1. The document contains solutions and marking schemes for questions in Module 3 Paper 1 of the 2015 Additional Mathematics JPNK examination. 2. It provides the full solutions or workings for many questions, along with brief answers for others, and outlines the marks awarded for different parts of solutions. 3. The questions cover a range of Additional Mathematics topics, including algebra, calculus, trigonometry, matrices, probability and statistics.

1. MODUL 3 [Kertas 1]: MATEMATIK TAMBAHAN JPNK 2015 Muka Surat: 1
JAWAPAN MODUL 3 KERTAS 1
No.
Soalan
Penyelesaian dan Skima Pemarkahan
Sub
Marks
Total
Marks
1 (a)
(b)
{2,4}
( )
3
x
f x =
1
1
2
2 (a)
(b)
4x-5
3
1
1
2
3
2
3 , 4 ( )
2: 3 4
1: ( 2)
a b both
B a or b
B a x b
= =
= =
− −
3
4
2
2
1
m , 5 ( )
3
1
B3: m 5
3
2
2: 3 5 2 0 2 2
3 3
1: 3( 2) ( 2) 2 ( 2) and ( )( 2)
n both
or n
n
B m m or m and m
B n or HTP m TDP m
= =
= =
+ − = − = − − = −
− + − − = + − = −
4
5
2
1 1
( )
3 2
2: (2 1)(3 1) 0
B1: 6x 1 0
f x x
B x x or
x
= − ≤ ≤ −
− + ≤
− − ≤
3
6 (a)
(b)
4
[ ] [ ]
15 5
400
15 5
2: 2(2) (15 1)(4) 2(2) (5 1)(4)
2 2
15 5
B1: [2(2) (15 1)(4)] [2(2) (5 1)(4)]
2 2
B
S S or or
+ − − + −
− + − + −
1
3
4
7 9, 3 ; 18, 12 ( )
B3: 9, 3 18, 12
B2: 27 x y 21
27
1: 27 x y 21
27
x y x y both
x y or x y
x y
and
x
x y
B or
x
= = = =
= = = =
+ + = =
+ + = =
4
1/2-1/3

2. MODUL 3 [Kertas 1]: MATEMATIK TAMBAHAN JPNK 2015 Muka Surat: 2
8 (a)
(b)
0.003,k 0.0003 ( )h both= =
12
0.03
1: 0.1 atau 0.08
1
p
B r
r
=
= +
−
1
2
3
9
B2: or equivalent
B1: ( changing to base m)
3
10
2
25
1: 5 2 @ (2 )x x
B =
2
11
8
3
3
2: 2 2
B1: (2 )(2 )
x
x
B = 3
12
B1: 2y
hx k
x
= +
B2: 2 2
5 (1) atau 1 (4)h k h k− = + = + h = 2 or k = -7
B3: h = 2 atau k = -7
B1: h = 2 dan k = -7 (kedua-duanya)
3
13 2 2
2 2 2 2
2 2 2 2
4 4 11 3 2 0
2: ( ( 2)) (y 3) 3 ( 1) (y 0)
1: ( ( 2)) (y 3) ( 1) (y 0)
x y x y
B x x
B x or x
+ − + − =
− − + − = − + −
− − + − − + −
3

3. MODUL 3 [Kertas 1]: MATEMATIK TAMBAHAN JPNK 2015 Muka Surat: 3
14 (a)
(b)
2
2
4
2
4
1: 2
2 2
k k
k k
B
−
 − 
     
2
1
1: 1 2sin
2
k
k
B θ
=
− =
2
2
4
15
2
33 41', 56 19', 213 41',236 19' (all)
3: 33 41' , 56 19'
2: (2tan 3)(3tan 2) 0
1: 6(1 tan ) 13tan 0
B
B x x
B x x
° ° ° °
° °
− − =
+ − =
3
16 2 , 4 ( )
2: 1, 2 atau ( 2)( 4) 0
1: 3 6 atau 4 (2 )
k k both
B k k
B k kλ λ
= = −
− − + =
= = +
3
17 (a)
(b)
3 5i j+
34
1
1
2
18
2 2
2 2
6.60
1 1
2: (7) (OP) (1.125)
2 2
1 1
1: (7) (OP) (1.125)
2 2
B
B or
π
π
=
3
19 (a)
(b)
0.0159
1
1: 0.1, (Perimeter 2 )
2
dr
B use r
dt
π
π
= × =
*
4.854
1: 5( 0.0159), (30 2 r)newB r r use π= + =
2
2
4

4. MODUL 3 [Kertas 1]: MATEMATIK TAMBAHAN JPNK 2015 Muka Surat: 4
20 49
atau 24.5
2
3: 7B x =
2: 7 0B x− =
1
1: ( )(14 )
2
B L x x= −
4
2
4
21 (a)
(b)
3t +
3, 4 ( )
1: 1 2 1 3
t t both
B t or t
= =
− = − =
1
2
3
22 (a)
(b) 2
33
1: 18 15 3 11
cm
B or+ ×
1
2
3
23 (a)
(b)
120
B1: 5! or
72
B1: 4! 2! or 48(seen)
2
2
4
24 (a)
(b)
8 8 0
8
0.05764
1: (0.7) (0.3)B C
8 0 8 8 1 7 8 0 8
0 1 0
8 1 7
1
0.9987
1: 1 (p) (q) (p) (q) (0.7) (0.3)
(0.7) (0.3)
B C C or C or
C
 − + 
2
2
4
x
2
0
X (3,11)
y

5. MODUL 3 [Kertas 1]: MATEMATIK TAMBAHAN JPNK 2015 Muka Surat: 5
25 r = 14.59
B2: = 1.53
B1: 1-0.937 or or 1.53 (seen)
3