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GCSEMathsEdexcelHigherExam Questions4. Geometry & MeasuresVectors

Vectors (Edexcel GCSE Maths: Higher): Exam Questions

Exam code: 1MA1

3 hours48 questions
Medium
Hard
Very Hard
1a1 mark

The vector bold a bold spaceand the vector bold b are shown on the grid.

1oyXjMTR_q1a-medium-4-13-vectors-edexcel-gcse-maths

On the grid, draw and label vector negative 2 bold a

How did you do?

1b2 marks

Work out bold a space plus space 2 bold b as a column vector.

How did you do?

2a1 mark
q2a-medium-4-13-vectors-edexcel-gcse-maths

O A B spaceis a triangle.

stack O A with rightwards arrow on top space equals space bold a stack O B with rightwards arrow on top equals bold b

Find  stack space A B space with rightwards arrow on top in terms of bold a bold spaceand bold b.

How did you do?

2b3 marks

P is the point on A B such that A P space colon space P B space equals space 3 space colon space 1

Find space stack O P with rightwards arrow on top spacein terms of bold a and bold space bold b.
Give your answer in its simplest form.

How did you do?

3a1 mark
q3a-medium-4-13-vectors-edexcel-gcse-maths

P Q R S spaceis a parallelogram.
N is the point on space S Q spacesuch that space S N space colon space N Q space equals 3 colon 2

stack P Q with rightwards arrow on top space equals bold a stack P S with rightwards arrow on top space equals space bold b

Write down, in terms of bold a bold spaceand bold b, an expression for stack S Q with rightwards arrow on top.

How did you do?

3b
Sme Calculator3 marks

Express stack N R with rightwards arrow on top in terms of bold a andbold space bold b.

How did you do?

43 marks
q4-medium-4-13-vectors-edexcel-gcse-maths

O A B space is a triangle.
M is the midpoint of O A.
N is the midpoint of O B.

stack O M with rightwards arrow on top space equals space bold m space stack O N with rightwards arrow on top space equals space bold n

Show that space A B spaceis parallel to M N.

How did you do?

5a1 mark
q5a-medium-4-13-vectors-edexcel-gcse-maths

O A Y B is a quadrilateral.

stack O A with rightwards arrow on top space equals 3 bold a stack O B space with rightwards arrow on top equals 6 bold b

Express stack A B with rightwards arrow on top in terms of bold a bold spaceand bold b.

How did you do?

5b4 marks

X is the point onspace A B spacesuch that space A X colon space X B space equals space 1 space colon 2 spaceand stack B Y with rightwards arrow on top space equals space 5 bold a minus bold b

Prove that  stack O X with rightwards arrow on top space equals fraction numerator space 2 over denominator 5 end fraction space stack O Y with rightwards arrow on top

How did you do?

62 marks

Here are two vectors.

stack A B with rightwards arrow on top equals open parentheses table row 5 row 3 end table close parentheses               stack C B with rightwards arrow on top equals open parentheses table row cell negative 2 end cell row 4 end table close parentheses

Find, as a column vector, stack A C with rightwards arrow on top

How did you do?

71 mark

A B C D E F and G H I J K L are regular hexagons each with centre O.

q14-4ma1-1h-qp-nov21-paper1-igcse-maths

G H I J K L is an enlargement of A B C D E F, with centre O and scale factor 2

stack O A with rightwards arrow on top space equals space bold a bold space space space space space space space stack O B with rightwards arrow on top space equals space bold b

Write the following vectors, in terms of bold space bold a bold spaceand bold b.
Simplify your answers.

i) stack A B with rightwards arrow on top

[1]

ii) stack K I with rightwards arrow on top

[2]

iii) stack L D with rightwards arrow on top

[2]

How did you do?

84 marks

A B C D E F is a regular hexagon.

q22-paper1hr-jan2019-edexcel-igcse-maths

A B X and D C X are straight lines.

stack A B with rightwards arrow on top equals space bold a bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space space stack B C with rightwards arrow on top bold space equals space bold b

Find stack E X with rightwards arrow on top in terms of bold a and bold b.
Give your answer in its simplest form.

How did you do?

91 mark

Here is vector bold a.

q2-paper-1h-nov-2020-aqa-gcse-maths

Choose the column vector that represents bold a.

