IGCSE Edexcel Maths 3 hours 50 questions

Exam Questions

Vectors
Introduction to Column Vectors / Representing Vectors as Diagrams / Magnitude of
a Vector / Position & Displacement Vectors / Finding Vector Paths / Vector Proof

Medium (16 questions) /47 Scan here to return to the course

or visit savemyexams.com
Hard (18 questions) /72

Very Hard (16 questions) /74

Total Marks /193

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Medium Questions
1 (a) The vector a and the vector b are shown on the grid.

On the grid, draw and label vector −2 a

(1 mark)

(b) Work out a + 2 b as a column vector.

(2 marks)

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2 (a)

OAB is a triangle.
⎯⎯⎯⎯
OA = a
OB = b
⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯
Find AB in terms of a and b .

(1 mark)

(b) P is the point on AB such that AP : PB = 3 : 1

OP in terms of a and b.
⎯⎯⎯
Find
Give your answer in its simplest form.

(3 marks)

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3 (a)

PQRS is a parallelogram.
N is the point on SQ such that SN : NQ = 3:2

PQ = a
⎯⎯⎯

⎯⎯⎯
PS = b
Write down, in terms of a and b , an expression for SQ
⎯⎯⎯
.

(1 mark)

⎯⎯⎯⎯
(b) Express NR in terms of a and b .

(3 marks)

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4

OAB is a triangle.
M is the midpoint of OA .
N is the midpoint of OB .

m
⎯⎯⎯⎯⎯
OM =
ON = n
⎯⎯⎯⎯

Show that AB is parallel to MN .

(3 marks)

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5 (a)

OAYB is a quadrilateral.

OA = 3a
⎯⎯⎯⎯

⎯⎯⎯⎯⎯
OB = 6 b
⎯⎯⎯⎯
Express AB in terms of a and b .

(1 mark)

5a − b
⎯⎯⎯⎯
(b) X is the point on AB such that AX : XB = 1 :2 and BY =

⎯⎯⎯⎯ 2 ⎯⎯⎯⎯
Prove that OX = OY
5

(4 marks)

6 Here are two vectors.

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 ⎯⎯⎯⎯ ⎛5 ⎞ ⎯⎯⎯⎯ ⎛ −2 ⎞⎟
AB = ⎜⎜⎜ ⎟⎟⎟ CB = ⎜⎜⎜ ⎟⎟
⎝3 ⎠ ⎝ 4 ⎠
⎯⎯⎯⎯
Find, as a column vector, AC

(2 marks)

7 ABCDEF and GHIJKL are regular hexagons each with centre O .

GHIJKL is an enlargement of ABCDEF , with centre O and scale factor 2

⎯⎯⎯⎯
OA = a ⎯⎯⎯⎯
OB = b
Write the following vectors, in terms of a and b . Simplify your answers.
⎯⎯⎯⎯
i) AB

[1]

⎯⎯
ii) KI

[2]

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 ⎯⎯⎯⎯
iii) LD

[2]

(1 mark)

8 ABCDEF is a regular hexagon.

ABX and DCX are straight lines.

⎯⎯⎯ =
AB
a ⎯⎯⎯ =
BC
b
Find ⎯⎯⎯ in terms of a and b .
EX
Give your answer in its simplest form.

(4 marks)

9 Here are two vectors.

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 ⎯⎯⎯⎯⎯⎛ 6 ⎞⎟ ⎯⎯⎯⎯ ⎛1 ⎞
AB = ⎜⎜⎜ ⎟⎟ CB = ⎜⎜⎜ ⎟⎟⎟
⎝ −9 ⎠ ⎝3 ⎠
⎯⎯⎯⎯
Find the magnitude of AC .

(3 marks)

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10 (a) Vectors a and 4b are drawn on the grid.

Write vector a as a column vector.

(2 marks)

(b) Find vector b as a column vector.

(2 marks)

⎛⎜ 3 ⎞⎟ ⎛1 ⎞⎟
a
11 Vector = ⎜⎜ ⎟⎟ and vector b = ⎜⎜⎜ ⎟⎟
⎝ −1 ⎠ ⎝3 ⎠

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 ⎛⎜ 5 ⎞⎟
Gavin starts to draw a diagram to show that + 2 = ⎜⎜ a b ⎟⎟
⎝5 ⎠

Complete Gavin’s diagram.

