practice

2 marks~3 minCriterion D

A homeowner plans to tile a rectangular floor with length

(x + 3)

metres and width

(x + 2)

metres. Tiles cost 15 dollars per square metre.

a

Show that the total tiling cost, in dollars, is

15x^2 + 75x + 90

. [1]

b

The homeowner has a budget of 300 dollars and estimates

x = 3

. Advise the homeowner whether to proceed with the purchase, giving one mathematical reason why the actual cost could differ from this estimate. [1]

Solutions

Algebraic Expressions and Identities — Free MYP5 Mathematics (Extended) Practice Questions