SPM Additional Mathematics 2019, Paper 2 (Question 6)


Question 6:
Solution by scale drawing is not accepted.
Diagram 3 shows the positions of jetty O and kelongs K, L, R, S and T in the sea.

Diagram 3

Kelong
L is situated 400 m from jetty O and kelong R is situated 600 m from jetty O in the direction of OL. Kelong S is situated 300 m from jetty O and kelong T is situated 600 m from kelong 5 in the directions of OS. Kelongs L, K and T are situated on a straight line such that the distance of kelong K from kelong T is 5 times its distance from kelong L.

(a) By using to represent 100 m in the direction of OR and  to represent 150 m in the direction of OT, express in terms of p ˜  and  q ˜ .
[3 marks]
(i)  OK ,        (ii)  RK .

(b) If Joe uses a binocular to observe kelong
R from kelong S, determine whether kelong R can be seen without being blocked by kelong K or otherwise.
Prove you answer mathematically.     [5 marks]


Solution:
(a)
KT=5KL KT KL = 5 1 KT:KL=5:1 OL =4 p ˜ ,  LR =2 p ˜ ,  OR =6 p ˜ OS =2 q ˜ ,  ST =4 q ˜ ,  OT =6 q ˜


(a)(i)
OK = OL + LK      =4 p ˜ + 1 6 LT      =4 p ˜ + 1 6 ( 4 p ˜ +6 q ˜ )      =4 p ˜ 2 3 p ˜ + q ˜      = 10 3 p ˜ + q ˜

(a)(ii)
RK = RO + OK      =6 p ˜ + 10 3 p ˜ + q ˜      = 8 3 p ˜ + q ˜

(b)
SR = OR OS      =6 p ˜ 2 q ˜      =2( 3 p ˜ q ˜ ) KR = RK      =( 8 3 p ˜ + q ˜ )      = 8 3 p ˜ q ˜ SR m KR ,m is a constant.
S
, K and R are not collinear.
Hence, Kelong R can be seen without being blocked by Kelong K.

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