Notesale: Turn your study into money

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---

Chapter 3

Decimals
3
...

There are a number of everyday occurrences in which we
use decimal numbers
...
5 MHz FM; 107
...

In a shop, a pair of trainers cost, say, £57
...
95 is
another example of a decimal number
...
95 is a decimal
fraction, where a decimal point separates the integer, i
...

57, from the fractional part, i
...
0
...
95 actually means (5 × 10) + (7 × 1)

 

1
1
+ 9×
+ 5×
10
100

3
...

Problem 1
...
375 to a proper fraction in
its simplest form
0
...
375 may be written as
i
...

1000
375
0
...
1016/B978-1-85617-697-2
...

3
Hence, the decimal fraction 0
...

Problem 2
...
4375 to a mixed number

0
...
4375 may be written as
i
...

10000
4375
0
...

7
(v) Hence, 0
...
4375 = 3
16
number
...
Express as a decimal fraction
8
To convert a proper fraction to a decimal fraction, the
numerator is divided by the denominator
...
8 7 5
8 7
...
Place the 0 above the 7
...
000
(iii) 8 into 70 goes 8, remainder 6
...

(iv) 8 into 60 goes 7, remainder 4
...

(v) 8 into 40 goes 5, remainder 0
...

7
Hence, the proper fraction = 0
...

13
Problem 4
...
Place the 5 above
the next zero
...
8125
16
13
Thus, the mixed number 5 = 5
...

Now try the following Practice Exercise
Practice Exercise 8 Converting decimals to
fractions and vice-versa (answers on
page 341)
1
...
65 to a proper fraction
...


Convert 0
...


3
...
175 to a proper fraction
...


Convert 0
...


5
...

(a) 0
...
84
(c) 0
...
282
(e) 0
...
Convert 4
...

7
...
44 to a mixed number
...


Convert 10
...


9
...
4375 to a mixed number
...
Convert the following to mixed numbers
...

 0
...
0 0 0 0

(a) 1
...
35

12
...
Place the 8 above
the first zero after the decimal point and carry the
2 remainder to the next digit on the right, making
it 20
...
Place the 1 above
the next zero and carry the 4 remainder to the next
digit on the right, making it 40
...
Place the 2 above
the next zero and carry the 8 remainder to the next
digit on the right, making it 80
...
125

5
as a decimal fraction
...

16
7
as a decimal fraction
...

16
9
Express as a decimal fraction
...
Express

(i) 16 into 13 will not go
...

(ii) Place the decimal point above the decimal point
of 13
...
275
(e) 16
...

14
...


3
...


18 Basic Engineering Mathematics
For example,
3

3
= 3
...
For
example,
5
1 = 1
...
is a non-terminating decimal
7

Note that the zeros to the right of the decimal point do
not count as significant figures
...
Express 14
...


The answer to a non-terminating decimal may be
expressed in two ways, depending on the accuracy
required:

2
...
7846 correct to 4 significant figures
...
Express 43
...


correct to a number of significant figures, or

(b) correct to a number of decimal places i
...
the
number of figures after the decimal point
...

For example,

3
...
3792 correct to 2 decimal places
...
Express 1
...

6
...
0005279 correct to 3 significant
figures
...
714285
...
714 correct to 4 significant figures
= 1
...

The last digit in the answer is increased by 1 if the next
digit on the right is in the group of numbers 5, 6, 7, 8 or
9
...
7142857
...
7143 correct to 5 significant figures
= 1
...

Problem 5
...
36815 correct to
(a) 2 decimal places, (b) 3 significant figures,
(c) 3 decimal places, (d) 6 significant figures
(a)

15
...
37 correct to 2 decimal places
...
36815 = 15
...

(c)

15
...
368 correct to 3 decimal places
...
36815 = 15
...

Problem 6
...
004369 correct to
(a) 4 decimal places, (b) 3 significant figures
(a)

0
...
0044 correct to 4 decimal places
...
004369 = 0
...


3
...
This is demonstrated in the
following worked examples
...
Evaluate 46
...
06 + 2
...
09
and give the answer correct to 3 significant figures
The decimal points are placed under each other as
shown
...

46
...
06
2
...
09
52
...
Place 5 in the hundredths column
...

