maths c semester 1 tutorial book.pdf

7/27/2019 Maths C Semester 1 Tutorial Book.pdf

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MathematicsCSemester1

TutorialBook2013


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MATHEMATICS C

Tutorial Book

Exercises

SEMESTER 1

Unit 1 Basic Number and Algebra Review

Unit 2 The Number Plane, Functions and Graphs

Unit 3 Differential Calculus

Unit 4 Logarithmic and Exponential Functions

Unit 5 Sequences and Series

Revision Exercises


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UNIT 1

Basic Number

andAlgebra Review

1-1 Practice in use of Casio Fx-82AU PLUSCalculators

1-2A Terms in Mathematics1-2B The Real Number System (Sets 1 & 2)

1-3A Sets

1-3B Further Sets Venn Diagrams and Word Problems

1-4A Linear Equations in One Variable

1-4B Linear Inequalities in One Variable

1-5 Operations with Surds (Radicals) (Sets 1 & 2)

1-6 Indices

1-7 Factoring of Algebraic Expressions1-8 Operations with Algebraic Fractions

1-9 Simultaneous Equations

1-10A Absolute Value Definition and Equations

1-10B Absolute Value Inequalities

1-11A Quadratic Equations (Sets 1 & 2)

1-11B Completing the Square (Sets 1 & 2)

1-12A Introduction to Polynomials

1-12B The Remainder and Factor Theorems

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EXERCISE 1-1PRACTICE IN USE OF Casiofx-82AU PLUS CALCULATORS

SET UP YOUR CALCULATOR to MATHS mode

Press:

1SHIFT SET UP

Math will appear on the screen.This mode is much better to use than LINE mode, especially when we want to workwith fractions.

2

THE CURSOR key

The cursor key is marked with four arrows

REPLAY

LEFT arrowRIGHT arrowUP arrowDOWN arrow

MEMORY

To put a number into the memory say 2505 press:M2505 will appear on the screen. This will go when you start to do a calculation.

2505 SHIFT STO M+

The symbol M will appear on the screen. This means that there is something in the mainmemory.

RCL M+To recall this number later press:

SHIFT STO0 M+To clear the main memory press:

M0 will appear on the screen. This will go when you start to do a calculation.

USING THE CALCULATOR

Example 1 To calculate42

5.3 4

press:

42 =x3.5 4 SHIFT

You will get 7.370512032

-2-


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878press:

8 87SHIFT SHIFT2 =You will get 0.278137814


4

3

5

1287

+ press:

SHIFT+43x87

=45 x )( 2 1


Example 4 To calculate press:)1025.1()1041.5( 315

=x10 x10 )( 3)( 1.25155.41


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Exercises

Calculate: Answers [not rounded]

1.73.3

1

373

100or 26809651470

2. 6)0775.1(974 2725541524

3. 2402 3386882297

4. ( ) ( )115 1097.4106.8 14107303822941

5.3.38.8

529.16.38

2765495871

6.

5.1

79587122110

7. 25

2 6568542495

8. 4 20 1147425272

9.3.4

8 65

701308599290

10.22 7.35.2

1

+

997

50or 505015045130

11.3

24

7

45

98

117or

98

191 or 1938775511

12.

3

2

5

13

3

121

3

214

132

2084266536

(ANSWERS to questions 13-20 are at the end of this exercise)

13. Write each of the following correct to 3 significant figures :

(a) 33 262 351 (b) 0 006 247 3

(c) 2 (d) the product of 436009 62 2 0 724 and

14. Write in scientific notation (standard form) correct to 3 significant figures:

(a) 220 000 (b) 9 630 000

(c) 0 004 (d)368 0 072 13

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EXERCISE 1-2A

TERMS IN MATHEMATICS

(The answers to these questions can be found in Part A, Part B and Part D of The Terms

in Mathematics booklet)

1. Complete these statements by inserting the correct word or words:(a) Numbers which are in the set consisting of positive whole numbers are called

or .

(b) Numbers which are in the set consisting of positive and negative whole numbersand zero are called .

(c) Numbers which can be represented in the form of a fraction are called .(d) Numbers which can not be represented in the form of fractions are called .(e) Rational and irrational numbers together are called .(f)

Consider the fraction 10

7

. The top number, 7 is called the and the

bottom number, 10 is called the .

(g) If one or more digits after the decimal point in a number is repeated endlessly thenthe number is called a .

(h) A fraction in which the numerator is greater than the denominator is called an fraction.

(i) A fraction in which the numerator is smaller than the denominator is called a fraction.

(j) The number ba

is called the of the number a

b,where a andb are integers.

(k) In the example 9327 = ,The divisor is the number ..

The dividend is the number ..

The quotient is the number ..

(l) When we divide 14 by 4 we get 3 and 2 is left over.The number 14 is called ..

The number 4 is called..

The number 3 is called ..The number 2 is called ..

(m) A number whose only factors are itself and 1 is called a .(n) Whole numbers that can be divided exactly by 2 are called (o) Whole numbers that do not have 2 as a factor are called (p) In the expression x7 , the number 7 is called the andx is called the (q) A comparison of quantities of the same kind in a given order is called a (r) Consider the example TD 5= . When two quantities D and T are related in this

way they are to one another.

(s) Consider the exampleT

S500

= . When two quantities SandTare related in this

way they are to one another.

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EXERCISE 1-2B

THE REAL NUMBER SYSTEM

Set 2

1. Indicate which of the following statements are true (T).

If the statement is false, give an example to show that it is false.(a) All real numbers are irrational(b) All rational numbers are real numbers

2. List the elements in the following sets:

(a) 5.and3betweennumberevenanis x (b) 6.and4betweennumberoddanis x (c) is an integer between and 2.x 1

3. To which sets [integers, rational, irrational, real or not real] do the following belong?

(a) 3 47

3

7

37

3 6

2(b) (c) (d) (e) (f) (g)

4. Insert the correct sign [ < , > , = ]

(a) 6 8 (b) 7 5 (c) 4 40 (d) 3

(e) 2 9 (f) 1

3

1

2(g)

1

2

1

3(h)

3

4

6

8

(i) 1 6 1 67 (j) 2 03 2 02 (k) 2 03 2 003

5. Find the reciprocals of the following:

(a) 3 (b)1

6(c) (d)

7

5x

6. List the following in ascending order (starting with the smallest):

(a) 9, , 5 8, 18. (b) 2 0 25, 13

11

54

1

3, , , , , ,

Set 2 ANSWERS

1. (a) F (b) T 2. (a) { } { } { }(c)5,3,1(b),4,2

3. (a) Integer, Rational, Real (b) Irrational, Real (c) Unreal (d) Irrational, Real

(e) Rational, Real (f) Irrational, Real (g) Integer, Rational, Real

4. (a) < (b) > (c) < (d) < (e) > (f) >

(g) > (h) = (i) < (j) < (k) =

xx

10. Express each of the following without negative indices.(a) (b)15 x ( ) 52 3 +ba (c) ( ) 111 + yx

11. Solve the following exponential equations :(a) 642 = (b)x

27

13 52 = x (c) 56262 =+ xx

(d)32

22 1 =x (e) ( )x

x

1624

12

=

+

12. In a NSW country town the population, P, is given by the formula 321000 tP = ,where t is the number of years elapsed since 1960.

(a) Find the population in 1960.(b) What is the population in 1969?(c) In what year is the population of the town 64 000 people?

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EXERCISE 1-6 ANSWERS

1. (a) (b) (c) (d)106a 6410 ba 26m 34a

(e) (f) (g) (h)64 11y 52n9

4 2b

(i) (j) 3 (k)62

4 yx pq

1(l) x

y

8

5

2. (a) 1 (b) 2

3. (a) a (b) 8 (c) b2

(d) (e) (f)2y x2 24xy

4. (a) 8 (b) 3 (c)25

1

(d) (e)4 5

1

(f) 8

1

(g)125

64(h)

9

4(i)

27

8

5. (a) (b)4=x3

1=x (c) 4=x (d)

2

3=x (e)

2

3=x

6. (a) 3 (b)54

9(c) 2

(d) 216 (e) 47. (a) 5 (b) 2 (c) 1

(d)2

1(e) 4

8. (a) 28 (b) 9 (c) 3

(d) 23 (e) (f) 461023

(g) 81 (h) 1 000 (i) 78 125

9. (a) (b)3=w 5=z (c) 512=y

(d) (e)16=p 8=k (f)100

1=y

(g)3

2=x

10. (a)x

5(b)

( )52

3+b

a(c)

yx

xy

+

11. (a) (b)6=x 1=x (c) 3=x

(d)2

7=x (e)

4

3=x

12. (a) (b) 8 000 (c) 19780001

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EXERCISE 1-7

FACTORING OF ALGEBRAIC EXPRESSIONS

1. Factorise the following by taking out a common factor:(a) (b)nm 55 + xx 63 2 +

(c) (d)yzxy 1612 + aa 2781 3

(e) (f)x63 + 22 53 xyyx +

(g) (h)baa 23 255 xxx 254 23 +

(i) (j)tatat 522 ++ bbcab 2177 +

(k) (l)xx 62 yy 124 2 +

(m) (n))(5 2 qpmm ++ )()( cbdcba +++

(o) (p))()( 2 yxzyx ++ ).()( qrsrqp

2. Factorise the following by grouping in pairs:(a) (b)cdbdacab +++ cabcab 1243 +++

(c) (d)zxyzxy 6432 + 623 23 + xxx

(e) (f)bcbcb 10808 2 + 22 393 yxyxyx +

(g) (h)cddbacab + 22 )()( abybax +

(i) (j)33 23 +++ yyy yxyxy + 2

(k) (l)1510128 + mnmn .104156 22 abbaaa +

3. Use the difference of two squares method to factorise the following:(a) (b)22 yx 12 a 6

1

8

(c) (d)2

249 m 816 2 a

(e) (f)2

2 94 nm 22

22 12149 ba 222(g) (h)425 cba

22

zyx 4(i) (j)

44

161 qp44

100x

(k) (l)22

ba 16dc 2

(m) (n) .)3( pnm +2

)2()4( ++ aa

4. Factorise the following completely:(a) (b)

2

182 2 a2

43 4 a22(c) (d)32242 yx

3

8045 yx22(e) (f)6015 aa rR

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6.(a) (b))1()23( ++ xx )3()12( ++ xx

(c) (d))4()32( xx )4()32( + xx

(e) (f))32()13( + aa )43()34( + xx

(g) (h))23()16( + bb )1()4( xx +

(i) (j))23()54( cc+

)3()4( yxyx

(k) (l))32()73( + yy )32()56( yxyx +

7.

