Grade 10 workbook paper 1 by GoMath Workbooks

Compiled by Chesley Nell

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Forward: Welcome to “ GO MATH WORKBOOKS”. This workbook is designed to be text book and class work book in one. There are sufficient exercises to ensure that learners get the required practice. A detailed memorandum booklet is available for each workbook. The statement “ You get out what you put in.” is very apt where maths is concerned. To succeed in mathematics one must be prepared to invest the time and effort to achieve that success. The partnership that you as a learner and this GOMATH WORKBOOK develop will be profitable if you allow it to be. Chesley Nell : Mathematics Educator.  Chesley Nell 2011

Grade 10 Core Mathematics

GOMATH WORKBOOKS

GRADE 10 CORE MATHEMATICS. PAPER ONE.

Topic:

Pages:

Number System

(4 – 6)

Mathematical Notation

(7 - 10)

Algebraic Products

(11 – 15)

Factors

(16 – 29)

Algebraic Fractions

(30 – 36)

Solution of Equations

( 37 – 72)

Basic Logarithm Simplification

( 72 – 74)

Exponents

( 75 – 90)

Number Patterns

(91 – 101)

10. Straight Line Graphs

(102 – 114)

11. Parabola Graphs

(115 – 130)

12. Hyperbola Graphs

(131 – 135)

13. Exponential Graphs

(136 – 152)

14. Financial Mathematics

(158 – 187)

15. Probability Theory

(188- 216)

Grade 10 Core Mathematics

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Rational Numbers and Irrational Numbers

A common fraction can be written in the form: a ; a, b  z; b  0 b

A mixed number is made up of two parts a whole number followed by a proper or decimal fraction. A terminating decimal is a decimal fraction that ends after a definite number of digits has been given, e.g. 1,23 To convert a terminating decimal into a common fraction change its denominator into 10, 100 etc. according to the number of decimal digits. E.g.: 1.23 

123 100

A recurring decimal is a decimal fraction or part thereof which goes on repeating itself without end. 

E.g.: 1. 2 3  1,23232323 We use dots to abbreviate recurring fractions e.g. 

 0,34 5  0,34555555 

 0,3 4 5  0,345454545 



 0, 3 4 5  0,345345345 To convert a recurring decimal into a common fraction Method 1: Multiply it by appropriate powers of 10, and then use subtraction to eliminate the repeating part, e.g.

x  1,2323... 100x  123,2323 100x  x  123,2323  1,2323 99x  122 122 99 23 1 99 x

Grade 10 Core Mathematics

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Method 2: In the denominator of the fraction to be formed write down a “9” for each of the recurring decimal and a “0” for each of the non- recurring decimal. In the numerator write down the decimal value as it is written and subtract the value of the non - recurring decimals from this. e.g. 0,3456 

3456  345 3111  9000 9000

Examples: 1. Convert 0,34 to fraction form:

34  3 31  . 90 90 2. Convert 1,34 5 to fraction form:

345  3 342 171 1,34 5  1 1 1 990 990 495 3. Convert 2,2 75 to fraction form:

275 2,2 75  2 999 EXERCISE 1: 1. Convert the following to decimals and whole numbers: 1 2 3 4 5 6 7 8 9 ; ; ; ; ; ; ; ; 2 2 2 2 2 2 2 2 2

Grade 10 Core Mathematics

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2. Convert the following decimal fractions into common fractions a)

0,45

b) 1,25

c) 6,125

d) 0,824

0,50

f) 2,5

g) 21,25

h) 0,05

3. Use dots to indicate the recurring digits in the following: e. g. 0,4444 = 0,4 a) 0,5555…. b) 0,454 454…. c) 0,714 1414…. d) 123,12333…

e) 123,12323… f) 123, 1231313 …

4. Convert the following recurring decimals into common fractions a)

0,6

e) 2,34 5

b) 0,7 8

c) 0,14

d) 45,345

f) 23,346 5

Grade 10 Core Mathematics

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2: Mathematical Notations: 1.

Set Builder Notation: Used with brackets and states the range of values including the section of the number system involved. < & > sign means â&#x20AC;&#x153;not incudingâ&#x20AC;? ď&#x201A;Ł & ď&#x201A;ł sign means : includingâ&#x20AC;? e.g. ď ťx : ď&#x20AC;5 ď&#x20AC;ź x ď&#x20AC;ź 6; x ď&#x192;&#x17D; ď&#x192;&#x201A;ď ˝ This is read as: the set of all values of x such that x ranges between -5 and +6 where x is a real number.

Interval Notation: Only used with Real numbers: Used with round and square brackets. ( ď&#x192;&#x17E; means â&#x20AC;?not includingâ&#x20AC;? [ ď&#x192;&#x17E; means â&#x20AC;&#x153; including. e.g. x ď&#x192;&#x17D; (ď&#x20AC;5;6) read as all the values of x that range between - 5 and + 6 but not including -5 & +6.

x ď&#x192;&#x17D; [ď&#x20AC;5;6) read as all the values of x ranging between -5 and 6 ; including -5 but not including 6. Exercise 2.1: 1. Express the following in interval notation: 1.1 .{đ?&#x2018;Ľ: đ?&#x2018;Ľ â&#x2030;¤ 2; đ?&#x2018;Ľđ?&#x153;&#x2013;â&#x201E;&#x153;} ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.2 {đ?&#x2018;Ľ: â&#x2C6;&#x2019;3 â&#x2030;¤ đ?&#x2018;Ľ â&#x2030;¤ 0; đ?&#x2018;Ľđ?&#x153;&#x2013;â&#x201E;&#x153;} ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.3

{x:đ?&#x2018;Ľ < 2; đ?&#x2018;Ľđ?&#x153;&#x2013;â&#x201E;&#x153;}

Grade 10 Core Mathematics

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8 1.4

 -1

 3

⃘ -2

 5

2. Express in set builder notation: 2.1 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2.2 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2.3 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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−4 to 2, including -4 and

2.4 not 2.

3. The set of integers (Z) . NB. Integers must be separately graphed on a number line as it is only the whole numbers involved and no fractions in between.

{ x : 3  x  2; x  }

     -4 -3 -2 -1 0 1

4. Number line graphs:



 means “ including.

O  means ”not including” A set of real numbers can be represented on a line graph as follows: 1. Interval notation: [2;9)

 2

o 9

5. Set Builder notation:

 2

o 9

Exercise 2.2 : 1. Represent the following on a number line graph: 1.1 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

1.2

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{𝑥: −2 ≤ 𝑥 ≤ 2; 𝑥 ∈ ℜ}

{𝑥: 𝑥 ≥ 6; 𝑥 ∈ ℜ}

1.1

{𝑥: 𝑥 < −6; 𝑥 ∈ ℜ}

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Algebraic Products : 1. Use the FOIL method for expanding product of binomials. ( x  2)( x  4)  x 2  2 x  4 x  8  x 2  2 x  8 Lasts

Firsts Combination of Outers & Inners 2. If the brackets have co-efficients multiply LASTLY by these.

