Mathematics for Technicians CD Blair Alldiss

CD Supplement to accompany

Mathematics for Technicians f i f t h e d i t i o n

Blair Alldis

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C HAPTER

1

Fractions and decimals

1.1 Integers Integers are whole numbers, either positive or negative.

Addition and subtraction Think of additions and subtractions as operations on a bank account, i.e. as deposits and withdrawals.

Example 23959

You commence with $2 in the account; you deposit $3, then

4

withdraw $9. Final balance $4, an ‘overdraft’ of $4.

Since the order of transactions does not affect the final result, it is often easier to change the given order.

Example 2 3 2 4 1 3 9 6

Total deposits $9 and total withdrawals $6.

3

When the same number is both added and subtracted, it is easiest to disregard it.

Example 2 3 2 4 1 3 4 1 3

Since a negative number has the opposite effect to a positive number, a double negative is equivalent to a positive. For example, 3 is the opposite of 3; also, 3 is the opposite of 3, and is equal to 3. Chapter 1 Fractions and Decimals

1

In this course we do not require a rigorous mathematical proof of this but will rely upon commonsense because it seems clear that in any language, a double negative gives a positive. For example: if I say then I am saying which means

‘I am not talking nonsense’ ‘I am not talking no-sense’ ‘I am talking sense’

(a double negative statement) (a positive statement).

Example Subtracting 2 from 5, we have: 5 2 5 2 7

Remember : Subtracting a negative quantity means adding that quantity.

Note: Do not use a calculator for Exercises 1.1 to 1.4.

Exercises 1.1 1

a d

2

3 1 5 2

Take: a 3 from 7 Take 5 from 2

26 17

c

e

f

5 2 235

b

8 from 6

c

3 from 8

b

Add 5 and 2

c

Add 8 and 8

b

3

a

4

In each case below write down the new temperature after the change described: Original temperature (°C)

5

Change in temperature (°C)

a

7°

rise of 3°

b

4°

rise of 9°

c

0°

drop of 6°

New temperature (°C)

What is the net result of: a a loss of 8 dB followed by a gain of 3 dB? b a loss of 6 dB followed by a gain of 8 dB?

Multiplication and division As with addition and subtraction, the operations may be performed in any order. 2

CD Supplement to accompany Mathematics for Technicians

Example 238694

           

296384 18 6 3 8 4

3

384

1

84

←This is a much more convenient order for computation, working from left to right

2

Exercises 1.1 (continued) 6

a

91 234 91 2

b

234568

When two numbers are multiplied or divided, the result is positive if the two numbers have the same sign and negative if the two numbers have different signs.

Examples 3 5 15

1

2

3 5 15 8 4 2 (or 8 4 2 (or

2 3 6 6

8 2) 4 8 2) 4

6 2 3 (or 2 3) 6

6 2 3 (or 2 3)

Signs the same, result is positive.

Note:

6 3 6 3 6 3

2 2 2

2 3 6

}

Signs different, result is negative.

A fraction is negative if it contains one negative sign.


a c

8

Multiply 3 by 5 Multiply 3 by 5

b d

Divide 8 by 4 Divide 16 by 2

Write down the missing number in each case below: a 4 . . . 12 b 8 . . . 4 c 4 . . . 20 d 16 . . . 4 Chapter 1 Fractions and Decimals

3

1.2 Order of operations By international agreement, when a string of numbers contains a variety of different operations, the order in which the operations are to be performed is: 1

remove brackets (i.e. evaluate the contents);

2

evaluate all and ;

3

evaluate all and .

Example

       

         

6 4 (2 1) 2 1 (5 1) 64

3

21

6

6

4

(brackets removed) ( and performed)

4

8

( and performed)

Note: With ‘nested’ brackets, the innermost brackets must be worked first.

Examples 1

17 [3 (2 3)] 17 [3 5] 17 15 2

2

2 [10 (3 1) 2] 2 [10 4 2] 2 [10 8] 22 4

Exercises 1.2 1

a

1 (3 2 4)

2

a

7 [2 (3 1) 1]

b

1 (2 3) 4 b

c

3 (1 2) (1 4)

2 [3 (4 1) 4 1]

Squares and square roots N 2 is the square of the number N; it is that number multiplied by itself. For example, 62 36.

N is the positive square root of the number N, i.e. the positive number that must be multiplied by itself in order to obtain that number. For example, 9 3 because 3 3 9. 4


Since 3 3 9 and also 3 3 9, the number 9 (and all other positive numbers) has two square roots. The positive square root of 9 is 9 ( 3). The negative square root of 9 is 9 ( 3). Note:

17 17 17 2 (41 ) 41 41

and

412 41 41

41

41

Numbers under a square root sign should be regarded as though they were within brackets. For example, 6 2 5 and (6 2 5) and (6 2 5) all mean the same. The value is 16 or 4.

Note:

(2 3)2 52

The operation inside the brackets must be evaluated first.

25 9 16 25 5

The square root sign acts as a bracket, so the operation under it must be evaluated first.


a

2 2 3 42 5 42

b

(3 4)2 102 62

1.3 Fractions In this section, we will revise fractions. There are three common notations for real numbers: fractions, decimals and percentages.

Example 23 100

0.23 23%

It is essential that you can perform operations on all such numbers and quickly convert between these different notations.

Equivalent fractions 2

You will remember that 3 of a quantity is the amount we obtain if we divide the quantity into 3 equal parts and take 2 of them. Chapter 1 Fractions and Decimals

5

To obtain the same amount, we could divide the quantity into 6 equal parts and take 4 of them, and so on.

Example 1 3 2 6

These are called equivalent fractions

3 9

It follows that if the numerator and denominator of a fraction are multiplied by the same number, the size of the fraction is unchanged; we obtain an equivalent fraction. Hence, also, the numerator and denominator may be divided by the same number to obtain an equivalent fraction. When as many as possible such divisions are made, the fraction is expressed in its simplest form and is said to be in its ‘lowest terms’.

Example 2250 225 45 9 3 5250 525 105 21 7

Divisibility rules Example Which of the numbers 2, 3, 4, 5 and 9 will divide into 31 836?

Solution First note that 3 1 8 3 6 21. Hence the answer is: 2 (because 2 divides into 6) 4 (because 4 divides into 36) 3 (because 3 divides into 21)

Exercises 1.3 1

Which of the numbers 2, 3, 4, 5 and 9 divide into a 123 456? b 234 567?

2

Reduce each of the following fractions to lowest terms: a

6

13 5 150

b

12 6 189


c

38 4 432

d

30 3 75 40 500

Cancelling Dividing a fraction above and below by the same number is often called ‘cancelling’ when this number occurs both in the denominator and in the numerator.

Example 7 5 13 5 13 3 11 7 3 11

Note: You can ‘cancel’ only when all the numbers in the numerator are multiplied and all the numbers in the denominator are multiplied.

Examples 1

7 5 13 3 11 7

2

7 5 13 3 11 7

The sevens cannot be cancelled here because of the ‘plus’ sign (in the denominator). The sevens cannot be cancelled here because of the ‘minus’ sign (in the numerator).

Exercises 1.3 (continued) Cancel first if possible, then express in simplest form: 234 321 2 3 a b 3 13 2 321

c

98 17 19 98

Mixed numbers A mixed number consists of an integer and a fraction. You should be able to convert mixed numbers to fractions and vice versa.

Example 1

2

2

33 3 3 9

2

3 3 11

3

2

19 4

16

3

4 4 3

4 4 3

44

Chapter 1 Fractions and Decimals

7


Convert the following improper fractions to mixed numbers (first reduce to lowest terms if necessary): a

5

13 5

39 4

b

139 12

c

d

78 30

1224 504

e

Convert the following mixed numbers to improper fractions: a

2

53

b

7

120

5

86

c

d

11

1012

5

1212

e

7

1111

f

Addition and subtraction of fractions Examples 1

2 threes 5 threes 7 threes

2

2 elevenths 5 elevenths 7 elevenths, i.e. 11 11 11

3

Similarly, 7 thirteenths 3 thirteenths 4 thirteenths, i.e. 13 13 13

2

5

7

7

3

4

To add or subtract fractions, we express them with the same denominator.

Example 5 7 5 5 20 14 24 12 24 6 24 24 11 24

5 7 5 20 5 14 24 12 6 24 11 24

or

Exercises 1.3 (continued) In the following exercises, all working is to be done in fractions and answers should be given as proper fractions or mixed numbers. a

3 25

100

e

1 3

4 12

3

7

a

1 2

5

8

a

1 2

3 4

6

7

5

2

1

1

b

3 8

12

5

f

2 3

12 24

b

3 4

6

11

1

1

b

3 4

1

c

1 9

6

g

3 8

12 3

c

2 3

6

2

10 5

5

5

2

1

c

1 4

d

3 5

20

h

3 4

8 5

d

1 30

1

7

5

2

11

20

1

3 5

Addition and subtraction of mixed numbers Express the fractional parts with the same denominator, then deal with the integral and fractional parts separately. 8


Examples 1

3

9

5

10

274 356 2712 3512 9

10

(27 35) (12 12 ) 19

6212 7

6312 2

1

3

4

15

24

15

165 124 1620 1220 1520 1220

(note this step)

9 320


a

1

2

482 363

b

4

1

405 204

c

2

2

715 483

d

7

4

318 495

Multiplication of fractions Multiply the numerators; multiply the denominators. ‘Cancel’ first, when possible.

Examples 1

16 2 4 2 242 105 3 5 7 357

3

5 52 122 10 3 13 13 39 318

2

7 93 9 3 27 10 10 5 5 7 50

Multiplication of mixed numbers Convert to fractions first.

Examples 1

5 105 1 1 14 33 3 24 25 6

2

11 22 5 2 1 45 14 5 42 11 2

1

46

1

52

Multiplication of a mixed number and an integer Multiply the integral and fractional parts separately. Chapter 1 Fractions and Decimals

9

Examples 1

3 3 5 574 (5 57) (5 ) 4 285

15 4

204

3

3

e 12 a

3 8 4 17 6 17 18 1 1 1 2 24

2

15 2

204 72 1

2884

11 a

5

3

1

285 34

Exercises 1.3 (continued) 2 10 a 3 7 b

12 8

5

12 178 (12 17)

2

2112

1

4 25

c

2

d

c

2 7

8

g

437 437

c

25 18

2

32 4 2 4

3 26 13 7 12 24 35 1 2 3 2 3

b f b

3

2225 3

34 3

7

123

2

7

898 2

e

5

51

d

98 98

h

8 9

d

34 13

15

28 3

1

Division of a fraction by a number Multiply by the reciprocal of that number.

Examples 1

7 8

7

1

2 8 2

1

7 7

7

1

2 4

16

7

8 3

6 7

3

2

6

4

4 7 3

1

2

8

7

8

3

3

2 8

1

9

17

Exercises 1.3 (continued) 3 13 a 1 4 b 1 1 14 a 23 12 b

3

12 23 2 3

4

16

3 4

2 1

247 3

c

3 8

1

c

1 32

2 1

d

2 3

d

288 8

5 7

7

3

e

4 7

e

44 14

14 1

Complex fractions These are most easily simplified by multiplying top and bottom by the same number. 10


3

Examples 1

2 3

3

1

12 49 17 25

2

6

24 33 64 34 1 5 53 26

13

33

17

30 6

11

10

(The original fraction was multiplied by 6.)

1

110

Exercises 1.3 (continued) 1 4

2

3 1

15 a

b

6

5 6

3 2

2

3 7

c

12

4

3

8 5 1 1 8 2 8

Squaring a fraction or a mixed number The square of a number is that number multiplied by itself.

Examples 1

3 4

2

2

3 3 4 4 9 16

A fraction is squared by squaring the numerator and squaring the denominator.

1 3 4

2

7 4

2

49

16

To square a mixed number, first convert it to a fraction.

1 316

Note: From the above, it can be seen that: 9 33 16 44 3

4

1 49 316 16 7

4 3

14 This method can be used only in special cases such as those shown above where the numerator and denominator are both perfect squares. Other cases require a calculator or tables to find the square root.

Exercises 1.3 (continued) 16 A wheel is rotating at a constant speed of 3 revolutions in each 2 s. a b

1

What fraction of a revolution is made in 4 s? 5 What fraction of a second does the wheel take to make 9 of a revolution? Chapter 1 Fractions and Decimals

11

17 A pendulum is timed to make 40 oscillations in 30 s. a b c

How long does it take to make 1 oscillation? 1 How long does it take to make 72 oscillations? 1 How many oscillations does it make in 52 s? 1

18 A turntable rotates at 333 rpm. How many revolutions does it make in: 1 1 a 20 min? b 42 min? c 12 h? 5

1

19 An alloy contains (by mass) 8 copper, 4 zinc and the remainder tin. In a sample of this

alloy, what fraction of the mass is tin? 1

1

20 A 1 L container is 2 full of liquid and a 2 L container is 3 full. What is the total number of

litres of liquid present? 1

21 How many dollars does a person earn by working for 43 h at $4.50/h? 22 Find the efficiency of a machine when its mechanical advantage (MA) is 5 and its velocity

MA 1 ratio (VR) is 72. (Use efficiency .) VR

1.4 Decimal fractions Addition and subtraction of decimals

Exercises 1.4 1

a

0.86 23.45

b

23.73 628.4

c

82.5 1.56

2

a

16 0.894

b

324.6 8.94

c

263 4.89

Multiplication and division by powers of 10 Exercises 1.4 (continued) 3

a

0.32 10

b

12.34 100

4

a

8 400

b

0.006 500

Multiplication of decimals To multiply decimals, follow these steps: 1

Multiply the numbers, ignoring the decimal points.

2

Insert the decimal point in your result, making sure that the number of decimal places in the result is the same as the total number of decimal places in the original numbers.

Examples 1

12

23.4 0.13 First multiply: 234 13 3042 Since there are three decimal places in the original numbers (one in 23.4 and two in 0.13), we must have three decimal places in the result. Hence, the answer is 3.042.


2

1.23 0.0024 123 24 2952 We must have six decimal places in the result. ∴ answer 0.002 952 (note two zeros inserted to give six decimal places)

Cases such as the above, where the multiplication cannot be done mentally, would normally be done using a calculator, but it is important that you should be able to use the method above if a calculator is not available.


a

0.023 0.45

b

0.034 1.6

Cases such as the following should be computed mentally: 6

a c

6.83 0.05 0.9 0.3

b d

0.004 0.6 1.1 0.12

Division by an integer Divide as usual and insert the decimal point when you come to it. Keep the decimal points aligned.

Examples

1

2

↓ 32.41 6194.4 6

194.46 6 32.41

↓ 3.082 2680.1 32 78

80.132 26 3.082

2 13 2 08 52 52 0


a

62.1 3

b

9.15 15

c

17.028 36

Converting a fraction to a decimal Write the numerator as a decimal, then divide in the usual way. Note: 7 7.000 000 . . .

Chapter 1 Fractions and Decimals

13

Examples 1

5 8

0.625

0.625 85.0 000 0. . .

2

7 16

0.4375

0.437 5 167 .0 000 0. . . 64 60 48 120 112 80 80 0


Convert the following fractions to decimals (exact answers required): a

5 8

b

3 16

c

13 40

d

17 80

e

13 32

Rounding up and down If the denominator has only 2 and 5 as its prime factors, as in the cases above, the decimal terminates, and the fraction can be expressed exactly as a decimal. However, if the denominator contains other prime factors, the decimal will not terminate and a recurring (or ‘repeating’) decimal will be obtained. These decimals are written with dots or bars over the recurring digits.

Examples 1 2

1 3 2114 990

. 0.333 333 3 … 0.3

.. –– 2.135 353 535 3 … 2.135 or 2.135

In such cases the results are given only to the accuracy required.

Examples 3 11

.. 0.27272727 . . . ( 0.27 )

When written correctly to one decimal place, the result is 0.3, since 0.3 is closer to the correct result than is 0.2. Written correct to two decimal places, the result is 0.27, since this is as close to the correct result as we can approximate using only two decimal places.

As can be seen from the above, if the next decimal place is 5 or greater, you should raise the last digit to obtain the more accurate approximation. 14


Division by a decimal Convert the fraction to an equivalent fraction such that the denominator is an integer, then proceed as before.

Examples 2.64 26.4 1 1.7 17 . . .

2

6.8 1.234

6800

1234 . . .


Evaluate the following (without using a calculator) and state each result exactly: a 91.8 1.35 b 15 0.625

10 Obtain each answer mentally:

0.84 30 b 0.4 1.5 11 Obtain each answer mentally: a 2 0.04 b a

c

0.3 0.15

0.9 0.003

d

c

14.4 0.12

2.4 0.008

Note: Do not use a calculator for the following exercises. It is important that this simple arithmetic can be performed without using any aid.

12 Voltage V volts (V) produces a current of I amperes (A) through the resistor of R ohms ().

Given that V I R, find the voltage needed R I to produce a current of 0.2 A through a resistor of 0.3 . b Given that V I R, find the voltage needed to produce a current of 0.4 A through a resistor V of 2.45 . V c Given that I , find the current produced through a resistor of 120 by 14.4 V. R V d Given that I , find the current produced through a resistor of 40 by 3.6 V. R e Given that R V I, find the size of the resistor such that 20 V produces a current of 0.4 A. 1 13 The ‘conductance’ of a resistor is given by G siemens (S), where R is the resistance in R ohms (). Find the conductance of a wire whose resistance is: a 0.2 b 50 c 20 d 0.04 e 0.001 f 2.5 g 400 a

14 The spot on an oscilloscope screen makes one sweep each 0.004 s. a b

Find the time taken for it to make 50 sweeps. What is the frequency of the sweep (in sweeps/s)? Chapter 1 Fractions and Decimals

15

V2 R R . Find the power dissipated when: a the voltage is 0.5 V and the resistance is 5 b the voltage is 1 V and the resistance is 0.2

15 The power dissipated by a resistor is watts, where V is the voltage across a resistance of

1.6 Significant figures Exercises 1.6

16

1

State the number of significant figures in each of the following measurements: a 7.0 g b 6.004 kg c 160 0.5 g d 0.430 t e 0.055 kg f 100.0 g

2

Express each of the following measured quantities correct to 3 significant figures in the same unit of measurement as given: a 50.012 3 mm b 0.000 803 2 kg c 1.000 826 m d 0.607 080 km

3

Perform the operations on the following measured quantities, expressing each result in square metres with the appropriate number of significant figures: a 2.00 m 1.43 m b 2.00 m 1.4 m c 2.0 m 1.43 m d 2 m 1.43 m e 20 m 3.4 m f 20.0 m 3.4 m


C HAPTER

2

Ratio, proportion and percentage

2.1 Ratio Exercises 2.1 1

The densities of silver and osmium are 10.5 Mg/m3 and 22.5 Mg/m3 respectively. a What is the ratio of the density of silver to that of osmium? b If the mass of a certain volume of osmium is 30 g, what is the mass of an equal volume of silver?

2

The coefficients of linear expansion for glass and invar are respectively 0.000 008 K1 and 0.000 001 6 K1. What is the ratio of the coefficient for invar to that for glass?

In the following exercises, unit conversions are required. 3

The masses of two castings are 625 g and 1.5 kg. a What is the ratio of the larger mass to the smaller mass? (1 kg 1000 g) b Is the mass of the smaller casting more than one-half the mass of the larger?

4

Two forces act upon a body, one of 3 MN and the other of 50 kN. What is the ratio of the smaller force to the larger? (1 MN 1000 kN)

5

A mass of 212.4 kg is divided into two parts, the masses being in the ratio 2 : 7. What is the mass of the larger part?

6

An object is divided into two parts, the masses being in the ratio 3 : 1. If the mass of the smaller part is 6.5 kg, find the mass of the original object.

2.2 Direct variation: the unitary method Exercises 2.2 It may be assumed in each case that the quantities involved do vary directly. 1

When 24 V is applied across a certain resistor, the current produced is 8 mA. What voltage is needed to produce a current of 13 mA?

Chapter 2 Ratio, Proportion and Percentage

17

2

The potential difference across 292 turns of the secondary winding of a transformer is 4 V. Across how many turns will the potential difference be 3 V?

3

A particular inductance has a reactance of 12 at a frequency of 100 Hz. At what frequency will it have a reactance of 156 ?

2.3 Direct variation: the algebraic method Exercises 2.3 1

A certain silicon junction diode passed a current of 20 mA with a potential difference of 0.6 V, and a current of 35 mA with a potential difference of 0.7 V. Does this verify that the current is proportional to the potential difference?

2

When a current of 50 mA passes through a certain moving-coil meter, the torque on the coil is 40 106 Nm. When the current is 30 mA, the torque is 24 106 Nm. Does this verify that the torque is proportional to the current?

3

At a frequency of 800 Hz, the reactance of a certain inductor was 140 . The reactance was 175 at a frequency of 1000 Hz. Does this verify that the reactance of an inductor is proportional to the frequency?

4

For a mild steel cable of length 20 m and diameter 8 mm, a load of 500 kg produces an extension of 10 mm and a load of 750 kg produces an extension of 15 mm. Does this verify that the expansion produced is proportional to the load?

Computations Exercises 2.3 (continued)

18

5

The mass of a piece of sheet metal of area 1350 mm2 is 23.7 g. Find the mass of a piece having area 855 mm2.

6

When the depth of water in a cylindrical tank is 1.27 m, the volume of water in the tank is 8.43 m3. Find the volume of water in the tank when the depth of the water is 2.58 m.

7

When a current of 30.0 mA passes through a certain moving-coil meter, the torque on the coil is 45.0 Nm. Given that the torque is proportional to the current, find the current needed to produce a torque of 35.0 Nm.

8

The magnetic field strength inside a certain long solenoid is 1.50 mT when it carries a current of 1.20 A. Given that the field strength is proportional to the current, find the current needed to produce a field strength of 1.25 mT.

9

When a heating coil is immersed in a given quantity of water in an insulated container, the rate at which the temperature rises varies directly as the square of the current passed through the coil. Given that for a certain coil immersed in a certain mass of water, the temperature rises by 9.7°C/min when the current is 3.4 A, find: a the rate at which the temperature rises when the current is 4.7 A


b c

the current required to raise the water temperature by 11.0°C/min the rise in temperature of the water when a current of 2.3 A is passed for 85 s

10 For ball-bearings made from any particular material, the mass of a ball varies directly as the

cube of its diameter. Given that for a particular material a ball of diameter 2.00 mm has mass 4.12 mg, find: a the mass of a ball whose diameter is 3.50 mm b the diameter of a ball whose mass is 7.35 mg c the number of balls of diameter 3.00 mm needed to make up a total mass of 10.0 g (correct to the nearest 10 balls) (l g 1000 mg)

2.4 Inverse variation Exercises 2.4 1

The frequency of oscillation, f, of a simple pendulum varies inversely as the square root of the length of the pendulum. Given that a pendulum of length 24.8 cm makes one oscillation per second, find: a the frequency of oscillation of a pendulum whose length is 10 cm (answer correct to 3 significant figures) b the number of oscillations made in 1 minute by a pendulum whose length is 1.50 m (answer correct to 3 significant figures) (1 m 100 cm) c the length of the pendulum that makes 10 oscillations per minute (answer in metres correct to 3 significant figures)

2.5 Joint variation Exercises 2.5 1

The pressure of a given mass of gas varies directly as the absolute temperature and inversely as its volume. For a certain mass of a gas, under a pressure of 0.730 kPa, the volume is 6.73 L when its temperature is 310 K. Find: a the pressure of the gas when its volume is reduced to 5.30 L and its temperature is increased to 345 K b the temperature of the gas when its volume is 8.63 L and its pressure is 1.82 kPa

2

When a uniform rectangular beam of given length and width is supported at each end and a load placed at its centre, the deflection produced by the load at the centre of the beam varies directly as the load and inversely as the cube of the thickness of the beam. Given that for a certain beam of thickness 5.5 mm, the deflection produced by a load of 35 kg is 2.5 mm, find the deflection produced by a load of 45 kg placed at the centre of a beam whose thickness is 6.0 mm, given that the length and width of the beam remain unchanged. Chapter 2 Ratio, Proportion and Percentage

19

Proportion exercises: miscellaneous Exercises 2.5 (continued) 3

The mass of 15.0 m of a particular wire is 126 g. Find: a the mass of 2.45 km of this wire (1 km 1000 m) b the length of this wire that has a mass of 53.4 mg (1g 1000 mg)

4

A constant potential difference of variable frequency is applied across a certain capacitor. When the frequency is 50 Hz, the current is 35 mA. Given that the current is proportional to the frequency, find the frequency that gives a current of 55 mA.

5

The energy stored in a given charged capacitor varies directly as the square of the voltage, V, across its plates. For a certain given capacitor the energy stored is 20 J when the voltage is 1000 V. Find: a the energy stored when the voltage is 300 V b the voltage across the plates when the stored energy is 15 J

6

Looking out to sea, the distance to the horizon varies directly as the square root of the height of the observer’s eye above sea level. Given that the distance to the horizon is 3.0 km when the observer’s eye is 5.0 m above sea level, find: a the distance to the horizon for a person whose eye is 7.2 m above sea level b how far above sea level a person’s eye must be in order to see a distance of 6.0 km

7

For an LC series circuit containing a given inductance, the resonance frequency, f, varies inversely as the square root of the capacitance C. For a particular such circuit the resonance frequency is 200 kHz when the capacitance in the circuit is 0.16 F. Find: a the resonance frequency when the capacitance is 0.30 F b the capacitance required to give a resonance frequency of 3.0 kHz

8

The volume of fluid that flows through a pipe each second varies directly as the pressure difference maintained between its ends, directly as the fourth power of the diameter of the pipe and inversely as the length of the pipe. For a given fluid at a given temperature it is found that 43 mL/s flows through a pipe whose diameter is 5.0 mm and length 3.5 m when a pressure difference of 100 kPa is maintained. Find the rate of flow of this fluid through a pipe whose diameter is 7.0 mm and length 4.5 m when a pressure difference of 130 kPa is maintained.

2.6 Percentages The following should be known: 1 2 1 5

20

50% 0.5 20% 0.2

1 4 1 3

25% 0.25 . 1 333% 0.3


1 10

10% 0.1

Example 8.37% of 16.2 t 0.0837 16.2 t 1.36 t

Expressing any number as a percentage N 100 N N 100 N% since N% . So, to express any number as a percentage, 100 100 multiply it by 100.

Examples 1

0.673 0.673 100% 67.3%

2

5 7

5

7 100% 71.4%

Exercises 2.6 1

Convert to decimals: a

2

7

8

0.61%

d

1.03%

e

0.034%

0.36

b

0.04

c

1.3

d

0.0023

e

17.25

3 5

b

17 20

c

11 25

d

39 50

e

425 1000

3 8

b

17 40

d

31 60

e

3 7

c

1 3

Express in the form x%, where x is a decimal (correct to 1 decimal place): a

6

c

Express in the form x%, where x is a mixed number: a

5

421%

Convert to percentages: a

4

b

Convert to percentages: a

3

63%

2 7

b

3 8

c

5 6

2

d

23

d

872%

i

333%

7

e

19

e

413%

j

647%

Express as fractions reduced to lowest terms: a

25%

f

163%

2

b

40%

g

833%

Express as fractions: 4 a 50% of 7 2

d

20% of 3

g

16% of 8

5

Express as decimals: a 3% of 1.2 d 29% of 10

1

3

c

34%

h

622%

1

3

b

25% of 5

e

40% of 14

h

b e

1 1

2 2

1

c

75% of 2

f

9% of 3

133% of 7

i

32% of 122

7% of 3.62 6.8% of 5

c

6% of 0.78 85% of 0.04

5

5

f

2

1

1

Chapter 2 Ratio, Proportion and Percentage

21

g 9

12% of 1.2

Find: a 17% of 88 kg

h

b

24% of 150

7.8% of 97 m

c

i

18% of 0.5

1

352% of 78.0 t

d

215% of 125 mm

brake power (BP) indicated power (IP) following cases, find the mechanical efficiency as a percentage: a a diesel engine whose BP 26.1 kW, IP 31.0 kW b a double-acting steam engine whose BP 133 kW, IP 150 kW

10 The mechanical efficiency of an engine is given by . In each of the

11 A compound belt and countershaft drive has input power 11 kW and output power 9.3 kW.

output Find the efficiency () of the drive as a percentage. input

22


C HAPTER

3

Measurement and mensuration

3.1 SI units Exercises 3.1 1

Express each of the following in terms of the basic unit, either as an integer or a decimal: a 1 km b 1 mg c 1 Mm d 1g e 1t f 1 m

2

Express: a 37 mm in metres c 63 m in kilometres

b d

0.52 t in kilograms 23 000 g in tonnes

3.2 Estimations Exercises 3.2 (The answers given are only approximate. Any answer of about the same magnitude will satisfy.) 1

A person in a library estimates that there are about 25 books on each metre of shelving, about 50 shelves each about 6 metres long, and 20 shelves that are about 20 metres long. What is her estimate of the total number of books on the shelves?

3.3 Approximations Exercises 3.3 In each case below approximate the value mentally to 1 decimal place. 3.28 2 2 1 (5.86 π 32.4) 2 7.63 19.2

3.4 Accuracy of measurement Exercises 3.4 1

State the MPE, correct to 1 significant figure, in each of the following measurements: Chapter 3 Measurement and Mensuration

23

a 2

72 m 0.5%

b

2.4 t 8%

State the MP%E, correct to 1 significant figure, in each of the following measurements: a 35 2 mm b 2.8 0.5 t c 56 500 200 km

3.6 Implied accuracy in a stated measurement Exercises 3.5 1

For each of the following stated measurements, find: (i) the MPE, (ii) the MP%E and (iii) the limits of accuracy of the measurement: a 0.123 g b 5.2 kg c 823 t

2

Write each of the following numbers correct to 2 decimal places: a 3.8449 b 12.994 c 12.999

3.7 Additional calculator exercises involving squares and square roots Exercises 3.6 Use a calculator to evaluate the following correct to 3 significant figures: 1

5.182 3.172 2.982

2

5.26 9.21

3

(1.12 2.13 2.09 )

4

5

45.6 67.9 1.26 8.72

6

95.9 73 .4

7

(3.172 2.272)3

8

65.2 18.7 3 (4.22 1.88)

2

2

2 2

2 8.212 5.14 2 1.26 2.99

3.8 Pythagoras’ theorem Exercises 3.7 1

24

Starting from point P, a person walks due north a distance of 250 m and then due east a distance of 180 m to point Q. What percentage of the distance walked would the person have saved by ‘cutting the corner’ (i.e. by walking in a straight line from P to Q)?


2

3

Pulley wheels having diameters of 84.0 mm and 226 mm are situated with their centres 855 mm apart. Find the length of the connecting belt between P and Q, the points where the belt contacts the wheels.

Q P

In a flight of steps, each tread is 287 mm and each riser is 213 mm. Find the length of the handrail required to be placed above the 13 treads.

handrail

13th tread 213 287

4

In the ideal figures below, the lengths of the sides of the triangles are labelled, in metres. Find the exact values of the pronumerals. (The marked angles are right angles.) a

b x 15

13

18

12 10 t

26

3.11 The area of a rectangle Exercises 3.9 1

A steel tape measure is 50 m long and 11.4 mm wide. What is the area of one side of this tape in square metres?

3.12 The area of a triangle Exercises 3.10 In the (ideal) figures below, find the shaded areas. Exercises requiring application of Pythagoras’ theorem are labelled (P). Chapter 3 Measurement and Mensuration

25

A

1

9m

4m 26 m

B A

3

E

B

10 m

2

C

D

4

1m

C

2m

AC 7 m AD 5 m BE 4 m Find: a the area of the shaded triangle b the length of segment BC

5m

5 6m

26 m

10 m C

6

(P)

7 4m

C

13 m

2.5 m 3m 3m

C is the centre of the circle.

3.13 The circle: circumference and area Exercises 3.11 State answers correct to 3 significant figures.

26

1

Find the area of a circle whose circumference measures 34.6 mm.

2

Find the circumference of a circle whose area is 2050 m2.

3

Find the circumference of a circle whose area is 0.644 m2.


4

Find the areas of the shaded regions. (The points labelled C mark the centre of a circular arc.) a

b

6m C 6m

6m

3m 5m

6m

c

d

e

f

30 m

8.4 m

C

C

8.4 m

20 m 5.3 m

g

C

64 mm

86 mm

3.14 Volumes of prisms Exercises 3.12 1

Find the volume of this symmetrical container in litres: 2.0 m 160 mm 110 mm 120 mm

Chapter 3 Measurement and Mensuration

27

2

A cylindrical rod has diameter 8.00 mm and length 3.50 m. Find its volume in cubic centimetres.

3

Find, in cubic metres, the volume of 650 km of wire of diameter 1.5 mm.

4

What length of copper rod of diameter 8.00 mm can be made from 1.00 m3 of copper?

5

A cylindrical can contains 1.0 L of liquid. Given that its internal diameter is 84 mm, find the depth of liquid in the can, in millimetres.

3.15 Surface areas of prisms Exercises 3.13 Find the surface areas of the following bodies described in Exercises 3.12. For the purposes of these exercises, assume that all dimensions given are correct to 0.1%. State all answers correct to 3 significant figures.

28

1

The internal surface area of the open container in question 1, in cm2.

2

The solid rod in question 2, in cm2.

3

The wire in question 3, in m2.

4

The internal surface area of the open can in question 5 in cm2, given that the depth of the can is 199 mm.


