Equations Inequalities and Formulae — Free IB MYP4 Mathematics (Extended) Questions

1QuestionGraphical Representation of InequalitiesConcept Practice

2 marks~3 minCriterion A

A delivery company charges a base fee of 1 dollar plus 2 dollars per kilogram. A customer's budget satisfies

y \leq 2x + 1

, where

x

is the mass (kg) of the parcel and

y

is the maximum total cost (dollars) the customer will pay.

a

On the coordinate grid provided, construct the graph of

y \leq 2x + 1

for

0 \leq x \leq 4

, showing clearly the boundary line and the solution region. [1]

b

State the coordinates of one point in the solution region. Justify whether a 2 kg parcel costing 6 dollars satisfies the customer's budget constraint. [1]

Solutions

2QuestionChecking Solutions for Both EquationsConcept Practice

2 marks~3 minCriterion A

A city planner models two proposed road alignments on a coordinate grid. Road 1 follows

y = 2x + 1

and Road 2 follows

y = -x + 4

. A junction is planned at point

P(1, 3)

.

Justify that

P(1, 3)

is the junction point of the two roads by verifying it satisfies both equations. [2]

Solutions

3QuestionRearranging to Make a Variable the SubjectConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a triangular garden bed. The area formula for a triangle is

A = \frac{1}{2}bh

where

b

is the base length and

h

is the perpendicular height.

The garden bed has a base of

b = 8

m and must cover an area of

A = 24

m

^2

.

Rearrange the formula to make

h

the subject, then calculate the height of the garden bed. [2]

Solutions

4QuestionMulti-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

5QuestionChecking Solutions and SubstitutionConcept Practice

4 marks~6 minCriterion A

A mobile data plan charges a monthly fee. The total monthly cost,

y

(in dollars), for using

x

gigabytes of data is modelled by the equation

y = 2x + 1

.

a

Show that the point

(2, 5)

lies on the line

y = 2x + 1

. [1]

b

A customer uses 3 gigabytes in a month. Calculate their total monthly cost. [1]

c

A customer's budget allows a maximum monthly spend of 6 dollars. Advise the customer whether this budget is sufficient for 3 gigabytes of data usage, justifying your answer with a numerical comparison. [2]

Solutions

6QuestionSolving by FactorizationConcept Practice

2 marks~3 minCriterion D

In a soccer match, a player kicks a ball from ground level. Its height,

h

metres, after

t

seconds is modelled by

h = -5t^2 + 20t.

a

Factorise

-5t^2 + 20t

and hence deduce the time at which the ball returns to the ground. [1]

b

A second player claims the same model applies to her kick, but that her ball stays in the air for 5 seconds. Deduce whether this is possible, and justify your answer using the structure of the factorised expression. [1]

Solutions

7QuestionGraphical 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

8QuestionRepresenting Solutions on a Number LineAssessment Practice

4 marks~6 minCriterion C

A building's emergency generator activates when the outdoor temperature

T

(in °C) satisfies the inequality shown on the number line below.

a

Write the inequality represented by the number line in the form

a \leq T < b

. [1]

b

Justify whether

T = -2

and

T = 4

each belong to the solution set, referring to the number line notation. [2]

c

The generator's manual states it must activate "at any temperature strictly below 4°C, including −2°C." Assess whether the inequality you wrote in part (a) fully satisfies this requirement. [1]

Solutions

9QuestionGraphical Representation of InequalitiesAssessment Practice

2 marks~3 minCriterion D

A student plans a party with a budget of 30 dollars and can carry at most 10 items. Let

x

represent the number of snacks and

y

the number of drinks, where each snack costs 2 dollars and each drink costs 3 dollars. This situation is modelled by the system:

2x + 3y \leq 30, \quad x + y \leq 10, \quad x \geq 0, \quad y \geq 0

Justify whether this system of linear inequalities is a reliable model for the student's party-planning decisions. [2]

Solutions

10QuestionSolving 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

11QuestionSolving GraphicallyAssessment Practice

4 marks~6 minCriterion C

A city planner models two proposed road alignments as straight lines on a coordinate grid. The graph shows both lines.

Line 1 passes through

(0, 1)

and

(2, 5)

.
Line 2 passes through

(0, -3)

and

(2, 1)

.

a

Deduce the gradient of each line from the graph. [1]

b

Explain why the two roads will never intersect, using the properties of both lines. [2]

c

Advise the city planner whether a single crossing point between the two roads can ever be achieved by extending them indefinitely, and justify your advice. [1]

Solutions

12QuestionChecking Solutions for Both EquationsAssessment Practice

4 marks~6 minCriterion D

A city planner models two road-widening proposals. Proposal A follows the cost equation

y = x - 1

and Proposal B follows

y = -x + 5

, where

x

is the number of months into the project and

y

is the projected budget surplus (in millions of dollars). The graph shows both lines intersecting at point

P

.

a

State the coordinates of

P

from the graph. [1]

b

Show that

P

satisfies both equations simultaneously. [2]

c

Interpret what the coordinates of

P

mean in this context, and advise the city planner whether

P

should be used as the basis for deciding when both proposals are financially equivalent. [1]

Solutions

13QuestionSolving Using the Quadratic FormulaAssessment Practice

7 marks~11 minCriterion B

A pattern of growing L-shaped figures is made from unit squares. The first three figures are shown in the diagram.

