Algebraic Expressions and Identities — Free IB MYP4 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

4 marks~6 minCriterion A

A biology researcher models bacterial decay using the function

y = 2^x

, where

x

represents time in hours (negative values indicate hours before a reference point) and

y

represents the relative bacterial population in millions.

The graph shows integer values of

x

from

-3

to

3

, but the

y

-values for

x = -1

,

x = -2

, and

x = -3

are missing.

a

Calculate the missing

y

-values for

x = -1

,

x = -2

, and

x = -3

using

a^{-n} = \dfrac{1}{a^n}

. [3]

b

The researcher claims the bacterial population was never zero in the hours before the reference point. Justify this claim using your results from part (a) and the behaviour of

y = 2^x

. [1]

Solutions

3QuestionDeriving Identities from Expansion and FactorizationConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a square patio with side length

(x + 3)

metres, where

x > 0

.

a

Show that the area of the patio can be written as

x^2 + 6x + 9

square metres. [1]

b

The architect has 26 square metres of paving material available. Given that

x = 2

, advise the architect whether the available material is sufficient to complete the patio, justifying your answer. [1]

Solutions

4QuestionExpanding Double Brackets BinomialsConcept Practice

2 marks~3 minCriterion A

A garden designer plans a rectangular patio with side lengths

(x + 3)

m and

(x + 5)

m, where

x > 0

. The rectangle is divided into four smaller rectangles by splitting each side at the integer term.

a

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

x^2 + 8x + 15

. [1]

b

The designer has 40 m² of paving material available. Given that

x = 2

, advise the designer whether the available material is sufficient to complete the patio. [1]

Solutions

5QuestionCombining 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

6QuestionMultiple Substitutions in Complex ExpressionsConcept Practice

4 marks~6 minCriterion A

The graph shows a parabola modelled by

y = ax^2 + bx + c

, opening upwards, with vertex at

(0, -1)

. The parabola passes through

(-2, 3)

,

(0, -1)

, and

(2, 3)

.

a

State the value of

c

by identifying the

y

-intercept from the graph. [1]

b

Substitute the points

(-2, 3)

and

(2, 3)

into

y = ax^2 + bx + c

to construct two equations in

a

and

b

. [1]

c

Deduce the values of

a

and

b

by solving the system of equations from part (b). [1]

d

The parabola models the cross-sectional profile of a satellite dish. The dish is functional only if its profile is symmetric about the central axis. Justify whether this dish meets the symmetry requirement, using the values of

a

,

b

, and

c

. [1]

Solutions

7QuestionFactoring Difference of SquaresConcept Practice

2 marks~3 minCriterion B

Consider the following factorised expressions:

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

4x^2 - 25 = (2x-5)(2x+5)

9y^2 - 16 = (3y-4)(3y+4)

25m^2 - 49 = (5m-7)(5m+7)

a

Identify the structural pattern linking each expression to its factors. [1]

b

Deduce a general rule for factoring any expression of the form

a^2 - b^2

. [1]

Solutions

8QuestionFactoring Difference of SquaresConcept Practice

2 marks~3 minCriterion A

An architect designs a square marble tile with side length

a = 9

cm. A square notch of side length

b = 4

cm is cut from one corner, creating an L-shaped tile. The area of the L-shaped tile is given by

a^2 - b^2

.

a

Show that

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

, and hence calculate the area of the L-shaped tile. [1]

b

A box holds exactly 10 L-shaped tiles. Justify whether the total tiled area is sufficient to cover a surface of

640 \text{ cm}^2

. [1]

Solutions

9QuestionAdding 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

10QuestionAdding and Subtracting with Common DenominatorsAssessment Practice

4 marks~6 minCriterion D

A bakery models its average cost per batch of cupcakes (in dollars) using the expression

\frac{1200}{x} + \frac{800}{x}

where 1200 dollars represents ingredient costs, 800 dollars represents labour costs, and

x

is the number of batches produced.

a

Simplify

\dfrac{1200}{x} + \dfrac{800}{x}

to a single rational expression. [1]

b

The bakery owner uses

\dfrac{2000}{x}

to decide how many batches to produce each week. Interpret this expression in the context of the bakery: explain what it reveals about cost behaviour, and advise the owner whether this model alone is sufficient to support production decisions. [3]

