Equations Inequalities and Formulae — Free IB MYP5 Mathematics (Extended) Questions

1QuestionGraphical Representation of InequalitiesConcept Practice

2 marks~3 minCriterion A

A city planner models a budget constraint for two infrastructure projects. The total spending on Project A (

x

million) and Project B (

y

million) must remain below a fixed limit. The boundary line of the feasible region passes through

(0, 2)

and

(2, 0)

and is shown as a dashed line on the graph. The region below this line is shaded.

a

Write the inequality that represents the feasible spending region. [1]

b

The planner proposes spending

x = 0.8

million on Project A and

y = 1.4

million on Project B. Justify whether this combination is a feasible spending plan. [1]

Solutions

2QuestionSolving by Elimination MethodConcept Practice

4 marks~6 minCriterion A

A city planner models two pedestrian pathways on a coordinate grid. Pathway P follows the line

2x + y = 8

and Pathway Q follows the line

x - y = 1

. The pathways intersect at a single point, which marks the location of a proposed information kiosk.

a

Use the elimination method to find the exact coordinates of the intersection point. [2]

b

The kiosk must be placed at least 2.5 units from the origin to avoid underground utilities. Justify whether the proposed location satisfies this requirement. [2]

Solutions

3QuestionMulti-Step Problems with VariablesConcept Practice

2 marks~3 minCriterion B

A community centre arranges daily volunteer shifts. The shift lengths (in hours) follow a pattern of consecutive integers:

Day 1:

1 + 2 = 3

hours
Day 2:

2 + 3 + 4 = 9

hours
Day 3:

3 + 4 + 5 + 6 = 18

hours
Day 4:

4 + 5 + 6 + 7 + 8 = 30

hours

On Day 5, shifts run for 6 consecutive integers starting at 5.

a

Deduce the pattern connecting the sum of each set to its terms. [1]

b

A volunteer coordinator claims Day 5 totals exactly 45 hours of coverage. Justify whether this claim is correct. [1]

Solutions

4QuestionEquations Involving Brackets and FractionsConcept Practice

2 marks~3 minCriterion A

A personal trainer designs a 45-minute session split into three identical blocks. Each block includes a 5-minute warm-up plus

2x

minutes of circuit training, where

x

is the number of circuit rounds.

a

Show that the total session time gives the equation

3(2x + 5) = 45

, and solve for

x

. [1]

b

The trainer states: "Each block has time for at least 4 circuit rounds." Using your value of

x

, justify whether this statement is correct. [1]

Solutions

5QuestionSolving by FactorizationConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a rectangular garden with a length of

(x + 4)

m and a width of

(x + 2)

m. The total area of the garden must be exactly

35 \text{ m}^2

.

a

Show that the area condition gives the equation

x^2 + 6x - 27 = 0

. [1]

b

Solve

x^2 + 6x - 27 = 0

by factorization. Hence, deduce the actual dimensions of the garden and justify which solution is valid in this context. [1]

Solutions

6QuestionGraphical Representation of InequalitiesAssessment Practice

6 marks~9 minCriterion B

A city planner is designing a pedestrian zone. The safe walking corridor must satisfy two constraints modelled by linear inequalities. Three trial corridor systems are tested:

System 1:

y \geq x + 1

and

y \leq -x + 5

System 2:

y \geq x + 3

and

y \leq -x + 3

System 3:

y \geq x + 5

and

y \leq -x + 1

A proposed corridor uses the system

y \geq x + 2

and

y \leq -x + 4

.

a

Analyse the three trial systems. For each, identify whether the feasible region is bounded, a single point, or empty, and explain how the relationship between the constants

c_1

and

c_2

determines this outcome. [2]

b

Deduce the nature of the feasible region for the proposed corridor system, justifying your answer using the pattern identified in part (a). [1]

c

Construct a graph of the proposed corridor system, clearly showing the boundary lines, the feasible region, and the coordinates of any intersection points. Hence advise the city planner whether the proposed corridor should be approved, justifying your recommendation with reference to the feasible region. [3]

