Millburn Academy
Maths Department
National 5 Maths
Exam
Revision
ANSWERS

Q
1)
3
16
Paper 1 Revision
2)
3)
4)
5)
𝐴 𝑟 = √
4𝜋
2 x 3 + 6 x 2 – 14 x – 5 x 2 – 15 x + 35 = 2 x 3 + x 2 – 29 x + 35
( a ) 2 g + 5 s = 125
( b ) 4 g + 3 s = 145
( c ) Gold costs £25 and Silver costs £15 𝑥 =
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎 𝑥 =
2 ± √(−2) 2 − 4 × 2 × (−1)
2 × 2 𝑥 =
2 ± √4 + 8
4
6) 𝑥 =
2 ± √12
4 𝑥 =
2 ± √4 × √3
4 𝑥 =
2 ± 2√3
4 𝑥 =
2(1 ± √3)
4 𝑥 =
1 ± √3
2
( a ) 64
( b ) n = -2

7) ( a ) A(-1,0), B(3,0)
( b ) x = 1
8)
9)
10)
(3 x
– 2)(2 x 2 𝑚 = (𝑘𝐿) 2
+ x + 5) = 6
12 2 = QR 2 + 10 2
144 = QR 2 + 100
QR 2 = 44
QR = √44
QR = √4√11
QR = 2√11 x 3 +3 x 2 +15 x – 4 x 2 – 2 x – 10 = 6 x 3 – x 2 + 13 x – 10
11)
12)
13)
14)
2 m 2 – 18 = 2( m 2 – 9) = 2( m + 3)( m – 3)
3𝑥 + 1 = 𝑥−5
2
6𝑥 + 2 = 𝑥 − 5
5𝑥 + 2 = −5
5𝑥 = −7 𝑥 = −
7
5
( a ) x = 2
( b ) y = 5 + 4 × 2 – 2 2 = 5 + 8 – 4 = 9
( a ) 2 a + 4 c = 56
( b ) a + 3 c = 36
( c ) (i) Child’s ticket costs £8
(ii) Adult’s ticket costs £12
15) ( a ) 2 a
–3
.
( b ) √𝑥 + √18 = 4√2

16)
17)
√𝑥 + √9 × √2 = 4√2
√𝑥 + 3√2 = 4√2
√𝑥 = 4√2 − 3√2
√𝑥 = √2 𝑥 = 2
( a ) 𝑦 = −
3
2
3
( b ) 8 = − 𝑥 + 12 𝑥 + 12
2
3
−4 = −
2 𝑥
3
2 𝑥 = 4
3𝑥 = 8 𝑥 = 8 3
(3 x + 1)( x 2 – 5 x + 4) = 3 x 3 – 15 x 2 + 12 x
– x 2 – 5 x + 4 = 3 x 3 – 16 x 2 + 7 x + 4
Q Paper 2 Revision Marks

1) 16days = 16 × 24 hours = 16 × 24 x 60 mins = 16 × 24 x 60 x 60 secs
= 1 382 400 secs
S = D/T = 3000/1 382 400 = 4·3 x 10 -3 m/s
2) 𝑥 =
−7 ± √7 2 − 4 × 2 × (−3)
2 × 2 𝑥 =
−7 ± √73
4 𝑥 = 0 · 8 𝑥 = −3 · 9
3) ( a ) 𝐴 =
1
(22 + 32) × 20 = 540 𝑐𝑚 2
2
( b ) 𝑉 = 𝐴ℎ = 540 × 60 = 32 400 𝑐𝑚 3
4) 72% = 1296
1% = 1296/72 = 18
28% = 28 × 18 = 504 failed their test.
5)
𝐴 =
1
2
1
9 =
2
9 =
1
2 𝑎𝑏 sin 𝐶 𝑥 × 𝑥 sin 30
× 𝑥 2 ×
1
2
18 = 𝑥 2 ×
1
2
36 = 𝑥 2 𝑥 = 6𝑐𝑚
6) 19 2 = r 2 + 18·2 2
361 = r 2 + 331·24 r 2 = 29·76 r = 29·76 r = 5·5cm
4
3
3
4
3
1
3

C = π × 11 = 34·6cm
7) 1st Feb – 0·93 × 94 = 87·42kg
8)
1st Mar – 0·93× 87·42 = 81·3kg
1st Apr – 0·93 × 81·3 = 75·61kg
1st May – 0·93 × 75·61= 70·32kg
He will reach his target weight in April.
𝑥
36 ∙ 7 =
360
× 𝜋 × 100 𝑥 = 42 · 1°
9) 𝑇𝐺 sin 32
=
46 sin 25
TG = 57·7m sin 57 =
𝑇𝐵
57 · 7
𝑇𝐵 = 48 · 4𝑚
10) ( a ) 30 2 = b 2 + 27·5 2
b 2 =143·75
b = 12cm
d = 30 – 12 = 18cm
( b ) 60 – 18 = 42cm
11) 2·69 × 1·04 3 = 3·03 million
12) 1 + sin x = 1·7 sin x = 0·7 sin -1 (0.7) = 44·4° x = 44·4°, 180° - 44·4° x = 44·4°, 135·6°
13) 𝑎 2 = 𝑏 2
(2𝑥) 2
+ 𝑐
= 6 2
2 − 2𝑏𝑐 cos 𝐴
+ 𝑥 2 − 2 × 6 × 𝑥 × 0 · 5
4𝑥 2
3𝑥 2
= 36 + 𝑥 2 − 6𝑥
+ 6𝑥 − 36 = 0 𝑥 2 + 2𝑥 − 12 = 0
14) A = lb = 20 × 28 = 560cm 2
4
3
4
4
4
1
4
4
3

