1 Past Year SPM Questions

4 VECTORS

1. SPM 2003 P1 Q12

2.SPM2003 P1Q13

(ans : -13)

3. SPM 2003 P1 Q 14

(ans: -6p+4q, 2p+ 8q/3}

4 SPM 2003 P2 Q 6

Given that , and , find

(a) the coordinate of A [2 marks](b) the unit vector in the direction of

[2marks](c) the value of k, if [2marks]

(ans:( -3,-4), , )

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Use the information given to find the values of h and k when r = 3p -2q [3marks]

Diagram shows two vectors, .Express

(a) in the form

(b) in the form [2marks]

( , -8i+4j)

p = 2a +3 bq = 4 a – br = ha + (h-k) b, where h and k are constants

Diagram shows a parallelogram ABCD with BED as a straight line.

Given that and DE =2EB,

express in terms of

(a) [4marks]

2 Past Year SPM Questions

5 SPM 2004 P1 Q16Given that O(0,0) A(-3,4) and B(2,16), find in terms of unit vectors, ,

(a) (b) the unit vector in the direction of [4marks]

(ans : )

6 SPM 2004 P1 Q17Given that A(-2,6), B(4,2) and C(m,p), find the value of m and of p such that

[4marks]

(ans : m=6, p=-2 )

7 SPM 2004 P2 Q 8

(ans : -2y+6x, 3y/2 +9x/2, h(6x-2y), k(9x/2 +3y/2, k=1/3, h=1/2)

8 SPM 2005 P1Q15

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Diagram shows triangle OAB. The straight line AP intersects the straight line OQ at R. It is given that

OP=

(a) Express in terms of x and y(i) [4marks]

(b) (i) Given that state in terms of h,

(ii) Given that state in terms of k, [2marks]

(c) Using from (b) , find the value of h and of k. [4marks]

Diagram shows vector drawn on a Cartesian plane.

(a) Express in the form

(b) Find the unit vector in the direction of [2marks]

(ans : )

3 Past Year SPM Questions

9 SPM 2005 P1 Q 16

10 SPM 2005 P2 Q 6

(ans : -20x+32y, 25x, 104 )

11. SPM 2006 P1 Q 13

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Diagram shows a parallelogram, OPQR, drawn on a Cartesian plane.

It is given that and

Find [3marks]

(ans : -10i+j)

Diagram shows a quadrilateral ABCD. AED and EFC are straight lines.It is given that

and

(a) Express in terms of

(i) [3marks] (b) Show that the points B,F and D are collinear

[3marks]

(C) if , find [2 marks]

Diagram shows two vectors, Express

(a)

(b) [2marks]

(ans : -4i-8j )

4 Past Year SPM Questions

12. SPM 2006 P1 Q 14The point P,Q, and R are collinear. It is given that , where k is a constant. Find

(a) the value of k,(b) the ration of PQ : QR [4marks]

(ans : -5/2, 4:3 ]

13. SPM 2006 P2 Q 5

14.SPM 2007 P1 Q16The following information refers to the vectors

, find

(a) the vector ,

(b) the unit vector in the direction of [4 marks]

Ans: (a) (b)

15. SPM 2007 P1 Q15

Ans:

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Diagram shows a trapezium ABCD.

It is given that and

(a) Express in terms of [2marks]

(b) Point F lies inside the trapezium ABCD such that 2 and m is a constant. (i) Express , in terms of m, (ii) Hence, if the points A,F and C are collinear, find the value of m [5marks](ans : 5x+2y, 4x+my, m=8/5]

Diagram shows a rectangle OABC and the point D lies on the straight line OB. It is given that OD=3DB. Express , in terms of [3marks]

5 Past Year SPM Questions

16,SPM 2007 P2Q8

[(a)(i) (ii) (b) (c) 24.08 unit]

17.SPM 2008 P1 Q15The vectors are non zero and non parallel.It is given that (h+3) = (k-5) , where h and k are constants.Find the value of(a) h (b) k [2marks][ans ; -3, 5]

18. SPM 2008 P1 Q16Diagram below shows a triangle PQRThe point T lies on QR such that QT: TR =3:1Express in terms of and : (a) (b) [4marks][ans: 4a-6b, 3a+3b/2 ]

19. SPM 2008 P1 Q6Diagram below shows a quadrilateral ABCD. The diagonals BD and AC intersect at point R. Point P lies on AD.

It is given that

(a) Express in terms of :(i) (ii) [3 marks]

(b) Given that and , where h and k are constants, find the value of h and of k. [4 marks]

[Answer : (a)(i) (ii) (b)

20.SPM 2009P1 Q14

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Diagram shows triangle AOB. The point P lies on OA and the point Q lies on AB. The straight line BP intersects the straight line OQ at the point S.It is given that

OA: OP = 4:1, AB : AQ = 2 : 1,

(a) Express in terms of :

(i)

(ii) [3marks]

(b) Using and , where h and k are constants, find the value of h and of k. [5marks]

(c ) Given that =3 units and AOB =90 , find [2marks]

P

Q

R

T

4 a

6 b

R

S

QP

6 Past Year SPM Questions

Diagram shows a triangle PQR.

