Vectors and Transformations — Free IB MYP4 Mathematics (Extended) Questions

1QuestionVector Arithmetic Word ProblemsConcept Practice

5 marks~8 minCriterion A

A delivery drone travels between three checkpoints in a city grid. Starting from point

A

, it flies 5 km due east to point

B

, then 3 km due north to point

C

, then 4 km due west to point

D

.

The diagram shows the drone's path on a coordinate grid, with

A

at the origin.

a

Write down the coordinates of

D

relative to

A

. [1]

b

Find the straight-line distance

DA

, giving your answer in exact form. [2]

c

The drone's battery allows a maximum straight-line return distance of 4 km. Advise the drone operator whether the drone can safely return directly from

D

to

A

, justifying your advice using your result from part (b). [2]

Solutions

2QuestionDescribing and Performing TranslationsConcept Practice

2 marks~3 minCriterion A

A city planner is mapping a park renovation. On the coordinate grid, a triangular flower bed

ABC

has been repositioned to a new location

A'B'C'

using a single translation.

Point

A

has coordinates

(1, 2)

and its image

A'

has coordinates

(5, -1)

.

a

Determine the translation vector that maps triangle

ABC

to triangle

A'B'C'

. Give your answer in the form

\dbinom{x}{y}

. [1]

b

The planner states: "The flower bed has moved further east than it has moved south." Justify whether the planner's statement is correct. [1]

Solutions

3QuestionIdentifying Single Equivalent TransformationConcept Practice

4 marks~6 minCriterion A

The diagram shows a coordinate grid with point

P(2, 3)

, its image

P'

after reflection in the

x

-axis, and its image

P''

after a further reflection in the

y

-axis.

a

Write down the coordinates of

P'

, the image of

P(2, 3)

after reflection in the

x

-axis. [1]

b

Write down the coordinates of

P''

, the image of

P'

after reflection in the

y

-axis. [1]

c

Justify, using correct transformation notation and by referring to the coordinates of

P

and

P''

, that the combined effect of the two reflections is equivalent to a single rotation of

180°

about the origin. [2]

Solutions

4QuestionSolving Complex Vector Path ProblemsConcept Practice

8 marks~12 minCriterion A

The diagram shows quadrilateral

ABCD

on a coordinate grid with vertices

A(1, 2)

,

B(5, 4)

,

C(7, 1)

, and

D(3, -1)

.

a

Calculate the gradient of side

AB

and the gradient of side

DC

. Hence justify whether

AB

is parallel to

DC

. [3]

b

Find the exact length of diagonal

AC

, giving your answer in the form

\sqrt{k}

where

k

is an integer. [2]

c

A student claims that

ABCD

is a parallelogram. Verify this claim by finding the midpoints of both diagonals

AC

and

BD

, and justify your conclusion. [3]

Solutions

5QuestionDescribing Combined Transformations in Real-Life ContextsConcept Practice

3 marks~5 minCriterion A

A logo designer places a triangle on a coordinate grid with vertices at

A(1, 3)

,

B(4, 1)

, and

C(2, -2)

. The designer first translates the triangle by the vector

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

, then reflects the result over the

y

-axis.

a

Calculate the coordinates of

A'

,

B'

, and

C'

after the translation. [1]

b

Calculate the final coordinates of

A''

,

B''

, and

C''

after the reflection. [1]

c

The designer requires the final logo to sit entirely in the third quadrant. Justify whether this condition is met. [1]

Solutions

6QuestionVector Arithmetic Word ProblemsAssessment Practice

7 marks~11 minCriterion B

The diagram shows a growing pattern of dot-and-line figures. In each figure, dots are arranged in rows and connected by horizontal line segments.

Figure 1	1 row	2 dots	1 segment.
Figure 2	2 rows	6 dots	4 segments.
Figure 3	3 rows	12 dots	9 segments.

Figure number $n$	1	2	3
Number of dots $D$	2	6	12
Number of segments $S$	1	4	9

a

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

b

Find a formula for

S

in terms of

n

. Hence find the figure number that contains exactly 225 segments. [3]

c

A student claims that

D = n(n+1)

.

Verify this formula for

n = 3

, and justify why the factor

(n+1)

appears by referring to the structure of the figures. [3]

Solutions

7QuestionScalar Multiplication of VectorsAssessment Practice

6 marks~9 minCriterion C

A delivery drone is launched from a depot at the origin

O

. Its position is tracked on a coordinate grid where each unit represents 1 km. At time

t = 0

minutes, the drone is at point

A(4, 2)

. The drone flies in a straight line so that at

t = 6

minutes it reaches point

B

, which is exactly 3 times as far from the depot as

A

is, in the same direction.