C HAPTER

4

Introduction to algebra

4.1 Substitutions Exercises 4.1 1

2

3

4

Given that a 2, b 3 and c 0, evaluate: 2 2 2 a a b c b bc Given that a 2 and b 3, evaluate: a 1 a b 2

b

c

3a 2b

c

mk2 mk

a 2a 2 3b b

Given that m 2, k 1 and t 3, evaluate: a mkt b ktm If y 2 3x, evaluate y when: a x3 b x 2

c

x0

d

x 1

4.2 Addition of like terms Exercises 4.2 1

Simplify: 2 2 a 2x 3x 5x x b 3ab 2a ab 3b a 2 2 2 2 c 5x y 2x 2xy 3x y x 4xy 2 2 2 2 2 2 2 2 2 d 2a b 2a 3b b 2a a b ab e pq p q 3p pq 2q 3pq

2

Simplify: 2 2 a 3x 5x 5x 4x c 2x 3 7x 1 6x 2

b d

5x2 3x 2x2 2x pq p q 3p 2q 5pq

Chapter 4 Introduction to Algebra

29

4.3 Removal of brackets Exercises 4.3 1

2

Express without brackets, then simplify: a 3x 2 (1 2x) b 3 (k 1) d (x 3) (3 x) e (2m 3) (m 7) Express without brackets, then simplify: a ab (a b) b (p q) (p q)

c

5a (3 2a)

c

(ab a) (ab a) a

4.4 Multiplication and division of terms Exercises 4.4 1

2

Write down the following products in simplest form: a 2a 4b b xx d xy y 2 e 2x ax 1 g 3a 2b 4ab h 3xy y 6x Simplify: a 8ab 4a d 5(a b) (a b)

b e

5a2b 5b 12(x y) 6(x y)

f

3xx 3m 2km

c

3abc 2ac

c

1

4.5 The distributive law Exercises 4.5 1

Multiply out so that the following are expressed without brackets: a 2(3a 2b) b 4(5 2a) c 2x(3x a) d 3ab(a 2b) e 5(1 2x) x(3x 10)

2

Multiply out to express without brackets: a (x 2)(x 3) b (x 4)(x 3) d (a 3)(p q) e (2 5b)(b a)

c

Multiply out to express without brackets: 2 2 a (b 4) b (k 2) 2 d (m 4) e (3 x)(3 x)

c

3

30


f

f

(3 x)(1 2x) (a 3)(a 3) (a 2)(a 2) (2a 3b)2

4.6 HCF and LCM Exercises 4.6 1

2

Write down the HCF of: a 6ab and 3a 2 3 c 12ax and 8a bx

b

Write down, in factored form, the HCF of: a a(b c) and x(b c) c (p q)(a b) and q(a b)

b

d

d

6ab and 9bc 2 3 3 2 12a b c and 18a bc (a b) (x y) and p(a b) 2 2 3 12k (x y) and 16k (x y)

3

Reduce the following fractions to lowest terms by dividing the numerator and denominator by their HCF: (m t)(3k y) 6ab a(b c) a b c (3k y)(a b) 9bc x(b c)

4

Write down the LCM of each pair of terms in question 1.

5

Write down, in factored form, the LCM of each pair of expressions in question 2.

4.7 Algebraic fractions Exercises 4.7 1

2

3

Simplify: 2 3 a b 2 7ax 7ax2 Express as a single fraction: 3 2 a b 2a a

c

a b ab ab

1 1 R 3R

c

5 3 2 x x

Express as a single fraction: 1 2x 2 3x a 4 6

b

4

a

3 t m 3

5

a

973 973

b

6

a

a x b y

b

7

a d

8

5k 2k y y

a

b 85

a 7 b a a m 3 ab ab 7

b e

a1 1b a ab

2x 3y 5m 2x

c

17m 97a2b 2 97a b m k t t x 5 y m x x

c c c

b

6a 10c 5c 18ab 2 13n k 2y 2 6x y m 3xy x m

m m t

xy xy k 2m Chapter 4 Introduction to Algebra

31

9

Express as single fractions: k a k m

b

2 3m 3 m

c

1x x x1

c

1 k k 1 m k

Complex fractions Exercises 4.7 (continued) 10 Express as single fractions in simplest form: a

1 k

1 1 k

b

a b b a

m

1 ab

4.8 Algebraic fractions of the form Exercises 4.8 1

2

3

Simplify: 8 4n a 3 4 Express as single fractions: n 1 2n a 3 5 Express as single fractions: 2C 1 C1 a 4 2

b

10 15C 5 C 5

c

12 6L 7 3

b

2C 1C 3 2

c

1 7x 5 6 12

b

k5 k5 6 2

c

x 2y 2x y 3 8

4.9 Two important points Exercises 4.9 1

2

3

Simplify: xy a xy

32

p q qp

Write as a positive fraction in simplest form: a2 a b 3

c

5R R5

x3 x 2

If P 3 2n and Q 2 5n, express the following in terms of n: a

4

b

3P 4Q

b

P 2Q

c

1 P 5

2

3Q

2t 1 3t If A and B , express the following in terms of t: 3 2 1 1 a 6A 2B b AB c 2A 3B


ab c

4.10 Solving linear equations Exercises 4.10 1

2

3

4

Solve: a 5x2x c 3 2x 3 e 1 4x 2 g 3RR4 i 2W 3 5W 1 W k K 3 4K 3

6

d f h j l

Solve: a 2(x 3) 5 c x 2(1 x) e 3x 1 3(2x 1) g 7 3(R 2) 2(1 3R) i 2a (a 2) 0 Solve: 3d 7d a 2 3 5 10 2 x 1 c 12 x 4 3 1 7x x3 e 2 3 2 2k 5k g 1 3 3 6 x 3 1 i 3x 4 2 2

2a 5a 1 2a 1L1L x 3 4x 2 2d 3 1 2d 5 3x 5 6x 3 1

j

4 3(2 x) 3x 1 (2 x) 3(1 W) 1 (W 2) 3(l 1) 3 (l 2) R 2(1 2R) 4(R 1)

b

2 l 3

d

3 22a 52 4a

f

m 3 3m 1

h

b b 2 1 3 2

j

22y 1 3y

c

3 2 1 k k

b d f h

5

2 6l 1

1

1

1

1

1

2

Solve: a

5

b

1 2 5 x

b

1 2 x 3

Solve for the pronumeral: 1k k2 a 3 2 3 t c 4 t2 Solve: m3 a 0 2

b d

b

(3t 1) 0 t3

d

2 3 2x 1 3x 2x

2a 3 3a 4 3 2 4 k3 2 3k c

2x 4 0 x4

Chapter 4 Introduction to Algebra

33

4.11 Solving more difficult linear equations Exercises 4.11 1

2

Solve: C 3 5C a 3 4 2 2E 3 5E c 1 4 6

d

V1 1 1V 2 3 6 3T 5 3 3T 4 4

Solve: a c

3

b

x1 3 0 2 1W 1 W 5

b d

2E 5 E 0 3 x 3x 18 2 2 4

Solve: 3

a

(n 2) n 3(2n 1) 6 n 1n

b

2(3 x) x 5 4 3( x 4)

4.12 Using linear equations to solve practical problems Exercises 4.12 1

The mass of one casting is 2.50 times the mass of another. If the total mass of the two castings is 16.7 kg, find the mass of the lighter casting.

2

It is known that resistor R2 is 18.7 greater than resistor R1. The two resistors in series have a resistance of R1 R2 56.3 . Find the value of R1 by setting up a single equation containing R1.

3

A machine can produce article A in 12 minutes or article B in 16 minutes. If it is planned to produce twice as many of A as B in a working week of 40 hours, how many of each will be produced, assuming that the machine works non-stop?

4.13 Simultaneous linear equations Exercises 4.13 1

a b c

34

Does (x 1, y 5) satisfy the equation y 2x 3? Does (x 1, y 5) satisfy the equation y 3x 1? Does (x 1, y 5) satisfy the equation y 7x 2?


d e f

{ yy 2x3x 31 y 3x 1 Is (x 1, y 5) the solution to these simultaneous equations? { y 7x 2 y 2x 3 Is (x 1, y 5) the solution to these simultaneous equations? { y 7x 2 Is (x 1, y 5) the solution to these simultaneous equations?

4.14 The substitution method Exercises 4.14 Use the substitution method to solve the following pairs of simultaneous equations: 1

a

{

2 E 3 2 E 3

1

4R 2

b

1

3 2R

{

1 W 2

2

3F 4 0 1

W 3F 5 0

4.15 The elimination method Exercises 4.15 1

Solve for the pronumerals, using the elimination method (giving answers as integers, fractions or mixed numbers): 2a 3t 0 3I1 2I2 4 3E 8R 0 a

2

4a 7t 1

b

{

c

5I1 6I2 26

{

1

2E 4R 3

Solve for the pronumerals, using the elimination method (giving answers as integers, fractions or mixed numbers): a

3

{ {

1

4R1 13R2 4 3

3R1 4R2 6

b

{

1

22x 3y 3 3

2x 24y 1

Solve for the pronumerals, using the elimination method, giving all values as decimal fractions: a

{

5V1 19V2 3.5 2V1 7V2 8.2

b

{

7C1 2C2 7 9C1 5C2 13

4.16 Practical problems Exercises 4.16 1

When a quantity of liquid is heated in an insulated container, it is known that its final temperature (T°C) is given by T T0 kt, where T0°C is the original temperature, t min is the duration of heating and k is a constant. For a particular mass of a particular liquid, it is Chapter 4 Introduction to Algebra

35

found that after heating for 3 min its temperature is 40°C, and after heating for a total time of 7 min its temperature is 50°C. a Form a pair of simultaneous equations and solve them to find the original temperature of the liquid and the value of the constant k. b Find the total heating time required to raise the temperature to 65°C. c Find the temperature of the liquid after heating it for a total time of 10 min. 2

In the tuned circuit shown, it is required that the ratio of the inductances L1 : L2 2 : 3 and that the sum of the inductances is 420 H.

L1

Form two equations in L1 and L2 and solve them simultaneously to evaluate L1 and L2. L2

3

The resistance of a wire at temperature t°C is given by R1 R0(1 t), where R0 is the resistance at 0°C and is the temperature coefficient of resistance of the material. The resistance at 20.0°C is 26.82 and the resistance at 32.0°C is 28.10 . a Evaluate and R0. b Hence find the ‘inferred zero-resistance temperature’ of the material (the temperature at which the resistance would be zero on the assumption that this linear relationship continued to hold down to this low temperature). I1

4

I2

E1

R1

E2

36

R2

R3

E1

E2

Equations obtained by applying Kirchhoff’s laws to this circuit are: I1R1 (I1 I2)R2 2E1 E2 I1R1 I2R3 E1 2E2 a Given that R1 2 , R2 3 , R3 4 , I1 4 A and I2 2 A, evaluate E1 and E2. b Given that R1 3 , R2 4 , R3 1 , E1 6 V and E2 9 V, evaluate I1 and I2 correct to 3 significant figures.


C HAPTER

5

Formulae: evaluation and transposition

5.1 Evaluation of the subject of a formula Exercises 5.1 Evaluate the following: n 1 when k 6.83, n 263, t 11.4 k t r2 w2 2 when c 86.5, r 32.8, m 1.26, v 92.1, w 27.3 mcv 3

(a2 b2)3 where a 6.92, b 7.05

4

k2 m2 where k 9.42, m 6.32

5

2 2 q s where q 6.39, s 4.17

6

1 3 where d 4.93, f 7.62 (d f)

7

(c2 d3)2 where c 1.27, d 1.85

8

j k where j 5.26, k 9.21

9

(q2 w2 e2)2 where q 1.12, w 2.13, e 2.09

10

2 3 x z where x 2.97, z 2.68

11

ab where a 45.6, b 67.9, c 1.26, d 8.72 cd

5.3 Evaluation of a formula Exercises 5.2 1

If at some instant, a body moving in a straight line with a velocity of u is given a constant acceleration of a, the time it takes from that instant to travel a distance s is given by: 2 u 2 as u t . a

Chapter 5 Formulae: Evaluation and Transposition

37

Find the time taken in the following cases: 2 a u 26.8 m/s, a 5.52 m/s , s 14.2 m 2 b u 28.4 km/h, a 8.13 m/s , s 860 mm 2

The power dissipated by resistance R1 when a potential difference of V is maintained across V2R1 resistances R1 and R2 in series is given by: P 2 . Find the power dissipated by (R1 R2) R1 in the following cases: a R1 8.63 , R2 11.4 , V 15.8 V b R1 1.42 k, R2 2.08 k, V 843 mV

3

At a frequency of f, the impedance of a circuit which consists of a resistance R in series 1 2 2 with a capacitance C is given by: Z R . 2π fC Find the impedance in the following cases: a R 1.86 , f 63.0 Hz, C 0.002 55 F b R 54.5 , f 1.24 kHz, C 2.74 F

4

2 V u 2as , the final velocity of a body, where u initial velocity, a its acceleration, s the distance travelled.

Evaluate v when u 20 km/h, a 8 m/s2, s 685 mm. 5

(m1 m2)g a , the acceleration when masses m1 and m2 hang m1 m2 over a smooth pulley. Evaluate a when m1 1.28 kg, m2 683 g, g 9.80 m/s2. m1

6

7

m2

l T 2 , the period (time of oscillation) of a simple pendulum. g 2 Evaluate T when l 893 mm, g 9.80 m/s .

l2 r2 , the radius of gyration of a solid cylinder 4 12 about the axis shown, where l the length and r the radius.

k

Evaluate k when l 1.28 m, r 93.2 mm.

5.4 Transposition Exercises 5.3 1

a b

38

5 Solve for x: 7 x k Make x the subject of the formula: w x


2

a b

x2 4 Solve for x: 4 5 xa m Make x the subject of the formula: b k

Transpositions involving squares and square roots Exercises 5.3 (continued) Make x the subject of the following formulae: xa a 2b 2 3 5k 2ax 2k 4 n y 5 k m 3 x Miscellaneous In each of the following exercises, for mechanical students, a formula is given followed by a pronumeral in parentheses. In each case, make this pronumeral the subject of the formula (express the given pronumeral in terms of the other variables and constants). 100(m1 mm) 6 Hs cts L (c) 7 P (m1) mm mv mu 8 F (v) 9 E aW b (W) t 2 10 T mg m r (r)

These exercises, intended for electrical students, also provide additional practice for nonelectrical students. I 11 B (a) 12 H (d) a 2d l 13 T F d (F) 14 R (a) a l1 F 15 R (F) 16 V1 E (l) l Questions 1720 involve taking a square or square root. 17 E I Rt 2

1

18 W 2CV

(I)

B (B) 2 Questions 2125 are more difficult. V 21 I (R2) R1 R2 R2 T t2 23 (t1) R1 T t1

(V) 2

N

2

19 W

2

N1

20 R RL

(N2)

22 R2 R1(1 t)

(t)

2

Q C

24 e iR

(C)

The total resistance of this circuit is given by:

25

rΩ

rΩ

4Ω

1 R 2 1 r 4 Make r the subject of this formula.


39

5.5 Transpositions in which grouping is required Exercises 5.4 In each exercise below, a formula is given followed by a pronumeral in parentheses. In each case make the pronumeral in parentheses the subject of the formula: 1 3

3 my ky E n kE E K t

(y)

2

(E)

4

x(k n) a b tx x x a kx a b

(x) (x)

Exercises 5.5 Mechanical In each case, make the pronumeral in parentheses the subject of the formula (express the given pronumeral in terms of the other variables and constants). SI units are used unless otherwise stated. W 1 a (F) Efficiency of a lifting machine, where W is the load and F is the WF friction load. W b (V) Efficiency of a lifting machine where W is the load, V is the E V velocity ratio and E is the effort force. p.l.a.n. 2 P (p) Indicated power of a cylinder, in kW, were p is the mean 1000 effective pressure, l is the length of stroke, a is the crosssectional area of the piston and n is the number of working strokes per second. na Tb 3 For two gear wheels in mesh, where na, nb are the speeds of the nb Ta wheels (rev/s) and Ta, Tb are the numbers of teeth on the wheels. a 4

40

(d2)

b

(d1)

Dryness fraction of steam where m mass of dry steam and ms mass of moisture in suspension.

(m)

Hs cts L

(nb)

Velocity ratio of a Weston differential pulley block, where d1 and d2 are the respective diameters of the larger and smaller pulleys in the top block.

ms q m ms a

6

b

2d1 V d1 d2

a 5

(Tb)

b

(c)

(ms)

The specific enthalpy of dry steam where c the specific heat capacity of water in kJ/kg K, ts the saturation temperature and L the latent heat of water, in kJ/kg.


7

100(m1 mm) P mm

a 8

(m1)

b

T mg m r 2

a 9

Percentage of excess air supplied to a combustion chamber, where m1 total mass of air supplied and mm minimum mass of air required for total combustion of the fuel. A mass m attached to the end of a light rod rotates in a vertical circle. T is the tension in the rod when the mass is at the lowest point of its path. g acceleration due to gravity angular velocity and r radius of circle.

(r)

b

(H h)Ws n Wf C

a

(m)

Thermal efficiency of a boiler, where H enthalpy of steam produced h enthalpy of water supplied Ws mass of steam produced Wf mass of fuel fired C calorific value of fuel

(C)

b

P M C

i 10 i

(Wf)

c

(h)

For an engine, where i indicated thermal efficiency

Pb b M C

b brake thermal efficiency m mechanical efficiency

b m i

a

(mm)

Pi indicated power Pb brake power M rate of fuel consumption C calorific value of the fuel Express M in terms of b, Pb and C. b Express m in terms of Pi and Pb.

Electrical In each exercise below, a formula is given followed by a pronumeral in parentheses. In each case, make the pronumeral in parentheses the subject of the formula. R2 R1

T t2 T t1

11

E1 E 2

E2 E 3

E 4

12

R1 R2

r(2l d ) Rs rd

13

(T)

Increase in resistance with temperature where T is the ‘inferred zero-resistance temperature’.

(E)

Current equation arising from a node-voltage solution of a network.

(d)

The Varley Loop.

14 E Ra(I1 I2) Rc(I1 I3) (I1)

Equation arising from application of Kirchhoff’s laws. Chapter 5 Formulae: Evaluation and Transposition

41

R2 1 bt2 R1 1 bt1

(b)

Increase in resistance with temperature.

16 R1

(Ra)

Delta network equivalent to a star network.

17

The total capacitance of this circuit is given by 5C1 C C1 5

15

RaRb RbRc RcRa Rb C1 F

5 F

Make C1 the subject of this formula.

18 L2 L1 L3

The total inductance of this circuit is L, where 1 1 1 L L1 L2 L3 Make L3 the subject of this formula.

5.6 More transposition and evaluation Exercises 5.6 In these exercises a formula is given. a Evaluate the subject using the data provided. b Change the subject to the pronumeral named. c Evaluate the new subject using the data provided. 1

2

Work, W mgh 2 a g 9.81 m/s , m 5.50 g, h 875 mm b h 2 c g 9.81 m/s , m 2.65 t, W 27.5 MJ Q emf, e iR C a i 3.50 A, R 6.80 k, Q 1.65 C, C 75.0 F b C c Q 1.50 mC, i 645 mA, R 4.70 , e 5.35 V

Squares and square roots mv2 3 Distance, s 2F a m 750 g, v 48.5 km/h, F 1.07 kN 42


b c 4

5

v F 98.5 N, s 1.42 km, m 1.23 t

Force, T (mg)2 P2 2 a g 9.81 m/s , m 1.38 t, P 7.85 kN b m 2 c T 3.50 N, P 950 mN, g 9.80 m/s

1 L Frequency, f R2 2L C a L 7.00 mH, C 8.50 F, R 18.0 b R c L 185 mH, C 15.5 F, f 50 Hz

Grouping of like terms 6

7

8

s vt1 Time interval, t2 v a s 3.62 km, v 50 km/h, t1 1 min 17 s b v c t1 860 s, t2 1.50 ms, s 2.14 mm emf, E I1R1 (I1 I2) R2 a I1 850 mA, I2 1.20 A, R1 1.50 k, R2 850 b I1 c E 240 V, I2 14.5 A, R1 8.20 M, R2 5.60 M I1(R1 R2) Current, I R2 a R1 680 , R2 2.20 k, I1 340 mA b R2 c I 1.05 A, I1 980 mA, R1 82.0 k

Exercises 5.7 Miscellaneous transpositions 1

Make the given pronumeral the subject of the formula: W a (a) b L L1 L2 2M (aW b) V c

2

2 Z R 2 L2

(L)

d

Z

1 R2 C

2

(M) ()

The total capacitance of this circuit is C, where: 1 1 1 1 C C1 C1 5 a Make C the subject of this formula. b Make C1 the subject of this formula.


43

3

When an inductance of L henries is in series with a resistor of R , the total impedance, Z, is given by the diagonal of the rectangle which has sides of lengths R and XL, where the reactance of the inductor is given by XL 2f L . Evaluate Z correct to 3 significant figures when: a R 200 and XL 150 b R 5.50 k and XL 3.60 k c R 2.00 k, L 15.2 mH and f 15 kHz R

L

XL

Z

XL R

44


C HAPTER

6

Introduction to geometry

6.2 Angles Exercises 6.1 In questions 1 and 2 below, find in each case the measure of angle A and state which type of angle it is (acute, right, obtuse, straight or reflex). 1

a c e

2

a c e

A 52°38' 67°43' A 136°38' 67°53' A 83°37' 96°23'

b

A 117°23' 68°56' A 273°36' 93°36' A 127°43' 37°43'

b

d

d

A 37°26' 52°34' A 37°48' 26°35' A 203°38' 18°59' A 131°13' 53°37'

6.3 Complements and supplements Exercises 6.2 1

2

3

Find the: a supplement of 139°41' c complement of 17°22'

b

Find the: a supplement of 2x° c complement of 90°

b

d

d

complement of 88°53' supplement of 38°51' complement of 3y° supplement of k°

In each of the following cases, evaluate the pronumeral: a

b

Chapter 6 Introduction to Geometry

45

c

d

6.5 Parallel lines and a transversal Exercises 6.3 In each case below, evaluate the pronumeral. (Parallel lines are indicated by arrows.) 1

2

3

6.6 The angles of a triangle Exercises 6.4 Evaluate the pronumerals in the following: 1

2

3 110

6.8 Geometrical constructions using a compass Exercises 6.6 1

46

Draw two intervals AB and CD that bisect each other. Check with rule and set square that ADBC is a parallelogram (a quadrilateral whose opposite sides are parallel).


C HAPTER

Geometry of triangles and quadrilaterals

7

7.1 Classification of triangles Exercises 7.1 Evaluate the pronumerals: 1

2 x

3

4

7.2 Detecting equal angles Exercises 7.2 For each of the following figures, state which angles must be equal to each other. It is suggested that in each case you make a quick copy of the diagram you are working on. 1

2

b a c

a

b c

d f e

Chapter 7 Geometry of Triangles and Quadrilaterals

47

3

4

a b d

b

c c

f

d

a e

7.3 The incentre, the circumcentre, the centroid and the orthocentre Construction of triangles and quadrilaterals

Exercises 7.3 1

Using only a straight edge and compass, construct the triangle PQR whose sides have lengths PQ : QR : RP 1 : 2 : 2. By construction, locate the incentre and the circumcentre of this triangle and check that they both lie on the perpendicular bisector of side PQ. Draw both the incircle and the circumcircle.

2

Construct the triangle ABC in which BC 80 mm, ABC BCA 50°. P is the point where the perpendicular bisector of side AB intersects the bisector of CAB. Locate the incentre, I, and measure the length of the interval IP correct to the nearest millimetre.

7.4 Congruent triangles Exercises 7.4 1

In each case below, sketch the two triangles and label the given equal sides and angles. Then state whether the two triangles must be congruent and if so state the test that proves the congruency: a PQT and KLR, given P R, t k, q l b ANK and PLT, given N P 90°, k t, n p

7.5 Similar triangles Exercises 7.5 Evaluate each pronumeral in the figures below. All interval lengths are in metres. It is recommended that, in each case, a proportion be first set up and then solved. Angles and sides marked in the same way are equal. Sides with arrows are parallel. 48


1

2

2 x

3

8 6

2 x

3

4 2.15 6

x

4

3.58

3

2.41 d

5

x

6 k

0.57 3

k+2

0.21 5

7

3.5 3.0

2.0

x

7.6 Special quadrilaterals and their properties Exercises 7.6 1

ABCD is a quadrilateral in which each side is 5 m long. a If one diagonal has length 52 m, what type of quadrilateral is this? What is the length of the other diagonal? b If one diagonal has length 8 m, what type of quadrilateral is this and what is the length of the other diagonal?

2

EFGH is a quadrilateral in which EF is parallel to HG and the perpendicular distance between these two sides is 12 m. EF 30 m, FG 20 m and HE 15 m. FGH and GHE are both acute. a What type of quadrilateral is this? Chapter 7 Geometry of Triangles and Quadrilaterals

49

b c d

Do the diagonals bisect each other? Find the length of side GH. Find the perimeter of this quadrilateral.

7.7 Areas of triangles and quadrilaterals Exercises 7.7 1

In each case below, the lengths of the three sides of a triangle are given. Find the areas of these triangles: a 3.80 m, 4.30 m, 5.10 m b 5.39 m, 6.46 m, 7.83 m

7.8 Areas of the parallelogram and the rhombus Exercises 7.8 In each of the following exercises, M is the point of intersection of the diagonals and h is the distance between the sides AB and CD. D

D

C

A

M

h

M

C

h

A

B

B

All answers are to be given correct to 3 significant figures. Figure 1

2

3

50

parallelogram ABCD

parallelogram ABCD

rhombus ABCD

Data

Find

AB 8.000 m BC 5.000 m h 4.000 m

a

AM 6.000 m BC 5.000 m h 3.000 m

a

AC 13.60 m BD 9.760 m

a


b c

b

b

area AC BD AB area DA area

C HAPTER

8

Geometry of the circle

1

2

x° 70°

C

C

k°°

70°

Evaluate x.

Evaluate k.

3

4

h° C

C

x°

t°

y°

Given that h t 78, evaluate h and t. 5

Express y in terms of x. 6 c° a°°

a° 70°

e°

b°

d°

d° c°

C

2 ° 20

f°

C

80° f°

b°

e°

Evaluate a, b, c, d, e and f.

Evaluate a, b, c, d, e and f. Chapter 8 Geometry of the Circle

51

7

8

k° C

x°

C 30°

70° 70

Evaluate x.

52


50°

Evaluate k.

C HAPTER

Straight line coordinate geometry

9

9.1 The number plane Exercises 9.1 1

In each case below the coordinates of two points are given. State (i) the distance between the two points, and (ii) the coordinates of the midpoint. a P(2, 4) and Q(8, 4) b K(6, 3) and T(1, 9) c D(1, 2) and F(5, 1) d A(3, 2) and B(6, 2)

2

C is the midpoint of the interval AB where B is the point (4, 5) and C is the point (2, 4.5). What are the coordinates of point A?

9.2 The gradient of a line Exercises 9.2 For each of the graphs below, find: a the (horizontal) run from P to Q; c the gradient of the line.

b

1

2

the (vertical) rise from P to Q;

P

10

Q

10 5

5 P

Q 0

1

–4

3

2

3

Q

–2

0

4 Q

5

7

P P –1

0

1

2 –6

–4

–2

0

Chapter 9 Straight Line Coordinate Geometry

53

5

In each case below, find the gradient of the line that passes through the given pair of points: a (2, 0) and (3, 3) b (1, 5) and (1, 11) c (2, 7) and (3, 2) d (1, 2) and (3, 3)

9.4 The straight line equation y mx b Exercises 9.3 1

For each of the lines given by the following equations, write down (i) the gradient, (ii) the y-intercept, (iii) the x-intercept: a 2x y 5 b x 2y 3 0 c 3x 2y 1 0 d y 7 3x

2

Which of the following points lie on the line y 3x 4? A(1, 1) B(2, 2) C(1, 1) E(0, 4) F(3, 13) G(2, 10)

D(2, 10) H(1, 1)

9.5 Finding the equation for a particular straight line Exercises 9.4 1

By substituting into the equation y mx b, find the equation of the line in each case below, given that: a the gradient is 4 and (1, 7) lies on the line b the gradient is 3 and the x-intercept is 6 c the point (2, 5) lies on the line and the gradient is 4 1 1 1 d the y-intercept is 22 and (2, 22) lies on the line 1 e the x-intercept is 2 and the y-intercept is 3

2

Find the equation of the line that passes through the points: a (2, 5) and (4, 1) b (2, 3) and (1, 9)

3

Find the equation for each of the lines in Exercises 9.2, questions 14, assuming in each case that y is plotted against x.

9.7 Plotting and interpolation Line or curve of best fit

Exercises 9.5 1

54

The rate of steam flow through an impulse steam turbine was measured for various power outputs, as shown:


Power output (kW)

760

780

800

820

840

Rate of steam flow (kg/s)

5.52

5.65

5.78

5.90

6.03

Plot the power output (vertical axis) against the rate of steam flow, and from the line of best fit find: a the power output given by a steam flow of 5.73 kg/s b the steam flow needed for a power output of 810 kW c the rate at which the power output increases with the steam flow 2

An insulated liquid is heated and its temperature at various times is read during the process: 0

1

2

3

4

5

20.4

23.5

25.9

29.2

31.6

34.6

Time (min) Temperature (°C)

Plot the temperature (vertical axis) against time, and from the line of best fit find: a how many minutes it took to raise the temperature from 20°C to 30.4°C 1 b the rise in temperature of the liquid during the first 22 min of the heating c the rate at which the temperature increased 3

The following table shows readings taken for the elastic limit load L of a mild steel cable for various diameters d of the cable: d (mm)

5

6

7

8

9

L (t)

0.90

1.30

1.75

2.30

2.90

Plot L (vertical axis) against d2 and from the line of best fit find: a the elastic limit load for a cable having diameter 6.5 mm b the maximum load that may be lifted by a mild steel cable of diameter 7.5 mm, given that the load must not exceed one-fifth of the elastic limit load 2 c the rate at which L increases with d . 4

The emf of a particular thermocouple was measured for different temperatures of the hot junction, the cold junction being maintained at 0°C. Temperature (°C)

500

550

600

650

700

emf (mV)

37.7

40.6

42.8

45.8

48.6

Plot the emf (vertical axis) against the temperature and draw the line of best fit. From your graph determine: a the emf that would be generated for a hot junction temperature of 520°C b the hot junction temperature at which an emf of 45.0 mV would be generated c the rate at which the emf increases with the temperature of the hot junction 5

(This graph is a curve.) For a particular specimen of iron, the relative permeability () was measured when the sample was placed in various magnetic field strengths (H): Chapter 9 Straight Line Coordinate Geometry

55

0

1000

2000

3000

4000

5000

6000

7000

8000

1200

1980

2000

1850

1660

1480

1320

1190

1080

H (A/m)

Plot the relative permeability (vertical axis) against H and draw the curve of best fit. From your graph, estimate: a the relative permeability when H is 500 A/m b the magnetic field strength that gives the maximum relative permeability c the maximum value of the relative permeability

9.8 Plotting graphs from equations Exercises 9.6 1

For a certain lifting machine, the effort force (E kN) required to raise a load (W kN) is given by E 0.056W 0.09. Plot a graph showing the effort required (vertical axis) to lift loads of from 0 to 15 kN. From your graph, determine the effort required to raise a load of 12.5 kN and check your result by substitution into the given equation.

2

A manufacturer’s net annual profit is given by I 43N 3500 dollars, where N is the number of articles produced in the year. Graph I (vertical axis) against N for 0 N 500. a What is the value of the I-intercept, and what is its significance? b What is the value of the N-intercept, and what is its significance? c What is the gradient of this graph, and what does this represent? d From the graph, read the net income that results from the production of 330 articles, and check this reading by substitution into the given equation.

9.9 Graphical solution of linear equations and systems of two (simultaneous) linear equations Exercises 9.7 1

Solve the following pairs of simultaneous equations by plotting the graphs of the two equations on the same set of axes for the values of x given. (Evaluate y for the extreme values of x before drawing the axes and affixing scales to them.) y 2x 5 2x y 4 0 a b y x 3 x 2y 4 0

{

Plot for 2 x 2. 2

56

{

Plot for 4 x 0.

Two thermometers A and B are placed in a beaker of water that is gradually heated and the lengths of the mercury columns LA and LB are measured at different temperatures, the readings being given in the table below:


Temperature (°C)

a b c d e

0

20

40

60

80

100

Thermometer A, LA (mm)

40

90

175

225

310

415

Thermometer B, LB (mm)

85

135

185

205

250

270

On the same axes, plot the graphs of the mercury lengths against temperature and draw the straight lines of best fit. From the graphs, find the temperature at which the mercury columns have the same length and state this length. For each thermometer, write down the equation that connects the length of the mercury column with the temperature. Solve the two simultaneous equations obtained in part c above, using the method of elimination. Compare your answers to b and d above.

Chapter 9 Straight Line Coordinate Geometry

57

C HAPTER

10

Introduction to trigonometry

10.1 Conversion of angles between sexagesimal measure (i.e. degrees, minutes and seconds) and decimal degrees Exercises 10.1 1

Express the following angles in sexagesimal measure (i.e. in degrees, minutes and seconds): a 26.874° b 263.437° c 82.109°

2

Express the following angles in decimal degrees correct to 3 decimal places: a 49°21'52" b 80°46'21" c 168°17'34"

10.2 The tangent ratio: construction and definition Exercises 10.2 1

58

Construct a large right-angled triangle. Mark an acute angle in this triangle. a By measuring the appropriate sides and evaluating a ratio, determine the value of the tangent of this angle as accurately as possible. b Measure the angle using a protractor and use your calculator to determine the value of the tangent of this angle. c Calculate the approximate percentage error in the value you obtained by measurement.


10.3 The tangent ratio: finding the length of a side of a rightangled triangle Exercises 10.3 1

In each of the following triangles, find the length of the side marked with a pronumeral: a

b x 6.24 m 72°

2

For each of the following triangles, find the length of the side marked by a pronumeral, correct to 3 significant figures: a

b

c

3.83 m 53°27'

x

284 mm t

33°52'

10.4 The tangent ratio: evaluating an angle Exercises 10.4 1

Find the acute angles that have the following values for their tangents (i) in decimal degrees correct to 4 significant figures, (ii) in degrees and minutes: a 3.1468 b 23.56 c 2.3347 d 0.008 312

2

Find the angles marked by pronumerals in the following triangles. Record answers both in decimal degrees and in degrees and minutes. a

b 2.13 m

12 mm 5 mm

1.48 m w

k

13 mm Chapter 10 Introduction to Trigonometry

59

3

a b c d

In PQR: Q 90°, P 23°48' and r 35.7 mm; evaluate p. In LNK: N 90°, l 0.681 km and k 0.528 km; evaluate K. In LTM: M 90°, T 63°13', and t 234 mm; evaluate l. In DPT: T 90°, D 38°31' and d 86.3 mm; evaluate p.

10.5 The sine and cosine ratios Exercises 10.5 1

c A

D

B

a

b

e

f C d

E

F

In the triangles above, the sides have lengths, a, b, c, d, e and f metres. It can be seen that f tan F . In the same way, write down the following ratios: d a sin A b cos D c tan B d cos F e sin D 2

Evaluate the pronumerals in the following figures: a

b

100 mm x

x 2m 1.648 m

53.26 mm

3

Use your calculator for these exercises. a In KTW: K 90°, T 51°43' and k 42.6 mm; evaluate w. b In ABE: B 90°, a 514 m and e 732 m; evaluate A.

4

A circle has radius 38.4 mm. Find the angle subtended at the centre by a chord 51.8 mm long.

5

Given that ABC 90°, BCD 116° and BC 40 mm: a Find the measure of OCT. b Express TC in terms of r, the radius of the circle. c Find the measure of r, using the fact that OK TC BC.

A D

r

B 60


O

K

T

C

Remember : The tangent to a circle is perpendicular to the radius drawn to the point of contact.

6

ABC is an isosceles triangle in which AB AC 80.00 mm and ABC ACB 50.00°. The incentre is I (the point where the bisectors of the angles intersect). The orthocentre is O (the point where the perpendicular bisectors of the three sides intersect).