Given , and point S lies on QR

such that QS:SR = 2:1, express in terms of and :

a)

b) [4 marks]

[Answer: (a) , (b) ]

21. SPM 2009P1 Q13

Given that and , find

a) in the form

b) The values of k if [4 marks]

[Answer: (a) ;(b) k=−9, 7]

22. SPM 2009 P2 Q5Diagram shows a triangle ABC. The straight line AQ intersects the straight line BR at P.

It is given that AR=3RC, BQ = , and .

a) Express in terms of and :

i) ii) [4 marks]

b) It is given that and ,

where h and k are constants. [5 marks]

[Answer: (a)(i) (ii) (b) ]

23. SPM 2010 P1 Q15

Diagram shows the vector . [3 marks]

Express in the form

a)

b) The unit vector in the direction of

[Answer: (a) ; (b) ]

24. SPM 2010 P1 Q16Diagram shows a triangle OAB and M is a point on AB.

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C

R

P

AB

Q

0

y R(3,4)

x

C

OB

M

7 Past Year SPM Questions

Given , and 2AM=3MB, find

a)

b) [4 marks]

[ Ans: (a) (b) ]

25. SPM 2010 P2 Q9Diagram shows a triangle OAB. The point C lies on OA and the point D lies on AB. The straight line OD intersects the straight line BC at the point E.

It is given that and

a)Express in terms of and

i)

ii) [4 marks]

b)It is given that and ,

where h and k are constants. Express

i) In terms of h, and

ii) in terms of k, and [3 marks]

Hence, find the value of h and of k. [3 marks]

[ Ans: (a) i : ii: (b) i: ii: (c) , ]

26. SPM 2011 P1 Q16Diagram shows a parallelogram ODEF drawn on a Cartesian plane.

It is given that and .

Find . [ 3 marks]

[Ans: ]

27. SPM 2011 P1 Q17

It is given that vector and vector ,

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E

B

D

MO C

O

E

F

D

y

x

8 Past Year SPM Questions

where h is a constant.

a) Express the vector in terms of h.

b) Given that , find the positive value of h. [ 3 marks]

[Ans: (a) , (b) h = 4]

28. SPM 2011 P2 Q10Diagram shows a parallelogram ABCD. Point P lies on the straight line AB and point Q lies on the straight line DC. The straight line AQ is extended to the point R such that

AQ=2QR. It is given that AP:PB = 3:1, DQ: QC =3:1, and .

a) Express, in terms of and

i)

ii) [ 6 marks]

Hence, show that the points P, C and R are collinear.

b) It is given that and .

i) Express in terms of and .

ii) Find the unit vector in the direction of . [4 marks]

29. SPM 2012 P1 Q15

Diagram shows the vector drawn on

a grid of equl squares with sides of 1 unit. [ 2 marks]Determine

a)

b) in terms of and .

[Ans: (a) 3 (b) ]

30. SPM 2012 P1 Q16

The following information refers to the vectors and .

,

It is given that , where is parallel to and k is a constant.

Find the value of a) kb) m [ 3 marks]

[Ans: (a) k=3, m= 19]

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P

D

B

Q

R

C

A

●

●

Ans: (a)i: + = 6 , ii) + )= (b) i: ii:

A

BP

O

9 Past Year SPM Questions

31. SPM 2012 P2 Q5

It is given that and .

a) Find

i)

ii) The unit vektor in the direction . [ 4 marks]

b) Given , where p is a constant and is parallel to , find the value

of p. [ 3 marks]

[Ans: (a)i: , ii: (b) k=-5, p =20]

32. SPM 2013 P1 Q15

Given and , find

a) The unit vector in the direction

b) The value of k such that and are parallel. [ 4 marks]

[Ans: (a) ]

33. SPM 2013 P1 Q16

Diagram shows two vectors and on a

Cartesian plane.

a) State in the form of .

b) Express in the form of

[ 3 marks]

[Ans: (a) , (b) ]

34. SPM 2013 P2 Q3Diagram shows quadrilateral PQRS. The straight line PR intersects the straight line QS at

point T. It is given that QT: TS = 2:3, , and .

a) Express in terms of and

i)

ii) [ 3 marks]

b) Find the ration PT: TR [5 marks ]

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O

B(-3, 2)

A (5, 12)

y

x

P

TR

Q

S

10 Past Year SPM Questions

[Ans:(a) i: ii: (b) 2: 1]

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