The diagram shows the depot at

O

, point

A(4, 2)

, and point

B

.

a

Write down the coordinates of

B

. [1]

b

Find the straight-line distance

OB

. Give your answer in km, correct to 3 significant figures. [2]

c

The drone travels from

O

to

B

at constant speed. A logistics planner claims: "Since $B$ is 3 times as far as $A$ from the depot, the drone must have taken 3 times as long to reach $A$ as it did to reach $B$ ."

Justify whether the planner's claim is correct, showing any necessary calculations. [3]

Solutions

8QuestionVector Arithmetic Word ProblemsAssessment Practice

7 marks~11 minCriterion D

A harbour master records the journey of a sailboat that travels 12 km due east, then 8 km due north, then 6 km due west. A rescue boat travels in a straight line from the sailboat's starting point to its finishing point.

The diagram shows the sailboat's path and the direct route of the rescue boat.

a

Calculate the straight-line distance the rescue boat travels. Give your answer in kilometres. [2]

b

The sailboat consumes fuel at a rate of 0.5 litres per km. The rescue boat consumes fuel at a rate of 0.4 litres per km. Find the difference in total fuel used between the two boats for this journey. Give your answer in litres. [3]

c

The harbour master claims: "The rescue boat will always use less fuel than the sailboat on any journey of this type, regardless of the consumption rates."

Advise the harbour master whether this claim is reliable, by finding the condition on the rescue boat's fuel consumption rate

r

(litres per km) for which the rescue boat uses less fuel than the sailboat on this specific journey. [2]

Solutions

9QuestionFinding Displacement Between Two PointsAssessment Practice

8 marks~12 minCriterion C

The diagram shows three line segments plotted on a coordinate grid.

Segment

AB

:

A(1,\,2)

to

B(4,\,6)

Segment

CD

:

C(0,\,0)

to

D(3,\,4)

Segment

GH

:

G(2,\,5)

to

H(5,\,9)

a

Show that segments

AB

,

CD

, and

GH

all have the same gradient and the same length. Give lengths in exact form. [3]

b

A fourth segment connects

P(k,\,3)

to

Q(k+3,\,7)

. Find the value of

k

for which

Q

lies on the line through

C

and

D

, and hence deduce whether segment

PQ

is parallel to, or collinear with, segment

CD

. Justify your answer. [3]

c

A student claims: "Any segment with gradient

\dfrac{4}{3}

and one endpoint on the line

y = \dfrac{4}{3}x

must be collinear with

CD

."

Justify whether this claim is correct or incorrect, referring to the structure of a straight line. [2]

Solutions

10QuestionPosition Vectors in Geometric DiagramsAssessment Practice

8 marks~12 minCriterion D

A surveyor is mapping a triangular plot of land with vertices at

A(2, 1)

,

B(8, 3)

, and

C(4, 7)

, where coordinates are given in metres. A fence post is to be placed at point

M

, the midpoint of side

AC

.

The diagram shows the triangular plot

ABC

with

M

marked on

AC

.

a

Find the coordinates of

M

and hence calculate the length

BM

, giving your answer in metres to 3 significant figures. [3]

b

The GPS device used to record each vertex has a possible error of

\pm 2\%

in each coordinate. Complete the table below to show the maximum and minimum possible

x

-coordinates of

A

and

C

, and hence determine the maximum possible error in the

x

-coordinate of

M

. Give your answer in centimetres.

Vertex — Nominal

x

(m) — Max

x

(m) — Min

x

(m)

A

—

2

— —

C

—

4

— — [3]

c

The surveyor claims the error in locating

M

on the ground is negligible compared to the length

BM

. Advise the surveyor whether this claim is acceptable for legal boundary placement, justifying your advice with the values calculated in parts (a) and (b). [2]

Solutions

11QuestionDefinition and Use of Position VectorsAssessment Practice

7 marks~11 minCriterion A

Point

A

has coordinates

(3, -2)

and point

B

has coordinates

(-1, 4)

.

a

Calculate the exact length of

AB

, giving your answer as a simplified surd. [2]

b

Find the coordinates of the point

M

that divides

AB

in the ratio

1:3

from

A

to

B

. [2]

c

A circle has centre

M

and passes through

A

. Justify whether point

B

lies inside, on, or outside this circle, giving all distances in exact form. [3]

Solutions

12QuestionFinding Displacement Between Two PointsAssessment Practice

5 marks~8 minCriterion B

The diagram shows the first four figures of a pattern made from dots arranged in an L-shape.