(3 marks)

12 b is a vector.
⎛ ⎞ ⎛ ⎞
Given that b + ⎜⎜⎜ 52 ⎟⎟⎟ is parallel to ⎜⎜⎜ 21 ⎟⎟⎟ , find two possible answers for b .
⎝ ⎠ ⎝ ⎠

⎛ ⎞⎟ ⎛⎜ ⎞⎟
b = ⎜⎜⎜ ⎟⎟ or ⎜⎜ ⎟⎟
⎝ ⎠ ⎝ ⎠

(3 marks)

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13 Here is vector . a

Circle the column vector that represents . a

⎛⎜ 3 ⎞⎟ ⎛⎜ −3 ⎞⎟ ⎛⎜ 3 ⎞⎟ ⎛⎜ −3 ⎞⎟
⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟
⎝2 ⎠ ⎝ 2 ⎠ ⎝ −2 ⎠ ⎝ −2 ⎠

(1 mark)

⎛ ⎞ ⎛ ⎞
14 a = ⎜⎜⎜ −23 ⎟⎟⎟ and b = ⎜⎜⎜ −15 ⎟⎟⎟
⎝ ⎠ ⎝ ⎠

Work out a – 3 b Circle your answer.

⎛⎜ −6 ⎞⎟ ⎛⎜ −6 ⎞⎟ ⎛⎜ 0 ⎞⎟ ⎛⎜ 0 ⎞⎟
⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟
⎝ 17 ⎠ ⎝ − 13 ⎠ ⎝ 17 ⎠ ⎝ − 13 ⎠

(1 mark)

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 ⎛⎜ 2 ⎞⎟ ⎛ −2 ⎞⎟
a
15 (a) Vector = ⎜⎜ ⎟⎟ , vector b = ⎜⎜⎜ ⎟⎟ .
⎝1 ⎠ ⎝ 1 ⎠

On each grid below, draw a vector to represent

(2 marks)

(b) Emma says that if she draws vector a and vector b they will be the same.
Explain why this is incorrect.

(1 mark)

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

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

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

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

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

(3 marks)

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Hard Questions
1 (a)

APB is a triangle.
N is a point on AP .
⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯ ⎯⎯⎯⎯
AB = a AN = 2b NP =b
PB , in terms of a and b .
⎯⎯⎯
Find the vector

(1 mark)

(b) B is the midpoint of AC .

M is the midpoint of PB .

Show that NMC is a straight line.

(4 marks)

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2 (a)

OPTR is a trapezium.

OP = a
⎯⎯⎯

PT = b
⎯⎯⎯

OR = 3 b
⎯⎯⎯⎯

⎯⎯⎯⎯
i) Find OT in terms of a and b
[1]

⎯⎯⎯
ii) Find PR in terms of a and b
Give your answer in its simplest form.

[1]

(2 marks)

(b) S is the point on PR such that PS : SR = 1 : 3

⎯⎯⎯
Find OS in terms of a and b .
Give your answer in its simplest form.

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 (2 marks)

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

(2 marks)

OAB is a triangle.
⎯⎯⎯⎯⎯
OA = 5a
OB = 2 b
⎯⎯⎯⎯

T is the point on AB such that AT : TB = 5 : 1

Show that OT is parallel to the vector a + 2b

(4 marks)

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4 (a)

OA = 6a and OB = 6 b
⎯⎯⎯⎯ ⎯⎯⎯⎯

M is the midpoint of AB .
⎯⎯⎯⎯⎯⎯
Write OM in terms of a and b.
Give your answer in its simplest form.

(2 marks)

(b) N is the midpoint of OB .

G is the point on OM such that OG : GM = 2 : 1

Show that AGN is a straight line.

(4 marks)

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5

CAYB is a quadrilateral.

CA = 3a
⎯⎯⎯⎯

CB = 6 b
⎯⎯⎯⎯

BY = 5a − b
⎯⎯⎯⎯

X is the point on AB such that AX : XB = 1:2

⎯⎯⎯⎯⎯ 2 ⎯⎯⎯⎯
Prove that CX = CY
5

(5 marks)

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6

OAB is a triangle. P is the point on AB such that AP : PB = 5:3

OA = 2a
⎯⎯⎯⎯

OB = 2 b
⎯⎯⎯⎯

OP = k (3a + 5 b)
⎯⎯⎯
where k is a scalar quantity.

Find the value of k .

(4 marks)

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7

ABC is a triangle.
The midpoint of BC is M.
P is a point on AM .

