SPM Additional Mathematics 2018, Paper 1 (Question 8 – 10)


Question 8 (3 marks):
Diagram 3 shows vectors OP ,  OQ  and  OM drawn on a square grid.

Diagram 3

( a ) Express  OM  in the form h p ˜ +k q ˜ where h and k are constants. ( b ) On the diagram 3, mark and label  the point N such that  MN + OQ =2 OP .


Solution:
(a)
OM = p ˜ +2 q ˜

(b)

MN + OQ =2 OP MN =2 OP OQ        =2 p ˜ q ˜


Question 9 (4 marks):
A( 2, 3 ) and B( 2, 5 ) lie on a  Cartesian plane. It is given that 3 OA =2 OB + OC . Find ( a ) the coordinates of C, ( b ) | AC |

Solution:
Given A( 2,3 ) and B( 2,5 ) Thus,  OA =2 i ˜ +3 j ˜  and  OB =2 i ˜ +5 j ˜

(a)

3 OA =2 OB + OC OC =3 OA 2 OB  =3( 2 i ˜ +3 j ˜ )2( 2 i ˜ +5 j ˜ )  =6 i ˜ +9 j ˜ +4 i ˜ 10 j ˜  =10 i ˜ j ˜ Thus, coordinate of C is ( 10, 1 )


(b)
AC = AO + OC  = OA + OC  =( 2 i ˜ +3 j ˜ )+10 i ˜ j ˜  =2 i ˜ +10 i ˜ 3 j ˜ j ˜  =8 i ˜ 4 j ˜ | AC |= 8 2 + ( 4 ) 2    = 80  units    = 16×5  units    =4 5  units


Question 10 (3 marks):
The following information refers to the equation of two straight lines, AB and CD.

   AB:y2kx3=0   CD: x 3h + y 4 =1 where h and k are constants.

Given the straight lines AB and CD are perpendicular to each other, express h in terms of k.

Solution:
AB:y2kx3=0 y=2kx+3 m AB =2k CD: x 3h + y 4 =1 m CD = 4 3h m AB × m CD =1 2k×( 4 3h )=1 8k=3h h= 8 3 k

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