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Chapter 7

Powers, roots and laws of
indices
7
...
In this chapter, powers and
roots of numbers are explained, together with the laws
of indices
...


7
...
2
...
When written as 24 , 2 is
called the base and the 4 is called the index or power
...

Similarly, 35 is read as ‘three to the power of 5’
...
e
...

Thus,
42 is called ‘four squared’ rather than ‘4 to the power
of 2’ and
53 is called ‘five cubed’ rather than ‘5 to the power of 3’
When no index is shown, the power is 1
...

Problem 1
...
e
...
e
...
1016/B978-1-85617-697-2
...
e
...
e
...
Change the following to index form:
(a) 32 (b) 625
(a) (i) To express 32 in its lowest factors, 32 is initially
divided by the lowest prime number, i
...
2
...

(iii) 16 is also divisible by 2, i
...
16 = 2 × 8
...

(iv) 8 is also divisible by 2, i
...
8 = 2 × 4
...

(v) 4 is also divisible by 2, i
...
4 = 2 × 2
...

(vi) Thus, 32 = 25
...
e
...
The next prime number is 3 and 625
is not divisible by 3 either
...

(ii) 625 ÷ 5 = 125, i
...
625 = 5 × 125
...
e
...

Thus, 625 = 5 × 5 × 25
...
e
...
Thus,
625 = 5 × 5 × 5 × 5
...


48 Basic Engineering Mathematics
Problem 3
...


= 108

7
...
2 Square roots

1
...


27

When a number is multiplied by itself the product is
called a square
...

A square root is the reverse process; i
...
, the value of the
base which when multiplied by itself gives the number;
i
...
, the square
√ root of 9 is 3
...
Thus,
9 = 3
...

√
√
Because −3 × −3 = 9, 9 also equals √
−3
...
Simi√
larly, 16 = ±4 and 36 = ±6
...

√
1
9 2 ≡ 9 = ±3

3
...


24 × 32 × 2 ÷ 3

5
...


6
...


64 2

8
...


102 × 103
105

√
32 × 23 × 36
Problem 4
...
3

by cancelling

1

10
...


Laws of indices

There are six laws of indices
...
Evaluate
taking the
103
positive square root only

1

1

Hence,

22 × 23 = 25

or

22 × 23 = 22+3

This is the first law of indices, which demonstrates
that when multiplying two or more numbers
having the same base, the indices are added
...


(3) (35 )2 = 35×2 = 310 and (22 )3 = 22×3 = 26
This is the third law of indices, which demonstrates that when a number which is raised to
a power is raised to a further power, the indices
are multiplied
...

1
1
and −3 = 23
4
3
2
This is the fifth law of indices, which demonstrates
that a number raised to a negative power is the
reciprocal of that number raised to a positive
power
...

Here are some worked examples using the laws of
indices
...
Evaluate in index form 53 × 5 × 52
3

2

3

1

5 ×5×5 =5 ×5 ×5
=5

2

1

(Note that 5 means 5 )

3+1+2

Problem 9
...

2
2 5
But

Problem 10
...
001
= 10−3 =
8
10
1000

from law (2)

102 = 100, 103 = 1000, 104 = 10 000,
105 = 100 000, 106 = 1 000 000

=3

10−1 =

24
Problem 8
...
1, 10−2 = 2 =
= 0
...

= 24−4
=2

Hence,

from law (2)

Understanding powers of ten is important, especially
when dealing with prefixes in Chapter 8
...
Evaluate

But

3 × 32
34

3 × 32 31 × 32 31+2 33
=
= 4 = 4 = 33−4 = 3−1
34
34
3
3
from laws (1) and (2)

= 56

24

49

from law (2)

0

2 × 2 × 2 × 2 16
=
=1
2 × 2 × 2 × 2 16
20 = 1

Problem 11
...
For
example, 60 = 1, 1280 = 1, 137420 = 1 and so on
...
Simplify (a) (23 )4 (b) (32 )5 ,
expressing the answers in index form
From law (3):
(a)

(23 )4 = 23×4 = 212

(b)

(32 )5

= 32×5

Problem 13
...

(b)

Find the value of (a)

(32 )3
3 × 39

23 × 24
27 × 25

From the laws of indices:
(a)

(b)

23 × 24 2(3+4)
27
=
=
= 27−12
27 × 25 2(7+5) 212
1
1
= 2−5 = 5 =
2
32
(32 )3
32×3
36
= 1+9 = 10 = 36−10
9
3×3
3
3
= 3−4 =

Problem 15
...

1
...


35 × 33 × 3
in index form

3
...


33
35

5
...


23 × 2 × 26
27

7
...


104 ÷ 10

9
...


56 × 52 ÷ 57

11
...


(33)2

13
...


(9 × 32 )3
in
(3 × 27)2
index form

15
...


5−2
5−4

17
...


72 × 7−3
7 × 7−4

19
...


5−7 × 52
5−8 × 53

Here are some further worked examples using the laws
of indices
...


√
4 = ±2
√
4
3/4
(b) 16 = 163 = (2)3 = 8
(Note that it does not matter whether the 4th root
of 16 is found first or whether 16 cubed is found
first – the same answer will result
...
Grouping terms having the same base and
then applying the laws of indices to each of the groups
independently gives
33 × 57 33 57
=
×
= 3(3−4) × 5(7−3)
53 × 34 34 53
= 3−1 × 54 =

54
625
1
=
= 208
1
3
3
3

Powers, roots and laws of indices

Problem 17
...
Find the value of

32 × 55
2
3
32 × 55
= 4 34 × 5 3
4
4
3
3
3 ×5 +3 ×5
3 ×5
3 × 53
+
32 × 53 32 × 53

√
√
3
41
...
5 × 81/3
8×2
16
= 16
=
=
2
−2/5
1
1
2 × 32
4×
4
Alternatively,
Hence,