Figure 1 has 2 dots, Figure 2 has 5 dots, Figure 3 has 10 dots, and 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 = n^2 + c

, where

c

is a constant to be determined. [2]

c

Verify your rule for Figure 3. Justify why the rule must contain an

n^2

term by referring to the structure of the L-shape. [2]

Solutions

13QuestionEnlargement with Positive and Negative Scale FactorsAssessment Practice

6 marks~9 minCriterion D

An architect is designing a mirrored facade for a building. A decorative triangular panel has vertices at

A(2,\ 1)

,

B(5,\ 1)

, and

C(3,\ 4)

on a blueprint (units in metres). The triangle is enlarged by a scale factor of

-0.5

with centre of enlargement at the origin.

The construction team marks points with a tolerance of

\pm 0.05\ \text{m}

in each coordinate, independently.

a

Determine the coordinates of the image vertices

A'

,

B'

, and

C'

. [2]

b

Calculate the maximum possible area of the constructed triangle, and hence find the maximum area error compared to the theoretical image. [3]

c

Advise the architect whether this enlargement transformation is appropriate for large-scale construction, addressing: the effect of tolerances on accuracy; one practical limitation of using a negative scale factor on site; and one broader limitation of applying geometric transformations directly to construction. [1]

Solutions

14QuestionRotations About a Point and AngleAssessment Practice

8 marks~12 minCriterion B

A video-game developer is programming a camera rotation feature. A triangular obstacle has vertices at

A(2,3)

,

B(5,1)

, and

C(4,6)

. After a

90°

clockwise rotation about the origin, the image vertices are

A'(3,-2)

,

B'(1,-5)

, and

C'(6,-4)

.

a

Describe the relationship between the coordinates of each original vertex and its image. [2]

b

Deduce a general rule for the image of any point

(x, y)

under a

90°

clockwise rotation about the origin. [2]

c

A sensor is located at

D(-3, 4)

. Apply your rule to find the image

D'

. [2]

d

The developer claims that rotating the sensor's position does not change its distance from the camera (origin). Justify whether this claim is consistent with your result for

D

and

D'

. [2]

Solutions

15QuestionReflections in Lines x-axis y-axis y=x y=-xAssessment Practice

5 marks~8 minCriterion C

A graphic designer uses coordinate geometry to plan a logo. Triangle

PQR

is reflected in the line

y = x

to produce triangle

P'Q'R'

, with

P'(3, 1)

,

Q'(5, 2)

, and

R'(2, 4)

.

a

Find the coordinates of

P

and

R

. [1]

b

Reflect segment

PR

in the line

y = -x

. Write the coordinates of the image endpoints

P''

and

R''

, and find the equation of segment

P''R''

. [3]

c

The designer claims that point

S(1, 3)

lies on segment

P''R''

and therefore falls within the logo boundary. Justify whether this claim should be accepted before the designer finalises the logo. [1]

Solutions

16QuestionUsing Matrices for Simple Transformations IntroductoryAssessment Practice

9 marks~14 minCriterion D

A game developer is designing a 2D platformer. The hero sprite starts as a unit square with vertices at

A(0,0)

,

B(1,0)

,

C(1,1)

and

D(0,1)

. To fit a widescreen display, the developer applies a stretch mapping each point

(x, y)

to

(3x, 2y)

.

The diagram shows the original sprite

ABCD

and the image

A'B'C'D'

after the stretch.

a

Write down the coordinates of

A'

,

B'

,

C'

and

D'

. [1]

b

The developer changes the stretch so that the horizontal factor is

k

and the vertical factor is

(k-1)

, where

k > 1

. The stretched sprite must fit exactly inside a screen region of area 30 square units. Find the value of

k

, giving your answer to 3 significant figures. [3]

c

The developer tests three stretch settings and records the predicted sprite area alongside the actual rendered area:

k

: 3, 4, 6

Predicted area (

k(k-1)

): 6, 12, 30

Actual area: 5, 11, 28

The percentage error is defined as

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

Calculate the percentage error at each value of

k

, giving your answers to 1 decimal place. [3]

d

Using your results from part (c), justify whether the developer should use the model

\text{Area} = k(k-1)

to predict rendered sprite areas in production. [2]

Solutions

17QuestionApplying Successive TransformationsAssessment Practice

11 marks~17 minCriterion B

The diagram shows the first three figures of a pattern made from unit squares arranged in an L-shape.

Figure 1 has 3 unit squares, Figure 2 has 8 unit squares, and Figure 3 has 15 unit squares.

a

Show that the number of unit squares in Figure 5 is 35 and find the number of unit squares in Figure 6. [3]

b

A student claims the number of unit squares

S

in Figure

n

is given by

S = n^2 + 2n.

