A65 VECTORS
ADVANCED
QUESTIONS
1
In the diagram,
Y
→
OP = p,
→
OQ = q,
→
PY = kq,
→ 1→
PX = PQ.
3
O
p
P
→
(i) Express PX in terms of p and q.
→
(ii) Express OX in terms of p and q.
→
(iii) Express QY in terms of k, p and q.
[1]
(iv) Given that OX is parallel to QY, find the value of k.
[2]
(v) The line OX, when produced, meets PY at Z.
→
Express PZ in terms of q.
[2]
[1]
[1]
4024/02/M/J/04
2
C
Z
B
X
b
D
O
E
A
a
F
Y
A regular hexagon, ABCDEF, has centre O.
→
→
OA = a and OB = b.
(a) Express, as simply as possible, in terms of a and/or b,
→
(i) DO,
→
(ii) AB,
→
(iii) DB.
(b) Explain why
|a| = |b| = |b–a|.
[1]
[1]
[1]
[1]
(c) The points X, Y and Z are such that
→
→
→
OX = a + b, OY = a – 2b and OZ = b – 2a.
(i) Express, as simply as possible, in terms of a and/or b,
→
(a) AX,
→
(b) YX.
[1]
(ii) What can be deduced about Y, A and X?
[1]
[1]
→
(d) Express, as simply as possible, in terms of a and/or b, the vector XZ.
[1]
(e) Show that triangle XYZ is equilateral.
[2]
(f) Calculate
Area of triangle OAB
.
Area of triangle XYZ
[2]
4024/02/O/N/05
3
C
B
F
E
D
A
→
→
→
In the diagram, AB = 2b, AD = 3a and DF = b – a.
E is the midpoint of AB and F is the midpoint of DC.
(i) Express, as simply as possible, in terms of a and/or b,
→
(a) EA ,
→
(b) DC ,
→
(c) EF ,
→
(d) BC .
[1]
[1]
[1]
[1]
(ii) (a) Give the special name of the quadrilateral ABCD.
Give your reason.
→ → →
(b) Find the ratio 冏 BC 冏 : 冏 EF 冏 : 冏 AD 冏 .
[2]
[1]
4024/02/O/N/07
4 In the diagram,
R
OT = 3OP, RS = 16RT and
Q is the midpoint of PR.
→
→
OP = p and PQ = q.
S
Q
q
O
p
P
T
(i) Express, as simply as possible, in terms of p and q,
→
(a) OR ,
→
(b) RT ,
→
(c) QS .
(ii) Write down the value of QS .
OR
[1]
[1]
[2]
[1]
4024/02/M/J/08
5
R
S
Q
O
p
T
q
P
In the diagram, OQ = QS , QR = 2PQ and ST = 2RS.
OP = p and PQ = q .
(i) Express, as simply as possible, in terms of p and/or q ,
(a) OQ,
[1]
(b) RS ,
[1]
(c) OS ,
[1]
(d) OT.
[1]
(ii) Hence write down two facts about O, P and T.
[2]
4024/O2/M/J/09
6(a)
冢 冣 and QR = 冢– 14冣 .
→
12
PQ =
5
→
→
(i) Calculate 兩 PQ 兩.
Answer ........................................ [2]
→
(ii) Find PR .
Answer
[1]
(b) You may use the grid below to help you answer this question.
T is the point (13, 7) and U is the point (8, 9).
→
(i) Find TU .
Answer
[1]
(ii) TUV is an isosceles triangle with TU = TV .
The y-coordinates of the points U and V are equal.
Find the coordinates of V.
Answer
(........... , ...........)
[1]
(iii) W is the point (1, 3).
Calculate the area of triangle TUW.
Answer .............................. units2 [3]
y
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 x
4024/21/M/J/11
7
ABCDEF is a regular hexagon with centre O.
A
B
F
C
O
E
D
(a) (i) Find AÔ B.
Answer ......................................... [1]
(ii) Explain why AO = BC.
Answer ......................................................................................................................... [1]
→
→
(b) OA = a and OB = b.
G is the point on AB such that AG : GB is 1 : 3.
H is the midpoint of BC.
A G
B
a
F
H
b
C
O
E
D
Express, as simply as possible, in terms of a and b,
(i)
→
AB ,
Answer ......................................... [1]
(ii)
→
FB ,
Answer ......................................... [1]
(iii)
→
OG ,
Answer ......................................... [2]
(iv)
→
OH ,
Answer ......................................... [1]
(v)
→
GH .
Answer ......................................... [2]
4024/21/M/J/12
8
OAB is a triangle and OBDC is a rectangle where OD and BC intersect at E.
3
F is the point on CD such that CF =4 CD.
OA = a, OB = b and OC = c.
A
a
b
O
B
c
E
C
F
D
(a) Express, as simply as possible, in terms of one or more of the vectors a, b and c,
(i)
AB ,
Answer ........................................ [1]
(ii)
OE ,
Answer ........................................ [1]
(iii)
EF .
Answer ........................................ [2]
3
2
(b) G is the point on AB such that OG = 5 a + 5 b.
(i) Express AG in terms of a and b.
Give your answer as simply as possible.
Answer ........................................ [1]
(ii) Find AG : GB.
Answer .................. : ................. [1]
(iii) Express FG in terms of a, b and c.
Give your answer as simply as possible.
Answer ........................................ [2]
4024/22/M/J/12
9
B
C
A
E
D
BADandCAEarestraightlinesandBCisparalleltoED.
1
12
1
BA = c m, ED = c m and BA = BD .
-2
4
-3
(i) DescribefullythesingletransformationthatmapstriangleABContotriangleADE.
Answer..............................................................................................................................
...................................................................................................................................... [2]
(ii) Calculate BA .
Answer ............................................... [1]
(iii) FindCD.
Answer
 