[(2)2 ]3/2 × (23 )1/3
41
...
Evaluate

32 × 55 + 33 × 53
34 × 54

Dividing each term by the HCF (highest common factor)
of the three terms, i
...
32 × 53 , gives
32 × 55 33 × 53
+
32 × 55 + 33 × 53
32 × 53 32 × 53
=
34 × 54
34 × 54
32 × 53
=

3(2−2) × 5(5−3) + 3(3−2) × 50
3(4−2) × 5(4−3)

=

30 × 52 + 31 × 50
32 × 51

=

1 × 25 + 3 × 1 28
=
9×5
45

32 × 55
34 × 54 + 33 × 53

To simplify the arithmetic, each term is divided by the
HCF of all the terms, i
...
32 × 53
...
5 × 81/3
Problem 18
...
Simplify

Since 7−3 =

1
1
1
,
= 32 and −2 = 52 , then
73 3−2
5

34 × 32 × 52
7−3 × 34
=
3−2 × 75 × 5−2
73 × 75
=

3(4+2) × 52 36 × 52
=
7(3+5)
78

162 × 9−2
4 × 33 − 2−3 × 82
expressing the answer in index form with positive
indices

Problem 22
...
e
...
Simplify
giving
 −3
2
5
the answer with positive indices
Raising a fraction to a power means that both the numerator and the denominator of the fraction are raised to that
 3
4
43
power, i
...

= 3
3
3
A fraction raised to a negative power has the same value
as the inverse of the fraction raised to a positive power
...

1
...


7−2 × 3−2
35 × 74 × 7−3

3
...


8−2 × 52 × 3−4
252 × 24 × 9−2

In Problems 5 to 15, evaluate the expressions given
...

6
...
25
32
1
 1/2
−
4
4
7
...

9
92 × 74
33 × 52
9
...

3 × 74 + 33 × 72
23 × 32 − 82 × 9
33 × 72 − 52 × 73
32 × 5 × 72
 3  −2
1
2
−
2
3
13
...


15
...
 2
2
9
12
...

The n’th term is: a + (n − 1)d
n
Sum of n terms, Sn = [2a + (n − 1)d]
2

Geometric progression:
If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 ,
...

4
...

10
...

16
...


Exercise 5 (page 11)

19 kg
2
...
479 mm
−66
5
...
−225
−2136
8
...
£10 7701
−4
11
...
5914
189 g
14
...
$15 333
89
...

3
...

5
...

9
...

9
...

17
...

(a) 8613 kg (b) 584 kg
(a) 351 mm (b) 924 mm
(a) 10 304 (b) −4433
6
...

(a) 8067 (b) 3347
10
...

4
...

8
...


(a) 48 m (b) 89 m
(a) 1648 (b) 1060
18 kg

1
...
14
7
...
88
8
...
1016/B978-1-85617-697-2
...


3 4 1 3 5
, , , ,
7 9 2 5 8
9
10
...
1
15
16
18
...


6
...

21
...
2
3
...
11
8
...

13
...

18
...

11
...

19
...

8
...

16
...


4
...
2

5
...
5

5
12
1
9
...

23
4
...
15

3
28
8
10
...


15
...
400 litres
22
...

15

1
...
59
6
...
−1

2
5

Exercise 6 (page 13)

11
...
7

(a) £1827 (b) £4158

Exercise 3 (page 6)
1
...

5
...

9
...
2

11
...
4
20
17
12
...
−

3
...
2

1
6

3
4
1
9
...
4

13
20
1
10
...


Answers to practice exercises
Exercise 14 (page 25)

Chapter 3

1
...
571
5
...
96
8
...
0871

Exercise 8 (page 17)
1
...
23
8
...


13
20
21
6
...
(a) 1
50
5
...
0
...


4
...
6

7
16

(e) 16

17
80

1
...

7
...

13
...
182
2
...
122
3
...
82
0
...
0
...
2
...
273
8
...
256
9
...
30366
6
1
...
3
...
37
...
2 × 10
14
...
767 ×10
15
...
32 ×106

12
...
6875 13
...
21875 14
...
1875

Exercise 16 (page 27)
1
...
4667

Exercise 9 (page 18)
1
...
18
5
...
297

2
...
785
3
...
38
6
...
000528

2
...
3
6
...
3

4
...
27

3
...
54
7
...
52 mm

4
...
83

13
14

3
...
458

6
...
7083

7
...
2
...
3

1
3

10
...
0776

1
...
9205
5
...
4424
9
...
6992

2
...
7314
6
...
0321
10
...
8452

3
...
9042
7
...
4232

4
...
2719
8
...
1502

Exercise 18 (page 28)

4
...
47
...
385
...
582
...
9 6
...
82
7
...
1
8
...
6
0
...
0
...
1
...
53
...
84 14
...
69
15
...
81 (b) 24
...
00639 (b) 0
...
(a) 8
...
6˙
2400

1
...
995
5
...
6977
9
...
520

Exercise 12 (page 23)
3
...
62
7
...
330

4
...
832
8
...
45

Exercise 13 (page 24)
1
...
25
2
...
0361 3
...
923 4
...
296 × 10−3
5
...
4430 6
...
197 7
...
96 8
...
0549
9
...
26 10
...
832 × 10−6

2
...
782
6
...
92
10
...
3770

3
...
72
7
...
0

4
...
42
8
...
90

Exercise 19 (page 29)
1
...

7
...


Chapter 4

2
...
1
...
12
...


Exercise 17 (page 27)

Exercise 11 (page 20)

1
...
797
5
...
42
9
...
59

1
21
9
...
567
5
...

5
...

13
...

18
...
40
3
...
13459
4
...
9
6
...
4481 7
...
36 × 10−6
9
...
625 × 10−9
10
...
70

Exercise 15 (page 25)

3
125

15
...
28125

1
...
3
5
...
3

341

A = 66
...
144 J
14 230 kg/ m3

2
...

8
...


C = 52
...
407 A
628
...
1 m/s

3
...

9
...


R = 37
...
02 mm
224
...
526 

Exercise 20 (page 30)
1
...