Solutions

7QuestionSolving Linear Inequalities in One VariableAssessment Practice

4 marks~6 minCriterion C

A city regulation states that a food truck must be located more than 4 metres from the nearest fire hydrant. A number line is provided showing all permitted distances

x

(in metres) from a hydrant.

a

Interpret the number line: explain what the open circle at

x = 4

and the arrow pointing right each indicate about the solution set. [1]

b

Write an inequality in the form

ax + b > c

, where

a

,

b

, and

c

are integers and

a \neq 1

, that represents the same solution set. [1]

c

A rival truck parks at

x = 4.0

m. Justify whether this truck complies with the regulation. [2]

Solutions

8QuestionGraphical Representation of InequalitiesAssessment Practice

4 marks~6 minCriterion D

A farmer has 100 hectares of land and a budget of 120000 dollars to plant two crops: wheat and corn. Planting one hectare of wheat costs 1000 dollars, while one hectare of corn costs 1500 dollars. The profit per hectare of wheat is 5000 dollars, and the profit per hectare of corn is 8000 dollars.

a

Let

x

be the number of hectares of wheat and

y

be the number of hectares of corn. Write down two inequalities that represent the constraints on the land and budget.

b

Sketch a graph showing the feasible region defined by these inequalities, with

x

on the horizontal axis and

y

on the vertical axis. Label the axes clearly.

c

The farmer wants to maximize their profit. Discuss the strengths and limitations of using this linear programming model to determine the optimal number of hectares of each crop to plant. Consider factors such as unpredictable weather patterns, fluctuating market prices, and the assumption of constant profit per hectare.

Solutions

9QuestionSolving by Substitution MethodAssessment Practice

4 marks~6 minCriterion B

A logistics company tracks two delivery routes. Each day, the total distance covered by both routes is

a

km, and the difference between the longer and shorter route is

b

km. The distances

x

(longer) and

y

(shorter), in km, satisfy:

x + y = a \qquad x - y = b

a

Deduce the values of

x

and

y

for each system. [1]

System 1:

a = 5,\ b = 1

\quad

System 2:

a = 7,\ b = 3

System 3:

a = 10,\ b = 4

\quad

System 4:

a = 12,\ b = 2

b

Deduce general formulas for

x

and

y

in terms of

a

and

b

. [1]

c

Prove that your formulas satisfy both original equations. [1]

d

The company requires the shorter route to be at least 30% of the total daily distance. For

a = 40

and

b = 10

, advise the company whether this scheduling requirement is met, justifying your answer using your formulas. [1]

Solutions

10QuestionSolving by Elimination MethodAssessment Practice

4 marks~6 minCriterion D

A small business owner models monthly revenue and cost (in dollars) as:

R = 15x \qquad C = 2000 + 8x

where

x

is the number of units sold, the selling price is 15 dollars per unit, the fixed monthly rent is 2000 dollars, and the variable material cost is 8 dollars per unit.

a

Use the elimination method to find the break-even value of

x

, where

R = C

. State your answer as a whole number of units and justify your rounding. [2]

b

The business introduces a bulk discount for large orders and experiences seasonal demand fluctuations. Advise the business owner whether this linear model should be used to predict profit under these conditions, citing the specific assumptions that are affected. [2]

Solutions

11QuestionSolving GraphicallyAssessment Practice

4 marks~6 minCriterion C

A city planner models two road-cost proposals using the equations

y = 2x - 1

and

y = -x + 5

, where

x

is years after the project starts and

y

is cost in millions of dollars. The graph of both lines is shown.

a

State the coordinates of the intersection point of the two lines. [1]

b

Interpret what the intersection point represents in terms of the two cost models. [1]

c

The planner states: "The two proposals reach equal cost at exactly year 2." Justify whether this claim is correct, showing your algebraic reasoning. [2]