[2]
Answer
 
[2]
(iv) FisthemidpointofBD.
Find EF .
4024/21/M/J/13
10 (a) ABCD is a parallelogram.
C
D
B
A
J1N
J- 4 N
AB = K O and BC = K O .
L4P
L 2P
(i) Find BD.
Answer
(ii) Calculate AC .
Answer
 
[1]
................................................ [2]
(iii) The parallelogram ABCD is mapped onto the parallelogram PBQR.
J 3N
J-12 N
PB = K
O and BQ = K O .
L12P
L 6P
(a) Describe fully the single transformation that maps the parallelogram ABCD onto
the parallelogram PBQR.
Answer ......................................................................................................................
.............................................................................................................................. [2]
4024/22/M/J/13
11 (a) In this question you may use the grid below to help you.
J4N
J 8N
K
O
The point P has position vector K O and the point Q has position vector KK OO .
-3
2
L P
L P
(i) Find PQ.
Answer
(ii) Find PQ .
J
K
K
KK
L
N
O
O
OO
P
[1]
Answer
.................................................... [1]
Answer
.................................................... [2]
(iii) Find the equation of the line PQ.
(iv) Given that Q is the midpoint of the line PR, find the coordinates of R.
Answer ( ...................... , ...................... ) [2]
(b)
D
B
b
O
a
C
A
In the diagram triangles OAB and OCD are similar.
OA = a, OB = b and BC = 4a - b.
(i) Express, as simply as possible, in terms of a and/or b
(a) AB,
Answer
.................................................... [1]
Answer
.................................................... [1]
Answer
.................................................... [2]
(b) AC ,
(c) CD.
(ii) Find, in its simplest form, the ratio
(a) perimeter of triangle OAB : perimeter of triangle OCD,
Answer
......................... : ........................ [1]
(b) area of triangle OAB : area of trapezium ABDC.
Answer
......................... : ........................ [1]
4024/22/M/J/14
(a) Thediagramshowsthevectors PQandQR.
12
a
5
PQ = c m and QR = c m.
2
b
Q
P
R
(i) Findaandb.
(ii) Calculate
Answer
a=............... b=...............[2]
Answer
............................................ [2]
PQ .
(b) OACBisaparallelogram.
OA = a, OB = b andDisthepointsuchthat2OB = BD.
EisthemidpointofCD.
A
C
a
F
O
b
B
E
D
(i) ExpressCE ,assimplyaspossible,intermsofaandb.
Answer
............................................ [1]
(ii) ExpressOE ,assimplyaspossible,intermsofaandb.
Answer
............................................ [1]
Answer
.....................:....................[2]
(iii) FisapointonBCsuchthatOF = kOE .
Find BF:FC.
4024/22/M/J/14
13
(a)
Q
NOT TO
SCALE
P
U
R
T
S
In the diagram, PQ = 4p , QR = 3q and PT = p + 2q .
QU =
2
2
QR and PT = PS .
3
3
(i)
Express, as simply as possible, in terms of p and/or q,
(a)
PS ,
PS = ................................................... [1]
(b) SR.
SR = ................................................... [2]
(ii)
State the name of the special quadrilateral PQRS.
Using vectors, give a reason for your answer.
................................... because ..................................................................................................
.................................................................................................................................................... [2]
(iii)
Find, in its simplest form, the ratio
PQ | SR .
......................... : ......................... [2]
3
AB = e o
2
(b)
(i)
6
BC = e o
-2
-7
CD = e o
-3
Find AD.
AD = f
(ii)
Find
p
[1]
BC .
.................................................... [2]
(iii)
Given that E is the midpoint of BC, find AE .
AE = f
p
[2]
4024/21/M/J/19
14(a) f = e o
-3
4
(i)
1
g=e o
-5
Find g - 2f.
f
(ii)
Petra writes
p
[1]
f 2 g.
Show that Petra is wrong.
[3]
(b)
A
a
P
NOT TO
SCALE
Q
O
R
b
B
O, A and B are points with OA = a and OB = b .
1
P is the point on OA such that OP = OA .
3
O, Q and R lie on a straight line and Q is the midpoint of PB.
(i)
Find PB in terms of a and b.
PB = ................................................... [1]
(ii)
Find OQ in terms of a and b.
Give your answer in its simplest form.
OQ = ................................................... [2]
(iii)
QR = 2OQ.
Show that AR is parallel to PB.
[3]
4024/22/M/J/19