A

T I

O

By symmetry it can be seen that points C B M A, I and O are colinear and the line AIO passes through M, the midpoint of BC. Find, correct to 4 significant figures, the distance that separates the incentre and the orthocentre. Hint: ◗ Use ABM to find BM and AM. ◗ Use IBM to find IM. ◗ Use ATO to find AO. Note: In Exercises 7.3, question 2 of the text (p. 111) you constructed this figure and measured the distance IO correct to 2 significant figures.

10.6 Applications Exercises 10.6 1

The top of a 40° wedge is rounded off by a radius of 15 mm. Calculate the dimension x.

2

The angle subtended by a tower at a point 50 m from its base is 38°27'. Find the height of the tower.

Chapter 10 Introduction to Trigonometry

61

3

Find the measure of BCD.

Eight holes are to be drilled, equally spaced on a circle of diameter 240 mm. Calculate the distance between two adjacent holes, centre to centre.

4

240 mm

5

A guy wire stretches from a point on the ground 21.5 m from the base of an antenna to the top of the antenna. The wire makes an angle of 53.7° with the ground. Find the length of the wire.

6 I

Z XL – XC

V

R

This impedance triangle shows the relationship between: R, the resistance of the circuit; XL, the inductive reactance; XC, the capacitative reactance; Z, the impedance of the circuit; , the angle by which the current input lags the applied AC voltage, V. a b c d e

Given that R 524 , XL 113 and XC 87 , evaluate . Given that Z 83.6 and 14.8°, evaluate R. Given that R 2.20 k, XC 1.50 k and 63.0°, evaluate XL. Given that 26.3°, Z 115 and XL 68.0 , evaluate XC. Given that R 685 and 36.2°, evaluate Z, and hence the current produced by an V applied voltage of 12 V use I . Z

62


10.7 How to find a trigonometrical ratio from one already given Exercises 10.7 1

In the following exercises, find the required trigonometrical ratio by using Pythagoras’ theorem. Do not use a calculator or tables. 12 a Given that sin A 1 3 , evaluate tan A. b Given that tan P 2.4, evaluate cos P.

2

a b

3 4

Given that tan Q m, express sin Q in terms of m. Given that sin A x, express tan A in terms of x.

Given that sin A 0.4128, evaluate cos A. 0.6314 a Given that sin A , evaluate tan A. 0.9520 1 1 b Given that 0.2517, evaluate . tan cos

10.8 Compass directions; angles of elevation and depression Exercises 10.8 1

On a ship that is sailing due south at a speed of 7.50 km/h, the navigator observed that the bearing of a light on the shore was S38°26'E. He also observed that the ship was due west of the light 144 min later. How far was the ship from the light when the second observation was made?

2

From the top of a cliff 125 m above sea level, two boats were observed on the water both on the same bearing from the observer, and their angles of depression were measured as being 27°35' and 53°28'. What was the distance that separated the two boats?

Chapter 10 Introduction to Trigonometry

63

C HAPTER

11

Indices and radicals

11.1 Radicals Exercises 11.1 1

Evaluate mentally: a 13 13 b

3 12 c

18 2

8 2

e

12 3

e

(1)2

2

d

2

11.2 Positive integral indices Exercises 11.2 1

Evaluate mentally: 2 a 3 b

(3)2

c

1

2

3

d

12

11.3 Substitutions and simplifications Exercises 11.3 1

2

3

4

Given that m 2 and k 3, evaluate: 2 a m k b k mk

k2(k m)2

d

(m 2k)(k m)

Given that a 1, b 2 and c 3, evaluate: 5 2 2 a 4a b (a b c) c (3b 2a)

d

a2(c b)2

Given that n 1, k 2, t 2, x 3, evaluate: 2 2 2 2 a 3k t b (tx) c t x

d

ntkx2

Express without brackets: 2 a (4t) d

64

(3t)2

c

b

2x(2x)2

c

(2k2)3

e

x3 x4 2 x

f

2x3 x4 x2


11.5 Zero, negative and fractional indices Exercises 11.5 1

Expand (i.e. multiply out): a

2

4n–

23

n 2n – 12

b

Multiply and divide out, expressing in simplest exponential form:

4

Evaluate mentally: a 8 1 3

2x 3x 6x0 3x 1 2

ex ex ex Evaluate mentally: 0 0 a 2x 4y a

3

(et et)(et et)

b

b

b

4

1 2

(m0 3k0)2 c

1 2

m0 3k0

c

27

2 3

d

25

3 2

11.6 Scientific notation 11.7 Engineering notation Exercises 11.7 1

Use your calculator to perform the given operation expressing the result in (i) scientific notation, (ii) engineering notation: a 23.40 5.670 b 23.45 34.56 c 2345 345.0 3 d (234) e 0.086 37 0.002 403 f 0.003 456 0.001 234

11.8 Transposition in formulae involving exponents and radicals Exercises 11.8 In each of the following exercises a formula is given. a Make the variable in part a of the question the subject of the formula. b Evaluate this new subject using the data provided. P d4 1 V k , where d is the diameter of a pipe. l a d b V 43 mL, l 3.50 m, P 100 kPa, k 2410 Chapter 11 Indices and Radicals

65

2

3

4

PV1.4 k, where V is the volume of a gas. a V b P 240 kPa, k 34.5 R2 t2 273 1.2 , where t2 is a temperature in degrees Celsius. R1 t1 273 a t2 b t1 1450°C, R1 825 , R2 1.15 k

Z a b

66

1 L , the impedance of a series RLC circuit. R C 2

2

C 390 rad/s, Z 78 , R 65 , L 230 mH, C 55 mF


C HAPTER

12

Polynomials

12.2 Multiplication of polynomials Exercises 12.1 1

2

3

Expand: 3 2 a (x 2)(x 2x 4x 8)

b

Find the coefficient of x3 in the expansion of: 3 3 a (x 3)(x 1) b

(2x3 x 2)(x2 x 3) x (x2 1)(x2 2)

Evaluate 3.6x3 1.9x2 2.7x 1.3 when x 2.014, correct to 4 significant figures.

12.4 Common factors Exercises 12.2 1

Factorise the following expressions: a 6x 8x2 b a2b ab2 2 2 3 2 d x 3xy e 3ax y ax y

c f

2p 4pq 6q 2x 2ax 6bx2

12.5 The difference of two squares Exercises 12.3 Factorise: 1

a

(a b c)2 (a b c)2

b

(p q r)2 (p q r)2

12.6 Trinomials Exercises 12.4 1

Factorise the following: 2 a F 13F 12 2 d L 16L 39

b e

C2 6C 5 r2 13r 36

c f

e2 6e 9 C2 29C 100 Chapter 12 Polynomials

67

2

3

4

5

6

Factorise the following: 2 a L 7L 12 2 c E 16E 39 Factorise: 2 a 4a 11a 6 2 d 8x 17x 9

b d

T 2 16T 48 V2 25V 100 c

e

4a2 20a 9 8x2 27x 9

f

8x2 18x 9 8x2 38x 9

Factorise: 2 a 6x 37x 6

b

9x2 19x 2

c

10k2 31k 3

Factorise: 2 a 4a 28a 15

b

6n2 11n 2

c

6n2 n 2

b

4k2 12k 5 10t2 101t 10

c f

8x2 13x 6 10t2 21t 10

Factorise: 2 a 6x 7x 3 2 d 8x 47x 6

b

e

12.7 Composite types Exercises 12.5 Factorise the following: 1

4R2 20R 24

2

10C 2 20C 30

3

32V 2 200

4

4R2 32R 60

5

18V2 32E 2

6

10C 2 130C 120

12.8 Algebraic fractions (simplification by factorising) Exercises 12.6 Factorise where possible and hence simplify: V 2 4V 4 6 3C 1 2 V2 2C 3 3 2V 3V 18M 2M 4 5 2 2 9 6V 6M 2M

3 6

R2 4 2R 2 2R 3R 9 6R

2

12.11 Solution of the equation x C Exercises 12.8 1

68

Solve the following (a calculator is not necessary): 2 2 2 2 a 12 x 3 b C 3 4 2 2 d 2(x 3) 12 e 3(x 2) 6


c f

x2 3 12 4(1 x 2) 12

2

3

4

Solve correct to 3 significant figures: 31.2 a 5(M 4) M(5 M) b R 0 R 2 (x 3) 2 c 3 d (2V 3) (V 4)(V 8) 2x 5 Solve the following (a calculator is not necessary): 2 2 a (2Q 1) 16 b 49 (3 2L) 0 3 1.68 x2 2M 1 c d x2 2M 1 3 1.68 4 t3 e t3 4 Using a calculator, solve, correct to 3 significant figures: 2 a 3(2t 1) 50 b (2x 1)(3x 4) 5 14.6 2.36 x4 2k 3 c d 2x 8 8k 12 11.2 5.12

12.12 Solution of the equation 2 ax bx 0 Exercises 12.9 Solve questions 12 by factorising. 1

a

4F F2 0

2

a

2 Q2 3

b

1

4Q

1 t 2

3

4t 0 b

c 2 (L 5

50E2 20E 0 3

2L2) 4L(L 2)

12.13 Solution of the equation 2 ax bx c 0 Exercises 12.10 Solve questions 13 by factorising. 1

2 3

c

L2 7L 18 0 x2 x 12 0

a

Q(Q 1) 12

a

6x x 2x 0

a

3

2

M2 5(5 2M ) 0 d x(x 3) 4 0 x2 2 C 2(3C 4) c 2 x4 3 2 b 20x 65x 15x 0 b

b

12.14 ‘Roots’ and ‘zeros’ Exercises 12.11 1

What are the zeros of the expression 5V 2 5V 60? Chapter 12 Polynomials

69

2

Find the roots of the equation 2E 2 4E 48 0.

3

Find a quadratic expression that has zeros of 3 and 7 (write the expression in the form x2 bx c).

12.15 The method of ‘completing the square’ Exercises 12.12 Solve the following quadratic equations using the method of ‘completing the square’. 1

State the roots correct to 3 significant figures. 2 a x 3x 15 17 b

2

a

2x2 4x 5 0

b

x1 2x 3 x2 x4 2 3x 2x 4

12.16 The quadratic formula Exercises 12.13 2 b ±b 4ac Given that the roots of the quadratic equation ax2 bx c 0 are x , 2a solve the following equations giving the roots correct to 3 significant figures:

1

a c

V 2 2V 7 0 1 1 5 M2 M 3 6 6

b d

k3 5k2 k 0 (C 2)(C 3) (3C 2)(C 4)

Mechanical and general applications

Exercises 12.14 1

A rectangle has perimeter 10 m. a If the length of the rectangle is l m, (i) express the breadth in terms of l, (ii) express the area in terms of l. b By putting the above expression for the area equal to 6, find the length of the rectangle if the area is 6 m2. Hence also find the breadth of the rectangle.

2

A rectangle of area 2 m2 has perimeter 8 m. Find the length and breadth of the rectangle.

3 6m

70

A square of side x m is surmounted symmetrically by an isosceles triangle of altitude 6 m. a Express the area of the figure in terms of x. 2 b Evaluate x, given that the area is 28 m . 2 c Evaluate x, given that the area is 13.0 m .


Electrical applications

Exercises 12.15 In this set of exercises for electrical students, all quantities and answers are in SI units, hence no units are actually stated. R

1 I

E

r

A battery having an emf of E V and an internal resistance of r drives a current I A through a load resistance of R . The power dissipated in the load resistance is given by P EI I2r, where I is positive.

In each of the following cases, given the values of r, E and P, solve the quadratic equation to find the size of the current. (In parts (a) and (b), the equation can be solved by factorising.) a r 1, E 8, P 12 b r 5, E 30, P 40 c r 1, E 6, P 2 d r 8, E 12, P 3

Exercises 12.16 1

When a constant force F is applied to a body of mass m, initially at rest, for time t, the body Ft will acquire a velocity given by v . m a How long does it take to give a body of mass 16 t a velocity of 81 km/h using a force of P N? b How long does it take to achieve the same result if the force is increased by 1 kN? c If the above increase in force reduces the time required by 1 min, find the force P and the time taken when using this force.

2

Given that one of the capacitors in this circuit has capacitance C farads, it can be shown that Q C when a total energy of W joules is stored in the capacitors by the application of V volts as shown, the charge in the other capacitor will be 2W given by Q CV coulombs. V 2 Write this quadratic equation in the form aV bV c 0. Find the voltage given that C 500 F, Q 500 C and W 3.00 mJ.

V

a b c

Q Hence find the capacitance of the other capacitor . V

Chapter 12 Polynomials

71

12.17 Graphical solution of a quadratic equation Exercises 12.17 1

72

Plot the graph of the following function f(x) for integral values of x over the domain specified. Solve the equation f(x) 0. Then compare your solution with that obtained using the quadratic formula. 2 a 2x 6x 3 0, for 1 x 4


C HAPTER

13

Functions and their graphs

13.1 Function notation Exercises 13.1 1

2

If f(x) 2x 1, evaluate: a f(3) b f(5) 1 If h(x) 1: x a evaluate h(1) c

3

evaluate h(2) h(3)

If f(x) x x, evaluate: a f(3) b f(0)

c

f(0)

b

evaluate h(2)

d

1 express h() in simplest form x

c

f(1)

2

d

f(2)

1

d

1 f() 2

4

If f(x) 2x 1 and g(x) x 3, evaluate: a g(1) b f(4)

c

f(2) g(0)

5

If k(x) 3x and m(x) x , evaluate: a m(2) b k(3)

c

k(0) m(1)

c

m(x) 0

6

7

2

3

If h(x) 2x 3 and m(x) 6 x, solve for x: h(x) a h(x) m(x) b 3 m(x) f(2 k) 12 If f(x) 1 3x, solve: f(k) f(k 3)

13.2 The parabola: graphing the curve from its equation Exercises 13.2 1

For the graphs of each of the following parabolas: i state the equation for the axis of symmetry; ii state the coordinates of the turning point (vertex); iii state the value of the y-intercept; iv sketch the curve, showing the turning point and any intercepts;

Chapter 13 Functions and Their Graphs

73

v a

draw an accurate graph of the curve (plot about 79 points). 2 y x2 4x 6 b y 5x 20x 32

2

For the parabola y 3x2 12x 15: a state the equation for the axis of symmetry; b state the coordinates of the turning point (vertex); c state the value of the y-intercept; d state the values of the x-intercepts; e sketch the curve, showing the turning point and any intercepts; f draw an accurate graph of the curve (plot about 79 points). (Note: The x-intercepts are computed in this case because the function factorises easily.)

3

For each of the following graphs: i state the x-intercepts; ii state the equation for the axis of symmetry; iii state the coordinates of the turning point; iv sketch the curve. 2 2 a y 2x 7x b y 5x 2x

13.3 The parabola: finding the equation knowing the coordinates of three points Exercises 13.3 1

Find the equation for the parabola that has: a y-intercept of 10 and x-intercepts of 1 and 5 b y-intercept of 30 and x-intercepts of 3 and 5

2

Find the equation for the parabola that passes through the three given points: (0, 8), (4, 0) and (2, 0).

13.4 The circle Exercises 13.4 1

74

Describe the graph that represents each of the following equations and draw a sketch-graph. 2 2 a (x 7) (y 5) 16 b x2 (y 3)2 25


13.6 Algebraic solution of simultaneous equations that involve quadratic functions Exercises 13.6 Solve the following simultaneous equations:

{ yy x6 3xx 1 2

1

{ yy 2x6x 25x 13 2

2

{ yy 6x13x4x5 7 2

3

13.7 Graphical solution of simultaneous equations that involve quadratic functions Exercises 13.7 For each pair of simultaneous equations below, plot on the same axes an accurate graph of each of the functions and hence find approximate values of the solutions: 1

{

y 3x2 3x 1 y 2x2 2x 5

2

{

y 7x2 3x 1 y x2 4x 1

3

{

y 5x2 x 1 y 3x2 10x 8

13.8 Verbally formulated problems involving finding the maximum or minimum value of a quadratic function by graphical means Exercises 13.8 1

PQRS is a rectangle in which PQ 3 m and QR 6 m. T is a point on PQ and K is a point S on SP such that SK 2 PT. Let the length of PT be x metres. R a Express A, the area of the triangle PTK, in terms of x. 2x b Draw a sketch-graph of A against x. K c From your sketch-graph, determine the maximum possible area of the triangle and the value of x for which this occurs. x P

T

Q

Chapter 13 Functions and Their Graphs

75

C HAPTER

14

Logarithms and exponential equations

14.1 A quick revision of exponents and exponential form Exercises 14.1 1

2

Evaluate, giving each value as an integer or a fraction: a

9

a

Write the number 4 in exponential form, using base 2.

b c

Write the number 1 in exponential form, using base 2. Write the number 27 in exponential form, using base 9.

d

Write the number 32 in exponential form, using base 4.

12

8

23

b

3 4

16

c

d

2 3

27

e

90

1

1

14.2 Definition of a logarithm: translation between exponential and logarithmic languages Exercises 14.2 1

2

Write in exponential language (‘exponential form’): a log2 8 3 b log2 1 0

c

log3 9 2

c

8 4

1

Write in logarithmic language (‘logarithmic form’): a

9 3 1 2

b

at x

2 3

1

14.3 Evaluations using the definition (logs and antilogs) Exercises 14.3 1

76

Evaluate: a log3 27

b

1

log4 2


c

log10 1013

14.6 The three laws of logarithms Exercises 14.6 1

Evaluate: a log2 3 log2 24 log2 9 b log3 12 log3 54 log3 6 c log5 3 log5 2 log5 8 log5 12 d log3 4 log3 2 log3 6 e log7 8 log7 6 log7 4 log7 3

2

Write each of the following in the form a log x b log y . . . P x a log b log y c log Qn y 5x ab d log e log f log 4x3 c y g

log Kt x

h

log (xy)n

14.10 Exponential equations Exercises 14.10 1

Solve: k k3 a 8 4

b

9x 272 x

c

32k 5 1

f

164 x 8

g

e 2

4

d

61 x 63

Solve: a

3

4t 1 82 t 2 81 2n 2

1 n

2

1

Solve: 2x x3 a 10 10 100

b

3

81 x 2 1

1 8 4x 1 Solve the following exponential equations for the pronumerals, giving answers correct to 3 significant figures: m m1 x 1 3x 1 k a e 2 10 b 10 e c 3.18 (4 e)2 k b

14.11 Change of subject involving logarithms Exercises 14.11 1

For each of the following formulae, make the variable in brackets the subject of the formula: t log C xy a kt (y) b ta (y) c k (Q) log Q xy

Chapter 14 Logarithms and Exponential Equations

77

14.12 Applications Exercises 14.12 1

2

R1 T1 For a tungsten lamp filament, R2 T2 temperatures T1 and T2 kelvin. R1 5 Show that: log T2 log T1 6 log . R2

1.2

, where R1 and R2 are the resistances at

Po The power gain of an amplifier, in decibels, is given by G 10 log10 , where Pi is the Pi input power and Po is the output power. a Make Pi the subject of this formula. b Hence find the power input required in order to obtain an output power of 6.00 W from an amplifier having a 45 dB gain. (Answer correct to 2 significant figures.)

14.13 The curve y k logb Cx Exercises 14.13 Remember : If no base is specified, then base 10 is implied.

Sketch the following curves, showing the value of the x-intercept and the scales on both axes: 1

78

1

y log (2x 1)


2

y 12 log8 2x 3

C HAPTER

Non-linear empirical equations

15

15.2 Conversion to linear form by algebraic methods Exercises 15.1 1

The velocity v of a body as measured at times t: t (s) v (m/s)

20

40

60

80

100

120

25.0

29.5

33.5

38.5

41.5

43.5

By plotting v against t, verify that the relationship is of the form v Ct K. Draw the line of best fit and hence: a determine the values of C and K b estimate the velocity when t 49 s c estimate the value of t when v 20 m/s

15.3 The use of logarithms to produce a linear form Exercises 15.2 1 x

0.2

0.5

1.5

2.5

4.0

6.0

10

y

1.1

1.6

2.3

2.5

3.0

3.4

14.2

The table shows measured corresponding values of x and y. Given that the relationship is of the form y C xn, plot a linear graph and, by drawing the line of best fit, determine the values of the constants C and n. 2

t

0

1

2

3

4

5

6

7

8

v

2.75

3.63

6.03

10.0

16.6

21.9

45.7

57.5

110

The table shows measured corresponding values of two variables t and v. Given that the relationship is of the form v K C t, plot a linear graph and, by drawing the line of best fit, determine the values of the constants K and C. Chapter 15 Non-linear Empirical Equations

79

C HAPTER

16

Compound interest: exponential growth and decay

16.2 Compound interest Exercises 16.1 1

How much total interest is paid on an investment of $30 000 for 5 years if the interest rate of 6% is paid: a annually? b monthly? c daily?

2

A person borrows $5000 for 30 weeks at a compound interest rate of 2% per fortnight. How much money will he have to repay at the end of the 30 weeks if the interest is compounded: a each fortnight? b daily?

3

What is the interest rate required in order that an investment will double in 10 years if the interest is paid: a annually? b daily?

4

How much money, to the nearest dollar, would a person have to invest for 3 years in order to obtain a total interest of $1000 if he pays compound interest of 4% p.a.: a compounded annually? b compounded daily?

16.3 Exponential growth Exercises 16.2 1

What is the difference between the total interest paid on an investment of $5000 for 4 years at a compound interest rate of 6.5% p.a. paid continuously and the same investment with the interest paid daily? (State the answer to the nearest cent.)

2

What interest rate per annum is required for an investment to double itself in 9 years if the interest is paid continuously? (State the answer correct to 3 significant figures.)

Exercises 16.3 1

80

When an iron casting 750°C above room temperature is removed from an oven, it cools exponentially at a rate of 3.00% per minute. How high above room temperature will it be after: a 10 minutes? b 1 hour? c 2 hours? d 3 hours?


2

The mass of a bacterial culture grows exponentially from 1.836 g to 3.274 g during a period of 8 hours. What will be its mass at the end of a further 24 hours? (State the result correct to 4 significant figures.)

3

A tank contains 145 L of water. When a tap at the bottom of the tank is opened, at any instant water flows out at a rate that is proportional to the volume of water in the tank at that instant. a Given that 3.86 L flow out during the first 2 minutes after opening the tap, how much water will remain in the tank 5 minutes after the tap is opened? b Starting with the 145 L in the tank, for how long must the tap be opened for 120 L to run out? State the answers correct to 3 significant figures.

16.4 Graphs of exponential functions Exercises 16.4 1

Draw sketch-graphs of the following functions: 14x a y 8e , for 0 ≤ x ≤ 0.15 500t b v 12e volts, where t is in seconds 3.5x c y 80(1 e )

2

When power is connected to an electrically driven flywheel, its rotational speed after t seconds is given by v 8(1 e0.5t) revs/s, where t is in seconds. Sketch a graph showing its approximate speed during the first 5 seconds.

Chapter 16 Compound Interest: Exponential Growth and Decay

81

C HAPTER

17

Circular functions

17.1 Angles of any magnitude Exercises 17.1 1

Given that each of the following is 1, 0.2, 0, 0.2 or 1 (correct to 1 significant figure), select the correct value for each. (Do this mentally, without a calculator.) a sin 348° b sin 191° c sin 168° d sin 270°

2

Evaluate mentally: a sin 90° sin 270°

b

sin 180° sin 270°

c

sin 0° sin 270°

3

Given that the value of each of the following is 1, 0.3, 0, 0.3 or 1 (correct to 1 significant figure), select the correct value for each. (Do this mentally, without a calculator.) a cos 180° b cos 0° c cos 73° d sin 343° e cos 287°

4

Evaluate mentally: a cos 270° cos 90°

5

6

b

cos 0° cos 180°

c

cos 270° cos 180°

Simplify: tan (x) a b sec sin () cos () sin (180° x) From my calculator I find that tan1 (11.43) 85°. From this information write down the tangents of four angles between 0° and 360°.

17.2 The reciprocal ratios Exercises 17.2 1

2 3

Using your calculator, evaluate: a cosec 26° b

c

Given that cot B 0.684, evaluate B in degrees and minutes. a b

82

sec 43.46°

Given that cot M 3.412, evaluate sec M. Given that cosec T 1.555, evaluate cos T.


cot 80°14'

17.3 The co-ratios Exercises 17.3 1

2

Write down the missing angles: a tan 10° cot . . . b cos 85° sin . . . c

sin ° cos . . .

Write down the missing trigonometrical ratios: a sec 10° . . . 80° b

sin 72° . . . 18°

d

sec x° cosec . . .

17.4 Circular measure Exercises 17.4 1

2

3

4

Convert the following angles into degrees, mentally: 5 2 a b c d e 3 6 6 3 Express the following angles in circular measure in terms of , mentally: a 270° b 150° c 180° d 20° e Answer mentally (do not use a calculator): a tan 2 b sin 3

5 cos 2 Evaluate cos 2.340 71 correct to 4 significant figures. c

d

2 10°

tan

17.6 The circle: length of arc, area of sector, area of segment Exercises 17.6 1

When the radius of a circle (r) and the angle () subtended at the centre C by a chord PQ are r 1.46 m, 72.0°, find: a the length of the minor arc PQ b the area of the minor sector CPQ

2

PQ is a chord of a circle whose length is 4.82 m and whose distance from the centre C is 3.17 m. Find: a the radius of the circle b the angle subtended by the chord PQ at the centre of the circle c the length of the minor arc PQ d the area of the minor sector PCQ

17.7 Graphs of a sin b, a cos b Exercises 17.7 1

Sketch one cycle of the curve y 27 cos 20A. State the amplitude and the period of the function.

Chapter 17 Circular Functions

83

Phase angles; more graphs of trigonometric functions

C HAPTER

18

18.1 Phase angles Exercises 18.1 For each of the following pairs of functions, state the function that leads and state the phase difference: 1 sin ( 20°) and cos ( 50°) 2 cos and cos 3

3 3 cos and sin 8 4

sin and cos 4 3 For each of the following pairs of functions, state which function leads and state the phase difference , such that 0° 180°: 5 sin and sin 6 cos and sin 2 4 2 4 3

7 cos and sin 2 2 18.2 Sketch-graphs of a sin ( ), a cos ( ) Exercises 18.2 For each of the following functions, state the amplitude and the period. Sketch two cycles of each function, marking the values where the curve crosses each axis: 1

y 12 cos ( 130°)

2

y 24 cos ( 90°)

18.4 The function A sin t Exercises 18.4 For definitions of the prefixes p, n and G, see section 3.1 of the main text. 1

84

State the amplitude (A), the period (T) and the frequency (f ) of each of the following periodic functions, giving answers correct to 2 significant figures: 6 6 a 2500 sin (3 10 t) b 17 sin (1.3 10 t) mV


18.8 ◗ Phase shift as a fraction of the period ◗ Advice concerning the rapid sketching of sinoidal curves Exercises 18.7 In each case sketch one cycle of the curve and state the frequency, marking the values on the horizontal axis where the curve crosses it. (All values are correct to 3 significant figures.) 1

i 13.6 cos (2.84 106t 1.24) A

2

i 18.4 cos (534t 60°) mA

Chapter 18 Phase Angles; More Graphs of Trigonometric Functions

85

C HAPTER

19

Trigonometry of oblique triangles

19.1 The sine rule Exercises 19.1 1

2

In each triangle below, find the length of the side labelled S. a

b

c

d

Find the size of the angle labelled A in degrees and minutes correct to the nearest minute:

3

86


Find the length of the side labelled S.

19.2 The ambiguous case Exercises 19.2 1

In each case below state i which angle you could find directly, using the sine rule, and ii whether or not the triangle is ambiguous: B 127° A 38° R 21° a ABD b 37 m b NTA t 17 m c RGH r 8 km d 23 m a 14 m h6m

{

{

{

19.3 The cosine rule Exercises 19.3 1

Evaluate the pronumerals in the following figures: a

b

c

6.9 m 6.0 m

A 2.2 m

d

e

f

A 2.21 m

1.84 m

2.43 m 1.81 m

2.03 m

1.52 m

19.4 The use of the sine and cosine rules Exercises 19.4 Applications 1

A surveyor takes readings on three objects P, Q and R situated on level ground. The surveyor measures the distance PQ to be 274.8 m and the distance PR to be 183.6 m. From P the surveyor measures the angle RPQ to be 68.17°. Find a the distance QR, b the angle PQR.

2

A pilot lodges a flight plan to fly due east for 72.0 km and then on a course with a direction bearing of 304° and then to return to base on a steady course, this final leg being 49.0 km. a Why is this flight plan unacceptable? Chapter 19 Trigonometry of Oblique Triangles

87

b

88

The pilot intends to fly on one of two possible bearings on the last leg of the flight. What are these two possible bearings?

3

An aircraft is flying due north at a ground speed of 736 km/h. The navigator reads the bearing of a radio station to be 34.2° at 1400 h and to be 71.8° at 1430 h (i.e. half an hour later). Find the distance of the aircraft from the station and the bearing of the station at 1500 h.

4

When observed from a point P on the same horizontal level as the base, the angle of elevation of a spire is 39.6° and, when observed from a point 27.3 m above P, the angle of elevation is 30.9°. Find the height of the spire.

5

A ship moving at a speed of 21.7 km/h steers a course N26.0°W for 2 hours and then alters course to S41.0°W for 1 hour. a Calculate the distance and the bearing of the ship from its original position at the end of this time. b Calculate how much further the ship will have to travel on its second course to be due west of its original position.


C HAPTER

20

Trigonometric identities

20.1 Definition Exercises 20.1 1

Solve the following equations and then state which one is an identity: 2 2 a (x 3)(x 2) x x 6 b (x 3)(x 2) x 2x 3 2 2 c (x 3)(x 2) x 2x 4 d (x 3)(x 2) x x 6

20.4 Summary: trigonometric identities Exercises 20.3 1

Simplify the following, expressing each as a single trigonometric ratio: a sin 53° sec 53° b sec 17° cot 17° c cosec A cot A d sin sec e cos A tan A f sec A cosec A

2

Write down the missing trigonometrical ratios: a cos 17° sec 73° . . . 17° b c tan 69° cosec 69° . . . 21° d

3

4

Simplify: 2 a 2 2 sin C

Prove the following identities: 1 2 a 2 tan x cosec x 1

tan2 K sec2 K 1 b

c

(tan 1)2 sec2

(1 sin2 x)(sec2 x 1) sin2 x

1 1 1 cos2 x cos2 x cot2 x cosec2 x Simplify, expressing each in terms of a single trigonometrical ratio, or as a constant: sin 1 cos2 tan2 x 1 a b c 2 2 sec tan cos tan x c

5

b

cos 40° cot 50° . . . 40° sin 34° cot 34° . . . 56°

tan 1 si n2 sin

d

Chapter 20 Trigonometric Identities

89

Exercises 20.4

sin Applications of tan cos 1 If 7 sin t 8 cos t C cos (t ), then it may be shown that C sin 7 and C cos 8. a Evaluate tan . b Evaluate . c Evaluate C.

Exercises 20.5 Applications of the Pythagorean identities 1

I V

If an AC voltage, V, supplied to a circuit produces a current I A, which lags the supply voltage by angle , then R Z cos and X Z sin , where R and X are the resistive and reactive components of the impedance Z .

Show that Z2 R2 X2, and hence evaluate Z when R 645 and X 582 . 2

If 2 sin t 4 cos t K cos (t ), it may be shown that K sin 2 and K cos 5. Evaluate (K sin )2 (K cos )2 and hence: a evaluate K b evaluate (Note: sin is negative and cos is positive.)

20.5 Other trigonometric identities There are many other identities that are useful when simplifying or otherwise manipulating trigonometric expressions. Below is a list of 16 identities that you should be able to use. Note: Unless you were using these formulae frequently, it would not be worthwhile trying to memorise them, but you should be aware of their existence and be able to locate them and use them when required. You will establish all these identities in the next set of exercises.

The addition formulae

90

1

sin (A B) sin A cos B cos A sin B

2

sin (A B) sin A cos B cos A sin B

3

cos (A B) cos A cos B sin A sin B

4

cos (A B) cos A cos B sin A sin B

5

tan A tan B tan (A B) 1 tan A tan B

6

tan A tan B tan (A B) 1 tan A tan B


The double-angle formulae 7

sin 2A 2 sin A cos A

8

cos 2A cos2 A sin2 A ( 1 2 sin2 A 2 cos2 A 1) 2 tan A tan 2A 1 tan2 A

9

The sum-to-product formulae AB AB 2 2 AB AB 11 sin A sin B 2 cos sin 2 2 AB AB 12 cos A cos B 2 cos cos 2 2 AB BA 13 cos A cos B 2 sin sin 2 2 10 sin A sin B 2 sin cos

The product-to-sum formulae 1

1

14 sin A sin B 2 cos (A B) 2 cos (A B) 1

1

15 cos A cos B 2 cos (A B) 2 cos (A B) 1

1

16 sin A cos B 2 sin (A B) 2 sin (A B)

Note: Some of these formulae are so similar that special care is needed when copying one down. Note especially the B A term in identity 13 .

Exercises 20.6 1 2 3

1

Evaluate a if sin (x 30°) 2 (a sin x cos x). x x Simplify sin cos , given that sin 2 2 sin cos . 2 2 Express cos 2A in terms of cos2 A. (Hint: Express cos 2A as cos (A A).)

4

Given that sin 2A 2 sin A cos A and cos 2A 2 cos2 A 1, express sin 3A in terms of sin A. (Hint: sin 3A sin (2A A).)

5

Simplify sin (A B) sin (A B). B is the angle shown in the diagram.

6

4

a

Evaluate sin B and cos B.

b

Evaluate the size of the angle B in degrees and minutes.

c

Using the results of a above, convert the expression 0.8 cos A 0.6 sin A into the form sin (A ), where is an angle in degrees and minutes.

d

Hence, solve the equation 0.8 cos A 0.6 sin A 0.5 for 0 A 360°.

B 3


91

Exercises 20.7 You have derived identity 4 in Exercises 20.6 in the text (p. 335), and by working through the following exercises you will see how all the other identities in the list can be derived from that one result. 1

Starting from identity 4 , prove identity 3 by substituting B for B and remembering that sin (B) sin B, but cos (B) cos B.

2

Using identity 3 , prove identity 2 , commencing with the steps sin (A B) cos [90° (A B)] cos [(90° A) B] and then expanding this expression using identity 3 .

3

4

5 6

Starting from identity 2 , derive identity 1 by the same method that you used in question 1 above. sin (A B) tan (A B) . Expand the numerator and denominator (using identities 1 cos (A B) and 3 ) and then divide each by cos A cos B to obtain identity 5 . In identity 1 , let B A, and hence derive identity 7 1 :

sin (P Q) . . .

2 :

sin (P Q) . . .

1 2 :

7

... ...

AB AB Now let P Q A and P Q B. Show that P and Q. 2 2 Substitute for P and Q, hence obtaining identity 10 . PQ QP From 13 : 2 sin sin cos P cos Q 2 2 PQ QP 1 sin sin 2 (cos P cos Q) 2 2 We now have an identity for the product of two sines. PQ QP Let A and B 2 2 A B Q and A B P. Hence derive identity 14 .