2( x  3)( x  6)  2( x 2  3x  18)  2 x 2  6 x  36 3. If there is a co-efficient with a binomial squared multiply LASTLY be the co-efficient. 2( x  3) 2

3.1

 2( x  6 x  9) 2

Multiply by 2 lastly

 2 x 2  12x  18

Multiply by -2 lastly

 2( x  3) 2

3.2

 2( x 2  6 x  9)  2 x 2  12x  18

N.B Signs change!

4. The product between a binomial and trinomial must be carried out separately and collect like terms where possible “unless a pattern is reflected.” (a  2)(a 2  2a  3) No pattern .  a 3  2a 2  3a  2a 2  4a  6  a 3  4a 2  7 a  6

4.1 If the trinomial part is formed by the square of each term and the product of the two terms in the binomial then the answer is simply the sum or difference of cubes. ( x  2)( x 2  2 x  4) Trinomial is formed  x3  2x 2  4x  2x 2  4x  8  x3  8

from the squares of each term and product of two terms

In this case leave out the second line and go straight to the answer. (2 x  3 y )(4 x 2  6 xy  9 y 2 )  8 x 3  27 y 3

Simply write down the cube of each term. NB: the sign of the last term takes the sign from the binomial

Grade 10 Core Mathematics

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Exercise 3.1: 1.

Simplify the following :

1.1

2( x  3)( x  4)

1.2

–3(3 – 5x) + 3(3-5x)

1.3

–2a2(a – b)2

1.4

(x – 3)2 –2(x– 1)2 + 2x

1.5

(a – 2b)( a2 + 2ab + 4b2)

Grade 10 Core Mathematics

1.6

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( x  2) 2  3( x  2)( x  2)

1.7

3( x  2)(x  3)  2( x  4) 2  6  x

1.8

4 x  3 y 2 xy  y 2 

1.9

32 x  y   2x  y x  5 y  2

p





 4 p 2 q  4 p q  16

Grade 10 Core Mathematics

1.11

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(ЁЭСе тИТ 2)(ЁЭСе 2 + ЁЭСе + 4)

1.15 (2 ЁЭСе тИТ 3 ЁЭСж) (4 ЁЭСе 2 + 6 ЁЭСеЁЭСж + 9 ЁЭСж 2 ) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.16

(3 ЁЭСе + 4 ЁЭСж) (9 ЁЭСе 2 тИТ 6 ЁЭСеЁЭСж + 16 ЁЭСж 2 )

Grade 10 Core Mathematics

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1.18 −2(2𝑥 − 5𝑦)(3𝑥 2 − 𝑥 + 2𝑦) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1

1.19 − 5 𝑥(25𝑥 − 10𝑦) + 3 (12𝑥 − 3𝑦) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1

1.20 5 (5 𝑥 − 1) (5𝑥 − 5) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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Factors :

4.1. Factors of Algebraic Expressions To factorise is the process of reducing many terms to one term. It is the opposite action of distribution. Procedure: Step 1 : First look for a HCF (if possible): 1.1 Look at the constants and choose the highest possible value, that can divide equally without a remainder, into the constant values in the expression. 1.2 Look at common variables and choose the lowest power (exponential value) of these as HCF. Step 2. Place the HCF outside a bracket and then divide the HCF into each term of the expression placing the answer inside the bracket. NB there must be the same number of terms inside the bracket as there is in the original expression. Step 3. Look at the expression inside the bracket and ascertain whether it can be factorised further. It could be one of the following; 3.1 The difference of two perfect squares. 3.2 A trinomial. 3.3 A quadrinomial ( 4 terms) Examples: HCF plus other expressions. 1. 2. 3.

2 x  4  2( x  2) 2 x  4 x 2  6 x 3  8 x 4  2 x(1  2 x  3 x 2  4 x 3 ) 2ax  4abx  6abcx)  2ax(1  2b  3bc)

4. 5.

2ax 2  8ay 2  2a( x 2  4 y 2 )  2a( x  2 y )( x  2 y ) ab4  ac 4  a(b 4  c 4 )  a(b 2  c 2 )(b 2  c 2 )  a(b 2  c 2 )(b  c )(b  c ) NB: The sum of two squares cannot be factorised at all.

4.2. Difference of two squares: Method: Write down the product ( two brackets) of the summand difference of the roots of each term. NB look at the bracket with the difference of the 2 terms and see if it can factorise further; NB: The sum of two squares cannot be factorised at all. Examples: 1. 2.

a 2  b 2  (a  b)(a  b) (b 4  c 4 )  (b 2  c 2 )(b 2  c 2 )  (b 2  c 2 )(b  c )(b  c )

Grade 10 Core Mathematics

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NB DO NOT FORGET TO BRING DOWN THE SUM OF 2 SQUARES DOWN TO THE NEXT LINE Exercise 4.1: 1.

x2  4

x2  9

x 2  16

2 x 2  50

4x 2  9 y 2

x 4  16

Grade 10 Core Mathematics

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( x  y) 2  ( x  y) 2

4( x  y) 2  9( x  y) 2

____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 9. 4(2𝑥 − 1)2 − 36(𝑥 − 3)2 ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 10. 9𝑚2 (2𝑥 + 3) − 4(2𝑥 + 3) _____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 11. 3𝑥(𝑥 2 − 𝑦 2 ) − 3𝑥(𝑦 − 𝑥) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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4.3. Sum & Difference of two Cubes: Method: Open two sets of brackets viz. for a binomial and a trinomial. If factorizing the sum of cubes then the binomial will have a plus sign and the middle term of the trinomial will be negative (minus sign) Binomial bracket: In the binomial bracket place the cube root of each term (NB Divide an exponent by 3 to find its cube root.) Trinomial bracket: Square each term in the binomial to get the first and last terms the middle term is the product of the two term in the binomial. Examples: Factorise: a3  b3 = 1. =

Open two brackets with the correct signs. (

)(

)

(a  b)(a  ab  b ) 2

In the binomial bracket enter the cube roots of the terms

a3  b3 = (

In the trinomial bracket enter the square of each term as Ist & last and then the product of the terms as the middle term. -

)(

)

= (a  b)(a 2  ab  b 2 )

8 x 3  27 y 6  ( 

)(





)

 (2 x  3 y 2 )(4 x 2  6 xy 2  9 y 4 )

Exercise 4.2: 1.

x3  y3

x3  y3

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8 x 3 ď&#x20AC; 27 y 3

64x 3 ď&#x20AC;Ť 125y 3

6. 9đ?&#x2018;Ľ 6 đ?&#x2018;? â&#x2C6;&#x2019; 3 đ?&#x2018;Ś 3 đ?&#x2018;? ______________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 7. 4đ?&#x2018;Ľ 2 (8đ?&#x2018;Ľ 6 â&#x2C6;&#x2019; 1) â&#x2C6;&#x2019; (8đ?&#x2018;Ľ 6 â&#x2C6;&#x2019; 1) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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4.4. Trinomials: The method used here is called “trial & error” In products of binomials we use the FOIL method to expand . The middle term of the expression formed is a combination of the products of the inner terms and the outer terms. i.e. the ‘OI’ of FOIL. The reverse procedure is used to factorise trinomials. We find out the correct combination of factors of the First and Last terms of the trinomial. e.g. x 2  2 x  3  ( x  3)( x  1) Method: Draw a table and use the factors of the Ist and Last terms. Cross multiply them and either subtract or add to get the middle term. The sign of the last term informs one whether to add or subtract 1x 1x

x2  2x  3  ( x  3)( x  1)

