1.
ABCDEF is a regular hexagon with centre O.
OA = a and AB = b
C
B
Diagram drawn accurately
b
O
D
E
(a)
a
A
F
Find expressions, in terms of a and b, for
(i)
OB
Answer ..................................................................
(1)
(ii)
AC
Answer ..................................................................
(1)
(iii)
EC
Answer ..................................................................
(1)
(b)
The positions of points P and Q are given by the vectors
OP = a – b
(i)
OQ = a + 2b
Draw and label the positions of points P and Q on the diagram.
(2)
(ii)
Hence, or otherwise, deduce an expression for PQ .
............................………....................................................................................
............................………....................................................................................
Answer ...............................................................
(1)
(Total 6 marks)
The Robert Smyth School
1
2.
PQRSTU is a regular hexagon and O is the centre of the hexagon.


OP = p and OQ = q
Express each of the following vectors in terms of p and q
P
U

(a) PQ
p
T
O
......................................................................................................
Q
q
(1)

(b) SP
S
......................................................................................................
R
(1)

(c) SQ
.....................................................................................................................................
(2)
(Total 4 marks)
3.
The diagram shows two sets of parallel lines.
Vector PQ = a and vector PS = b
PR = 3PQ and PU = 2 PS
U
V
W
Not to scale
S
T
b
P
(a)
a
Q
R
Write the vector PV in terms of a and b
.....................................................................................................................................
Answer ..................................................................
(1)
(b)
Write the vector RU in terms of a and b
.....................................................................................................................................
Answer ..................................................................
(1)
(c)
Find two vectors that can be written as 3a – b
.....................................................................................................................................
Answer ............................ and ............................
(2)
(Total 4 marks)
The Robert Smyth School
2
4.
In the diagram OACD, OADB and ODEB are parallelograms.
OA  a and OB  b
E
D
B
C
b
O
(a)
a
A
Express, in terms of a and b, the following vectors.
Give your answers in their simplest form.
(i)
OD
...........................................................................................................................
(1)
(ii)
OC
...........................................................................................................................
(1)
(iii)
AB
...........................................................................................................................
Answer...............................................................................
(1)
(b)
The point F is such that OCFE is a parallelogram.
Write the vector CF in terms of a and b.
.....................................................................................................................................
.....................................................................................................................................
Answer...............................................................................
(2)
(c)
What geometrical relationship is there between the points O, D and F? Justify your
answer.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 7 marks)
The Robert Smyth School
3
5.
OACB is a parallelogram and M is the mid-point of BC.
OA = a and OB = b
C
A
Not drawn accurately
a
M
O
(a)
N
B
b
Express the following vectors in terms of a and b
(i)
Answer ..................................................................
BA
(1)
(ii)
Answer ..................................................................
AM
(1)
(b)
AM is extended to N, where AN  2 AM .
Show that BN = b
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(c)
What does this tell you about the position of N?
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(1)
(Total 5 marks)
6.
OP = –4a + 5b and OQ = 5a – b.
P
–4a + 5b
O
5a – b
Q
R is a point on PQ such that PR : RQ = 1 : 2.
The Robert Smyth School
4
(a)
Express OR in terms of a and b.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
Answer ....................................................
(3)
(b)
PS = a + 4b
Express OS in terms of a and b.
.....................................................................................................................................
