Vectors
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Transcript of Vectors
A star is made up of a regular hexagon, centre X, surrounded by 6 equilateral triangles.{O|A # a and {O|B # b.
(b) When | a | # 5, write down the value of
(i) | b |, [1]
(ii) | a 0 b |. [1]
0580/4, 0581/4 Jun/03
6
O
P
Q
T
S
R
F
E
DC
B
A
X
a
b
7
(a) Write the following vectors in terms of a and�or b, giving your answers in their simplest form.
(i) {O|S, [1]
(ii) {A|B, [1]
(iii) {C|D, [1]
(iv) {O|R, [2]
(v) {C|F. [2]
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� UCLES 2004 0580/2, 0581/2 Jun/04
For
Examiner's
Use16
C B
O A
M
a
c
OABC is a parallelogram. = a, = c and M is the mid-point of CA.Find in terms of a and c
(a) ,
Answer(a) [1]
(b) ,
Answer(b) [1]
(c) .
Answer(c) [2]
4
© UCLES 2005 0580/04, 0581/04 Jun 05
5 C
D
E
O
A
B
cdNOT TO
SCALE
OABCDE is a regular hexagon. With O as origin the position vector of C is c and the position vector of D is d. (a) Find, in terms of c and d,
(i) , [1]
(ii) , [2]
(iii) the position vector of B. [2]
3
© UCLES 2010 0580/42/M/J/10 [Turn over
For
Examiner's
Use
2 (a) p = 3
2
and q = 6
3
.
(i) Find, as a single column vector, p + 2q.
Answer(a)(i)
[2] (ii) Calculate the value of | p + 2q |. Answer(a)(ii) [2]
(b)
In the diagram, CM = MV and OL = 2LV.
O is the origin. = c and = v. Find, in terms of c and v, in their simplest forms
(i) , Answer(b)(i) [2]
(ii) the position vector of M, Answer(b)(ii) [2]
(iii) . Answer(b)(iii) [2]
C
O VL
M
NOT TOSCALE
11
© UCLES 2006 0580/02, 0581/02 Jun 06
For
Examiner's
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23
V W
S R
U P
QT
O
a b
The origin O is the centre of the octagon PQRSTUVW.
= a and = b. (a) Write down in terms of a and b
(i) ,
Answer(a)(i) [1]
(ii) ,
Answer(a)(ii) [1]
(iii) ,
Answer(a)(iii) [2] (iv) the position vector of the point P.
Answer(a)(iv) [1] (b) In the diagram, 1 centimetre represents 1 unit.
Write down the value of a – b.
Answer(b) [1]
16
© UCLES 2011 0580/43/M/J/11
For
Examiner's
Use
10 (a)
D
A
C
Bp
q
L
N
M
NOT TOSCALE
ABCD is a parallelogram. L is the midpoint of DC, M is the midpoint of BC and N is the midpoint of LM.
= p and = q. (i) Find the following in terms of p and q, in their simplest form.
(a) Answer(a)(i)(a) = [1]
(b) Answer(a)(i)(b) = [2]
(c) Answer(a)(i)(c) = [2]
(ii) Explain why your answer for shows that the point N lies on the line AC. Answer(a)(ii) [1]
10
© UCLES 2012 0580/42/M/J/12
For
Examiner's
Use
7 (a) P is the point (2, 5) and =
− 2
3 .
Write down the co-ordinates of Q. Answer(a) ( , ) [1]
(b)
D
CE
M
B
AO
c
3a
NOT TOSCALE
O is the origin and OABC is a parallelogram. M is the midpoint of AB.
= c, = 3a and CE = 3
1CB.
OED is a straight line with OE : ED = 2 : 1 . Find in terms of a and c, in their simplest forms
(i) , Answer(b)(i) = [1]
(ii) the position vector of M,
Answer(b)(ii) [2]
(iii) , Answer(b)(iii) = [1]
(iv) . Answer(b)(iv) = [2]
(c) Write down two facts about the lines CD and OB.