(a) (b))3()4(4 + aa 2)6(2 +m

(c) (d))2()7(3 + yy )7()32(3 + tt

(e) (f))7()3(4 yxyx )1()6(6 + dd

(g) (h))2()5( + aab )23()92(2 + xxx

(i) (j))13(4 2 +yy )32()34(5 nmnm +

8. (a) ( ) )422 2 ++ xxx (b) ( ) )933 2 ++ aaa (c) ( ) )25255 ttt ++ (d) ( ) )22 9121634 nmnmnm ++ (e) ( ) )12412 2 ++ aaa (f) ( ) )115 2 ++ aaa (g) ( ) )4224 2 ++ xxx (h) ) )632432 16442 bbaaba ++ (i) ( ) )3666 2 ++ aaab (j) ( ) )117 22 ++ xyyxxy (k) ( ) )( ) ) ( )( ) )42242222 yyxxyxyxyxyxyxyxyxyx +++=++++ (l) )32 2 +xx

9.

(a) (b)( )( 222 + xxx ) ( )( )3523 xx

(c) (d))81()91( xx + )7()1( 2 zz

(e) (f))2()( ++ baba )423()23( ++ yxyx

(g) (h))652()52( ++ baba )31()31( yxyx +++

(i) )562()562( nmnm + (j) )35()35( ++ xyxy

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1. (a)1

2

x(b)

2

2+x(c) 23 +x

(d)

2

5

x

x(e)

( )

93

32

++

+

aa

aa(f)

42

52

+

tt

t

2. (a)2

3 2yx(b) (c)26a

22

1

x(d)

5

1

+

a

a(e)

( )337

+

x

x

3. (a) a (b)6

7b(c)

6

x(d)

3

2a(e)

4

5e

(f)8

11y(g)

5

4r(h)

4

5s(i)

30

23x

4. (a)3

43 x(b)

6

83 a(c)

10

97 y

(d)6

715 x(e)

10

1130 t(f)

8

3312 c

5. (a)( )( )yxyx

x

+

2(b)

( )( )bababa

+

+ 5(c)

( )( )( )1314

+ xxx

(d)

2

2

+a

(e)

( )( )( )321 xxx

x

6. (a)5

1

+

x

x(b)

3

6

a(c)

273 m

m(d)

1

33 y

7. (a)y x

y x

+(b)

t+ 1

2(c) 2 (d) x + 1 (e)

2y

x y

(f)ba

ba

+

1

1(g)

( )

+

1

x x h(h)

x y

x y

+

(i)

x

x

+ 2.

8. (a)( )

+

1

xy x y(b)

ab ba

b a

2 2

2 2

+

+(c)

xy

y x+

(d)b a

ab

2 2(e)

y x

y x

+.

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EXERCISE 1-10A

ABSOLUTE VALUE DEFINITION and EQUATIONS

1. Evaluate the following:

95(ix)95(viii)9(vii)

525(vi)25(v)95iv)(

95(iii)95(ii)9(i)

2+

2. Simplify:

057293(iv)43625(iii)

32(ii)25392(i)

2

33

++

+

3. (i) (a) ?7then,7If = xx ?7then,7If(b) =< xx

(ii) Simplify 10)c(,0)(b,1)a(for1


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EXERCISE 1-10A ANSWERS

1. (i) (ii) 4 (iii) 14 (iv) (v) 10 (vi) 5 (vii) 819 4

45(ix)14(viii)

2. (i) 5 (ii) (iii) 25 (iv) 13 19

3. (i) (ii) (a)2xx 7(b)7(a) 1x (b) +2 1x (c)1

3,0(ix)

5,3

13(viii)

2

15,3(vii)8(vi)8,5(v)

7,13(iv)2(iii)6,6(ii)3,3(i)4.

2

12,4(viii)1,

5

16(vii)

5

13,5(vi)5,21(v)

5,21(iv)2(iii)4,6(ii)2,4(i).5

6. (i) No solution (ii) x = 1

2(iii) 13 4

1

3, (iv) 1, 7

(v) 7 only (vi) No solutions ( )x x 3 7, (vii) 3,1

(viii) 2 only

EXERCISE 1-10B ABSOLUTE VALUE INEQUALITIES

1. Solve the following absolute value inequalities:

(i) x 3 (ii) x >3 2 (iii) x


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EXERCISE 1-10B ANSWERS

1. (i) { } x x: 3 3 (ii) { } { }5:1: >< xxxx (iii) { }x x: <


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EXERCISE 1-11A

QUADRATIC EQUATIONS

Set 1

1. Solve the following:

(a) (b)0)3()6( =+ xx 0)4()13( =+ xx

(c) (d)0)34()7( = yy 0)53( =aa

(e) 0)6(2 = mm (f) .0)52( 2 =x

2. Solve the following quadratic equations by factorising:

(a) (b)01272 =+ xx 02142 = aa

(c) (d)072172 =+ yy 032 =+ mm

(e) (f)0962 =+ bb 063 2 = xx

(g) (h)0673 2 =+ xx 094 2 =x

(i) (j)xx 566 2 = 203112 2 = xx

(k) (l) .05296 2 =+ bb 0246 2 =+ gg

3. Use the quadratic formula to solve the following, giving your answer correct to two

decimal places:

(a) (b) (c)0142 = xx 0532 =+ yy 0372 = xx

(d) (e) (f) .0425 2 = xx 0272 2 =++ xx 147 2 += xx

4. Use the quadratic formula to solve the following, giving your answer in simplest surd

form:

(a) (b) (c)0162 = mm 010102 =++ xx 0522 =+ xx

(d) (e) (f) .0372 =+ bb 0274 2 =++ xx 0423 2 =+ xx

5. Solve the quadratic equations by the method you think is the simplest in each case. Ifthe roots are irrational, give your answer correct to two decimal places.

(a) (b)0)3( 2 =x 322 2 =y

(c) (d)01252 2 = xx 09124 2 =+ xx

(e) (f)252 = xx 0364 2 = yy

(g) (h)0)32()25( =+ xx 81)52( 2 =+m

(i) (j)02185 2 = aa 032418 2 = pp

(k) (l) .0295 2 =+ xx )3()4()32( 2 ++=+ xxx

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6. The sum of a number and its reciprocal is12

12 . Find the number.

7. The sum of the first n counting numbers is given by ( 12

1+= nnS ) . Find the value of

n when 105=S .

8. The length of a rectangle is 5cm longer than its width. If the area is66cm, find the width.

9. Find two consecutive even integers if the sum of their squares is 164.

10. What must be the dimensions of a rectangle to have a perimeter of 60 metres and anarea of 125 square metres?

11. A ball is thrown vertically upward with its height h metres after time t seconds,

given by . Find when the ball is first at a height of 12 metres.28 tth =

12. The sides of a right-angled triangle are given by x cm, ( )12 x cm and cm.

Find the length of the hypotenuse.

( 12 +x )

13. Solve the following pairs of simultaneous equations:

324532

3(c)0113(b)0(a)22

2

+=++=+==+==

xyxxxyxy

yxyxyx

60231e)(13(d)

2

22

=+===+

xyxyx

yxyx

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EXERCISE 1-11A Set 1 ANSWERS

1. (a) (b)3,6 4,3

1 (c)

4

3,7

(d)3

5,0 (e) 6 (f),0

2

5

2. (a) 4, 3 (b) 3,7 (c) 8, 9

(d) (e) 3 (f)3,0 2,0

(g) 3,3

2 (h)

2

3 (i)

2

3,

3

2

(j)3

4,

4

5(k) 5,

6

1 (l) 4,0

3. (a) 240,244 (b) 194,191 (c) 410,417

(d) 720,121 (e) 193,310 (f) 190,760

4. (a) 103 (b) 155 (c) 61

(d)2

377 (e)

8

177 (f)

3

131

5. (a) 3 (b) 4 (c) 4,2

3

(d) 2

3

(e) 30,375 7

(f) 0, 9

(g)2

3,

5

2 (h) 7,2 (i) 3,

5

7

(j) 2, 4 (k) 260,541 (l) 142,470

6. 34

,4

3

7. n = 148. 6 cm9. 8, 1010. 5m x 25m11. 2 seconds12. 17 cm13. (a) 1,1;9,3 ==== yxyx (b) 4,5;2,3 ==== yxyx

(c) 0,3;3,0 ==== yxyx (d) 3,2;3,2 ==== yxyx

(e) x=2,y=1 ; 2

5,2

3

=

= yx

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EXERCISE 1-11B

COMPLETING THE SQUARE

Set 1

1. For each of the following, complete the perfect square.

(a) (b)( )22 ...................6 +=+ xxx ( )m m m22

10 =............ .........

(c) (d)( )t t22

100+ + =.......... ......... ( )x x x2 25 =............ .........

(e) ( )a aa222

3+ =............ ......... (f) y y y2 21

2 + = ........... ( ........)

2. Express each of the following in the form ( ) :x h k +2

5(f)1(e)

93(d)206(c)

114(b)62(a)

22

22

22

++

++

+++

xxxx

xxxx

xxxx

3. Express each of the following quadratic functions in the form k x h ( )2 :

22

22

22

92

3(f)62(e)

51(d)3(c)

512(b)49(a)

xxxx

xxxx

xxxx

+

+

4. (a) Find the minimum value of 2)1( x

(b) For what value ofx does the minimum value of ( )x 12 occur?

(c) Find the minimum value of the expression ( )x +1 42

(d) Find the minimum value of the expression 2 1 32( )x .

5. (a) Find the greatest value (maximum value) of .( )x 5 2

(b) Find the maximum value of which this

maximum value occurs.

forofvaluetheand)5(7 2 xx

6. By completing the square, find

(i) the maximum or minimum value of each of the following quadratic expressions.

(ii) the value ofx for which the maximum or minimum occurs.

3212(d)84(c)

542(b)36(a)

22

22

+

+++

xxxx

xxxx

7. The minimum value of x x k k2 4 + is 5. Find the value of .

8. For what values ofx is positive?44 2 xx +

9. Show that is always positive for all real values ofx.1062 ++ xx

10. Find the minimum value of the function 2 .f x x x( ) = + +3 82

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EXERCISE 1-11B

Set 2

1. The daily profit (in dollars) from the sa e el of a product is giv n by

, where x products are sold.

any products b axim m profit?