1 3

x 2  8 x  12  ( x  2)( x  6)

1x 1x

6 12

NB Trinomial + + +

Brackets ( + )( + ) ( - )( - )

+ -

( + )( - ) ( + )( - )

Not necessary to work out

Must work out which bracket is negative and which is positive

Exercise 4.3: 1.

x 2  2x  3

x 2  6x  5

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x 2  6x  8

x 2  2x  8

x 2  7 x  12

x 2  8 x  12

2 x 2  5x  3

2 x 2  5 x  12

Grade 10 Core Mathematics

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3x 2 яАл 7 x яАл 2

12ЁЭСе 2 + 8ЁЭСе тИТ 15

(ЁЭСе + 3)2 тИТ 3(ЁЭСе + 3) тИТ 4

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4.5. FOUR OR MORE TERMS : Grouping into Brackets This involves a series of HCF’s: Method: 1. Group pairs of terms with common variables into brackets. NB don’t change the operation from addition and subtraction to multiplication. 2. Continue with HCF until fully factorised. Examples: 1.

ab  ac  db  dc  (ab  ac)  (db  dc)  a(b  c )  d (b  c )  (b  c )(a  d ) 3. x 2 (a  b)  y 2 (a  b)

3 px  3 py  x  y  3 p( x  y )  ( x  y )  ( x  y )(3 p  1)

 (a  b)( x 2  y 2 )  (a  b)( x  y )( x  y )

Exercise 4.4: Factorise completely. 1.1 x(a-b) + y(a-b) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2

1.2 p ( x + y) - q ( y + x ) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 1.3 (m  n)  ( pn  pm) ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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1.4 a  b  ax  bx ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.5

a2  b2  a  b

4x 2  2x  9 y 2  3 y

Grade 10 Core Mathematics

1.8

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(2 x  y ) 2  ( x  2 y ) 2

5 x 2  10xy  5 y 2

1.3

2x2 + 6x – 8x3

Grade 10 Core Mathematics

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1.4 a2 – 4 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.5

2x2 – 32

(m  n)  ( pn  pm)

Grade 10 Core Mathematics

1.9

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a  b  ax  bx

a2  b2  a  b

1.11

(2 x  y ) 2  ( x  2 y ) 2

4( x  y) 2  9( x  y ) 2

1.13

4x 2  2x  9 y 2  3 y

Grade 10 Core Mathematics

1.14

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x2  x  6

x 2  7 x  12

2 x 2  24x  70

9 x 2  42x  45

x3  y3

Grade 10 Core Mathematics

GOMATH WORKBOOKS

1.19 8x 3  27 y 3 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

5. Algebraic Fractions: Multiplication & Division: NB. ALL TERMS MUST BE FULLY FACTORISED BEFORE ANY SIMPLIFICATION IS ATTEMPTED. Examples: 1. =

x2 x2  4

Terms are factorised and simplified

( x  2) ( x  2)( x  2)

1 x2

x 2  4 x  1 x 2  2x  4 x x x 2x  4 x3  8

All terms factorised first. And then common values are simplified.

( x  2)( x  2) x  1 ( x 2  2 x  4) x x x 2( x  2) ( x  2)( x 2  2 x  4) x 1  2x 

4  x 2 x 2  8x  4 x 2  16  ( x  4) 4   ( x  4)( x  4 2 x( x  4) 4  2 x ( x  4) 2

Terms all fully factorised first and a –ve sign is used to reverse ( 4 – x ) as well as inverting the fraction after the division sign and changing the operation to multiplication.

Grade 10 Core Mathematics

5.1

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Algebraic Fractions Addition & Subtraction

1. NB Before you can find a LCD (lowest common denominator) you must ensure that all denominators are fully factorised. 2. The next step is to get the LCD. Write down the product of the highest power of each type of factor. Remember look at each factor in each separate term to ascertain its power. 1 2 3   Example: 2 ( x  1)( x  2) ( x  2) ( x  1) 2 The highest power of ( x  2) is in the second term and is ( x  2) 2 and the highest power of ( x  1) is in the third term and is ( x  1) 2 thus the LCD = ( x  2) 2 ( x  1) 2 3. Divide each denominator into the LCD and multiply the answer by the numerator of each fraction. 4. Simplify to get the final answer.

Examples.

2 3  x3 x2 2( x  2)  3( x  3)  ( x  3)( x  2) 2 x  4  3x  9  ( x  3)( x  2) 5x  5  ( x  3)( x  2)

LCD

Final Answer

Highest power of (x+3)

1 3 2   2 ( x  3) ( x  3) ( x  3) 3 ( x  3)( x  3)  3( x  3)  2  ( x  3)( x  3)( x  3) x 2  6 x  9  3x  9  2  ( x  3) 3 

x 2  3x  2 ( x  3) 3

LCD Final Answer.

Grade 10 Core Mathematics

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2 3 4  2  x  2 x  1 x  1 ( x  1) 2 2 3 4    ( x  1)( x  1) ( x  1)( x  1) ( x  1)( x  1) 2( x  1)( x  1)  3( x  1)( x  1)  4( x  1)( x  1)  ( x  1)( x  1)( x  1)( x  1)

Denominators fully factorised.



2 x 2  4 x  2  3x 2  3  4 x 2  8 x  4 ( x  1) 2 ( x  1) 2



3x 2  4 x  5 ( x  1) 2 ( x  1) 2

LCD = product of the highest power of each type of factor Expanded form of previous line Final answer.

Examples involving the change of signs.

2 3 4   (b  1) (1  b) (b  1)(1  b) 2 3 4    (b  1) (b  1) (b  1)(b  1) 2(b  1)  3(b  1)  4  (b  1)(b  1) 2b  2  3b  3  4  (b  1)(b  1) b5  (b  1)(b  1)

3 2 1 4    2 2 2 x 1 ( x  1) 1 x (1  x) 3 2 1 4   2   2 2 x 1 ( x  1) x  1 ( x  1) 3 2 1 4     2 2 ( x  1)( x  1) ( x  1) ( x  1) ( x  1) 

3( x  1)  2( x  1)  ( x  1)  4( x  1)( x  1) ( x  1)( x  1)( x  1)



3x  3  2 x  2  x  1  4 x 2  8 x  4 ( x  1)( x  1)( x  1)



 4 x 2  10x  2 ( x  1)( x  1)( x  1)

Use a negative to reverse the order of the denominator (1 – b) So just change the middle sign (1 + b) is the same as (b + 1) so no sign change needed.