Answer (c)
[2]
6
© UCLES 2010 0580/23/M/J/10
For
Examiner's
Use
15
G
O
H
N
g
h
NOT TOSCALE
In triangle OGH, the ratio GN : NH = 3 : 1.
= g and = h. Find the following in terms of g and h, giving your answers in their simplest form.
(a) Answer(a) = [1]
(b) Answer(b) = [2]
10
© UCLES 2011 0580/21/M/J/11
For
Examiner's
Use
For
Examiner's
Use
18
Q
P
M
X
S
R
NOT TOSCALE
In the diagram, PQS, PMR, MXS and QXR are straight lines.
PQ = 2 QS.
M is the midpoint of PR.
QX : XR = 1 : 3.
= q and = r.
(a) Find, in terms of q and r,
(i) ,
Answer(a)(i) = [1]
(ii) .
Answer(a)(ii) = [1]
(b) By finding , show that X is the midpoint of MS.
Answer (b)
[3]
19
0580/41/M/J/14© UCLES 2014 [Turn over
(b)
A
N
O
Y
C
Ba
b
NOT TOSCALE
OACB is a parallelogram. = a and = b. AN : NB = 2 : 3 and AY = 5
2 AC.
(i) Write each of the following in terms of a and/or b. Give your answers in their simplest form.
(a)
Answer(b)(i)(a) = ................................................ [2]
(b)
Answer(b)(i)(b) = ................................................ [2]
(ii) Write down two conclusions you can make about the line segments NY and BC.
Answer(b)(ii) ...............................................................................................................................
..................................................................................................................................................... [2]__________________________________________________________________________________________
© UCLES 2011 0580/22/M/J/11
16
C
O
B
Aa
c
P
MQ
NOT TOSCALE
O is the origin and OABC is a parallelogram. CP = PB and AQ = QB.
= a and = c . Find in terms of a and c, in their simplest form,
(a) ,
Answer(a) = [2]
(b) the position vector of M, where M is the midpoint of PQ. Answer(b) [2]
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0580/43/M/J/14© UCLES 2014
5 (a)
5
4
3
2
1
01 2 3 4 5 6 7 8
x
y
A
B
(i) Write down the position vector of A.
Answer(a)(i) f p [1]
(ii) Find ì ì , the magnitude of .
Answer(a)(ii) ................................................ [2]
(b)
p
q
O
Q
S
P R
NOT TOSCALE
O is the origin, = p and = q. OP is extended to R so that OP = PR. OQ is extended to S so that OQ = QS.
(i) Write down in terms of p and q.
Answer(b)(i) = ................................................ [1]
(ii) PS and RQ intersect at M and RM = 2MQ.
Use vectors to fi nd the ratio PM : PS, showing all your working.
Answer(b)(ii) PM : PS = ....................... : ....................... [4]__________________________________________________________________________________________
10
© UCLES 2012 0580/21/M/J/12
For
Examiner's
Use
19
O
T
P
Q
RS
t
p O is the origin and OPQRST is a regular hexagon.
= p and = t. Find, in terms of p and t, in their simplest forms,
(a) , Answer(a) = [1]
(b) , Answer(b) = [2]
(c) the position vector of R.
Answer(c) [2]
9
© UCLES 2012 0580/23/M/J/12 [Turn over
For
Examiner's
Use
18 Q R
O P
MXq
p
NOT TOSCALE
O is the origin and OPRQ is a parallelogram. The position vectors of P and Q are p and q. X is on PR so that PX = 2XR. Find, in terms of p and q, in their simplest forms
(a) , Answer(a) =
[2] (b) the position vector of M, the midpoint of QX. Answer(b) [2]
0580/21/M/J/13© UCLES 2013
Answer t = ............................................... [3]_____________________________________________________________________________________
20R Q
S
Mr
pO P
NOT TOSCALE
OPQR is a parallelogram, with O the origin. M is the midpoint of PQ. OM and RQ are extended to meet at S. = p and = r.