Example 1

Expand sin (x – 60°) sin (x 60°) sin x cos 60° cos x sin 60° 3 1 (sin x) (2) (cos x) 2 3 1 2 sin x cos x 2

92


2

Simplify sin (45° ) sin (45° ) Expression (sin 45° cos cos 45° sin ) (sin 45° cos cos 45° sin ) 2 cos 45° sin 1 2 sin 2 2 sin

3

4

5

tan x tan y sin (x y) Prove the identity 1 tan x tan y cos (x y) We expand the RHS to remove the brackets, and obtain: sin x cos y cos x sin y sin (x y) cos x cos y sin x sin y cos (x y) sin x sin y cos x cos y —————–— — Dividing numerator and denominator by cos x cos y sin x sin y 1 cos x cos y tan x tan y 1 tan x tan y Find the exact value of tan 12

tan tan 15° tan (45° 30°) 12 tan 45° tan 30° 1 tan 45° tan 30° 3 1 ... 3 1 Express 3 cos x 2 sin x in the form A sin (x ) where A 0 and 0 180° A sin (x ) A (sin x cos cos x sin ) (A cos ) sin x (A sin ) cos x

{

A cos 2 A sin 3

2

A 13

(by squaring each side of the simultaneous equations and then adding)

A 13 A sin 3 tan A cos 2 56.3° or 123.7°

2 3 But since A is to be positive and cos and sin A A cos 0 and sin 0. Hence, is an angle in the second quadrant, 123.7° 3 cos x 2 sin x 13 sin (x 123.7°) (This result, like any other identity, can be verified by substituting any value you like to choose for the variable and then evaluating each side of the equation using your calculator.)


93

6

Simplify cos 9x cos 4x sin 9x sin 4x Checking through the list of identities, we see that this expression is in the form cos A cos B sin A sin B. cos 9x cos 4x sin 9x sin 4x cos (9x 4x) cos 5x

7

Find the exact value of sin 75° cos 15° 1

1

Value 2 sin (75° 15°) 2 sin (75° 15°) (from identity 16 ) 1

1

2 sin 90° 2 sin 60° 1 1 3 (2) (1) (2) 2 3 1 2 4

2 3 4

Exercises 20.8

1

Given that sin sin 2 cos sin , show that 2 2 AB B A cos A cos B 2 sin sin . 2 2 (Hint: substitute 90° A and 90° B.)

2

3

4

Expand: a cos (A 45°)

sin 3

b

c

Simplify:

a cos sin 3 6 c cos (30° x) sin (60° x)

b d

cos ( 120°)

cot A tan A 4 4 sin (x y) sin y cos (x y) cos y

Prove the identities: a cos ( ) cos ( ) 2 sin sin 2 2 b sin (x y) sin (x y) sin x sin y sin (x y) tan x tan y

cos sin c d tan sin (x y) tan x tan y 4 cos sin

5

b a

h x

94

tan x 4

d

y

If and are any angles such that < 180°, they may be placed in a triangle as shown in the diagram. Use the cosine rule to express cos ( ) in terms of , , x, y and h. Hence prove that for < 180°, cos ( ) cos cos sin sin .


6

Draw any triangle ABC in which B 90°. Mark any point D on AB. Label ACB , DCB and ACD . Label the lengths of the intervals BC, BD and DA with any numerical values you like to choose. a By expanding tan ( ), evaluate tan without calculating the sizes of any angles. b Use trigonometry to evaluate , and . Hence find tan and check that your result is the same as you obtained using the expansion of tan ( ).

7

Express 4 sin t 3 cos t in the form A sin (t ) where A 0 and 0 180°. State the value of correct to 2 decimal places. (Hint: Study example 5 above.)

8

a

b

Study example 5 (p. 93) where it was proved that 3 cos x 2 sin x 13 sin (x 123.7°) and use this result to solve the equation 3 cos x 2 sin x 0.440, evaluating x in degrees correct to 4 significant figures. Solve the equation 21.8 sin x 14.3 cos x 25.6, evaluating x in degrees correct to 4 significant figures. E

9

D

C

B

P

B, C and D are marks on a vertical pole AE such that AB BD CD DE. P is a point on level ground such that the distance of P from the foot of the pole is one-quarter the height of the pole AE. CPD . a Without using a calculator find the exact value of tan . (Hint: Let APC and expand tan ( ).) b Now check your value for tan by using a calculator to find the sizes of the angles and hence the value of tan .

A


95

C HAPTER

21

Introduction to vectors

Note: In Appendix B of this supplement there is a brief explanation of the meanings of the terms acceleration, force, mass and weight. If you do not understand the difference between the ‘mass’ and the ‘weight’ of a body and why a body of mass m has a weight of m g where g 9.8 m/s2, it is recommended that you study this appendix.

21.5 Resolution of a vector into two components at right angles Exercises 21.2 1

A conveyor belt is travelling at the rate of 1.5 km/h, its angle of inclination to the horizontal being 25°. Through what vertical distance will it raise an object in 30 seconds?

21.7 Addition of vectors graphically Exercises 21.3

96

1

An aircraft flies a distance of 195 km on a compass bearing of 289° and then 225 km on a compass bearing of 152°. What distance and on what bearing must it then fly in order to return in a straight line to its starting point?

2

A river is flowing due north at a speed of 5.0 km/h. A boat that travels at a speed of 6.4 km/h in still water is heading directly across the river in direction due east (i.e. its compass is reading 90°). Find the speed of the boat in respect to the land and its actual direction of travel.

3

A ship is travelling at its cruising speed of 23 km/h. If the navigator knows that there is a current running at 6 km/h in the direction S30°W, on what compass bearing should he head the ship in order to travel directly towards a point that is due north of him?


21.8 Equilibrium Exercises 21.4 1

T

m

A trolley of mass 71 kg is held stationary by a mass m kg hanging from a cable which is attached to the trolley over a smooth pulley as shown in the diagram. a Find the tension T in the cable. b Evaluate m.

38°

2

A mass m of 0.50 t is suspended from a cable. The mass is pulled by a horizontal force F so that the cable makes an angle of 40° with the vertical. If this force is supplied by a mass M hanging over a frictionless pulley, find the: a size of the mass M b tension T in the cable

40° T

F

m

mg

Chapter 21 Introduction to Vectors

M

97

Euler’s constant, e, and the exponential growth formula

A PPENDIX

A

A1 Euler’s constant, e 1 The proof that the expression h approaches a limiting value as h becomes larger and h h

larger (i.e. as h → ) is beyond the scope of this course. However, you can verify the truth of this statement by evaluating the expression for larger and larger values of h. You should understand that no amount of verification of a statement constitutes a proof because you have not tested the statement for all possible cases. No matter how many tests you perform for different values of h, there is still the possibility that for some untested value the statement does not hold true. 1 h For example, using your calculator, evaluate the expression h when h 1, 10, 100, h 1000, . . .

In the eighteenth century, the Swiss mathematician Leonhard Euler discovered that although the value of this expression becomes larger and larger as h increases, this value does not increase without limit but approaches a value that is known as e. We say that the limit of the value of 1 h h as h becomes larger and larger is e. Mathematicians abbreviate this statement to h 1 h lim h e. h h→

This value, like the value of , has been calculated to thousands of decimal places and 2.718. Its value is such a useful number in mathematics that your calculator has keys on it that enable you to find the value of ex, e–x and logarithms to the base e.

A2 Proof of the formula for exponential growth: Q Q0 ekt The amount after 1 year of compound interest at r% per year when the interest is paid and compounded n times per year is given by: nr n AP 1 100

Appendix A Euler’s Constant, e, and the Exponential Growth Formula

99

r which P 1 100n

1 P 1 100n r

P

1 1 h

n

1 P 1 100n r r 100n 100 r

P

1 1 100n r

n

100n r

r 100

r h 100

100n , where h r

1 h As n → , h → and so 1 → e. h A→Pe r i.e. A → P ek, where k , the interest rate expressed as a decimal. 100 During the second year we have $(P ek) invested under the same conditions. r 100

at the end of the second year:

A $(P ek) ek $P e2k

at the end of the third year:

A $(P e2k) ek $P e3k

At the end of t years:

A P ekt

This formula does not, of course, apply to only a sum of money but to any quantity that grows exponentially. The general formula for exponential growth is: Q Q0 ekt

where

100

Q is the quantity originally present (i.e. the value of Q when t 0) k is the percentage rate of increase per some specified period of time expressed as a decimal t is the number of these growth periods

Mathematics for Technicians

A PPENDIX

B

Mass and weight

An acceleration of a m/s per second means that the velocity increases by a m/s each second. For example, a body that starts from rest and has an acceleration of 7 m/s per second (i.e. 7 m/s/s, 7 m/s2) will have velocities on each successive second of 0 m/s, 7 m/s, 14 m/s, 21 m/s, etc. A body that has no resultant force acting upon it will continue to remain at rest or will continue to move with a constant speed in a straight line (Newton’s first law of motion). In order to give a body an acceleration (i.e. a change in speed or direction of motion, or both), a resultant force must act upon it. The force required to give a body a particular acceleration is proportional to the mass (‘inertia’) of the body and to the magnitude of the acceleration. The force required to give a mass of 1 kg an acceleration of 1 m/s2 is called a ‘newton’ (N). The force required to give a mass of m kg an acceleration of a m/s2 is given by the formula F ma. When a body is in free fall in a vacuum (i.e. with no resistance from air, water etc.), the only force acting upon it is the gravitational attraction towards the centre of the earth, which will give it an acceleration that is independent of its mass and is known as ‘g’. Because the earth is not a perfect sphere this force and acceleration depend on its location on the earth (i.e. on how far it is from the centre of gravity of the earth). The value of g throughout Australasia has the value of 9.80 m/s2 correct to 3 significant figures but in other places can vary from this value by about 0.2%. In our work we will take the value of g to be 9.80 m/s2. The force of gravitational attraction that produces this acceleration in free fall is commonly called the ‘weight’ of the body and is given by given by F mg. For example, a body of mass 13.7 kg has a weight of approximately 13.7 9.80 N 134 N. A mass of 1.649 t (1649 kg) has a weight of approximately 1649 9.80 N 16 200 N (i.e. 16.2 kN). Remember : ‘Weight’ is a force—the force with which the earth attracts the body (for a reason still undiscovered by scientists). This is the force required to support the body or to lift the body.

Isaac Newton (1642–1727) is commonly associated with the concept of the falling apple and gravity so it is appropriate that the unit of force is called the ‘newton’ (N) and that 1 N is approximately the weight of a medium-size apple. Remember when you hold an apple in your hand that the force on your hand is approximately 1 newton (1 N).

Appendix B Mass and Weight

101

Note: In some exercises you will need to use the fact that on any body of mass m kg, there is a vertically downward force acting upon it of m 9.80 N (correct to 3 significant figures). This is the ‘weight’ of the body—the vertical force required to lift it or to prevent it from falling.

102


A PPENDIX

C

2 2 determinants and their use in solving simultaneous equations

There is another method for solving simultaneous linear equations that is applicable to all such equations and often saves a lot of time. Once learnt, this method is very concise, very simple to use and provides less opportunity for careless error. However (there is always a catch, of course), there are a few facts that you will have to learn first.

C1 Definition and evaluation of a 2 2 determinant

a c is called a two-by-two determinant. It is an array of numbers having two rows b d and two columns enclosed between vertical bars. It is shorthand notation for ad – bc.

Note:

a b

c d

bc ad

Examples 1

2

2

3

4

5

(2 5) – (4 3) –2

–2

3

–4

–5

(–2 –5) – (–4 3) 10 12 22

Exercises C1 Evaluate the following 2 2 determinants: 1

3

4

2

5

2

2

0

6

1

3

2

6

5

3

Appendix C 2 2 Determinants and their Use in Solving Simultaneous Equations

103

4

3

–2

5

4

3 –1

5

6

–4 –2

6.72 3.91 4.84 5.06

C2 Solution of simultaneous equations using 2 2 determinants If we have two simultaneous equations, for example:

{5x2x 3y6y 47 is the determinant we obtain from the coefficients on the left-hand sides, in the same order and arrangement as they appear in the equations. In this case, x

2

3

5

6

12 – 15 –3

x is the same determinant as above, except that the x-coefficients are replaced by the righthand numbers of the equations. In this case, x

4

3

7

6

24 – 21 3

y is the same determinant as but with the y-coefficients replaced by the right-hand numbers of the equations. In this case, y

2

4

5

7

14 – 20 –6

Examples 1

If

{6x3x 4y5y 72

then

104

3

5

6

4

x

2

5

7

4

y

3

2

6

7

12 30

8 35

21 12

18

27

9


2

If

4x 3y 5 {2x y6

then

4 3

x

2 1

5 3

y

6 1

4 5

2

(4) (6)

(5) (18)

24 10

4 6

5 18

14

10

23

6

Exercises C2 1

{ 4xx 2y3y51

Given that evaluate:

2

a

b

x

c

y

b

m

c

t

b

V

c

e

{ –2mm– 5t4t 3–1


3

a

2.8e 1.8 { 3.6V –1.2V – 5.7e –3.3


a

Solution of simultaneous equations Consider any two simultaneous linear equations: a1x b1y c1

➀ ➁

{axbyc 2

2

2

a1b2x b1b2y b2c1 a2b1x b1b2y b1c2

➀ b2 ➁ b1

Subtracting: a1b2x – a2b1x b2c1 – b1c2 x(a1b2 – a2b1) b2c1 – b1c2 b2c1 – b1c2 x a1b2 – a2b1 c1 b1 c2 b2 a1 b1 a2 b2 x y Similarly, it may be shown that y . Appendix C 2 2 Determinants and their Use in Solving Simultaneous Equations

105

Summary The solution of two simultaneous linear equations: a1x b1y c1

{axbyc 2

2

2

x y is : x , y This is known as ‘Cramer’s Rule’. To apply this rule, both equations must be expressed with the constants, c1 and c2 on the right-hand sides.

Example Solve: 16.7I 58.7 0 { 23.5I 81.2I – 34.2I 13.9 0 2

1

1

2

We first rearrange the equations:

{

16.7I1 23.5I2 58.7 81.2I1 34.2I2 13.9

16.7 81.2

23.5 571.14 1908.2 2479.34 –34.2

I1

58.7 23.5 2007.54 326.65 1680.89 –13.9 –34.2

I2

16.7 81.2

58.7 232.13 4766.44 4998.57 –13.9

I1 1680.89 I1 0.678 2479.34

I 4998.57 I2 2 2.02 2479.34

Answer: I1 0.678, I2 2.02 Note the advantage of this method. The substitution or elimination method would be very tedious to apply for such equations.

Exercises C2 (continued) Solve the following simultaneous equations correct to 3 significant figures: 4

6

106

7E 19V 24 {13E – 5V 9 2.70x – 1.40y 3.40 {3.80x 4.60y 1.30


5

7

13L – 15W 87 {17L – 11W 231 29.3I – 37.8I 83.7 {41.2I – 26.4I –16.3 1

2

1

2

A PPENDIX

Matrices and 3 3 determinants

D

D1 Matrices: introduction Matrices can be used, among other purposes, to solve simultaneous equations provided we define their operations (e.g. addition and multiplication) in special ways. The interest in and use of matrices has increased greatly since the introduction of computers because their operations are easy to program on a computer. Manually, for example, it is usually quicker to solve simultaneous equations using determinants. However, with a computer, matrices are much easier to use, regardless of how many variables are involved.

Definitions A matrix is a set of numbers (called elements) arranged in a rectangular pattern (or array) of rows and columns. A determinant, as we have seen in Appendix C, is such an array distinguished by vertical bars at each side. To distinguish a matrix, the array is enclosed in parentheses, either round or square. For example,

23 14 is a determinant, which has the value 5, but 23 14 or 23 14 is a matrix,

which does not have a value. A matrix does not represent a number. If a matrix has a rows and b columns, it is said to be an ‘a b matrix’ or to ‘have an order of a b’. A matrix of order ‘a 1’ is called a column matrix. A matrix of order ‘1 b’ is called a row matrix.

Examples

1

2

3 A 2 5

D

5 3 1 2

0 4 is a matrix of order 3 2. 3

is a matrix of order 4 1, a column matrix.

Appendix D Matrices and 3 3 Determinants

107

3

K (5

4

P

5

T

4

–2 is a 2 2 matrix, a square matrix of order 2. 3

0

2) is a matrix of order 1 3, a row matrix.

–3

1 2 4

0 3 3

3 1 is a 3 3 matrix, a square matrix of order 3. 0

We identify a matrix by a capital letter and its order can be shown under this letter. For example, K is a matrix that we call K and its order is 3 2 (i.e. 3 rows and 2 columns). 32

Exercises D1 Below is a set of matrices that are also referred to in following exercises. 3 1 0 2 2 1 B C 1 A 3 2 1 3 2 2

2 D 1

G

1 4

3 2

2 1 3

1 3 2

1 2 4

3 5 1

1 H 2

3 1

E

4 2 3

F (3

K

1 3 0 1

2) 1 2 0

2 3 2

1

Using the above set of matrices, state the order of: a D, b G, c C, d A, e F.

2

Which of the matrices in the above set is: a a 2 3 matrix, b a 3 2 matrix, c a square matrix, d a row matrix, e a column matrix?

D2 Some definitions and laws Equal matrices Two matrices are said to be equal if, and only if, they are identical in every respect—that is, the elements of each are the same and in the same positions.

Example If

108

c d 1 3, then a 2, b 4, c 1, d 3. a b

2

4


The sum or difference of two matrices These are found by adding or subtracting the corresponding elements of each matrix.

Example a b c g h i ag bh ci e f j k l dj ek fl

d 5

2

3 4

1 3 0 5

3 1

0 2 0 2 3 3

1 2

Note: Two matrices can be added or subtracted only when they have the same order (i.e. they must have the same number of rows and the same number of columns; they must have the ‘same shape’). The resulting sum or difference will also have the same order.

From the definition, it can be seen that A B B A (the commutative law for addition).

The zero matrix For matrix A the zero matrix, O, is defined to be the matrix such that A O O A A (the law of addition of zero). The letter O is used to denote a zero matrix.

Example The zero matrix of

a b c 0 is e f 0

d

0 0

0 . 0

The zero matrix of

0 2

–3 0 is 5 0

0 . 0

Note: The matrix O is not the number zero but is the matrix of the same order as A, which has the number 0 for each of its elements.

Multiplication by a constant By definition, a matrix is multiplied by a constant by multiplying every element of the matrix by that constant. Appendix D Matrices and 3 3 Determinants

109

Example

3 2 1 0 6 3 0 0 3 2 0 9 6

Exercises D2 1

Solve the following matrix equations: x 7 a y 3

c

2

2y 3 y 9 x – 1

3–x

x2

5

xy

3

b

y – 3 6

d

x – y 5

Write the single matrix equation 3x 2y – 5z 8 4x – 3y 2z 7 5x 5y – 3z 9

as three separate simultaneous equations. 3

Using the set of matrices (A, B, C . . . K) in Exercises D1: a state which pairs of those matrices can be added or subtracted b write down the matrix K E c state the zero matrix of D d state the zero matrix of E e write down the matrix 3 A f write down the matrix 2E 3K

D3 Multiplication of matrices Since a matrix is simply an array (arrangement) of numbers in a rectangular pattern and does not have a value, we can define the product of two matrices in any way we choose. For the purposes of explanation, the rows and columns of a matrix will be designated as shown in the matrix below: the rows being called R1, R2, R3 . . . and the columns C1, C2 . . . . Any particular element of a matrix can be identified by stating its row and its column. C1 C2 C3 ↓ ↓ ↓ R1 → 3 6 7 , R1C3 7, R1C1 3 and R2C2 5. For example, in the matrix R2 → 2 5 4

By definition, when two matrices are multiplied, the product is another matrix and regardless of how many rows and columns the matrices may possess, when the elements in Rn of the first 110


matrix are multiplied in succession by the elements of Cm of the second matrix and these products are added, this gives element RnCm of the product matrix. When put into words this definition seems very complicated but some illustrations and some practice should enable you to gain facility with this process. Ignoring all the other rows and columns that may be present:

1

R3 → 5 2 7

C2 ↓ 4 0

1

R3 →

C2 ↓ 27

In the product matrix, element R3C2 (5 4) (2 0) (7 1) 27 2

3 4 6 1

2

5

7 8

9 (1 5) (2 6) (1 7) (2 8) (1 9) (2 10) 10 (3 5) (4 6) (3 7) (4 8) (3 9) (4 10)

5 12

15 24

39 17

23 53

7 16 9 20 21 32 27 40

29 67

You are advised to practise this process until it becomes quite familiar to you. Below are some exercises to enable you to practise the multiplication of two matrices. You will quickly discover the benefit of using fingers or a pen to obscure the rows and columns not being used to obtain a particular element of the product matrix.

Example (2 4) (3 1) (2 2) (3 3) (4 2) (5 3)

4 5 1 3 (4 4) (5 1) 2

3

4

2

83

16 5

21 23 11

49 8 15

13

When the elements are small you should be able to obtain the product matrix without needing to write down the intermediate steps. e.g.

2 4 1 3 8 12 3

1

2

0

7

3


111

Exercises D3 1

a

You are given the following matrices: 3 1 5 1 1 2 A , B , C , 2 4 3 2 4 3

D

2 1,

Write down the following matrices: i AB ii A C v BC vi B D ix C E x DE b

You are given the following matrices: –1 3 3 –2 4 –3 P , Q , R , –2 4 6 –4 2 –1

3

E

0 3 2

0

AD vii B E

AE viii C D

iii

S

Write down the following matrices: i PQ ii P R v QR vi Q S ix R T x ST 2

5

–3 0, –2

0

iv

–6 3 –4

T

2

PS vii Q T

PT viii R S

iii

iv

As always in algebra, if there is no operation sign between two terms, multiplication is to be assumed. AK means A K, i.e. matrix A matrix K. Find the product of the matrices in each case below:

112

1 0 2

3 2 0

0 1 3

c

2 1 3

1 0 1

0 3 1

e

1 3

0 2

5 6

4 7

8 9

g

1 0

2 3

1 3

2 0

i

(0 3

k

a

2 4

1 2 0

2 0 3

3 2 1

0 1 2

1 1 3

2 0 1

d

f

0 1 1 1 3 3

4 1

h

3 0 1 0 2 1

j

l

(7

5 6


2 2 1

1 2) 1 4

1 3

1 2 1 0 0 3 2 0 1

b

1 1

3 2 1 5

0 2

2 1 (2 3

2

1 0 3 0 0 1

2 1 1 4 3 0 0 1 5

1 4 6

3

5 2 0

1)

6 5) 2 8

3

m

o

2 1 1 3

2 1 2

If A

0 3 4

3 0

1 2

3 1 0 2 1 3

3 1

2 –1 1 and B 2 0

5

0 1) 2 3

5 1 3

0 1 4 2 2 0

n

(4

p

4 0 3 2 0 1

2 , find a A2, b B3. 0

D4 Compatibility By now it has probably become clear to you that multiplication of two matrices is only possible when there are the same number of elements in any row of the first matrix as there are in any column of the second matrix, i.e. the product mn A pq B C exists only when n p, that is, only when the number of columns in the first matrix equals the number of rows in the second matrix. When, and only when, this is so, the matrices are said to be ‘compatible’ for this multiplication. If n p, the matrices cannot be multiplied and are said to be ‘incompatible’ for this operation.

Examples 1

The product M N does exist and has order 2 5.

M

N

2

3

3

5

compatible order of product

2

The product P Q does not exist.

P 2

Q 2

3

2

incompatible

3

The product R T does exist and has order 4 3.

R 4

T 3

3

3

compatible order of product

It is quite common for a product A B to exist but for the product B A not to exist. For example: V W exists (and has order 2 3), but 23 33 W V does not exist. 33 23 Appendix D Matrices and 3 3 Determinants

113

It is easy to show that A B and B A both exist only for mn A nm B :

A

B

B

e.g.

A

and 23

32

both exist. 32

23

Summary Facts about the product of two matrices:

◗

A B may not exist.

◗

Even if A B does exist, it is possible that B A does not exist.

◗

Even when both products exist, in general, A B B A.

◗

It is possible that, A B O even though neither A nor B is the zero matrix O.

Example

6 4 6 3

2

4

2 0 3 0

0 O 0

You can verify this after studying the next section.

Exercises D4 1

Using the set of matrices in Exercises D1, state whether the given product exists in each case (answering ‘yes’ or ‘no’) and, if it does exist, state its order. a AB b BA c AC d DA e FE f EF g CF h AE i BD j CD k DC l CE

D5 The identity matrix, I Exercises D5 1

Write down the product matrix: a b 1 0 a c d 0 1

2

0 1 c d 1

0

a

b

b

0 1 0

0 0 1

Write down the product matrix: a

114

b

a b c d e f g h i


1 0 0

0 1 0

0 0 1

1 0 0

a b c d e f g h i

The principal diagonal of a square matrix is the diagonal that runs from the top left-hand corner to the bottom right-hand corner. The square matrix, which has the number 1 for each element on the principal diagonal and all other elements zero, plays a very special role in the theory of matrices. 1 0 is called the identity matrix of order 2, and is specified as I2. 0 1

1 0 0

0 1 0

0 0 is called the identity matrix of order 3, and is specified as I3. 1

For any square matrix An (i.e. of order n n), An In In An. In plays the same role in matrix theory as unity does in arithmetic (e.g. 7 1 1 7 7), and so it is called the unit matrix, Un or the identity matrix, In. We use the latter name and symbol in this book. We use capital letters to identify matrices but the capital letter I is reserved for the identity matrix and the capital letter O is reserved for the zero matrix. Note: For a matrix A which is not square, A Im A (but Im A does not exist) and nm

nm

nm

nm

In A A (but A In does not exist). nm

nm

nm

Remember : I stands for the Identity matrix, not the numeral 1.

Therefore, non-square matrices do not have an identity matrix.

Exercises D5 (continued) 3

For the following, write down the product matrix if it exists. If it does not exist, write ‘incompatible’. 1 0 0 1 0 0 1 2 3 2 1 3 0 1 0 a 0 1 0 b 4 5 6 1 3 2 0 0 1 0 0 1

4

c

0 1 4

a

If

1

0

1

34

2 5

3 6

d

4 1

2 5

3 6

0 1 1

0

17 –13 19 17 –13 19 A , write down the matrix A. 28 15 34 28 –15

Remember: an identity matrix is always square.

b

If B

c

Does

34

17 –13 19 17 –13 19 , write down the matrix B. 28 15 34 28 15

34

17 –13 19 have an identity matrix? State the reason for your answer. 28 15 Appendix D Matrices and 3 3 Determinants

115

5

For each of the following matrices, state whether an identity matrix exists (answering ‘yes’ or ‘no’) and, if it does exist, write it down. 2 8 3 2 7 3 a b 7 5 1 0

c

6

3 6

If M

4 2 1 5

10

d

3 2 5 4 1 6 7 9 8

1 , show that a M2 –I, b M3 –M. 0

D6 The inverse matrix, A

–1

1

The numbers 13 and 13 are said to be multiplicative inverses of one another because in multiplication one undoes what the other does. 1

1

For example, 957 13 13 957, 7 13 13 7 . 1

1

This occurs because 13 13 1 and 13 13 1. Now we will see how this applies to matrices.

Exercises D6 1

If A

5 3

1 3

2 1 and B 2 5, write down a the matrix AB, b the matrix BA.

Since both products in the exercise above are the identity matrix, you can probably guess that A and B are said to be inverses of each other. The inverse of matrix M is written as M–1, so you have proved for the above matrices A and B that AB BA I, that is, that B A–1 and A B–1. Definition: Two matrices, A and B, are said to be inverses of one another (i.e. A B–1 and B A–1) if AB BA I.


Given that C a b c

116

1 1 1

1 1 2

1 2 and D 1

3 1 1 1 0 1 : 1 1 0

write down the matrix CD write down the matrix DC what have we proved about the matrices C and D?


3

Given that matrix pq A has an inverse rs B: a b c

d

what is the order of matrix AB? what is the order of matrix BA? Since these matrices are inverses of each other, by definition AB BA nn I Therefore, the orders of AB and BA are both n n. What can you deduce about the values of p, q, r, s and n? Hence, if matrices A and B are inverses of each other, what can you deduce about the shapes of the matrices A and B?

Most square matrices have an inverse, but not all of them. (Actually it can be shown that all square matrices have an inverse except those for which the determinant A 0.) A matrix that has an inverse is said to be invertible.

Summary ◗

If A B B A I, then A and B are inverses of each other (i.e. A B–1 and B A–1).

◗

A non-square matrix cannot have an inverse.

◗

Most, but not all, square matrices have an inverse (i.e. they are invertible).

Note: You are not required to be able to find the inverse A–1 of a given matrix A, but you should be able to determine whether or not two given matrices A and B are inverses of each other by testing whether AB BA I.


If A

D a i v b

5

1 1 1

B

1 0 1

2 1 0

0 2 2

3 4 2 1 0 : E 2 1 1 1

1 2 2 5 3 7

write down the products: AB ii AC iii AD BC vi BD vii BE hence, write down: (i) matrix A–1 (ii) matrix B–1

b

1 1 0 C 1 0 1 0 2 1

1 1 2

1 2 1 Given that J 0 3 2 0 0 1 a

2 4 5

2 1 2

iv

AE

3 2 1 and K 0 1 2 : 0 0 3

write down the matrix JK write down the matrix J–1 Appendix D Matrices and 3 3 Determinants

117

6

1 2 2 Given that P 1 3 1 3 2 0

2 4 4 and Q 3 6 1 : 7 4 1

write down the matrix PQ write down the matrix Q–1

a b

Note: Although you are not required to be able to find the inverse of a given matrix, you may be interested in a quick way to write down the inverse of a 2 2 matrix: If A

a b 1 , then A1 A c d

d b c a .

To obtain A–1: a interchange the elements on the principal diagonal; b reverse the signs of the elements on the secondary diagonal; c divide by the determinant of the original matrix.

Example

If A

4

4

2 1, then A 2 1 3

3

(4) (6) A

–1

2 1 3 2 2 4

0.5 1.5 1 2

Note:

◗

This method does not work for 3 3 matrices or higher orders.

◗

If A 0, the matrix A has no inverse (because division by zero is not defined). Therefore, the matrix is not invertible.

D7 The algebra of matrices As a result of the definitions of the operations of matrices, the laws for the algebra of matrices are mostly the same as the laws for the algebra of real numbers. 118


Name of law

Real numbers

Matrices

Commutative law for addition

xyyx

ABBA

Associative law for addition

(x y) z x (y z)

(A B) C A (B C)

Identity law for addition

x00xx

AOOAA

(0 is the identity element for addition)

(O is the identity matrix for addition, the zero matrix of A)

x11xx

AIIAA

(1 is the identity element for multiplication)

(I is the identity matrix for multiplication for matrix A)

xx x x1

AA A AI

Identity law for multiplication

Law of multiplicative inverse

–1

–1

–1

Distributive law for multiplication

–1

–1

(x is the multiplicative inverse of x)

(A–1 is the multiplicative inverse of A)

C(x y) Cx Cy (x y)C xC yC

k(A B) kA kB (A B)k Ak Bk

x(y z) xy xz

A(B C) AB AC

(y z)x yx zx

(B C)A BA CA

Remember:

◗

O, the zero matrix for matrix A, has been defined as the matrix with the same order as A but having all its elements zeros.

◗

I, the identity matrix for matrix A, has been defined for square matrices only, being the matrix having the same order as A, with all the elements on the principal diagonal being 1 and all the other elements being zero.

◗

A–1, the inverse of matrix A, has been defined for square matrices only, being the matrix such that AA–1 A–1A I. The only matrices that have an inverse are square matrices whose determinant 0.

Hence, most algebraic operations with matrices are already quite familiar to us.

Examples 1

If

3A 4B 5C

then

3A 5C – 4B 1

A 3(5C – 4B) 2

(A B)(C D) AC AD BC BD


119

3

(A B)(A I) A2 AI BA BI A2 A BA B

4

A(A–1 I) AA–1 AI I A, or A I –1

5

A(A B) (AA–1)B IB B

6

AB A AB AI A(B I), not A(B 1) because we cannot add a real number to a matrix

7

–1

A (A I) A–1A A–1I I A–1, or A–1 I

However, there is a difference between the algebra for matrices and the algebra for real numbers when we are multiplying because, as we have already noted, in general: A B B A. (You are reminded that this statement does not mean that they can never be equal but that we cannot assume they are equal, because usually they are not equal.) Hence, care must be taken to maintain the correct order of matrices when multiplying. For example, A(B C) (B C)A and ABA–1 A–1AB (which would equal B). The only occasion when multiplying matrices is commutative is when they are inverses of one another (AA–1 A–1A [ I]) or when one of them is the identity matrix (AI IA [ A]).

Example Make K the subject of the equation AK B. We proceed thus: AK B A–1(AK) A–1B (not BA–1) (A–1A)K A–1B K A–1B

Note: We cannot add a matrix and a real number (e.g. A 2 makes no sense). But we can always replace A by AI or by IA.

120


Example Solve the matrix equation AF 2F B, for F. AF 2F B AF 2IF B (A 2I)F B –1

(A 2I) (A 2I)F (A 2I)–1B F (A 2I)–1B

Exercises D7 1

2

Simplify, removing all brackets: –1 a A(A B) –1 –1 c A(BA ) A(A B) 2 e I 2 g (A I) i A(I – A–1) I2 Solve the following matrix equation for X: a 2X – A B c

3

3(A – 2X) 2(3A – X)

Solve the following matrix equations for X: a AX B c 2A AX B –1 e X A g AX X B i X XA B

4

If AB AC: a does it follow that B C (‘yes’ or ‘no’)? b and B C, what is B equal to?

5

If AB CA: a does it follow that B C (‘yes’ or ‘no’)? b and B C, what is B equal to?

b d f h j

b d

b d f h j

A(A–1I) AB(A–1B B–1A) I2 I3 (A I)(B A) A(A–1 I) – B–1B – IA C – 3X D XA X – 3B 3 2 XA B AX I B 3X – AX B AX B C – X 2X B C – 2XA


121

Note:

◗

The matrix, A B C (A B) C A (B C), the associative law, but we must not change the order of the matrices when they are being multiplied.

◗

The matrices ABC, ACB, BCA, BAC, CAB and CBA are probably all different matrices.

◗

However, consideration will show you that if k is a constant (i.e. a real number), then k (A B) (k A) B A (k B). Which matrix we multiply by k, either A B or A or B, makes no difference to the final result. Although we must not change the positions of the matrices, we may change the position of a constant, k.

◗

Note also that k A k AI A kI.


