VECTORSFURTHER PRACTICE WITH PAST YEAR SPM QUESTIONS – PAPER 1 1. SPM 2003 P1 Q12 2. SPM2003 P1Q13 p = 2a +3 b q=4a–b r = ha + (h-k) b, where h and k are constants Use the information given to find the values of h and k when r = 3p -2q [3marks] uuur uuur Diagram shows two vectors, OP and QO . Express uuur x y uuur (b) OQ in the form xi yj % % (a) OP in the form [2marks] 5 , -8i+4j) 3 ( 3. SPM 2004 P1 Q16 Given that O(0,0) A(-3,4) and B(2,16), find in terms of unit vectors, i and j , uuur % ( h = –2 , k = –13) 4. % (a) AB uuur (b) the unit vector in the direction of AB [4marks] 5 AB 2BC 10i 12 j % % [4 marks] 1 5 (b) (a) 13 12 12 Vectors SPM 2004 P1 Q17 Given that A(-2,6), B(4,2) and C(m,p), find the of uuurvalueuu urm and of p such that m = 6 , p = –2 15 (b) the ration of PQ : QR . SPM 2005 P1 Q 16 uuur Diagram shows vector OA drawn on a Cartesian plane. uuur on a Cartesianuu ur It is given that uuur x (a) Express OA in the form y OP 6i 4 j and PQ 4i 5 j % % uuur% % Find PR [3marks] (b) Find uuur the unit vector in the direction of [2marks] OA 12 1 12 (b) 13 5 5 (a) -10i+j 8. Express uuur x . OA and AB . y (a) OA in the form uuur (b) AB in the form xi yj . It is given uuur P. % % % % where k is a constant. [2marks] % % 4 3 (a) . drawn plane. SPM 2005 P1Q15 6. Diagram shows a parallelogram. OPQR. [4 marks] 7. (b) -4i-8j Vectors 16 (a) 5 2 (b) 4 : 3 .Q.5. and R are u uur that PQ 4a 2b and QR 3a (1 k )b . SPM 2006 P1 Q 14 The point collinear. SPM 2006 P1 Q 13 uuur uuur Diagram shows two vectors. Find (a) the value of k . SPM 2008 P1 Q15 27 4 12. It is given that (h + 3) a = (k – 5) b . [3 marks] (b) the unit vector in the direction of 2a b [4 marks] 5 1 5 (b) (a) 13 12 12 11. b 8 4 Find (a) the vector 2a b . Express in terms of a and b .9. SPM 2007 P1 Q15 The following information refers to the vectors a and b . 2 1 a . Diagram shows a rectangle OABC and the point D lies on the straight line OB. It is given that OD=3DB. SPM 2008 P1 Q16 Diagram below shows a triangle PQR. in terms of x and y . [4 marks] (a) 4a – 6b (b) 3a + 3 2 b (a) –3 (b) 5 Vectors x 154 y 17 . where h and k are constants. (a) QR . (b) k . SPM 2007 P1 Q16 10. [2 marks] P 4a R The point T lies on QR such that QT : TR = 3 : 1. Q 6b T The vectors a and b are non-zero and nonparallel. uuur Express OD . (b) PT . Find the value of (a) h . find BD [2 marks] % % (ans : -20x+32y. 104 ) Vectors 18 . AE 8 y . x and y % uuur uuur uuur % (ii) Given that RQ k OQ . AED and EFC are straight lines. k=1/3. 1 3 25 ) . 5 4 7 2.F and D are collinear [3marks] uuur (C) if x 2 and y 3 . k(9x/2 +3y/2.OP 6x and OA 2 y % 3 4 % (a) Express in terms of x and y uuur uuur (i) AP (ii ) OQ [4marks] uuur uuur uuur (b) (i) Given that AR h AP . SPM 2004 P2 Q 8 Diagram shows triangle OAB. [2marks] [4marks] (ans : -2y+6x. find the value of h and of k. It is given that OP= uuur uuur 1 1 OB . h=1/2) 3. h(6x-2y). The straight line AP intersects the straight line OQ at R.FURTHER PRACTICE WITH PAST YEAR SPM QUESTIONS – PAPER 2 1. 25x.-4). DC 25x 24 y . AQ AB . state RQ in terms of k. find 5 (a) the coordinate of A uuur (b) the unit vector in the direction of OA uuur uuur (c) the value of k. SPM 2003 P2 Q 6 uuur 5 uuu r Given that AB . state AR in terms of h. 3y/2 +9x/2. x and y % uuur uuur % (c) Using AR and RQ from (b) . AE AD and % % 4 % % 3 EF EC 5 (a) Express in terms of x) and y uuur uuur % (i) BD (ii ) EC [3marks] (b) Show that the points B. if CD is parallel to AB [2 marks] [2marks] [2marks] (ans:( -3. SPM 2005 P2 Q 6 Diagram shows a quadrilateral ABCD. Ob 7 uuur 2 and CD 3 k . It is given that uuur uuur uuuu r 1 AB 20x . 4. x and y % % (ii) Hence. where h and k are constants. [5marks] uuur (c ) Given that x 2units . y =3 units and AOB =90 . It is given that uuur uuur OA: OP = 4:1. AE AD and % 3 % uuur 5 uuuu r BC AD . if the points A. find AB [2marks] % % [(a)(i) Vectors BP 2 x 6 y (ii) OQ 4 x 3 y 19 (b) h 25 . find the value of m [5marks] (ans : 5x+2y. OB 6 y % % (a) Express in terms of x and y : % uuur % (i) BP uuur (ii)OQ [3marks] uuur uuur uuur uuur (b) Using OS hOQ and BS k BP . in terms of x and y [2marks] % % uuur uuur (b) Point F lies inside the trapezium ABCD such that 2 EF m AB . 6 uuuur (a) Express AC . OA 8x . SPM 2007 P2Q8 Diagram shows triangle AOB. The straight line BP intersects the straight line OQ at the point S.08 unit] . AB : AQ = 2 : 1. The point P lies on OA and the point Q lies on AB. 4x+my. and m is a constant. in terms of m. AD 6x . SPM 2006 P2 Q 5 Diagram shows a trapezium ABCD. uuur uuuu r uuur 2 uuuu r It is given that AB 2 y .F and C are collinear. m=8/5] 5. uuur (i) Express AF . k 4 5 (c) 24. find the value of h and of k. where h and k are constants. y and [Answer : (a)(i) Vectors [3 AR h AC . k 43 ] . The diagonals BD and AC intersect at point R. find the [4 marks] DB x 3 y 20 (ii) AR 23 x y (b) h 1 2 . Point P lies on AD. . ABCD is a quadrilateral. (a) Express in terms of and : (i) (ii) marks] B Given that DC k x value of h and of k. and . SPM 2008 P2Q6 In the diagram.6. D C R A (b) It is given that .