AB = 4a
⎯⎯⎯⎯

AC = 2 b
⎯⎯⎯⎯

3 3
AP = a + b
⎯⎯⎯
2 4

Find the ratio AP : PM

(3 marks)

8 OAB is a triangle.
⎯⎯⎯ = a ⎯⎯⎯ = b
OA OB

The point C lies on OA such that OC : CA = 1 : 2

The point D lies on OB such that OD : DB = 1 : 2

Using a vector method, prove that ABDC is a trapezium.

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 (3 marks)

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9 (a) The diagram shows parallelogram ABCD .

⎯⎯⎯⎯ ⎛2 ⎞ ⎯⎯⎯⎯ ⎛ 10 ⎞
AB = ⎜⎜⎜ ⎟⎟⎟ AC = ⎜⎜⎜ ⎟⎟⎟
⎝7 ⎠ ⎝ 11 ⎠

The point B has coordinates (5, 8)

Work out the coordinates of the point C.

(3 marks)

(b) The point E has coordinates (63, 211)

Use a vector method to prove that ABE is a straight line.

(2 marks)

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10 (a) The diagram shows triangle ABD .

N is the midpoint of BC .

C is the midpoint of AD .

M is the point on AB such that AM: MB = 3 :1

⎯⎯⎯⎯
AB = p and
⎯⎯⎯⎯
AC = q
Express, in terms of p and q,
⎯⎯⎯⎯
i) BD
⎯⎯⎯⎯⎯
ii) MN

[3]

(3 marks)

(b) State, giving reasons, two different geometric facts relating MN and BD .

(2 marks)

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11 In triangle JKL

M is the midpoint of JK
JN : NL = 3 : 2
KL = 7a NL = 4 b
⎯⎯⎯⎯ ⎯⎯⎯⎯

⎯⎯⎯
Work out JM in terms of a and b.
Give your answer in its simplest form.

(3 marks)

12 PQR is a straight line.

PQ : QR = 3 : 1
⎯⎯⎯
PQ = a

⎯⎯⎯⎯
Circle the vector RQ

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1 1 1 1
3
a 4
a −
3
a −
4
a

(1 mark)

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 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
13 (a) a = ⎜⎜⎜ −610 ⎟⎟⎟ b = ⎜⎜⎜ −21 ⎟⎟⎟ c = ⎜⎜⎜ −74 ⎟⎟⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Work out a+b+c

(2 marks)

(b) Show that a + 2c is parallel to b

(2 marks)

14

Is BCD a straight line?

Show working to support your answer.

(3 marks)

15 Given that

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 ⎛4 ⎞ ⎛ 5 ⎞ ⎛ 12 ⎞
m ⎜⎜⎜ ⎟⎟⎟ + n ⎜⎜⎜ ⎟⎟⎟ = ⎜⎜⎜ ⎟⎟⎟
⎝1 ⎠ ⎝2 ⎠ ⎝ 6 ⎠

find the value of m and the value of n .

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

n = .......................

(5 marks)

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

Show that the vector a + 2b has length 7cm.

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 You may use the grid below.

(3 marks)

⎛⎜ 2 ⎞⎟ ⎛ −2 ⎞⎟
a
17 Vector = ⎜⎜ ⎟⎟ , vector b = ⎜⎜⎜ ⎟⎟ .
⎝1 ⎠ ⎝ 1 ⎠

⎛ ⎞
c = ⎜⎜⎜ −012 ⎟⎟⎟
⎝ ⎠

Find the value k so that k ( − a b) = c

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

(2 marks)

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

⎯⎯⎯
OP = a , XQ
⎯⎯⎯
= b and OS = c .
⎯⎯⎯

Show that PQ is parallel to SR .

(5 marks)

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Very Hard Questions
1 (a) OACB is a parallelogram.

⎯⎯⎯⎯
OA = a and ⎯⎯⎯⎯
OB = b
⎯⎯⎯⎯ ⎯⎯⎯⎯
D is the point such that AC = CD
The point N divides AB in the ratio 2: 1

⎯⎯⎯⎯
Write an expression for ON in terms of a and b.

(3 marks)

(b) Prove that OND is a straight line.

(3 marks)

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2

ABC is a straight line.

AB : BC = 2:5
⎯⎯⎯⎯⎯
OA = 2a + b
OB = 3a + 2 b
⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯
Express OC in terms of a and b .
Give your answer in its simplest form.

(4 marks)

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3

ACEF is a parallelogram.
B is the midpoint of AC .
M is the midpoint of BE .
⎯⎯⎯⎯
CB = a
ED = b
⎯⎯⎯⎯

⎯⎯⎯⎯⎯
DC = 2b

Show that AMD is a straight line.

