Vectors and Transformations — Free IB MYP5 Mathematics (Extended) Questions

1QuestionDescribing and Performing TranslationsConcept Practice

4 marks~6 minCriterion A

A city planner uses a coordinate grid to map a triangular green space with vertices at

A(1, 2)

,

B(3, 5)

, and

C(4, 1)

. The entire triangle is translated so that

A

maps to

A'(5, 3)

.

a

Calculate the translation vector that maps triangle

ABC

to triangle

A'B'C'

. Give your answer in the form

\begin{pmatrix} x \\ y \end{pmatrix}

. [1]

b

Determine the coordinates of

B'

and

C'

. [1]

c

A second translation,

\begin{pmatrix} -2 \\ 4 \end{pmatrix}

, is then applied to the whole triangle. The planner claims that vertex

A

now lies within the region

x > 2

and

y > 6

. Justify whether this claim is correct. [2]

Solutions

2QuestionProving Collinearity Using VectorsConcept Practice

5 marks~8 minCriterion A

The diagram shows three points

A

,

B

, and

C

plotted on a coordinate grid.

The coordinates of the points are

A(1, 2)

,

B(3, 5)

, and

C(7, 11)

.

a

Write down the gradient of the line segment

AB

. [1]

b

Find the equation of the straight line passing through

A

and

C

, giving your answer in the form

y = mx + c

. [2]

c

A fourth point

D

has coordinates

(d, 20)

and lies on the line through

A

and

C

. Find the value of

d

, then justify whether

D

lies between

C

and the origin or beyond

C

on the same line. [2]

Solutions

3QuestionDescribing Combined Transformations in Real-Life ContextsConcept Practice

2 marks~3 minCriterion A

A graphic designer is positioning a logo on a digital canvas. The logo is modelled by triangle

ABC

with vertices

A(1, 1)

,

B(3, 1)

, and

C(2, 4)

.

The designer applies two transformations in sequence:

1

Translate the triangle 4 units to the right.

2

Reflect the translated triangle in the

y

-axis.

The canvas has a vertical centre line along the

y

-axis.

Determine the final coordinates of vertex

C

after both transformations, and justify whether the designer's requirement that

C

lands in the negative

x

-region is met. [2]

Solutions

4QuestionAdding and Subtracting Column VectorsAssessment Practice

6 marks~9 minCriterion A

The diagram shows the graph of

f(x) = x^2 - 4x + c

for

0 \le x \le 5

, where

c

is a constant.

a

Given that the graph passes through the point

(5, 6)

, find the value of

c

. [1]

b

Hence write down the coordinates of the

y

-intercept and find the coordinates of the vertex of

f(x)

. [3]

c

State the values of

k

for which

f(x) = k

has exactly one solution in the domain

0 \le x \le 5

, justifying your answer. [2]

Solutions

5QuestionAdding and Subtracting Column VectorsAssessment Practice

6 marks~9 minCriterion B

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

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 to be found. [3]

c

A student claims that no figure in this pattern can contain exactly 100 unit squares. Justify whether this claim is correct, using your rule from part (b). [2]

Solutions

6QuestionAdding and Subtracting Column VectorsAssessment Practice

8 marks~12 minCriterion D

A set of 30 students each recorded the number of hours,

x

, they spent on social media in one week and their score,

y

, on a wellbeing survey (scored out of 100). The mean number of hours is

\bar{x} = 14

and the mean wellbeing score is

\bar{y} = 62

.

A student suggests modelling the relationship with the linear equation

y = -2.5x + c

where

c

is a constant.

a

Show that

c = 97

, given that the line of best fit passes through

(\bar{x},\, \bar{y})

. [2]

The table below gives data for three students and the scores predicted by the model

y = -2.5x + 97

.