Figure number ( $n$ )	1	2	3
Number of squares ( $S$ )	3	7	13

a

Write down the number of unit squares in Figure 4. [1]

b

Find a rule for the total number of unit squares

S

in Figure

n

. Give your rule in the form

S = an^2 + bn + c

, where

a

,

b

and

c

are integers. Show your method clearly. [3]

c

A student claims that every figure in the pattern contains an odd number of unit squares. Justify whether this claim is correct for all positive integer values of

n

. [3]

Solutions

14QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion C

A quadratic function is given by

f(x) = 2x^2 + 5x + c

, where

c

is a constant.

a

Given that

f\!\left(\dfrac{1}{2}\right) = 0

, show that

c = -3

. [1]

b

Find the other solution of

f(x) = 0

. Give your answer as an exact value. [2]

c

A engineer states that the equation

f(x) = k

has no real solutions for

k < -6

. Justify whether the engineer's claim is correct. [3]

Solutions

15QuestionNature of Roots Using DiscriminantAssessment Practice

10 marks~15 minCriterion D

A basketball player shoots a free throw. The height

h

(in metres) of the ball

t

seconds after release is modelled by

h(t) = -5t^2 + 4t + 2.1.

The hoop is at a height of

3.05

m. The table below shows the height predicted by the model and the actual measured height of the ball at three moments during a real shot.

Time

t

(s):

0.2

,

0.4

,

0.6

Predicted

h

(m):

2.70

,

2.90

,

2.70

Actual

h

(m):

2.15

,

2.05

,

1.78

a

Show that

h(t) = 3.05

can be written as

5t^2 - 4t + 0.95 = 0,

and hence calculate the discriminant of this equation. [2]

b

The player adjusts their release so that the new model is

h(t) = -5t^2 + bt + 2.1, \quad b > 0.

Find the minimum value of

b

for which the ball can just reach the hoop height of

3.05

m. Give your answer to 3 significant figures. [3]

c

Percentage error is defined as

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Calculate the percentage error at each of

t = 0.2

,

t = 0.4

, and

t = 0.6

. Give each answer to 1 decimal place. [3]

d

Using your percentage errors from part (c), justify whether this quadratic model is suitable for predicting the height of the ball throughout the shot. [2]

Solutions

16QuestionSolving Using the Quadratic FormulaAssessment Practice

5 marks~8 minCriterion A

A quadratic function is defined by

f(x) = 2x^2 + bx + 18

, where

b

is a constant.

The diagram shows the parabola

y = f(x)

crossing the

x

-axis at

x = \dfrac{3}{2}

and at a second point labelled

p

.

a

Write down the

y

-intercept of

y = f(x)

. [1]

b

Given that

x = \dfrac{3}{2}

is a root of

f(x) = 0

, calculate the value of

b

and hence the value of

p

. [3]

c

The equation

f(x) = k

has exactly one solution. Justify why exactly one value of

k

produces this result and state that value of

k

. [1]

Solutions

17QuestionApplications 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

18QuestionApplications in Geometry and Physics FormulasAssessment Practice

4 marks~6 minCriterion C

A photographer uses a thin lens with a focal length of

f = 50

mm. The lens formula

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

relates object distance

u

and image distance

v

.

a

Show that

v = \dfrac{uf}{u - f}

. [1]

b

An object is placed at

u = 200

mm from the lens. Calculate the image distance

v

. [1]

c

A sharp image requires

v

to fall within 2 mm of the calculated value. Assess whether the thin-lens model is reliable enough to guarantee this precision for a professional camera lens, justifying your answer with two distinct physical limitations of the model. [2]

Solutions

19QuestionApplications in Geometry and Physics FormulasAssessment Practice

6 marks~9 minCriterion D

A civil engineer is designing a crash barrier for a highway. The barrier must stop a test vehicle of mass 1500 kg traveling at 20 m/s within a stopping distance such that the average force on the vehicle does not exceed 50 kN. The work-energy principle states that the work done by the stopping force equals the change in kinetic energy: $F \cdot d = \frac{1}{2}mv^2$ , where $F$ is the average force, $d$ is the stopping distance, $m$ is the mass, and $v$ is the speed.

a

Rearrange the formula

F \cdot d = \frac{1}{2}mv^2

to express

d

in terms of

F

,

m

, and

v

.

b

Using the given values, calculate the minimum stopping distance required so that the average force does not exceed 50 kN.

c

Suppose the actual mass of the test vehicle could be up to 5% higher than stated (i.e., 1575 kg). Evaluate how this measurement error affects the calculated stopping distance. Specifically, calculate the new stopping distance if the mass is 1575 kg and comment on the impact on the safety margin.

d

The work-energy model assumes constant deceleration (i.e., constant force throughout the stopping process). Discuss one real-world limitation of this assumption and explain how it could affect the actual safety of the barrier. In your response, evaluate whether the model is still useful for engineering design despite this limitation.