  • open parentheses table row 3 row 2 end table close parentheses

  • open parentheses table row cell negative 3 end cell row 2 end table close parentheses

  • open parentheses table row 3 row cell negative 2 end cell end table close parentheses

  • open parentheses table row cell negative 3 end cell row cell negative 2 end cell end table close parentheses

Choose your answer

10
Sme Calculator1 mark

bold a equals open parentheses table row cell negative 3 end cell row 2 end table close parentheses and bold b equals open parentheses table row 1 row cell negative 5 end cell end table close parentheses

Work out bold a space – space 3 bold b

  • open parentheses table row cell negative 6 end cell row 17 end table close parentheses

  • open parentheses table row cell negative 6 end cell row cell negative 13 end cell end table close parentheses

  • open parentheses table row 0 row 17 end table close parentheses

  • open parentheses table row 0 row cell negative 13 end cell end table close parentheses

Choose your answer

11a2 marks

Vectors bold a and 4 bold b are drawn on the grid.

q7-paper6-nov2020-ocr-gcse-maths

Write vector bold a as a column vector.

How did you do?

11b2 marks

Find vector bold b as a column vector.

How did you do?

123 marks

Vector bold a equals open parentheses table row 3 row cell negative 1 end cell end table close parentheses and vector bold b equals open parentheses table row 1 row 3 end table close parentheses

Gavin starts to draw a diagram to show that bold a plus 2 bold b equals open parentheses table row 5 row 5 end table close parentheses

q9b-paper6-nov2019-ocr-gcse-maths

Complete Gavin’s diagram.

How did you do?

133 marks

bold b is a vector.

Given that bold b space plus space open parentheses table row 5 row 2 end table close parentheses is parallel to open parentheses table row 2 row 1 end table close parentheses , find two possible answers for bold b.

bold b bold space bold equals space stretchy left parenthesis table row blank row blank end table stretchy right parenthesis or open parentheses table row blank row blank end table close parentheses

How did you do?

14a2 marks

Vector bold a equals open parentheses table row 2 row 1 end table close parentheses , vector bold b equals open parentheses table row cell negative 2 end cell row 1 end table close parentheses.

On each grid below, draw a vector to represent

q11a-paper5-june2017-ocr-gcse-maths

How did you do?

14b1 mark

Emma says that if she draws vector bold a and vector bold b they will be the same.

Explain why this is incorrect.

How did you do?

153 marks

Vectors a, b, c, d and e are drawn on an isometric grid.

q1b-paper5-spec2015-ocr-gcse-maths

Write each of the vectors c, d and e in terms of a and/or b.

c = ....................

d = ....................

e = ....................

How did you do?

1a1 mark
q1a-hard-4-13-vectors-edexcel-gcse-maths

A P B is a triangle.
N is a point on A P.

stack A B space with rightwards arrow on top space equals bold a bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space stack A N space with rightwards arrow on top bold space bold equals 2 bold b bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space stack N P space with rightwards arrow on top bold space bold equals bold b

Find the vector stack P B with rightwards arrow on top, in terms of bold a and bold b.

How did you do?

1b4 marks

B is the midpoint of A C.
M is the midpoint of P B.

Show that N M C is a straight line.

How did you do?

2a
Sme Calculator2 marks
q2a-hard-4-13-vectors-edexcel-gcse-maths

O P T R spaceis a trapezium.

stack O P with rightwards arrow on top equals bold a space space stack P T with rightwards arrow on top equals bold b space stack O R with rightwards arrow on top space equals space 3 bold b

i) Find stack O T with rightwards arrow on top in terms of bold a andbold space bold b

[1]

ii) Find stack P R with rightwards arrow on top in terms of bold a and bold b Give your answer in its simplest form.

[1]

How did you do?

2b2 marks

S is the point on P R such that P S space colon space S R = 1 : 3

Find stack O S with rightwards arrow on top in terms of bold a and bold b.
Give your answer in its simplest form.

How did you do?

2c2 marks

What does your answer to part (b) tell you about the position of point S?

How did you do?

3
Sme Calculator4 marks
q3-hard-4-13-vectors-edexcel-gcse-maths

O A B spaceis a triangle.

stack O A space with rightwards arrow on top equals 5 bold a bold space stack O B with rightwards arrow on top space equals 2 bold b

T is the point on space A B spacesuch that A T colon space T B equals 5 space colon space 1

Show that space O T spaceis parallel to the vector bold a bold space plus space 2 bold b

How did you do?

4a2 marks
q4a-hard-4-13-vectors-edexcel-gcse-maths

stack O A with rightwards arrow on top space equals bold space 6 bold a and stack O B with rightwards arrow on top space equals space 6 bold b
M is the midpoint of A B.

Write space stack O M space with rightwards arrow on topin terms of bold a bold space and bold b.
Give your answer in its simplest form.

How did you do?

4b4 marks

N is the midpoint of O B.
G is the point on O M such that O G space colon space G M= 2 : 1

Show that A G N is a straight line.

