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Chapter 28

Irregular areas and volumes,
and mean values
28
...
1

(c) the mid-ordinate rule, or
(d) Simpson’s rule
...

(a) A planimeter is an instrument for directly measuring small areas bounded by an irregular curve
...
A pointer on the
planimeter is used to trace around the boundary
of the shape
...


(iii) Area PQRS


y1 + y7
=d
+ y2 + y3 + y4 + y5 + y6
...
1,

D

A

(i) Divide base PS into any number of equal
intervals, each of width d (the greater the
number of intervals, the greater the accuracy)
...

DOI: 10
...
00028-4

C

d

d

d

d

d

d

Figure 28
...
2,

258 Basic Engineering Mathematics
30

(i) Divide base PS into an even number of intervals, each of width d (the greater the number
of intervals, the greater the accuracy)
...

d
(iii) Area PQRS = [(y1 + y7 ) + 4(y2 + y4 + y6 )
3
+ 2(y3 + y5 )]
In general, Simpson’s rule states

 

1 width of
first + last
Area =
ordinate
3 interval




sum of even
sum of remaining
+4
+2
ordinates
odd ordinates

0

Problem 1
...

3

4

5

6

1

2

24
...
25

17
...
0

12
...
75

5
...
0

3
4
5
Time (seconds)

6

Figure 28
...
Mid-ordinates are erected as shown in
Figure 28
...
The length of each
mid-ordinate is measured
...
25 + 4
...
0 + 10
...
0
+ 20
...
25 m
Simpson’s rule (see (d) above)
The time base is divided into 6 strips each of width
1 s and the length of the ordinates measured
...
0) + 4(2
...
75
3

Speed v (m/s) 0 2
...
5 8
...
5 17
...
0
Determine the distance travelled in 6 seconds (i
...

the area under the v/t graph), using (a) the
trapezoidal rule (b) the mid-ordinate rule
(c) Simpson’s rule

4
...
25
2
...
75

Area = (width of interval)(sum of
mid-ordinates)
(d) Simpson’s rule
To determine the area PQRS of Figure 28
...

(ii) Erect ordinates in the middle of each interval
(shown by broken lines in Figure 28
...

(iii) Accurately measure ordinates y1 , y2, y3, etc
...


+ 17
...
5 + 12
...
33 m
Problem 2
...
Soundings of
the depth are made at equal intervals of 3 m across
the river and are as shown below
...
2 3
...
5 4
...
4 0

A graph of speed/time is shown in Figure 28
...

(a)

Trapezoidal rule (see (b) above)
The time base is divided into 6 strips, each of
width 1 s, and the length of the ordinates measured
...
0
area = (1)
+ 2
...
5
2

+ 8
...
5 + 17
...
75 m

Calculate the cross-sectional area of the flow of
water at this point using Simpson’s rule
From (d) above,
1
Area = (3)[(0+0) + 4(2
...
5+2
...
3+4
...
4 + 15] = 51
...
2

Practice Exercise 110 Areas of irregular
figures (answers on page 352)
1
...


3
...

Determine the area enclosed by the curve,
the x-axis and ordinates x = 0 and x = 3 by
(a) the trapezoidal rule (b) the mid-ordinate
rule (c) Simpson’s rule
...
0 4
...
0 14
...
0 29
...
e
...

The shape of a piece of land is shown in
Figure 28
...
To estimate the area of the land,
a surveyor takes measurements at intervals
of 50 m, perpendicular to the straight portion
with the results shown (the dimensions being
in metres)
...


140 160 200 190 180 130

50 50 50 50 50 50

Figure 28
...


If the cross-sectional areas A1 , A2 , A3 ,
...
5), by
Simpson’s rule
Volume, V =

A1

The velocity of a car at one second intervals is
given in the following table
...


Volumes of irregular solids

Plot the graph of y = 2x 2 + 3 between x = 0
and x = 4
...