Sign change to reverse term (1- x2)

Use a double sign change to change both brackets sign stays a +

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 5.1: 1

6 x. 12

4x 2 y 3 8x 3 y

2x  4 4

xy  y y __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

8x 2  4 x 4x __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

x2 1 ( x  1) 2

x 2  x  12 x 2  7 x  12 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

a b ba

Grade 10 Core Mathematics

GOMATH WORKBOOKS

a2 a a2 2

10.

x 2  x  12 4 x __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

11.

ab  a 2 b 2  ab X b2  a2 a2 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

12.

x 2  x  6 x3  2x 2 1  2  2 3x  12x x  16 x  4 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2x 2  8 4x  2x 2 4x  3  2 2 x 8 x  2x  4 13. x  4 x  4 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ . 14.

x x2  2 x  y y  x2

7x 3x  2 x  2 y 5 y  5x

15. __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

16.

GOMATH WORKBOOKS

 5x 3x 2   2 x 1 x  2x  1 1  x 2

6. EQUATIONS: 6.1

SOLUTION OF LINEAR EQUATIONS:

The method used to solve linear equations is as follows: Remove all variables to the same side of the equal sign and all constants to the opposite side. Solve for the variable concerned

NB: these are equations thus what you do to one side you must do to the other to keep the balance. Examples: 2 x  4  10

2x  6 x3 4 x  5  10  x

4 x  x  10  5 5 x  15 x3 5( x  2)  2( x  1)

5 x  10  2 x  2 3 x  12 x  4

Remove the +4 to the RHS by subtracting 4 from both sides.

Remove the -x to the LHS by adding +x to both sides and the -5 from LHS by adding +5 to both sides4 from both sides. Distribute to remove the brackets. Get the x’s to one side the contstants to the other side and solve for x

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 6.1: 1.

2x  14  5  x

x  5  2x  3

5  3x  6  4x  5  2x

3  2x  6x  1

3( x  2)  2( x  1) ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

1  8( x  5)  2( x  3)  10  4(2 x  3) ____________________________________________________________________

4(2 x  7)  3(2 x  4)  8(5  x)  5( x  7) ____________________________________________________________________

6.2

Linear Equations with Fractions:

Method: Multiply each term by the LCD( lowest common denominator) to convert the question to whole numbers and then solve in the normal way. Examples: 1.

x2 x2 3   3 4 2

LCD= 12

( x  2) ( x  2) 3   3 4 2 4( x  2)  3( x  2)  18

 ( x  2) 12   ( x  2) 12  3 12       1  3 1 2 1  3

4 x  8  3 x  6  18 x  20 2.

3 2  3 x   2 1  7 x  LCD = 20 4 5

152  3 x   81  7 x  30  45 x  8  56 x  45 x  56 x  8  30 11x  22 x  2

 3 20   2 20    2  3 x     1  7 x  4 1  5 1 

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 6.2.1: 1.

y y   1. 2 4

x x  3 5 10

x x 1   4 3 2

x 1 x 3    2 4 4 2

Grade 10 Core Mathematics

GOMATH WORKBOOKS

3x 2 x 1   0 5 5 10

2x  7 x  5  0 6 3

x 1 x 1  40 4 5

Grade 10 Core Mathematics

GOMATH WORKBOOKS

7x  2 9x  2  2 3 5

3x  4 2x  3  1 2 4

10.

3 x  5 2( x  5)  2 3

Grade 10 Core Mathematics

11.

GOMATH WORKBOOKS

x 1 x  2 x 1 x  2    3 4 2 3

12.

1 3  1 2 x  x  2 10  3  3

1 3  LCD = 2x x 2

2  3x x

2 3

5 7  x x4

LCD = x(x-4)

5( x  4)  7 x 5 x  20  7 x 2 x  20 x  10

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 6.2.2: 1.

4 1  0 3 3x

2 4 2   0 x 3 3x

3 1 9 2  x 2 2x

Grade 10 Core Mathematics

3

GOMATH WORKBOOKS

2 3  4 x 2x

1 1 2 x   2 3x 6x

5 2 x3   4 4 3x 12 x

x 3 2 x3 x3

Grade 10 Core Mathematics

GOMATH WORKBOOKS

6.3 Quadratic Equations: 6.3.1 Solve by factorizing: Method: 1. Equate to zero, factorise and solve: x2  3x  4 2 e.g. . x  3 x  4  0 ( x  4)( x  1)  0

x  4 or x  -1 2. If already factorized simply solve the equation e.g.

( x  4)( x  1)  0 x  4 or x  -1

3. If not in factorized form do the necessary steps to get the equation into factorized form before solving. x ( x  4)  12

x 2  4 x  12  0 e.g.

( x  6)( x  2)  0 x  6 or

x2 Exercise 6.3.1: Solve the following equations:

1. ( x  5)( x  2)  0 __________________________________________________________________ __________________________________________________________________

2. (a  6)(a  1)  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

3. x( x  1)  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

4. ( x  2)( x  3)( x  5)  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

5. x(2 x  5)(3x  2)  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ 6. y 2  3 y  10  0

7. x 2  5 x  6  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

8. x 2  7 x  6  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

9. x( x  1)  6 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

10. ( x  3)( x  2)  12 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

11. x 2 ď&#x20AC;Ť 2 x ď&#x20AC; 3 ď&#x20AC;˝ 12 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ _________________________________________________________________

12. x( x ď&#x20AC; 16) ď&#x20AC;˝ 3(24 ď&#x20AC; 5 x) __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ _________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

13.

(2đ?&#x2018;Ľ + 1)2 â&#x2C6;&#x2019; 4 = 0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ _________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

14.

đ?&#x2018;Ľ(2đ?&#x2018;Ľ + 1) = 6 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ _________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

6.3.2

GOMATH WORKBOOKS

SOLVING QUADRATIC EQUATIONS USING A FORMULA: Extension Work (Optional)

The formula is derived by completing the square with the general quadratic equation: ax 2  bx  c  0  b  b 2  4ac General Quadratic Formula: x  : 2a

NB: a ; b ; & c are constant values WHERE: a = coefficient of x2 ; b = coefficient of x ; c is the constant in equations written in the form: ax 2  bx  c  0 NO x –values must be substituted into the formula. i.e. only the constant values are used. Examples: Solving quadratic equations using the general quadratic formula. Solve: 1. x 2  x  12  0 x

 b  b 2  4ac 2a

 1  1  4(1)(12 2  1  1  48 x 2  1  49 x 2 1 7 x 2 8 6 x OR 2 2 x  4 OR 3 x

2. 2x2  7x  6  0  b  b 2  4ac x 2a x

7  49  4( 2)(6) 2( 2)

7 1 4 8 6 x  or 4 4 3 x  2or 2 x

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 6.3.2: Solve using x 

 b  b 2  4ac ; 2a

(Answers rounded to 2 decimal places where necessary) NB: First expand if necessary and equate to zero before using the formula. 1.

x2  4x  3  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

2 x 2  x  10 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

3x2  x  2  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

x2  6x  4  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

2x2  4  7x __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

2 x ( x  3)  3  0 __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________

Grade 10 Core Mathematics

6.4 A.

GOMATH WORKBOOKS

Simultaneous Equations: 2 linear equations:

Method: 1. 2. 3. 4.