(b) What is the maximum possible profit?

2. A company manufactures and sells x televisions weekly. The total cost ofproducing x televisions is C dollars where

50108)( 2 = xxxP

(a) How m should e sold to achieve the m u

xC 10000016 += . Each television has

a selling price ofp dollars where x2p 700 = .

(a) Show that the profit P dollars when x televisions are manufactured and sold

is given by the equation

(b) Express P in the form

000166002 2 += xxP .

( ) cbxa + 2 , where a, b and c are constants.

(c) Find the number of televisions which should be manufactured and sold in aweek to achieve the maxim

Fin e for each television toac

aximum profit

st C(x) in dollars of producing a certain product is

here x is the number of employees. Find:

(a) the cost if the manufacturer employs 4 people.

(b) the cost if the manufacturer employs 12 people.

imise the cost, and the minimum cost.

A farmer wishes to enclose a rectangular vegetable garden with 40 metresof wire netting.

Let the length of the rectangle be x metres and find an expression for the area A interms ofx. Find:

(a) the largest rectangular area that he can enclose.

(b) the dimensions of this rectangle.

5. A manufacturer produces x hundred solar heaters at a total cost of

hundred dollars where

um profit.

(d) d the selling price the company should charghieve the maximum profit.

(e) Find the m .

3. A manufacturer is told that the co2given by C x x x( ) = +14 55, w

(c) the number of employees needed to min

4.

2001180 +x

300 x .

He sells the solar heaters at a price of x2300 dollars each.

(a) Find how many solar heaters the manufacturer should sell to break even.

(b) Show that the profit is hundred dollars.

(c) What is the maximum profit?

2)30(2600 x

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6. A car manufacturer produces and sells x thousand cars per week where, due to the

availability of staff and manufacturing constraints , 150 x .The cost of production (thousand dollars) is given by .

uadratic function

.

ollars).

cars are sold and explain the

significance of your answer.

turer.

. m

produc thousand dollars.

)(xC ( ) xxC 1002002 +=

The revenue from sales )(xR (thousand dollars) is given by the q

( ) 2501001 xxxR =

(a) Write down an expression for the weekly profit )(xP (thousand d

(b) Find the value of )(xP in a week when 0002

(c) By completing the square, or otherwise, find the maximum weekly profitwhich can be made by the car manufac

7 A co pany produces and sells x thousand Notebooks where 256 x . The total

tion cost is x1464320 +

dollars.The selling price per notebook is x36p 1190 =

(a) Find the revenue function )(xR thousand dollars.

Find the profit function )(xP thousan(b) d dollars.

(c) Express )(xP in the form ( ) CBxa + 2 .

(d) What is the maximum profit?

(e) Find the break-even point.

. m . The weekly cost and price-and equations are

(f) Find the price which should be charged to maximize profit.

8 A co pany manufactures and sells x brooms per weekdem xxC 20005)( xp 001010 =+= and (in dollars)

respectively, for domain 000100 x .

(a) Find the revenue function.

(b) Find the profit function.

achieveprofit.

(c) Find the mmaximum

aximum profit and the price which should be charged to

(d) Find the number of brooms which should be produced if a profit is to bemade.

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EXERCISE 1-11B Set 2 ANSWERS

1. (a) 40 (b) $110

. (b) (c

. (a) (b)

2 ( ) 000291502 2 += xP ) 150 (d) $400 (e) 00029$

3 $15 $31 (c) 7 employees. Minimum cost =$6

4. 220x (a) Maximum area = 100 2 xA = m (b) Dim ns : 10m 10mensio. (a)

. (a)

5 2681 (c) 00060$

2002000150)( 2 += xxxP (b) 4006 (loss of

(c)

. (a)

)000400$

0008002$

( ) 200136 2 += xx 7 2361901)( xxxR = (b) 3444 xP

( ( )2514362493 xxP (d) 0003$(c)

(e) 668

8. (a) (b)

) = 249

024 notebooks (f) $00

( ) 2001010 xxxR = ( ) 000580010 += xxxP 2

(c) Maximum profit of 00011$ at $6 per broom

(d) eeBetw n 684 and 3167 brooms

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54/244


(eYes(d)No(c)No)

2(vii)1(vi)

0(v)no(iv)4(iii)4(ii)6(i)(f)

3(vii)49(vi)

1(v)yes(iv)1(iii)(ii)2(i)(e)

4(vii)12(vi)

0(v)yes(iv)1(iii)(ii)4(i)(d)

2(vii)1(vi)

0(v)yes(iv)1(iii)(ii)8(i)(c)

3(vii)0(vi)

-3(v)no(iv)7(iii)7x(ii)7(i)(b)

3(vii)7(vi)

0(v)no(iv)1(iii)(ii)5(i)(a)2.

No)(bYes(a).

6

2

4

8

7

5

x

x

x

x

x

1

3. (a) x x x4 33 4+ 2 (b) + +x x4 33 2

(c) 2 6 5 5 3 17 6 4 3x x x x x + 3 (d) 41 (f) 10(e)

5=Degree,6If

,6=If

10=Degree,0If

9=Degree,0=If(b)4=Degree,6=If)

a

a

a

aa

(a4.

5=Degree,6If a

4=Degree(c)

( )( )( )

1846)124()72(1648(h)

)1()12(122(g)

14)4()1(583(f)

4

21

4

31

4

9

2

33)12(346(e)

02242(d)

0123)32(3856(c)

33)62)(5(342(b)

5)2)(3(15(a)5.

243247

322345

23225

224

2234

223

2

2

+++=+

++=+++

+++++=+++

+

++=+

++=

++=+

++=+

=+

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxx

xxxxxx

xxxxxx

xxxx

xxxx

6. ( ) ( ) xxRxxxQ 6and232 ==

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EXERCISE 1-12B

THE REMAINDER AND FACTOR THEOREMS

1. Use the remainder theorem to find the remainder when :

(a) is divided by873 2 + xx 1x

(b) is divided by42 34 ++ xx 1+x

(c) is divided by253 234 + yyy 4y

(d) is divided by60132 24 + xxx 32 +x

2. Use the factor theorem to show that ( )2x is a factor of the polynomial

3. (a) , find the value of

( ) 1074 23 += xxxxP .

If ( ) 187 24 = xxxP ( )3P .(b) What does this mean?

4. (a) Show is a zero of the polynomialthat 3=x ( ) 64 23 ++= xxxxP .(b) What does this mean?

(c) Hence express as a product of linear factors.

(d) Hence write down all the zeros of

( ) 64 23 ++= xxxxP( )xP .

is divided by5. When x + 33 mxx + 72 2x he remainder is 20. Findm, t .

Find the value of the constant. k if 1x is a factor of ( ) 5 22 2 += kxxxP6 .

7. Express each of the following polynomials as a product of linear factors.(a) (b) (c)

(d) (e) (f)

. If is divisible by

33 23 + xxx 673 xx 18923 + xxx 2

15892 23 + xxx 42412 23 + xxx 4034356 23 ++ xxx

8 baxx ++3 ( )3+x and ( )4x , find the values of a and b .

9. The factors of the polynomial are , and( )2x ( )1+x4423 + xxx ( )2+x .

Write down the zeros of the polynom

10. The zeros of the polynomial are

ial 23 + xx 4x .4

( ) 1074 23 += xxxxP 5and2,1 . What arethe factors of ?

11. Use the factor theorem to find a factor and hence find all real solutions to the followingpolynomial equations.

(a) (b)

(c) (d)

(e) (f)

12. When a polynomial is divided by

( )xP

0652 23 =+ xxx 0673 =+ xx

0223 = xxx 0863 23 =+ xxx

012 23 =+ xx 045 24 =+ xx

( )xP ( )( )31 + xxinder when

the quotient is and theremainder is . Find the rema

( )xQ13 x ( )xP is divided by ( )1x .

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1. (a) 4 (b) 3 (c) 142 (d)16

495

2. ( ) ) factorais22 P = (and0 x .

(b)

3. (a) ( )P 03 =

( ) ( )3soandpolynomialtheofzeroais3 = xxPx is a factor of

4. (b) is a factor of

( )xP

( )3+x ( )x P

(c) ( ) ( )( )( )123 ++= xxxxP

(d) 1,2,3 = x

5 4m 6.. 1=2

k3

=

7. (a) )( )( )311 +( + xxx (b) ( )( )( )23 (c) )( )( )( 332 ++ xxx 1 ++ xxx

( )( )( 3251 )++ xxx(d) (e) ( )( )( )671 ++ xxx ( )( )( 52234 )++ xxx (f)

8. 12and13 == ba

9. 1and2,2

10 ) , )( +x 1 ( 2x an1. (a) (b)

d ( )+x .5

3or2,1 =x 3or2,1 =x (c) 2or,1,0 =x 1

(d) (e)2

512,1 ,1=x (f) =x 4or,2,1 =x

Remainder = 212.

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UNIT 2

The Number Plane

Functions and Graphs

2-1 Gradient of a Straight Line

2-2 Equation of a Straight Line (Sets 1 & 2)

2-3 Distance and Midpoint

2-4A Circles2-4B Semicircles

2-5 Co-ordinate Geometry Problems

2-6A Functions and Relations (Sets 1 & 2)

2-6B FFuunnccttiioonn NNoottaattiioonn

2-6C Even and Odd Functions

2-7 Transformation of Functions (Sets 1 to 5)

2-8A Linear Functions and Their Graphs

2-8B Absolute Value Graphs Involving Straight Lines (Sets 1 & 2)

2-9A Basic Parabolas

2-9B Basic Cubic Curves

2-9C Rectangular Hyperbolas (Sets 1 & 2)

2-10A Quadratic Functions and Parabolas

2-10B Quadratic Inequalities

2-11 Polynomial Functions

2-12A More on Domain

2-12B Piecemeal Functions

2-13 Interpretation of Graphs (Sets 1 & 2)

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EXERCISE 2-1

GRADIENT OF A STRAIGHT LINE

1. The gradient of the line l is 2

3.

(a) Find the gradient of a line parallel to the line l.(b) Find the gradient of a line perpendicular to the line l.

2. Two lines which are perpendicular have gradients .21 and mm

Complete the following table:

m1 m2

2

31

2

3

4

a

3. Consider the points andA( , )3 2 B( , )1 6 . Find the gradient of

(a) the lineAB.

(b) a line parallel to the lineAB.

(c) a line perpendicular to the lineAB.