Solutions

12QuestionNature of Roots Using DiscriminantAssessment Practice

10 marks~15 minCriterion C

Consider the quadratic function

f(x) = x^2 + kx + 9

, where

k

is a real constant.

a

Find the set of values of

k

for which

f(x) = 0

has two distinct real roots, giving your answer in set notation. [3]

b

For the specific case

k = 8

, show that the vertex of the graph of

y = f(x)

lies below the

x

-axis, and hence state the number of

x

-intercepts of the graph. [3]

c

A student claims: "Because the coefficient of

x^2

is positive and

f(0) = 9 > 0

, the graph of

y = x^2 + kx + 9

can never cross the

x

-axis for any real value of

k

."

Critique this claim. Use the discriminant to identify a specific value of

k

for which the graph crosses the

x

-axis at two distinct points, and explain why the student's reasoning is flawed. [4]

Solutions

13QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion B

The diagram shows Figures 1 to 4 of a pattern made from small squares arranged in an L-shape.

Figure 1: 3 squares, Figure 2: 7 squares, Figure 3: 13 squares, Figure 4: 21 squares.

a

Write down the number of small squares in Figure 5. [1]

b

Find a rule for the number of small squares

S

in Figure

n

, giving your answer in the form

S = n^2 + bn + c

, where

b

and

c

are integers to be found. [3]

c

A student claims that no figure in this pattern can contain exactly 100 small squares. Justify whether the student is correct. [2]

Solutions

14QuestionNature of Roots Using DiscriminantAssessment Practice

5 marks~8 minCriterion A

A quadratic function is given by

f(x) = x^2 + bx + c

, where

b

and

c

are integers.

The graph of

f(x)

passes through the point

(1, 0)

and has its axis of symmetry at

x = 3

.

a

Write down the two roots of

f(x) = 0

. [1]

b

Find the values of

b

and

c

. [2]

c

The equation

f(x) = k

has no real solutions. Justify whether a value of

k = -5

would satisfy this condition, using the vertex of the parabola. [2]

Solutions

15QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion D

A basketball player shoots a free throw. The height

h

metres of the ball above the ground,

t

seconds after release, is modelled by

h(t) = -4.9t^2 + 8t + 2, \quad t \geq 0.

A motion sensor records the following observed heights.

t

(seconds):

0.4

\quad

0.8

\quad

1.2

Observed

h

(m):

4.1

\quad

5.0

\quad

3.6

a

Calculate the predicted heights given by the model at

t = 0.4

,

t = 0.8

, and

t = 1.2

. Give each value to 2 decimal places. [1]

b

The axis of symmetry of the parabola is

t = \dfrac{8}{9.8}

. Calculate the maximum height of the ball. Give your answer in metres to 3 significant figures. [2]

c

The player claims the ball reaches a maximum height of more than 5 metres. Assess whether the model supports this claim, and justify your reasoning using your answers to parts (a) and (b). [3]

Solutions

16QuestionApplications in Geometry and Physics FormulasAssessment Practice

4 marks~6 minCriterion B

A construction team is checking whether a triangular brace is truly right-angled before installing it. They measure the three sides as 2.5 m, 6 m, and 6.5 m.

Before applying this to the brace, investigate the relationship between side lengths in right triangles using the four Pythagorean triples below.

Triangle 1:

(3, 4, 5)

— Triangle 2:

(5, 12, 13)

— Triangle 3:

(6, 8, 10)

— Triangle 4:

(7, 24, 25)

a

For each triangle, calculate the sum of the squares of the two shorter sides and the square of the longest side. [1]

b

Deduce a general equation relating the three side lengths of any right triangle. [1]

c

Using your equation from (b), advise the construction team whether the brace with sides 2.5 m, 6 m, and 6.5 m should be approved for installation, justifying your answer with full working. [2]

Solutions

17QuestionCommon Mistakes in RearrangingAssessment Practice

2 marks~3 minCriterion C

A household toaster is rated at

P = 920\ \text{W}

and has a heating element with resistance

R = 28.8\ \Omega

.