Number the equations as 1 and 2. Rewrite one of them in terms of x or y . ( try not to have fractions involved) Substitute the new x or y value into the other equation and solve. Substitute this value into the changed equation to get the value of the other variable.

Example: Solve for x and y simultaneously if x  2 y  5 and x  y  1 Change x  y  1 Number (1) / x  2 y  5 (2) the the x  y 1 equation equations to the x 1&2 formSubstitute 1 into 2 y 1 2y  5 Change one of the Substitute the value for equations to the x – 3y  6 x in the changed form or y - form y2 equation for y in the other equation x = 1 {from x  2  1} Substitute the value from above into the changed equation and solve for the second variable Exercise 6.4: Solve the following equations simultaneously.

x y 5 1. and x  y  3 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

x  3 y  5 and 2 x  3 y  1 2. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

x y 8 3. and 3 x  2 y  21 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3x  2 y  60 and 3 x  3 y  45 4. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

x  y  36 5. and x  2 y  12 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

x  2y  5 6. and 3x  y  1 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

6 x  y  22 and 4 x  y  8 7. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2y  x  3 8. and 4 x  3 y  10 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3x  y  5  0 and 7 x  3 y  1  0 9. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

10.  m  1 2k and 2k  m  3 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2 x  3 y  14 and x  5 y  0 11. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3 x  4 y  24 and 7 x  4 y  16 12. ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3x  y  2 13. and 6 x  y  25 ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2x  9  y 14. and x  36  4 y ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

Level of difficulty increased: Example: Find x and y if y 

x  11 4 -----(1) x y  11  4 Substitute 1 into 2:

x x  11 and 3 y   14 4 5

y

3y 

x  14 -----(2) 5

x x 3(11  )   14 4 5 3x x 33    14 4 5 LCD  20 660  15x  4 x  280 19x  380 x  20 y  11 

20 4

y  16

x y   9 and 2 3

x y 1   0 3 2 2

LCD = 6 LCD = 6 3 x  2 y  54 3 x  2 y  54 3 x  2 y  54 --------(1) and 2 x  3 y  3  0 -------(2) 2 x   y  18 3 Substitute 1 into 2: 2 y  18)  3 y  3  0 3 4y   36  3 y  3  0 3 LCD  3 2( 

 4 y  108  9 y  9  0  13 y  117

y9 x  12

GOMATH WORKBOOKS

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise continued: 15.

x y  5 3

and

x y  1 4 2

16.

x8  2 y  8 and 3

x y x y  2 2 3

Grade 10 Core Mathematics

6.5

GOMATH WORKBOOKS

PROBLEMS INVOLVING SIMULTANEOUS EQUATIONS:

Example: 1. The sum of two numbers is 4 and their difference is 6. 1.1 Let their numbers be x and y . Write down two equations in x and y. 1.2 Solve the equations and find the two numbers. Let one number be x and the other y x  y  4 and x  y  6

x y  4

----(1) and y  4 x Substitute 1 into 2

x  y  6 -------(2)

x  (4  x)  6 x4 x  6 2 x  10 x5 y  1

Exercise 6.5: 1.

The sum of two numbers is 54 and their difference is 6. Find the numbers.

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

The sum of two numbers is 35 and their difference is 19. Find the numbers

In a two digit number, the sum of the digits is 12 and their difference is 4. Find the number if the tens digit is larger than the units digit.

Grade 10 Core Mathematics

GOMATH WORKBOOKS

The length of a rectangle is twice the breadth, while the perimeter is 6m. Find the length and breadth of the rectangle. { Hint: P  2(l  b) .}

The perimeter of a rectangular flower bed is 26m. If the length exceeds the breadth by 3m, find its dimensions.

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

A number consisting of two digits has the following properties. When the number is added to twice the tens digit the answer is 33. If the digits are reversed, the number obtained exceeds the original number by 63. What is the original number?

A boy is 6 years older than his sister. In three years time he will be twice her age. What are their present ages?

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

Tumi is twice as old as John. Two years ago she was three times as old as John was then. What are their present ages?

____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

6.6

GOMATH WORKBOOKS

LINEAR INEQUALITIES :

Linear inequalities are solved in the same manner that linear equalities are . Examples: 1.

Solve the following inequality and illustrate the answer on a number line.

2x  4  6

2 x  10

Solution:

 5 2.

2x  4  6 x5

Solve the following inequality and represent the answer on a number line:

 4  2x  2  8

 4  2x  2  8  2  2 x  10

 -1

1 x  5

 5

Exercise 6.6.1: Solve the following inequalities and illustrate the answers on a number line: 1. 2x  6  8 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3x  6  x  14

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2x  7  5x  14

3x  7  3 x  14 2

 6  2x  14

Grade 10 Core Mathematics

GOMATH WORKBOOKS

 5  2x  3  15

Exercise 6.6.2: 1. Find the possible solutions of the following linear inequalities and illustrate the answers on a number line . 1.1

3x  2  x  14

5x  3  3x  15

Grade 10 Core Mathematics

1.3

GOMATH WORKBOOKS

1  2x  x  2

4( x  4)  7( x  2)  1 1.4 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4(2 x  1)  5 x  2 1.5 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2( x  1)  2(2 x  1)  2  x 1.6 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

 2( x  3)  5 x  78 1.7 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

1.8

3( x  2) 7( x  3)  3 2 4

1.9

x2 7 3 x  3 8 2

Grade 10 Core Mathematics

1.10

GOMATH WORKBOOKS

4  x 2x  1   x2 2 3

1.11

3 x  1 3 x  3 19   4 8 8

1.12

x

5  3 x  3( x  5) 2

Grade 10 Core Mathematics

1.13

3

GOMATH WORKBOOKS

x  2 1  3x  2 5 3

___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 1.14  1  x  3  5 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 1.15 1  1  2x  7 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

1.16

3

x3 4 2

Grade 10 Core Mathematics

1.17

2

GOMATH WORKBOOKS

2x  3  4 3

1.18

2

2  2x 3 4

LOGARITHMS:

Calculations using basic logarithms : Calculator work: NB a logarithm is an exponent: 3 2  9 In logarithm form this is written as log3 9  2 and is read : the logarithm of 9 to the base 3 is 2 ( i.e. the exponent of the base 3 is 2 to give the number 9)

Example: Solve for x in the following: : x 

log 5 log 2

log 5 log 2 0,6989700043 Answer: x  Answer to 2 decimal digits. 0,3010299957 x  2,32 (Simply enter values as given into your calculator using the fraction and log facility) x

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Exercise 7.1: Solve for x giving answers to 2 decimal digits.

log1000 . log100 ___________________________________________________________________ 1.

x

log12 log 4 ___________________________________________________________________ 2.

x

log 300 log 200 ___________________________________________________________________ 3.

x

Grade 10 Core Mathematics

x

GOMATH WORKBOOKS

log15,5 log 3

x

2 log 25 3 log15

x

5 log 50 2 log 8  2 log 25 6 log 6

Grade 10 Core Mathematics

x

GOMATH WORKBOOKS

10 log 450 log 500  12 log 60 6 log 65

Exponents:

N.B. The basic laws must always be applied. 1.