Solutions

11QuestionMultiplying and Dividing Algebraic FractionsAssessment Practice

4 marks~6 minCriterion C

A two-stage water filtration system removes contaminants at each stage. The fraction of contaminants removed by stage one is

\dfrac{x}{x+2}

and by stage two is

\dfrac{x+2}{x+4}

, where

x > 0

represents the initial contaminant level. The combined removal fraction is the product of the two stage fractions.

a

Deduce a simplified expression for the combined removal fraction after both stages. [1]

b

The initial contaminant level is

x = 4

. Calculate the combined removal fraction and interpret what this value means for the filtration system. [2]

c

A filtration engineer claims that this two-stage system is sufficient to meet a safety standard requiring a combined removal fraction greater than

\dfrac{3}{4}

for all

x > 0

. Justify whether the engineer's claim is correct. [1]

Solutions

12QuestionProduct Quotient and Power LawsAssessment Practice

4 marks~6 minCriterion C

The functions

y = 2^x

and

y = 2^{x+3}

are defined for all real

x

.

a

State the transformation that maps the graph of

y = 2^x

onto the graph of

y = 2^{x+3}

. [1]

b

The point

(0, 1)

lies on

y = 2^x

. Show that the image of this point under the transformation confirms your answer to part (a). [1]

c

A student claims: "Adding 3 inside the exponent is equivalent to multiplying the original function by a constant, so the transformation could also be described as a vertical stretch." Justify whether this claim is correct, using algebraic reasoning. [2]

Solutions

13QuestionNegative 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

14QuestionExponents in Algebraic EquationsAssessment Practice

4 marks~6 minCriterion D

A doctor prescribes a 200 mg dose of a drug. The amount remaining in a patient's bloodstream after

t

hours is modelled by

A = 200(0.85)^t

where

A

is in mg.

a

Calculate the amount of drug remaining after 4 hours. [2]

b

The drug is considered safe to re-dose when less than 50 mg remains. Using your answer to part (a), explain whether a second dose could be given after 8 hours. [1]

c

Advise the doctor whether this model alone is sufficient to determine a safe re-dosing time for all patients. [1]

Solutions

15QuestionDeriving 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

16QuestionDifference of Squares IdentityAssessment Practice

2 marks~3 minCriterion D

A contractor is designing a square patio of side length

(n + 2)

m, from which a square tree well of side length

(n - 2)

m will be removed from one corner, where

n = 4

.

a

Show that the remaining patio area can be written as

(n+2)^2 - (n-2)^2

, and use the difference of squares identity

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

to calculate this area. [1]

b

The contractor's measurements are recorded to the nearest centimetre. Advise the contractor whether this level of precision is adequate for a construction project requiring an accuracy of

\pm 0.01\ \text{m}^2

, justifying your answer with a calculation of the resulting uncertainty in area. [1]

Solutions

17QuestionDeriving Identities from Expansion and FactorizationAssessment Practice

4 marks~6 minCriterion C

A graphic designer creates a square logo with intended side length

(x + 3)

cm, where

x

is a positive integer. Due to manufacturing tolerance, the printed side length is

(x + 3 \pm 0.1)

cm. The designer uses the identity

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

to model the logo's area.

a

Expand

(x + 3)^2

to find the intended area of the logo in the form

ax^2 + bx + c

. [1]

b

When

x = 5

, calculate the difference between the area at the upper tolerance side length and the intended area. [2]

c

The designer must decide whether to present the logo's area as

x^2 + 6x + 9

or as

(x + 3)^2

when communicating the effect of manufacturing tolerance to a client. Advise the designer which form to use, justifying your answer with reference to the structure of each expression. [1]

Solutions

18QuestionCombining Like Terms After ExpansionAssessment Practice

6 marks~9 minCriterion B

The product

(x+1)(x+2)(x+3)\cdots(x+n)

generates a family of polynomials. A cryptographer uses the constant term of this product to count the number of ways to assign distinct integer keys

1, 2, \ldots, n

to

n

slots.

a

Expand

(x+1)

,

(x+1)(x+2)

,

(x+1)(x+2)(x+3)