Solve the following matrix equations for X: a 3X 2XA C – B b

A 2X B 3XC

D8 Expressing simultaneous equations in matrix form Exercises D8 1

Find the following products and state their order: 3 5 2 2 3 x 1 4 3 a b 5 4 y 2 3 4

2

3x 4y 2x – 5y

b

Solve for x and y: 2x 8 a 3y 9

4

2x – 3y 4z x 2y – 5z 3x – y 2z

b

c

3a – 5b 2c 2a 4b 5a – 7c

12 1 y 27 13

1

x

29

Express as a system of simultaneous equations: a

c

122

Express as the product of two matrices: a

3

x y z

2x 3y

5x – 2y 4 3

3x – 2y 4z 6 2x 3y – 5z 7 x – 4y 2z 8


b

2

d

3

5 4

y 7

1 2 0 1 1 2

x

3 2 0

6

x 4 y 5 z 6

5

Express each of the following systems of simultaneous equations as a single matrix equation: 5a 7b – 3c 17 4x – 5y 6z 9 7a – 2b – 8c 13 7x 3y 5 a b 3a 5b 5c 19 3x – 8z 7

D9 Solving simultaneous linear equations using matrices A simple example is probably the best way to demonstrate how to solve simultaneous linear equations using matrices.

Example Solve the simultaneous equations

{ 5x3x––2yy 65 given that the inverse of

5 2

1

3 1 is 3 5. 2

Steps 1

Express the equations as a single matrix equation: 5 2

3 1 y 5 2

x

6

Multiply both sides by the inverse matrix: 1

5 2 1

3 5 3 3

2

1

y 3 5 5 x

2

6

Multiply out the matrices on each side—we know the product on the left-hand side because: 1 0 A–1A I . 0 1

0 1 y 7 1

0

x

4

y 7 x

4

x 4, y 7

1 Warning: The error most often made is in step 3. Remember that 3 6 1 2 because for matrices A B B A. 5 3 5

5 is not the same as

2 5

6

This method involves knowing the inverse of the matrix formed by the coefficients in the equation. You are not expected to be able to find the inverse of a given matrix, so either you Appendix D Matrices and 3 3 Determinants

123

would be given the required inverse or you would be expected to deduce it by showing that the product of two given matrices I.

Example 1 If M 3 2

2 5 –1

–1 11 1 –1 and N –8 2 –2 –7

–5 4 3

–3 2 : 1

find the matrix MN write down the matrix M–1 express the given system of linear equations as a single matrix equation and use the result of b above to solve these simultaneous equations:

a b c

{

x 2y z –1 3x 5y z 2 2x y 2z 9

Solutions a

2 1 MN 0 2 0

b

11 –5 3 1 M –8 4 2 2 –7 3 1

–1

0 2 0

0 1 0 0 2 0

0 1 0

0 0 1

Note: So that the matrix M will match the coefficients of the third equation, we write the third equation as – 2x – y – 2z –9. 1 2 –1 x –1 c 3 5 –1 y 2 –2 –1 –2 z –9

x –1 M y 2 z –9

x 11 –5 –3 1 M M y –8 4 2 2 z –7 3 1 –1

6 1 –2 2 –4

x 3 y –1 z 2

x 3, y 1, z 2

124


–1 2 –9

Exercises D9 1

a

Express the simultaneous equations 2x – 3y 9 5x – 7y 22 as a single matrix equation. Solve the above equations using matrices, given that the inverse of the matrix 2 3 –7 3 is 5 7 –5 2

{

b

2

You are given that if A a

If P

c d then A a

b

–1

d –b 1 –c a A

5 4: 3

2

i

evaluate P

ii

write down the matrix P

–1

Use the result for P–1 above to solve the system of equations 3x 2y 4 5x 4y 10 4 3 2 3 If M and N : 1 –2 1 –4 b

{

3

a

find the product MN

b

write down the matrix M–1

c

use the result of a above to solve the simultaneous equations below, showing each step of the working: 4x 3y 7 x – 2y 10

{

4

4 1 1 1 2 3 2 5 7 . Use this result to solve the system The inverse of 3 2 1 is 1 1 1 1 3 5

4a – b c 15 of equations 3a – 2b – c 13 . a–b–c3

5

11 5 3 1 2 1 1 4 2 and Q If P 8 3 5 1 : 2 7 3 1 2 1 2

a b

write down the matrix PQ write the system of equations below as a single matrix equation:


125

c

6

11x – 5y – 3z –12 –8x 4y 2z 10 7x 3y z 10

use the result of a above to solve the simultaneous equations in b using matrices.

3 4 If P 2 1 1 1

a b c

2 0 ,Q 0

1 2 2 5 3 7

2 3 4 2 4 , R 2 1 0 : 5 1 1 1

write down the matrices (i) PQ (ii) PR (iii) QR which two of these three matrices are inverses of each other? express the simultaneous equations a 2b – 2c 3 2a 5b – 4c 7 3a 7b – 5c 8 as a single matrix equation solve the above equations using matrices

d 7

Solve the following systems of simultaneous equations using matrices: 3x 2y – 2z –3 3 2 2 2x 2y – z 1 2 2 1 is a given that the inverse of –4x – 3y 2z 0 4 3 2 b

c

d

e

f

g

126

2a – 4b – c 0 5a – 10b – 3c 1 15a – 29b – 9c 5

given that if K

2p 3q 3r –2 3p 5q 5r –4 5p 3q 4r 0

2 given that 3 5

3 5 3

3 5 4

2x 3y 2z 1 3x 4y 3z 1 5x y 4z –1

2 given that 3 5

3 4 1

2 3 4

2 4 1 3 7 5 10 3 , then K–1 0 3 15 29 9 5 2

a 2b 3c –2 2a 3b 4c 0 3a b – 2c 17

1 2 3 given that 2 3 4 3 1 2

2a 2b –6 3a 2b c 0 7a 5b 2c –1

given that if M

5 3 0 1 13 7 1 0 16 9 1 0

1 1 1

2 1 0

0 1 0

0 0 1

0 1 0

0 0 1

6 3 2 4 2 , then M–1 3 11 5 7

3 2 5

13 10 1 3 2 0 17 13 1

–1

23x 5y – 35z –2 7 5 5 13x 3y – 20z –1 given that 3 2 5 19x 4y – 29z –2 5 3 4


1 2 2 0 2 1 2 1 2

23 13 19

5 3 4

–35 1 –20 0 –29 0

10 7 1 16 11 2 I 7 5 1

0 1 2

h

i

j

k

l

m

n

o

32x – 71y – 6z –1 38x – 84y – 7z –1 5x – 11y – z 0

32 71 6 given that 38 84 7 5 11 1

7p – 18q 3r 5 5p – 13q 2r 3 11p – 29q 5r 8

7 given that 5 11

18 13 29

3 2 5

3k – 2n 2t 0 5k – 3n 3t 1 7k 5n 4t 2

3 2 given that 41 26 46 29

0 1 1

2x 3y 2z 2 5x 2y 6z 11 3x 5y 3z 4

2 3 2 given that the inverse of 5 2 6 is 3 5 3

2a 3b 2c 0 3a 4b 3c 1 5a 5b – 5c 55

2 3 2 3 given that 3 4 5 5 5

3x 2y 6z 1 4x 3y 5z 1 5x 5y 2z 0

3 given that 4 5

4p 5q 7r 21 7p – 2q – 5r 0 3p 3q 4r 13

4 5 7 given that 7 2 5 3 3 4

2x 3y 2z 1 3x 2y 3z –6 5x 5y 4z 0

2 given that 3 5

7 5 7 3 2 4 2 3 10

2 3 5

3 2 5

2 3 4

–1

1 10

–1

1 7

36 1 22 3 0 2 31 1 19

35 25 1 30 20 0 5 5 1

35 5 –55 –215 –25 345 135 15 –215

–1

1 10

19 26 8 17 24 9 5 5 1

6 5 2

0 0 1

3 2 2 5 3 3 I 7 5 4

0 1 0

7 3 3 3 2 1 2 5 1

–1

1 0 0

–14 6 10

10 0 0 10 0 0

0 0 10

–4 10 –4 0 10 –10

By now you will have realised that the solution to any system of simultaneous equations is: U C–1 K where: U is the matrix formed by the unknowns; C is the matrix formed by the coefficients; K is the matrix formed by the constants.

Hence, matrices provide a general method for solving any number, n, of simultaneous equations with n unknowns. A computer program can easily be designed for this purpose. Note: Although a solution can be set out very concisely using the above formula, in an examination a student should show every step of the solution.


127

D10 3 3 determinants: definition and evaluation a b c d e f is a square array of ‘elements’ having three rows and three columns. It is called a g h i ‘3 3 determinant’ or a ‘determinant of third order’.





In Appendix C the value of a second order determinant was defined so that it provided a shorthand method of solving two simultaneous equations in two unknowns. The value of a third order determinant is defined so that it provides a shorthand method of solving three simultaneous equations in three unknowns. The value of the above determinant is defined as aei bfg cdh – gec – hfa – idb. There are several ways of obtaining this result without having the very difficult task of committing it to memory. We will use the method called the Rule of Sarrus. This method is the simplest but it applies only to determinants of order three. (Later, when you study determinants of higher orders, you will learn other methods and ones that are easier to program for a computer.)

The Rule of Sarrus 1

Write down the determinant, repeating the first two columns.

2

Obtain the products on the diagonals as shown below.

3

Add the lower products and add the upper products.

Subtract the sum of the upper products from the sum of the lower products. a b c Follow the application of this rule as we apply it to the general determinant d e f g h i a b c a b d e f d e 1 g h i g h 4



(Sum gec hfa idb)

2, 3 gec

hfa

idb

a

b

c

a

b

d

e

f

d

e

g

h

i

g

h

aei

bfg

cdh

(Sum aei bfg cdh) 4

128

Value: (aei bfg cdh) – (gec hfa idb)




Example 2 0 3 0 Evaluate: 1 4 3 1 2 Method:





(Sum 36) –36

0

0

2

0

–3

2

0

–1

4

0

–1

4

3

1

–2

3

1

–16

0

3

(Sum 13) Value: (13) (36) 23

Hint: When copying down a determinant be very careful to include any negative signs. Check your copy before working on it. If you copied it row by row, check it column by column. It is very annoying to work on data that is later discovered to have been copied down incorrectly.

Exercises D10 1

Evaluate: (Note: All the elements are exact numbers.) a

c

e

g

   

2 3 5

1 2 0 1 0 1



1 1 1 1 1 1

b

0 2 2



d

3 0 2

1 4 3

2 1 2

f

42 36 37

28 29 30

62 91 47

h

 

   

4 0 4

3 5 5 0 4 5

  

1 2 3 2 3 8 5 2 1 0 t t 0 n x

n x 0

20 24 28 31 47 64 83 51 86




129

2

In each case below, evaluate the pronumeral: x 2x 3x 1 2 0 9 a b 0 1 3 c

e

  

2 1 1

n n n

1 1 k

 

1 2 5 1 0 2 2

d

k 1 2 1 0 2



k 6 0 2 1 k



 

1 2 3 t 4 2

x 0 1

2 x 10 1 t 1 t

 

1 0 3 t

D11 Solutions of simultaneous linear equations using 3 3 determinants A system of three linear equations in three unknowns may be solved algebraically by the elimination method. However, this method is usually very tedious and the equations are more easily solved using determinants. a1x b1y c1z d1 The general equations are: a2x b2y c2z d2 a3x b3y c3z d3

If we solve these equations by elimination we obtain the solution: x y z x , y , z a1 where: a2 a3

b1 b2 b3

c1 c2 , which is the determinant formed by using the coefficients on the c3 left-hand side of the equation;

d1 x d2 d3

b1 b2 b3

c1 c2 , which is the same determinant but with the coefficients of x c3 replaced by the constants (i.e. the numbers on the right-hand side);

a1 y a2 a3

d1 d2 d3

c1 c2 , which is again but with the coefficients of y replaced by the c3 constants;

a1 z a2 a3

b1 d1 b2 d2 , which is again but with the coefficients of z replaced by the b3 d3 constants.

   

130


   

Example

Given:

x 2y 3z 4 3x y 2z 7 4x 4y 3z 3

12

1 2 3 (17) 2 3 1 2 19 4 4 3





8

1

–2

3

1

–2

3

1

–2

3

1

4

–4

3

4 3





1 4 3 (80) (42) y 3 7 2 38 4 3 3





32

–2

3

4

–2

–7

1

–2

–7

1

–3

–4

3

–3

–4

Solution

19 x x –1 19 y 38 y 2 19 z 57 z 3 19



42

(65)

12

–12

84

(84)

–84

6

36

(–42)

1

4

3

1

4

3

–7

–2

3

–7

4

–3

3

4

–3 –32

16



–36 (–17)

4

–21

1 2 4 (5) (62) 1 7 z 3 57 4 4 3

(2)

–4 16

–9

4 2 3 (84) (65) x 7 1 2 19 3 4 3

–18

–27 (–80)

28

1

–2

4

1

–2

3

1

–7

3

1

4

–4

–3

4

–4

–3

56

18

(62)

–48

(5)


131

Points to note •

After much practice, the value of a determinant can be found on a calculator, using the Rule of Sarrus, showing no intermediate results. The diagonal products can be summed using the M and M– keys. However, for the time being, you are advised to write down the product of each diagonal before adding it to the calculator memory, as in the examples given. This allows you to check your work more easily and also allows for credit to be given in an examination for knowledge of the method even if an error is made during the computation.

•

When you have solved a set of simultaneous equations, check your result by substituting your values back into the original equations. You will be surprised how often careless errors are made during a long series of calculations.

•

When using a calculator, always work with all the significant figures in the data. State your result to the appropriate number of significant figures, that is, the number required or justified.

Hints:

a

Before using the Rule of Sarrus to solve a set of simultaneous equations, make sure that: i

all the equations have the pronumerals in the same order;

ii

all the pronumerals are on the left-hand sides of the equations and all the constants are on the right-hand sides;

iii if a pronumeral is absent from an equation, write it in with a zero coefficient—for

example, if there is no y in an equation, write it in as 0y. b

When copying down an array to work on, be careful to place the elements in a neat rectangle so that the diagonal elements are approximately in straight lines. An untidy array leads to errors in computation.

c

Before working on your arrays, check that you have made no errors, especially by omitting any negative signs. Discover any errors before you start to multiply.

d

When multiplying out diagonals, either mentally or with a calculator, ignore any negative signs present. Decide after the multiplication whether the product should be positive or negative.

Exercises D11 1

Solve for the pronumerals using determinants (do not use a calculator): a

132

2x y 3z 4 3x – y – 4z 5 4x 3y 2z 1


b

2p – 4q 3r 8 0 p 3q – 2r 2 0 3p – 5q – 4r – 4 0

c

2

3x 2y 4z –7 2y – 3z 4x 6 4z – 5x – 2y –7

d

4n – 3p 5t –3 3n 4p –6 5p – 4t –4

Solve for the pronumerals using a calculator:

a

37x 49y 76z 98 68y – 34x – 93z 136 82z – 36y 29x –72

b

0.002x 0.003y 0.001z 0.01 0.3x – 0.4y 0.2z 0.8 4000x 5000y – 3000z 4000

Hint: In b, multiply or divide both sides of each equation by a constant so as to obtain simpler numbers. State results correct to 3 significant figures.

c

2(3.7k – 1.4t) 3(4.3t 2.6w) 8.7 0 4(2.3k 1.7w) – 5.3w – 6.1 0 3(3.9k 4.2t) 0

3

F2

F3

P

37°

53°

F1 1m

1m

1m 2 kN

The equations for equilibrium of this beam are: Vertical forces:

0.6F2 – 2 0.8F3 0

Horizontal forces:

F1 0.8F2 – 0.6F3 0

Moments about point P: 0.6F2 – 4 2.4F3 0 Using determinants, solve for F1, F2 and F3, stating the results correct to 3 significant figures. Explain the meaning of any negative values. 4

6V

4

4

12 V I1

2

I2

3

I3 12 V


133

Applying Kirchhoff’s laws to this network: 4I1 2(I1 – I2) 12 2(I2 – I1) 3(I2 – I3) 6 3(I3 – I2) 4I3 12 Use determinants to solve for I1, I2 and I3, stating their values correct to 3 significant figures. 5

A developer receives council permission to divide his land of area 44 ha into 100 blocks, the only areas allowed for a block being 2 ha, 0.5 ha and 0.2 ha. He prices the 2 ha blocks at $100 000 each, the 0.5 ha blocks at $40 000 each and the 0.2 ha blocks at $20 000 each. He sells all the blocks, the total gross income from the sales being $3 200 000. How many blocks of each size did he sell?

6

A firm has a stock of three different bronze alloys. Alloy A consists of 95% copper, 3% tin and 2% zinc. Alloy B consists of 90% copper, 9% tin and 1% zinc. Alloy C consists of 80% copper, 15% tin and 5% zinc. How many kilograms of each of these alloys must be melted and mixed in order to produce 100 kg of a new alloy that consists of 87% copper, 9.6% tin and 3.4% zinc?

134


CD Supplement Answers 1 Fractions and decimals

2 Ratio, proportion and percentage

Exercises 1.1

Exercises 2.1

1 b c b c

a –2

b –4

c –7 d –3 e –6 f –6 2 a –10 –2 c 11 3 a 3 b –7 c 0 4 a –4°C b 5°C –6°C 5 a loss of 5 dB b gain of 2 dB 6 a 468 10 7 a 15 b –2 c –15 d –8 8 a –3 b –2 –5 d –4

b –19

c 0

2 a 2

b 32

3 a 8 b 41

Exercises 1.3 1 a 2, 3, 4

19 24

17 18

9 2 8 2 a 1 0 b 3 c 9 d 3 3 3 7 a 25 b 94 c 111 2 d 25 53 131 149 128 c 6 d 1 e 12 f 1 2 1

b 3, 9

323 17 b 642 c 19 4 17 27 5 a 3 b 2 0

19 20

1 12

5 124

3 4

10 3 a 3 9 3

e 27 19 6 a 100

31 b c d e f g h 140 1 31 1 17 1 7 7 8 a b c – 7 a 1 4 60 0 b 12 c 2 d – 60 12 5 11 27 6 4 1 9 a 116 b 6120 c 2215 d 814 10 a 7 b 85 0 1 1 3 1 c 6675 d 114 e 1794 11 a 6 b 6 c 4 d 51 1 1 10 3 1 9 12 a 3 b 2 c 3 e 3 f 1 0 g 123 h 21 8 3 10 1 3 3 2 2 5 d 5 13 a 13 b 8 c 4 d 15 e 23 14 a 19 1 2 3 1 2 3 15 a 5 b c b 82 1 c 7 d 33 e 2 7 2 7 7 3 10 3 5 1 2 16 a 8 b 27 17 a 4 s b 58 s c 73 18 a 6663 1 1 1 2 b 150 c 3000 19 8 20 16 21 192 22 3

11 124

Exercises 1.4 1 b b b c e d b b

b no

Exercises 2.2 1 39 V

2 219

3 1300 Hz

Exercises 2.3

Exercises 1.2 1 a 6

1 a 7 : 15 b 14 g 2 1 : 5 3 a 12 : 5 4 1 : 60 5 165.2 kg 6 26 kg

a 24.31

b 652.13 c 84.06 2 a 15.106 315.66 c 258.11 3 a 0.032 b 1234 4 a 0.02 3 5 a 0.010 35 b 0.0544 6 a 0.3415 0.0024 c 0.27 d 0.132 7 a 20.7 b 0.61 0.473 8 a 0.625 b 0.1875 c 0.325 d 0.2125 0.406 25 9 a 68 b 24 10 a 2.1 b 20 c 2 120 11 a 50 b 300 c 300 12 a 0.06 V 0.98 V c 0.12 A d 0.09 A e 50 13 a 5 S 0.02 S c 0.05 S d 25 S e 1000 S f 0.4 S

g 0.002 5 s 14 a 0.2 s b 250 sweeps/s 15 a 0.05 watts b 5 watts

Exercises 1.6 1 a 2 b 4 c 3 d 3 e 2 f 4 2 a 50.0 mm b 0.000 803 kg c 1.00 m d 0.607 km 2 2 2 2 2 3 a 2.86 m b 2.8 m c 2.9 m d 3 m e 70 m 2 f 68 m

1 6 9 b

no 2 yes 3 yes 4 yes 5 15.0 g 17.1 m3 7 23.3 mA 8 1.00 A a 18.5°C/min b 3.62 A c 6.3°C 10 a 22.1 mg 2.43 mm c 720

Exercises 2.4 1 a 1.57 Hz b 24.4

c 8.93 m

Exercises 2.5 1 b 6 8

a 1.03 kPa b 991 K 2 2.5 mm 3 a 20.6 kg 6.36 mm 4 78.6 Hz 5 a 1.8 J b 866 V a 3.6 km b 20 m 7 a 150 kHz b 710 F

167 mL/s

Exercise 2.6 1 2 3 4 5

a a a a a

0.63 b 4.21 c 0.0061 d 0.0103 e 0.000 34 36% b 4% c 130% d 0.23% e 1725% 60% b 85% c 44% d 78% e 42.5% 1 1 1 2 6 372% b 422% c 333% d 513% e 427% 28.6% b 37.5% c 83.3% d 266.7%

e 177.8% h

5 8 1 10

i

1 3

1

2

3

7

5

1

5

6 a 4 b 5 c 8 0 d 8 e 12 f 6 g 6 9 2 3 3 2 1 3 j 14 7 a 7 b 20 c 8 d 15 e 7 f 50

19

7

8 a 0.036 b 0.2534 c 0.0468 g h 2 0 i 16 d 2.9 e 0.34 f 0.034 g 0.144 h 36 i 0.09 9 a 15 kg b 7.6 m c 27.7 t d 269 mm

10 a 84.2% b 88.7%

11 a 84.5%

3 Measurement and mensuration Exercises 3.1 1 a 1000 m b 0.000 001 kg c 1 000 000 m d 0.001 kg e 1000 kg f 0.000 001 m 2 a 0.037 m b 520 t c 0.063 km d 0.023 t

Exercises 3.2 1 18 000

CD Supplement Answers

135

Exercises 3.3

Exercises 4.4

1 8000

1 a 8ab b x c 3x d 2xy e 2ax f 6km 2 2 2 2 2 g 24a b h 2x y 2 a 2b b a c 6b d 5 e 2

2

2 0.1

Exercises 3.4 1 a 0.4 m (40 cm, 400 mm) 2 a 6% b 20% c 0.4%

Exercises 4.6

1 27.9 2 2.31 3 2.03 6 1.11 7 117 8 0.532

4 3.21

5

1 a 3a

1.91

1 28% 2 852 mm b x 30

2 a bc

3 a b

Exercises 4.7

2

1 0.57 m

7 2 5 3k c 1 2 a b 2a 3R y 7ax t 3x 5 7 12x ab 1 3 a b 4 a 2 m 12 ab x 2 3y 2 13n k c 5 a 85 b 17m c 2 5m 3b 3x m2 a ay m 5y 1 a b c 7 a b c 7b bx k x t 3x2y x2 2 m mk k 2 3 m e 8 a 7 b 9 a b m k m m x2 1 1 a2 b2 km m2 10 a b c 2 x1 k1 t k m

1 a b 2

Exercises 3.10 2 14 m 2 5 36 m

2

2

3 a 14 m b 5.6 m 2 2 6 6m 7 36 m

c b

Exercises 3.11 2

2

1 95.3 mm 2 161 m 3 2.84 m 4 a 18.3 m 2 2 2 2 b 1.93 m c 8.38 m d 6.23 m e 8.02 m 2 2 f 463 m g 3900 mm

6 d

Exercises 3.12 1 31 L 2 176 cm 5 180 mm

3

3

3 1.1 m

c

4 19.9 km

Exercises 4.8

Exercises 3.13 2

2 881 cm

2

2

3 3060 m

4 581 cm

2

7C 3 7x 3 c – 6 12 2x 13y c 24

Exercises 4.1 1

b –3 c 0 2 a 16 b 8 c 2 d 5

2 b 9

3 a 0 b 4

c 0

Exercises 4.2 a 6x 5x b 4ab 3a 3b 8x2y 6xy2 3x d 3a2b2 ab2 4a2 4b2 5pq 4p 3q 2 a 2x2 x b 7x2 5x x 2 d 2p 3q 4pq 2

Exercises 4.3 1 a 5x 3 b 4 k c 3a 3 d 2x 6 e 3m 4 2 a ab a b b 2q c a Mathematics for Technicians

1 a 1 n b 2C 3 b

4 Introduction to algebra

136

d 6a bc

2a a 3c x mt 3 2 c 4 a 6ab b 18abc c 24a bx ab 3 3 2 d 36a b c 5 a ax(b c) b p(a b)(x y) 3 2 c q(a b)(p q) d 48k (x y)

Exercises 3.9

1 c e c

2

c 4ax

2

4 a t 14

3 4.65 m

b 3b

b a b c a b d 4k (x y)

Exercises 3.7

1 a 13 4 a –7

2

2

Exercises 3.6

1 7180 cm

2

1 a 6a 4b b 20 8a c 6x 2ax 2 2 2 2 d 3a b 6ab e 5 20x 3x 2 a x 5x 6 2 2 b x x 12 c 3 5x 2x 2 d ap aq 3p 3q e 2b 2a 5b 5ab 2 2 2 f a 6a 9 3 a b 8b 16 b k 4k 4 2 2 2 c a 4 d m 8m 16 e 9 x 2 2 f 4a 12ab 9b

1 a 0.0005 g 0.5 mg 500 g, 0.4%, 0.1225–0.1235 g 122.5–123.5 mg b 0.05 kg 50 g, 1%, 5.15–5.25 kg c 0.5 t 500 kg, 0.06%, 822.5–823.5 t 2 a 3.84 b 12.99 c 13.00

2

2

Exercises 4.5

b 0.2 t (200 kg)

Exercises 3.5

1 117 m 2 4 60 m

2

c 3 2L 3 3 a 4

n3 15 k 10 b – 3

2 a –

Exercises 4.9

2a 3x 3 x2 44n 11 3 a 14n 1 b 8n 1 c 15 7t 11 3t 4 4 a 5t 5 b c 6 6 1 a 1

b –1

c –1

2 a b

Exercises 4.10 1

1 a x 12 b a –1 c x 0 d L 0 1 1 1 e x –4 f x 3 g R 32 h d 1 i W 1

1

j x0

1

k K 0 l x – 3 2 a x 52 1 2 c x –2 d x – 2 e x 3 f W 0 2 g R 13

2

b x 3

8

2

h l 2 i a –2 j R – 7

10

1

10

3 a d 13 b l –12 c x –72 d a –11 1 1 e x – 17 f m 1 g k 13 h b 18 1

6

c t6

3 d k 15

1

10

1

4 a x – b x 1 i x 1 0 j y 19 3 2 4 5 1 c k 5 d x – 12 5 a k –5 b a 21 0

13

19

1 a C –2.1 b V –1 2 a x7 b E5 3 a n

16

1 6 a m 3 b t 3 c x 2

Exercises 4.11

1 12

H L mm (P 100) 7 m1 ts 100 Ft mu Eb v 9 W m a T mg

I r 11 a 12 d B 2H m 2 T l F 14 a 15 F R

d R E El 2W l 1 17 I 18 V Rt V1 C RL B 2 W 20 N2 N1 R V IR1 V R2 R1 R2 or R1 22 t I I R1 R1(T t2) R2T Q t1 24 C e iR R2 8R r 4R

s 6 c

b x

c E

3 16

d T

c W 1 d x –2

2 – 9

21

1 16

23

Exercises 4.12 2 18.8

1 4.77 kg

25

3 120 of A and 60 of B

Exercises 4.13 1 a yes

b no

2

1 a E – 2, R 63

e no

f yes

4

b W 4, F –3

1

1

2

1 a a –12, t 1 b I1 –1, I2 –32 c E – 3, 1 4

R 2 a R1 5, R2 12 b x 6, y 4 3 a V1 43.8, V2 –11.3 b C1 1.15, C2 0.528

Exercises 4.16 1 a T0 32.5°C, k 2.5 b 13 min

3E ba 2 x mk kn t a2b x a b abk

1 y

Exercises 4.15

c 57.5°C

2 L1 168 H, L2 252 H 3 a 0.004 32, R0 24.7 b –231°C 4 a E1 4 V, E2 6 V b I1 6.16 A, I2 5.53 A

W(1 )

1000P l.a.n d 1(V 2) naTa b n Tana 4 a d a Tb b 2 V nb Tb dV ms(1 q) d1 2 5 a m V2 q qm Hs L ms 6 c 1q ts 100m1 mm(P 100) a m1 b mm P 100 100

3 b b 7

Exercises 5.1

(H h)Ws 9 a C nWf

Exercises 5.2 1 a 504 ms b 103 ms 2 a 5.37 W b 82.4 W 2 3 a 2.11 b 71.9 4 6.47 m/s 5 2.98 m/s 6 1.90 s 7 372 mm

Exercises 5.3 k 5 b x 2 a x 4.4 w 7 bm ka bm 9k2 b x or a 3 x k k 4a a 2b 2 4 x 3(n –y) a 5 x k m

W E

1 a F b V

8 a r 2

1 1.59 2 16.2 3 –5.99 4 6.99 5 4.84 6 –6.35 7 63.1 8 2.31 9 2.03 10 5.30 11 1.42

n kt kt

3 E

Exercises 5.5

5 Formulae: evaluation and 5 transposition

1 a x

Exercises 5.4 c yes d no

Exercises 4.14 1

2 p

T T mg b m 2r g m

nWf C Ws

c h H

(H h)W nC

b Wf s

P b C

b 10 a M

P R2t1 R1t2 m b 11 T Pi R1 R2 6E1 4E2 2rR2l R1Rs 12 E 13 d 13 r(R1 R2) E RaI2 RcI3 R1 R2 14 I1 15 b Ra Rc R2t1 R1t2 Rb(R1 Rc) 5C 16 Ra 17 C1 Rb Rc 5C LL1 LL1 L1L2 LL2 18 L3 L2 or L1 L L1 L b


137

Exercises 5.6

Exercises 7.2

W 1 a 47.2 mJ b h c 1.06 km 2 a 45.8 mV mg Q b C c 647 F 3 a 63.3 mm e iR 2Fs b v c 15.1 m/s 4 a 15.6 kN m

1 c d, a b e f 2 a c 3 b c d f, a e 4 b c, a d

T2 P2

b m c 344 g

g

5 a 508 Hz

L 2 2 2 4 L f c 92.5 6 a 184 s C s b v c 907 mm/s 7 a 3.02 kV t1 t2 E I2R2 b I1 c 11.5 A 8 a 445 mA R1 R2 I1R1 b R2 c 1.15 M I I1 b R

b 1 V W L L L2 b M 1 2 1 d 2 C Z R2 10C ii C1 3 a 5C

W Vb WV 1 2 2 c L Z R 5C1 2 i C C1 10

250 b 6.57 k c 2.46 k

6 Introduction to geometry Exercises 6.1 1 a 120°21, obtuse

b 90°, right

c 204°31, reflex

e 180°, straight

2 a 48°27, acute

b 184°39, reflex c 180°, straight acute e 90°, right

d 77°36,

Exercises 6.2 1 a 40°19 b 1°7 c 72°38 d 141°9 2 a (180 2x)° b (90 3y)° c 0 d (180 k)° 1 3 a t 72 b x 20 c x 31 d t 38

Exercises 6.3 2 m 40

1 a yes, SAS

5 x 1.5

3 x 1.45

6 k3

4 d8

7 x 6.3

Exercises 7.6 1 a square, 52 m b rhombus, 6 m 2 a trapezium b no c 55 m d 120 m

Exercises 7.7 2

b 17.2 m

Exercises 7.8 2

1 a 32.0 m b 11.7 m c 6.40 m 2 a 7.62 m 2 2 b 22.9 m 3 a 8.37 m b 66.4 m

8 Geometry of the circle 1 x 55 2 k 20 3 h 52, t 26 1 4 y 180 2x 5 a 70, b 40, c 20, d 40, e 90, f 50 6 a 40, b 20, c 50, d 50, e 70, f 30 7 x 110 8 k 100

9 Straight line coordinate geometry Exercises 9.1 1 a i 10 units ii (5, 0) b i 13 units ii (3.5, 6) c i 5 units ii (–3, 1.5) d i 5 units ii (–4.5, 0) 2 (–8, 14)

Exercises 9.2 1 a 2

b –10

b 5

2 13

c

3 m 150

c –5

c

3 14

1

c 22 3 a 3 5 a 3 b –3 c 9

2 a 4 b 10

4 a 4 b 7

1

1 a i –2 ii 5 iii 22 1

7 Geometry of triangles and 7 quadrilaterals

iii –3

1

1

b i –2 ii 12 iii 3 1

d i –3 ii 7 iii 23

2 x 20


3

60

1 1 c i 12 ii 2

2 B, D, F, G, H

Exercises 9.4 1 a y 4x 3 b y 3x 18 c y –4x 3 1

Exercises 7.1

138

1 2 x 13

1 x 4.5

Exercises 9.3

2 w 130

1 x 40, y 100 4 x 20

b yes, RHS

Exercises 7.5

1 d – 4

3 p 43

Exercises 6.4 1 x 20

Exercises 7.4

2

1 a a

1 x 110

2 12 mm

1 a 7.98 m

Exercises 5.7

d 64°23, acute

Exercises 7.3

d y 10x 22 e y –6x 3 2 a y –2x 9 b y 2x 7 3 1 y –5x 15 2 y 2.5x 10 2

2

3 y 13x 13

4 y 1.75x 10.5

Exercises 9.5 (The values indicate roughly the expected accuracy.) 1 a 793 kW ( 2) b 5.84 kg/s ( 0.01) c 157 kW/kg/s ( 5) 2 a 3.7 min ( 0.1) b 7.1°C ( 0.2) c 2.82°C/min ( 0.05) 3 a 1.52 t ( 0.03) b 0.40 t ( 0.02) 2 c 36 kg/mm ( 1) 4 a 38.8 mV ( 0.2) b 635°C ( 2) c 54 V/°C ( 1) 5 a 1700 ( 200) b 1600 A/m ( 200) c 2030 ( 20)

x 3 0.9108 4 a 0.8862 b 4.097 1 x2

b

Exercises 10.8 1 14.3 km

2 147 m

11 Indices and radicals Exercises 11.1 1 a 13

b 6 c 3

Exercises 9.6

Exercises 11.2

1 790 N

1 a –9

2 a –$3500 net profit if no articles produced b More than 81 articles must be produced in a year in order to make a net profit c $43 increase in net annual income per article produced d $10 700

Exercises 11.3

Exercises 9.7

Exercises 11.5

1 a x –0.67, y 3.67 b x –1.33, y 1.33 2

b 39°C (1), 168 mm ( 10)

c LA 3.7 25, LB 1.8 96

10 Introduction to trigonometry Exercises 10.1 1 a 26°5227 b 263°2613 c 82°0632 2 a 49.364° b 80.773° c 168.293°

b 19.2 m

1

b 9 c –8

e 27

d 1

e 1

1 a –12 b 3 c 225 d 20 2 a –4 b 0 c 16 2 d 25 3 a 48 b 36 c –12 d –36 4 a 16t 3 6 2 2 5 b 8x c 8k d 9t e x x f 2x