4. Find the gradient of the lines joining the given points:

(a) (1, 2) and (5, 7) (b) ( , ) ( , ) 3 2 7 2and

(c) (a, b) and ( , (d)) a b ( , ) ( , )2 22 2ap ap aq aqand

5. Sketch the line passing through (1, 2) and (1, 3). What can you say about its gradient ?

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6. Find the gradients of the following lines:

(a)

x

y

O

(-2,2) (3,2)

(b)

1

1x

y

0

(c)

(-2,5)

x

y

O

(d)

(a,2a)

x

y

O

7. In each case find the value of k ifAB is parallel to CD.

(a) ( ) ( ) ).,0(),4,5(,3,2,5,3 kDCBA

(b) ( ) ( ) ).5,1(),5,0(,2,1,2,0 + kDCBA

(c) ( ) ( ) ).7,1(),1,2(,3,3,1,2 kDCBA

8. Repeat question 7 ifAB is perpendicular to CD.

9. Show that the points (3, 5), (1, 1) and ( , ) 2 5 are collinear.

10. The points )23,43(and)7,1(),3,1( kkFED + are collinear. Find the value of k.

11. are the vertices of the parallelogram ABCD.Find the value of k.

( ) ( ) ),0(and)5,3(,5,1,0,4 kDCBA

12. Consider the points ( ) ( )A B x1 2, and y, . If the gradient ofAB is 4, express y interms ofx.

13. A ladder, placed on horizontal ground, reaches 55 m up a vertical wall. Find thegradient of the ladder if its foot is m from the base of the wall.5.1

14. The gradient of the line isl1

1

5and the gradient of the line isl

2

k 1

5. Find

The value ofkif the lines and l arel1 2

(a) parallel; (b) perpendicular.

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63/244

EXERCISE 2-2 Set 1 ANSWERS

1. (a)2

5

2

3

2

5

2

3,;b)(3,2;32 =====+= cmxycmxy

(c)2

1

2

3

2

1

2

3,;(d)2,3;23 ==+===+= cmxycmxy

(e) .,;f)(5,1;54

15

4

3

4

15

4

3

==+==== cmxycmxy

3. 4(c)1(b)2(a) === aaa

2intercept,3interceptb)( == y-x- 4. 2intercept,2intercept(a) == y-x-

5

2

3

2intercept,interceptc)( == y-x- .interceptintercept(d)

4

5

3

5== y-x-

5. (a) (b)0155 = yx 01832 =++ yx (c) 02 =+ yx

(d) (e)032 =+ yx 0162 =+ yx (f) 033 =+ yx

(g) 02 =+y

6. 7.m = 1. A =

2

30, , B = ( , ).0 2

8. .7

10(b)2)a( == kk

9. 042(b)023a)( ==+ yxyx

.035(d)022(c) ==+ yxyx

10. ).3,9((b))2,3((a) == BA

12. (a) 12=

0222isofEquation.32isofEquation(b) =++= yxDCxyAD .

(c)

=

5

2

5

19,3D

y

A(0,3)

C

xy +

(4,9)

D

B

x

1

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EXERCISE 2-2

EQUATION OF A STRAIGHT LINE

Set 2

1. An economist, studying the supply and demand for bathroom tiles, has determined that

the price per unit ($p) and demand (x), are related by the linear equation xp4360 = .

(a) Find the price per unit if the demand is 32 units.

(b) Find the demand at a price of $40 per unit.

2. A company sells memory chips for computers. The price demand equation iswhere the price charged is per chip and the number which can be

sold is x million chips.

xxp 6119)( = p$

(a) Find the demand when the price charged is $65 per chip.

(b) Find the price charged per chip when the demand is 15 000 000 chips.

3. A company that manufacturers electric can openers determined that at a price of $45each, the demand would be openers, and at $20 each the demand would be

openers.

5001

5006

(a) Find a linear function that models the price-demand relationship in the formbmxp += , where p is the price andx is the number of openers sold (in units

of one thousand).

(b) What would be the price if the demand is can openers?0006

4. A company manufacturing and selling DVDs finds the profit on sales of

DVDs is and the profit on sales of DVDs is

00020

00012$ 00035 00048$ .

The graph below represents this information, where the number of DVDs sold is xthousand and the profit is $P thousand.

)12,20(

)48,35(

)(thousands$P

)(thousandsx

Diagramnot to scale

(a) Assuming that the equation for P in terms ofx is linear, find an expressionfor P.

(b) Hence find the number of DVDs which must be sold for the company to breakeven.

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67/244

EXERCISE 2-3

DISTANCE AND MIDPOINT

1. Given the points ).5,5(),3,0(),0,5(,)0,1( ==== DCBA

(a) Plot the points on a number plane

(b) Find the lengths of AB AC CD, , BD and .

M

N

(c) If M A Cis the midpoint of find the coordinates of, .

(d) If N CDis the midpoint of find the coordinates of, .

(e) Show that MN ADis parallel to .

2. Show that ( is the centre of the circle which passes through the points 1 2, )

( , ), ( , ),11 3 1 15 ( 13, 7), (4, 14). [Note: show that the distance of each point from the centre is the same, (equal radii)]

3. Find the length of the interval AB where

(a) and (d))2,1(=A )10,7(=B )2,3( aaA = and )3,( aaB =

(b) )2,5( =A and )2,3( =B (e) ),( baA = and ),( abB =

(c) and)3,3( =A )2,2(=B (f)

=

baA

1,

1and

=

abB

1,

1

4. Prove that are the vertices of an isosceles)6,14(and,)6,2(,)10,10(

triangle, and find the length of the base.

5. If the distance between the points possible(1,3) and (5, is 5 units, find thea)

values of a.

6. If the distance between the points and( , )k 4 ( , )2 5 is 10 , find the possible

values of k.

7. Find the midpoint of the interval AB where

(a) and)7,4(=A )5,6(=B

(b) and)3,1(=A )3,2( =B (c) ),( babaA += and ),( babaB +=

8. (a) The midpoint of ( , ) )12 5 and (4, is (8,3). Find the value ofk k.

(b) The midpoint of .( , ) , )4 2 and ( is (2,1). Find the values of anda b a b

(c) The midpoint of and)1,(if,ofvaluestheFind5).,(6is cAdc-AB =

).,1( dB =

9. axis.on theliesandaxison thelieswhereintervalanis xByAAB

The midpoint of AB is the point (4 1, ). Find the coordinates of A Band .

10. If , find the coordinates of if the midpoint of isA B= ( , ) ( , )4 3 2 7AB .

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11. ABC A B b C a c Mis a triangle, where and is the= = =( , ), ( , ) ( , ).0 0 2 0 2 2

midpoint of .2thatShow.ofmidpointtheand, MNABBCNAC =

12. ly.respective)1,4(),4,5(),1,3(,)2,2(pointstheare ABCD

Show that AB CDand bisect each other.

13. ABCD A B Cis a parallelogram. Use the fact that= = = ( , ), ( , ), ( , ).4 0 1 5 3 5

the diagonals of a parallelogram bisect each other to find the coordinates of D.14. Find the equation of the perpendicular bisector of the interval PQ if:

(a) (b))0,1(,)2,3( QP )5,4(,)3,2( QP

15. The diagonals of a rhombus bisect each other at right angles.

(a) Show that this statement is true for the rhombus PQRSwhere

).1,2(and)6,3(,)1,4(,)6,1( SRQP

(b) The equation of the diagonalACof the rhombusABCD is 72 + .= xy

Find the coordinates of the vertices CandD if ).1,1(and)3,2( BA


1.(a)

0 A(1,0)

C(0,3)

M

B(5 ,0)

D(5,5)

N

x

y

(b) AB AC

BD CD

= =

= =

4 1

5 2

0

9

( ,c) M=

1

2

3

2

(d) N =

5

24,

(e) and m mMN AD= =5

4

5

4

MN is parallel to (equal gradients)AD

2. The length from every point to the centre is 13 units.

3. (a) 10 (b) 8 (c) 26 (d) 5 a (e) 2 22a b+ 2 (f)2 22 2

a b+

4. ),((c)0,2

1(b)6),(5(a).71,5.66,05.32 aaka

==

8. (a) k= 1 (b) a (c)b= =8, 0 c d= = 13 11,

9. A B= =( , ) ( , ).0 2 8 0,

)11,0(.10 =B 13. ).10,0( D

14. (a) (b)32 += xy 73 = xy

15. (a) , (b))0,1(M 1= QSPR mm ( ) ( )3,7,1,4 DC

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EXERCISE 2-4A

CIRCLES

1. Find the centre and radius of each of the following circles.

0222(e)

10)1()2((d)4)1((c)

5(b)1(a)

22

2222

2222

=++=++=+

==+

yxyx

yxyx

xyyx

2. For each of the following circles, state the centre C, the radiusR, and draw a sketch .

(a) x y (b) 12 (c)2 2 25+ = 12 32 2x y+ = ( )x y+ + =1 22 2

(d) ( ) ( )x y + + =2 32 2

16 (e) ( )x y22

7 4+ = 9 0(f) x y x y2 2 4 4+ + =

3. Give the equation of each the following circles.

y

x

(a)

3

3

-3

-3 C(1,2)

x

y

3

(b)

(c)

C(3,0) x

y

y

x

(d)

-3

y

x

(e)

5

y

x

(f)

(-2,1)

4. Find the equation of each circle with features as given

(a) Centre ( )3,3 and radius 5

(b) Centre ( )4,1 and passing through the point (0,0)

(c) Endpoints of the diameter are (6,8) and ( )2,2

(d) Centre (1,3) and touches thex-axis.

(e) Touches thex-axis at (4,0) and they-axis at (0,4)

5. Describe the relation whose equation is .x y x y2 2 6 2 10+ + + = 0

6. The equation of a circle is . A tangent is drawn from thepoint (5,2) to this circle. Find the distance from the point of contact to the point (5,2).