(5 marks)

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4

OABC is a parallelogram.
⎯⎯⎯⎯
OA = a and ⎯⎯⎯⎯⎯
OC = c

X is the midpoint of the line AC .

OCD is a straight line so that OC : CD = k : 1

1
3c − a
⎯⎯⎯⎯
Given that XD =
2

find the value of k .

(4 marks)

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5

OAN , OMB and APB are straight lines.

AN = 2OA .
M is the midpoint of OB .

OA = a OB = b
⎯⎯⎯⎯ ⎯⎯⎯⎯

⎯⎯⎯ ⎯⎯⎯⎯⎯
AP = k AB where k is a scalar quantity.

Given that MPN is a straight line, find the value of k .

(5 marks)

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6

OMA , ONB and ABC are straight lines.

M is the midpoint of OA .
B is the midpoint of AC .

OA = 6a OB = 6 b ON = k b where k
⎯⎯⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯⎯
is a scalar quantity.

Given that MNC is a straight line, find the value of k .

(5 marks)

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7 (a) CDEF is a quadrilateral.

CD = a , DE = b and a − b.
⎯⎯⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯
FC =
⎯⎯⎯⎯⎯
Express FE in terms of a and/or b.
Give your answer in its simplest form.

(2 marks)

(b) M is the midpoint of DE .

X is the point on FM such that FX : XM = n : 1
CXE is a straight line.

Work out the value of n .

(4 marks)

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8 The diagram shows triangle OAB

OA = 8a OB = 6 b
⎯⎯⎯⎯ ⎯⎯⎯⎯

M is the point on OB such that OM: MB = 1 :2

N is the midpoint of AB
P is the point of intersection of ON and AM
⎯⎯⎯
Using a vector method, find OP as a simplified expression in terms of a and b
Show your working clearly.

⎯⎯⎯
OP = ...................................

(5 marks)

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9 OXYZ is a trapezium.

OX = a
⎯⎯⎯⎯

XY = b
⎯⎯⎯⎯

OZ = 3 b
⎯⎯⎯⎯

M is the midpoint of OX
N is the point such that MNZ and ONY are straight lines.

Given that ON : OY = λ : 1

use a vector method to find the value of λ

λ= ....................................................

(5 marks)

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10 OACB is a trapezium.

OA = 2a OB = 5 b AC = 3 b
⎯⎯⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯⎯

The diagonals, OC and AB, of the trapezium intersect at the point P .

Find and simplify an expression, in terms of a and b, for OP

⎯⎯⎯
Show your working clearly.

⎯⎯⎯
OP = ...............................

(5 marks)

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11 (a) In the diagram

DE = a
⎯⎯⎯⎯

DH = b
⎯⎯⎯⎯

HG = 8 b
⎯⎯⎯⎯

EX : XH = 3:1
EF: FG = 1:3

1 3
Show that
⎯⎯⎯⎯
DX =
4
a + b
4

(2 marks)

(b) Is DXF a straight line?

Show working to support your answer.

(4 marks)

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12 Here is quadrilateral PQRS .

PS = a SR = b RQ = c
⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯⎯

X is a point on PS where PX : XS = 1 : 2

Y is a point on RQ where RY : YQ = 2 : 1

Is XY parallel to PQ

Show working to support your answer.

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 (3 marks)

13 ABCDE is a pentagon.

Show that BCDE is a parallelogram.

(3 marks)

⎛⎜ 2 ⎞⎟ ⎛ 3 ⎞⎟
14 Vector m= ⎜⎜ ⎟⎟ and vector n = ⎜⎜⎜ ⎟⎟
⎝k ⎠ ⎝ 11 ⎠

⎛⎜ 1 ⎞⎟
Vector 2m + n is parallel to ⎜⎜ ⎟⎟
⎝ −1 ⎠

Find the value of k .

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

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 (4 marks)

15 The diagram shows triangle OAB and points C and D.

⎯⎯⎯
a ⎯⎯⎯
OA = 3 and OB = 3b.
C lies on AB such that AC = 2CB.
⎯⎯⎯
a
D is such that BD = + 2 + b.
Show, using vectors, that OCD is a straight line.

(5 marks)

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16 ABCD is a parallelogram.

BD = a and AD = b .
⎯⎯⎯ ⎯⎯⎯

F is the midpoint of BC .
G is the midpoint of DC .
AE = 3EB.

⎯⎯⎯ ⎯⎯⎯⎯
Prove that EF and AG are parallel.

(3 marks)

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