How did you do?

55 marks
q5-hard-4-13-vectors-edexcel-gcse-maths

C A Y B is a quadrilateral.

stack C A with rightwards arrow on top space equals space 3 bold a bold space stack C B with rightwards arrow on top space equals space 6 bold b bold space stack B Y with rightwards arrow on top space equals 5 bold a bold minus bold b

X is the point on space A B spacesuch that A X colon X B space equals space 1 colon 2

Prove that stack C X space with rightwards arrow on top equals space 2 over 5 stack C Y with rightwards arrow on top

How did you do?

6
Sme Calculator4 marks
q6-hard-4-13-vectors-edexcel-gcse-maths

O A B spaceis a triangle.
P is the point on space A B spacesuch that space A P space colon space P B space equals space 5 colon 3

space stack O A with rightwards arrow on top equals space 2 bold a space stack O B with rightwards arrow on top space equals space 2 bold b

stack O P with rightwards arrow on top space equals space k left parenthesis 3 bold a space plus space 5 bold b right parenthesis space where k is a scalar quantity.

Find the value of k.

How did you do?

7
Sme Calculator3 marks
q23-4ma1-2h-qp-jan20-paper2-igcse-maths

A B C spaceis a triangle.
The midpoint of space B C spaceis M.
P is a point on A M.

table row cell stack A B with rightwards arrow on top space end cell equals cell space 4 bold a end cell row cell stack A C with rightwards arrow on top space end cell equals cell space 2 bold b end cell row cell stack A P with rightwards arrow on top space end cell equals cell space 3 over 2 bold a bold space plus space 3 over 4 bold b end cell end table

Find the ratio A P space colon space P M

How did you do?

83 marks

O A B is a triangle.

stack O A with rightwards arrow on top equals bold a

stack O B with rightwards arrow on top equals bold b

The point C lies on O A such that O C space colon space C A space equals space 1 space colon space 2
The point D lies on O B such that O D space colon space D B space equals space 1 space colon space 2

Using a vector method, prove that A B D C is a trapezium.

How did you do?

9a
Sme Calculator3 marks

The diagram shows parallelogram A B C D.

q17-paper1h-june2018-edexcel-igcse-maths

stack A B with rightwards arrow on top equals open parentheses table row 2 row 7 end table close parentheses       stack A C with rightwards arrow on top equals open parentheses table row 10 row 11 end table close parentheses

The point B has coordinates (5, 8)
Work out the coordinates of the point C.

How did you do?

9b
Sme Calculator2 marks

The point E has coordinates (63, 211)
Use a vector method to prove that A B E is a straight line.

How did you do?

10a3 marks

The diagram shows triangle A B D.

q19-paper2h-spec2016-edexcel-igcse-maths

N is the midpoint of B C.

C is the midpoint of A D.

M is the point on A B such that A M colon M B space equals space 3 space colon 1

stack A B with rightwards arrow on top space equals space bold p     and    stack A C with rightwards arrow on top space equals space bold q

Express, in terms of bold p and bold q,

i) stack B D with rightwards arrow on top

ii) stack M N with rightwards arrow on top

How did you do?

10b2 marks

State, giving reasons, two different geometric facts relating M N and B D.

How did you do?

113 marks

In triangle J K L

M is the midpoint of
J K J N space colon space N L space equals space 3 space colon space 2
stack K L with rightwards arrow on top equals 7 bold a      stack N L with rightwards arrow on top equals 4 bold b

q23-paper-3h-nov-2020-aqa-gcse-maths

Work out stack J M with rightwards arrow on top in terms of bold a and bold b.

Give your answer in its simplest form.

How did you do?

121 mark

P Q R is a straight line.

P Q space colon space Q R space equals space 3 space colon space 1

stack P Q with rightwards arrow on top space equals space bold a

q23-paper-2h-nov-2018-aqa-gcse-maths

Choose the vector stack R Q with rightwards arrow on top

  • 1 third bold a

  • 1 fourth bold a

  • negative 1 third bold a

  • negative 1 fourth bold a

Choose your answer

13a2 marks

bold a space equals space open parentheses table row 6 row cell negative 10 end cell end table close parentheses       bold b space equals space open parentheses table row cell negative 1 end cell row 2 end table close parentheses      bold c space equals space open parentheses table row cell negative 4 end cell row 7 end table close parentheses

Work out bold a bold space bold plus bold space bold b bold space bold plus bold space bold c

How did you do?

13b2 marks

Show that bold a space plus space 2 bold c is parallel to bold b

How did you do?