Time t (s) 0 1

The deck of a ship is 35 m long
...

Width (m) 0 2
...
2 6
...
8 4
...
0 2
...


259

d

d
[(A1 + A7 ) + 4(A2 + A4 + A6 )
3
+ 2(A3 + A5 )]

A2

A3

d

d

A4

d

A5 A6

d

A7

d

Figure 28
...
A tree trunk is 12 m in length and has
a varying cross-section
...
52, 0
...
59, 0
...
72, 0
...
97 m2
...
5 above, where d = 2 m, A1 = 0
...
55 m2 , and so on
...
52 + 0
...
55 + 0
...
84)
3
+ 2(0
...
72)]
2
= [1
...
08 + 2
...
13 m3
3
Problem 4
...

Calculate the capacity of the reservoir in litres
Using Simpson’s rule for volumes gives
Volume =

=

10
[(210 + 170) + 4(250 + 350 + 230)
3
+ 2(320 + 290)]
10
[380 + 3320 + 1220] = 16 400m3
3

260 Basic Engineering Mathematics
16 400 m3 = 16 400 × 106 cm3
...
6
=
7
y=

= 16 400 000 = 16
...
The areas of equidistantly spaced sections of
the underwater form of a small boat are as follows:
1
...
78, 3
...
12, 2
...
24 and 0
...

Determine the underwater volume if the
sections are 3 m apart
...
To estimate the amount of earth to be
removed when constructing a cutting, the
cross-sectional area at intervals of 8 m were
estimated as follows:
0, 2
...
7, 4
...
1, 2
...

Estimate the volume of earth to be excavated
...
The circumference of a 12 m long log of timber
of varying circular cross-section is measured
at intervals of 2 m along its length and the
results are as follows
...

Distance from
one end (m)

0

2

4

6

y

y1 y2 y3 y4 y5 y6 y7
d

d

d

d

d

d

d

b

Figure 28
...
7(a)),

cycle

is

zero

(see

(b) over half a cycle is 0
...
7(b)) is 0
...
7(c)) is 0
...

π

Circumference (m) 2
...
25 3
...
32
Distance from
one end (m)

8

Circumference (m)

5
...
82

10

12
6
...
3 Mean or average values of
waveforms

area under curve
length of base, b

(b)

V
Vm
0

The mean or average value, y, of the waveform shown
in Figure 28
...
7

Irregular areas and volumes, and mean values

20

0

1 2 3 4 t (ms)

220

3
2
1
0

1 2 3

21
22
23

4 5 6 t (s)

Problem 6
...
9
Current (mA)

Current (A)

Voltage (V)

Problem 5
...
8

261

5

0

4

8 12 16 20 24 28 t (ms)

2

4

(b)

Current (A)

Voltage (V)

(a)
10

0

2 4 6 8 t (ms)

210

2

0

6

8 10 12 t (ms)

Figure 28
...
8

(a) One cycle of the trapezoidal waveform (a) is completed in 10 ms (i
...
the periodic time is 10 ms)
...
33 A
3s

Average value of waveform =

(c) A half cycle of the voltage waveform (c) is
completed in 4 ms
...
5 V
4 × 10−3s

=

area under curve
length of base
30 × 10−6 As
10 × 10−3 s

= 3 mA
(b) One cycle of the saw-tooth waveform (b) is completed in 5 ms
...
6 A

262 Basic Engineering Mathematics
Problem 7
...

0

1

2

3

4

5 6

Power (kW) 0 14 29 51 45 23 0
Plot a graph of power against time and, by using the
mid-ordinate rule, determine (a) the area under the
curve and (b) the average value of the power
The graph of power/time is shown in Figure 28
...

Graph of power/time

10

0 308608908
␲
2

1808
␲

2708
3␲
2

3608
2␲

␪

Figure 28
...
The base is divided into 6 intervals, each of
width 30◦
...


