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Additional Mathematics Form 5 Answers

Question Bank

Form 4

Chapter 1 Functions

1 1. (a) a = –1, b = — 4

(b) (i) 2 (ii) 2

2. (a) 2 (b) 8

3. (a) g(x) = x 2 (b) 49

4. 2(x – 1)————

x – 2 2 – x 1 – x 2 5. (a) ——— (b) ——— 3 3

6. (a) (i) {a, c, d} (ii) Many-to-one relation

(b) {(1, c), (2, a), (3, a), (4, d)}

Chapter 2 Quadratic Equations

1. p , –1 or p . 2

2. p = 10, x = 3 or 7 49 3. (a) p = 6, q = –20 (b) – —– 2 4. p . 5

5. (a) α is a root of the equation,∴ α2 – 2α – 4 = 0 α2 = 2α + 4 α3 = 2α2 + 4α = 2(2α + 4) + 4α = 8(α + 1)

(b) α2 – 2α– 4 = 0 α(α – 2) = 4

α– 2———2

= 2—α

6. m = –1, n = 5 or m = 2, n = 2x 2 – 3x – 18 = 0

Chapter 3 Quadratic Functions

1. (a) –7 < x < 3(b) x , –2, x . 2

2. (a) a = 4, b = 2 (b) y = 9, x = –2(c) x = –2

1 3. (a) p = —, r = 1, q = –2 2

1(b) y = – —(x – 2)2 – 1 2

4. 4 < x < 6

5. (a) p = 8, q = –4(b) 16, 32

6. (a) p , – 1—2

(b) x + y = q y = q – x x 2 – 2x + 2y 2 = 3 x 2 – 2x + 2(q – x)2 = 3 3x 2 – (2 + 4q)x + 2q 2 – 3 = 0 b 2 – 4ac = 0 (2 + 4q)2 – 4(3)(2q 2 – 3) = 0 (1 + 4q + 4q 2) – (6q 2 – 9) = 0 –2q 2 + 4q + 10 = 0 q 2 = 2q + 5

Chapter 4 Simultaneous Equations

1. x = 13, y = 4 or x = – 4, y = –13

2. x = 5, y = 7 or x = 7, y = 5

3. x = 1—2

, y = 2 or x = –2, y = – 1—2

4. (7, 8), (–8, –7)

x + (4 + x) 5. ————— (y) = 18 2 xy + 2y = 18 x + y + (4 + x) + 5 = 20 2x + y = 11 1 x = – —, y = 12 or x = 4, y = 3 2

6. 30 cm2

Chapter 5 Indices and Logarithms

1. 4

2. 4 log10 3 + 4—5

log10 2

3. (a) 3.096 (b) 0.834 4. x = –7 or 3 5. x = –1, y = 1 6. (a) x = 1 or –2

(b) x = 3 or –2

Chapter 6 Coordinate Geometry

1. 2y = –x + 19, 4y = –x + 7, P(31, –6) 1 2. y = – —x + 9, 8 : 1 2 3. (a) –2

(b) y = –2x + 5(c) A(1, 3), B(5, 5), C(3, –1)(d) 10 unit2

4. (a) 7 (b) 5y = 2x + 10(c) y 2 + x 2 – 4y – 25 = 0

5. (a) A(0, 2 – 3m), B(3 + 2m, 0)(b) ±2

Answers

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6. (a) (i) B1 4—3

, 4—3 2

(ii) 2 unit2

(b) h = 4(k – 1)

(c) x 2 – 8y + 16 = 0

Chapter 7 Statistics

1. (a) k = 5, m = 9 (b) 30.29

2. (a)

30

Time (hours)

Num

ber

of s

tude

nts

20

10

4.5 14.5 24.5 34.5 44.5 54.50

Mode = 30.0 hours(b) 30.8 hours

3. (a)

120

40

60

80

100

20

10 20 30 400

Volume (Litres)

Cum

ulat

ive

frequ

ency

(b) (i) 20.0 (ii) 12

4. (a) 10(b) 6

5. (a) 63.1 mm(b)

20

30

40

50

10

34.524.5 44.5 54.5 64.5 74.5

43

84.5 94.50

Length (mm)

Cum

ulat

ive

freq

uenc

y

14%

4x + 3 6. (a) ———– 2

(b) 2(c) 5.07

Chapter 8 Circular Measure

1. If ∠ODB = 90°,

then DB—–OB

= sin �—6

rad.