Show that this formula is consistent with Figures 1, 2, and 3, and find the figure number that first contains more than 120 unit squares. [4]

c

Justify why the formula

S = n^2 + 2n

must contain an

n^2

term by referring to the structure of the figures, and prove algebraically that the difference

S_{n+1} - S_n

is always an odd number. [4]

Solutions

18QuestionApplying Successive TransformationsAssessment Practice

5 marks~8 minCriterion C

The diagram shows a coordinate grid with point

P(2, 3)

.

Point

P

is first rotated

90^\circ

anticlockwise about the origin to give image

P'

, then

P'

is translated by

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

to give final image

P''

.

a

Write down the coordinates of

P'

. [1]

b

Find the coordinates of

P''

. Show your working clearly. [2]

c

A classmate claims that applying the translation first and then the rotation gives the same final image

P''

. Justify whether this claim is correct, showing all working. [2]

Solutions

19QuestionUnit Vectors and Position Vectors IntroductoryAssessment Practice

6 marks~9 minCriterion D

A right-angled triangle has legs of length

a

cm and

b

cm, and hypotenuse of length

c

cm, where

a

,

b

,

c

are positive integers satisfying

a^2 + b^2 = c^2

.

The triangle below has legs

a = 5

cm and

b = 12

cm.

a

Write down the value of

c

. [1]

b

A second right-angled triangle is similar to the first, with hypotenuse

c' = 39

cm. Find the lengths of its two legs. [2]

c

Show that the perimeter of the second triangle is

3

times the perimeter of the first triangle, and hence justify whether the second triangle could be used to represent the first triangle at a scale of

1:3

on an architectural plan. [3]

Solutions

20QuestionNotation and Labeling for Displacement VectorsAssessment Practice

8 marks~12 minCriterion B

The diagram shows four figures in a growing pattern made from small equilateral triangles.

Figure 1 has 1 shaded triangle. Figure 2 has 4. Figure 3 has 9. Figure 4 has 16. In each figure, some triangles point upward (△) and some point downward (▽).

Figure $n$	1	2	3	4
$U$	1	3	6	10
$D$	0	1	3	6
$T$	1	4	9	16

a

Write down the number of upward-pointing triangles

U

and downward-pointing triangles

D

in Figure 5. [2]

b

Find a rule for the total number of triangles

T

in Figure

n

. Hence find the figure number in which the total number of triangles first exceeds 200. [3]

c

The number of downward-pointing triangles in Figure

n

is given by

D(n) = \frac{(n-1)n}{2}.

Verify this formula for

n = 3

, and justify why

D(n)

must always be a whole number for any positive integer

n

, by referring to the structure of the expression. [3]

Solutions

21QuestionUnit Vectors and Position Vectors IntroductoryAssessment Practice

8 marks~12 minCriterion A

The diagram shows the graph of

f(x) = \sqrt{x^2 + k}

for a positive constant

k

, passing through the point

A(4, 5)

.

a

Show that

k = 9

. [2]

b

Find the coordinates of the two points where

f(x) = \sqrt{34}

. [3]

c

A second function is defined by

g(x) = f(x - 3) + 1

. Determine the coordinates of the image of point

A

under this transformation, and hence state the equation of

g(x)

in the form

g(x) = \sqrt{x^2 + bx + c} + 1

, identifying the values of

b

and

c

. [3]

Solutions

22QuestionNotation and Labeling for Displacement VectorsAssessment Practice

8 marks~12 minCriterion C

The diagram shows triangle

PQR

on a coordinate grid with vertices

P(1, 1)

,

Q(7, 1)

, and

R(7, 9)

.

a

Calculate the length of

PR

, giving your answer in exact form. [2]

b

Find the bearing of

R

from

P

, giving your answer to the nearest degree. [3]

c

A point

S

lies on

PR

such that

QS

is perpendicular to

PR

. The length

QS

represents the shortest distance from vertex

Q

to the opposite side of the triangle. Justify whether

QS

is sufficient to confirm that

Q

lies closer to

PR

than to either

PQ

or

QR

. [3]

Solutions

23QuestionMidpoints and Parallel VectorsAssessment Practice

10 marks~15 minCriterion B

The diagram shows four line segments on a coordinate grid.