02024

22 =++yxyx

7. Determine whether the circles and overlap.7422 =++ yyx 0132822 =+++ yxyx

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EXERCISE 2-4B

SEMICIRCLES

1. Sketch each of the following semicircles, and give the domain and range:

2

22

2(c)25b)(4(a)

xy

xyxy

===

2. Give the equation of each of the following semicircles:

y

x

(a)4

4-4 0 5-5

-5

x

y

(b)

y

x

(c)

-1/2

1/2-1/2

3. Sketch the following semicircles. Write down the domain (X) and range (Y)for eachfunction:

(a) y = 36 2x (b) y x= 49 2


2

2

-2 0

x

y1. (a)

(b)

5

-5

-5 0

x

y

(c)

x

y

2

2

-2 0

D: 22 xR: 20 y

D: 55 x R: 05 y

D: 22 x

R: 20 y

2. (a) y x= 16 2 225(b) xy = (c) y x= 1

4

2

3. (a) y x= 36 2

{ }

{ }

X x x

Y y y

=

=

:

:

6 6

0 6

y

x

6

6-6

(b) y x= 49 2

{ }

{ }

X x x

Y y y

=

=

:

:

7 7

7 0

y

x

-7

7-7

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8. A(1,4), B(1,2) and C(5,2) are the vertices of a triangle.

(a) Find the length ofAB , as a surd.

(b) Write down the length ofBC.

(c) Find the co-ordinates of P , the midpoint ofAC .

(d) What is the equation ofBC?

(e) Find the equation of OP and the co-ordinates of Q , the point of intersection

of the lines BC and OP [O is the origin].

(f) Find the gradient ofAB.

(g) Explain whyAB is parallel to OP.

9. In the diagram on the right, ADCO is a parallelogram.

(a) Find the length of the interval AC as a surd.

(b) Find the co-ordinates ofEthe midpoint ofAC.

(c) What is the length of the interval OE?

(d) Find the gradient (slope) of the intervalAC.

(e) Find the equation of the line passing throughA andC.

(f) What is the equation ofDC?

(g) What are the co-ordinates ofD?

(h) Find the area of the parallelogramADCO.

10. Given A(4,6),B(5,3) andC(1,2). y(a) Find the length of the intervalAB.

(b) Find the slope of the lineAB.

(c) Find the equation of the lineAB.

(d) Find the co-ordinates of themidpoint,M, of the intervalAC.

(e) Find the co-ordinates of a point D sothatABCD forms a parallelogramwithACas a diagonal.

11. A(2,0), B(2,0) andC(1, 3 ) are the vertices of a triangle. P is the foot of the

perpendicular from Cto thexaxis.

(a) Calculate the length of the intervalBC.

(b) Find the slope (gradient) of the lineBC.

(c) Prove that ACB is a right angle.

(d) What is the equation of the circle that passes throughA,B, andC?

(e) Find the co-ordinates ofP.(f) What is the equation of the line OC?

D

C

B

A

x

( )0,2A 0 x

C( )4,2

y

D

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12. A(2,2),B(2,1) andC(2,1) are the vertices of a triangle. Eis the point (1,0).

(a) Find the equation ofAB.

(b) D is the foot of the perpendicular fromEtoAB.Find the equation ofED.

(c) Prove that the co-ordinates ofD are2

5

4

5, .

(d) Calculate the length ofDE.

(e) What is the equation of the circle centreEand radius 1 unit?


1. 5)2()1( 22 =+ yx

2.(a)

A(-4,2)

B(4,6)

C(8,-2)

x

y

(d) 40 units of area.

(e) 63 = xy (f) (2,0)

(g) 40)2( 22 =+ yx

3.(a)

C(0,2)A (5,3)

O

B

x

y

45

(b) 2= xy

(c) 25

3+= xy

(d) )8,10(=B

4. 54 units

5. (a) 2 5 (b) x y+ =2 1 0 (c) 2 1 0x y = .

6. (b) 4 2 (c) (1,5) (d) 06 =+ yx

7. (a) x y+ + =2 2 0 (b)2

5

6

5,

8. (a) 2 2 (b) 6 (c) (3,3) (d) y = 2 (e) y x= ; ( , )2 2

(f) 1 (g) Both lines have the same slope of 1.

9. (a) 4 2 (b) (0,2) (c) 2 (d) 1 (e) x y + =2 0

(f) y = 4 (g) (0,4) (h) 8 unit2.

10. (a) 3 10 (b)3

1(c) x y + =3 14 0 (d) 2

1

22,

(e) (10,1).

11. (a) 2 (b) 3 (d) x y (e) (1,0)2 2 4+ =

(f) y x= 3 .

12. (a) 3 4 2 0x y + = (b) 4 3 4 0x y+ = (d) 1 (e) ( )x y + =1 12 2

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EXERCISE 2-6A FUNCTIONS AND RELATIONS Set 1

1. Complete the following :

The set of ordered pairs {(1,3),(2,5),(3,1),(5,0)} form a relation.

This __________ is also a ___________.

The set of ordered pairs {(1,3),(1,5),(5,0)} also form a relation.

This ___________ is _________________.

2. (i) State which of the following sets of ordered pairs are functions.

(ii) Give the domain and range of each function.

(a) {(0,2), (0,4), (1,3), (2,5)}

(b) {(1,4), (1,4), (3, 4), (4,1)}

(c) {(Emma,162cm), (Jan,175cm), (Ali,172cm)}

(d) {(January,31), (April,30), (June,30), (July,31)}

(e) {(31,January), (30,April), (30,June), (31,July)}

(f)

x 2 2 2

y 1 2 3

(g)

x 1 2 3

y 2 2 2

3. (i) Which of the following graphs represent functions?

(ii) State the domain and range of each of the functions.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

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x cm

x cm y cm

7. A parcel delivery service willdeliver packages only if the sum ofthe height, length and depth doesnot exceed 260 cm.

The package shown is a square-basedrectangular prism.

If the full 260 cm is used,

(a) Find an expression for the volume, V, of the package in terms ofx.

(b) Find the domain of the function V.

120 cm

90 cm

8. A packing box is to be made using apiece of cardboard that measures 90 cmby 120 cm. Squares of side x cm arecut from each corner. The remainingshape is then folded into a box.

(a) Express the volume as a function ofx.)(xV

(b) Write down the domain of the function .)(xV

EXERCISE 2-6A Set 2 ANSWERS

1. (a)

+=

wwP

302 (b) }0:{ >ww

2. , Domain =)100( bbA = }1000:{


79/244

EXERCISE 2-6B FFUUNNCCTTIIOONN NNOOTTAATTIIOONN

1. If ( ) ( ) ( ) ( ) ( ).0and3,2,,1find,13 ffffxxf +=

2. Given ( ) ( ) ( ).and2

1,1gfind,25 agg-xxg

=

3. If ( ) ( ) ( )f x x x f f f x=

2 2

1

3

2 , ,find and .

4. If ( ) ( ) ( ) ( )f x x f fx= 2 2 0, , ,calculate f 1 1 .

5. If ( ) ( ) ( ) ( )f x x x f f f= + + 2 3 5 3 1 02 , , ,calculate and .

6. Given ( ) ( ) ( )g uu u

u

g z g x=+

2

1

3

, , ( )find g -2 and .

7. If ( ) ( ) ( )P x x x x x= + +4 3 27 6, verify that P 1 and P -1 are both zero.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

8 16 2 0 1 21

2

1

4

1

2

3

2. , , , , , , ,

.

If find

Show that

f u f f f f f f f f

f a f b f a b

u=

= +

, .

9. If ( ) ( )f uu u

uf u f

u=

+ +=

1 12, .prove that

10. Calculate Also obtain an expression for( ) ( )g x g x x x+ = 1 2if .

( )g x1 .+2 3

( )g x +

11. If ( )( ) ( )

f t tf f

h=

16 2 , .obtain an expression for

t + h t

Evaluate this expression when (i) t h= =2 0 01, . (ii) t h= =2 0 0001, . .

12. Solve ( ) ( )f x f= 2 wherefis the function ( )

{ }x y y x x, : .= 2 3 4

13. If ( ) ( ) ( )g t t t t g t g t = + = +3 2 7 12 , .find if

14. If ( )h xx

=+

1

1:

(i) evaluate ( ) ( )h h h1 01

2, , and

(ii) evaluate h h

1

3

1

4

+

(iii) express in simplest form. (iv) express(h x 3) hx

1

in simplest form.

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EXERCISE 2-6C EVEN AND ODD FUNCTIONS

1. Determine algebraically whether the following functions are even, odd, orneither even nor odd:

xxxfxxxf

xxfxxf

==

==224

32

)((d))((c)

)((b))((a)

2. Complete each given portion of the curve drawn below to make it into

(a) an even function (b) an odd function

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EXERCISE 2-6C ANSWERS

1. (a) even (b) odd (c) even (d) neither.

2.

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y f x= ( ) y f x= ( ) y f x= ( )

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y f x= ( ) y f x= ( ) 2 y f x= +( ) 1

x

y

x

y

x

y

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y f x= ( ) y f x= ( )1 y f x= + ( )2 1

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EXERCISE 2-7 TRANSFORMATION OF FUNCTIONS Set 4

1. Using the graph of drawn below, draw the graphs of:y x= 2

(a) y (b)x= 2 2 y x=1

2

2 (c) yx

=2

4

2. Shown below is the graph of y x= . Draw the graphs of y C x= where:

(a) C= 2 (b) C=1

2

(c) C= 4 (d) C=1

3

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3.

4.

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EXERCISE 2-7 TRANSFORMATION OF FUNCTIONS Set 5

1. Using the graph of sketch the graphs of :y x= 2

(a) y x (b)= 2 2 3 ( )y x= +1

22 1

2(c) y x= +

1

252

on separate number planes.

2. Using the graph of y x= sketch the graphs of :

(a) y x= 2 3 (b) y x= +1

21 (c) y x= +2 5


3. Using the graph of y = x sketch the graphs of :

(a) y x= +3 2 (b) y x= 4 1 2 (c) y x= +3 1



1. (a) (b) (c)

2. (a) (b) (c)

3. (a) (b) (c)

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EXERCISE 2-8A

LINEAR FUNCTIONS AND THEIR GRAPHS

1. Complete the tables below and sketch the graph.

(a) 2+= xy

(b)3 2 6x y =

x 0 1 2 x 3 0 2 4y y

2. Set up tables and sketch the following:

(a) y x= 2 1 (b) y x= 1 (c) yx

= 43

3. Rewrite withy as the subject of the equation and sketch the graph of:

(a) =2 1y (b) 02 = yx

4. Sketch the following by using the intercept method (i.e. findx andyintercepts first):

(a) 3 4 12x y = (b) x y+ =4 8 0 0(c) 2 5 15x y+ + =

5. State the domain and range of the functions in Q4. above.

6. Sketch the following straight lines:

(a) (b)7=y 03 = yx (c) 42 = yx

(d) (e)01 =+x x y+ =1 0 (f) y x=

(g) 12

+=x

y (h) x y= 2 (i) 2 3 9x y+ =

7. A chain-saw rental firm charges plus per hour for renting one chain-saw.Sketch a graph of the cost function for renting a chain-saw for up to 8 hours. Findthe equation of the cost function.