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

2 marks~3 minCriterion A

A ladder leans against a vertical wall. The ladder is 8 m long and its base rests on horizontal ground 6 m from the wall. The angle

\theta

is formed between the ladder and the wall at the top.

Deduce the size of angle

\theta

, giving your answer in degrees to one decimal place. [2]

Solutions

22QuestionChoosing Appropriate MethodsAssessment Practice

4 marks~6 minCriterion D

A cyclist's journey is modelled by a straight-line graph of distance (km) against time (hours). The line passes through the points

(1, 15)

and

(3, 45)

.

a

Calculate the cyclist's speed. [1]

b

Deduce the equation of the line in the form

d = mt

, where

d

is distance in km and

t

is time in hours. [1]

c

The cyclist's destination is 70 km away. Using your equation, advise whether the cyclist should plan to complete the journey within a 5-hour window. Justify your answer. [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

24QuestionSolving One-Step and Two-Step EquationsAssessment Practice

4 marks~6 minCriterion C

A mobile data plan charges a fixed monthly fee of 4 dollars plus a cost per gigabyte used. In January, a customer used 6 GB and paid a total usage fee modelled by:

x + 4 = 10

In February, the customer doubled their data usage, giving a total usage fee modelled by:

2x + 4 = 16

where

x

is the cost per gigabyte in dollars.

a

Calculate the value of

x

for each equation. [2]

b

Explain why Equation B requires one more algebraic step than Equation A to isolate

x

. [1]

c

Justify whether the February plan offers the customer better value per gigabyte than the January plan. [1]

Solutions

25QuestionSolving One-Step and Two-Step EquationsAssessment Practice

6 marks~9 minCriterion D

A student tracks the battery percentage of a phone over time. The data collected is:

Time (h): 0, 1, 2, 3

Battery (%): 100, 82, 64, 46

The student models the battery drain with the equation $B = 100 - 18t$ , where $B$ is the battery percentage and $t$ is the time in hours.

a

[2 marks] Use the model to predict the battery percentage after 4 hours.

b

[2 marks] Use the model to find the time when the battery reaches 0%.

c

[2 marks] Discuss two limitations of using this linear model to represent the phone's battery drain over time.

Solutions

26QuestionPractice Quadratic Equations solvingAssessment Practice

5 marks~8 minCriterion B

A projectile is launched from a raised platform. Its height

h

(in metres) above the ground is recorded at selected times

t

(in seconds):

$t$ (s)	0	1	2	3	4
$h$ (m)	10	25	30	25	10

a

Deduce the quadratic function

h(t) = at^2 + bt + c

that models the height of the projectile. [2]

b

Determine the maximum height reached by the projectile. [1]

c

The projectile is considered safe to retrieve only if it lands more than 4.0 s after launch. Using your quadratic model, calculate the time at which the projectile hits the ground, then justify whether the retrieval condition is met. [2]

Solutions

27QuestionPractice Quadratic Equations solvingAssessment Practice

4 marks~6 minCriterion A

A ball is launched upward from a platform. Its height

h

(in metres) at time

t

(in seconds) is modelled by

h(t) = -5t^2 + 20t + 1

The graph of

h(t)

has vertex

(2,\ 21)

and positive

t

-intercept at approximately

(4.05,\ 0)

.

a

State the initial height of the ball and explain what the coefficient

-5

indicates about the shape of the graph. [1]

b

Use the quadratic formula to determine the time at which the ball hits the ground. [2]

c

A safety barrier is positioned 20 m above the launch platform. Justify whether the ball clears the barrier. [1]

Solutions

28QuestionPractice Quadratic Equations solvingAssessment Practice

6 marks~9 minCriterion C

A basketball player shoots a free throw. The height

h

(in metres) of the ball

t

seconds after release is modelled by

h(t) = -5t^2 + 6t + 2.2

The basket is at a height of

3.05

m above the floor.

a

Calculate the maximum height reached by the ball. [2]

b

Determine the two values of

t

at which the ball is at the height of the basket. [2]

c

The ball passes through the basket height twice. Justify which value of

t

represents the ball passing through the basket. [2]

Solutions

Equations Inequalities and Formulae — Free MYP4 Mathematics (Extended) Practice Questions