Law: a m .a n  a m n

When multiplying like bases you must add the exponents. Do not multiply the bases except in the following case: 2 3.33  8.27  216. e.g. BUT 2 3.33  6 3  216

am 2. Law: n = a m  n a When dividing the bases subtract the exponents: the following exception. 3

63  6      33  27 3 2 2

3. Law: (a m ) n  a mn When raising a power to a power you must multiply the exponents. 4. 5.

Law: (a m ) 0  1 Any value raised to the power of “zero” will equal 1.

N.B. 1. Never multiply the bases 2. Never divide the bases 3. Never multiply a base by an exponent. 4. If bases are separated by plus or minus signs you MUST FACTORISE before simplifying.

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

Prime Base Factorising It is important to remember to use prime bases in simplifying with exponents ( especially if no calculators are allowed. NB when reducing bases by factorizing, always use the lowest possible bases. e.g for 16 rewrite as 2 4 and not as 4 2 . Use prime numbers for bases and not composite ones. Once you have factorised using prima bases then the normal laws apply. Example 1:

ď&#x20AC;¨ ď&#x20AC;Š

4 3

Simplify 125 ď&#x20AC;˝ 53 3 (Now use law 3 and raise a power to a power by multiplying the exponents:)

1253 ď&#x20AC;˝ ď&#x20AC;¨53 ď&#x20AC;Š3 ď&#x20AC;˝ 5 1 4

3 4 ď&#x201A;´ 3

ď&#x20AC;˝ 5 4 ď&#x20AC;˝ 625

Example 2: Simplify without the use of a calculator. 5 a ď&#x20AC;Ť 3.5 2 a ď&#x20AC;1 Use law 1 25a ď&#x20AC;1 in the Answer:

5 a ď&#x20AC;Ť 3.5 2 a ď&#x20AC;1 5 3a ď&#x20AC;Ť 2 ď&#x20AC;˝ 2aď&#x20AC; 2 ď&#x20AC;˝ 5 aď&#x20AC;Ť 4 2 a ď&#x20AC;1 (5 ) 5

Prime base factorize and use law 3 to simplify

numerator

Use law 2 to get the answer.

Follow the procedures as set out for any of the following types of simplifications with exponents. Exercise 8.1: 1.

đ?&#x2018;&#x17D;3 Ă&#x2014; đ?&#x2018;&#x17D;2

___________________________________________________________________ ___________________________________________________________________ 2.

đ?&#x2018;?2 Ă&#x2014; đ?&#x2018;?4

___________________________________________________________________ ___________________________________________________________________ 3.

đ?&#x2018;&#x17D;3 đ?&#x2018;? 2 Ă&#x2014; đ?&#x2018;&#x17D;2 đ?&#x2018;? 4

___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

2đ?&#x2018;&#x17D;đ?&#x2018;? 2 Ă&#x2014; 3đ?&#x2018;&#x17D;4 đ?&#x2018;? 6

____________________________________________________________________ ____________________________________________________________________ 5.

2đ?&#x2018;&#x17D;(3đ?&#x2018;&#x17D; + 2đ?&#x2018;?)

____________________________________________________________________ ____________________________________________________________________ 6.

2đ?&#x2018;&#x17D;2 đ?&#x2018;?(3đ?&#x2018;&#x17D;đ?&#x2018;? 3 â&#x2C6;&#x2019; 2đ?&#x2018;&#x17D;4 đ?&#x2018;?)

____________________________________________________________________ ____________________________________________________________________ 7.

3(2đ?&#x2018;&#x17D;2 đ?&#x2018;? 3 )3

____________________________________________________________________ ____________________________________________________________________ 8.

3đ?&#x2018;&#x17D;0 (2đ?&#x2018;&#x17D;2 đ?&#x2018;? 3 )0

___________________________________________________________________ ___________________________________________________________________ NB. Prime base factorizing first then apply the laws.

2 xď&#x20AC;Ť3 2 x 2 x ď&#x20AC;1

Grade 10 Core Mathematics

10.

GOMATH WORKBOOKS

2 x 1 2 x  3 2 x  2 .2 x

4 x 1 8 x  1 32 x 1 ___________________________________________________________________ 11.

5 x 25 x 1 12. 5.125 x ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

13.

GOMATH WORKBOOKS

7 x  2 49 x  2 7 3 x2

6 n1 .12 n1 .2 n 18 n 2 .8 n1

15.

6 n 212 2 n1 4 2 n 3 8 3 n1 9 n1 3 n

Grade 10 Core Mathematics

GOMATH WORKBOOKS

If the bases are separated with plus or minus signs one must FACTORISE FIRST.. It is easier to split the bases as shown because it is easier to see the HCF. Factorise and simplify. NB the base with the variable exponent should always cancel leaving pure numerical values. Exercise 8.2:

2 x3  2 x 2 x 1

2 x 1  2 x  3 2 x2  2 x

Grade 10 Core Mathematics

GOMATH WORKBOOKS

3 x  3 x2 3 x 1  3 x

5 2  x  4 .5  x 5  x  2 .5  x  1

3 n .3 4  6.3 n .3 1 7.3 n .3 2

Grade 10 Core Mathematics

GOMATH WORKBOOKS

2 n .2 5  3.2 n .2 2 5.2 n 2 3

Variable in the exponent: Method: 1.1 Equate the bases using prime base factorizing. 1.2 If the bases are now equal then the exponents are also equal, thus simply equate the exponents and simplify further.

2 2 x  16 Example 1.

22x  24

Rename 16 to the base 2 and equate the bases.

2x  4 x2 2.3 x ( x  3)  54 3 x ( x  3)  27

First divide by the coefficient ‘2’

3 x ( x  2)  33

Example 2:

x 2  2x  3  0 ( x  3)( x  1)  0

Rename 27 to the base 3 and equate the bases.

x3 or x  1

Equate the exponents and solve for x.

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

Variable in the base: Method: 2.1 Raise the power of the exponent of the variable base to its multiplicative inverse. This will give a new exponent with the value of 1. As it is an equation you must do the same to both sides of the equal sign. Example. 2 3

x ď&#x20AC;˝ 16 ď&#x192;Ś ď&#x192;§x ď&#x192;§ ď&#x192;¨

2 3

3 2

ď&#x192;ś ď&#x192;ˇ ď&#x20AC;˝ 24 ď&#x192;ˇ ď&#x192;¸

x ď&#x20AC;˝ ď&#x20AC;¨2 4 ď&#x20AC;Š 2 3

x ď&#x20AC;˝ 26 x ď&#x20AC;˝ 64

NB. You must first remove any coefficient values prior to solving for the variable:

More advanced equations: Extension Work Variables separated by + or â&#x20AC;&#x201C; signs: This involves factorizing. 2 x ď&#x20AC;Ť 3.2 x ď&#x20AC;˝ 16

2 x (1 ď&#x20AC;Ť 3) ď&#x20AC;˝ 16

1.1

2x ď&#x20AC;˝ 4

Use 2 x as the HCF

2 x ď&#x20AC;˝ 22 xď&#x20AC;˝2

Equate the bases thus and then the exponents.