50
40
Power (kW)

Voltage (V)

Time (h)

Problem 8
...
11 shows a sinusoidal
output voltage of a full-wave rectifier
...
588 V,

30

At 45◦ the height of the mid-ordinate is
10 sin 45◦ = 7
...

The results are tabulated below
...
0 21
...
0 49
...
0 10
...
588 V

Figure 28
...
071 V

(a)

75◦

10 sin 75◦ = 9
...
659 V

135◦

10 sin 135◦ = 7
...
588 V

The time base is divided into 6 equal intervals, each
of width 1 hour
...
10) and measured
...
10
...
636 V

= (1)[7
...
5 + 42
...
5 + 37
...
0]
= 167 kWh (i
...
a measure of electrical energy)
(b) Average value of waveform =

area under curve
length of base

167 kWh
6h
= 27
...
636
sum of mid-ordinates
=
= 6
...
)
For a sine wave the actual mean value is
0
...
37 V
...
An indicator diagram for a steam
engine is shown in Figure 28
...
The base line has

2

0
22

20 t (ms)

10

3
...
0

3
...
9

2
...
0 cm

(c)

Figure 28
...
0
cm
...
0)[(3
...
6) + 4(4
...
9 + 1
...
5 + 2
...
2 + 34
...
4] = 34 cm2
3
area of diagram
(b) Mean height of ordinates =
length of base
34
= 2
...
83 cm × 100 kPa/cm = 283kPa

Now try the following Practice Exercise
Practice Exercise 112 Mean or average
values of waveforms (answers on page 352)
1
...
13 over a half
cycle
...
Find the average value of the periodic waveforms shown in Figure 28
...


Voltage (mV)

Figure 28
...
7 1
...
14

3
...


Voltage (V)

been divided into 6 equally spaced intervals and the
lengths of the 7 ordinates measured with the results
shown in centimetres
...
9 2
...
9 5
...
5 0
Plot a graph of current against time and estimate the area under the curve over the 30 ms
period, using the mid-ordinate rule, and determine its mean value
...
Determine, using an approximate method, the
average value of a sine wave of maximum
value 50 V for (a) a half cycle (b) a complete
cycle
...
An indicator diagram of a steam engine is
12 cm long
...
90, 5
...
22, 3
...
32, 3
...
16 cm
...


Revision Test 11 : Volumes, irregular areas and volumes, and mean values
This assignment covers the material contained in Chapters 27 and 28
...

1
...
Calculate the volume
of the alloy in cubic centimetres
...
Determine how many cubic metres of concrete are
required for a 120 m long path, 400 mm wide and
10 cm deep
...
Find the volume of a cylinder of radius 5
...
5 cm
...

(3)
4
...
35 m wide and has a diameter of 0
...
What area will it roll in making 40
revolutions?
(4)
5
...
5 cm and
base diameter 6
...

(3)
6
...
1
...
70 cm

4
...
Find (a) the volume and (b) the surface area of a
sphere of diameter 25 mm
...
A piece of alloy with dimensions 25 mm by
60 mm by 1
...
Assuming no wastage, calculate the height of the
cylinder in centimetres, correct to 1 decimal
place
...
Determine the volume (in cubic metres) and the
total surface area (in square metres) of a solid
metal cone of base radius 0
...
20 m
...

(6)
12
...
2 m by 90 cm by 60 cm
...

(4)
13
...

(8)
14
...
0 m long and diameter 1
...
2
...
(1 litre = 1000 cm3
...
6 cm

3
...
0 m

Figure RT11
...
1

7
...
4 cm3
...
(4)
8
...
It has a bore of 80 mm
and an outside diameter of 100 mm
...
(4)

(5)
15
...
0 cm and end diameters 60
...
0 cm
...
A boat has a mass of 20 000 kg
...
If the model is
made of the same material as the boat, determine
the mass of the model (in grams)
...
Plot a graph of y = 3x 2 + 5 from x = 1 to x = 4
...