7
...

12
...
27
2
...
1 W
3
...
61 V
F = 854
...
I = 3
...
T = 14
...
96 J
8
...
77 A 9
...
25 m
A = 7
...
V = 7
...
53 h (b) 1 h 40 min, 33 m
...
h
...
02 h (d) 13
...
£556 2
...
264 kg 4
...
14
...
(a) 0
...
(a) 440 K (b) 5
...
0
...
173
...
5
...
37
...
128
...
0
...
0
...
0
...
38
...
(a) 21
...
2% (c) 169%
13
5
9
11
...

13
...

20
16
16
1
15
...
25, D = 25%, E = 0
...
60, H = 60%, I = 0
...
(a) 2 mA (b) 25 V 2
...
685
...
83 lb10 oz
5
...
1 litres (b) 16
...
29
...
584
...
$1012

Exercise 28 (page 46)
Exercise 22 (page 36)
1
...

5
...

10
...


21
...
9
...
4 t (b) 8
...
67%
14 minutes 57 seconds
37
...
39
...
7%
15
...
60 m

(c) 20
...

(c) 5
...

8
...

12
...

16
...
5%

2
...
8 g
£611
38
...
3
...
20 days
3
...
18 (b) 6
...
3375 4
...
(a) 300 × 103 (b) 0
...

5
...

13
...

15
...


2
...
18%
3
...
£175 000
£260
6
...
£9116
...
£50
...
60 10
...
70 11
...
7
...
6 kg, B 0
...
5 kg
54%, 31%, 15%, 0
...
5 mm, 11
...
600 kW

Chapter 6

1
...
±5

2
...
±8

3
...
100

5
...
64

4
...
1

Exercise 30 (page 50)
1
5
...
8
3
...

9
1
or 0
...
5 11
...
100 8
...

100
12
...
36
14
...
1 16
...
5 or
18
...

or 0
...
1
3
243
2
1
...
39

Exercise 24 (page 41)
1
...
3
...
47 : 3
1
4
...
5 hours or 5 hours 15 minutes
4
6
...
12 cm
8
...


1
3 × 52

5
...
1 : 15 2
...
25% 4
...
6 kg
5
...
3 kg 6
...
−
18
9
...


1
3
7 × 37

6
...
−1
14
...
±
2

3
...
−3
15
...


1
210 × 52

8
...

45
9

2
3

344 Basic Engineering Mathematics
Chapter 10

Chapter 11

Exercise 39 (page 69)
1
...

5
...

9
...

13
...

17
...

21
...

25
...


x 2 + 5x

+6
+9
4x 2 + 22x + 30
a 2 + 2ab + b2
a 2 − 2ac + c2
4x 2 − 24x + 36
64x 2 + 64x + 16
3ab − 6a 2
2a 2 − 3ab − 5b2
7x − y − 4z
x 2 − 4x y + 4y 2
0
4ab − 8a 2
2 + 5b2
4x 2 + 12x

Exercise 42 (page 75)

2
...

6
...

10
...

14
...

18
...

22
...

26
...


2x 2 + 9x

+4
− 12
2 pqr + p2 q 2 + r 2
x 2 + 12x + 36
25x 2 + 30x + 9
4x 2 − 9
r 2 s 2 + 2rst + t 2
2x 2 − 2x y
13 p − 7q
4a 2 − 25b2
9a 2 − 6ab + b2
4−a
3x y + 9x 2 y − 15x 2
11q − 2 p
2 j2 +2 j

Exercise 40 (page 71)
2(x + 2)
p(b + 2c)
4d(d − 3 f 5)
2q(q + 4n)
bc(a + b2 )
3x y(x y 3 − 5y + 6)
7ab(3ab − 4)


2x y x − 2y 2 + 4x 2 y 3
3x
17
...
(a + b)(y + 1)
22
...

3
...

7
...

11
...

15
...
0 19
...
( p + q)(x + y)
23
...

4
...

8
...

12
...

16
...
1

2
...
1

7
...
2
16
...
−4

3
...

2
13
...
−3

12
...
6

9
...
−2

2
...
4 + 3a
6
...
10y 2 − 3y +
9
...


1
− x − x2
5

1
7

1
4

8
...
5

2
...
−4

6
...
−4

8
...
−10

12
...
9

17
...
±12

22
...
−15t
12
...
2
...
2

5
...
2

10
...
3

14
...
−6

18
...
4

20
...
±3

24
...

4
...

6
...
8 m/s2
3
...
472
(a) 1
...
30 m/s2

Exercise 45 (page 80)
1
...
45◦ C
7
...
0
...
50
8
...
30
6
...
3
...
d = c − e − a − b

1
− 4x
3

10
...
2x + 8x 2
3
...

− 4x
2

5
...
R =
I
c
7
...
v =

y
7
v −u
4
...
y = (t − x)
3
y−c
8
...
x =

Answers to practice exercises
I
PR
E
11
...
C = (F − 32)
9
9
...
L =

XL
2π f

12
...
x = a(y − 3)
14
...
64 mm

1
2π CX C

*

1
√

Z2 −

14
...
1 × 10−6

aμ
ρCZ 4 n

2

Chapter 13
Exercise 47 (page 87)

Exercise 49 (page 92)

S −a
a
1
...
x =

yd
d
(y + λ) or d +
λ
λ

3
...
D =

AB 2
5E y

5
...
R2 =

R R1
R1 − R

E −e
E − e − Ir
or R =
−r
I
I

y

ay
8
...
x = 
2
4ac
(y 2 − b2 )

7
...
R =
πθ
√
Z 2 − R2
, 0
...
L =
2π f
10
...
u =




xy
1
...
r =
(1 − x − y)
c
5
...
b =
2( p2 + q 2 )
9
...
L =

8S 2
3d

Q
, 55
mc
+ d, 2
...
L =
μ−m


x−y
6
...
R = 4

uf
, 30
u− f


2dgh
, 0
...
v =
0
...
v =

x = 4, y = 2
x = 2, y = 1
...
5, n = 0
...