, and

(x+1)(x+2)(x+3)(x+4)

, writing each result in descending powers of

x

. [2]

b

Deduce the constant term and the coefficient of

x^{n-1}

for each expansion in part (a), expressing both as general formulas in

n

. [2]

c

Justify why the constant term always equals

n!

and the coefficient of

x^{n-1}

always equals

1+2+\cdots+n

, using the structure of polynomial multiplication. Then advise the cryptographer whether

5

distinct integer keys provide enough unique assignments to serve as a secure one-time code, given that a code is considered secure only if more than

100

distinct assignments exist. [2]

Solutions

19QuestionExpanding Double Brackets BinomialsAssessment Practice

10 marks~15 minCriterion D

A concert venue has a maximum capacity of 4000 seats. When the ticket price is 50 dollars, 2000 people are expected to attend. Market research shows that for every 2 dollar decrease in ticket price, 120 more people are expected to attend.

Let

x

represent the number of 2 dollar decreases in ticket price.

Revenue is modelled by

R(x) = (50 - 2x)(2000 + 120x)

.

a

Expand and simplify

R(x)

into the form

R(x) = ax^2 + bx + c

. [2]

b

Deduce the value of

x

that maximises revenue, and hence find the corresponding ticket price. [3]

c

The venue manager claims that the model predicts a maximum revenue above 105 000 dollars. Assess whether this claim is correct, and explain one reason why the model may not reflect actual revenue at this ticket price. [5]

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

21QuestionEvaluating 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

22QuestionEvaluating with Negative Numbers and FractionsAssessment Practice

4 marks~6 minCriterion D

A meteorologist uses the following formula to calculate wind chill temperature

T_{wc}

(in °C), based on air temperature

T_a

(in °C) and wind speed

v

(in km/h):

T_{wc} = 13.12 + 0.6215T_a - 11.37v^{0.16} + 0.3965T_av^{0.16}

a

Calculate

T_{wc}

when

T_a = -15

°C and

v = 30

km/h. [1]

b

Explain why

T_{wc}

is lower than

T_a

when wind speed increases, referring to the structure of the formula. [1]

c

A public safety advisory warns of dangerous conditions whenever

T_{wc} < -35

°C. A forecast gives

T_a = -25

°C and

v = 50

km/h. Calculate

T_{wc}

for these conditions, then advise whether the advisory should be issued, justifying your decision with reference to the threshold and the risk to public safety. [2]

Solutions

23QuestionFactoring Trinomials Quadratic FormAssessment Practice

4 marks~6 minCriterion D

A factory produces rectangular metal sheets. The length of each sheet is

(x + 4)

cm and the width is

(x + 1)

cm, where

x

is a positive integer. The area

A

cm

^2

of each sheet is modelled by

A = x^2 + 5x + 4

.

Due to a material constraint, each sheet must have an area of exactly 40 cm

^2

.

a

Factorise

x^2 + 5x + 4

. Hence solve

x^2 + 5x + 4 = 40

, stating both solutions. [2]

b

Deduce which value of

x

is valid in this context and state the corresponding length and width of the sheet. [1]

c

Discuss two limitations of using this algebraic model to represent real-world sheet production. [1]

Solutions

24QuestionChoosing Appropriate Factorization MethodsAssessment Practice

4 marks~6 minCriterion C

A rectangular solar panel has an area modelled by

A(x) = 3x^2 - 12

square metres, where

x

is a design parameter in metres. Engineers require the panel to have zero net area at the boundary values of

x

.

The graph of

y = 3x^2 - 12

crosses the

x

-axis at

x = -2

and

x = 2

.

a

Show that

3x^2 - 12 = 3(x-2)(x+2)

by first extracting the common factor, then applying the difference of squares identity

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

. [2]

b

Explain how the

x

-intercepts of the graph confirm that the factored form

3(x-2)(x+2)

correctly represents

A(x)

. [1]

c

The engineers state: "Only positive values of $x$ are physically meaningful, so only the boundary value $x = 2$ is relevant to our design." Assess whether this statement is mathematically valid. [1]

Solutions

Algebraic Expressions and Identities — Free MYP4 Mathematics (Extended) Practice Questions