(16)
18
...
The inside
diameter of the tower at different heights is given
in the following table
...
0 10
...
0 20
...
0 13
...
7

8
...
0

Determine the area corresponding to each diameter and hence estimate the capacity of the tower in
cubic metres
...
A vehicle starts from rest and its velocity is
measured every second for 6 seconds, with the
following results
...
2 2
...
7 5
...
0 9
...
e
...

(6)

List of formulae
Laws of indices:

Areas of plane figures:

a m × a n = a m+n
a m/n =

√
n m
a

am
an

= a m−n (a m )n = a mn

a −n =

Quadratic formula:
If ax 2 + bx + c = 0

1
an

Area = l × b

(i) Rectangle

a0 = 1
b

√
−b ± b2 − 4ac
x=
2a

then

Equation of a straight line:

l

(ii) Parallelogram Area = b × h

y = mx + c

Definition of a logarithm:
If y = a x

then

h

x = loga y

Laws of logarithms:
log(A × B) = log A + log B
 
A
= log A − log B
log
B

b

(iii) Trapezium

log An = n × log A

1
Area = (a + b)h
2
a

Exponential series:
ex = 1 + x +

x2 x3
+
+···
2! 3!

h

(valid for all values of x)
b

Theorem of Pythagoras:
b 2 = a 2 + c2

(iv) Triangle

Area =

1
×b×h
2

A

c

B

b
h

a

C
b

List of formulae
Area = πr 2 Circumference = 2πr

(v) Circle

(iii) Pyramid
If area of base = A and
perpendicular height = h then:

r

s

␪

Volume =

1
× A×h
3

r

2π radians = 360 degrees

Radian measure:

h

For a sector of circle:
θ◦
(2πr) = rθ
360

(θ in rad)

1
θ◦
(πr 2 ) = r 2 θ
360
2

(θ in rad)

s=

arc length,

shaded area =

Equation of a circle, centre at origin, radius r:
x 2 + y2 = r 2
Equation of a circle, centre at (a, b), radius r:

Total surface area = sum of areas of triangles
forming sides + area of base
(iv) Cone
1
Volume = πr2 h
3
Curved Surface area = πrl

(x − a)2 + (y − b)2 = r 2

Total Surface area = πrl + πr2

Volumes and surface areas of regular
solids:
l

(i) Rectangular prism (or cuboid)

h

Volume = l × b × h
Surface area = 2(bh + hl + lb)

r

(v) Sphere

l

h

4
Volume = πr3
3
Surface area = 4πr2

b

(ii) Cylinder
Volume = πr2 h
Total surface area = 2πrh + 2πr2
r

r

h

337

338 Basic Engineering Mathematics
Areas of irregular figures by approximate
methods:
Trapezoidal rule


 
width of 1 first + last
Area ≈
interval
2 ordinate

+ sum of remaining ordinates
Mid-ordinate rule
Area ≈ (width of interval)(sum of mid-ordinates)
Simpson’s rule



1 width of
first + last
Area ≈
ordinate
3 interval




sum of even
sum of remaining
+4
+2
ordinates
odd ordinates

For a general sinusoidal function y = A sin (ωt ± α),
then
A = amplitude
ω = angular velocity = 2π f rad/s
ω
= frequency, f hertz
2π
2π
= periodic time T seconds
ω
α = angle of lead or lag (compared with
y = A sin ωt )

Cartesian and polar co-ordinates:
If co-ordinate (x, y) = (r, θ) then

y
r = x 2 + y 2 and θ = tan−1
x
If co-ordinate (r, θ) = (x, y) then
x = r cosθ and y = r sin θ

Mean or average value of a waveform:
area under curve
length of base
sum of mid-ordinates
=
number of mid-ordinates

mean value, y =

Triangle formulae:
Sine rule:
Cosine rule:

b
c
a
=
=
sin A sin B
sin C
a 2 = b2 + c2 − 2bc cos A

B

b

a

If a = first term and d = common difference, then the
arithmetic progression is: a, a + d, a + 2d,
...