(ii) 8 + 0 + 4 + 0 + 1 (carried) = 13
...
Carry the 1 into the units
column
...
Place the 2 in
the units column
...


Decimals
(iv) 4 + 1(carried) = 5
...

Hence,
46
...
06 + 2
...
09 = 52
...
4, correct to 3
significant figures

19

Now try the following Practice Exercise
Practice Exercise 10 Adding and
subtracting decimal numbers (answers on
page 341)
Determine the following without using a calculator
...
Evaluate 64
...
77 and give the
answer correct to 1 decimal place
As with addition, the decimal points are placed under
each other as shown
...
46
−28
...
69
(i) 6 − 7 is not possible; therefore ‘borrow’ 1 from
the tenths column
...
Place
the 9 in the hundredths column
...
This gives 13 − 7 = 6
...

(iii) 3 − 8 is not possible; therefore ‘borrow’ from the
hundreds column
...
Place
the 5 in the units column
...


Evaluate 37
...
6, correct to 3 significant figures
...


Evaluate 378
...
85, correct to 1 decimal
place
...


Evaluate 68
...
84 − 31
...


4
...
841 − 249
...
883, correct
to 2 decimal places
...


Evaluate 483
...
44 − 67
...


6
...
22 −349
...
336 +56
...


7
...
1
...

82
...
Place the 3 in the hundreds column
...
41

8
...
67

Hence,
64
...
77 = 35
...
7 correct to 1 decimal place
Problem 9
...
64 − 59
...
66 +
38
...
64 + 38
...
14
...
826 + 79
...
486
...
140
−139
...
654
Hence, 351
...
486 = 211
...
7, correct to 4 significant figures
...
1

3
...

This is demonstrated in the following worked examples
...


Evaluate 37
...
4

(vi) 6 × 12 = 72; place the 72 below the 82
...

(viii) Bring down the 5 to give 105
...
500
(x) 8 × 12 = 96; place the 96 below the 105
...


(i) 376 × 54 = 20304
...


(ii) As there are 1 + 1 = 2 digits to the right of
the decimal points of the two numbers being
multiplied together, 37
...
4, then

(xiii) 12 into 90 goes 7; place the 7 above the first
zero of 442
...
6 × 5
...
04

(xiv) 7 × 12 = 84; place the 84 below the 90
...

(xvi) Bring down the 0 to give 60
...
Evaluate 44
...
2, correct to
(a) 3 significant figures, (b) 2 decimal places
44
...
2 =

44
...
2

The denominator is multiplied by 10 to change it into an
integer
...
Thus,
44
...
25 × 10 442
...
2
1
...

 36
...
500
36
82
72
105
96
90
84
60
60
0
(i) 12 into 44 goes 3; place the 3 above the second
4 of 442
...
500
(iii) 44 − 36 = 8
...
500
442
...
875
(xviii) Hence, 44
...
2 =
12
So,
(a)

44
...
2 = 36
...


(b) 44
...
2 = 36
...

2
Problem 12
...
666666
...
666666
...
667 correct to 4 significant figures
...
6666

...
6 recurring and is

...
6
Now try the following Practice Exercise
Practice Exercise 11 Multiplying and
dividing decimal numbers (answers on
page 341)
In Problems 1 to 8, evaluate without using a
calculator
...


1
...
57 × 1
...
500

2
...
92 × 0
...


Evaluate 167
...
3

4
...
6 × 1
...


Evaluate 548
...
2

6
...
3 ÷ 1
...


7
...
48 ÷ 0
...


8
...
4 ÷ 1
...

4
, correct to 3 significant figures
...

10
...
1 , correct to 4 significant figures
...


5
, correct to 3 decimal places
...
13 , correct to 2 decimal places
...
8 , correct to 3 significant figures
...
53

15
...
8 ÷ 17, (a) correct to 4 significant figures and (b) correct to 3 decimal
places
...
0147
, (a) correct to 5 decimal
2
...


16
...



...
6
(b) 5
...
Evaluate (a)
1
...
A tank contains 1800 litres of oil
...
75 litres can be filled from
this tank?

21

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
...


£58

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
...

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
...
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, 2

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