DB = 141 1—2 2

= 7 cm

= Radius of sector CBD

25.66 cm2

1 2� � 2. ACB = —1—–2 = — 2 3 3 AC = 2 × r cos 30° 2r3 = ——— 2 = 3 r

�Length of arc APB = AC × — 3 3 = —— �r 3 2�Length of arc AQB = r × —– 3 2 3Total length = —�r + ——�r 3 3 � = —r (2 + 3 ) 3

3. (a) 4 cm (b) 9.26 cm (c) 2.9 cm2

4. (a) 2.35 rad. (b) 11.75 cm (c) 30.63 cm2

5. (a) 0.58 rad. (b) 6.13 cm2

6. Shaded area= Area of ∆OAC – Area of sector OAB 1 1= —r × r tan θ – —r 2θ 2 2 1= —r 2(tan θ – θ) 2

Chapter 9 Differentiation

1. (a) 4 (b) 1

2. (a) 2(3x – 2)(6x 2 – 2x – 3)

(b) 2(6 – x)————(x + 1)3

(c) 2—x 3

(d) –10(4x + 1)——————3(2x 2 + x)3

3. 1, y = –x + 1

�— rad. 6



4. –0.24 y 5 – y 5. (a) ——— = ——— 12 – x x xy = 60 – 12y – 5x + xy 60 – 5x y = ———— 12

x(60 – 5x) (b) ————— 12

(c) x = 6, y = 2 1—2

r 18 – h 6. (a) —– = ——— 12 18 3(12 – r) h = ———— 2

(b) 8

Chapter 10 Solution of Triangles

1. (a) 62°5 (b) 6.26 cm(c) 145.8 cm2 (d) 13.25 cm

2. (a) 8.38 cm (b) 10.3 cm(c) 34°23 (d) 25.88 cm2

3. (a) 1.63 m (b) 35°8(c) 84°53

4. (a) 5.13 cm (b) 28°16(c) 20.84 cm2

5. (a) 7.81 cm (b) 34°56(c) 75°15

6. (a) 10.88 cm (b) 72°25

Chapter 11 Index Number

1. x = 179, y = 140, z = 288

2. x = 1, y = 2(a) RM0.65 (b) RM10.45

3. 115, 120, 106, 113

4. (a) 120 (b) RM1080

5. x = 130, y = 3

6. (a) 129 (b) RM806.25

Form 5

Chapter 1 Progressions

1. (a) a = 3, d = 2 (b) 12(c) 25

2. 3 cm

3. (a) 3—2

, 3 (b) –1, 2

(c) 1—4

4. (a) 9 11—16

(b) 5

(c) 5—16

(d) 1—2

5. 1—2

6. (a) Areas of squares are

x 2, 1 x—2

22, 1 x—

4 22, …

\ x 2, x 2—4

, x 2—16

, … forms a geometric

progression with r = 1—4

.

(b) 4—3

x 2 cm2

Chapter 2 Linear Law

1. a = 2, b = – 1—2

2.

4

6

8

10

2

1

5 10 15 20 250

x + y Graph (x + y) against x 2

x2

a = 2.5, b = 1

3. p = 2, q = –4

4. (a) yx = ax + b

(b)

10

–5

20

30

40

01 2 3 4 5

x

Graph against xy x√ y x√

a = 7.80, b = –5

➤ Straight line Y = mX + c.



1 b 1 1 5. — = —1—2 + — y a x a

0.4

0.2

0.6

0.8

1.0

0.16

0.1 0.2 0.3 0.4 0.5 0.60

Graph against1y––

1y––

1x––

1x––

a = 6.25, b = 8.5

1 6. (a) log10 y = (–b log10 3)1—2 + log10 A x

(b)

1.0

1.5

2.01.75

0.5

0.02 0.04 0.06 0.080

log10 y

1x

––

Graph log10 y against 1x—

(c) A = 56.2, b = 31.44

Chapter 3 Integration

3 1. a = —, n = –1 2

2. (a) 11 (b) 14

1 1 1 3. (a) A1—, —2 (b) —– unit2

2 2 12 3(c) —–� unit3

80 1 4. (a) A(2, 3), B(5, 0) (b) 11— unit2

6

2 1 5. (a) k = –2, y = 5 — (b) –4 — 3 2

3 2 6. (a) A(1, 0), C12, —2 (b) 3—� unit3

2 3

7. (a) y = –x + 2, B(2, 0) (b) �—3

unit3

Chapter 4 Vectors

9 1. (a) 1 2 (b) 9i + 11j 11 ~ ~

2. (a) – 4i~ + j~

(b) 313 units

(c) m = 4, n = 7 (d) 1——W34

(5i~ + 3j~

)

→ 3. PS = 2a + b ~ ~→

XY = ma + nb ~ ~ → →Since XY // PS, then n 1 — = — m 2 m : n = 2 : 1

→ → → 4. (a) OC = 2i + 2j, OD = 4i + 5j, OE = 5i + 7j ~ ~ ~ ~ ~ ~

4 2(b) m = —, n = —, 4 : 1 5 5

1 1 2 5. (a) (i) —pa + —pb (ii) —qa + (1 – q)b 2 ~ 2 ~ 3 ~ ~

4 3(b) p = —, q = — 5 5(c) 4 : 1, 3 : 2

6. (a) A→C = –5i~ + 5j

~ ; B

→C = – 6i~ + 2j

~

(b) 5(c) 1 : 1

Chapter 5 Trigonometric Functions

1. (a) – 1—–3

(b) – 2—–3

(c) – 1—2

(d) –2 (e) 1

2.