The n’th term is: arn−1
Sum of n terms, Sn =

A
c

Arithmetic progression:

a (1 − r n )
a (r n − 1)
or
(1 − r )
(r − 1)

If − 1 < r < 1, S∞ =

a
(1 − r )

C

Area of any triangle
1
= × base × perpendicular height
2
1
1
1
= ab sin C or
ac sin B or
bc sin A
2
2
2

a +b+c
= [s (s − a) (s − b) (s − c)] where s =
2

Statistics:
Discrete data:

#
mean, x¯ =

x

n
)
(
 #

(x − x¯ )2

standard deviation, σ =
n

List of formulae
Grouped data:

#
fx
mean, x¯ = #
f

)
 #$
%(

f (x − x)
¯ 2

#
standard deviation, σ =
f

Standard integrals
y
axn
cos ax

Standard derivatives
sin ax

y or f(x)

dy
= or f (x)
dx

axn

anxn−1

sin ax

a cos ax

cos ax

−a sin ax

eax

aeax

ln ax

1
x

eax
1
x

&

a

y dx
x n+1
+ c (except when n = −1)
n +1

1
sin ax + c
a
1
− cos ax + c
a
1 ax
e +c
a
ln x + c

339

Answers

Answers to practice exercises
Chapter 1

Chapter 2

Exercise 1 (page 2)
1
...

7
...

13
...

17
...
16 m
3
...
£565
6
...
−36 121
9
...
1487
12
...
−70872
15
...
25 cm
d = 64 mm, A = 136 mm, B = 10 mm

1
...

4
...

7
...


5
...

13
...


Exercise 2 (page 5)
(a) 468 (b) 868
2
...

(a) 259 (b) 56
8
...


2
...

6
...

10
...


(a) 12 (b) 360
(a) 90 (b) 2700
(a) 3 (b) 180
(a) 15 (b) 6300
(a) 14 (b) 53 900

Exercise 4 (page 8)
2
...
68

3
...
5

DOI: 10
...
00040-5

3
...

10
1
14
...
1
21
6
...


16
...


2
6
8
2
...

35
9
11
3
1
7
...

5
13
5
3
2
12
...
3
2
5
12
4
3
1
17
...
13
4
9
(a) £60, P£36, Q£16

7
...

15
...


22
9
8
25
3
16
4
27
17
60

4
...

12
...

20
...
5
9
...
−33
10
...
1
2
12
14
...


71
8
11
15
3
16
51
8
52
17
20

19
...

15
5
...
4
20
...
2880 litres

Exercise 7 (page 14)
1
18
7
6
...
2
1
...
22
11
...

(a) 4 (b) 24
(a) 10 (b) 350
(a) 2 (b) 210
(a) 5 (b) 210
(a) 14 (b) 420 420

2
...

3
...

7
...


1
7
4
11
17
30
43
77
9
1
40

1
...
4

1
9
19
7
...
2
20
2
...
1
8
...
7
3

4
...

15
5
...
0
...
14
...
1
...


13
20

7
40
21
1
141
(b)
(c)
(d)
(e)
25
80
500
11
3
7
...
10
25
200
11
1
7
(b) 4
(c) 14
(d) 15
40
8
20
2
...
4
40
41
10
...
(a)

11
...
625

9
250

3
...


6
125

9
...

4
...

10
...