4
...

8
...

12
...

16
...
a = N 2 y − x

Exercise 48 (page 89)

1
...

5
...

9
...

13
...


1
...

5
...


p = −1, q = −2
a = 2, b = 3
x = 3, y = 4
x = 10, y = 15

2
...

6
...


x = 4, y = 6
s = 4, t = −1
u = 12, v = 2
a = 0
...
40

Exercise 51 (page 96)
1
1
1
...
p = , q =
4
5
5
...
x = 5, y = 1
4

1
1
2
...
x = 10, y = 5
1
6
...
1

Exercise 52 (page 99)
1
...

5
...

8
...
2, b = 4
u = 12, a = 4, v = 26
m = −0
...
00426, R0 = 22
...
I1 = 6
...
62
4
...
a = 12, b = 0
...
F1 = 1
...
5

Exercise 53 (page 100)
1
...
x = 5, y = −1, z = −2

2
...
x = 4, y = 0, z = 3

346 Basic Engineering Mathematics
5
...

9
...

11
...
x = 1, y = 6, z = 7
x = 5, y = 4, z = 2 8
...
5, y = 2
...
5
i1 = −5, i2 = −4, i3 = 2
F1 = 2, F2 = −3 F3 = 4

Exercise 57 (page 109)
1
...

6
...

10
...
191 s 2
...
345 A or 0
...
619 m or 19
...
066 m
1
...
165 m
12 ohms, 28 ohms

3
...

7
...


7
...
0133
86
...
4 or −4
4
...
5 or 1
...

10
...

16
...
−2 or −

2
3

2
...
0 or −
3
8
...
−3 or −7
14
...
−3
20
...
5

1
...
2 or −2

2
1
2
1
2
...

12
...

18
...
−1 or 1
...

or −
2
5
2
or −3
28
...
1 or −
3
7
4
1
29
...

2
3
27
...

4
21
...
4 or −7

31
...
x 2 + 5x + 4 = 0
35
...
2 or −6

or −
or

or −

Chapter 15
Exercise 59 (page 112)

1
3

1
3
1
1
6
...
2
8
...
1 10
...
2
12
...
100 000 14
...

32
1
16
...
01 17
...
e3
16
1
...
x 2 + 3x − 10 = 0
34
...
x 2 − 1
...
68 = 0

2
...
3

4
...


Exercise 60 (page 115)

Exercise 55 (page 106)
1
...
732 or −0
...
1
...
135
5
...
443 or 0
...
x = 0, y = 4 and x = 3, y = 1

2
...
137 or 0
...
1
...
310
6
...
851 or 0
...
log 6
5
...
log 15
6
...
log 2
7
...
log 3
8
...
log 10 10
...
log 2
12
...
log 16 or log24 or 4 log2
14
...

3
...

7
...

11
...


0
...
137
2
...
719
3
...
108
0
...
351
1
...
081
4 or 2
...
562 or 0
...

4
...

8
...

12
...
296 or −0
...
443 or −1
...
434 or 0
...
086 or −0
...
176 or −1
...
141 or −3
...
0
...
1
...
b = 2 20
...
x = 2
...
t = 8
21
...
x = 5

Exercise 61 (page 116)
1
...
690 2
...
170 3
...
2696 4
...
058 5
...
251
6
...
959 7
...
542 8
...
3272 9
...
2

Answers to practice exercises
Chapter 16

Chapter 17

Exercise 62 (page 118)
1
...

3
...

5
...
1653
(a)
5
...
55848
(a) 48
...
739

(b)
(b)
(b)
(b)
6
...
4584
0
...
40444
4
...
7 m

(c)
(c)
(c)
(c)

22030
40
...
05124
−0
...
2
...
(a) 7
...
7408
8 3
3
...
2x 1/2 + 2x 5/2 + x 9/2 + x 13/2
3
1 17/2
1
+ x
+ x 21/2
12
60

1
...
1 V
(c) Horizontal axis: 1 cm = 10 N, vertical axis:
1 cm = 0
...
(a) −1 (b) −8 (c) −1
...
14
...
(a) −1
...
4
5
...
3
...
05
3
...
1
...
30
4
...

2
...

7
...

14
...


(a) 0
...
91374 (c) 8
...
2293 (b) −0
...
13087
−0
...
−0
...
2
...
816
...
8274 8
...
02
9
...
522 10
...
485
1
...
3
13
...
9 15
...
901 16
...
095
a
t = eb+a ln D = eb ea ln D = eb eln D i
...
t = eb D a
 
U2
18
...
W = PV ln
U1

Exercise 68 (page 140)
1
...
75, 0
...
75, 2
...
75;
1
Gradient =
2
2
...
(a) 6, −3 (b) −2, 4 (c) 3, 0 (d) 0, 7
3
...
(a) 2, − (b) − , −1 (c) , 2 (d) 10, −4
2
3
3
18
3
3
5
6
...
(a) and (c), (b) and (e)
8
...
(1
...
(1, 2)

11
...
4 (d) l = 2
...
P = 0
...
5

13
...
(a) 40◦ C (b) 128 
2
...
5 V

Exercise 66 (page 127)
1
...
5◦C

3
...
25 (b) 12 (c) F = 0
...
99
...
(a) 29
...
31 × 10−6 s
4
...
993 m (b) 2
...
(a) 50◦ C (b) 55
...
30
...
(a) 3
...
46 s
8
...
45 mol/cm3
10
...
(a) 7
...
966 s

(d) 89
...
−0
...
73
5
...
5 m/s (b) 6
...
7t + 15
...
m = 26
...
63
7
...
31 t (b) 22
...
09 W + 2
...
(a) 96 × 109 Pa (b) 0
...
8 × 106 Pa

348 Basic Engineering Mathematics
1
1
(b) 6 (c) E = L + 6 (d) 12 N (e) 65 N
5
5
10
...
85, b = 12, 254
...
5 kPa, 280 K
9
...
(−2
...
2), (0
...
8); x = −2
...
6
10
...
2 or 2
...
75 and −1
...
3 or −0
...
(a) y (b) x 2 (c) c (d) d

2
...
(a) (b) x (c) b (d) c
x
x
1
y
5
...
a = 1
...
4, 11
...
y = 2x 2 + 7, 5
...
x = 4, y = 8 and x = −0
...
5
2
...
5 or 3
...
24 or 3
...
5 or 3
...
(a) y (b)

8
...
a = 0
...
6 (i) 94
...
2

Exercise 75 (page 162)
1
...
0, −0
...
5
2
...
1, −4
...
8, 8
...
x = 1
4
...
0, 0
...
6
5
...
7 or 2
...
x = −2
...
0 or 1
...
x = −1
...