Student A: hours

x = 8

, actual score

= 79

, predicted score

= 77

Student B: hours

x = 14

, actual score

= 62

, predicted score

= 62

Student C: hours

x = 20

, actual score

= 41

, predicted score

= 47

b

Calculate the percentage error for Student A and for Student C, using the formula

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

Give each answer correct to 2 decimal places. [3]

c

A fourth student, Student D, spent 28 hours on social media. Using the model, determine the predicted wellbeing score for Student D. Advise whether this prediction should be trusted, referring to both the percentage errors found in part (b) and the position of

x = 28

relative to the data. [3]

Solutions

7QuestionAdding and Subtracting Vectors GraphicallyAssessment Practice

6 marks~9 minCriterion C

The diagram shows a right triangle

ABC

drawn on a coordinate grid, where

A = (0, 0)

,

B = (6, 0)

, and

C = (6, 8)

.

a

Write down the length of

BC

. [1]

b

Find the length of

AC

, giving your answer in exact form. [2]

c

A second triangle

PQR

is similar to triangle

ABC

, with

AC

corresponding to

PQ

. Given that

PQ = 15

, the area of triangle

PQR

is calculated to be

54

square units. Justify whether this area is correct. [3]

Solutions

8QuestionDefinition and Use of Position VectorsAssessment Practice

6 marks~9 minCriterion B

The diagram shows Figures 1, 2 and 3 of a pattern made from dots. In each figure, the dots form an

n \times n

square with one additional dot placed immediately to the right of the top-right corner dot. Figure 4 has 17 dots.

a

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

b

Find a rule for the number of dots

D

in Figure

n

, in the form

D = an^2 + b,

where

a

and

b

are integers to be found. [2]

c

Verify your rule for Figure 3, and justify why the rule must contain an

n^2

term by referring to the structure of the pattern. [3]

Solutions

9QuestionDefinition and Use of Position VectorsAssessment Practice

6 marks~9 minCriterion C

The diagram shows a coordinate grid with three vertices of a parallelogram

ABCD

, where

A = (1, 2)

,

B = (4, 3)

, and

C = (6, 7)

.

a

Find the coordinates of the fourth vertex

D

, using the property that the diagonals of a parallelogram bisect each other. [2]

b

Show that the midpoints of diagonals

AC

and

BD

coincide, and state the coordinates of the common midpoint

M

. [2]

c

A student claims: "The common midpoint

M

is equidistant from all four vertices

A

,

B

,

C

, and

D

."

Justify whether this claim is correct by calculating the exact distances

MA

,

MC

,

MB

, and

MD

, and comparing them. [2]

Solutions

10QuestionDefinition and Use of Position VectorsAssessment Practice

8 marks~12 minCriterion D

A drone delivery company plans routes on a coordinate grid where each unit represents 1 km. The launch point is at the origin

O(0, 0)

. Waypoint

A

is at coordinates

(3, 4)

and waypoint

B

is at coordinates

(7, 1)

.

The diagram shows the positions of

O

,

A

and

B

and the two-leg route

O \to A \to B

.

The company models energy cost as proportional to total distance flown and charges customers 12 USD per km.

a

Calculate the total distance flown along the two-leg route

O \to A \to B

. Give your answer in km. [2]

b

Find the percentage saving in distance if the direct route

O \to B

is used instead of the two-leg route. Give your answer to 3 significant figures. [3]

c

The straight-line distance from

O

to

B

is

\sqrt{50}

km. A rival company claims that flying the direct route always gives the shortest possible path between two points and should therefore always be used as a flight-planning rule. Assess whether this claim should be adopted as a general flight-planning rule, supporting your answer with both a mathematical argument and one real-world consideration. [3]

Solutions

11QuestionDefinition and Use of Position VectorsAssessment Practice

6 marks~9 minCriterion A

Points

P(3,\, 4)

and

Q(-1,\, 7)

are shown in the diagram.

a

Write down the distance

OP

, where

O

is the origin. [1]

b

Find the coordinates of the midpoint

M

of

PQ

, and hence calculate the exact distance

OM

. [3]

c

Point

R

lies on segment

PQ

such that

PR : RQ = 1 : 2

. Determine the coordinates of

R

, then justify whether

O

,

R

, and

M

are collinear. [2]

Solutions

12QuestionEnlargement with Positive and Negative Scale FactorsAssessment Practice

4 marks~6 minCriterion D

An architect designs a building with a rectangular footprint. The corners on the blueprint are:

A(2,\ 1),\quad B(6,\ 1),\quad C(6,\ 4),\quad D(2,\ 4)

A scale model is produced by applying an enlargement with scale factor

k = 0.02

, centred at the origin. A mirror-image wing is then modelled by applying an enlargement with scale factor

k = -1

, centred at the origin, to the original coordinates.

a

Calculate the coordinates of

A'

,

B'

,

C'

,

D'

after the enlargement

k = 0.02

. [1]

b

Calculate the coordinates of

A''

,

B''

,

C''

,

D''

after the enlargement

k = -1

. [1]

c

Advise the architect whether the mirror-image wing produced by

k = -1

can be directly used as a construction plan, referring to at least two real-world constraints in your response. [2]

Solutions

13QuestionReflections in Lines x-axis y-axis y=x y=-xAssessment Practice

4 marks~6 minCriterion B

A graphic designer is creating a logo using reflected line segments on a coordinate grid. The segment

MN

has endpoints

M(-4,\, 2)

and

N(-1,\, 5)

. The designer reflects

MN

across the line

y = x

to produce image segment

M'N'

.

a

Calculate the gradient of segment

MN

. [1]

b

Deduce the coordinates of

M'

and

N'

, and hence calculate the gradient of segment

M'N'

. [1]

c

The designer claims that any line segment parallel to

y = x

is unaffected by reflection in

y = x

. Analyse the gradients of

MN

and

M'N'

to justify whether this claim is mathematically valid, with reference to the general effect of this reflection on gradients. [2]

Solutions

14QuestionReflections in Lines x-axis y-axis y=x y=-xAssessment Practice

4 marks~6 minCriterion C

A graphic designer is creating a logo by reflecting a triangular motif. Triangle

ABC

has vertices

A(1, 2)

,

B(3, 5)

, and

C(4, 1)

. The triangle is reflected in the

x

-axis to produce image triangle

A'B'C'

.

a

State the coordinates of

B'

. [1]

b

Determine the coordinates of

A'

and

C'

. [1]

c

The designer claims the reflected motif is identical to the original and can replace it in any orientation. Justify whether this claim is correct, using the coordinates of the triangle and its image. [2]

Solutions

15QuestionApplying Successive TransformationsAssessment Practice

5 marks~8 minCriterion A

The diagram shows a coordinate grid with point

P(2, 3)

.

Point

P

is first reflected in the

y

-axis to give image

P'

, then

P'

is translated by the vector

\begin{pmatrix}4 \\ -1\end{pmatrix}

to give image

P''

.

a

Write down the coordinates of

P'

. [1]

b

Find the coordinates of

P''

. [1]

c

A third transformation maps

P''

back to the original point

P(2, 3)

. The transformation is a reflection in the line

y = c

, where

c

is a constant.

Find the value of

c

and justify whether this mirror line lies closer to

P''

or to

P

. [3]

Solutions

16QuestionApplying Successive TransformationsAssessment Practice

6 marks~9 minCriterion C

The diagram shows the graph of

f(x) = x^2 - 4x + 3

for

0 \le x \le 5

.

a

Write down the coordinates of the

y

-intercept of

f(x)

. [1]

b

Find the coordinates of the vertex of

f(x)

by completing the square, giving your answer in exact form. [2]

c

Justify the set of values of

k

for which

f(x) = k

has exactly two distinct solutions in the domain

0 \le x \le 5

. [3]

Solutions

17QuestionApplying Successive TransformationsAssessment Practice

7 marks~11 minCriterion B

The diagram shows the first four points of a pattern. Starting from

P_0(1, 2)

, each new point is obtained by applying the same translation to the previous point. The translation moves each point 3 units in the positive

x

-direction and 2 units in the positive

y

-direction.

a

Write down the coordinates of

P_1

,

P_2

,

P_3

, and

P_4

. [2]

b

Find a general expression for the coordinates of

P_n

in terms of

n

, where

n

is a non-negative integer. [2]

c

Justify why both coordinate expressions are linear in

n

, and verify your expression by substituting

n = 3

. [3]

Solutions

18QuestionApplying Successive TransformationsAssessment Practice

6 marks~9 minCriterion D

A delivery drone operates inside a large warehouse. Its position is tracked on a coordinate grid where each unit represents one metre. The drone starts at point

A(1, 2)

.

