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

1QuestionAdding and Subtracting with Common DenominatorsConcept Practice

4 marks~6 minCriterion A

A water treatment plant monitors chemical concentration using the function

y = \frac{2}{x+1} + \frac{3}{x+1}

where

y

is the concentration (mg/L) and

x

is the time elapsed (hours) after treatment begins,

x > 0

.

a

Show that

\dfrac{2}{x+1} + \dfrac{3}{x+1} = \dfrac{5}{x+1}

. [1]

b

Deduce the concentration at

x = 3

hours. [1]

c

The plant requires a minimum concentration of 1.5 mg/L to be effective. Advise the plant operator whether the treatment is effective at

x = 3

hours, justifying your answer with reference to the threshold. [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 B

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

9QuestionMultiplying 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

10QuestionRestrictions 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

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

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

13QuestionNegative 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

14QuestionPerfect Square IdentitiesAssessment Practice

4 marks~6 minCriterion D

A landscape architect models the cross-section of a decorative water channel using the function

y = x^2 + 6x + 9

, where

x

is the horizontal distance (metres) from a reference point and

y

is the height (metres) above ground level.

a

Show that

x^2 + 6x + 9 = (x + 3)^2

. [1]

b

Hence deduce the coordinates of the

x

-intercept of the graph. [1]

c

The channel is designed so that water flows only where

y = 0

. Justify whether this design is suitable for carrying water along the channel. [2]

Solutions

15QuestionRecognizing and Using Identities in ProblemsAssessment Practice

4 marks~6 minCriterion A

An architect models the curved profile of a pedestrian bridge using a parabola. The lowest point of the arch is at coordinates

(3, -2)

relative to a reference point, and the arch passes through

(4, -1)

. The

y

-intercept of the model represents the height of the bridge support at the reference edge.

a

Deduce the equation of the parabola in vertex form

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

, showing how the value of

a

is determined. [2]

b

Show that the standard form of the equation is

y = x^2 - 6x + 7

, applying the perfect square identity in your expansion. [1]

c

Interpret the

y

-intercept of the model, and justify whether this value confirms the arch is structurally above the reference level at the support edge. [1]

Solutions

16QuestionDifference of Squares IdentityAssessment Practice

4 marks~6 minCriterion C

A square tile of side length

a = 5

cm has a smaller square of side length

b

cm removed from one corner, leaving a shaded L-shaped region. The graph shows the area of this shaded region,

A

cm

^2

, as a function of

b

, for

0 \leq b \leq 5

.

a

Write down an expression for

A

in terms of

b

when

a = 5

, and state the value of

A

when

b = 3

. [1]

b

Show that

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

, and verify this identity using

a = 5

,

b = 3

. [2]

c

A manufacturer requires the shaded region to have area at least 9 cm

^2

. Using your result from (b), advise the manufacturer on the largest value of

b

that meets this requirement, justifying your answer in context. [1]

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

5 marks~8 minCriterion B

A tile manufacturer cuts square ceramic tiles of side length

a

cm, then removes a smaller square of side length

b

cm from one corner to create an L-shaped tile for border edging.

a

Show that

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

by expanding the left-hand side algebraically. State the property used at each step. [2]

b

Construct an area model for the L-shaped tile. Show that the remaining area can be expressed both as

a^2 - b^2

and as

(a+b)(a-b)

. [2]

c

A tile is described as "large" if its L-shaped area exceeds

300\ \text{cm}^2

. Given

a = 20

and

b = 5

, justify whether this tile qualifies as large. [1]

Solutions

23QuestionChoosing 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

24QuestionGrouping and Advanced PatternsAssessment Practice

2 marks~3 minCriterion C

A landscape architect models a rectangular garden plot with area given by

x^3 + 3x^2 + 2x + 6

square metres, where

x > 0

.

a

Show that

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

using the grouping method. [1]

b

Given that

x = 3

, the architect claims one side length exceeds 11 m. Justify whether this claim is correct. [1]

Solutions

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