Algebraic Expressions and Identities — Free IB MYP4 Mathematics (Standard) 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 microchip component has a rectangular cross-section with width

2^{-3}

m and length

2^{-2}

m. An engineer must confirm the component meets a size specification requiring an area less than

2^{-4}

m².

a

Calculate the area of the cross-section. [1]

b

Justify whether the component meets the size specification. [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

4QuestionPerfect 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

5QuestionSimplifying Like TermsConcept Practice

2 marks~3 minCriterion C

A landscape architect is designing a triangular garden bed. The three side lengths are

2x

cm,

3x

cm, and

4x

cm, and the perimeter of the garden bed is 36 cm.

a

Calculate the value of

x

. [1]

b

The architect states: "The longest side of this garden bed exceeds half the perimeter, so the design is impractical." Justify whether this statement is correct, supporting your answer with calculations. [1]

Solutions

6QuestionExpanding 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

7QuestionMultiple 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

8QuestionFactoring Difference of SquaresConcept Practice

4 marks~6 minCriterion A

A landscape architect models a rectangular garden bed. Its area in square metres is described by the function

y = x^2 - 9

, where

x

is a design parameter.

a

Deduce the factored form of

x^2 - 9

. [1]

b

Using your answer to part (a), determine the values of

x

where

y = 0

. [2]

c

The design parameter

x

must be positive. Justify which x-intercept is valid for this context. [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

10QuestionRestrictions and Undefined Values in ExpressionsAssessment Practice

4 marks~6 minCriterion D

A car engine's fuel efficiency is modelled by the rational expression

E(v) = \frac{120v}{v^2 + v - 12}

where

E

is efficiency in km/L and

v

is speed in km/h.

a

Show that the expression is undefined at

v = 3

km/h and find any other value of

v

for which it is undefined. [2]

b

Discuss the strengths and limitations of this model in the context of engine calibration, referring to the undefined value found in part (a) and at least one other feature of the model. [2]

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

12QuestionNegative 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

13QuestionProduct 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

14QuestionWord Problems Involving ExponentsAssessment Practice

4 marks~6 minCriterion D

A student invests 5000 dollars in a savings account at a 6% annual interest rate compounded monthly. The account balance after

t

years is modelled by

A = P\!\left(1 + \frac{r}{n}\right)^{nt}

where

P

is the principal,

r

is the annual interest rate,

n

is the number of compounding periods per year, and

A

is the balance.

a

Show that the predicted balance after 5 years is approximately 6744.25 dollars. [1]

b

The student's goal is to reach a balance of 7000 dollars. Deduce the number of complete years required for the balance to exceed 7000 dollars. [1]

c

Advise the student whether this exponential model alone is a sufficient basis for making an investment decision. Refer to at least two specific limitations in your response. [2]

Solutions

15QuestionPerfect Square IdentitiesAssessment Practice

2 marks~3 minCriterion C

A gardener designs a square flower bed of side length

x

metres, then adds a 2-metre-wide border path along two adjacent sides, forming an L-shaped region. The combined area satisfies

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

, where

x^2

is the flower bed area and

4x + 4

is the path area.

a

Given

x = 5

metres, calculate the area of the path alone. [1]

b

The measurement of

x

has an error of

+0.1

metres. Analyse how this error affects the path area, and explain why the amplification of the error depends on the border width. [1]

Solutions

16QuestionPerfect Square IdentitiesAssessment Practice

6 marks~9 minCriterion D

A contractor is tiling a square patio of side length

(x + 3)

m, where

x

is a positive integer. Each square tile has side length

0.5

m and area

0.25

m². The client requests the patio be enlarged by adding

2

m to each side, giving a new side length of

(x + 5)

m. The contractor uses the perfect square identity

(x + 5)^2 = x^2 + 10x + 25

to find the new area, then calculates the number of tiles as

\dfrac{x^2 + 10x + 25}{0.25}

, assuming tiles fit the patio exactly with no cuts, gaps, or wastage.