(a) Find, in terms of p and r, in its simplest form,
(i) ,
Answer(a)(i) = ............................................... [1]
(ii) the position vector of S.
Answer(a)(ii) ............................................... [1]
(b) When = – 2
1 p + r , what can you write down about the position of T ?
Answer(b) ................................................................................................................................. [1]_____________________________________________________________________________________
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0580/23/M/J/13© UCLES 2013 [Turn over
ForExaminer′s
Use
19C
O
E A
D B
c
b
OABCDE is a regular polygon.
(a) Write down the geometrical name for this polygon.
Answer(a) ............................................... [1]
(b) O is the origin. = b and = c.
Find, in terms of b and c, in their simplest form,
(i) ,
Answer(b)(i) = ............................................... [1]
(ii) ,
Answer(b)(ii) = ............................................... [2]
(iii) the position vector of E.
Answer(b)(iii) ............................................... [1]_____________________________________________________________________________________
Question 20 is printed on the next page.
0580/22/M/J/14© UCLES 2014 [Turn over
14R
O P
Q
Mr
p
NOT TOSCALE
OPQR is a trapezium with RQ parallel to OP and RQ = 2OP. O is the origin, = p and = r. M is the midpoint of PQ.
Find, in terms of p and r, in its simplest form
(a) ,
Answer(a) = ................................................ [1]
(b) , the position vector of M.
Answer(b) = ................................................ [2]__________________________________________________________________________________________
In the diagram and .# p and # q.
(a) Find in terms of p and q
(i) ,
Answer (a)(i)� # ........................... [2]
(ii) .
Answer (a)(ii)� #........................... [2]
(b) AQ and BP intersect at T..
Find in terms of p and q, in its simplest form.
Answer (b)� # ............................... [2]{Q|T
{Q|TBT = �13 ���BP
{B|P
{B|P
{A|Q
{A|Q
{O|Q{O|POB = �34 ���OQOA = �23 ���OP
0580/02/0581/02/O/N/03 [Turn over
9 ForExaminer’s
use
A
Q
P
B
NOT TOSCALE
OT
20
9
© UCLES 2008 0580/21/O/N/08 [Turn over
For
Examiner's
Use
17
A B
C
O
q
p
O is the origin. Vectors p and q are shown in the diagram. (a) Write down, in terms of p and q, in their simplest form (i) the position vector of the point A, Answer(a)(i) [1]
(ii) BC ,
Answer(a)(ii) [1]
(iii) BC − AC .
Answer(a)(iii) [2]
(b) If | p | = 2, write down the value of | AB |.
Answer(b) [1]
6
© UCLES 2009 0580/21/O/N/09
For
Examiner's
Use
15
M
O R
P Q
r
p
O is the origin and OPQR is a parallelogram whose diagonals intersect at M.
The vector OP is represented by p and the vector is represented by r.
(a) Write down a single vector which is represented by (i) p + r, Answer(a)(i) [1]
(ii) 2
1 p – 2
1 r.
Answer(a)(ii) [1] (b) On the diagram, mark with a cross (x) and label with the letter S the point with position vector
2
1 p + 4
3 r. [2]
8
© UCLES 2011 0580/21/O/N/11
For
Examiner's
Use
13 CA B D
O
ab
A and B have position vectors a and b relative to the origin O.
C is the midpoint of AB and B is the midpoint of AD.
Find, in terms of a and b, in their simplest form
(a) the position vector of C,
Answer(a) [2]
(b) the vector .
Answer(b) [2]
11
© UCLES 2011 0580/22/O/N/11 [Turn over
For
Examiner's
Use
17
O
C B
Aa
c M
4a
O is the origin, = a, = c and = 4a. M is the midpoint of AB. (a) Find, in terms of a and c, in their simplest form
(i) the vector , Answer(a)(i) = [2]
(ii) the position vector of M. Answer(a)(ii) [2]
(b) Mark the point D on the diagram where = 3a + c. [2]
Answer(b) C = [3]
Question 19 is printed on the next page.
