SPM Matematik Tambahan

1. Additional Mathematics Marking Scheme – Paper 1 Set 1
1 (a) one to one
(b) 𝑓: → 𝑥2
1
1
5 𝑎 = 15, 𝑏 = −4, 𝑐 = −4
15𝑥2
− 4𝑥 − 4 = 0
(3𝑥 − 2)(5𝑥 + 2) = 0
3
B2
B1
2
−
9
2
−6𝑥−3)
2𝑥+2
or
−3(−3)
−3+1
−3(2𝑥+1)
2𝑥+1+1
or -3 seen
3
B2
B1
6
a) 3𝑥2
− 7𝑥 − 2 = 0
b) 2.59, −0.26
−(−7)±√(−7)2
−4(3)(−2)
2(3)
𝑎 = 3, 𝑏 = −7, 𝑐 = −2
1
3
B2
B1
3
(a) 𝑘 = 3
(b)
3𝑥
𝑥−1
, 𝑥 ≠ 1
𝑥 =
3𝑦
𝑦 − 1
𝑦 =
𝑥
𝑥 − 3
1
3
B2
B1
7 (a)
(
3
4
, −
11
4
)
−4(𝑥 −
3
4
)2
−
11
4
−4[(𝑥 −
3
4
)2
−
9
16
+
5
4
]
(b) 𝑥 =
3
4
3
B2
B1
1
4
−
4
3
< 𝑚 < 4
(7𝑚 + 4)( 𝑚 − 4) < 0 & graf
(−3𝑚)2
− 4(2𝑚 + 1)(2𝑚 − 4) < 0
3
B2
B1
8
𝑥 ≤ −
1
3
, 𝑥 ≥
1
2
(3𝑥 + 1)(2𝑥 − 1) > 0 & graf
6𝑥2
− 𝑥 − 1 > 0 or
equivalent
3
B2
B1
9
4
3𝑥 = 4𝑥 − 4
3
B2
12
𝑚 = 9, 𝑘 = 3 (both)
𝑚 = 9 or 𝑘 = 3
4
B3

2. 23𝑥
= 24𝑥−4 B1 0.27
1 − 0.01
𝑎 = 0.27 , 𝑟 = 0.01
B2
B1
10
𝑦 =
𝑥2
4096
log4
𝑥2
𝑦
= 6
2log4 𝑥 − log4 𝑦 = 6
3
B2
B1
13 (a)
𝑢 = 1, 𝑣 =
1
2
(both)
𝑢 = 1 or 𝑣 =
1
2
−3𝑢 =
4(−4)+1(𝑢)
5
or
2 =
4𝑣 + 8
5
(b) (−
3
2
,
17
4
)
3
B2
B1
1
11
28
𝑥 = 5
3𝑥 − 2 − 8 = 18 − (3𝑥 − 2)
3
B2
B1
14 12 , 9k  
8 12 84k   
3
B2
20 1 11
4 2
y x  
3

3. ( 2 6 8) ( 24 2 2 ) 84k k        B1
21
4 2 8x  or 4y 
1
4 ( 6)
4
y x   
1 4m  or 2
1
4
m  
0.09y  
2
3(3) 6(3) 9 
2
3 6
dy
x x
dx
 
B2
B1
3
B2
B1
15 x = 146. 31o
, 326.31o
, 153.43o
,
333.43o
tan x =
2
3
 or tan x =
2
1

2cot2
x + 7 cot x + 6 = 0
2 ( 1 + cot2
x )
4
B3
B2
B1
16 18p , 3q
18p or 3q
186 2
 x
x
y
3
B2
B1
22
23
(a) 25.70cm
9 sin20.5OM  
(b) 2
19.69cm
21
(25.70) (0.7157 sin 41)
2

(a)
20
3
(b)
5
3
5
2
1
2
B1
2
B1
1
2
B1
17 (a) 5 3w u
% %
(b) 9 10u w
% %
15 9BA w u  
uuur
% %
1
1
B1
18 2 , 14h 
2 2 2
(8 ) 8 10h  
8
8
h 
 
 
3
B2
B1
24 240
2 5
1 4P P
2 5
1 4P or P
3
B2
B1
19 3
1
1
2 or
3
2
2
14
2




 x
x
3
B2
B1
25 0.8427
0.5+0.3427
( 0) 0.5p z  
3
B2
B1