N
12/26/2018
15° N
A
40°
70° E
E
35 °S
B
C
S
𝑦
1
𝑦 = cos 𝜃
90°
270°
180°
360°
-1
MATHEMATICS
PAPER 2
SUGGESTED SOLUTIONS
b (i)
Z
b (ii)
5.5 cm
T
7 cm
P (c)
38 °
X
9 cm
Y
FULL SOLUTIONS | 2018
QUESTION ONE
Question
Number
Solution
suggest
|83 𝑦5| = |41
1 (a) (i)
|
‒5
2
(40) - (3𝑦) = 8 ‒ ( ‒ 5)
40 ‒ 3𝑦 = 13
‒ 3𝑦 = 13 ‒ 40
‒ 3𝑦 = ‒ 27
𝑦=9
(a)(ii)
𝑩=
(𝟖𝟑 𝟗𝟓)
|𝑩| = 𝟏𝟑
𝑴
1 (b) (i)
‒𝟏
(
𝟏
𝟓
= 𝟏𝟑 ‒ 𝟑
‒𝟗
𝟖
)
P(both black) = P(ww )
𝟔
𝟓
= 𝟏𝟓 × 𝟏𝟒
𝟏
=𝟕
P(different colours) = P(WG or GW )
𝟔
𝟗
𝟗
𝟔
= 𝟏𝟓 × 𝟏𝟒 + 𝟏𝟓 × 𝟏𝟒
=
𝟓𝟒 + 𝟓𝟒
𝟐𝟏𝟎
𝟏𝟖
= 𝟑𝟓
2
By T.H Musondela
QUESTION TWO
Question
Number
Solution
2 (i)
𝟐
(𝐚)
𝒙=
𝒙=
𝟑𝒙 ‒ 𝒙 ‒ 𝟓 = 𝟎
‒ ( ‒ 𝟏) ± ( ‒ 𝟏)𝟐 ‒ 𝟒(𝟑)( ‒ 𝟓)
𝟐(𝟑)
𝟏 ± 𝟔𝟏
𝟔
𝒙 =1.47 or -1.14
𝐌𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐬
𝐂𝐡𝐞𝐦𝐢𝐬𝐭𝐫𝐲
𝟒
𝟎
𝟓
3
𝟐
0
𝟔
𝐏𝐡𝐲𝐬𝐢𝐜𝐬
or
M
P
C
4
5
3
2
6
(b) (ii)
(a) 6
(𝒃) 𝟓 + 𝟐 + 𝟎 = 𝟕
(c) 𝟐
3
By T.H Musondela
QUESTION THREE
Question
Number
3 (a)
Solution
Your Comments
∫𝟐 (𝟐 + 𝒙 ‒ 𝒙𝟐)𝒅𝒙
‒𝟏
=
[
𝟏 𝟐 𝟏 𝟑
𝟐𝒙 + 𝟐𝒙 ‒ 𝟑𝒙
[
𝟏
𝟐
𝟐
]
‒𝟏
𝟏
= 𝟐(𝟐) + 𝟐(𝟐) ‒ 𝟑(𝟐)
= (𝟒 + 𝟐 ‒
𝟑
] ‒ [𝟐( ‒ 𝟏) + 𝟏𝟐( ‒ 𝟏)𝟐 ‒ 𝟏𝟑( ‒ 𝟏)𝟑]
𝟖
𝟏
𝟏
)
‒
(
‒
𝟐
+
+
)
𝟑
𝟐
𝟑
=
𝟏𝟎
𝟕
‒
(
‒
)
𝟑
𝟔
=
𝟏𝟎
𝟕
+
𝟑
𝟔
𝟗
=𝟐
𝟏
= 4.5 or 𝟒𝟐
(a) ((ii)
(b) 𝒚 = 𝒙 + 𝟒𝒙
𝒅𝒚
=
𝒅𝒙
1-
‒𝟏
𝟒
𝟐
𝒙
𝟒
𝟑
𝒎 = 𝟏 ‒ 𝟏𝟔 = 𝟒
𝒎𝒎𝟐 =‒ 𝟏
𝟒
∴ 𝒎𝟐 =‒ 𝟑
𝒎𝒙 + 𝒄 = 𝒚 at (𝟒,𝟓)𝒙 = 𝟒 and y= 5
𝟒
‒ 𝟑(𝟒) + 𝒄 = 𝟓
‒ 𝟏𝟔 + 𝟑𝒄 = 𝟏𝟓
𝒄=
𝟑𝟏
𝟑
𝟒
∴ 𝒚 =‒ 𝟑 +
𝟑𝟏
𝟑
or 3y + 𝟒𝒚 = 𝟑𝟏
4
By T.H Musondela
QUESTION FOUR
Question
Number
Solution
Your Comments
b (i)
Z
b (ii)
5.5 cm
T
7 cm
P (c)
38 °
9 cm
X
Y
QUESTION FIVE
Question
Number
Solution
5 (a)
You Comments
𝒃‒𝒂
𝟐
𝟏(𝒃 ‒ 𝒂)
𝟐
𝒂 ‒𝒃
= (𝒂 ‒ 𝒃)(𝒂 + 𝒃)
‒ 𝟏(𝒂 ‒ 𝒃)
= (𝒂 ‒ 𝒃)(𝒂 + 𝒃)
‒𝟏
= 𝒂+𝒃
5(b)(i)
𝟐
(𝟐𝒌 ‒ 𝟏𝟓)(𝒌 + 𝟒) = 𝐤
𝟐
𝟐
𝟐𝒌 + 𝟖𝒌 ‒ 𝟏𝟓𝒌 ‒ 𝟔𝟎 = 𝒌
𝟐
𝒌 ‒ 𝟕𝒌 + 𝟔𝟎 = 𝟎
𝟐
𝒌 ‒ 𝟏𝟐𝒌 + 𝟓𝒌 ‒ 𝟔𝟎 = 𝟎
𝒌(𝒌 ‒ 𝟏𝟐) + 𝟓(𝒌 ‒ 𝟏𝟐) = 𝟎