1 a 4n 8n– –1 b 2x 4x– d –125 1 3

7 6

1 2

b e e 3 a 8 b 4 2t

–2t

2 a e 1 c 4 4 a 2 b –2 –2x

c 9

Exercises 11.7 1 ii d ii

a i 1.327 10 ii 132.7 b i 8.104 10 810.4 c i 8.090 105 ii 809.0 103 7 6 –4 i 1.281 10 ii 12.81 10 e i 2.075 10 –6 –7 –9 207.5 10 f i 4.265 10 ii 426.5 10 2

2

Exercises 11.8

Exercises 10.3 1 a 162 mm c 5.17 m

d 2

2 a 237 m b 423 mm

Exercises 10.4 1 a 72.37°, 72°22 b 87.57°, 87°34 c 66.81°, 66°49 d 0.4762°, 0°29 2 a 34.79°, 34°48 b 22.62°, 22°37 3 a 15.7 mm b 37°47 c 118 mm d 108 mm

Exercises 10.5

Vl

0.25

kP

1 a d

b 5.00 mm

k

5 7

P

2 a V

R 56 273 R1 1 b 55 F b 2270°C 4 a C ( L Z2 R2) b 1.80 L

2 3 a t2 (t1 273)

12 Polynomials

a f b d d b c d e 2 a 34°31 c e a e e b 32°11 3 a 26.4 mm b 35°05 4 84°50 5 a 58° b TC 0.625r c r 24.6 mm 6 14.91 mm

Exercises 12.1

Exercises 10.6

d x(x 3y) e ax y(3 xy) f 2x(1 a 3bx)

1 a

1 28.9 mm 2 39.7 m 3 109°39 4 91.8 mm 5 36.3 m 6 a 2.84° b 80.8 c 5.82 k d 17.0 e Z 849 , I 14.1 mA

Exercises 10.7 1

5 m a 2.4 b ( 0.3846) 2 a 13 m2 1

1 a x 16 b 2x 2x 7x 3x 5x 6 2 a 2 b 1 3 –43.85 4

5

4

3

2

Exercises 12.2 1 a 2x(3 4x) b ab(a b) c 2(p 2pq 3q) 2

Exercises 12.3 1 a 4a(b c) b 4p(q r)

Exercises 12.4 1 a (F 1)(F 12) b (C 1)(C 5) c (e 3)(e 3) d (L 3)(L 13) CD Supplement Answers

139

e 2 c 3 c e 4 c b 6 c e

(r 4)(r 9) f (C 4)(C 25) a (L 3)(L 4) b (T 4)(T 12) (E 3)(E 13) d (V 5)(V 20) a (4a 3)(a 2) b (2a 9)(2a 1) (2x 3)(4x 3) d (8x 9)(x 1) (8x 3)(x 3) f (2x 9)(4x 1) a (6x 1)(x 6) b (9x 1)(x 2) (10k 1)(k 3) 5 a (2a 15)(2a 1) (6n 1)(n 2) c (3n 2)(2n 1) a (3x 1)(2x 3) b (2k 5)(2k 1) (8x 3)(x 2) d (8x 1)(x 6) (10t 1)(t 10) f (5t 2)(2t 5)

Exercises 12.14 1 a i b 5 l ii A 5l l b l 3 m, b 2 m 2 l 3.41 m, b 586 mm 2 2 3 a x 3x m b x 4 c x 2.41 2

Exercises 12.15 1 a I 2, 6 b I 2, 4 c I 0.354, 5.65 d I 0.317, 1.18

Exercises 12.16 100 P

1 a h

100 P 1000

b h

c 2 kN for 3 min

2 a CV QV 2W 0 b 3.00 V c 167 2

F

Exercises 12.5 1 4(R 2)(R 3) 2 10(C 1)(C 3)

13 Functions and their graphs

3 8(2V 5)(2V 5) 4 4(R 3)(R 5) 5 2(3V 4E)(3V 4E) 6 10(C 12)(C 1)

Exercises 12.6 1 V2 5 3M 1

3 –R 2

2 3

R 3

V 3

1 3

4 – or –V

1 3

6 – or –R

1 3 b 7

a 7 a 6

1

b 11 c 1 d –3 2 a 2 b 3 c 6 d x 1 3 b 0 c 2 d 4 4 a –2 b –7 c 8 5 a –8 c 1 6 a x 3 b x 4.2 c x 6

27 k 1, 3

Exercises 13.2 1 a i x 2 ii (2, 2) iii 6

Exercises 12.8 1 e b 3

Exercises 13.1

a x 3 b C 5 c x 3 d x 3 x 2 f x 2 2 a M 4.47 R 5.59 c x 2.45 d V 2.77 a Q –1.5, 2.5 b L –2, 5 c x –1, 5

iv y

d M –0.34, 1.34 e t 5 4 a t –2.54, 1.54 b x –0.0868, 1.92 c x –5.04, 13.04 d k 0.631, 2.37

6

2

Exercises 12.9

x

2

1 a F 0, 4 b t 0, 1.5 c E –0.4, 0 2 a Q 0, 0.375 b L –38, 0

Exercise 12.10 1 a L –2, 9 b M –5 c x –4, 3 d x – 4, 1 2 a Q –3, 4 b C 2, 4 c x –2, 4

2 1 3 a x – 3, 0, 2

b i x –2 ii (–2, –12)

iii –32

iv

1 b x 0, 4, 3

Exercises 12.11 1 –4, 3 2 –6, 4 3 x 4x 21 2

y x

–2 –12

Exercises 12.12 1 a –3.56, 0.562 b –0.732, 2.73 2 a x –0.871, 2.87 b x –1.535, 0.869

Exercises 12.13 1 a V –1.83, 3.83 b k 0, 0.209, 4.79 c M –1.35, 1.85 d C –8.34, 0.839 140


–32

2 a x 2 b (2, –27) e

c –15

b circle with centre at (0, 3) and radius 5 units

d –1, 5

y

y 8

–1

2

x

5

3 C –15 x –27

3 a i 0, –3.5

–2 ii x –1.75

iv

iii (–1.75, –6.125) y

–3.5

Exercises 13.6 1 x –5, y 21; x 1, y 3 1

2 x 22, y 13; x 3, y 16 1 1 1 1 3 x 13, y 123; x 12, y 142

x

–1.75

Exercises 13.7 1 x –2, y 17; x 3, y 17 1

1

2

1

2 x 2, y 14; x 3, y 29 1 3 3 x 3, y 49; x 12, y 134 –6.125

Exercises 13.8 1 2 c maximum area 24 m ,

1 a A x(3 x) ii x –0.2

b i 0, –0.4 iv

1

which occurs when x 12

iii (–0.2, –0.2) y

–0.4

x

–0.2

14 Logarithms and exponential 14 equations Exercises 14.1 1 1 a 3

–0.2

1

b 4

c 8

–2

d 9 e 1

2 a 2

0

b 2

3

c 92

– 52

d 4

Exercises 14.2 1 a 2 8 b 2 1 c 3 3

Exercises 13.3 1 a y 2x 12x 10 b y 2x 16x 30 2

2

0

1

Exercises 14.3

Exercises 13.4

1 a 3 b – 2

1 a circle with centre at (7, 5) and radius 4 units

Exercises 14.6

1

1

2 a log9 3 2

2

c 13

1 a 3 b –3

c 0 d –1 e 0 1 1 b 2 log y c log x 2 log y

y

1

9

b loga x t c log8 4 – 3

2 y x 2x 8 2

–2

2 a log P n log Q

1 d log 5 log x 2 log y e log a log b log c

5

C

f log 4 3 log x g log K x log t

4

h n log x n log y

Exercises 14.10 7

x

1 a k6

b x 1.2 c t 1.6 d x 4 CD Supplement Answers

141

1

1

7

e k 22 f x 34 g n 12 2 a n 0 2 1 b x 2 3 a x 13 b x –22 4 a m 1.24 b x 0.189 c k 0.727

2 a $6729.34 b $6747.85 4 a $8009 b $7844

Exercises 14.11

1 15 cents 2 7.70%

x log t log t t 1 a y b y x c Q C k log k log a 2 a Pi Po 10

b 190

Exercises 16.2

1 a 556°C b 124°C c 20°C d 3°C 3 a 136 L b 130 min

W

1 a y

y

64

log 3 (≈ 0.48) 0

2 18.57 g

Exercises 16.4

Exercises 14.13 1

b 6.93%

Exercises 16.3

Exercises 14.12 –0.1G

3 a 7.18%

2

4

6

8

x

32 16 8 0.05

0 b

2 y

0.1

0.15

x

v (v) 12

y = 12 log8 2x + 3

15 12 9

6

y = 12 log8 2x

6 3

3 0

1

2

3

4

x 0 c

1.4

2.8

t (ms)

4.2

y 80 70 60

15 Non-linear empirical equations

40

Exercises 15.1 1 a C 3.0, K 11.2 b 32 m/s

c 8.8 s

Exercises 15.2 1 C 1.95 0.05, n 0.31 0.01 2 K 2.5 0.1, C 1.58 0.01

0 y 2 (revs/min) 8

0.02

0.04

0.06

x

6

16 Compound interest: exponential 16 growth and decay

4 2

Exercises 16.1 1 a $10 146.77

142

b $10 465.50


c $10 494.77

0

1.4

2.8

4.2

5.6

t (s)

17 Circular functions

Exercises 18.2 1

Exercises 17.1 1 a –0.2 b –0.2 c 0.2 d –1 2 a –1 b –1 c 1 3 a –1 b 1 c 0.3 d –0.3 e 0.3 4 a 0 b 2 c 1 5 a –sec x b –sin x 6 tan 85° 11.43,

tan 95° –11.43, tan 265° 11.43, tan 275° –11.43

Exercises 17.2 1 a 2.28 b 1.38 b 0.7658

c 0.172

2 55°38

3 a 1.042

a 12

T 360°

a 24

T 360°

2

Exercises 17.3 1 a 80° b 5° c (90 )° d (90 x)° 2 a cosec b cos

Exercises 17.4

3 2 5 b c d e 3 a 0 b 0 c 0 d 0 9 18 6 4 –0.6961

Exercises 18.4

Exercises 17.6

T 4.8 s, f 210 kHz

1 a 60° b 30° c 150° d 120° e 360°

2 a

2

1 a A 2.5 kV, T 670 ns, f 1.5 MHz b A 17 mV,

1 a 1.83 m b 1.34 m 2 a 3.98 m 2 b 1.30 radians (74°29) c 5.18 m d 10.3 m

Exercises 18.7

Exercises 17.7

i ( A) 13.6

1

y 27

1

18° A

0

amplitude 27 360° period 18° 20

–13.6

–27

18 Phase angles; more graphs of 18 trigonometric functions Exercises 18.1

2 i (mA) 18.4 9.2

0.98 6.86

1 cos ( 50°) leads by 60° 2 3 4 5 6 7

cos 3 cos 3 3 sin 4 sin 4 cos 2 sin 2

leads by 3 leads by 6 5 leads by 8 3 leads by 4 3 leads by 4 leads by 2

2.10 t 2.21 ( s)

0.990

t (ms)

19 Trigonometry of oblique triangles Exercises 19.1 1 a 1.31 m b 19.1 mm 2 15°29 3 87.8 mm

c 7.97 m d 66.3 mm

Exercises 19.2 1 a D, no

b T, yes

c H, no


143

Appendix C

Exercises 19.3 1 a 104°52 b 150.9 mm e 5.89 m f 93°19

c 239 mm

d 59°20

1 7 2 2 3 –24 4 22 5 –10 6 15.1

Exercises 19.4 1 a 267.8 m b 39.53°

Exercises C2

2 a Because it is

ambiguous. There are two different flight paths that follow this plan. b 179°, 249° 3 415 km, 129° 4 98.7 m 5 a 40.2 km, N55.8°W b 30.0 km

20 Trigonometric identities 1 a x0 b x9 c x2 Equation d is an identity.

d x any number.

Exercises 20.3 1 a tan 53° b cosec 17° c sec A d tan e sin A 2 f tan A 2 a cot b sin c cosec d sin 3 a 2 cos C 2 2 2 b 0 c 2 tan 5 a tan b cosec x c cos

Exercises D1 1 a 2 3 b 3 2 c 3 1 d 2 3 e 1 3 2 a A, D b G c B, E, H, K d F e C

Exercises D2 1 a x 7, y 3 b x 3, y 9 c x 2, y 6 d x 4, y –1

{

b –41.19° c C 10.6

Exercises 20.5

5x 5y – 3z 9

3 a A and D, B and H, E and K b

2 a 5.39 b 338.2° (or –21.8°)

Exercises 20.6 1 4 6 c

3x 2y – 5z 8

2 4x – 3y 2z 7

Exercises 20.4

1 869

1 a 5 b –13 c 19 2 a –6 b 7 c 5 3 a –17.16 b –1.02 c –9.72 4 E 1.03, V 0.883 5 L 22.4, W 13.6 6 x 0.984, y –0.530 7 I1 –3.61, I2 –5.01

Appendix D

Exercises 20.1

1 a –0.875

Exercises C1

c

00

f

1

a 3 2 2 sin x 3 2 cos2 A 1 3 sin A 4 sin3 A 5 2 sin B cos A a sin B 0.8, cos B 0.6 b B 53°08 sin (A 53°08) d A 96°52, 336°52

Exercises 20.8

1 3 1 cos 2 2 3 t a n x 1 1 c sin 2 cos d 3 a cos 2 tan x 1 b 0 c sin x d cos x 7 5 sin ( t 36.87°) 2 a (cos A sin A) b 2 sin

8 a x 63.32°, 243.3° b 225.8°, 247.7°

18 22

iv

64

3 12

vii

106 36

b i

15 18

1 5.3 m

3 7.5°

144


2 a M 320 kg

iii

17 18

vi 27 19

16 11

0 0 0

ii 17 18

9 16

d

14 13 12

0 0 0

e

0 0 0

39

0 6

6 3

6 5 5

5 10

v 119 269

13 12

10 10

ix

28 69 x 104 93

ii 20 02

iii

0 7 8 0

viii

5 15

10 12

iv

7 8 v 14 16 8 16

vii

00 00

Exercises 21.4 1 a T 430 N b m 44 kg b T 6.4 kN

9 16 2

1 a i

1

Exercises 21.3

0 0

4 7 1

Exercises D3

21 Introduction to vectors

1 156 km on a bearing of 30° 2 8.1 km/h in direction N52°E

11 4 11

9 a 7

Exercises 21.2

0 0

4 2 5

viii

7 14

11 00

vi 00 00

ix

22

1 1

x

128

4 6

c

3 10 7

f

144

j

1 6 3

2 a

4 5 7

d

7 4 2

4 6 2 2 3 1 6 9 3

3 a

k

0 7 5

1 7 5

5 2 5

b

g 79

11 1 3

9 5 9

3 2 1 12

2 5 13 16

n (7) o

2 3 13

b

p

e

5 27

6 3

16 38

4 26

h 92

2 0

5 1 0 3 2 1

5 3 5

l (86)

m

8 42

15 4

i (5)

6 3 2

20 1 8 6 6 2

2

4 a i

iii

910 10

vii

b

c

b

a b c d e f g h i

14

2 5

3 a

b d

ii

14

2

2 5

a

c no

0 1 0

5

0 0 1

a

10 01 i.e. I

2

b

0 1 0

a yes,

4 4 6

1 0 0

10 01

0 1 0

0 0 i.e. I3 1 –1

3 0 1

0 1 0

8 4 7

5 3 5

1 2 2

4 9 13

vi

0 0 1

4 9 12

1 0 0

ii

0 0 1

b i

0 2 2

1 1 0 1 0 1 0 2 1

0 1 0

3 4 2 2 1 0 1 1 1

0 3 0 or 0 1 0

2 1 1 2 or 0 3

1

a 10 0

0

1 10

0 1 0

a 3 0

6

13 11 17

iv

1 5 3 10 4 13

3 1 b 0 3 0

0 0 1

1 0 0

0 1 0

0 3 0

1 0 0

0 10 0 or 0 1 0

0 0 i.e. 3I3 3 23 1 3

0 0 10 0

1 3 2 3

0 0 i.e. 10I3 10

2 2 0.1 0.2 0.2 3 1 or 0.1 0.3 0.1 3 2 0 0.3 0.2 0

1

b 1

Exercises D7 1 a B b I c ABA B d ABA B A e I 2 2 f 2I g A 2A I h AB A B A i A 1 1 j O 2 a X 2 (A B) b X 3(C D) –1

Exercises D6 1

1 0 0

5 3 6

0

c no, because A B

1 d yes, 0 0

5

3 b incompatible 6

a

16 14 21

1

a b c d e f g h i

3 d incompatible 4 6

10 01

b no

a b c

0 0 i.e. I3 b 1

7 6 9

3 2 3

Exercises D5 b d

0 1 0

–1

Exercises D4

a 1 a c

1 0 0

c CD DC I, D C and C D . C and D are inverses. 3 a p s b r q c p q r s n d both are square matrices

v

1 a no b yes, 2 3 c yes, 2 1 d no e yes, 1 3 f no g yes, 3 3 h yes, 2 3 i yes, 2 3 j no k yes, 2 1 l no

a

10 01 i.e. I

2

3

–1

2

c X 4A d X 2A 9B 3 a X A B –1


145

b X BA

c X A (B – 2A) or A B – 2I

–1

–1

–1

d X A (B – I) e X A f X (3I – A) B –1 –1 g X (A I) B h X (A I) (C – B) –1

–1

1

i X B(A I)

j X 2(C – B)(A I) –1 b B C 5 a no b B A CA –1

–1

4 a yes –1 6 a X (C – B)(2A 3I) b X (A – B)(3C – 2I)

–1

–1

or (B – A)(2I – 3C)

Exercises D8 1

2x5x3y4, order 2 1

a

order 3 1 2

b

2 3 4 1 2 5 3 1 2

x y z

c

{5x2x 2y3y 43

{

3x 2y 4z 6 2x 3y 5z 7 x – 4y 2z 8

5 a

b

x y

3 5 2 2 4 0 5 0

a b c

5 7 3 7 2 8 3 5 2

{ 3x2x– 4y5y 76 d

{

x 2y 3z 4 y 2z 5 x 2y 6

4 5 6 7 3 0 3 0 8

a 17 b 13 c 19

25

a

3 x 9 7 y 22

1 4 2 2 5 3

ii

3 1 2 11 1 4

b

146

a

b x –2, y 5 3

c x 4, y –3

1 0 0

11 5 3 8 4 2 7 3 2

0 1 0

0 0 1

b

c x –1, y 2, z –3 6

ii PR

a i PQ

15 18 8 8 9 4 5 5

c

1 2 2 5 3 7

x 12 10 y 10 z

iii QR

2 4 5

a 3 b 7 c 8

1 0 0 0 1 0 1 3 2

1 0 0

0 1 0

0 0 1

d a –3, b 1, c –2 7 a x 1, y 2, z 5 b a 3, b 2, c –2 c p 2, q 2, r –4 d x 2, y 1, z –3 e x 1, y 2, z 1 f a 3, b 2, c –3 g a 3, b –4, c –1 h x 2, y 1, z –1 i p 2, q 1, r 3 j k 2, n –24, t –27 k x 5, y –2, z 3 l a 3, b 2, c –6 m x 1, y –1, z 0 n p 2, q –3, r 4 o x 1, y 3, z –5

Exercises D10 1 a –8 b 0 c –8 d 0 e 1 f 0 g –10 080 h –19 608 2 a x 3 b x –1, 2 c n –5 1 d t 3, 1 e k –2, –4

1 a x 2, y –3, z 1 b p –3, q –1, r –2 c x –1, y 2, z –2 d n –2, p 0, t 1 2 a x 0, y 2, z 0 b x 3.04, y 0.896, z 1.23 c k 0.81, t –0.75, w –0.91 3 F1 –0.583 kN, F2 1.67 kN,

b x 3, y –1 2


5

Exercises D11

x 9 y 5 z 7

Exercises D9 1

a a –2, b –14, c 9

b Q and R

4 a

b

3x 5y – 2z x 4y 3z , 2x – 3y 4z

a x 4, y –3 b x 2, y 3

3

c

a

b

3 4 2 5

4

–1

a i 2

10 01

a 11

F3 1.25 kN. F1 acts in the opposite direction to that shown in the diagram. 4 I1 3.66 A, I2 4.97 A, I3 3.84 A 5 10 of 2 ha, 20 of 0.5 ha, 70 of 0.2 ha 6 A: 40 kg, B: 10 kg, C: 50 kg

Textbook Answers 1 Fractions and decimals Exercises 1.1 1 a 8 b 4 c 2 d 2 e 4 f 2 2 a 3 b 10 c 3 3 a 0 b 0 c 2 4 a 2°C b 6°C 5 a contraction of 3 mm b contraction of 5 mm 6 a loss of 2 dB b loss of 8 dB c gain of 4 dB 7 a 2 b 14 8 a 6 b 8 c 4 d 4 e 6 f 12 9 a 4 b 2 c 6 d 4 e 8 f 1

Exercises 1.2 1 a 14 b 1 c 26 2 a 12 b 5 4 a 40 b 6 c 9 d 2

b 7

c 29

3 a 10

Exercises 1.3 1 a 2, 3, 5

b 3, 9

c 2, 3, 4

d 3, 5, 9

e 2, 3, 9

7 3 f 3, 5 g 2, 4 h 3, 9 2 a b c 4 d 1 9 3 4 2 3 8 3 997 1 4 a 2 e 4 f 7 g 3 h 5 3 a 1 b 5 c 3 7000 2 7 2 1 1 7 11 37 b 47 c 1111 d 42 e 44 5 a 2 b 4 c 5 103 124 137 11 7 31 17 d d 8 e 1 f 1 6 a 1 b 112 c 1 2 2 200 20 7 53 1 11 1 3 11 g 1 h 7 a b c e 3 f 1 6 200 12 40 2 4 18 13 1 1 5 1 23 2 8 a b 1 c d e f d 100 18 50 6 8 30 3 1 2 1 1 4 9 a 58 b 963 c 645 d 832 10 a 602 b 315 3 9 4 1 1 11 c 137 d 164 11 a 674 b 211 c 8610 d 291 4 2 3 3 1 1 1 10 12 a 4 b 27 c 312 d 37 e 1033 13 a 19 b 2 1 2 1 2 1 3 c 16 d 33 e 3 f 8 g 26 h 9 i 2 j 50 k 0 2 1 2 1 1 l 21 14 a 15 b 43 c 1 d 23 15 a 12 b 18 3 3 1 1 2 1 7 c 7 d 6 e 18 16 a 21 b 7 c 29 d 23 e 18 3 7 1 3 4 1 9 1 17 a 5 b 2 c 75 18 a 9 b 49 c 2 d 4 e 19 5 1 1 1 4 7 2 1 f 204 g 51 19 a 9 b 59 c 29 20 a 3 b 2 6 9 5 1 3 1 1 2 1 c 9 d 3 21 a 4 b 5 c 12 d 13 e 23 22 a 10 1 1 2 2 7 3 b 1 c 2 A 23 a 3 b 13 c 48 m 24 a 134 0 3 11 7 5 2 b 10 m 25 1 h 26 12 27 6 h 28 3 h 29 a 0 8 13 15

b U c U d 0

e U f 0

16 17

g U h 0

i U j 0

k U l 0

Exercises 1.4 1 c 4 c

a 21.124 b 239.58 c 30.008 2 a 63.68 b 3.05 11.618 3 a 0.043 b 0.63 c 2500 d 0.006 a 0.23 b 69 c 60 d 0.02 5 a 22.78 b 16.38 4.046 d 7.752 6 a 0.006 b 1.15 c 25.2 d 0.1

e 0.02 f 0.003 7 a 0.024 b 0.0006 c 0.002 88 d 1.8 e 0.6 f 0.000 06 8 a 0.09 b 0.0004 c 0.36 d 1.44 e 0.0144 f 0.000 121 g 0.0009 h 1.21 9 a 0.2 b 0.9 c 0.07 d 1.2 10 a 26.08 b 1.203 c 230.7 d 0.0521 11 a 2.03 b 732.8 c 6.003 d 1.703 e 0.0732 f 0.0039 12 a 0.68 b 5.33 c 62.15 d 234.20 e 0.04 f 0.00 g 0.01 h 0.00 13 a 400 b 0.567 c 0.234 14 a 30 b 0.4 c 0.2 d 0.03 15 a 20 b 20 c 0.3 16 a 3.6 b 200 m 17 a 0.12 b 0.16 c 0.024 d 0.48 18 a 150 b 0.06 s 19 a 1.46 mm b 43.8 mm c 0.584 mm 2 20 3066 mm 21 3.4 m/s 22 4000 W 23 19.8 J 24 20 s 25 a 1.5 s b 45 s 26 4.8 min 3 27 a 7000 kg b 0.002 m

Exercises 1.5 1 a 2500 b 2400 c 2500 d 600 e 700 f 800 g 1000 2 a 74 b 73 c 74 d 99 e 100 f 1000 3 a 1.36 b 1.36 c 1.35 d 0.03 e 0.06 f 0.01

Exercises 1.6 1 b 3 e

a 3 b 2 c 3 d 4 e 2 f 1 2 a 0.0346 kg 34.0 m c 0.0490 g d 601 m e 4.01 t f 781 km a 520 g b 52 mm c 0.542 kg d 16.1 t 650 m f 0.024 m2

Exercises 1.7 1 a 1.00 b 6.45 c 0.02 d 0.92 e 1.34 f 10.01 g 21.26 h 1.00 i 2.14 j 19.67 k 18.83 l 16.21 m 1.81 n 2.45 o 8.25 p 4.03 q 1.62 r 11.31 s 16.21 t 1.40 u 9.36 v 2.78 1 1 7 2 w 8.27 x 6.72 2 a 49 b 235 c 38 d 25 222

e 25 245

113

f 10 120

Exercises 1.8 1 9.37 2 6.39 3 0.28 4 0.72 5 3.88 6 0.37 7 6.25 8 7.30 9 0.02 10 0.77 11 3.23 12 3.04

1.9 Self-test 1 3 b 9

a 2 b 2 2 a 0 b 8 c 0 d 2 e 4 a 5°C rise b 2°C drop 4 a 4 b 4 5 a 0 9 6 a 5 b 3 7 a 4 b 2 8 a 7 b 11 1 11 3 1 3 4 a 62 b 1612 10 a 437 b 79 11 a 48 b 9 7

3

13

2

12 a 1 b 5 13 a 15 b 1 14 a 8.10 b 4.37 5 5 c 3.216 d 3.17 15 a 0.06 b 0.0252 c 0.132 d 30 e 3.2 f 0.2 g 0.000 144 h 0.11 i 1.21 Textbook Answers

147

16 a 1.06 g 4.36 h 17 a 62.1 g 82.9 h

b 0.99 c 1.81 d 0.97 e 3.65 f 0.17 1.53 i 0.69 j 9.20 k 2.54 l 1.87 b 55.4 c 27.5 d 1291.4 e 5.1 f 1.1 44.7 i 199.9 j 189.3 k 0.8 l 3.3

b 2.8 m

Exercises 2.2 1 $1.60 2 1400 3 70 km 4 340 5 39 km

Exercises 2.3 1 yes 2 no 3 a no b no 4 yes 5 a no b yes c no d yes e no 6 a no b yes c no 7 a no b yes 8 12.1 m/s 9 500 g 10 18.8 mN 11 84 12 a i 32 ii 5 b i 27.4 ii 199 c i 66.8 ii 154 13 a 0.56 t b 20 mm 14 a 18.7 m/s b 19.5 N

Exercises 2.4 1 a 30.9 b 30.5 c 2.53 2 a 21 s b 3.0 A 3 a 0.506 revs/hour b 119 min c 42 300 km d 384 000 km

Exercises 2.5 1 a 0.244 b 24.2

2 7.9 mm

3 24.7

Exercises 2.6 1 a 0.38 b 0.02 c 0.0163 d 0.000 14 2 a 0.501 b 2.46 c 10.356 d 25.2 e 426 3 a 0.09 b 0.6 c 0.21 d 0.2 4 a 3.80 b 90 c 2.03 d 0.050 e 0.273 5 a 35c b 50.0 c 5 min 39 s d 3.70 t e 47.3 m 6 a 43.1% b 55.5% c 236% d 83.3% e 400% f 66.7% g 0.318% h 37.5% i 2680% j 91.7% 7 a 21.1% b 27.5% 8 a 140.4 kg 1

9 22% 10 4.3% 11 a 7.21 b 11.61 c 20.52 d 2.79 e 5.17 12 a 19.9 b 6.37 c 101 d 0.730

2.7 Self-test 1 $102, $170 2 82% 3 a 327 Hz b 12.9% decrease 4 a 2.96 mL b $2.82 5 0.05% 6 20% 7 a 128.5 kg b 0.7 kg c 147 kg

3 Measurement and mensuration Exercises 3.1 1 a 0.001 kg b 1000 kg c 0.001 m d 0.000 001 m e 0.01 m f 0.000 001 kg 2 a 2.63 g b 75.4 km

148


1 1600

2 50 km

2 600

Exercises 3.4

1 15:17 2 17:48 3 30:1 4 7:10 5 9:40 6 3:25 7 1.54:1 8 1:2:80 9 a 0.810:1 b 1:1.24 10 6 m 1

m f 800 g

Exercises 3.2

1 30

Exercises 2.1

1

e 81.6

Exercises 3.3

2 Ratio, proportion and percentage

11 10 L, 172 L, 222 L 12 a 14 m

c 0.5 t d 350 mm

1 a 0.1 kg (100 g) b 40 mg 2 a 3% c 1%

b 0.7%

Exercises 3.5 1 a 0.5 km, 0.64%, 77.578.5 km b 0.0005 kg, 0.6%, 0.08250.0835 kg c 0.0005 m, 0.013%, 3.99954.0005 m d 0.000 000 5 t, 0.8%, 0.000 062 50.000 063 5 t e 0.005 t, 1.1%, 0.4650.475 t f 0.5 km, 0.076%, 653.5654.5 km 2 a 354.11 b 0.01 c 20.00 d 0.15 e 0.00 f 6.30

Exercises 3.6 1 9.14 7 6.35

2 6.99 8 2.45

3 4.84 9 24.8

4 15.8 5 6.90 6 7.82 10 270 11 1.01 12 13.5

Exercises 3.7 1 5.39 m 2 2.65 m 3 2.75 m 4 1.987 km 5 170 mm 6 0.62 m 7 17.4 mm 8 1180 mm 9 24.5 mm 10 13.1 m 11 1.8 m

Exercises 3.8 1 a 5 b 16 c 1 d 13 e 36 2 2 :1 3 a k 13 b t 15 c n 26

Exercises 3.9 2

1 a 0.211 m

2

b 211 000 mm

2 $1.60

Exercises 3.10 2

2

2

2

1 30 m 2 12 m 3 6m 4 6.02 m 5 a 6m 2 2 2 b 2.4 m 6 a 6 m b 2.4 m 7 8 m 8 6m 2 2 2 2 9 20 m 10 12.3 m 11 25 m 12 21 cm

Exercises 3.11 1 c e c 4 8 e

a 126 mm, 1260 mm

2

2

b 7.54 m, 4.52 m 400 mm, 12 700 mm d 2.59 m, 0.536 m2 1.02 m, 83 500 mm2 2 a 15.7 m2 b 93.9 mm2 9570 mm2 d 505 mm2 e 5.74 m2 3 549 mm 165 mm 5 1.39 m 6 174 mm 7 0.209 m2 2 2 2 2 a 24.6 m b 2340 m c 57.9 m d 16.1 m 2 2 20.6 m f 10.3 m 2

Exercises 3.12 1 d b 3

3

3

a 0.000 008 63 m b 69 mL c 4.5 cm 0.027 m3 e 23 000 mm3 2 a 376 000 mm3 376 mL c 0.000 376 m3 d 0.376 L e 376 cm3 3 3 3 a 156 cm b 156 mL 4 1.75 m 5 1560 cm

2

3

c k 6k 9 d x 9 e 4x 12x 9 2 2 2 f h 1 g 9t 12t 4 h 25b 10b 1

3

2

6 1.9 m 7 302 mL 8 47.2 cm 9 a 1.26 kL 3 2 2 b 0.707 m 10 a 4.81 m b 9.08 m c 5.78 m 3 d 83.2 m

2

2

1 0.0177 m 2 62.7 m 2 2 5 283 m 6 110 m

3 176 mm

2

4 126 cm

2

2

3

1 a 0.0567 km b 2.700 g c 54 cm d 120 cm 2 e 0.086 00 t f 4.83 mL 2 a 64.0 cm b 144 cm 2 2 3 a 70.7 mm b 2500 mm c 3930 mm 3 2 3 d 55 300 cm e 2.57 m 4 3530 cm , 27.6 kg 2 5 11.9 mm, 446 mm 6 a 7.23 b 4.64 c 1.91 d 29.3 e 195 f 17.0 g 4.83 h 2.39 i 0.263 j 4.89

Exercises 4.1 a 6 b 18 c 11 d 1 e 5 2 a 9 b 5 18 d 36 e 12 3 a 64 b 49 c 4 d 16 a 9 b 9 c 9 d 1 e 25 f 5 g 2 3 I 18 j 12 k 36 l 36 5 a 0 b 5 1 1 c 5 d 12 6 a 1 b 26 c 6 7 a 8 b 2 c 1 d 6 e 1 f 5 1 c 4 h

Exercises 4.2 a i 6 ii 7 iii 6 iv 7 b i 9 ii 3 iii 9 iv 3 i 3 ii 8 iii 3 iv 8 d i 6 ii 2 iii 6 iv 2 a 3a 2b b 3x 7y c 2g 4f 2 2 2 2ab 5a b e 13xy 4x 6y f 2a b

Exercises 4.3 1 d b 3

3x y 3 x(p q) x cd 3(x y) b c d e f y 2a2b 2(p q) 3 4k 2 2 2 2 4 a 6x b 6xy c 6xy d x y e 24p q 2 3 2 f 12a b 5 a 6(a b) b 18(x y) 2 c 12a b(p 3q) d 3p(m 2k) e 6a(b 2x) 2 f a (a b)(p q)

a abc b abc c a b c a bc e a b c f a 2 a x 1 m 3 c 3k 6 d 4 e 8 5b f 2t 2 2 2 a 2a b b x c 2q