NB Clues to let you know when a simple HCF must be used are: 1. Terms separated by plus and minus signs and 2. the variable exponents have the same value. i.e the coefficients of the variables exponents are equal in value. 2.

Variables separated by + or â&#x20AC;&#x201C; signs BUT the values of the coefficients of the variables exponents are not equal. i.e. one is double the other. This involves a trinomial and needs to be factorised accordingly. 2.1 22đ?&#x2018;Ľ â&#x2C6;&#x2019; 6. 2đ?&#x2018;Ľ + 8 = 0 (2đ?&#x2018;Ľ â&#x2C6;&#x2019; 4)(2đ?&#x2018;Ľ â&#x2C6;&#x2019; 4) = 0 The exponent of the 1st term is double that of the 2nd . i.e The 2x ď&#x20AC;˝ 4 2x ď&#x20AC; 2 ď&#x20AC;˝ 0 expression is a 2 x ď&#x20AC;˝ 2 2 or 2 x ď&#x20AC;˝ 2 trinomial. xď&#x20AC;˝2 x ď&#x20AC;˝1 The variable base is 2 x

Grade 10 Core Mathematics

84 2x  4

GOMATH WORKBOOKS

2x  2  0

2 x  2 2 or 2 x  2 x2 x 1

2.2

2.3 2 x  12.3 x  54  0 (2.3 x  6)(3 x  9)  0

2 .3 x  6  0 3x  3

3x  9

3 x  32 x2

x 1

The exponent of the 1st term is double that of the 2nd . i.e The expression is a trinomial. The variable base is 3 x

NB a substitution method can be utilized here: Let 3 x = k [ thus 3 2 x  k 2 ] 3 x 2k 2  12k  54  0 2(k 2  6k  27  0 (k  3)(k  9)  0 k  3 or k  9 3x  9 3x  3 x 1

Substitute 3 x for k at this point and solve for x

or 3 x  3 2 x2

Exercise 8.3: 1. Solve for x without the use of a calculator:

1.1

2x = 8

Grade 10 Core Mathematics

1.2

GOMATH WORKBOOKS

3 x  81

1.3

x3 = 27

1.4

4

= 80

Grade 10 Core Mathematics

1.5

GOMATH WORKBOOKS

2 x  16 x 1

1.6

3.2x = 48

1.7

5.4 2 x  40

Grade 10 Core Mathematics

GOяВ╖MATH WORKBOOKS

1.8 2 x яАл 2 x яАл3 яА╜ 18 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 1.9 2ЁЭСе+1 тИТ 2ЁЭСе = 4 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

1.10

2ЁЭСе + 2ЁЭСетИТ2 = 5

Grade 10 Core Mathematics

1.11

GOď&#x201A;ˇMATH WORKBOOKS

22đ?&#x2018;Ľ+1 â&#x2C6;&#x2019; 22đ?&#x2018;Ľ = 2

1.12

3 2 x ď&#x20AC; 5.3 x ď&#x20AC;Ť 6 ď&#x20AC;˝ 0

1.13

22đ?&#x2018;Ľ â&#x2C6;&#x2019; 2. 2đ?&#x2018;Ľ + 1 = 0

Grade 10 Core Mathematics

1.14

GOď&#x201A;ˇMATH WORKBOOKS

4.22đ?&#x2018;Ľ â&#x2C6;&#x2019; 5. 2đ?&#x2018;Ľ + 1 = 0

1.15

9. 32đ?&#x2018;Ľ â&#x2C6;&#x2019; 10. 3đ?&#x2018;Ľ + 1 = 0

1.16 đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 2đ?&#x2018;Ľ 3 + 1 = 0 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics 1

GOď&#x201A;ˇMATH WORKBOOKS

1.17 đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 5đ?&#x2018;Ľ 4 + 6 = 0 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

1.18 đ?&#x2018;Ľ 5 + 4đ?&#x2018;Ľ 5 â&#x2C6;&#x2019; 12 = 0 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

Number Patterns:

Number patterns are limited to a first order difference and are basically linear in character. General Formula is: Tn  an  c where d1  a  T2  T1  common difference Tn  an  c General formula for any sequence. To calculate the components of a sequence and its nth formula: 1.

Given a sequence as follows: 4;7;10.....

Tn  an  c

T1  4

d1  a  3

T2  4  3  7

T3  10  3(3)  c c 1

T3  4  3  3  10

Nth formula for the sequence is Tn  3n  1 General formula for the specific sequence: 4;7;10..... Tn  3n  1 th

The 10 term of the sequence is:

T10  3(10)  1 T10  31

Given the 1st term and the common difference: T1  5 and d1  a  6 T1  5 T2  5  6  11 T3  5  6  6  17 T3  17  6(3)  c c  1 Tn  6n  1 Tn  6n  1

The 10 term of the sequence is: T10  6(10)  1 th

T10  59

Given the sequence below , Find an expression for the general term. 1;6;11;16;21........

d1  a  6  1  5 T4  16  5( 4)  c c  16  20  4 Tn  5n  4

The General term is Tn  5n  4

Grade 10 Core Mathematics

GOMATH WORKBOOKS

In the sequence below , Find the common difference and the general (nth) term. 4; 9; 14 ; 19…… The common difference is d1  T2  T1  9  4  5 Next 3 terms are 24 ; 29 ; 34. Tn  an  c Tn  an  c T1  4 T3  5(3)  c

d1  a  5

T2  4  5  9 T3  4  5  5  14

14  15  c c  1

Tn  5n  1

Exercise 9.1: 1. Mina invests R8000,00 in a plan where the growth is as follows: 0 1 2 3 4 5 8000 8440 8880 9320

No of yrs passed :n Value of investment (in R) :A

1. Find the rule for this investment plan. 1.1 Use the rule to find the values for the next 3 years. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 1.2

Use the rule to find the value after 23 years.

Grade 10 Core Mathematics

1.3

GOMATH WORKBOOKS

If the value after n years was R13 280 , what is the value of n?

A class is given the sequence 3; 5; 7;….. to continue for 4 more terms. 2.1 Explain in a sentence how you would complete it.

Find a rule in the form of Tn  an  c for the given sequence.

Grade 10 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

Find the 20th term using the formula in 2.2

2.3

3. 3.1

Find the 50th term

Grade 10 Core Mathematics

GOMATH WORKBOOKS

4. Find a rule or general term for each of the following sequences and then find the 10th term for each one. 4.1 1; 6; 11; 16; 21;…. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.2 13; 23; 33; 43;…. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.3 6; 9; 12; 15;…. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.4 22; 20; 18; 16;…. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

4.5 7; 2; -3 ;-8;….. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.6 7; 11; 15; 19;…. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5. Use the general terms below to find the first 5 terms of each sequence and then find the 20th term for each one. 5.1 Tn = n – 5 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5.2

Tn = 3n – 5

Grade 10 Core Mathematics

5.3

GOMATH WORKBOOKS

Tn = -2n – 5

Tn = 4n + 2

Tn = 2 – 3n

The following sequences are given: A. For each one write down the next 3 terms B. The nth term ( i.e. Tn = an + c) C. The 50th term. 6.1

3; 9; 15; 21;…..