1

0

2

90° 180° 270° 360°

y = 2

y

x

y = 2 – cos x

k = 2

3. (a) – 1 – p 2

(b) 1———–1 – 2p2

(c) p

4. LHS = (cosec x + sec x)(cosec x – sec x) = cosec2 x – sec2 x 1 1 = ——— – ——— sin2 x cos2 x

cos2 x – sin2 x = ——————— sin2 x cos2 x 4 cos 2x = ——————— 4 sin2 x cos2 x 4 cos 2x = ———— sin2 2x = 4 cot 2x cosec 2x = RHS



sin 2θ 5. (a) LHS = —————– 1 + cos 2θ 2 sin θ cos θ = ————————– 1 + (2 cos2 θ – 1) = tan θ = RHS

(b) (i) x = 45°, 75°58, 225°, 255°58 (ii) x = 180°, 80°24, 279°36

6. (a) x = 19°28, 30°, 150°, 160°32

(b) (i) 5—–3

(ii) 5—6

1 (iii) —– 5

Chapter 6 Permutations and Combinations

1. 120

2. (a) 2880 (b) 1152

3. (a) 1440 (b) 720

4. (a) 10 (b) 10

5. (a) 31 (b) 35

6. (a) 84 (b) 1260

Chapter 7 Probability

1. (a) 3—5

(b) 3—5

(c) 9—20

2. (a) 3—5

(b) 7—10

9 3. —– 25

4. (a) 5—12

(b) 1—84

(c) 1—84

5. (a) 1—14

(b) 1—14

6. x = 4, y = 3

Chapter 8 Probability Distributions

1. (a) 0.2605 (b) 0.3953 (c) 0.8652

2. (a) 0.1536 (b) 0.0256 (c) 0.9744

3. µ = 3, σ = 1.64

4. (a) 0.7692 (b) 0.484

5. (a) 0.1360 (b) 0.6977

6. µ = 53.9, σ = 16.48

Chapter 9 Motion Along a Straight Line

1. (a) v = 2 m s–1, a = –2 m s–2

(b) 1.5 s(c) 2.25 m

2. (a) t = 3 s, s = –69 m (b) –36.75 m s–1

3. (a) –1.125 m s–1

(b)

20

v

t

23–

(c) 1 7—12

m

4. (a) 4 seconds (b) 38 m(c) 11 m s–1

5. (a) 80 m (b) 50 m s–1

(c) 4.5 seconds, 78.75 m from O

6. (a) v = 2t 2 + 4t – 6 (b)

0–6

v

t12

(c) 8 m

Chapter 10 Linear Programming

5 3 1. x > 3, y . – —x + 5, y , —x + 3 4 8

2. (a) x + 3y > 10 (b) x < y(c) y – x > 5

3. (a) x + 2y < 3000, 2x + y > 2400, y – x < 300(b)

1000750

1500

2000

2500

500

500 1000 1500 2000 2500 30000

y

x

R

y – x = 300

2x + y = 2400

x + 2y = 3000

3.2x + 2y = c

(3000, 0)

(c) (i) RM9600(ii) 750



4. (a) x + 2y 12, x + 4y 20, 5x + 2y 40(b)

10

15

20

(7, 2.5)5

5 10 15 200

y

xR

5x + 2y = 405x + 2.5y = c

x + 4y = 20

x + 2y = 12

(c) Maximumprofit =RM41.25

5. (a) 12x + 13y 2000, 20x + 17y 500, 2x y(b)

100

150

(167, 0)

200

50

50 100 150 2000

y

x

12x + 13y = 2000

y = 2x

80x + 40y = c

20x + 17y = 500R

(c) Optimum function=80x + 40y Total profit =RM13360

6. (a) 2x + y 600, 2x + 3y 900, x + y 350(b)

200

300

(250, 100)

(175, 175)

400

500

600

100

100 200 300 400 5000

y

x

2x + 3y = 900

2x + y = 600

2x + 1.5y = c

x + y = 350

R

(c) (i) 175 (ii) RM650

additional mathematics answers - pelangi books ace add... · ©penerbitan pelangi sdn. bhd. 2...

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