Exercises 4.4 1 a 12pq b 42lm c 32a d 15tx e 5x 2 2 2 2 2 2 f 8x g 6x h 3a b i 8a b j 2ax k 15k m 2 2 l 15x y 2 a x b 3x c 3t d 18t e 4sy f a b g 3(x y) h a(2b c) i 3a 2

b 1 3x

1 3y 2 a 4y

c 1 3a

d 3ax

3 2k d 2t 2t 1 5 3 a 4x 1 b 3b 1 c 4 a 7 t 5 3 ab 4 7 7x b c d e f 5 a x m m x ab 6 2 5R 5C 3m 4x 4C b c 6 a b 12 8 mx 2C 2 2M 9 5m 1 x 6 1 13 c 7 a b c d 2 6 6 6M2 3x2 2t 3t ax a 5a2 8 a b c 9 a 6396 b c 3m 2b by 3 3t k 9 n a 15 10 a b c 11 a b 2 3y 10m 2b k2 b 3k2 a 5 ab c 12 a b c 3b 2m 2a 3b 10t2 k 2a t2 x d 13 a b c 3bc ab t1 y x2 8 2t2 yx 14 a b c 1 2x 3 yx 3x b 2

2

9k c t

Exercises 4.8 1 a 2x 2 e

Exercises 4.5 1 a 3x 6y b 2a 2b c 6x 12y 2 2 d 2ab 3ax e a 2ab f 3m 2m g 8a 8b h 2mp 3pq i 5 2 a ax ay bx by b at 2a 3t 6 2 c mt mf kt fk d x 5x 6 2 2 e 6x 13x 6 f 2x 7x 3 2 g mk 3m 2k 6 h ab ax bx x 2 2 2 2 i x a 3 a x 6x 9 b 4a 16a 16

f a(a b) 3 a

Exercises 4.7 1 a 15x

4 Introduction to algebra

2

1 a 3x b 3 c 2x d xy e 2pq f 2ab 2 2 a a b b 3(x y) c 2ab(p 3q) d p(m 2k) e a(b 2x)

3.16 Self-test

1 c 2 d

2

Exercises 4.6

Exercises 3.13 2

2

c 3 e b

b 3T

c 3 3R

d 7 2R x 2x 16E 1 2V 3 f 6t 2 a b 6 12 5 3C 3V 4 9L 5 5k 15 d e f 6 12 12 18 5x 1 5x 8R 5 10n 1 a b c d 6 6 6 12 2x 25 26G 5R 11 f 4 a 1x 24 18 4 3m 2 2m

Textbook Answers

149

Exercises 4.9

c C1 4, C2 8

b 1 1 2 a b yx R b c 1 6R

bT

c 1 d 1 e 1 f 1 3k 3 2x ba c 3 a k3 x2 3

1 a 1

d 2

4 a 3n 1

13 16n 6

d 3 13n e

b x2

e x 13 f W 1 1 1 i t 12 j x 5

f

4 m E 7 4 c x 27 1 g W 12

c x 10 g x0

1

1 2 3 b c

d x1

n l 1

2 a x3

bx

135

d x 9 e x 24 f d 9 3 h x 1 i L 4 j x 3 1 1 3 a x 6 b x 15 c a 6 d k 42 3 2 e m 7 f x 4 g n 2 h W 3 i y6

1

j x 2

c a 2

1

4 a

d a1

5 a x1

e x

x 13 17

b f

b x 24 c x 5 2 f x 15 6 a x 2 1

e x 42 1

c x0

3

x 15 m 257

4.17 Self-test 1 a 10

b n 1.1 c R 2

c k 2

d E 2 e x 3 1

1

bL3 b t 1 0

f W 35 2

3n (n 5) 28 2n 5 28 2n 23 n 1112 2 2 a 2.50 m b 18.8 m 3 20.5 m 4 2.05 2 5 BC 27 mm 6 a 5 m b 12 m 7 36 km/h and 44 km/h 8 72 s 9 a at rate A: $18; at rate B: $25 b 286

Exercises 4.13 c yes 2

a yes

b no

c no

W 13 150

2 a x 1, y 5

c F 3,

b E 3, V 2


2

8 $62 500 9 a k 6, m 2 b I1 11 , 2 1 I2 32 c V 5, t 2 10 a F 2, m 5 b a 2, k 0 11 a a 1, t 2 b S 10.6, t 1.85 12 41 L of Full-cream, 59 L of Slim 13 21 kg/min and 14 kg/min 1

5

5 Formulae: evaluation and transposition Exercises 5.1 1 0.386 2 0.398 3 14.1 4 0.988 5 93.8 6 1.01 7 15.8 8 6.90 9 0.128 10 26.1 11 13.5 12 25.8 13 1.91 14 0.903 15 8.39 16 1.49 1 a 56.3 kJ

b R 3, r 1

f 11

Exercises 5.2

Exercises 4.14 1 a x 2, y 3

e 15 2

1

3

1 Let n be the number

b yes

d 6

4a 1 3LV 2 9x y b c 2 3 RL 2a b 6x y 2 4tx 3p 1 7L 7 d 4 a b 5 a 2 2 3t 1 x 24 12p t 7V 5 1 b 6 a a 3 b b 4 c k 6 d E 24 1 2 e t 3 f x 3 g a 57 h W 113 3 2 1 i x 3 j t 32 7 a 850 kg/m b 5.1 kg

Exercises 4.12

1 a yes

c 9 2

3 a 2

2

d L3

8

1

2 a 4x 9 b 4t 12t 9 c x 6tx 9t 2 2 2 2 d 4a 9b e 25k 20km 4m

d x 11 1 b x 5

e R 21 f M 1 2 a x 3 1 c V 6 d n 10 3 a x 4.6

b 32

2

Exercises 4.11 1 a x 7

a k 0.012, de 6.3 mm b 7.86 mm a k 0.25, b 0.2 kN b 2.7 kN m 20 kg, F 8 N 4 a k 8, C 200 mm 272 mm c 12.5 kg 5 a 7.5 m/s2 b 4.5 m/s 57 m/s 6 R1 17 500 , R2 2500

7 a R 1.5 , E 5.5 V b R 235 , E 53.1 V 8 a R 10 , r 2 b R 4.58 , r 0.387

l x 3

2

c V 3, E 2

Exercises 4.16

h R 2

k x 7

l 2

1 a x 2, y 3 b W 4, d 2 c I1 3, I2 5 d t 2, x 23 e L1 3, L2 57 1 1 f i 3, V 5 2 a a 5, b 22 b E 1, 1 R 2 c v 3 2 , d 1 3 a x 1, b 3 b V1 2, V2 3 c S 5, v 6 4 a a 1.35, b 0.435 b x 0.857, y 2.71

b 8 9n

Exercises 4.10 1 a x4

3 a x 2, y 5

Exercises 4.15

8n 23 6 31 17t 5 a t 2 b 4t 3 c 12 9t d 6 17t 11 19 13t e f 4 15 c n7

1 , 3

2

b 459 kJ 2 a 10.3 N

b 43.0 N

3 a 259 mm b 560 mm 4 a 16.6 m/s b 186 m/s 5 a 799 mF b 366 F

6 a 51.6 mHz b 1.16 kHz b 478 Hz 8 4.24 s

Exercises 5.3 3kn ay 3kn a a b(m t) a x 15 b x a nbt na na a x 8.5 b x or nt b b h mt h a x 3.5 b x or m t t 3a m 2 x (m t) 6 x k 2a 3 5t x 8 x a 3t 9 x 3ky 2

1 a x 13 2 3 4 5 7

b x or y

a 2b n Ta T na 11 a Tb a b nb a n Tb (k t) b d1(V 2) T2v 1000P 12 d2 13 a T1 V v 10 x 2

T v 1000P v

1 b T2

2 FS u m

14 v

2

V I

Q C

IN H

Fd CI1I2

21 l 22 l

M2 k L2

W b s F R Bl

20 I

23 Q 2WC

2W

1 4 f C EV R2(I I1) 27 a l G 28 I 29 R1 Ra I1 R R2 LL1 L2 30 R 1 31 L3 R1 R2 L1 L 24 L1 2

25 H

26 L 2 2

Exercises 5.4 5 k t 2 x 3 x a4 3t m am 5 k 4 n 5 w b k 3m 1 ma b b ma 6 t or m1 1m 1 x

Exercises 5.5 vu 1 t

a

PVM RT

2 a m

mRT PV

b M

Q2

) b Q1 1

T2v 1000P T1v 1000P 4 a T1 b T2 v v 2FS WF 5 m b V 2 2 6 a F VE W E v u 3 a Q2 Q1(1

Exercises 5.6 1 a 259 mW

ms(1 q) 15 F VE W 16 m q 17 R 18 W Pt 19 V

Ft mu Ft Eb b m 8 W vu a m P P 2qlt 9 a q b b 2t(l b) 2qt Q mct2 Q 10 a c b t1 m(t2 t1) mc W 273(V Vo) 11 t 12 a V

(aw b) Vo 1 b W bV

bV b a c W

V W

WV 1 aV IRs RE 13 IF 14 V i Rs r Rx Ri eR Ra(R2 R3) 15 R1 16 r R2 Ra Ee I2R2 R3(R R1) 17 R1 18 R2 I I2 R1 R3 R C2(C1 C ) E I2Rb 19 I1 20 C3 or C C1 C2 Ra Rb C2(C C1) 8R 21 r C1 C2 C 4R 7 a v

7 a 1.84 Hz

b W Pt

c 1.36 kJ 2 a 36.3 J

E R R2(I I1) c 725 A 4 a 260 mA b R1 I1 2Fs 2 c 2.57 k 5 a 43.0 N b v u m 2w mv2 c 39.5 m/s 6 a 186 m/s b h 2mg c 2.80 km 3 a 1.22 kV

b I

P(R R ) R 2

c 98.2 mm c 875 mV c 16.0

7 a 773 mW 8 a 1.09 kHz

F 9 a 845 mJ b

c 833 mm

10 a 855 mN

c 954 mm

11 a 832

c 618 m

12 a 1.93 mF

A

b V

1

2

1

1 b C 2 4 Lf 2 W s FP W b h F mg Ir b R 1 I2 I1 C2(C C1) b C3 C1 C2 C

c 3.12 mF

5.7 Self-test 1 102 V 2 a 63.6

b 27.4 3 0.065 4 2 273(V Vo) Vt Ra 4 a t b R c l Vo E R1Rs CC1 d Ls LA 4M e R2 f C2 Rx C1 C IR i1R1 V g i2 h n 5 a 1.32 kN R2 E Ir Textbook Answers

151

mv F

Q C b C V 2 2 v u c 474 F 7 a 6.47 m/s b s 2a b t

c 541 ms 6 a 520

c 390 mm 8 a 103 ms b s ut 2 at 1

c 2.36 km 9 a 5.39 k

b 3.30 k

2

c 49.8 k

Exercises 7.2 1 b d, c e

2 a b, d f

4 a f, b e, c d 6 b c, f d

3 b c, a d

5 c f, b e, a d

Exercises 7.3 2 15 mm

Exercises 7.4

6 Introduction to geometry Exercises 6.1 1 a 159°51, obtuse b 76°38, acute c 88°45, acute d 189°41, reflex e 60°11, acute f 92°49, obtuse g 172°48, obtuse h 180°40, reflex i 172°47, obtuse j 78°21, acute

Exercises 6.2 1 e c 3 e

a 61°14 b 151°14 c 16°32 41°43 f 41°35 2 a (90 )° 38°26 d 67°18 e (133 t)° a x 30 b x 18 c m 36 k 36 f t 26

d 170°36 b (180 x)°

f (94 )° d y 15

Exercises 6.3 1 t 80

2 k 50

3 m 70

Exercises 6.4 1 a 30, b 80, c 70 2 p 46, q 72, r 62 3 k 76, l 50, m 54 4 x 70 5 b 107, g 128, p 73 6 x 80

Exercise 6.5 1 a 65 mm

Exercises 6.6 b 30°

6.9 Self-test c (200 k)° (t 90)° e (270 2x)° 2 a h 64 y 36 c u 208 d x 132 3 a x 70 p 49, q 15 c x 23 d x 36 a 122°32

b 46°09

7 Geometry of triangles and quadrilaterals Exercises 7.1 1 t 30 2 k 70 3 y 15 5 x 35, y 30 6 x 20 152


1 x 8, y 2

2 d 7.5 3 t 10 4 x 2.8 n 33.2 6 y 0.49 7 x 4.33 8 x 1.6 9 p 3.6, q 1.8 10 x 4.5 11 d 6.92 12 x 6 13 x 4.3 14 x 0.5 15 8.57 m 16 29.3 m 5

Exercises 7.6 1 a parallelogram b rhombus c no special type d trapezium e no special type f square g parallelogram h rhombus i rectangle j parallelogram k rectangle l parallelogram 2 a yes; PQBA will always be a parallelogram

because both pairs of opposite sides must always remain equal in length b 2 m c extremely small (approaching zero) as rope QB is made longer and longer d no 3 a parallelogram b yes c no d 70° 2

parallel to the third side divides those two sides in equal ratios. 3 The opposite sides of a parallelogram are equal in length.

1 d b b

Exercises 7.5

Exercises 7.7

b 81°

2 An interval joining two sides of a triangle and

2 a 60°

1 a yes, SSS b no c yes, AAS d no e yes, SAS f yes, RHS g no h no 2 a yes, SSS b no c no d yes, AAS

1 a 9.80 m 2 d 1380 mm

2

b 13.4 m

2

c 128 000 mm

Exercises 7.8 1 3 b c 7

2

28.8 m 2 a 7.00 m a 10.0 m b 7.07 m 2 14.4 m c 156 m 5 113 mm 6 a 6.45 m 2 a 1.73 m b 3.46 m

2

b 12.2 m c 84.0 m 2 c 50.0 m 4 a 21.6 m a 80.0 mm b 215 mm b 15.7 m c 7.07 m c 3.46 m

7.9 Self-test 1 a p 120 b m 30 c x 53 2 a b d, a e c b c d c b e, c g d a d, b c 3 a yes, SAS b no c no d yes, AAS e no f yes, SAS 4 a h 0.62 m b x 5.7 m, y 8.9 m c d 13 m 5 a x 10.5 b d 27 c t 1.7 d d 55.2 e n 1 f x 3 6 a rectangle

b yes

c no

d 8m

e 28 m

7 a yes; ADPQ always remains a parallelogram 4 h 115

since both pairs of its opposite sides remain equal, hence AD remains horizontal (parallel to PQ) and

hence AB remains vertical, since ADCB is a rectangle 2 b 424 mm c 88 mm 8 a 32.0 m b 11.7 m 2 c 6.40 m 9 a 641 mm b 276 000 mm

r 80, s 70, t 100

8 Geometry of the circle

9 Straight line coordinate geometry

Exercises 8.1 1 17.1 m

2 60°

Exercises 8.2 1 a 207 mm

b 80.9 mm

2

2 a 117°

b 0.237 m

1 a p 40, q 100, r 80

b p 25, q 130,

r 50 c p x, q 180 2x, r 2x 2 a i 40° ii 60° iii 100° b i 2x° ii 2y° iii 2(x y)° c The angle subtended at the centre of a circle is

twice the angle subtended at the circumference by the same arc. 3 a i 50° ii 50° b i 12x° ii 12x° c Angles in the same segment are equal. 4 a i 40° ii 60° iii 80° iv 90° b An angle in a semicircle is a right angle. 5 a 75° b 90° c 100° d 120° 1 1 6 a i 65° ii 115° iii 180° b i 2y° ii 2x° 1 1 1 iii 2(x y) ° 2(360°) 180° c i 2 ° 1 1 ii 360° ° iii 2(360° )° 180° 2 ° iv 180° d Opposite angles in a cyclic quadrilateral are supplementary. 7 m 74, n 42 8 g 31 9 k 45 10 p 55 11 m 52 12 k 36 13 x 100, y 92 14 m 25 15 h 67 16 a x 55, y 40 b x 24 17 t 23 18 x 105, y 105 19 q 20, r 30 20 x 52, y 38 21 m 136, t 22 22 p 30, q 70 23 x 28

Exercises 8.4 2 9.5 m, 10.5 m

Exercises 8.5 1 a 90°

b 50° 2 a 24°

4 a 102° 8 4.90 m

b 132°

c 66° 3 28°

b 129° 5 10.58 m 6 12 m 7 12 m

b x2 x1

c y2 y1

d (x2 x y2 y1 ) 2 a the AAS test 1) ( b AQ MP, QM PB, MA BM c they are the opposite sides of a rectangle. d Q is the midpoint of AK, P is the midpoint of BK e Q is the point 2

2

x1 x2

y1 y2

2, y , P is the point x , 2 x x y y f , 3 a i 5 units ii (3.5, 5) 2 2 1

1

2

2

1

2

b i 13 units ii (5.5, 10) c i 10 units ii (1, 5) d i 5 units ii (1, 0.5) e (i) 13 units ii (9, 10.5) f (i) 85 units ii (2.5, 4) 4 a (3, 9) b (11, 0)

Exercises 9.2 1 b b c

1

a 3

b 10 c 33 2 a 3 b 6 c 2 3 a 3 6 c 2 4 a 4 b 12 c 3 5 a 10 3 c 0.3 6 a 2 b 10 c 5 7 a 3 b 6 4 d 2 e 2 f 7

Exercises 9.3 1 a y 8 b y 11 c y b 20 2 d y 4x b e y mx b 2 a i 3 ii 5 iii 13 1 3 b i 2 ii 7 iii 32 c i 5 ii 3 iii 5 d i 2 2 1 ii 4 iii 2 e i 3 ii 6 iii 2 f i 3 ii 3 iii 42 3 a P, Q, S, V e m 3

c b 1

b M, P, Q

f a

d k1

212

Exercises 9.4 1 a y 2x 3 b y 3x 5 2 c y 3x 3 d y 3x 2 e y 3x 5 1 2 a y 3x 1 b y 22x 6

Exercises 8.6 1 x 74

2 m 40, n 62

Exercises 8.7 1 7.73 m

Exercises 9.1 1 a (x1, y1)

Exercises 8.3

1 7.5 m

b q x 40 c x 54 3 a 40 b 64 c 50 d 40 e 100 4 a 50 b 54 c 47 d 67 5 6.5 m 6 32.9 mm 7 a p 106 b p 60, q 30,

c y 3x 5 3 1 y 33x 63 2 y 2x 2 3 y 2x 12 4 y 3x 12 5 y 0.3x 1

2 273 mm

6 y 5x 15

Exercises 8.8 1 3.08 mm

2 9.17 mm

5 7.65 mm b 2.36 m

6 a i 433 mm ii 60° iii 2.23 m

3 93.4 mm

4 9.80 mm

8.9 Self-test 1 a 8.37 mm

2

b 119 mm

2

2 a p x 32

Exercises 9.5 (The values indicate roughly the expected accuracy.) 1 a 12.1 ( 0.2) b 6.3 ( 0.1) c 2.90 ( 0.05) 2 a 50.88 m/s ( 0.02) b 1.77 s ( 0.01) Textbook Answers

153

c 2.32 m/s ( 0.05) 3 a 15.53 mm ( 0.02) b 170 mm ( 2) c 6.4 m/mm ( 0.1)

e 0.2275 2 a 5.109 b 0.079 76 c 58.35 d 1.676 3 a 40.26°, 40°16 b 20.04°, 20°02

4 a 1.44 L ( 0.01) b 479 K ( 2) c 3.35 mL/K ( 0.05) 5 a 30.4 mA ( 0.5) b 82 V ( 2) c 301 A/V ( 10)

c 87.50°, 87°30 d 66.04°, 66°02

2

Exercises 9.6 1 b 106.5 5 a $9100

c 3.35 2 a 29.4 s 4 67.4 kW b 32 c 96 articles

b P

c Q

d U

e S

f V

g R

h R

2 a x 4 b x 6 c x 4 d x 4 e x 2 f x 2 3 a i R ii Q iii T iv S v U vi Q b U c V d Q e Q f (3, 42) 4 c (2, 5) d x 2, y 5 e . . . read off the coordinates of the point of intersection. 5 a x 112, y 3 b x 0.6, y 7.2 6 b t 1.28 h ( 0.05), D 83 km ( 1) c D 62t 2, D 49t 145 1 a i 3 ii 2 iii 6 b i 2.5 ii 0.8 iii 2 c i 2.5 ii 1.4 iii 3.5 d i 0.2 ii 6 iii 1.2 2 a y 2x 5 b y 3x 12 1 c y 12x 2 d y x 1 e y 3x 6 f y 2.5x 5 3 a 4.23 mA ( 0.01) b 0.3% ( 0.1) c 28 A/V ( 2) 4 a 244°C ( 10) b 0.49 /°C ( 0.01) 6 283 W 7 x 2.14, y 45.7 8 b t 2.4 h ( 0.1), 43°C ( 1) c A 9.4t 21, B 15t 80 b 49.4 mA ( 0.2)

Exercises 10.2 1 1.38 4 a 0.6009 b 1.428 c 0.7265 6 a 0.5048 b 3.149 c 0.2852 7 a 2.70 b 1.44 c 19.9 d 1.21

Exercises 10.3

Exercises 10.4

1 a 0.8

b 0.6 2 a 0.8660

w

n

2 a n2

b 0.6446 7 380

b

b 1.118 c 0.5

k2 w2

5 a 0.9454

4 a 0.8808

b 0.9600 6 29.6 N

Exercises 10.8 1 a 160° b 350° c 225° d 220° e 070° f 315° 2 31.1 km 3 94.8 km 4 82.0 km 5 a 64.66 m b 43.63° (43°38) c 75.99 m 6 147.8 m

Exercises 10.9 2

1 a 84.4 m

1 b 4 7 9

2

b 5.86 m

a 0.5543 b 12.3 mm 3 51.7 mm 5 a 654 m b 4.31 km 10

2

c 1.13 km

0.5095 c 2.356 2 a 47.0 m a 0.9413 b 0.3375 c 0.5416 a 750 mm b 56°55 6 5.27 3.46 m 8 a 23°32 b 33°44 3.706 m

11 Indices and radicals 1 a 9 b 7 c 163.72 d 893.4 e 6 f 35 2 1 2 2 a 9x b ab c a b d ab e 4k n f 48ab

tx a a b c mk b b a 1 d 5 a 3 b 3 c 4 d 2 e 6 b 2 f 9 g 6 h 2 i 4 6 a x 2x x x b 9m 12 m 4 c 25 10 a a d 1 4 x 4x e 4n2 12n n 9n 3 f x 2x x 1 7 a 3 b x2 3 c x 2 c b d e c a y2 3 a 4

2 a 35.70° b 7.102° c 2.090° 3 a 73.87° b 57.55° c 0.3679° 4 a 32°16 b 86°47 c 79°32 5 53.23° (53°14) 6 a 36.07°, 36°04 b 53.13°, 53°08 7 a 61°52 b 57°58 b 47°41

c 3.79 mm

d 34°01

Exercises 10.5

154

Exercises 10.7

Exercises 11.1

1 a 35.4 m b 7.48 m c 448 mm 2 a 99.2 mm b 8.61 m c 36.7 mm d 31.3 m

1 a 0.8387

a 6.86 kN b 28°14 2 a 96.2 km/h 18°34 3 9°23 4 41°07 5 196°00 130.8 m 7 81.6 mm 8 42.0 mm 9 a 2.75r 1.73r c 7.81 mm 10 a 77.5 V b 7.10 V 26.8 V d 44°52 11 a 1.39 A b 17.2 mA 23.6° d 34.0°

10.10 Self-test

10 Introduction to trigonometry

8 a 4.51 mm

1 b 6 b c c

3 a

9.10 Self-test

9 a 233 Hz ( 1)

12

Exercises 10.6

Exercises 9.7 1 a T

3

4 a 4 b 1 3 3 5 12 12 4 4 h 5 a 6.16 m c 5 d 13 e 5 f 1 g 3 5 5 b 7.88 m c 6.57 m d 18.7 mm e 223 mm f 11.6 m 6 a 25°50 b 42°16 7 a 24.7 m b 4.11 m 8 a 37°27 b 44°29 9 63°26 10 90°, 36°52, 53°08 11 14.8 m

b 0.7268

c 0.9781


d 0.5931

b 6

c 9

d 12

4 a

Exercises 11.2 1 h e c d

Exercises 11.6

b 1.44 c 0.008 d 9 e 16 f 9 g 9 8 i 8 j 196 2 a 4 b 8 c 16 d 32 64 3 a P b N c N d P e P 4 a 1 b 1 1 d 1 5 a 0.09 b 0.008 c 0.01 0.09 e 0.09 1

4

a 81

Exercises 11.3 1 a 18 b 36 c 72 d 1440 e 4 f 25 g 49 h 5 2 a 4 b 36 c 15 d 216 e 2 f 3 g 16 e 8x

1 b 4x

3

h 72 3

2

3 a 8x 2

f 12x

2

g 7x

2

d 9x

4

i 8x

Exercises 11.4 16 3

4

1 a x y 5

2

6

3

4

4 a a b 4

5

d 7m 5m

3

2

3

b a a 3a 3

2

8

5

3

c k 5k 6 d 25 m 4

2 a a b

8

c 8k 11k

2

5 3

d 36x

2

c 6a x y

b 6x 3x

6

c 24x

d 2k m 3 a a a

5 4 2

b k m t

8

b 12ak m

2

5 a a a

6

5

b k 3k 2k c 6m 3m 2m 1 d t t 4 2 4 1 6 6 a m b k c 6a d 16x 7 a 3ab b mt 2 2 2 7 c 8k d 3m e ab f kx 8 a x b 2m 4 2 3 15 6 6 2 6 c 2a bc d m t 9 a x b 2a c 8a d a b 6 15 12 18 6 9 12 3 6 e m b f x g m h 8m t w i 2x y 6 12 18 6 10 a 4x b 30x c 0 d 2m x 7

4

3

2

4

11 a p 6p 9

b 4t 12t 9t

c 1 6a b 9a b

12 a 8 1

6

3

2

5

16

4 2

3

e 4

1 b 32

9n2 25

c

f 13 a 24 6

n

b 4 c 16 15 d (2x 3y) 5 n 5 g (x y) 14 a 6 b n c f k

6

4

t4 4t3 4t2 9k

16a6 b

8t 27n

d 6

6

x

t

e 7 f (a b) k 2 2n d 8 xy e a

t

x

7

Exercises 11.5 5

1

1 a k

5

d e

2

b a

e

3 a a 2a 6

1 2

3

3 2

x

x

d a

a5

e e e

2x

2

c 3m 2

2 a x x x

6

b 8t

b 1k

3

c 6k 2 2k

0

3 2

x

c e e x

3 2

1 c 4

1 d 3

1 b 16

1 3

d x2 x

e t t f 3m 2m 4 a 1 b 3 d 1 e 1 f 1 g 12 h 1 i 0 j 72 k 5 a 4 b 3 c 1 d 2 e 2 f 2 g i 243 j 4 k 4 l 1000 6 a 9 b 2 e 3 f 2 g 3 h 27 7 a 2 b 7 c 1 1 e 3 8 a 8

7

f 4b 6

f 2k 5k 3k 2

2

1 2

5 4

e 4m

c 1 5 l 4 1 h 8 c 5 d 4 d 9

1 c 5

1 d 81

1 1 e 32 9 a 2

1

1 b 2 7

c 1

e 1 10 a 4

1 d 6 4

19 11 a 8 b 3 c 9 d 8 e 81 12 a 9x2 w 3x 2 1 x2 c d e f 2 3 2 x (w 1) x

1 b 2 1 e 8

8 b 3t

4

5

2

c 12x

h 3x

2

1

1 a 1.23 10 b 5.91 10 c 8.36 10 2 8 d 5.47 10 e 4.38 10 f 6.39 10 7 3 10 g 5.61 10 h 9.63 10 i 4 10 3 6 1 j 5.13 10 k 7 10 l 6.9 10 6 5 7 4 12 2 a 10 b 10 c 10 3 a 10 b 10 8 2 12 2 c 10 d 10 4 a 8 10 b 10 2 6 5 2 3 2 c 3 10 d 10 5 a 10 m b 10 m 2

7

2

c 10 m d 10 m 6 a 3 10 m/s 6 5 4 b 3 10 m/s c 2 10 km/h d 5 10 mm/s 8 6 4 7 a 3 10 J b 10 J c 6 10 J 8 5 3 d 4.8 10 J 8 a 10 b 10 c 10 6 8 4 2 d 10 9 a 10 W b 10 W c 10 W 8 4 5 4 d 10 W 10 a 10 W b 10 W c 10 W 3 2 1 3 d 10 W 11 a 10 A b 10 A c 10 A 2 24 6 d 10 A 12 a 1.36 10 b 1.77 10 15 16 12 c 1.68 10 d 7.60 10 e 9.33 10 7 5 17 f 2.00 10 g 9.11 10 h 1.50 10 1 2 i 4.29 10 j 4.42 10 2

2

Exercises 11.7 6

1 a i 26.8 10 g ii 26.8 10 kg iii 26.8 mg 3 3 b i 26.8 10 A ii 26.8 10 A iii 26.8 mA 3 6 c i 82.5 10 mm ii 82.5 10 m iii 82.5 m 3 6 d i 6.84 10 kW ii 6.84 10 W iii 6.84 MW 3 3 e i 637 10 MHz ii 637 10 Hz iii 637 kHz 3 3 f i 3.8 10 H ii 3.8 10 H iii 3.8 mH 3 3 g i 74.5 10 MV ii 74.5 10 V iii 74.5 kV 3 6 h i 62 10 nm ii 62 10 m iii 62 m 3 i i 13.8 10 g ii 13.8 kg iii 13.8 kg 3 6 j i 4.25 10 kHz ii 4.25 10 Hz iii 4.25 MHz 3 3 k i 830 10 kg ii 830 10 kg iii 830 g 3 l i 68 10 mm ii 68 m iii 68 m 6 6 3 2 a 11.32 10 b 94.65 10 c 181.3 10 6 4 3 d 429.5 10 3 a i 5.609 10 ii 56.09 10 13 12 7 b i 1.035 10 ii 10.35 10 c i 2.727 10 6 5 3 ii 27.27 10 d i 9.152 10 ii 915.2 10 5 6 7 e i 5.609 10 ii 56.09 10 f i 5.636 10 9 ii 563.6 10 3

Exercises 11.8 1 2

2

b 173 mm 2 a S (2 V) 3

1 a l (hd ) 3 2

b 1.37 m

3 a R 100

T2GM 4

5 a x

1 3

4 a r 2

P

n

A

1

b 14.5

b 384 Mm

R B 2R2NI 107

2 3

2

b 58 mm

11.9 Self-test 3

1 a 6a 2 2a

3

b x 2x 2 x 2

3

c 1 2m 2 m

Textbook Answers

3

155

1

d 2a 2 a 1 2 a 81 1 1 3 a 8 b 1 c 5 d 1 1

c 0.292 c k6 4 c 9

d 1.81 d

d 2

f 144 g 6 d k

e 4 4 a 6.69

c 9 4

6

f 2n

d 8

b n0

1

g (a b)

9

1

b 25 e 5

b 2m

1

c 4a 6

h a b 2

2

1

9 a mt 3m t b a a 1c n 4n 4n 2 4 2 d 3y y e y y f b b3 10 V

2

3

ER , 83.5 mV t V

1

V

1 11 t2 (t1 273) 2

273, t2 107°C

12 Polynomials Exercises 12.1 1 a 2x 11x 16x 6 b x 2x x 2x 2x2 x 1 c 3x4 2x3 4x2 2x 1 3 d x 1 2 a 4 b 8 3 51 4 7.01 5 4.088 3

2

6

5

4

3

Exercises 12.2 1 d g 2 c 3 c 4 c e 5 d

a 2(2x 3) b 3(3 4a) c a(b 1) 3x(2y 3x) e 5(2a 1) f 8a(2 3ax) 3a3(1 2b2) h 5x2(y 2a 1) i 3ab2(a 2) a (p q)(a b) b (b c)(a 2) (2a y)(x 3b) d (3 x)(2l m) a (b c)(a 1) b (m k)(l 1) (x y)(1 m) d (2 y)(x 1) a (q k)(p 1) b (q k)(p 1) (b 3)(a 1) d (x y)(y 1) (t m)(1 k) f (2x 1)(k m) k 2 x 2x x 2x a a (1 a ) b e (1 e ) c e (e 1) x x x 2x 2e (3e 2) 6 a e b e

Exercises 12.3 1 c e h c 3 c e g i

a (E V)(E V) b (L 3)(L 3) (3Q 5)(3Q 5) d (2a 3b)(2a 3b) x(x 2y) f (m 1)(m 9) g 3(2G 3) 3t(2m t) i 3e(2V e) 2 a 89 b 157 140 d 200 e 400 f 0.175 1 a (V 3)(V 3) b (2C 5)(2C 5) (3L M)(3L M) d (4 3Q)(4 3Q) (7R 1)(7R 1) f (ab 2)(ab 2) (1 r)(1 r) h (5E 4)(5E 4) (4a 5bc)(4a 5bc)

Exercises 12.4 1 a x 5x 6 b x 7x 12 2 2 2 c x 8x 15 d x 9x 20 e x 9x 8 2

156

2


f x 12x 35 g x 10x 21 2 2 h x 16x 60 i x 13x 40 2 2 2 a x 11x 24 b x 14x 24 2 c x 10x 24 3 a (x 1)(x 3) b (x 3)(x 5) c (R 1)(R 7) d (Q 2)(Q 6) e (L 2)(L 3) f (E 1)(E 4) g (Z 2)(Z 5) h (F 1)(F 16) i (Q 4)(Q 8) j (V 3)(V 4) k (E 3)(E 8) 2 l (R 2)(R 8) 4 a x 5x 6 2 2 b x 9x 20 c x 7x 12 2 d x 8x 7 5 a (x 3)(x 8) b (R 2)(R 4) c (C 1)(C 9) d (F 1)(F 36) e (r 3)(r 3) 2 f (Q 2)(Q 50) 6 a x 3x 10 2 2 2 b x 4x 21 c x 3x 4 d x 6x 16 7 a (x 2)(x 5) b (y 2)(y 6) c (R 3)(R 4) d (C 3)(C 4) e (E 4)(E 6) f (Q 2)(Q 3) g (t 2)(t 3) h (L 4)(L 6) i (V 3)(V 5) j (r 2)(r 5) k (Z 4)(Z 9) l (F 5)(F 6) 8 a (2x 1)(x 1) b (2x 3)(x 1) c (2x 1)(x 3) d (3a 1)(2a 1) e (6a 1)(a 1) f (4m 5)(m 1) 9 a (2x 1)(x 2) b (2x 3)(x 1) c (2a 1)(a 5) d (3k 1)(2k 2) e (3k 2)(2k 1) f (6k 1)(k 2) 10 a (3x 1)(2x 3) b (6x 1)(x 2) c (3x 2)(2x 1) d (18k 5)(2k 1) e (6m 5)(4m 1) f (2m 1)(12m 5) 2

b 0.996

212

e t 1 6 a 6 b 1

h 6 i 10 8 a 3t 4

1 e 4

c 1 d 3

e 7.17 5 a x

n 112 1 7 a 24

e 3t 4t 3 2

1 b 3

2

11 a (3b 4)(2b 1) b (6b 1)(b 4) c (4a 3)(a 3) d (6x 1)(x 6) e (2x 1)(4x 9) f (8x 1)(x 9) 12 a (3x 2)(2x 3) b (3x 8)(2x 1) c (3x 1)(3x 2) d (9x 2)(x 1) e (10k 3)(k 1) f (5k 1)(2k 3) 13 a (2a 5)(2a 3) b (2a 5)(2a 3) c (4a 5)(a 3) d (4a 5)(a 3) e (3n 2)(2n 1) f (6n 1)(n 2) 14 a (3n 2)(2n 1) b (6a 5)(a 2) c (4x 5)(x 1) d (4n 1)(n 5) e (8x 2)(x 3) f (5t 2)(2t 5)