(𝒌 ‒ 𝟏𝟐)(𝒌 + 𝟓) = 𝟎
𝒌 = 𝟏𝟐
5
By T.H Musondela
5(b) (ii)
𝑻𝟏 = 𝒌 + 𝟒 = 𝟏𝟐 + 𝟒 = 𝟏𝟔
𝑻𝟐 = 𝒌 = 𝟏𝟐
𝑻𝟑 = (𝟐𝒌 ‒ 𝟏𝟓) = 𝟐(𝟏𝟐) ‒ 𝟏𝟓 = 𝟗
∴ 16, 12, 9
5(b) (ii)
𝒂
𝑺∞ = 𝟏 ‒ 𝒓
𝑺∞ =
𝟏𝟔
𝟑
𝟏‒𝟒
= 64
QUESTION SIX
Q no
Solution
6 (i) (a)
(a)
𝟏
𝑨𝑬 = 𝟑 𝑨𝑪
You Comments
𝟏
= 𝟑(𝒂 + 𝟐𝒃)
(b) 𝑩𝑬 = 𝑩𝑨 + 𝑨𝑬
𝟏
𝟑
= ‒ 𝒂 + (𝒂 + 𝟐𝒃)
𝟏
= 𝟑 ( ‒ 𝟑𝒂 + 𝒂 + 𝟐𝒃)
𝟏
= 𝟑 ( ‒ 𝟐𝒂 + 𝟐𝒃)
𝟐
𝟐
= 𝟑 ( ‒ 𝒂 + 𝒃) or = 𝟑 (𝒃 ‒ 𝒂)
(c) 𝑩𝑫 = 𝑩𝑨 + 𝑨𝑫
= ‒𝒂+𝒃
𝟐
(b) 𝑩𝑬= 𝟑 ( ‒ 𝒂 + 𝒃) and 𝑩𝑫 =
‒𝒂+𝒃
𝟐
𝑩𝑬= 𝟑 𝑩𝑫 ∴ the points B, E and D are colinear
6
By T.H Musondela
6 (b)
Start
Enter x, y
M= sqrt (x^2+y^2)
Is M < 𝟎?
Yes
Error ‘M’ must be positive
no
Display M
Stop
7
By T.H Musondela
QUESTION SEVEN
Question
Number
7 (a) (i)
Solution
Comments
3
2
(a) 𝑦 = 2𝑥 ‒ 3𝑥 + 5
When 𝑥 =‒ 2,
3
2
𝑝 = 2( ‒ 2) ‒ 3( ‒ 2) + 5
𝑝 =‒ 23
7 (a) (ii)
7 (a) (iii)
𝑦=𝑥
X= -1.1
8
By T.H Musondela
(a) (1.5,5) and (2,7.3)
7 (a) (iv)
𝑦2 ‒ 𝑦1
M =𝑥
7.3 ‒ 5
= 2 ‒ 1.5
2 ‒ 𝑥1
=
2.25
0.5
= 4.5
7 (b)
3
2
‒
x+1 x‒1
=
3(x ‒ 1) ‒ 2(x + 1)
(x + 1)(x ‒ 1)
=
3x ‒ 3 ‒ 2𝑥 ‒ 2
(x + 1)(x ‒ 1)
x‒5
= (x + 1)(x ‒ 1)
QUESTION EIGHT
Question
Number
Solution
Comments
𝐾𝑅
8 (i)
80
(a) sin 60 = sin 52
𝐾𝑅 =
80sin 60
sin 52
= 87.92015
= 87.9°
(b) A=
1
(80)(50)𝑠𝑖𝑛 60
2
= 1732.0508
2
= 1700 𝑚 (3sf)
8 (ii)
𝟏
(iii) 𝟐 (sd)(KN) = Area of Δ KNR
𝐬𝐝 =
𝟐 × 3260
𝟖𝟎
= 81.5 (m)
9
By T.H Musondela
8 (B)
𝑦
1
𝑦 = cos 𝜃
90°
180°
270°
360°
𝑥
-1
8 (c)
3
12𝑑𝑛
3
15𝑐𝑑
3
÷
9𝑐 𝑛
2 2
10𝑐 𝑑
12𝑑𝑛𝑛𝑛
10𝑐𝑐𝑑𝑑
×
15𝑐𝑑𝑑𝑑
9𝑐𝑐𝑐𝑛
𝟐
=
𝟖𝒏
𝟐
𝟗𝒄
QUESTION NINE
Question
Number
8 (a)
Solution
Comments
𝒇
2
10
15
23
30
10
𝒙
15
25
35
45
55
65
∑𝒇 = 90
𝐌𝐞𝐚𝐧 (𝐗) =
∑𝑓𝑥
𝟐
𝒇𝒙
30
250
525
1035
1650
650
𝒇𝒙
450
6250
18375
46575
90750
42250
∑𝑓𝑥 = 4140
∑𝑓𝑥 = 204650
2
4140
∑𝑓 = 90
= 46
SD =
=
𝜮𝒇𝒙
∑𝒇
𝟐
- (𝐗)
𝟐𝟎𝟒𝟔𝟓𝟎
𝟗𝟎
𝟐
- (46)
𝟐
= 𝟏𝟓𝟕.𝟖𝟖𝟖𝟖𝟖𝟖𝟖
= 𝟏𝟐.𝟓𝟔𝟓𝟑𝟖
= 12.6
(3sf)
10