A student rearranges

P = I^2 R

incorrectly, writing

I = \dfrac{1}{2}\sqrt{\dfrac{P}{R}}

, and uses this to select a wire for the toaster circuit.

Explain why this algebraic error leads to an unsafe choice of wire. [2]

Solutions

18QuestionCommon Mistakes in RearrangingAssessment Practice

4 marks~6 minCriterion D

A civil engineer uses the formula

d = \frac{v^2}{2\mu g}

to model the braking distance

d

(in metres) of a car on a wet road, where

v

is the initial speed (in m/s),

\mu

is the coefficient of friction, and

g = 9.8 \text{ m/s}^2

.

a

Show that rearranging the formula for

\mu

gives

\mu = \frac{v^2}{2dg}.

[1]

b

Calculate

\mu

when

d = 50

m and

v = 20

m/s. Round your answer to two decimal places. [1]

c

A road safety standard requires

\mu \geq 0.45

on wet roads. Using your result from part (b), advise the engineer whether this road should remain open to traffic in wet conditions, and explain one limitation of using this model to support that decision. [2]

Solutions

19QuestionRearranging with Fractions and BracketsAssessment Practice

2 marks~3 minCriterion A

A surveyor models a triangular plot of land. The three sides measure

a = 5

m,

b = 7

m, and

c = 6

m, where angle

\theta

is opposite side

c

. The cosine rule states:

c^2 = a^2 + b^2 - 2ab\cos\theta

a

Rearrange the cosine rule to make

\cos\theta

the subject. [1]

b

Calculate

\theta

and hence justify whether the triangular plot contains any obtuse angle. [1]

Solutions

20QuestionChoosing Appropriate MethodsAssessment Practice

3 marks~5 minCriterion C

A car's distance from its starting point is recorded below.

Time (hours)	0	1	2	3	4
Distance (km)	0	40	90	120	200

a

Calculate the average speed of the car during the interval

t = 2

hours to

t = 4

hours, using

\text{average speed} = \frac{\text{total distance}}{\text{total time}}

Show your working and give your answer in km/h. [1]

b

The car travels the same total distance of 200 km in 4 hours on a second journey, but at a constant speed throughout. Calculate this constant speed. [1]

c

The road used for the first journey has a speed limit of 60 km/h. Justify whether your answer from part (a) alone is sufficient evidence to conclude that the speed limit was exceeded during the interval

t = 2

to

t = 4

hours. [1]

Solutions

21QuestionAge Distance and Geometry-Based ProblemsAssessment Practice

5 marks~8 minCriterion A

A rectangular park has length

(3x + 5)

m and width

(2x - 3)

m, where

x

represents the age in years of a student. The perimeter of the park is 124 m. A straight diagonal path connects two opposite corners of the park.

a

Show that the perimeter of the park gives the equation

10x + 4 = 124

, and hence deduce the value of

x

. [2]

b

Using your value of

x

, calculate the length and width of the park. [1]

c

The park's management states that any diagonal path longer than 45 m requires safety lighting. Using Pythagoras' theorem, calculate the length of the diagonal path correct to 2 decimal places, and hence justify whether safety lighting is required. [2]

Solutions

22QuestionChoosing Appropriate MethodsAssessment Practice

2 marks~3 minCriterion D

A small business pays a monthly electricity bill structured as follows: a fixed base charge of 50 dollars, a rate of 0.10 dollars per kWh for the first 500 kWh, and 0.15 dollars per kWh for any usage beyond 500 kWh.

A single linear equation

C = 50 + 0.10x

has been proposed to model the total monthly cost

C

(in dollars) for

x

kWh consumed.