2
...

4
...

6
...

9
...
0012 V2 , 6
...
0, b = 0
...
7, b = 2
...
53, 3
...
0, c = 1
...
y = 0
...
24x
T0 = 35
...
27, 65
...
28 radians

Exercise 72 (page 156)
x = 2, y = 4
x = 3
...
5
x = 2
...
2
a = 0
...
6

Exercise 76 (page 167)
1
...
27◦54
3
...
100◦6 52
◦


◦


5
...
86 49 1 7
...
55◦ 8
...
754◦
9
...
58◦22 52

Exercise 77 (page 169)
1
...
obtuse 3
...
right angle
5
...

3
...

7
...
x = 1, y = 1
4
...
x = −2, y = −3

Exercise 73 (page 160)
1
...
−0
...
6
3
...
9 or 6
...
−1
...
1
5
...
8 or 2
...
x = −1
...
75, −0
...
x = −0
...
6
8
...
63 (b) 1 or −0
...
(a) 102◦ (b) 165◦ (c) 10◦ 18 49
7
...
3◦ (h) 79◦ (i) 54◦
8
...
59◦ 20
10
...
51◦
12
...
326 rad 13
...
605 rad 14
...
(a) acute-angled scalene triangle
(b) isosceles triangle (c) right-angled triangle
(d) obtuse-angled scalene triangle
(e) equilateral triangle (f ) right-angled triangle

Answers to practice exercises
2
...
DF, DE
4
...
122
...
φ = 51◦, x = 161◦
7
...
a = 18◦ 50 , b = 71◦10 , c = 68◦ , d = 90◦,
e = 22◦ , f = 49◦, g = 41◦
9
...
17◦

11
...
sin A = , cos A = , tan A = , sin B = ,
5
5
4
5
3
4
cos B = , tan B =
5
3
8
8
3
...
sin X =
113
113
15
15
8
5
...
(a) sin θ =
(b) cos θ =
25
25
7
...
434 (b) −0
...
(a) congruent BAC, DAC (SAS)
(b) congruent FGE, JHI (SSS)
(c) not necessarily congruent
(d) congruent QRT, SRT (RHS)
(e) congruent UVW, XZY (ASA)
2
...

4
...

9
...


2
...
4
...
36
...
8660 (b) −0
...
5865
42
...
15
...
73
...
7◦56
◦

◦

◦
31 22 10
...
29
...
20◦21
0
...
1
...
x = 16
...
18 mm 2
...
79 cm
3
...
25 cm (b) 4 cm
4
...
(a) 12
...
619 (c) 14
...
349
(e) 5
...
275
2
...
831 cm, ∠A = 59
...
96◦
(b) DE = 6
...
634 cm, GH = 10
...
810 cm, KM = 13
...
125 cm, NP = 8
...
346 cm, QS = 6
...
Constructions – see similar constructions in
worked problems 30 to 33 on pages 179–180
...
6
...
9
...

4
...

9
...

13
...
36
...
48 m
3
...
5 m 4
...
1 m
5
...
0 m
6
...
50 m 7
...
8 m
8
...
43 m, 10
...
60 m

9 cm
2
...
9
...
81 cm 5
...
21 m
6
...
18 cm
24
...
82 + 152 = 172
(a) 27
...
20
...
35 m, 10 cm
12
...
7 nautical miles
2
...
24 mm

Chapter 22
Exercise 87 (page 198)

Exercise 83 (page 185)
40
40
9
9
1
...
(a) 42
...
22◦ (b) 188
...
47◦
2
...
08◦ and 330
...
86◦ and 236
...
(a) 44
...
21◦ (b) 113
...
12◦

350 Basic Engineering Mathematics
4
...
α = 218◦41 and 321◦19
6
...
5
2
...
30
4
...
1, 120
6
...
3, 90
8
...
, 960◦ 10
...
4, 180◦ 12
...
40 Hz
14
...
1 ms
15
...
leading 17
...
p = 13
...
35◦, R = 78
...
7 cm2
2
...
127 m, Q = 30
...
17◦ ,
area = 6
...
X = 83
...
62◦, Z = 44
...
8 cm2
4
...
77◦, Y = 53
...
73◦ ,
area = 355 mm2

Exercise 92 (page 210)
Exercise 89 (page 203)
1
...
04 s or 40 ms (c) 25 Hz
(d) 0
...
62◦) leading 40 sin 50πt

1
...

5
...


193 km 2
...
6 m (b) 94
...
66◦, 44
...
4 m (b) 17
...
163
...
9 m, EB = 4
...
6
...
37 m
32
...
31◦

2
...
37 Hz (c) 0
...
54 rad (or 30
...
(a) 300 V (b) 100 Hz (c) 0
...
412 rad (or 23
...
(a) v = 120 sin100πt volts
(b) v = 120 sin (100πt + 0
...
i = 20 sin 80πt −
6
i = 20 sin(80πt − 0
...
3
...
488) m
7
...
75◦ lagging
(b) −2
...
363 A (d) 6
...
423 ms

Chapter 23
Exercise 90 (page 207)
1
...
1 mm, c = 28
...
A = 52◦2 , c = 7
...
152 cm,
area = 25
...
D = 19◦48 , E = 134◦12 , e = 36
...
E = 49◦ 0 , F = 26◦ 38 , f = 15
...
6 mm2
5
...
420 cm,
area = 6
...
811 cm, area = 0
...
K = 47◦ 8 , J = 97◦ 52 , j = 62
...
2 mm2 or K = 132◦52 , J = 12◦8 ,
j = 13
...
0 mm2

Exercise 93 (page 212)
1
...
42◦, 59
...
20◦ 2
...
23 m (b) 38
...
40
...
05◦
4
...
8 cm 5
...
2 m
6
...
3 mm, y = 142 mm 7
...
13
...