The warehouse manager uses the following model to predict the drone's position after

n

complete delivery runs. After each run, the drone moves 3 metres east (positive

x

-direction) and 2 metres north (positive

y

-direction) from its previous position.

a

Write down the coordinates of the drone after run 1 and after run 2. [1]

b

The drone must stay within the rectangular boundary

0 \le x \le 16

and

0 \le y \le 12

. Find the maximum number of complete delivery runs the drone can make before it first exits this boundary. Show your reasoning clearly. [3]

c

The manager claims the model predicts the drone will always remain inside the boundary provided the number of runs does not exceed the value found in part (b). Identify one assumption of this translation model and advise the manager whether this claim can be relied upon in a real warehouse, justifying your answer. [2]

Solutions

19QuestionColumn Vector NotationAssessment Practice

6 marks~9 minCriterion B

The diagram shows a pattern of figures made from small equilateral triangles. Figure 1 has 1 shaded triangle, Figure 2 has 4 shaded triangles, and Figure 3 has 9 shaded triangles.

a

Write down the number of shaded triangles in Figure 4. [1]

b

Find a rule for the number of shaded triangles

T

in Figure

n

. [2]

c

A student claims that Figure

k

is the first figure with more than 200 shaded triangles. Justify whether the student's claim that

k = 15

is correct. [3]

Solutions

20QuestionColumn Vector NotationAssessment Practice

5 marks~8 minCriterion C

The diagram shows a coordinate grid with triangle

ABC

, where

A = (1, 4)

,

B = (3, 1)

, and

C = (5, 3)

.

a

Triangle

ABC

is translated by the vector

\begin{pmatrix} -4 \\ 2 \end{pmatrix}

. Write down the coordinates of

A'

,

B'

, and

C'

. [1]

b

Triangle

ABC

is then enlarged from the origin

O = (0, 0)

with scale factor

-2

. Find the coordinates of

A''

,

B''

, and

C''

. [2]

c

A student claims that the enlargement in part (b) is equivalent to a rotation of

180°

about the origin followed by an enlargement of scale factor

2

from the origin. Justify whether the student's claim is correct by comparing the image coordinates from both transformation sequences. [2]

Solutions

21QuestionColumn Vector NotationAssessment Practice

8 marks~12 minCriterion D

A school tuck shop models its weekly profit

P

(in dollars) from selling

x

snack packs using the quadratic function

P(x) = -2x^2 + 60x - 200.

The shop owner records the actual weekly profit at three different sales levels.

Sales level (snack packs)	10	20	25
Actual profit (dollars)	190	570	410

a

Calculate the predicted profit for each sales level using

P(x)

, and complete the table below by finding the percentage error for each sales level. Give each percentage error correct to 1 decimal place.

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

Sales level (snack packs)	10	20	25
Predicted profit (dollars)	?	?	?
Percentage error (%)	?	?	? [3]

b

Show that

P(x)

can be written in the form

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

, and state the value of

x

that maximises

P(x)

and the maximum predicted weekly profit. [3]

c

The owner claims the model is reliable for sales levels between 10 and 25 snack packs. Using your results from part (a), advise the owner whether this claim should be trusted, and suggest one limitation of using a quadratic model to predict the shop's profit. [2]

Solutions

22QuestionColumn Vector NotationAssessment Practice

6 marks~9 minCriterion A

Point

P

has coordinates

(3, -2)

and point

Q

has coordinates

(-1, 4)

.

a

Write down the midpoint

M

of segment

PQ

. [1]

b

Find the equation of the perpendicular bisector of

PQ

, giving your answer in the form

y = mx + c

. [3]

c

A circle passes through both

P

and

Q

. Its centre lies on the perpendicular bisector of

PQ

and also on the line

y = x - 1

. Justify whether the point

(4, 3)

is the centre of this circle. [2]