9
© UCLES 2012 0580/23/O/N/12 [Turn over
For
Examiner's
Use
20
O C
D
Ed
c
NOT TOSCALE
In the diagram, O is the origin.
= c and = d. E is on CD so that CE = 2ED. Find, in terms of c and d, in their simplest forms,
(a) , Answer(a) = [2]
(b) the position vector of E. Answer(b) [2]
15
0580/43/O/N/13© UCLES 2013 [Turn over
ForExaminer′s
Use
(b)
R
T
A
O B3b
4a
NOT TOSCALE
In the diagram, = 4a and = 3b.
R lies on AB such that = 5
1 (12a + 6b).
T is the point such that = 2
3 .
(i) Find the following in terms of a and b, giving each answer in its simplest form.
(a)
Answer(b)(i)(a) = ............................................... [1]
(b)
Answer(b)(i)(b) = ............................................... [2]
(c)
Answer(b)(i)(c) = ............................................... [1]
(ii) Complete the following statement.
The points O, R and T are in a straight line because ................................................................
........................................................................................................................................... [1]
(iii) Triangle OAR and triangle TBR are similar.
Find the value of area of trianglearea of triangle
OARTBR
.
Answer(b)(iii) ............................................... [2]_____________________________________________________________________________________
9
0580/41/O/N/13© UCLES 2013 [Turn over
ForExaminer′s
Use
(b)P Q
O S
Rp
s
NOT TOSCALE
In the pentagon OPQRS, OP is parallel to RQ and OS is parallel to PQ. PQ = 2OS and OP = 2RQ. O is the origin, = p and = s.
Find, in terms of p and s, in their simplest form,
(i) the position vector of Q,
Answer(b)(i) ............................................... [2]
(ii) .
Answer(b)(ii) = ............................................... [2]
(c) Explain what your answers in part (b) tell you about the lines OQ and SR.
Answer(c) .................................................................................................................................. [1]_____________________________________________________________________________________
10
© UCLES 2012 0580/42/O/N/12
For
Examiner's
Use
6 (a) Calculate the magnitude of the vector
− 5
3.
Answer(a) [2]
(b)
16
14
12
10
8
6
4
2
04 8 12 16 182 6 10 14
x
y
R P
(i) The points P and R are marked on the grid above.
=
− 5
3. Draw the vector on the grid above. [1]
(ii) Draw the image of vector after rotation by 90° anticlockwise about R. [2]
(c) = 2a + b and = 3b O a.
Find in terms of a and b. Write your answer in its simplest form. Answer(c) = [2]
11
© UCLES 2012 0580/42/O/N/12 [Turn over
For
Examiner's
Use
(d) =
−
5
2 and =
−1
5.
Write as a column vector.
Answer(d) =
[2]
(e)
A
B
CX
MNOT TOSCALE
= b and = c.
(i) Find in terms of b and c. Answer(e)(i) = [1]
(ii) X divides CB in the ratio 1 : 3 . M is the midpoint of AB.
Find in terms of b and c. Show all your working and write your answer in its simplest form. Answer(e)(ii) = [4]
10
© UCLES 2012 0580/42/O/N/12
For
Examiner's
Use
6 (a) Calculate the magnitude of the vector
− 5
3.
Answer(a) [2]
(b)
16
14
12
10
8
6
4
2
04 8 12 16 182 6 10 14
x
y
R P
(i) The points P and R are marked on the grid above.
=
− 5
3. Draw the vector on the grid above. [1]
(ii) Draw the image of vector after rotation by 90° anticlockwise about R. [2]
(c) = 2a + b and = 3b O a.
Find in terms of a and b. Write your answer in its simplest form. Answer(c) = [2]