2
...

4
...

6
...

8
...
83, 59
...
83, 1
...
61, 20
...
61, 0
...
47, 116
...
47, 2
...
55, 145
...
55, 2
...
62, 203
...
62, 3
...
33, 236
...
33, 4
...
83, 329
...
83, 5
...
68, 307
...
68, 5
...
294, 4
...
(1
...
960)
(−5
...
500)
4
...
884, 2
...
353, −5
...
(−2
...
207)
(0
...
299)
8
...
252, −4
...
04, 12
...
04, 12
...
51, −32
...
51, −32
...
47
...

3
...

7
...


Answers to practice exercises
Exercise 103 (page 234)

Chapter 25
Exercise 96 (page 221)
1
...
t = 146◦

351

2
...
(i) rhombus (a) 14 cm2 (b) 16 cm (ii) parallelogram
(a) 180 mm2 (b) 80 mm (iii) rectangle (a) 3600 mm2
(b) 300 mm (iv) trapezium (a) 190 cm2 (b) 62
...
35
...
(a) 80 m (b) 170 m 4
...
2 cm2
5
...
1200 mm
7
...
560 m2
2
9
...
4 cm
10
...
43
...
32

1
...

7
...

11
...

16
...

20
...
2376 mm2
3
...
1709 mm
6
...
(a) 106
...
9 cm2
2
21
...
17
...
07 cm2
(a) 59
...
8 mm
12
...
2 cm
8
...
48 cm 14
...
5◦ 15
...
698 rad (b) 804
...
10
...
24%
19
...
8 mm
7
...
(a) 2 (b) (3, −4)
2
...
Circle, centre (0, 1), radius 5
4
...

2
...

5
...


482 m2
(a) 50
...
9 mm2 (c) 3183 mm2
2513 mm2
4
...
19 mm (b) 63
...
01 cm2 (b) 129
...
5773 mm2
2
1
...
1932 mm2 2
...
(a) 0
...

4
...

8
...

12
...

15
...

19
...
2 m3
2
...
8 cm3
3
2
(a) 3840 mm (b) 1792 mm
972 litres
6
...
500 litres
3
9
...
3 cm3 (b) 61
...
44 m
(a) 2400 cm3 (b) 2460 cm2 11
...
04 m
1
...
8796 cm3
4
...
9 cm2
2
...
28060 cm3 , 1
...
22 m by 8
...
62
...
4
...
80 ha

2
...
3
...
45
...
259
...
2
...
47
...
38
...
12730 km 7
...
13 mm

Exercise 106 (page 246)
1
...
1 cm3 , 159
...
7
...
81 cm2
3
...
1 cm3 , 113
...
5
...
3 cm
6
...
(a) 268 083 mm3 or 268
...
06 cm2
8
...
53 cm
9
...
09 × 1012 km3 10
...
(a) 0
...
481 (c) 4
...
(a) 210◦ (b) 80◦ (c) 105◦
4
...

2
...

6
...
(a)

5890 mm2 or 58
...
55 cm3 (b) 84
...
13
...
393
...
32 cm3
(i) (a) 670 cm3 (b) 523 cm2 (ii) (a) 180 cm3
(b) 154 cm2 (iii) (a) 56
...
8 cm2
(iv) (a) 10
...
0 cm2 (v) (a) 96
...

9
...

13
...
5 cm3 (b) 142 cm2
(vii) (a) 805 cm3 (b) 539 cm2
(a) 17
...
0 cm
8
...
3 m , 25
...
6560 litres
12
...
7 cm3
657
...
77 m (c) £140
...
69 cm
5
...
72 N at −14
...
15 m/s at 29
...
28 N at 16
...
6
...
56◦
◦
15
...
33 to the 10 N force
21
...
22◦ S

Exercise 115 (page 276)

Exercise 108 (page 255)
1
...

5
...


6
...

8
...

11
...
403 cm3 , 337 cm2
4
...
55910 cm3 , 6051 cm2

1
...
0 N at 78
...
64 N at 4
...
(a) 31
...
81◦ (b) 19
...
63◦

Exercise 116 (page 277)
1
...
5 km/h at 71
...
4 minutes 55 seconds, 60◦
3
...
79 km/h, E 9
...
8 : 125

2
...
2 g

Chapter 28

Exercise 117 (page 277)

Exercise 110 (page 259)
1
...
5 square units 2
...
7 square units 3
...
33 m
4
...
70 ha
5
...
42
...
147 m3

3
...
42 m3

Exercise 112 (page 263)
1
...
5 A
3
...
093 As, 3
...
49
...
5 kPa

1
...

5
...

9
...
5j − 4k
3
...
4j − 6
...

4
...

8
...


4i + j − 6k
5i − 10k
−5i + 10k
20
...
(a) 2
...
(a) 31
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
A scalar quantity has magnitude only; a vector
quantity has both magnitude and direction
...
scalar
3
...
vector 5
...
scalar
7
...
scalar
9
...

2
...

4
...


17
...
00◦ to the 12 N force
13 m/s at 22
...
40 N at 37
...
43 N at 129
...
31 m at 21
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
11
...
324)
5
...
73 sin(ωt − 0
...
11 sin(ωt + 0
...
8
...
173)
i = 21
...
639)
v = 5
...
670)
x = 14
...
444)
(a) 305
...
2t − 0
...
21 sin(628
...
818) V (b) 100 Hz
(c) 10 ms
8
...
83 sin(300πt + 0
...
667 ms

1
...