25$ 5$

8. A factory incurs a total cost of to produce 50 items. The fixed cost is equal to

. Assuming that the total cost function, , is linear, sketch its graph as a

function ofx, the number of items produced where

6001$

100$ )(xC

500 x . Find the equation ofthe function .)(xC

9. A building is depreciated by the owners accountant. The value y of the

building (in dollars) afterx months of use is given by the equation

.

000400$

000400 = xy 6001

(a) Sketch a graph of this function for .0x

(b) How long is it until the building is completely depreciated (its value is zero)?

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10. The profit obtained by a company in producing x items is given by dollars.

The following are graphs of profit functions which are linear functions ofx.

)(xP

)(xP

In each case, . For each graph find80 x

(i) the number of items the company should produce to break even.

(ii) the maximum possible profit.

(iii) the profit obtained when the company produces 10 items (assuming that theprofit remains linear).

(iv) the equation for in terms ofx.)(xP

(a) (b)

(c)

11. The annual revenue and cost(in millions of dollars) for afactory are shown in thegraph.

(a) What are the annual fixed costs?

(b) What is the annual productionrequired to break even?

(c) What are the revenue and costs atthe level of production needed to

break even? 200 4000

1

2

3

4

600 800 x units

$y million

5

6Revenue

Cost

2 3 410

10

20

30

40

-10

-20

65 7 8 x

P

2 3 410

10

20

30

P

40

-10

-20

-30

-40

)30,8(

6 7 85 x

(0,-30)

(8,30)

2 3 41

P

0

10

2030

40

-10

-20

6 7 85 x

(5,0)

(8,10)

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200 4000

1000

2000

3000

4000

600 800 x

$y

5000

6000

)(xC

)(xR 12. The diagram shows graphs of the costfunction, in dollars, and the revenue

function, in dollars, for a commodity

(where x is the number of units sold).

)(xC

)(xR

(a) From the graph, write down thenumber of units sold to break even.

(b) Find the equation for the costfunction and the revenue function.

(c) Find the marginal cost and the marginalrevenue (the cost and revenue involved in selling one extra unit).

13. A company producing poultry feed finds that the total cost of producing x

units is given by

)(xC

5010)( += xxC . The company charges $12 per unit for the feed.

(a) Find the formula for the profit .)(xP

(b) What is the break-even point?

(c) Sketch the graph of for)(xP 300 x .

14. A cafe sells each cup of coffee for 503$ . The variable costs are for each cupand the monthly fixed costs for the cafe are . Let x represent the number of

cups of coffee sold in a given month.

501$ 0006$

(a) Find an expression for the profit made by the cafe in the given month.

(b) Sketch a graph of the profit function for 00050 x .

(c) How many cups in the given month should be sold to break even?


1. (a) y 2 3 4 (b) y 7 5 3 0 3.

311

210a)(.2

y

x (b)

x

y

0 1 2

1 0 1(c)

x

y

3 0 3

5 4 3

y

x1 2 3-1-2

1

2

3

-1

-2

y

x1 2-1

1

2

3

-1

y

x6 12

1

2

3

4

5

-1

y

x1 2-1-2-3

1

2

3

-1

y

x2 4

12

-1-2-3-4

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3. (a) y = 1

2

(b) y x= 2

4.(a)x y- intercept = 4, - intercept = 3

y

x1 2 3 4 5-1-2

1

2

-1

-2

-3

-4

(b)x y- intercept = 8, - intercept = 2

y

x2 4 6 8-2

1

2

3

-1

(c)x y- intercept = -7 , - intercept =1

23

y

x1-1-2-3-4-5-6-7-8

1

-1

-2

-3

-4

5. For all functions in Q.4:

Domain : { }X x x R= : and Range : { }Y y y R= :

6. (a)

y

x

7

(b)

y

x1 2 3-1-2-3

1

2

3

-1

-2

-3

(c)

y

x1 2 3-1

1

-1-2-3-4-5

(d)

y

x1-1-2

(e)

y

x1 2-1

1

2

3

-1

(f)

y

x1 2 3-1-2-3

1

2

3

-1

-2

-3

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EXERCISE 2-8B

ABSOLUTE VALUE GRAPHS INVOLVING STRAIGHT LINES

Set 1

1. For each of the following functions:

(a) sketch the graph

(b) write down the equations of the 2 straight lines which form the graph

(c) find the domain and range.

xy

xy

xy

xy

21(iv)

2(iii)

3(ii)

(i)

=

+=

+=

=

2. For each of the following, sketch the graph and write down the equations of the lineswhich form the graph:

12y+x(ii)

3(i)

=

=x


1.(i)(a)

f(x)

x

(b) y x y x= =,

) D:(c x

R:y 0

(ii)(a)

f(x)

x

3

(b) y x= + 3y x= + 3

(c) D: x

R: y 3

(iii)(a)

f(x)

x-2

2

0

(b) y x= + 2

y x= 2

(c) D: x

R: y 0

(iv)(a)f(x)

x

1

0

(b) y x= +2 1

y x= +2 1

(c) D: x

R: y 1

2.(i)

f(x)

x-3 0 3

x= -3 x=3

(ii)

x

f(x)

21

121

-1

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EXERCISE 2-8B

Set 2

Sketch the following graphs. For each function state the domain and range.

1. x = 2 2. y = 3 3. y x=

4. y x= + 2 5. y x= 3 6. y x=

7. y x= 4 8. y x= 2 9. y x= 2 3

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

10. y x= +2 3 11. x y= 12. x y+ =1

13. x y= 2 14. x y+ = 2 15. 2 3 6x y =

y

x

y

x x

y

x

y

x

y

x

y

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1. x = 2

NOT a function

2. y = 3

NOT a function

3. y x=

{ } {X x x Y y y= = : ; : }0

4. y x= + 2

{ } {X x x Y y y= = : ; : }2

5. y x= 3

{ } { }X x x Y y y= = : ; : 0

6. y x=

{ } {X x x Y y y= = : ; : }0

7. y x= 4

{ } {X x x Y y y= = : ; : }4

8. y x= 2

{ } { }X x x Y y y= = : ; : 0

9. y x= 2 3

{ } {X x x Y y y= = : ; : }3

10. y x= +2 3

{ } {X x x Y y y= = : ; : }0

11. x y=

NOT a function

12. x y+ =1

NOT a function

13. x y= 2

NOT a function

14. x y+ = 2

NOT a function

15. 2 3x y 6 =

NOT a function

y

x-2 2

y y

3

x x

-3

y y y

x

23

x x

3

yy y

x

4

-4 4

x 23

23 x

-3

-1

-1

1x

y

x

yy

3

x

23

2

3-3

-2

x

y

2x

y

2

2

2

-2

-2 x

y

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EXERCISE 2-9A

BASIC PARABOLAS

1. Sketch the following parabolas, showing the main features:

222

222

222

222

2

2

122

(o)*)12((n)*)3(2(m)

1)2(2(l)13(k))1(2(j)

51(i))1((h))2((g)

3(f)1(e)(d)

(c)2(b)a)(

yxxyxy

xyxyxy

xyxyxy

xyxyxy

xyxyxy

=+=+=

+=+==

=+==

=+==

===

2. Write down the equation of each of the following parabolas :

(a)

y

x1 2 3-1-2-3

1

2345

-1-2

(b)

y

x1 2-1-2-3

1

-1

-2-3

-4

(c)

y

x2-2

2

4

(1,2)

(d)

y

x1 2 3-1-2-3

1

2

3

-1

-2

-3

(f)

y

x1 2 3 4 5 6-1-2-3-4

12

3

-1

-2

(1,2)

(e)

y

x1-1-2-3

1

2

3

4

5

-1

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EXERCISE 2-9A ANSWERS1.

2-2

2

4

6

8

10 y

x

(a)

2-2

2

4

6

8

10 y

x

(b)

1 2-1-2

1

2

3

4

-1

y

x

(c)

2-2

2

-2

-4

-6

-8

-10

yx

(d)

2-2

2

4

6

8

10

-2

y

x

(e) y

x

-3

(f)

3-3

1 2 3 4-1

2

4

6

8 y

x

(g)

1-1-2-3

1

2

3

4

5

6

-1

y

x

(h) y

x

(i)

15

15-15

y

x

(j) (1,2)

1-2 1+2

2-2

5

y

x

(k)

x-1 1 2 3 4 5

y

2

4

6

8

10

(2, 1)

(l)

-3-6

2

-2

-4

-6

y

x

(m)

1 2-1-2

2

4

6y

x

(n)

2 4

2

-2

y

x

(o)

2. (a) y x (b)= 2 1 ( )y x= + 12

(c) y x = 2 2

(d) y x= +12

22 (e) ( )y x= +2 12

(f) ( )y x= +18

1 22

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1 3. y x=

y

x

2 3. y x=

y

x

3 13. y x= +

y

x1

-1

4 2 3. y x=

y

x

2

126

5 83. y x=

y

x

-8

2

6 43. y x=

y

x

-4

-159

( )7 13

. y x=

y

x1

-1

( )8 23

. y x=

y

x

8

2

( )9 23

. y x= + 4

y

x

4

-04

(-2,-4)

( )10 1 23

. y x= +

y

x1

-026

(1,2)

( )11 2 33

. y x= +

y

x(-3,2)

-174

-25

12 2 13. x y=

y

x1

08

13 2 13. y x=

y

x1

-05

14 1 2 1 3. (y x =

( ) +2 43

( )1 23

+

)

y

x

-1

(1,1)

15 3. x y=

y

x1

1

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EXERCISE 2-9C

RECTANGULAR HYPERBOLAS

Set 1

1. Find the equation of the vertical asymptote for each of the following functions:

(a) (b)yx

y x=+

=13

2 1 2( )

2. Find the equation of the horizontal asymptote for each of the following functions:

(a) (b)xy yx

= + =

2 21

4

3. Sketch each of the following, showing the main features.

State the domain (X) and the range (Y) for each function.

xy

xy

xy

11(c)

1

22(b)

1

4(a)

=+

==

EXERCISE 2-9C Set 1 ANSWERS

1. (a) = -3 (b)x x =1

2

2. (a) (b)y y= = 0 2

3.