Segment 1:

A_1(1,\ 2)

and

B_1(5,\ 8)

Segment 2:

A_2(3,\ -1)

and

B_2(7,\ 5)

Segment 3:

A_3(-2,\ 4)

and

B_3(4,\ -2)

Segment 4:

A_4(-5,\ -3)

and

B_4(1,\ 1)

a

Show your working to find the midpoint

M_1

of Segment 1. Write down the midpoints

M_2

,

M_3

, and

M_4

. [3]

b

Describe the pattern connecting the coordinates of

A

,

B

, and

M

for each segment, and hence state a general rule giving the coordinates of

M

in terms of

A(x_A,\ y_A)

and

B(x_B,\ y_B)

. [3]

c

A fifth segment has midpoint

M_5(2,\ -1)

and one endpoint

A_5(-3,\ 4)

. Use your rule from part (b) to find the coordinates of

B_5

. Then justify whether

M_5

is a valid midpoint of

A_5B_5

by comparing the distances

A_5M_5

and

M_5B_5

. [4]

Solutions

24QuestionSolving Complex Vector Path ProblemsAssessment Practice

11 marks~17 minCriterion C

The diagram shows three figures in a growing pattern made from small equilateral triangles.

Figure 1 contains 1 small triangle. Figure 2 contains 4 small triangles. Figure 3 contains 9 small triangles.

a

Find the number of small triangles in Figure 5 and Figure 6, and write down the first six terms of the sequence. [2]

b

A student claims the number of small triangles

T

in Figure

n

is given by

T = n^2.

Show that this rule is consistent with Figures 1, 2, and 3, and find the value of

n

for which

T = 225

. [3]

c

The total number of small triangles in Figures 1 through

n

is denoted

S_n

. The values of

S_n

for the first four figures are given below.

$n$	1	2	3	4
$S_n$	1	5	14	30

A second student proposes that

S_n

can be written as

S_n = \frac{n(n+1)(2n+1)}{6}.

Verify this formula for

n = 3

and

n = 4

. [2]

d

Using the formula

S_n = \dfrac{n(n+1)(2n+1)}{6}

, find the smallest value of

n

such that

S_n > 500

. Justify why no smaller value of

n

satisfies this inequality. [4]

Give all answers in exact form.

Solutions

25QuestionProving Collinearity Using VectorsAssessment Practice

9 marks~14 minCriterion D

A city planner models the monthly profit

P

(in thousands of dollars) from a new market district using the quadratic function

P(x) = -2x^2 + 20x - 32,

where

x

is the number of stalls (in tens) operating each month, and

1 \leq x \leq 8

.

The table below shows the model's predicted profit and the actual recorded profit for three values of

x

.

$x$ (tens of stalls)	2	5	7
Predicted $P$ (thousands of dollars)	0	18	6
Actual $P$ (thousands of dollars)	2	15	9

a

Show that the predicted profit when

x = 5

is 18 thousand dollars, and find the value of

x

for which the model predicts maximum profit, stating the maximum profit in thousands of dollars. [3]

b

The city planner claims the market is profitable (i.e.

P(x) > 0

) for more than half of the valid domain

1 \leq x \leq 8

. By solving the inequality

P(x) > 0

, determine whether this claim is correct. [3]

The percentage error formula is

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

c

Calculate the percentage error at each value of

x

in the table. Advise the city planner whether the quadratic model should be used to forecast profit across the domain

1 \leq x \leq 8

, and state one reason why the model should not be used for values of

x

outside this domain. [3]

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

27QuestionIdentifying and Describing Single TransformationsAssessment Practice

4 marks~6 minCriterion C

An architect designs a symmetrical building wing. The original section is a right-angled triangle with vertices at

A(2, 1)

,

B(5, 1)

, and

C(5, 4)

on a coordinate grid (units in metres). The wing is reflected across the vertical line

x = 6

to create a mirror-image section.

a

Describe the single transformation that maps triangle

ABC

to its image

A'B'C'

, stating the coordinates of

A'

,

B'

, and

C'

. [2]

b

A construction error causes the intended reflection axis

x = 6

to be rotated

3°

clockwise about the point

(6, 0)

. Explain how this error affects the position of the reflected section, and why the displacement is not the same for all three vertices. [1]

c

The architect claims the reflected design will remain a perfect mirror image of the original wing throughout the building's lifetime. Assess the validity of this claim, considering how real structures behave over time. [1]

Solutions

28QuestionPerforming Two Successive TransformationsAssessment Practice

2 marks~3 minCriterion D

A graphic designer places a triangular logo on a coordinate grid. The logo is first translated 3 units right and 2 units upward to centre it on the billboard layout, then reflected across the

y

-axis to produce a mirrored version for the opposite side of the billboard.

Critique the use of this coordinate grid method for positioning the logo on an actual billboard.

[2]

Solutions

Vectors and Transformations — Free MYP4 Mathematics (Extended) Practice Questions