143 marks
q23-paper2h-june2017-aqa-gcse-maths

Is B C D a straight line?
Show working to support your answer.

How did you do?

155 marks

Given that

      m open parentheses table row 4 row 1 end table close parentheses space plus space n open parentheses table row 5 row 2 end table close parentheses space equals space open parentheses table row 12 row 6 end table close parentheses

find the value of m and the value of n.

m = .......................

n = ....................... 

How did you do?

163 marks

Two vectors, a and b, are shown on the 1 centimetre grid below.

q10-paper6-june2018-ocr-gcse-maths

Show that the vector a + 2b has length 7cm.
You may use the grid below.

q10-2-paper6-june2018-ocr-gcse-maths

How did you do?

172 marks

Vector bold a equals open parentheses table row 2 row 1 end table close parentheses , vector bold b equals open parentheses table row cell negative 2 end cell row 1 end table close parentheses and vector bold c equals open parentheses table row cell negative 12 end cell row 0 end table close parentheses

Find the value k so that k open parentheses bold a minus bold b close parentheses equals bold c

k = .....................................................

How did you do?

185 marks

P, Q, R and S are the midpoints of OX, XY, YZ and OZ respectively.

q18-paper5-spec2015-ocr-gcse-maths

OP with rightwards arrow on top space equals space bold a , XQ with rightwards arrow on top space equals space bold b and OS with rightwards arrow on top space equals bold space bold c.

Show that begin mathsize 16px style PQ end style is parallel to SR.

How did you do?

1a3 marks

O A C B is a parallelogram.

q1a-vhard-4-13-vectors-edexcel-gcse-maths

stack O A with rightwards arrow on top space equals space bold a space and space stack O B with rightwards arrow on top space equals space bold b

D spaceis the point such that stack A C with rightwards arrow on top space equals space stack C D with rightwards arrow on top
The point N divides space A B spacein the ratio 2 colon space 1

Write an expression for stack O N with rightwards arrow on top in terms of bold a bold spaceand bold b.

How did you do?

1b
Sme Calculator3 marks

Prove that space O N D spaceis a straight line.

How did you do?

2
Sme Calculator4 marks
q2-vhard-4-13-vectors-edexcel-gcse-maths

A B C spaceis a straight line.

A B space colon thin space B C space equals space 2 colon 5

stack O A space with rightwards arrow on top equals space 2 bold a plus bold b stack O B with rightwards arrow on top space equals 3 bold a space plus 2 bold b

Express stack space O C space with rightwards arrow on top in terms of space bold a bold spaceand bold b.

Give your answer in its simplest form.

How did you do?

35 marks
q3-vhard-4-13-vectors-edexcel-gcse-maths

A C E F spaceis a parallelogram.
B is the midpoint of A C.
M is the midpoint of B E.

stack C B with rightwards arrow on top space equals space bold a stack E D with rightwards arrow on top space equals space bold b stack D C space with rightwards arrow on top equals space bold 2 bold b

Show that space A M D space is a straight line.

How did you do?

4
Sme Calculator4 marks
q4-vhard-4-13-vectors-edexcel-gcse-maths

O A B C spaceis a parallelogram.
stack O A with rightwards arrow on top space equals space bold a space and space stack O C space with rightwards arrow on top space equals space bold c

X is the midpoint of the line A C.

O C D spaceis a straight line so that O C space colon space C D equals space k space colon space 1

Given that stack X D with rightwards arrow on top space equals space 3 bold c space minus 1 half bold a find the value ofspace k.

How did you do?

55 marks
q5-vhard-4-13-vectors-edexcel-gcse-maths

O A N comma space O M B and space A P B spaceare straight lines.
A N space equals space 2 O A.
M is the midpoint of O B.

stack O A with rightwards arrow on top equals bold a space space space space space space space space space space space space space space space space stack O B with rightwards arrow on top space equals bold b

stack A P with rightwards arrow on top space equals space k stack A B space with rightwards arrow on top where k is a scalar quantity.

Given that space M P N spaceis a straight line, find the value of k.

How did you do?

6
Sme Calculator5 marks
q6-vhard-4-13-vectors-edexcel-gcse-maths

O M A comma space O N B and space A B C spaceare straight lines.
M is the midpoint of O A.
B is the midpoint of A C.

stack O A with rightwards arrow on top space equals space 6 bold a space space space space space space space space stack O B with rightwards arrow on top space equals space 6 bold b space space space space space space space space stack O N with rightwards arrow on top space equals space k bold b 
where k is a scalar quantity.

Given that space M N C spaceis a straight line, find the value of k.

How did you do?