.....................................................................................................................................
Answer ....................................................
(2)
(c)
What two facts do OR and OS indicate about the points O, R and S?
Give a reason for each of your answers.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 7 marks)
The Robert Smyth School
5
7.
In the diagram OP = 4a, PA = a, OB = 5b, BR = 3b and AQ = 2 AB
5
A
a
P
Q
4a
O
5b
B
R
3b
Not drawn accurately
(a)
Find, in terms of a and b, simplifying your answers,
(i)
AB
...........................................................................................................................
...........................................................................................................................
Answer ……………………………………………
(1)
(ii)
PQ
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
Answer ……………………………………………
(2)
(b)
Show clearly that points P, Q and R lie on a straight line.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(3)
(Total 6 marks)
The Robert Smyth School
6
8.
B
Not drawn accurately
Q
b
OAB is a triangle where M is the
mid-point of OB.
M
P
P and Q are points on AB such
that AP = PQ = QB.
b
A
a
O
(a)
OA = a and OB = 2b
Find, in terms of a and b, expressions for
(i)
BA
...........................................................................................................................
...........................................................................................................................
Answer ..........................................................
(1)
(ii)
MQ
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
Answer ..........................................................
(2)
(iii)
OP
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
Answer ..........................................................
(2)
(b)
What can you deduce about quadrilateral OMQP?
Give a reason for your answer.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(Total 7 marks)
The Robert Smyth School
7
9.
OPQR is a parallelogram.
M is the midpoint of the diagonal OQ.
OP = 2p and OR = 2r
R
Q
2r
M
O
(a)
2p
P
Express OM in terms of p and r.
.....................................................................................................................................
.....................................................................................................................................
Answer OM .......................................................
(1)
(b)
Use vectors to show that M is the midpoint of PR.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 4 marks)
The Robert Smyth School
8
The Robert Smyth School
9
1.
(a)
(i)
a+b
B1
(ii)
b–a
B1
(iii)
a + 2b
B1
or equivalent eg. b + a + b
(b)
(i)
P in correct position
Q in correct position
‘by eye’ judgement ... tolerance of 2mm in any direction (if no
labels then B1, B0)
B1
B1
(ii)
PQ = 3b
Accept PQ = PO + OQ = b – a + a + 2b = 3b
B1
[6]
2.
(a)
– p + q oe
B1
(b)
2p
B1
(c)
SQ = SO+ OQ
or SR + RQ or SP + PQ
M1
p+q
A1
[4]
3.
(a)
a + 2b
B1
oe
(b)
2b –3a
B1
oe
a 
 – 3a 
 correct scores SC1
Note:   and 
 2b 
 2b 
(c)
SR
UT
B1
B1
(a)
(i)
a+b
b+a
B1
(ii)
2a + b
2a + b
B1
–a+b
B1
[4]
4.
(iii) b – a
(b)
CF = OE
M1
a + 2b
(c)
a + b + b oe
A1
Straight line because
OD = a + b
B1
3 times bigger because
OF = 3a + 3b
B1
[7]
5.
(a)
(b)
(i)
– b + a or a – b
(ii)
b–
1
a
2
oe
BN  BM  MN  BM  AM
oe
1
1
=
a+b–
a
2
2
The Robert Smyth School
B1
B1
M1
A1
10
(c)
ON = 2 OB
B1
or OBN a straight line
or BN = OB
or B is midpoint of ON
[5]
6.
(a)
1
((4a – 5b) + (5a – b))
3
M1
(– 4a + 5b) + their (3a – 2b)
M1 dep
– a + 3b
A1
or 3a – 2b
(b)
(c)
(– 4a + 5b) + (a + 4b)
M1
– 3a + 9b
A1
ORS is a straight line
oe
B1
OS is 3 times the length of OR
oe
B1
[7]
7.
(a)
(i)
–5a + 5b
B1
(ii)
a+
M1
2
5
their(–5a + 5b)
condone one error (eg, missing brackets)
2b – a
(b)
A1
PR = –4a + 5b + 3b
M1
= – 4a + 8b or 4(2b – a)
alternatively: QR =
A1
3
5
(5a + 5b) +3b
= 6b – 3a = 3(2b – a)
PR is a multiple of PQ
hence points are co-linear
A1
QR is a multiple of PQ
hence points are co-linear oe
[6]
8.
(a)
(i)
BA = a – 2b
or equivalent
B1
(ii)
MQ = MB + BA = b + (a – 2b)
Attempt to set up a route, must include substitution of a and b,
condone one error
M1
MQ = a + b or (a + b)
Need not be simplified
OP = OA + AB = a + (2b – a)
or OP = OB + BA = 2b + (a – 2b)
A1
(iii)
OP = a + b
M1 as above, need not be simplified for A1
The Robert Smyth School
M1
A1
11
(b)
OP = 2 × MQ
B1
Trapezium
B1
only if accompanied by a sound reason
[7]
9.
(a)
p+r
B1
(b)
PM = – 2p + p + r
or MR = – (p + r) 2r or PR = – 2p + 2r
M1
PM = – p + r
or MR = – p + r
A1
PR = 2PM so M is mid-point
A1
The Robert Smyth School
[4]
12