a

Calculate the number of tiles required for the enlarged patio when

x = 4

. [2]

b

Identify two distinct real-world factors that would cause the actual number of tiles needed to exceed the model's prediction. Explain how each factor leads to underestimation. [2]

c

A supplier recommends ordering

12\%

more tiles than the model predicts to account for wastage. Using your answer from part (a), calculate the adjusted tile order for

x = 4

, then advise the contractor whether this adjusted figure should be used in place of the model's prediction for project planning. [2]

Solutions

17QuestionSimplifying Expressions with Multiple TermsAssessment Practice

6 marks~9 minCriterion B

A sound engineer balances two audio channels using the expression

a(x+y)+b(x-y)

, where

a

and

b

are integer gain values and

x

,

y

represent signal inputs.

a

Expand and simplify each expression. State the coefficient of

x

and the coefficient of

y

. [2]

a=2,\ b=3

:

\quad 2(x+y)+3(x-y)

a=4,\ b=1

:

\quad 4(x+y)+1(x-y)

a=5,\ b=2

:

\quad 5(x+y)+2(x-y)

a=3,\ b=5

:

\quad 3(x+y)+5(x-y)

b

Deduce a general rule for the coefficient of

x

and the coefficient of

y

in the simplified form of

a(x+y)+b(x-y)

. Justify your rule using the distributive property. [2]

c

A new channel setting uses

a=7

and

b=4

. The engineer needs the coefficient of

y

to be positive to avoid phase cancellation. Use your rule to predict the simplified form of

7(x+y)+4(x-y)

, then advise the engineer whether this setting should be approved for use. [2]

Solutions

18QuestionExpanding 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

19QuestionEvaluating with Negative Numbers and FractionsAssessment Practice

6 marks~9 minCriterion C

A structural engineer models the flexibility of a beam joint using the expression

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

, where

a

is a dimensionless stiffness ratio. A negative value of

a

indicates the joint resists in the opposite direction.

a

Evaluate

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

and

\left(a + \dfrac{1}{a}\right)^{-2}

for each value of

a

below. Show all working.

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

[2]

b

Show that

\left(a + \dfrac{1}{a}\right)^{-1} = \dfrac{a}{a^2+1}

, and hence justify why

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

always has the same sign as

a

, while

\left(a + \dfrac{1}{a}\right)^{-2}

is always positive. [2]

c

The engineer states: "For any negative stiffness ratio, the flexibility measure $\left(a + \dfrac{1}{a}\right)^{-1}$ is always negative and its magnitude is always less than $\tfrac{1}{2}$ ." Evaluate

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

for

a = -\dfrac{2}{3}

and

a = -4

, then critique the engineer's claim, using the expression

\dfrac{a}{a^2+1}

to support your reasoning. [2]

Solutions

20QuestionEvaluating 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

21QuestionMultiple Substitutions in Complex ExpressionsAssessment Practice

12 marks~18 minCriterion D

In electrical circuit analysis, the total resistance

R_T

(in ohms) of a parallel circuit with three resistors is given by:

\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}

A circuit is designed with

R_1 = 20\ \Omega

,

R_2 = 30\ \Omega

,

R_3 = 60\ \Omega

.

a

Calculate

R_T

for these values. [2]

b

Each resistor has a tolerance of

\pm 5\%

. Deduce the maximum and minimum possible values of

R_T

, showing clearly which resistor values you use in each case. [4]

c

A safety specification requires

R_T

to remain within

\pm 5\%

of its nominal value under all operating conditions. Temperature fluctuations cause each resistor's resistance to change independently of its stated tolerance. Advise the engineer whether the

\pm 5\%

tolerance alone is sufficient to guarantee the specification is met, and state what additional information is required to complete the analysis. [6]

Solutions

22QuestionFactoring Difference of SquaresAssessment Practice

4 marks~6 minCriterion D

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

23QuestionFactoring 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

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 (Standard) Practice Questions