GCSEMathsOCRExam Questions9. Congruence & SimilarityVectors

Vectors (OCR GCSE Maths)

Exam Questions

3 hours48 questions
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1a
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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

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1b
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Work out bold a space plus space 2 bold b as a column vector.

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2a
Sme Calculator1 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.

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2b
Sme Calculator3 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.

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3a
Sme Calculator1 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.

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3b
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Express stack N R with rightwards arrow on top in terms of bold a andbold space bold b.

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4
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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.

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5a
Sme Calculator1 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.

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5b
Sme Calculator4 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

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6a
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Vectors bold a and 4b are drawn on the grid.

q7-paper6-nov2020-ocr-gcse-maths

Write vector bold a as a column vector.

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6b
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Find vector b as a column vector.

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7
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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.

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8
Sme Calculator3 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 

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9a
Sme Calculator2 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

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9b
Sme Calculator1 mark

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

Explain why this is incorrect.

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10
Sme Calculator3 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 = ....................

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11
Sme Calculator2 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

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12
Sme Calculator1 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]

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13
Sme Calculator4 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.

rightwards arrow for A B of space 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 bold space rightwards arrow for B C of bold space equals space bold b

Find rightwards arrow for E X of in terms of bold a and bold b.
Give your answer in its simplest form.

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14
Sme Calculator1 mark

Here is vector bold a.

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

Circle 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

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15
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
Circle your answer.

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

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1a
Sme Calculator1 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.

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1b
Sme Calculator4 marks

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

Show that N M C is a straight line.

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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]

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2b
Sme Calculator2 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.

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2c
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What does your answer to part (b) tell you about the position of point S?

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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

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4a
Sme Calculator2 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.

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4b
Sme Calculator4 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.

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5
Sme Calculator5 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

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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.

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7
Sme Calculator5 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 = .......................

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8
Sme Calculator3 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

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9
Sme Calculator2 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.

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 = ..................................................... 

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10
Sme Calculator5 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 begin mathsize 16px style SR end style.

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11
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

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12
Sme Calculator3 marks

O A B is a triangle.

rightwards arrow for O A of equals bold italic a bold space bold space bold space bold space rightwards arrow for O B of bold equals bold italic 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.

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13a
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.

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13b
Sme Calculator2 marks

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

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14a
Sme Calculator3 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

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14b
Sme Calculator2 marks

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

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15
Sme Calculator3 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.

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16
Sme Calculator1 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

Circle 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

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17a
Sme Calculator2 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

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17b
Sme Calculator2 marks

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

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18
Sme Calculator3 marks

q23-paper2h-june2017-aqa-gcse-maths

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

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1a
Sme Calculator3 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.

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1b
Sme Calculator3 marks

Prove that space O N D spaceis a straight line.

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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.

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3
Sme Calculator5 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.

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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.

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5
Sme Calculator5 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.

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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.

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7a
Sme Calculator2 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.

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7b
Sme Calculator4 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.

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8
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........................................

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9
Sme Calculator5 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 = 3b.
C lies on AB such that AC = 2CB.
D is such that BD with rightwards arrow on top = + 2bold a + b.
Show, using vectors, that OCD is a straight line.

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10
Sme Calculator3 marks

ABCD 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.
straight G is the midpoint of DC.
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.

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11
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= ...................................

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12
Sme Calculator5 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 .................................................... 

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13a
Sme Calculator2 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

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13b
Sme Calculator4 marks

Is D X F a straight line?

Show working to support your answer.

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14
Sme Calculator3 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

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.

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15
Sme Calculator3 marks

A B C D E is a pentagon.

q26-paper2h-nov2017-aqa-gcse-maths

Show that B C D E is a parallelogram.

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