4
...

6
...


Chapter 31
Exercise 122 (page 288)
1
...
(a) discrete (b) continuous (c) discrete (d) discrete

Exercise 123 (page 292)
1
...
5, 4
...

2
...
5
...
6 equally spaced horizontal rectangles, whose
lengths are proportional to 35, 44, 62, 68, 49 and
41, respectively
...
5 equally spaced horizontal rectangles, whose
lengths are proportional to 1580, 2190, 1840, 2385
and 1280 units, respectively
...
6 equally spaced vertical rectangles, whose heights
are proportional to 35, 44, 62, 68, 49 and 41 units,
respectively
...
5 equally spaced vertical rectangles, whose heights
are proportional to 1580, 2190, 1840, 2385 and 1280
units, respectively
...
Three rectangles of equal height, subdivided in the
percentages shown in the columns of the question
...

8
...

Little change in centres A and B, a reduction of
about 8% in C, an increase of about 7% in D and a
reduction of about 3% in E
...
A circle of any radius, subdivided into sectors having angles of 7
...
5◦, 52
...
5◦ and 110◦,
respectively
...
A circle of any radius, subdivided into sectors having angles of 107◦, 156◦, 29◦ and 68◦ , respectively
...
(a) £495 (b) 88
12
...
There is no unique solution, but one solution is:
39
...
4 1; 39
...
6 5; 39
...
8 9;
39
...
0 17; 40
...
2 15; 40
...
4 7;
40
...
6 4; 40
...
8 2
...
Rectangles, touching one another, having midpoints of 39
...
55, 39
...
95,
...

3
...
5–20
...
0–21
...
5–21
...
0–22
...
5–22
...
0–23
...

4
...

5
...
95 3; 21
...
95 24; 22
...
95 46;
23
...
Rectangles, touching one another, having midpoints of 5
...
5 and 43
...
The
heights of the rectangles (frequency per unit class
range) are 0
...
78, 4, 4
...
33, 0
...
2
...
(10
...
45 9), (11
...
45 31), (12
...
45, 50)
8
...

9
...
05–2
...
10–2
...
15–2
...
20–2
...
25–2
...
30–2
...

(b) Rectangles, touching one another, having midpoints of 2
...
12,
...

(c)

Using the frequency distribution given in the
solution to part (a) gives 2
...
145 13;
2
...
245 37; 2
...
345 48
...


Chapter 32
Exercise 125 (page 300)
1
...
33, median 8, mode 8
2
...
25, median 27, mode 26

Answers to practice exercises
Exercise 138 (page 324)
1
...
(a) 0
...
5 V
3
...
9 V/s
4
...
635 Pa/m

Chapter 35

5 4
x +c
4
2 3
x +c
3
...
(a) x 5 − x 2 + c
5
2

2
...
(a)
6
...

8
...


3x 2
2

− 5x + c

2
u2
ln x + c
(b)
− ln u + c
3
2
√
√
18 √ 5
14
...
(a)

Exercise 139 (page 328)
1
...
(a) −6 cos x + c
2
3 2x
12
...
(a)

355

4t 3
1
+c
(b) − + 4t +
t
3

Exercise 140 (page 330)

θ3
3

+c

1
...
5 (b) 0
...
(a) 105 (b) −0
...
(a) 6 (b) −1
...
(a) −0
...
833

5
...
67 (b) 0
...
(a) 0 (b) 4

7
...
248

8
...
2352 (b) 2
...
(a) 19
...
457

10
...
2703 (b) 9
...
proof
5
...
5
8
...
67

2
...
7
...
2
...
32
7
...
140 m

4
...
33 Nm

Index
Acute angle, 165
Acute angled triangle, 171
Adding waveforms, 278
Addition in algebra, 62
Addition law of probability, 307
Addition of fractions, 10
numbers, 1, 18
two periodic functions, 278
vectors, 267
by calculation, 270
Algebra, 61, 68
Algebraic equation, 61, 73
expression, 73
Alternate angles, 165, 191
Ambiguous case, 207
Amplitude, 199
Angle, 165
Angle, lagging and leading, 200
types and properties of, 165
Angles of any magnitude, 196
depression, 191
elevation, 191
Angular measurement, 165
velocity, 202
Annulus, 226
Arbitrary constant of integration, 325
Arc, 231
Arc length, 233
Area, 219
Area of common shapes, 219, 221
under a curve, 330
Area of circle, 222, 233
common shapes, 219
irregular figures, 257
sector, 222, 233
similar shapes, 229
triangles, 205
Arithmetic, basic, 1
Average, 299
value of waveform, 260
Axes, 130
Bar charts, 289
Base, 47
Basic algebraic operations, 61
BODMAS with algebra, 71
fractions, 13
numbers, 6

Boyle’s law, 46
Brackets, 6, 68
Calculation of resultant phasors, 281,
283
Calculations, 22, 28
Calculator, 22
addition, subtraction, multiplication
and division, 22
fractions, 26
π and e x functions, 28, 118
reciprocal and power functions, 24
roots and ×10 x functions, 25
square and cube functions, 23
trigonometric functions, 27
Calculus, 313
Cancelling, 10
Cartesian axes, 131
co-ordinates, 214
Charles’s law, 42, 142
Chord, 230
Circle, 222, 230, 233
equation of, 236
properties of, 230
Circumference, 230
Classes, 293
Class interval, 293
limits, 295
mid-point, 293, 295
Coefficient of proportionality, 45
Combination of two periodic functions,
278
Common factors, 69
logarithms, 111
prefixes, 53
shapes, 219
Complementary angles, 165
Completing the square, 105
Cone, 245
frustum of, 252
Congruent triangles, 175
Construction of triangles, 179
Continuous data, 288
Co-ordinates, 130, 131
Corresponding angles, 165
Cosine, 27, 183
graph of, 195
Cosine rule, 205, 281
wave, 195