7a2 marks

C D E F spaceis a quadrilateral.

q7a-vhard-4-13-vectors-edexcel-gcse-maths

stack C D with rightwards arrow on top space equals space bold a comma space space space space stack D E with rightwards arrow on top space equals bold b space space space space and space space space space space stack F C with rightwards arrow on top space equals space bold a bold minus bold space bold b.

Express stack space F E space with rightwards arrow on top in terms of bold a and/or bold space bold b.
Give your answer in its simplest form.

How did you do?

7b4 marks

M is the midpoint of D E.
X is the point on space F M spacesuch that F X colon X M equals space n space colon space 1
C X E is a straight line.

Work out the value of n.

How did you do?

8
Sme Calculator5 marks

The diagram shows triangle O A B

q22-january-2022-1h-edexcel-igcse-maths

stack O A with rightwards arrow on top equals 8 bold a     stack O B with rightwards arrow on top equals 6 bold b
M is the point on O B such that O M colon M B space equals space 1 space colon 2
N is the midpoint of A B
P is the point of intersection of O N and A M

Using a vector method, find stack O P with rightwards arrow on top as a simplified expression in terms of bold a and bold b
Show your working clearly.

stack O P with rightwards arrow on top= ...................................

How did you do?

95 marks

O X Y Z is a trapezium.

q24-4ma1-1hr-qp-jan22-paper1-igcse-maths

stack O X with rightwards arrow on top equals bold a

stack X Y with rightwards arrow on top equals bold b

stack O Z with rightwards arrow on top equals 3 bold b

M spaceis the midpoint of O X
N is the point such that  M N Z and  O N Y are straight lines.
Given that O N space colon space O Y space equals space lambda space colon space 1
use a vector method to find the value of lambda

lambda space equals ....................................................

How did you do?

10a2 marks

In the diagram

table row cell stack D E with rightwards arrow on top end cell equals bold a row cell stack D H with rightwards arrow on top end cell equals bold b row cell stack H G with rightwards arrow on top end cell equals cell 8 bold b end cell row cell E X colon X H end cell equals cell 3 colon 1 end cell row cell E F colon F G end cell equals cell 1 colon 3 end cell row blank blank blank end table

q27-paper-3h-nov-2021-aqa-gcse-maths

Show that  stack D X with rightwards arrow on top equals 1 fourth bold a plus 3 over 4 bold b

How did you do?

10b
Sme Calculator4 marks

Is D X F a straight line?

Show working to support your answer.

How did you do?

113 marks

Here is quadrilateral P Q R S.

stack P S with rightwards arrow on top equals bold a      stack S R with rightwards arrow on top equals bold b     stack R Q with rightwards arrow on top equals text bold c end text

stack P S with rightwards arrow on top and stack Q R with rightwards arrow on top are not parallel.

q27-paper3h-nov2019-aqa-gcse-maths

X is a point on P S where    P X space colon space X S space equals space 1 space colon space 2

Y is a point on R Q where         R Y space colon space Y Q space equals space 2 space colon space 1

q27-2-paper3h-nov2019-aqa-gcse-maths

Is X Y parallel to P Q?

Show working to support your answer.

How did you do?

123 marks

A B C D E is a pentagon.

q26-paper2h-nov2017-aqa-gcse-maths

Show that B C D E is a parallelogram.

How did you do?

13
Sme Calculator4 marks

Vector m=open parentheses table row 2 row k end table close parentheses and vector n =open parentheses table row 3 row 11 end table close parentheses

Vector 2m + n  is parallel to open parentheses table row 1 row cell negative 1 end cell end table close parentheses

Find the value of k.

k equals.........................................

How did you do?

145 marks

The diagram shows triangle OAB and points C and D.

q15-paper4-june2019-ocr-gcse-maths

OA with rightwards arrow on top = 3bold a spaceand OB with rightwards arrow on top = 3bold b.
C lies on AB such that AC = 2CB.
D is such that BD with rightwards arrow on top = + 2bold a + bold b.
Show, using vectors, that OCD is a straight line.

How did you do?

15
Sme Calculator3 marks

begin mathsize 16px style ABCD end style is a parallelogram.

q16-paper6-nov2017-ocr-gcse-maths

BD with rightwards arrow on top space equals bold space bold a and AD with rightwards arrow on top space equals space bold b.
begin mathsize 16px style straight F end style is the midpoint of begin mathsize 16px style BC end style.
begin mathsize 16px style straight G end style is the midpoint of begin mathsize 16px style DC end style.
AE space equals space 3 EB.

Prove that space stack EF space with rightwards arrow on topand space stack AG space with rightwards arrow on top are parallel.

How did you do?