2
...
11
...
185
...
8307
5
...
1581
6
...
571
5
...
1
...
0
...
068
11
...
5 ×10
12
...
5 ×103
−6
−3
4
...
202
...
18
...
6
...
0
...
11
...
0
...
14
...
1
...
2
...
65
...
0
...
329
...
18
...
43
...
72
...
12
...
−124
...
4
...
0
...
−

9
10

4
...
732
8
...
0
...
0
...
0
...
−0
...
0
...
0
...
5
...
2
...
0
...
0
...
0
...
998 2
...
544
3
...
02 4
...
42
456
...
434
...
626
...
1591
...
444 10
...
62963
11
...
563 12
...
455
13
...
8
...
(a) 24
...
812
(a) 0
...
0064 17
...
4˙ (b) 62
...
4
...
0
...
3
...
13
...
50
...
53
...
36
...
12
...
0
...
46
...
1
...
2
...
2
...
30
...
0
...
219
...
5
...
5
...
52
...
0
...
25
...
591
...
69
...
17
...

4
...

10
...
11927
6
...
0944
10
...
325

2
...
30
...
84
...
10
...
2
...


Exercise 10 (page 19)

1
...

9
...

16
...


2
...
0
...
137
...
19
...
515
...
15
...
52
...
0
...
80
...
295
...
59 cm2
159 m/s
0
...

5
...

11
...
78 mm
0
...
8 m2
281
...

6
...

12
...
5
5
...
5
2
...

4
...

10
...


£589
...
508
...
V = 2
...
5
5
...
81 A 6
...
79 s
E = 3
...
I = 12
...
s = 17
...
184 cm2 11
...
327
(a) 12
...
p
...

(c) 13
...
15 h

342 Basic Engineering Mathematics
Exercise 26 (page 43)

Chapter 5

1
...
£66 3
...
450 g 5
...
56 kg
6
...
00025 (b) 48 MPa 7
...
76 litre

Exercise 21 (page 34)
1
...
32%
2
...
4% 3
...
7% 4
...
4%
5
...
5%
6
...
20
7
...
0125 8
...
6875
9
...
462% 10
...
2% (b) 79
...
(b), (d), (c), (a) 12
...

14
...
A = , B = 50%, C = 0
...
30,
2
17
3
F = , G = 0
...
85, J =
10
20

Exercise 27 (page 45)
1
...
170 fr
3
...
8 mm
4
...
(a) 159
...
5 gallons
6
...
4 MPa
7
...
2 mm 8
...

3
...

7
...

14
...
8 kg 2
...
72 m
(a) 496
...
657 g
(a) 14%
(b) 15
...
49% 11
...
2%
2
...
5
...
73 s 4
...
36% 6
...
76 g
9
...
17% 13
...
3
...
25%
37
...
7%

1
...
5 weeks
2
...
(a) 9
...
12 (c) 0
...
50 minutes
5
...
375 m2 (c) 24 × 103 Pa

Chapter 7
Exercise 29 (page 48)

Exercise 23 (page 38)
1
...

9
...

14
...

16
...
5%
2
...
£310
4
...
£20 000 7
...
45 8
...
25
£39
...
£917
...
£185 000 12
...
2%
A 0
...
9 kg, C 0
...
3 t
20 000 kg (or 20 tonnes)
13
...
5 mm 17
...
27
6
...
128
7
...
100 000
8
...
24
10
...
96
9
...
1 6
...
16 4
...
01 10
...
76
7
...
1000 9
...
36 13
...
34 15
...
25
1
1
1
17
...
49 19
...
5
20
...
128

2
...
36 : 1

2
...
5 : 1 or 7 : 2 3
...
96 cm, 240 cm 5
...
£3680, £1840, £920 7
...
£2172

Exercise 31 (page 52)
1
...
9

Exercise 25 (page 42)
1
...
76 ml
3
...
12
...
14
...
25 000 kg

147
148
17
13
...


2
...
±3
10
...
64

19
56

32
25
1
7
...


11
...
4

1
2

4
...
±
13
1
12
...

3
...

6
...

12
...

16
...

20
...


Exercise 35 (page 62)

m3

cubic metres,
2
...
metres per second, m/s
kilogram per cubic metre, kg/m3
joule
7
...
watt
radian or degree
10
...
mass
electrical resistance 13
...
electric current
inductance
17
...
pressure
angular velocity
21
...
m, ×10−3
−12
6
×10
24
...