(a)

f(x)

x

-4

0 1

X x xY y y

= =

{ ,{ ,

10}}

(b)

f(x)

x-1 0

2

X x xY y y

= =

{ ,{ , }

12

}

(c)

f(x)

x-1 0

1

X x x

Y y y

= =

{ ,

{ ,

0

1

}

}

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EXERCISE 2-10A ANSWERS1.(a) (i) x = 2 2(ii) ( , )1

(iii)

f(x)

x(2,1)

5

0

(b) (i) x = 1 1(ii) ( , )0

(iii)

f(x)

x1

-1 0

(c) (i) x = 1 1(ii) ( , )1

(iii)

f(x)

x0

(-1,1)

3

(d) (i) (ii) (1,1)x = 1

(iii)

f(x)

x20

(1,1)

(e) (i) (ii) (1,1)x = 1

(iii)

f(x)

x

(1,1)

-1

x intercepts are

2 2

2

2 2

2

+and

2.(a) y x= +( )(1 3 x)

x intercepts 1 3, .

f(x)

x10-3

(b) y x x= +(2 1)

x intercepts 0 12

,

f(x)

x-05 0

(c) y x x= +( )(1 4)

x intercepts 4 1,

f(x)

x-4 0 1

(d) y x x= + ( )(3 1 2 )

x intercepts 3 12

,

f(x)

x-3 0 05

3.

(a) (i) ( )1 2= xy(ii) )0,1(

(b) (i) ( ) 12 2 += xy (ii) ( )1,2

(c) (i) ( ) 41 2 += xy(ii) ( )4,1

-3

-3 10

(-1,-4)

1

0 x1

5

0 ( )1,2

x

x

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(d) (i) ( )213 = xy(ii) )3,1(

(e) (i) ( ) 312 2 ++= xy (ii) )3,1(

y1,3

2

5

4.(a)

f(x)

x

8

2 40

(b)

f(x)

x-3 0 05

3

(c)

f(x)

x

9

(d)

f(x)

x-3 0

-9

(e)

f(x)

x0 25

(f)

f(x)

x

-5

0

5. (a) (b) 15 cameras (c)0004$

x13 +31 (-1,3)x0

P

x

(25,4000)(30,3000)

15

00021

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EXERCISE 2-10B

QUADRATIC INEQUALITIES

1. (a) Draw a sketch of the parabola y x x= +( )(2 1) showing thex-intercepts.

(b) Using the parabola, solve the inequality ( )( )x x + (b) 3 12 15 02x x+

4. Sketch the parabola y x x= + +( )(1 3)

0

0

. Use your sketch to solve the inequalities.

34(b)0)3)(1((a) 2 ++>++ xxxx

5. By sketching an appropriate parabola, solve the following quadratic inequalities:

( ( )( ) ( )( )

(

a) (b)

(c) (d)

(e) (f)

(g) h)

(i) (j)

2

2 3 0 4 1 2

5 0 4 5 0

4 5 0

3 1 1 3 1 1

4 5 1 3 1

2 2

2

2 2

2 2

+

< >

+ >

+ < +

+ >

x x x x

x x x x

x x x x

x x x x

x x x x

6. Solve forx:

035(f)052x(e)

012d)(05(c)

0)3(5(b)9(a)

22

22

22

++>+

++

> 1 3 0 3for 1

(b)

f(x)

x

x x x2 3 0 3 1+ + for

2

53

2

53(j)2(i)

21(h)30(g)

10(f)(e)

51(d)50(c)

42

1(b)23a).(5

+>


114/244

EXERCISE 2-11

POLYNOMIAL FUNCTIONS

1. Sketch the following polynomial functions:

(a) ( )( )( )y x x x= + +1 2 3 (b) ( )( )( )y x x x x= + 1 2 5

)

(c) (d)(y x x= 32

( )y x x= +2 1

(e) (f)( ) (y x x x= + 2 32

) ( )( )( )y x x x= + 1 2 3 42

(g) (h)( ) (y x x= + 33

)1 ( ) ( )y x x= +3 12 2

(i) (j)( ) (y x x= +3 2 22

)2

( )( )y x= + 3 4 23

x

(k) ( )( )( )( )y x x x x= + + 1 2 3 2 3 (l) ( )( )( )( )y x x x x= + 4 2 3 1 2

2. Sketch the polynomial function ( ) ( ) ( )f x x x= + 1 3 3 and hence determine

the values ofx for which ( ) ( )x x+ >1 33

0 .

3. Sketch the polynomial function ( )f x x x x= + 3 2 2 and hence determine

the values ofx for which x x x3 2 2 0+ .

4. Determine the values of the zeros of the polynomial ( )P x x x x= 3 23 4 and

hence sketch the graph of ( )y P x= .

5. (a) Factorise x x4 213 36 + .

(b) Hence sketch the graph of .y x x= +4 213 36

(c) Find the values ofx for which x x4 213 36 0 +

6. (a) Factorise .xxx 4334

+(b) Hence sketch the graph of ( ) xxxxg 43 34 += .

(c) Find the values ofx for which .043 34 + xxx

7. Sketch the graph of and hence solve .42683 xxxy += 0683 42


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1. (a) x = 0 (b) x < 0 (c) x x< >3 3or (d) 1=x (e) (f)2xx (l) { }44:


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1. (a) (b) (c)f f f( ) ( ) ( )21

21 1 0= = 3=

a2+

2. (a) F F F a( ) ( ) ( ) ( )1 1 1 1 1 12 2= = + =(b) (c)

3. (a) 10 (b) (c) 0 (d)2 2 2a

4. (a) (b) 0 (c) 0 (d) 11

5. a b= =2 5,

6.

f(x)

x20

4

f(x)

x1-1

-2

-4

1-3

-16

(g) (h)

(i) (j)

{ }1: = yyRf { }Y y y= = : 1

{ }R y yg = : 1{ }

41: = yyRH

(e) (f)

{ }116: = yyyyRf

{ }2,1: == yyyR { }1: = yyRM

{ } { 1:0: = yyyyRL { }0: >= yyRT

}

(c)

(a) (b)

(d)

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EXERCISE 2-13 INTERPRETATION OF GRAPHS Set 1

1. The graph of is drawn. Plus signs indicate

where . Minus signs indicate where

y f x= ( )

0>)(xf f x( ) < 0 .

Using the graph write down values ofx for which

(i) (ii) f x f x( ) = 0 ( ) > 0

(iii) (iv) f x f x( ) < 0 ( ) 0

2. The graph ofy f x= ( ) is drawn.

Put plus signs and minus signs on the graph to indicatewhere andf x( ) > 0 f x( ) < 0 respectively.

Write down values ofx for which

(i) 0)( =xf (ii) f x( ) 0 (iii) f x ( ) > 0

3. For each of the following functions y f x= ( ) locate the points where f x( ) .= 0 Mark

these points with dots; then write plus signs on the part of the graph where f x( ) > 0 andminus signs on the part of the graph where f x( ) .< 0

Example (a) has been done.

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4. For each of the following sketches of functions y f x= ( ) state the values ofx for which

(i) f x( ) = 0

(ii) f x( ) > 0

(iii) f x ( ) < 0

Example (a) has been done.

(i) 0)( =xf when x = 1 or 2

(ii) f x( ) > 0 when x > 2 orx < 1

(iii) f x( ) < 0 when <


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5. For each of the following graphs y f x= ( ) state the values ofx for which

(i) is not definedf x( )

(ii) f x( ) = 0

(iii) f x ( ) > 0

(iv) f x ( ) 0

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6. The formula is given for each function which is drawn. Use the sketch to solve thegiven inequality.

y x= 2 1 y x x= 3 4

Solve x 2 1 0 > Solve x x3 4 0

c

y x x= + 4 25 4 y x= +

2 3Solve + x x4 25 4 0 Solve x 2 3 0+ >

2 3

y

x

e

2

x

1

y

21

y x x= +( )(3 2)2 yx

x=

+

+

2

1

Solve Solve( )( )3 2 2 + x x 0x

x

+

+>

2

10

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1. (i) x = ,1 x = 1 2. (i) x = 3 , x = 1, x = 2(ii) x > 1, x < 1 (ii) x 3, 1 2x (iii) < 2(iv) x 1, x 1

3.

4. (a) (i) x = 1 orx = 2 (b) (i) x = 1 orx = 1(ii) x > 2 orx < 1 (ii) < 2 or <


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EXERCISE 2-13 INTERPRETATION OF GRAPHS Set 2

1.

(a) Find ( ) ( ) (f f f9 0 1, , . )5

(b) Find ( )x f xif = 2

(c) Findx if ( )f x = 0

(d) Find the values ofx for which(i) (ii)( )f x > 0 ( )f x 0

(iii) (iv)( )f x 1 ( ) < 2 2f x

(v) ( )f x 2

(e) Find the values of for which( )f x

(i) 0 x 2 (ii) < (iv)0 x 0

(f) y is a line, where kis a constant.k=

Find the values ofkfor which the equation ( )f x k= has

(i) no solution

(ii) one solution

(iii) two distinct solutions

(iv) three distinct solutions

(v) four distinct solutions

x-4 -3 -2 -1

y

4

3

2

1

1 2 3 4 5 6 7 8 9 10-1

-2

-3

-4

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2. (i) Sketch the graph of 12

1+

=

xy .

(ii) Hence, use your graph or otherwise, to solve

012

1+

x.

3. (i) On the same set of axes, sketch the graphs of y x y= =2 2 and x .

(ii) Show that the graphs ofy x y x= =2 2 and intersect at the points (1,1)

and (1,1).

(iii) Use the graphs drawn, or otherwise, to solve

(a) 2 2 x x

(b) x x2 2 0+ > .

4.

Diagram not toscale

The graph ofy x x x= +3 22 2 is drawn above.

(i) Find the coordinates of A.

(ii) Use the diagram to find the solutions ofx x x3 22 2 0 + = .

(iii) Hence, find the values ofx for which x3 2 2 2x x+ .

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128/244


1. (a) 2 (b) (c)0 1, , 35 2 9. , , 3 0 4 8, , ,

(d) (i) x x x< < < >3 0 4 8, , (ii) 3 0 4 8x x,

(iii) 4 5.1,5.75. = x x (iv) < < 35 5 7 9. ,x x

(v) x x x =35 9 2. , ,

(e) (i) 0 (ii)( ) f x 2 ( ) < 2 2f x

(iii) f x (iv)( ) 3 ( ) f x 1

(f) (i) k< 3 (ii) k= 3

(iii) < < >3 1k k, 2 (iv) k k= =1 2,

(v) < xx .