Cross-multiplication, 75
Cube root, 23
Cubic equation, 161
graphs, 161
units, 240
Cuboid, 240
Cumulative frequency distribution,
293, 297
curve, 293
Cycle, 199
Cylinder, 241
Deciles, 304
Decimal fraction, 216
places, 13, 18
Decimals, 16
addition and subtraction, 19
multiplication and division, 19
Definite integrals, 328
Degrees, 27, 165, 166, 232
Denominator, 9
Dependent event, 307
Depression, angle of, 191
Derivatives, 315
standard list, 321
Derived units, 53
Determination of law, 147
involving logarithms, 150
Diameter, 230
Difference of two squares, 103
Differential calculus, 313
coefficient, 315
Differentiation, 313, 315
from first principles, 315
of ax n , 315
of e ax and ln ax, 320
of sine and cosine functions, 318
successive, 322
Direct proportion, 40, 42
Discrete data, 288
standard deviation, 302
Dividend, 63
Division in algebra, 62
Division of fractions, 12
numbers, 3, 4, 19
Divisor, 63
Drawing vectors, 266

358 Index
Parabola, 156
Parallel lines, 165
Parallelogram, 219
method, 267
Peak value, 199
Pentagon, 219
Percentage component bar chart, 289
error, 36
relative frequency, 289
Percentages, 33
Percentile, 304
Perfect square, 105
Perimeter, 171
Period, 199
Periodic function, 200
plotting, 238
Periodic time, 200
Phasor, 280
Pictograms, 289
Pie diagram, 289
Planimeter, 257
Plotting periodic functions, 238
Polar co-ordinates, 214
Pol/Rec function on calculator, 217
Polygon, 210
frequency, 293, 296
Population, 289
Power, 47
series for e x , 119
Powers and roots, 47
Practical problems
quadratic equations, 108
simple equations, 77
simultaneous equations, 96
straight line graphs, 141
trigonometry, 209
Precedence, 6, 71
Prefixes, 53
Presentation of grouped data, 292
statistical data, 288
Prism, 240, 242
Probability, 306
laws of, 307
Production of sine and cosine waves,
198
Proper fraction, 9
Properties of circles, 230
triangles, 171
Proportion, 40
Pyramid, 244
volumes and surface area of frustum
of, 252
Pythagoras’ theorem, 181
Quadrant, 230
Quadratic equations, 102

by completing the square, 105
factorization, 102
formula, 106
graphically, 156
practical problems, 108
Quadratic formula, 106
graphs, 156
Quadrilaterals, 219
properties of, 219
Quartiles, 303
Radians, 27, 165, 166, 232
Radius, 230
Range, 295
Ranking, 299
Rates of change, 323
Ratio and proportion, 40
Ratios, 40
Reciprocal, 24
Rectangle, 219
Rectangular axes, 131
co-ordinates, 131
prism, 240
Reduction of non-linear laws to linear
form, 147
Reflex angle, 165
Relative frequency, 289
velocity, 276
Resolution of vectors, 269
Resultant phasors, by drawing, 280
horizontal and vertical components,
283
plotting, 278
sine and cosine rules, 281
Rhombus, 219
Right angle, 165
Right angled triangle, 171
solution of, 188
Sample, 289
Scalar quantities, 266
Scalene triangle, 171
Scales, 131
Sector, 222, 230
area of, 233
Segment, 230
Semicircle, 230
Semi-interquartile range, 304
Set, 289
Short division, 4
Significant figures, 17, 18
Similar shapes, 229, 256
triangles, 176
Simple equations, 73
practical problems, 77
Simpson’s rule, 258

Simultaneous equations, 90
graphical solution, 155
in three unknowns, 99
in two unknowns, 90
practical problems, 96
Sine, 27, 183
graph of, 195
Sine rule, 205, 281
wave, 198, 260
mean value, 260
Sinusoidal form A sin(ωt ± α), 202
SI units, 53
Slope, 134
Solution of linear and quadratic
equations simultaneously, 110
Solving right-angled triangles, 188
simple equations, 73
Space diagram, 276
Sphere, 246
Square, 23, 219
numbers, 23
root, 25, 48
units, 219
Standard deviation, 302
discrete data, 302
grouped data, 303
Standard differentials, 321
form, 56
integrals, 326
Statistical data, presentation of, 288
terminology, 288
Straight line, 165
equation of, 135
Straight line graphs, 132
practical problems, 141
Subject of formulae, 83
Subtraction in algebra, 62
Subtraction of fractions, 10
numbers, 1, 18
vectors, 274
Successive differentiation, 322
Supplementary angles, 165
Surface areas of frusta of pyramids and
cones, 252
of solids, 247
Symbols, 28
Tally diagram, 293, 296
Tangent, 27, 183, 230
graph of, 195
Terminating decimal, 17
Theorem of Pythagoras, 181
Transposition of formulae, 83
Transversal, 165
Trapezium, 220
Trapezoidal rule, 257

Index
Triangle, 171, 219
Triangles, area of, 205
congruent, 175
construction of, 179
properties of, 171
similar, 176
Trigonometric functions, 27
Trigonometric ratios, 183
evaluation of, 185
graphs of, 195
waveforms, 195
Trigonometry, 181
practical situations, 209
Turning points, 156

Ungrouped data, 289
Units, 53
Upper class boundary, 293
Use of calculator, 22
Vector addition, 267
subtraction, 274
Vectors, 266
addition of, 267
by calculation, 267
by horizontal and vertical
components, 269
drawing, 266
subtraction of, 274
Velocity, relative, 276

Vertical axis intercept, 133
bar chart, 289
component, 269, 283
Vertically opposite angles, 165
Vertices of triangle, 172
Volumes of common solids, 240
frusta of pyramids and cones, 252
irregular solids, 259
pyramids, 244
similar shapes, 256
Waveform addition, 278
y-axis intercept, 135
Young’s modulus of elasticity, 143

359

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