3
...

7
...


Exercise 33 (page 56)
1
...
39 × 10 (b) 2
...
9762 × 102

4
...

11
...


3
...
401 × 10−1 (b) 1
...
23 × 10−3

15
...
(a) 5 × 10−1 (b) 1
...
306 × 102 (d) 3
...

4
...

8
...


1
...
(a) 2
...
317 × 104 (c) 2
...
(a) 1
...
004 × 10 (c) 1
...
8a 2
3
...
5
6
...

9
...
3x
14
...
3 p + 2q 17
...
z 8
4
...
a 8
5
...
(a) 1
...
1 × 105

7
...
x −3 or

9
...
(a) 2
...
4 × 10−1
(c) 3
...
11 × 10−1 MeV
(e) 9
...
241×10−2 m3 mol−1

Exercise 34 (page 58)
1
...

5
...

9
...

13
...

17
...


60 kPa
50 MV
100 kW
1
...
50 m
13
...

4
...

8
...

12
...

16
...

20
...
15 mW
55 nF
0
...
5 mV
15 pF
46
...
025 MHz
0
...
0346 kg
4 × 103

6q 2
1
−
2
5x z 2

2ab
2a 2 + 2b2

Exercise 37 (page 66)

6
...
7 (c) 54100 (d) 7
7
...
0389 (b) 0
...
008
1
...


2 × 102

343

1
x6
1
13
...
p6 q 7r 5
10
...
n 3
6
...
t 8

9
...
c14

1
15
...
s −9
x 12
y
18
...
x 5 y 4 z 3 , 13 20
...
b−12 or

Exercise 38 (page 67)
1
...
a −4 b5 c11
1
7
...


1+a
b

3
...


p2 q
q−p

6
...


a 11/6 b1/3 c−3/2

√
6

z 13

√
6 11 √
3
a
b
or √
c3

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

354 Basic Engineering Mathematics
Exercise 132 (page 314)

3
...
7225, median 4
...
72
4
...
2, median 126
...
16, 8

Exercise 126 (page 301)
1
...
85 kg
2
...
7 cm
3
...
5, median 89, mode 89
...
Mean 2
...
02152 cm, mode
2
...
28x 3

2
...
6x 2 − 5

5
...
1 +

Exercise 127 (page 303)
1
...
60
2
...
83μF
3
...
53 MPa, standard deviation 0
...
0
...
9
...
0
...
30, 25
...
5 days
2
...
Q 1 = 164
...
5 cm, Q 3 = 179 cm,
7
...
37 and 38; 40 and 41
5
...
(a)
2
...
(a)
5
...
(a)

8
...
4 − 8x
13
...


1
√
2 x

3
x4

(a) 12 cos3x (b) −12 sin 6x
6 cos 3θ + 10 sin 2θ
3
...
707
4
...
2 A/s
6
...
4 V/s
12 cos(4t + 0
...
72)

Exercise 135 (page 321)
1
...
2
...
664

2
...
0, 11, −10, 21

6
...
1 + √
17
...
6t − 12
20
...
− 3 + √
9
x
x
2 x
1
1
1
21
...
12x − 3 (a) −15 (b) 21
23
...
−6x 2 + 4, −9
...

2
...

7
...
6 (b) 0
...
15
2
...
64 (b) 0
...
0768
4
...
4912 (b) 0
...
38% (b) 10
...
0227 (b) 0
...
0169

1
...

2
9
...
2222 (b) or 0
...
1655 (b)
or 0
...
4964
139
1
1
1
5
(b) (c)
4
...

3
...