Exercises 12.5 1 3(1 x)(1 x) 2 5(x 2)(x 4) 3 w(t 2)(t 2) 4 4(E 3)(E 3) 5 6(Q 3)(Q 2) 6 5(L 5)(L 2) 7 7(E 4)(E 2) 9 2(v 6)(v 3)

8 3(Z 1)(Z 4)

Exercises 12.6

x2 2 1 2 3 E 1 4 2 C 5 R 2 3 x8 1 6 R 2 7 8 R 1 or (R 1) 2Q 1 9 3t

Exercises 12.7 1 4 7 9

x 0, 2 2 x 0, 6 3 k 2, 1 t 3, 4 5 E 0, 2.5 6 C 0, 3.5 Q 1, 1.5 8 x 1.25, 3.5 V 2.5, 1.2

1 a V 5

b R 7 c R 5 d L 4 x 2 f Q 6 2 a L 5.22 R 3.73 c x 1.25 d R 0.938 x 1.07 f x 11.5 3 a x 6 Q 9 c x 9 d L 4 e x 4 t 8 4 a m 3.87 b x 1.73 k 4.58 d x 2.65 5 a x 1, 5 R 4, 2 c C 4, 6 d R 8, 2 k 72.9, 74.9 f t 5.41, 1.41 a k 7.12, 1.12 b C 9.58, 13.6 c x 0.962, 6.96 d n 0.876, 2.21 e k 4.45, 1.55 f x 1.88, 5.88 e b e b f c b e 6

Exercises 12.9 a V 0, 6 L 1.5, 0 a R 0, 5 K 0, 0.2

b W 2, 0 e Q 0.5, b V 0, 0.6 e R 0, 0.5

c M 0, 2.5 f R 0, 5 c x 0, 0.3 f t 0.2, 0

0

Exercises 12.10 1 a R 2, 3 b t 2, 3 d d 4 2 a C 6, 2 c x6

d R 8, 3

b V 4, 2 3

4 a x 2, 6

c W 12, 2 b T 4, 9

3 a k 12, 22 1

1 2 c C 2, 3

1

b W 2, 18

6 a V 12, 0, 13 1

1 1 d x 12, 6

c x 1, 5

1

b C 22, 0, 4 1

3

c a 2, 0, 24 d x 3, 0, 12 1

1

1

Exercises 12.11 1 1, 2

2 2, 6

3 2, 3

4 x 7x 10 0 2

Exercises 12.12 1 a 3.41, 0.586 d 0.382, 2.62

c x 2.35, 0.851

b 0.796, 8.80

e 0.449, 4.45

c 1.38, 3.62

f 2.62, 0.382

g 2.73, 0.732 h 1.27, 10.3 2 a 3.24, 1.24 b 0.314, 3.19 c 0.768, 0.434 d 0.146, 2.75

b t 0.303, 3.30 d Q 0.434, 0.768

2 a x 2.69, 0.186

b R 0.425, 1.18

c C 2.62, 0.382

d E 3.56, 0.562

3 a x 0.732, 2.73

b C 3.00, 0.500

4 a x 1.30, 7.70

b d 0.303, 3.30

c V 1.78, 0.281

d E 3.19, 2.19

Exercises 12.14 1 a 4 m, 6 m

b 0 m, 10 m

c 2.76 m, 7.24 m

2 a 2b 12b m b 4 m c 2.77 m 2 2 3 a l 2 mm b l 2l mm c 6 mm 1

2

12 v

4 a h

d 4.74 mm e 1.83 mm c 10 km/h

d 6.81 km/h

200 v 10

ii h

12 v2

b h

200 v

5 a i h

b 40 km/h

Exercises 12.15 1 a i R 2R 1 0 ii R 1 2 b i R 13R 36 0 ii R 4, 9 2 c i R 15R 25 0 ii R 1.91, 13.1 2

2 a R2 e 3 ii d 4

b R 1, 4 c R 8 d R 4, 9 R 1.37, 46.6 f R 1.61, 22.4 a i V 1 ii r 0.333 iii I 1 b i V 3 r 3 iii I 0.333 c i V 6 ii r 3 iii I 3 i V 1.80 ii r 0.599 iii I 0.898 11.5 , 78.5

Exercises 12.16 5 p

18 000 p

1 1 s, 4 s 2 a h s

1

d d 4 e R 2, 3 f t 4, 8 5 a x 4, 2 b R 2, 10 c C 3, 0, 3 d E 4, 4 1

1 a x 3.45, 1.45

1 2

Exercises 12.8

1 d 2 d

Exercises 12.13

5 p 0.5

18 000

p 0.5

b h s

c 1.5 W for 3 h 20 min ( 12 000 s) 3 a Q CEQ 2CW 0 2

b 2.00 C

c 1000 V

12.18 Self-test 1 a x x 9x 3 b 2x x 6x 9 2 9 3 a 3x(1 4a 3ax) b 6q(2pq 3) c (k t)(a 3) d (t 3)(1 b) 5

2

3x

2x

5

4

3

4e (3e 2) f (x 3y)(2a b) (2a b)(p 1) 4 a (5a b)(3a 7b) (1 6R)(1 6R) 5 a (R 1)(R 32) (t 4)(t 6) c (Z 1)(Z 36) (Q 3)(Q 6) e (4k 3)(k 2) f (3n 5)(3n 2) g (2x 3)(2x 1) h (10t 1)(t 6) e g b b d

Textbook Answers

157

6 a 2(V 18)(V 1) c a(K 6)(K 1)

b 7(1 2R)(1 2R)

c i x 10

d 2k(Q 9)(Q 2)

iv

ii (10, 150)

C2 e 3a(b ac)(b ac) 7 a b 3(2y 1) C1 c 1 8 a R 0, 5 b d 0 c x 0, 9 5 d R 4, 3 e V 2, 0, 2 f I 2, 6 1 g E 4, 8 9 a F 1.72, 0.387 b d 0.740, 0.540 c V 0.349, 2.15 d F 0.539, 1.65 e k 0.221, 1.94 f C 8.34, 0.839 10 2.16 m, 1.25

iii 50

y x

10 –50

–150

13 Functions and their graphs Exercises 13.1

2 a

1 a 4x 3 b 2x 1 c x 3x 3 2 d 2x x 2 a 1 b 13 c 2 d 8 2

3 b b 6 c 9

Exercises 13.2

y 3

iv 1, 3

–3

–2

x

–1 –1

b y

iv

v

iii 3

1 x 1 a b 2 c d 1 4 a 0 a1 x1 1x 2x 1 c 1 d 5 a 3y 7 1x 2x 1 13 c 2 3x d 7 e 3x2 10 f 6x 2 1 1 a 1 b 13 c 3 d 4 7 a x 1 b x 12 x 4 8 a x 12 b x 0 c x 16 x 1, 123

1 a i x 3

i x 2 ii (2, 1)

i x 4

iv y

ii (4, 8)

ii (3, 5)

iii 24

4

iii 4

8

iv 2, 6 –6

–4

x

–2

x

–3

–24 –5 b

ix1

c i x3 iv

ii (1, 9)

y

v

ii (3, 12) y iii 15 12 iv 1, 5

9

iii 7 7

1

–15 –2 158

–1


1

2

3

x

3

5

x

d

ix4

4

v

ii (4, 8) iii 24

y

y

30 24

iv 2, 6 20

10

2

4

6

x

8

–3

–2

–1

0

1

2

x

3

–8 (sketch-graph only) 3 a i 0, 2

iv y

ii x 1 iii (1, 3)

5 1

y

x

2

–4

–3

–2

–1

0

1

2

3

4 x

–10

–20

–3

b

i 0, 5

–30

iv

ii x 2.5 iii (2.5, 62.5) y

(sketch-graph only)

62.5

Exercises 13.3 1 a y 2x 4x 6

b y 3x 6x 24

2

2

c y 2x 6x 8 2 a y x x 6 2 2 b y 3x 12x 9 c y 4x 16x 12 2

2

3 a y 2x 6x 4 2

b y 3x 6x 9 2

c y 2x 14x 12

d y 5x 45

4 a y x 4x 5

b y x 2x 3

2

2.5

c

i 0, 13

x

5

2

2

2

Exercises 13.4

iv

1 a circle with centre at

ii x 6.5

y

iii (6.5, 42.25)

(0, 0) and radius 3 units

y 3

42.25 C –3

3

x

–3 –13

– 6.5

x

Textbook Answers

159

b circle with centre at (2, 3)

Exercises 13.6

y

1 x 2, y 7; x 11, y 16

and radius 1 unit 4

2 x 3, y 93; x 3, y 93 1 1 3 x 2, y 42; x 6, y 96 1

3

C

1

4 k 6, m 177; k 2, m 9

5 n 1, g 11; n 2.5, g 17.7 x

–2

Exercises 13.7

c circle with centre at (4, 1)

y

and radius 1 unit

1 a to f : x 1, 4 (Note: all these equations are

reducible to the same equation and hence all have the same solutions.) g x 0.7, 4.3 h x 0.4, 4.6 i x 1.4, 3.6 2 a x 3, y 0; x 1, y 0 b x 2.4, y 2; x 0.41, y 2 c x 1, y 4 d no solutions 3 a x 3.7, y 16; x 4.7, y 9.0 b x 0.64, y 1.3; x 3.1, y 0.57 4 x 0, y 0; x 5, y 5 5 x 2.4, y 1.9; x 1.4, y 1.9 6 x 3.5, y 2.5; x 1.5, y 2.5

–4 x –1

C d circle with centre at (0, 0)

y

and radius 1 unit

1

–1

1 x

C

Exercises 13.8 1 a W 6 2h

–1 2 a x y 64 2

2

2

b (x 7) (y 3) 1 2

2

c (x 2) (y 6) 9

3

b (x 3) (y 3) 9

c x y 25

2

2

2

2

d (x 2) (y 4) 16 2

2

2 b i 1.8 m ii 5 m iii 18 m iv If this trajectory

a x y 16 2

2

2

had originated from a point at ground level, this point would have been 2 m behind its actual projection point. v If this trajectory had been allowed to continue, instead of the ball striking the ground, it would have been 2.2 m below ground level when its horizontal displacement was 20 m.

2

e x y 169 2

2

f (x 3) y 25 2

2

Exercises 13.5 1

2

y

–6

y

13.9 Self-test

10 10

6 6 –6

–10 –10

x

b A 2h(3 h)

d W 3 m, h 1.5 m, A 4.5 m

1 a 1 b 4 c 0 d 1 e 8t 4t 2 a 2 b 4 c 2 d 6 3 a t 2, 1 b k 2, 3 c m 1, 1.5 3

x

4 a y 3

4 y

y

– 2.5

– 1.25

x

6 6 4

160

x


–8

– 3.125 x

b

Exercises 14.3

y

1 a 4 –2

–1

1

2

4 x

3

b 2

c 1

h 3 i 8 f 0

d 2

g 1

f 1

e 1

1

2 a 2 b 0 h 7 i 15

c 2

g 4

d 1

1

1 e 3

Exercises 14.4 1 a 100, 115 b 0.1, 0.118 c 0.001, 0.001 35 d 2, 2.03 e 3, 2.99 f 2, 1.89 2 a 10.4 b 2.15 c 0.0141 d 2.75 e 29.9 f 0.884 g 27.2 h 15.1 3 a 104 b 16.0 c 2.45 d 21.2 e 9.39 f 1.34 g 2.13 h 10.9 i 22.3 j 46.0 k 5.82 l 0.361 4 a 0.434 b 0.809 c 2.29

–8 –9 c

y 8

Exercises 14.5 5

–5

–3

–1

1 35 dB 2 29 dB 3 10 4 32 W 5 7.6 W 6 a 37 dB b 15 dB c 9.3 dB 7 60 dB 2 8 1 W/m

x

Exercises 14.6 1 a log 2 log 15, log 3 log 10, log 5 log 6 b log 6 2 a log 2 log 10, log 4 log 5 b log 5 3 a 2 b 1 c 3 d 3 e 1 f 0

– 10

4 a 1 b 1 5 a y x 2x 8

2

c y 2x 2x 3 6 a circle with centre at (0, 3) and radius 2 units 1 2

d log k n log x

b rectangular hyperbola with its branches in the second and fourth quadrants

f log 5 3 log x

14 Logarithms and exponential equations

k log c n log x

1 a

1 5

b

1 k 4

j 3 s 1

t 27

1 2x

f k

1 2

d 4

d

1 m 10 5

2 a a

g b

6

h x 0

e 4

1 2

e 1

1 n 2

b b 8

2

f 4

1 a 3 9 e 9 b

32

f 3

o 1 7n

c x

i n

t2

3

g 2

g 5

p 4

log4 116

b 5 5

4

3 a 2

3

r 81 7

d V

e E 1

b 8

1 2

c 64

1 3

h 8

e log9 9 1 b 10 y k

f

4 a logt y 2 d log2 8 3 1

3

c 7 1

c log3 1 0

log4 312

ce M t

l log K x log e

b 6.64 c 8.58 d 2.43 e 38.4 1 2 (or loga b) loga b

52

da Q 2

b loge k m

Exercises 14.8 1 a 1.3 b 3.8 c 4.6 d 8.8 2 a 2 b 2.3 c 6.9 d 10.58

1

e 26

f 0.7

Exercises 14.9 d 2

2 a log5 25 2

1

2

1

Exercises 14.7

h 2 i 1 q 8

0

1 2

f 4 4

1 27

e log m 3 log w 1 g 2 log x log 3

1

Exercises 14.2 2

1

1 a 1.85 1 9

c

1 l 2

f 2

1 e 2

h log k 2 log t log m 1 i log 5 2 log x log y j x log e

Exercises 14.1 1 8

d 1

5 a 2 b 1 c 2 d 2 6 a 2.30x b 0.434x c x d x e 6.91x f 3.04x 7 a k log N b log M log N c log A log B

b y x 3x 10

2

c 1

1 d log16 4 2

1 8

1 d 2 d

a V 48 b x 0.364 c E 0.1 x 13 e x 1.75 f V 0.1 5 a x 12.2 b P 0.903 c C 2.45 10 3 x 3680 e P 2.02 10 f k 8.23

Exercises 14.10

y

1 a k4

e 10 x

2 a m0

3 a2 7 y

c log10 Q p

b n 5 b x1

c x 12 d t 4 1 c t 0 d x 2 1

3 a n 12 b x 3 c x 6 d t 8 1 1 1 4 a m 2 b k 14 c x 4 1

2

1

1

Textbook Answers

161

d 6 b 8 d

t 14 5 a x 13 b t 1 c m 5 1 a x 4 b t 0 c x 0 7 a x 1.77 x 2.59 c n 1.78 d x 11.3 a k 1.71 b n 0.783 c k 0.244 t 3.19 e x 1.25 f n 2.54

3, 4

y

Exercises 14.11 log y log F y b x 10 c x log 3 log k b log C m d a e b (a 10 ) c log x 1 y0 n f x y 10 g x y 2 a t log10 k y y log 1 y0 1 Q0 b t loge c t k Q k log A 1 a t

1 k

Q Q0

log 2 (≈ 0.30)

10 0

5

x

y = –log 0.2x + 0.3

d t loge 1

–log 2 (≈ –0.30)

Exercises 14.12 log S log 2

1 a b

b 9

y = –log 0.2x

3

p log p s h log 0.883

5

Q log Q 1 log Q log Q Q0 t ln 0 Q0 k k log e k log e

log p log ps

Z 126

7 a D 0.5d 10

b 27.8 mm

y 0.90

0.5

x

1

2 a 5.70 f 1.06

c 3 d 2 e 2 b 3.13 c 47.5 d 83.2

b 0

2

1

3 a 1

b 0.631 c k6

Exercises 14.13

0

1 a 3

A log log A log P P 6 n log R log R

1

14.14 Self-test

log 0.883

1

f 7 e 0.165

b 2 c 1 d 1.5 4 a 2.32 1 c 0.605 5 a x 22 b n 0 1

d m 12 e t 1 6 a k 6 b t 7 c x 3 d V 65.5 7 a t 1.79 b n 2.19 c k 0.462 d x 1.29 1

k log A k b x a(1 10 ) log b loge k b x c t d y 1a 1 ekt 8 a t

V log V0 log V log V0 9 a t log 2 log 2

b 11.7 ms

10 31.6

W/m2

15 Non-linear empirical equations 2

y

Exercises 15.1

log 2 (≈ 0.30)

–8

162

–4


1 K 5.0, b 1.5 c 12.2 m/s x

2 a C 16.6,

b 960 N

Exercises 15.2 1 K 48.4, a 0.122

2

3 K 2.5, N 0.49 0.744t b i 70.8 e

4 a K 70.8, b 0.475

K 3.15, N 1.32

2 (only sketch-graphs shown here)

15.4 Self-test 1 C 5.6, K 1.2

2 C 7.2, n 1.5

a

y

3 K 33.3, c 0.0117

40

16 Compound interest: exponential growth and decay

30

Exercises 16.1

20

1 a i 19.1% ii 59.4% b 12 2 a $2550 (1.007)5, $2640.51 b [(1.007)12 1] 100%, 8.73% 3 $106.55

10 5

Exercises 16.2 1 a $P 1.08 b $P (1.02) c $P e 2 a $10 400.00 b $10 406.04 c $10 407.42 d $10 407.95 e $10 408.08 4

0.08

–6 – 4 –2 0

x 6 4 2 y 1.51 2.25 3.35

Exercises 16.3 1 a 0.181 b 4.40 days 2 a 14 800 b 9.90 years c 4.29 years 3 a 47 mA b 9.6 mA 4 a i 9.2 g ii 1.9 kg b 29.2 g c 5.3 h 5 13 kg

b

4

6

x

8 10

0 2 4 6 8 10 5 7.46 11.1 16.6 24.8 36.9

x 0.1 0.2 0.3 0.4 0.5 y 13.2 19.1 21.8 23.0 23.6

Exercises 16.4 1 a, b

2

y y y = 2–x

y = 2x

8

24 18 12

4 0

0.1

0.2

0.3

0.4

1

2

0.5 x

2 3 a

1

y –3 –2 –1 0

1

2

3

x

8

c y 4

1 3 4 1 2

2 1

0 1

2

3

4

x

d Multiply the values on the y-axis scale for c by 327.

–2

–1

0

1 2

3

x

b i 0.5, 1, 1.5 ii 0.25, 0.5, 0.75 iii 0.69, 1.4, 2.1 iv 0.35, 0.69, 1.0 c 43, 86, 172, 344

Textbook Answers

163

4

y

16.5 Self-test

16

1 a $8508.54 b $8539.23 c $8549.70 d $8549.82 2 a 0.0715 b 17.0 g 3 a 85 V 4 a

b 5.9 s

y 80

8

4 2 – 2.1

–1.4 –0.7

0

40 0.7

1.4

2.1

2.8

x 20

(sketch-graph only) 5 a 0.35 min ( 21 s) b 0.7 min ( 42 s) c 1.05 min ( 63 s) d 54 s

10 0.1

0.2

x

N b 8000

i (mA)

6000

40

4000

2000 20

1000 0

21

42

63

10

t(s)

5

(sketch-graph only) 6 a 1.4 s, 2.8 s, 4.2 s b

7

14

1.4

2.8

21

t (ms)

c

I(mA) 200

(mC)

175 16

150

14 12

100

8

0

1.4

(sketch-graph only) c 80 mA 164


2.8

4.2 t(s) 4.2

t (ms)

5 a 7 years

b 14 years

c 21 years

d 35 years

e m 37 g M (g) 100

e c e c c

330.4800° 178°00 d 1 f 1 2.625 d 0.825 9

f 77.1429° 5 a 114°35 b 79°21 27°42 6 a 0 b 0 c 1 d 0 g 0 h 1 7 a 0.5720 b 0.8121 0.6613 8 a 0.743 b 0.891 a 4.87 b 10.6

Exercises 17.5

mass present after 10 years = 37 g

1 2 4 b

50 25 12.5

a 419 rad/s b 3.49 rad/s c 3140 rad/s a 2480 rpm b 3820 rpm 3 200°/s a 1.44 s b 628 ms c 4.44 s 5 a 37.2°

5.10°

Exercises 17.6

17 Circular functions Exercises 17.1

Exercises 17.7

1 a 0.2 b 0 c 0 d 0 e 1 f 0.2 2 a 0 b 0 c 2 d 2 3 a 0.3 b 0.3 c 0.3 d 0 e 0.3 f 0 g 0.3 h 1 4 a 2 b 1 c 0 d 1 5 a 1.2 b 1.2 c 0 d 1.2 e 0 f 1.2 6 a 0.98 b 0.98 c 6 d 0.98 e 0.2 f 0.2 g 6 h 0.2 i 6 j 0.98 k 0.2 l 0.98 m 0.98 n 6 o 0.98 7 a tan b 1 c cot i 8 a sin 150° 0.5,

1 a

7

14

21

28

35

t (y)

2

a i 4.19 m ii 16.8 m b i 157 mm 4.71 103 mm2 c i 249 mm ii 15.2 103 mm2 2 3 2 i 5.93 m ii 115 m 2 a 3.85 10 mm 3 2 13.1 10 mm 3 a 2.69 m b 1.51 m2

1 ii d b

0


sin 210° 0.5, sin 330° 0.5 9 tan 120° 3 , tan 240° 3 , tan 300° 3 10 sin 50° 0.766, sin 130° 0.766, sin 310° 0.766 11 cos 70° 0.342, cos 110° 0.342, cos 250° 0.342, cos 290° 0.342

b

Exercises 17.2

c


1 a 1.67 b 2.40 c 1.67 d 0.750 e 1.08 f 1.08 g 1.25 h 0.417 i 2.60 j 2.60 2 a 0.933 b 1.66 c 1.05 d 0.207 e 1.20 f 1.14 3 a 50.5 b 2.40 c 1.20 d 0.632 4 a 52.48° b 25°11 5 a 2.571 b 0.4660

amplitude 200

Exercises 17.3

period 180°

a b 1 a b c c c b 50° c 57° d 3 a sin b tan

a d b 41° c cot

b e a e 42°

c b

c f 2 a 70° a f (90 A)°

d

Exercises 17.4 1 a 45°

amplitude 24

2

period 90°

3 d 4 d

b 180° c 90° d 18° e 75° a b c d 2 e 2 3 4 6 a 0.820 30 rad b 0.017 453 rad c 5.9690 rad 4 0.415 04 rad e 3.3379 rad f 1.7453 10 rad a 171.8873° b 57.2958° c 116.5797° 68.1818°

2 0°, 45°, 90°, 135°, 180°

3 30°, 90°, 150° 15°, 75° 5 31, when A 36°, 108°, 180°, 252°, 324° 6 3 7 212

4 7, when

Textbook Answers

165

17.9 Self-test

Exercises 18.2

2 cos 110° 0.342, cos 250° 0.342, cos 290° 0.342 3 cos 38° 0.788, cos 142° 0.788, cos 218° 0.788, cos 322° 0.788 4 17, when 0°, 120°, 240°, 360° 5 25

1

a 18 T 360°

2

a 200 T 2

1 a sin x

b cos

6 a

amplitude 12 period 90° b

y 200 173 4 3

3 –200 amplitude 6 period 12° 7 K 30.35°, 120.35°, 210.35°, 300.35° 8 R 165.15° 9 2.59 10 a 3.72° b 7.65° 3 2 11 a 413 mm b 30.1 10 mm 3 2 3 2 c 50.8 10 mm d 139 10 mm

7 3

10 3

a 0.6 T 2

3

y

18 Phase angles; more graphs of trigonometric functions

0.6 0.42 3 4

Exercises 18.1 1 sin ( 30°) leads by 70°

5 2 sin leads by 6 18

7 4

– 0.6

3 cos ( 20°) leads by 20° 4 cos ( 10°) leads by 30° 5 cos ( 80°) leads by 100° 6 cos

7

6 leads by 12

7 cos ( 10°) leads by 40° 8 sin

leads by 30°

9 sin ( 60°) leads by 130° 10 sin ( 30°) leads by 170° 11 sin ( 85°) leads by 120° 12 cos

2

6 leads by 3

13 functions are in phase

166


4

a 50 T 360°

11 4

15 4

a 2.3 T 2

5

vi

y 2.3 2.12

13 8

5 8

21 8

29 8

– 2.3

7° c right 6 29° a V 40 cos 5 b c left 5 a i 17 sin 4 b 0.3 c left a V 35 cos 3 b 40° c right a i 23 sin 8 b c right 2 a V 240 cos 20 b 0.01 c left a i y 50 sin 2 ii y 50 sin (2 30°) 3 i y 18 cos 32x ii y 18 cos (32x ) 8

2 a y 12 sin 6 3 4

a6 T 2

6 y

5 6 7

6

8

2

3 2

5 2

b

7 2

b

9 a y

–6

40

Exercises 18.3 2

3

1 a i (2.09) ii 120° iii 477 10

3 3 iv 8.33 10 cycles/degree v 3

3

cycles/rad

0

9 24

24

2 3

–40

vi

b

y

37

3 cycles/rad 2 3 iv 11.1 10 cycles/degree v 4 b i (1.57) ii 90° iii 637 10 vi

0

0.075

0.16

x

– 37 10 a y 93 7 16

4

(3.14) ii 180° iii 318 103 cycles/rad 3 iv 5.56 10 cycles/degree v 2

3 16

0

c i

2

–93

vi b y 19 3 4

2 5 3 iv 13.9 10 cycles/degree v 5

3

d i (1.26) ii 72° iii 796 10

0

cycles/rad

4

2

–19

Textbook Answers

167

Exercises 18.4

b

1 a 5 Hz b 20 mHz c 250 Hz d 400 Hz e 200 kHz 2 a 20 ms b 500 ns c 50 s d 2 s e 100 ps

A 3.5 7.4 0

s

3 a A 240 V, T 20 ms, f 50 Hz b A 3.5 A, T 40 ms, f 25 Hz

A, T 2.9 ns, f 350 MHz A 6.7 mV, T 1.2 ns, f 815 MHz a A 6.0 V, T 20 ms, f 50 Hz A 3.5 A, T 7.4 s, f 140 kHz A 41 A, T 37 ps, f 27 GHz A 8.3 kV, T 9.7 ns, f 100 MHz

c A 13 d 4 b c d

c

A 41 37 0

d

Exercises 18.5

ps

kV 8.3 9.7 0

1 a

ns

V 240 20 0

ms

Exercises 18.6 3

1 a 24 10

s, to the left (i.e. leading) s, to the right (i.e. lagging) 3 c 2.34 10 s, to the right (i.e. lagging) 6 d 66.7 10 s, to the left (i.e leading) 12

b 556 10

b A 3.5

2 a

40 0

ms

y 75 126 0

c A 13

227

102

251

t (ms)

– 75 2.9

0

ns

b y 24

d 3.06

mV 6.7

0

10.00

7.50

2.50

8.06

1.2 0

ns

t (ns)

–24

c v (V)

2 a V 6.0

240 20

0

12.3 ms

0 –240

168

251


2.34

10.0

20.0

t (ms)

Exercises 18.8

d i (mA)

1

85 93.5

214

0 26.8

281

374

t ( s)

–85

Exercises 18.7 1 i ( A) 8.7

2 20.0

12.1 0

2.11

t (ms)

–8.7

2 v (V) 12 673 294

0

757

t (ms)

–12

18.10 Self-test 3

s ii 314 rad/s b i 15.4 106 s 9 ii 408 10 rad/s c i 35.7 10 s 6 3 ii 176 10 rad/s 2 a i 500 10 Hz 3 ii 3.14 rad/s b i 200 Hz ii 1.26 10 rad/s 3 3 c i 125 10 Hz ii 785 10 rad/s 3 3 3 a i 50.3 10 s ii 19.9 Hz b i 1.26 10 s 3 ii 796 Hz c i 3.88 s ii 258 10 Hz 4 a i 5.00 ii 3.00 rad/s iii 2.09 Hz 3 iv 477 10 Hz b i 240 V ii 314 rad/s iii 20.0 ms iv 50.0 Hz c i 17.0 A 6 ii 565 10 rad/s iii 11.1 ns iv 89.9 MHz 3 d i 3.80 mV ii 723 10 rad/s iii 8.69 s iv 115 kHz 5 a i 0 ii 99 iii 0 iv 99 v 0 1 a i 20.0 10

3

3

i (mA) 26.3 13.1 0

14.7

5.72

t ( s)

– 26.3

4 v (V)

b y

240

99 18.8 0 –240

8.75

t 20.0 (ms) 0

2

t (s)

Textbook Answers

169

– 99

2

c amplitude 99, period s

b v (V)

6 a

19 Trigonometry of oblique triangles Exercises 19.1

b

1 a 7.6 m b 590 mm c 2.86 m d 41.5 mm e 1.90 m f 32.9 mm 2 a 50°15 b 42°43 c 22°41 d 28°24 3 a 117°27 b 134°17,

34.0 mm

Exercises 19.2 1 a R, no b A, yes c H, yes d T, yes e K, no f N, no 2 a 52°49 b 92°45 c 272 mm or a 127°11 b 18°23 c 85.9 mm 3 a 41.34° b 112.23° c 45.5 m or a 138.66° b 14.91° c 12.7 m

c

Exercises 19.3 1 a m k t 2kt cos 2 2 2 b i t p q 2pq cos T 2

ii c ii b f

2

2

p2 q2 t2 2qt cos P 2 2 2 i x n v 2nv cos X 2 2 2 v n x 2nx cos V 2 a 11.9 m 30°52 c 1.70 m d 3.01 m e 119°28 25.3 m g 222 mm h 35°26 i 95°10

Exercises 19.4 d

1 28°57, 46°34, 104°29 2 23.3 cm 3 1.49 m, 2.39 m 4 a 893 mm b 14.5° 5 115.5° 6 18°35 7 a 836 km b 46.4° 8 199 mm, 319 mm 9 193 m 10 S69.2°E 11 97.7° 12 19.7 km

19.5 Self-test 1 a 30.8 mm b 1.14 m 2 75.1° 3 180 m 4 36°49 or 143°11 5 a 20.5 mm b 39°35 c 571 mm d 134°03 6 17.4 km from P and 11.5 km from Q 7 69.7 km on bearing 123.8°

7 a

20 Trigonometric identities Exercises 20.1 1 a x 1, 0

b x0

c x any number

d x 0. Equation c is an identity.

Exercises 20.2 1 a 0.2191

170


b 0.2051

c 0.6545

d 0.9231

Exercises 20.3 1 e 2 3 b h e

a tan 47° b cos 23° c cot 38° d cot cot A f cosec x g cos h sec i cot a tan b tan c sec d sec e tan f cosec 2 a cot b sin A c tan x d cos 4 a sin 2 2 1 c cos A d 1 e 1 f sin A g 1 3 sin2 K 7 a cos2 b 1 c cot d sin x tan2 f tan2

Exercises 20.4 1 b F2 646 N 2 b R 540 N 3 b 9.88 kN c 34.2° 4 b R 3.46 k c 16.7° 5 a 0.6 b 30.96° c A 5.83

Exercises 20.5 1 29.8 N b 60.3°

2 28.4 N

3 5.44 kN

4 a 8.06

Exercises 20.6

3 1 1 (, ) 2 (cos , sin ) 3 a P(cos , sin ), 2 2 2 Q(cos , sin ) b PQ 2 2 cos ( )

cos )2 (sin sin )2 d cos ( ) cos cos sin sin 2 4 sin 2x 2 sin x cos x 5 cos 2x 2 cos x 1 6 cos 3x 7 sin y 8 cos x 9 tan y (tan x 1) tan A tan B 2 2 10 11 sin A sin B 1 tan A tan B 7A 1 12 a tan y b 0 13 sin x 14 sin 2A A7 3 1 2 2 15 a tan b 2 sec 16 a 22 3 1 3 1 3 1 b i ii iii 22 22 3 1 1 3 iv 22 20.6 Self-test 2 2 2 1 a cos A b sec c cosec x d sec A e 1 2 2 2 a 1 b tan x c 1 d cot x 4 A 18.61, 3 127.71° 5 a 12 cos sin 2 tan x 1 b 6 a cos b tan tan x 1 9 4.96 sin ( 56.4°) c PQ (cos 2

5 a downwards making angle 28.0° with the vertical b 11.1 m s2 6 9.8 m s1 making an

angle 18.4° with direction of car’s motion

Exercises 21.2 1

1

1 a 103 m s b 292 m s b 5 h 19 min c 6 h 32 min e 28 h 48 min

2 a 5 h 00 min d 10 h 00 min

3 a 8.61 s b 6.75 s c 14.4 s d 23.8 s 4 F1: 6.13 N to the right, F2: 2.39 N to the right,

F3: 9.40 N to the left, net horizontal force 874 mN to the left 5 1.98 N upwards 6 a 75.1 N 2 b 7.51 m s 7 a 177 N down the plane 12 b 157 N down the plane 8 a 0 b 7.0 10 N 12 12 c 12 10 N d 14 10 N

Exercises 21.3 1 2.6 kN in direction N5°E (i.e. in direction bearing 005°) 2 134 km from P in direction bearing 123° from P 3 447 kN in direction N7°E 4 a 122 N directed between the directions of the two given forces, making an angle of 32° with the 83 N force (i.e. making an angle of 40° with the 67 N force) b 5.9 kN directed between the directions of the two given forces, making an angle of 88° with the 4 kN force (i.e. making an angle of 35° with the 7 kN force) 5 25.6 km

Exercises 21.4 1 a R 43.5 N, F 20.4 N b R 39.6 N, 41.3° c T 3.13 kN, P 2.81 kN d T 389 N, 55.4° 2 a 3.51 kN b 7.2:1 3 1.43 kN

21.9 Self-test 1

1 a 22.5 m s upwards at angle 27.5° with horizontal b 27.6 m s1 downwards at angle 43.5° with horizontal 2 0.652 kN in direction between

F1 and F4 making an angle of 19.9° with force F1 3 T1 320 N, T2 240 N 4 291°

21 Introduction to vectors Exercises 21.1 1 5.00 km from starting point in direction bearing 53.1° 2 1.30 kN upwards making an angle of 22.6° with the horizontal 3 2.44 kN in direction S61.7°W 4 608 km in direction N54°E Textbook Answers

171

Mathematics for Technicians CD Blair Alldiss

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