Calculate the cost predicted by the linear model and the actual cost when

x = 800

kWh, then justify whether the single linear equation is a suitable model for this billing structure. [2]

Solutions

23QuestionChecking Solutions and SubstitutionAssessment Practice

6 marks~9 minCriterion B

A water treatment plant adjusts chemical dosing using equations of the form

ax + 2a = 5a

, where

a

is a non-zero real number and

x

represents the dosing multiplier.

The following equations and proposed solutions are given:

x + 2 = 5, \quad x = 3

2x + 4 = 10, \quad x = 3

3x + 6 = 15, \quad x = 3

4x + 8 = 20, \quad x = 3

a

Deduce the solution to

10x + 20 = 50

. [1]

b

Show that every equation of the form

ax + 2a = 5a

, where

a \neq 0

, has solution

x = 3

. [3]

c

The plant requires a dosing multiplier strictly greater than 3 to safely treat a new water source. The engineer models this situation with the equation

100x + 200 = 500

. Solve this equation and advise the engineer whether the plant can safely treat the new water source under this model. [2]

Solutions

24QuestionWord Problems Involving Linear EquationsAssessment Practice

4 marks~6 minCriterion D

A school fundraiser sells adult tickets for 12 USD and student tickets for 7 USD. A total of 20 student tickets are sold alongside

x

adult tickets, generating total revenue

R

(in USD).

a

Write the equation that represents

R

in terms of

x

. [1]

b

The total revenue collected was 284 USD. Calculate the number of adult tickets sold. [2]

c

The organiser claims that selling twice as many adult tickets would double the total revenue. Justify whether this claim is correct, and interpret what your answer means for the fundraiser. [1]

Solutions

25QuestionSolving One-Step and Two-Step EquationsAssessment Practice

4 marks~6 minCriterion C

A taxi company charges a flat fee of 3 USD plus 2 USD per kilometre travelled. A passenger pays a total fare of 13 USD.

a

Write an equation in one variable to represent this situation. [1]

b

Determine the number of kilometres travelled. Show your working clearly. [2]

c

A second passenger uses a different taxi company that charges a flat fee of 6 USD and 2 USD per kilometre, paying a total fare of 26 USD. Justify whether the two passengers travel the same number of kilometres. [1]

Solutions

26QuestionSolving by FactorizationAssessment Practice

2 marks~3 minCriterion B

Examine the following factorized quadratics:

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

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

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

x^2 + 11x + 30 = (x+5)(x+6)

Analyse the relationship between the constant term, the coefficient of

x

, and the integers in the brackets across all four expressions.

Deduce the factorized form of

x^2 + 13x + 42

, clearly stating your conjecture in written form. [2]

Solutions

27QuestionPractice Quadratic Equations solvingAssessment Practice

4 marks~6 minCriterion D

A basketball player shoots a free throw. The height of the ball above the ground, in metres, after

t

seconds is modelled by

h = -4.9t^2 + 8t + 2.1.

a

Calculate the time at which the ball reaches its maximum height and the maximum height itself. [2]

b

Determine, to two decimal places, the time at which the ball hits the ground. [1]

c

The regulation basketball hoop height is 3.05 m. Justify whether this model provides sufficient evidence that the free throw could be successful, identifying one limitation that affects the reliability of your conclusion. [1]

Solutions

28QuestionPractice Quadratic Equations solvingAssessment Practice

2 marks~3 minCriterion C

A small business owner models weekly profit (in dollars) using

P(x) = -x^2 + 40x - 300

where

x

is the number of items sold that week.

a

Solve

-x^2 + 40x - 300 = 0

to find the break-even points. [1]

b

The owner claims that profit is possible for any sales volume above 30 items. Using your results from part (a), advise the owner whether this claim is correct, justifying your answer in context. [1]

Solutions

Equations Inequalities and Formulae — Free MYP5 Mathematics (Extended) Practice Questions