6
...
2x − 1

3
...
(a) −1 (b) 16
4
2
6
2
...
(a) 36x 2 + 12x (b) 72x + 12
2
...
(a) − 5 + 3 + √ (b) −4
...
−12 sin 2t − cos t
5
...
16

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

Index
Elevation, angle of, 191
Engineering notation, 57
Equation of a graph, 135
Equations, 73
circles, 236
cubic, 161
indicial, 115
linear and quadratic, simultaneously,
110
quadratic, 102
simple, 73
simultaneous, 90
Equilateral triangle, 171
Evaluation of formulae, 28
trigonometric ratios, 185
Expectation, 306
Exponential functions, 118
graphs of, 120
Expression, 73
Exterior angle of triangle, 171
Extrapolation, 133
Factorial, 119
Factorization, 69
to solve quadratic equations, 102
Factors, 5, 69
False axes, 142
Formula, 28
quadratic, 106
Formulae, evaluation of, 28
list of, 336
transposition of, 83
Fractions, 9
addition and subtraction, 10
multiplication and division, 12
on calculator, 26
Frequency, 200, 289
relative, 289
Frequency distribution, 293, 296
polygon, 293, 296
Frustum, 252
Full wave rectified waveform, 260
Functional notation, 313, 315
Gradient, 134
of a curve, 314
Graph drawing rules, 133
Graphical solution of equations, 155
cubic, 161
linear and quadratic, simultaneously,
110
quadratic, 156
simultaneous, 155
Graphs, 130
exponential functions, 120
logarithmic functions, 116

reducing non-linear to linear form,
147
sine and cosine, 199
straight lines, 130, 132
trigonometric functions, 195
Grid, 130
reference, 130
Grouped data, 292
mean, median and mode, 300
standard deviation, 302
Growth and decay, laws of, 125
Half-wave rectified waveform, 260
Hemisphere, 249
Heptagon, 219
Hexagon, 219
Highest common factor (HCF), 5, 51,
66, 69
Histogram, 293–296, 300
Hooke’s law, 42, 142
Horizontal bar chart, 289
component, 269, 283
Hyperbolic logarithms, 111, 122
Hypotenuse, 172
i, j,k notation, 277
Improper fraction, 9
Indefinite integrals, 328
Independent event, 307
Index, 47
Indices, 47, 64
laws of, 48, 64
Indicial equations, 115
Integers, 1
Integral calculus, 313
Integrals, 325
definite, 328
standard, 326
Integration, 313, 325
of ax n , 325
Intercept, y-axis, 135
Interest, 37
Interior angles, 165, 171
Interpolation, 132
Inverse proportion, 40, 45
trigonometric function, 185
Irregular areas, 257
volumes, 259
Isosceles triangle, 171
Lagging angle, 200
Laws of algebra, 61
growth and decay, 125
indices, 48, 64, 316
logarithms, 113, 150

precedence, 6, 71
probability, 307
Leading angle, 200
Leibniz notation, 315
Limiting value, 314
Linear and quadratic equations
simultaneously, 110
graphical solution, 160
Logarithms, 111
graphs involving, 116
laws of, 113, 150
Long division, 4
Lower class boundary, 293
Lowest common multiple (LCM), 5,
10, 75
Major arc, 231
sector, 230
segment, 231
Maximum value, 156, 199
Mean, 299, 300
value of waveform, 260
Measures of central tendency, 299
Median, 299
Member of set, 289
Mid-ordinate rule, 257
Minimum value, 156
Minor arc, 231
sector, 230
segment, 230
Mixed number, 9
Mode, 299
Multiple, 5
Multiplication in algebra, 62
law of probability, 307
of fractions, 12
of number, 3, 19
Table, 3
Napierian logarithms, 111, 122
Natural logarithms, 111, 122
Newton, 53
Non right-angled triangles, 205
Non-terminating decimals, 18
Nose-to-tail method, 267
Numerator, 9
Obtuse angle, 165
Obtuse-angled triangle, 171
Octagon, 219
Ogive, 293, 297
Ohm’s law, 42
Order of precedence, 6, 13, 71
with fractions, 13
with numbers, 6
Origin, 131

357

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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