3.

(iii) (a) 11 x (b) 11 >< xx

4. (i) A(0,2) (ii) x = 1,1,2 (iii) 1 1 2x x .

(i)y

x2 4 6 8 10-2-4-6-8-10

2468

10

-2-4-6-8

-10

(i)

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130/244

EXERCISE 3-1A

LIMITS

1. Use the graph of the function f below to evaluate the limits.

(a) (b)( )xfx

1lim ( )xf

x+1

lim (c) ( )xfx 1lim

(d) ( )1f

x-4 -2 2 4 6

y

-2

2

4

(e) (f)( )xfx

3lim ( )xf

x+3

lim (g) ( )xfx 3lim

(h) ( )3f

(i) (j)( )xfx 4lim ( )xf

x +4lim (k) ( )xf

x 4lim

(l) ( )4f

2. Given and( ) 3lim5

=

xgx

( ) 4lim5

=

xhx

, find the value of( ) ( )

( )xhxhxg

x 2

3lim

5

3. Find

++

3

1

1lim 2

2

1x

x

x

x

4. Find the following limits as 0x

(a)

97

23

+

x

x(b)

115

64

2

+

x

xx(c)

( )2

2

qpx

cbxax

+

++

5. Evaluate:

(a) )322

4321lim xxxx

+

(b) xx

14lim2

(c) ( )( )43lim1

+

aaa

(d) )44lim 22

+

hhh

(e)x

x

x

2lim

(f)1

+

++

2

32

lima

x

a

xx

ax

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3. The graph below shows the function ( )


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P

S

R

T

Q

1.

(a) P, T (b) R (c) Q, S

2. gradient =3

2, ( )

3

23 =f

3. (a) 5 (b) (c) 1 (d) 0 (e) 0, 29

4. (a) 4 (b) 2

5. (a) 5 (b) (c) 3 (d)3+h 13 = xy

6. (a) 22 +x (b)2

1

x (c) x23 (d)

x2

1

7. (a) 0 (b)

8. (a) (b) tangents are parallel( ) 3)1(1 == ff

x

y

2

tangent

x

y

tangents

(c)

9. (a) (b) (c) 3 (d) 75 6

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EXERCISE 3-2B

NONEXISTENCE OF THE DERIVATIVE

1. State whether or not each of the following functions is continuous.

Give thex-coordinate of any point at which the function is not continuous.

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139/244

EXERCISE 3-2C

BASIC DIFFERENTIATION

1. Find

dy

dx if:

(a) y (b) y (c)x= 7 . x= 5 y x=3

4

(d) y x= (e) y = 6 (f) y x=

9

5

2. Find if:( )f x

(a) (b)( )f x x= 3 4 ( )f x x= 5 5 (c) ( )f x = 7

(d) ( )f x x=4

5

10 (e) ( )f x x= 4 1 (f) ( )f x x=

3. Find the derivatives of the following with respect tox:

(a) x3 (b)1

7x

(c) x 25

(d)110

x(e) x (f) x56

(g)1

x(h)

15

x(i)

15

3

x

4. Find y if:

(a) y (b)x= 3 2 yx

=3

2(c) y

x=

1

3 2

(d) yx

=2

3(e) ( )y x= 3

2(f) y

x=

32

5. (a) IfA = 6 xdA

dx, find (b) If find z

y

dz

dy=

5

2,

(c) If W= 44 M

dW

dM, find (d) If C =5

3

t d

dt, findC

(e) IfN=8

4q

dN

dq, find (f) If P = 6 4x

dP

dx, find

(g) IfD = 8, finddD

dz(h) If ( ) ( )f c c f c=

4

9

10 , find

(i) If ( ) ( )h q q h q= 5

8 , find (j) If W( ) ( )dd

d= 7

5, .find W

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[ ]

[ ] [ ]

4(n)

3

2(m)

4(l)3

1(k)4(j)

3(i)

5

4(h)

5

1(g)

4(f)

5(e)4(d)

4

5(c)4(b)

8

5a)(

:Find.6

3

2

3

3

34

g

dg

d

cdc

d

fdf

d

wdw

di

di

d

t

dt

d

rdr

d

pdp

d

ndn

d

qdq

dm

dm

d

z

dz

dx

dx

dy

dy

d

82

3(l)119(k)47(j)

35(i)1(h)

26g)(

34(f)79(e)75(d)

723(c)82(b)7(a)

:respect towithateDifferenti.7

5

4223

2

2

2524

3422

++

+

++

xxxxxx

xxxx

xx

xxxxx

xxxxxxx

x

1724

10302547

(f)y(e))6((d)

(c))(b)((a):atedifferentithenandfollowingheSimplify t.8

xxyx

xxxyy

.)1)(28(i)()42)(3((h))43(g)(

)9((f))2+)(13((e))5)(2((d)

)43(c)()3(2(b))32((a)

:atedifferentithenandfollowingtheExpand.9

222

2

22437

++

++

+

xxxxxx

xxxxx

xxxxxxxx

10. Simplify and then differentiate (x 0)

(a)x x

x

x x

x

x

x

2 2

2 2

3 7 3+ + (b) (c)

4x3

((dd))9 7 22

3

4 2 3

4

x

x

x x x

x

x

x

+ + +(e) (f)

5x5 4+

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141/244

EXERCISE 3-2C ANSWERS

1. (a)7 6x (b)5

6x

(c)3

4

1

4

x

(d)1 (e)0

(f)9

514

5x

2. (a)12 3x (b) 25 4x (c)0 (d)8 9x (e)4

2x

(f)1

3. (a)1

3

2

3x

(b)7

8x

(c)2

5

3

5x

(d)10

11x

(e)1

2 x

(f)5

61

6x

(g)1

23

2x

(h)5

27

2x

(i)15 14x

4. (a)6x (b)6

3

`

x(c)

2

3 3x(d)

2

3

x(e)18x

(f)2 3

3x

5. (a) 3x

(b) 52 2y

(c) 1 34M

(d) 53

(e) 325q

(f)24 3x (g)0 (h)40

9

9c

(i)5

83

8q

(j)7

56

5d

6. (a)5

2

3y

(b)2

x(c)

15

4

2z

(d)1

8 m(e)

154

q

(f)2

n n

(g)3

54

p

(h)2

5r r

(i)1

6 t

(j)4

(k)1

6w w(l)8f (m)

1

3c c(n)

3

4

2g

7. (a)2 7x (b) 2x2 (c)12 63 2 7x x+ (d)20 73x

(e)18x (f) 20 64x x (g) 62

2+

x(h) 2 1x (i) 5

63

+x

(j) 21 82x x (k) 18 44 3x x+ (l) 310

6+

x

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143/244

EXERCISE 3-2D

THE SECOND DERIVATIVE

.find,123If.4

)1((c))((b))0((a)

:find,132)(If.3

2

1

3

1(c)53(b)12(a)

Find.2

2(c)3(b)(a)

:Find.1

2

2

23

3

232

310342

2

2

dtsdttts

fxff

xxxf

xxxyxyxy

y

xxyxxyxy

dx

yd

+=

+=

+==+=

===

( )( )

( )4

2

2

2

2

23

2

22

3

3

434(d)42

1(c)

3

1(b)53(a)

Find.9

ruleschainquotient /product /require12to9Questions

.)()(solve,32)(If.8

.0)(solve,62

5

3

1)(If.7

.find,3If.6

)1()1()1(evaluate,43)(If.5

=

=

=++=

=+=

==

+

=

++++=

xyx

y

x

xyxxy

dx

yd

xyxyxxxy

xfxxxxf

dx

yd

dx

dyxxy

fffxxxf

( ) .find13If.102

262

dx

ydxy +=

.2

1)(when)2(Find.11

+

+=

x

xxff

12. ( ) ( )xfx

xxf +

= find1

If 2 .

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EXERCISE 3-2D ANSWERS

( )( )

( )( )

( )322

242

2

33

24

82

1

32.12

32

1.11

)133()13(36.10

43432(d)2

1(c)

3

2(b)66(a).9

2.8

6,1.7

96189.6

20.5

418.4

12(c)12(b)3(a).3

12(c)6(b)0(a).2

1290(c)636(b)2(a).1

x

xxxf

xx

xxx

x

x

x

xxx

t

x

x

xxxx

+

=

++

+

=

=

++

+

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EXERCISE 3-3A

THE PRODUCT AND QUOTIENT RULES

1.

If )73()52(+=

xxy , find dx

dy

by:

(a) expanding and then differentiating (b) using the product rule.

2. Given that ),32( find y)6( 2 += xxy by:

(a) expanding and then differentiating (b) using the product rule.

3. Given the differentiable functions f and g, find ( )1h if ( ) ( ) (fxh = )xgx and( ) ( ) ( ) 31,11,51 === gff , ( ) 21 =g .

4. Use the product rule to differentiate each of the following:

(a) )( )252 2 += xxxy (b) )xxxy 323 = (c) ) )435 22 += xxxy (d) ) )xxxxxy ++= 232 213

5. If ) )53 22 = xxy , finddx

dyand find the values ofx for which 0=

dx

dy.

6. Find if( )2f ( ) ( ) )13 2 += xxxf .

7. If2

34 3

x

xxy = , finddxdy by:

(a) simplifying and then differentiating (b) using the quotient rule.

8. Using the information given in Question 3 above, find ( )1h if ( )( )( )xgxf

xh = .

9. Use the quotient rule to find the derivatives of the following:

(a) 1+x

x

(b) 53

72

+

x

x

(c) 3

532 +

x

x

(d)52

2

+x

x(e)

x

x

3

2 2(f)

1

132

3

+

x

xx

10. Find if( 2g ) ( )x

xxg

+=

2

32

11. Find the gradient of the tangent to the curve12

3

=

x

xy at the origin.

12. Find the points on the curve ( )1

=x

xxf for which ( ) 1= xf .

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146/244


1. (a) 112;356 2 == xdx

dyxxy (b) 112 x

2. (a) xxdx

dy

xxxy 6612;321812223

=+=

(b) 1266 2 + xx

3. ( ) 71 =h

4. (a) (b)1026 2 + xx 34 125 xx

(c) (d)15294 23 ++ xxx 1101845 234 + xxxx

5. 2,0;164

3

== xxxdx

dy

6. ( ) 232 =f

7. (a) 32;32 == xdx

dyxxy (b) 32

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