Solutions

23QuestionApplications in Navigation and DesignAssessment Practice

5 marks~8 minCriterion D

The diagram shows a right-angled triangle

OAB

where

O

is the origin. A ship leaves

O

, sails 12 km due east to point

A

, then 5 km due north to point

B

.

a

Write down the distance

OB

, the straight-line distance from

O

to

B

. [1]

b

Find the bearing of

B

from

O

, giving your answer to the nearest degree. [2]

c

A second ship travels directly from

O

to

B

. It arrives at the same time as the first ship, which sailed via

A

at a constant speed of 10 km/h. The port authority requires all vessels to travel at no less than 7.5 km/h in this zone. Justify whether the second ship complies with this regulation. [2]

Solutions

24QuestionApplications in Navigation and DesignAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first four figures of a pattern made from dots arranged in rows.

Figure 1 has 2 dots, Figure 2 has 6 dots, Figure 3 has 12 dots, and Figure 4 has 20 dots.

a

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

b

Find a rule for the number of dots

D

in Figure

n

, giving your answer in the form

D = n(n + 1)

. Show that your rule is consistent with Figure 3. [3]

c

A student claims that no figure in this pattern can contain exactly 50 dots. Justify whether the student is correct. [2]

Solutions

25QuestionApplications in Navigation and DesignAssessment Practice

5 marks~8 minCriterion C

A surveyor walks 7 km due east from point

A

to point

B

, then 24 km due north from

B

to point

C

.

a

State the straight-line distance from

A

to

C

. [1]

b

Find the bearing of

C

from

A

, giving your answer to the nearest degree. [2]

c

A second surveyor claims that point

D

, located on the direct path from

A

to

C

at a distance of 13 km from

A

, is exactly 5 km east of

A

. Justify whether this claim should be accepted or rejected. [2]

Solutions

26QuestionDescribing Combined Transformations in Real-Life ContextsAssessment Practice

6 marks~9 minCriterion B

A graphic designer uses a repeating tile pattern. A triangle with vertices

A(0,0)

,

B(2,0)

,

C(1,3)

is transformed repeatedly to produce a sequence of images.

Image 1	$A_1(0,0)$	$B_1(2,0)$	$C_1(1,3)$
Image 2	$A_2(2,2)$	$B_2(4,2)$	$C_2(3,5)$
Image 3	$A_3(4,0)$	$B_3(6,0)$	$C_3(5,3)$
Image 4	$A_4(6,2)$	$B_4(8,2)$	$C_4(7,5)$

a

Construct Image 5 and Image 6 on the coordinate grid provided. Describe the pattern you observe in the sequence. [2]

b

Deduce the general rule for the transformation that maps each image to the next, expressing it as a combination of translations. [2]

c

The designer considers extending the tile pattern using a new triangle with vertices

D(0,0)

,

E(3,0)

,

F(1,2)

. Apply your rule to find the vertices after one transformation, then justify whether this new triangle would produce a valid repeating tile pattern consistent with the original sequence. [2]

Solutions

27QuestionDescribing Combined Transformations in Real-Life ContextsAssessment Practice

2 marks~3 minCriterion C

A graphic designer maps a logo element using triangle

ABC

with vertices

A(1, 2)

,

B(4, 2)

, and

C(4, 6)

. The triangle is reflected in the

y

-axis and then translated by the vector

\begin{pmatrix} 3 \\ -5 \end{pmatrix}

to produce triangle

A'B'C'

.

Identify one geometric property that is preserved under this combined transformation, and justify why both rigid transformations guarantee this property is unchanged. [2]

Solutions

28QuestionDescribing Combined Transformations in Real-Life ContextsAssessment Practice

2 marks~3 minCriterion D

A graphic designer applies a translation of

5 \text{ cm}

to the right to a triangular logo element, then reflects it across a vertical line of symmetry to produce a mirrored pattern.

a

Describe one real-world context, other than logo design, in which a translation followed by a reflection is used to create a symmetrical pattern. [1]

b

The designer measures the translation distance imprecisely. Explain how this error affects the symmetry of the final pattern. [1]

Solutions

Vectors and Transformations — Free MYP5 Mathematics (Extended) Practice Questions