By T.H Musondela
(b)(i)
Marks(x )
≤10
≤ 20
≤ 30
≤ 40
≤ 50
≤ 60
≤ 70
CF
RCF
0
0
2
0.02
12
0.13
27
0.3
50
0.56
80
0.89
90
1
𝟏
0.𝟗
0.𝟖
0.𝟕
0.𝟔
0.𝟓
0.𝟒
0.𝟑
0. 𝟐
0. 𝟏
𝟎
𝟏𝟎
(iii)
65
× 1 = 0.65
100
𝟐𝟎
𝟑𝟎
𝟒𝟎
,
∴ 65 percentile 53
11
By T.H Musondela
𝟓𝟎
𝟔𝟎
𝟕𝟎
QUESTION 10
Question
Number
Comments
Solution
(ii) Un anticlockwise Rotation of 270 about (0,0)
10 (a)
Or
clockwise Rotation of 90 about (0,0)
10 (b)
it is enlargment, centre (0,0) and scale factor 2
10 (c)
This is a stretch in which y axis is invariant and k = -2
∴ the required matrix is
( ‒02 01)
Or
(
)( 21
𝑎 𝑏
𝑐 𝑑
4
1
)
=
( ‒14
‒8
1
)
Solving the simultaneous equation
𝑎 =‒ 2 , b = 0 , c = 0 and d = 1
∴ the required matrix is
10 (d)
(i)
(
)( 21
1 0
‒2 1
( ‒02 01)
2 4
4 1
)
(
2 2
= ‒3 0
∴ the coordinates are
(2,-3) ,
(2,0) ,
( 4,-7)
12
By T.H Musondela
4
‒7
)
QUESTION ELEVEN
Question
Number
11 (a)
Solution
Comments
}
𝑥 + 𝑦 ≤ 60
𝑥 + 𝑦 ≥ 35
𝑖𝑛 𝑎𝑛𝑦 𝑜𝑟𝑑𝑒𝑟
𝑦≥6
𝑦 ≤ 14
11 (b)
70
𝒚 ‒ 𝒂𝒙𝒊𝒔
60
50
𝒙 + 𝒚 = 𝟔𝟎
40
𝒙 + 𝒚 = 𝟑𝟓
30
20
𝒚 = 𝟏𝟒
Solution
set
10
𝒚=𝟔
0
11 (c)
10
20
30
40
(i) 10 teachers
(ii) 52 learners
d
K 30.00
13
By T.H Musondela
50
60
70
QUESTION 12
Question
Number
Solution
Comments
1
3 × 11.4 × [14 × 10 + 8 × 4 +
14 × 10 × 8 × 4 ]
1
(3 possible answers do exist here)
= 3 × 11.4 × [140 + 32 + 4480 ]
1
= 3 × 11.4 × [2389328 ]
= 907.94484
N
12 (b)
15° N
A
40°
70° E
E
C
B
35 °S
S
(i)
Differ in latitude = 15+35 = 50°
XZ =
=
50
× 2 × 3.142 × 6370
360
2,001,454
360
= 5,559.5944444 km
= 5,560 km (3sf)
(ii) (a)
𝜃
× 2 × 3.142 × 6370cos 35
360
= 900
900 × 360
𝜃= 2 × 3.142 × 6370cos 35
= 9.881088 (3sf)
= 9.9° (1dp)
(ii) (b)
Longitude of Q is (70-9.881088) E = 60.1
∴ Q (35 °S, 60.1 °E)
14
By T.H Musondela
This not a frustum. Me I don’t know
teacher