15)
A = ½ bh = ½ × 20 × 4.5 = 45cm 2
Total Area = 560 + 45 = 605cm 2
V = Ah = 605 × 9 = 5445cm 3 𝑥
= 5445ml = 5·445l = 5litres
𝐴𝑟𝑐 = × 𝜋𝑑
360
65
𝐴𝑟𝑐 = × 𝜋 × 4 · 6
360
𝐴𝑟𝑐 = 2 · 6𝑚
16) 104% - 894·40
1% - 894·40/104 – 8·60
17)
100% - 8·60 × 100 = £860 sin 59 = 𝑥
8 x = 8 × sin 59 = 6·9 sin 22 =
6 ∙ 9 𝑑 d =
6∙9 sin 22
= 18·4m
18) Scale Factor = 4 / 0·8 = 5
Area Scale Factor = 5 2 = 25
Area of Larger Bead = 0·6 × 25 = 15 cm 2
19) Area of triangle = ½ a b sin C
= ½ × 40 × 40 × sin 110
= 751·8cm 2
Area of Paving Stone = 751·8 × 2 = 1503·6cm 2
20) a) m = (6·60 – 1·80) / (4 – 0) = 4·80 / 4 = 1·20
c = 1·80
y = 1·20 x + 1·80
f = 1·20 d + 1·80 b) f = 1·20d + 1·80
= 1·20 × 7 + 1·80 = 10·20
Fare = £10·20
21) a) AC 2 = 4 2 + 6.2
2
3
3
4
3
4
3
2
2

AC 2 = 54·44
AC = 7·4m b) cos 𝐴 =
5
2
+7
2
− 7∙4
2
2×5×7 cos 𝐴 = 0 ∙ 275
A = 74°
22) a) f ( x ) = 3 sin x ° f (270) = 3 sin 270 = -3 b) f ( t
) = 0·6.
3 sin t = 0·6
sin t = 0·2
sin -1 (0.2) = 11·5 t = 11·5, 180 – 11·5 t = 11·5°, 168·5°
23) a) 2 x × 3 = ( x 2 + 5)
6 x = x 2 + 5
0 = x 2 + 5 – 6 x
x 2 – 6 x + 5 = 0 b) ( x – 1)( x – 5) = 0 x – 1 = 0 or x – 5 = 0 x = 1 or x = 5 x cannot equal 1 as this would make PR the shortest side of the triangle therefore x = 5.
24) 1.1
4 × 28 = 40.9948
In the 4 th week she will run 41 miles
25) 𝑥 =
−3 ± √(3 2 − 4 × 2 × (−7))
2 × 2 𝑥 =
−3 ± √65
4 𝑥 = 1 ∙ 3 𝑜𝑟 − 2 ∙ 8
26) 84% - £3780
4
3
4
4
1
3
2
3

27)
1% - £45
100% - £4500
42
𝐴𝑟𝑐 = × 𝜋 × 2 ∙ 4 = 0 ∙ 88𝑚
360
The arc is less than 0·9m therefore the staircase will not pass safety regulations.
28) a ) 108° b ) AC 2 = 1 2 + 1 2 – 2 × 1 × 1 × cos 108
AC 2 = 2.6
AC = 1.6cm
29) r 2 = (r – 5) 2 + 9 2 r 2 = r 2 – 10r + 25 + 81
10r = 106 r = 10·6cm
30) a ) A = ½ × 24 × (32 + 68)
A = 1200cm 2 b ) V = Ah
156000 = 1200 × l
l = 130cm
31) A of paper = 3000
A of paper for 1 wrap = 1000
A of paper = C × l
1000 = C × 70
C = 14.29
C = πd
14.29 = π × d d = 4.55cm
32) a ) (90,1) b ) 4 sin x – 3 = 0
sin x = ¾ x = 48·6, 180 – 48·6 x = 48·6°, 131·4°
33) h = 48 + 8 t – t 2
4
1
4
4
3
2
3
4
1
3

0 = 48 + 8 t – t 2 t 2 – 8 t + 48 = 0
( t – 12)( t + 4) = 0 t – 12 = 0 or t + 4 = 0 t = 12 or t = -4
The flare entered the sea after 12 seconds.