revision 13 vectors - math medicine
TRANSCRIPT
145 SEKOLAH BUKIT SION - IGCSE MATH REVISION
MATHEMATICS (EXTENDED) 0580 IGCSE MAY/JUNE 2020
REVISION 13
VECTORS
146 SEKOLAH BUKIT SION - IGCSE MATH REVISION
NOTES:
147 SEKOLAH BUKIT SION - IGCSE MATH REVISION
1.
O is the origin and OPQRST is a regular hexagon. ππ#####β = π© and ππ#####β = π
Find, in terms of p and t, in their simplest forms (a) ππ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(b) ππ #####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(c) the position vector of R.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2] 2. P is the point (2, 5) and ππ#####β = + 3β2/. Write down the co-ordinates of Q.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
148 SEKOLAH BUKIT SION - IGCSE MATH REVISION
3.
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) ππ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(b) the position vector of M, the midpoint of QX.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
149 SEKOLAH BUKIT SION - IGCSE MATH REVISION
4.
O is origin and OABC is a parallelogram. M is the midpoint of AB. ππΆ#####β = π, ππ΄#####β = ππ,and CE = 8
9CB.
OED is a straight line with OE : ED = 2 : 1.
Find in terms of a and c, in their simplest forms (i) ππ΅#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1] (ii) the position vector of M
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2] (iii) ππΈ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1] (iv)πΆπ·#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2] (c) Write down two facts about the lines CD and OB.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
150 SEKOLAH BUKIT SION - IGCSE MATH REVISION
5.
OPQR is a parallelogram with O the origin. M is the midpoint od PQ. OM and RQ are extended to meet at S. ππ#####β = p and ππ #####β = r.
(a) Find, in terms of p and r, in its simplest form, (i) ππ######β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) the position vector of S
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(b) When ππ#####β = β 8>π© + π«, what can you write about the position of T?
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
6. OABCDE is a regular polygon.
(a) O is the origin. ππ΅#####β = b and ππΆ#####β = c. Find, in terms of b and c, in their simplest form,
(i) π΅πΆ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1] (ii) ππ΄#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2] (iii) the position vector of E.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
151 SEKOLAH BUKIT SION - IGCSE MATH REVISION
7.
OPQR is a trapezium with RQ parallel to OP and RQ = 2OP. O is the origin, ππ#####β = p and ππ #####β = r. M is the midpoint of PQ. Find, in terms of p and r, in their simplest form
(a) ππ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(b) ππ######β , the position vector of M.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2] 8. (a) ππ#####β = +β34 /. (i) P is the point (β2 , 3).
Write down the co-ordinates of Q.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) Work out Bππ#####β B, the magnitude of ππ#####β .
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
152 SEKOLAH BUKIT SION - IGCSE MATH REVISION
9.
(i) Write down the position vector of A.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) Work out Bπ΄π΅#####β B, the magnitude of π΄π΅#####β .
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
10. ππ#####β = + 5β8/. (i) Q is the point (β2 , 3).
Write down the co-ordinates of P.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) Work out Bππ#####β B, the magnitude of ππ#####β .
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
153 SEKOLAH BUKIT SION - IGCSE MATH REVISION
11.
OABC is a parallelogram. ππ΄#####β = a and ππ΅#####β = b. AN : NB = 2 : 3 and AY = >
E AC.
(i) Write each of the following in terms of a and b.
(a) ππ######β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(b) ππ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(ii) Write down two conclusions you can make about the line segments NY = BC.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
154 SEKOLAH BUKIT SION - IGCSE MATH REVISION
12. O is the origin, ππ#####β = p and ππ######β = q. OP is extended to R so that OP = PR. OQ is extended to S so that OQ = QS. (i) Write down π π#####β in terms of p and q.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) PS and RQ intersect at M and RM = 2MQ. Use vectors to find the ratio of PM : PS, showing all your working.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [4]
155 SEKOLAH BUKIT SION - IGCSE MATH REVISION
13.
PQRS is a quadrilateral and M is the midpoint of PS. ππ#####β = a, ππ #####β = b and ππ#####β = a β 2b.
(a) Show that ππ####β = 2b. [1]
(b) Write down the mathematical name for the quadrilateral PQRM, giving reasons for your answer.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ because β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
156 SEKOLAH BUKIT SION - IGCSE MATH REVISION
14.
OAPB is a parallelogram. O is the origin, ππ΄#####β = a and ππ΅#####β = b. M is the midpoint of BP.
(a) Find, in terms of a and b, giving your answer in its simplest form, (i) π΅π΄#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(ii) the position vector of M
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1]
(b) X is on BA so that BX : XA = 1 : 2. Show that X lies on OM. [4]
157 SEKOLAH BUKIT SION - IGCSE MATH REVISION
15.
In the diagram, M is the midpoint of AB and L is the midpoint of OM. The lines OM and AN intersect at L and ON = 8
9 OB.
(i) Find in terms of a and b, in its simplest terms.
(a) ππ######β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(b) ππΏ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [1] (c) π΄πΏ#####β
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [2]
(ii) Find the ratio of AL : AN in its simple form.
Answer: β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦ [3]