Grade 10 Core Mathematics

6.2

GOMATH WORKBOOKS

2; 9; 16 ; 23; …..

6.3

2; -3; -8; -13;….

3; -1; -5; -9;…

Grade 10 Core Mathematics

GOMATH WORKBOOKS

20; 27; 34; 41;…….

6.5



Find the rule for the nth term Tn



Find the value of the 50th term.



Graph the relationship.

Grade 10 Core Mathematics

GOMATH WORKBOOKS

100

2. 



 



 



 



     

 

 





       ___________________________________________________________________ 



Grade 10 Core Mathematics

101

GOď&#x201A;ˇMATH WORKBOOKS

4. From the diagram below: 4.1 4.2 4.3

fig1

Determine the next 3 terms. Determine the nth term formula Determine the 20th term.

Fig 2

Fig 3

Fig 4

From the diagram below: 5.1 5.2 5.3

Determine the next 3 terms. Determine the nth term formula Determine the 20th term.

Grade 10 Core Mathematics

102

GOď&#x201A;ˇMATH WORKBOOKS

From the diadram below: 6.1 6.2 6.3

Determine the next 3 terms. Determine the nth term formula Determine the 20th term.

Grade 10 Core Mathematics

10.

GOMATH WORKBOOKS

103

STRAIGHT LINE GRAPHS:

General Equation : y  mx  c Written in this form : y  mx  c

y - intercept

Gradient The Table Method: Draw a table using x-values of -2 ; 0;& +2. Change the equation to the y-form and by substitution calculate the corresponding y-values. Example: Sketch the graph of

X Y

-2 -1

0 3

2 5

Plot the above points on a Cartesian plane. Lable and name the graph. 4

fx = 2x+3

-5

-2

Exercise 10.1: Change the following equations to the y-form. 1.1

Grade 10 Core Mathematics

104

GOMATH WORKBOOKS

1.3 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.

Sketch the following linear functions using the table method:

y  x3 2.1 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

105

GOMATH WORKBOOKS

y  2x  4 22. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2 y  4x  8 2.3 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

106

GOMATH WORKBOOKS

3y  6x  3 2.4 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2y  x  6 2.5 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

107

GOMATH WORKBOOKS

4 y  8x  8  0 2.6 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

108

Method 2 : Dual Intercept Method. Sketching linear graphs ( straight line) the best method to us is the Dual Intercept method. This method finds the values where the line cuts ( intercepts) the x- axis and y-axis. NB The only time that this method is not used is sketching any line passing through the origin. Use substitution to find another point or use the gradient. METHOD: To find the x – intercept substitute zero for x – intercept substitute zero for y and To find the y – intercept substitute zero for x. Example: Sketch the graph of 2 y  3x  6 using the dual intercept method.

2 y  3(0)  6 Let x = 0  2 y  6

y3

2(0)  3x  6 and Let y = 0 3x  6

x y 2

2y + 3x = 6

Grade 10 Core Mathematics

109

GOMATH WORKBOOKS

Exercise 10.2: Use the” dual intercept method” to sketch the following:

y  x3 1. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

y  2x  4 2. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

110

GOMATH WORKBOOKS

2 y  4x  8 3 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3y  6x  3 4. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

111

GOMATH WORKBOOKS

2y  x  6 5. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 6.

4 y  8x  8  0

Grade 10 Core Mathematics

GOMATH WORKBOOKS

112

Gradients of straight lines: 1. 2.

In the y- form of the equation simply use the m- value( co-efficient of x) as this represents the gradient . Given at least 2 points that lie on a given line, then calculate the gradient y  y1 y y1  y 2  or 2 using: m  x x1  x 2 x 2  x1 i.e the difference between the y – values divided by the difference between the x – values.

Parallel lines have the same gradients i.e. their m – values are the same i.e y  2 x  6 . Is parallel to y  2 x  10

Perpendicular lines have inverse gradients. i.e. m1  m2  1

1 2

y  2 x  c is perpendicular to y   x  c

m1  m2  1 2 1    1 1 2 Domain & Range Domain represents all the x – values Range represents all the y – values. The domain is also referred to as the independent value and is made up of all the numbers in the number system. The range is referred to as the dependent value because you can only find why if you know x. The x and y values together form a ordered pair and represent a point on a Cartesian plane. x is always written first.

Grade 10 Core Mathematics

113

GOď&#x201A;ˇMATH WORKBOOKS

Functions and Relations: A function occurs when the x-value is not repeated i.e there is one x-value for one y-value. e.g. 1. (3 ; 4) ; (4 ;5) ; (5;6) 2. (3;4) ; (4;4) ; 5;4) [the y-value can be repeated in a function. A non- function occurs when the x-value is repeated. i.e more than one x-value for a y-value. e.g. 1. (2;3) ; (2;4); (3;5) 2. (3;4) ; (3;5); (3;6) Graph representations of functions and non- functions. Graphs of Functions:

Graphs of Non- Functions:

Grade 10 Core Mathematics

GOMATH WORKBOOKS

114

A pencil test can be used to see if a particular graph is function or a non- function. i.e. A vertical line test: 1. If the line cuts the graph at only once then the graph represents a function. 2. If it cuts more than once then the graph represents a non-function. The reason however is due to the fact that the x-value is not repeated for a function and is repeated for a non-function .

Domain and Range of graphs: It is easy to read the domain and range from graph representations.

(-2;4)

Domain: x  R Range: y  R

Domain: { x  2 ; x  R } Range: { y  4 ; y  R }

(4 ; 6)

Domain: { x  4 ; x  R }

{Domain: x  R }

Range: { y  6 ; y  R }

Range: { y  8 ; y  R }

-6

Domain:{  6  x  6 ; x  R .} Range: {  6  y  6 ; y  R }

-6

Grade 10 Core Mathematics

11.

GOMATH WORKBOOKS

115

Parabola Graphs:

General Equation: y  ax 2  c OR y  ax 2  q Two methods can be utilized to sketch a parabola graph: 1. Table method. 2. A ‘5 point’ method. Method: Table: A. Sketch the graph of y  x 2  4 X Y

-3 5

-2 0

-1 -3

0 -4

1 -3

2 0

3 5

y 4

f x  = x 2-4

-10

-5

-2

-1

-2

-3 -4

Grade 10 Core Mathematics

116

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Exercise 11.1: Sketch the following graphs using a table method: Use x –values of -3 to 3 for each table.( as in the example above) 1. y  x2 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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2. y  x2  1 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

118

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3. y  x2  2 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

119

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4. y  x2  1 ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

120

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y  x2  2

___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

121

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Exercise 11.2: A. Sketch the following graphs on the same set of axes: y  x2 1. y  x2  1 2. y  x2  1 3. What deduction can be made? ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

122

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B. Sketch the following graphs on the same set of axes: 1. y  x2 2.

y  2x 2

y

1 2 x 2

What deduction can be made? ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

123

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C. Sketch the graphs of the following equations: 1. y  x2 2.

y  x2

Grade 10 Core Mathematics

124

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D. Sketch the graphs of the following on the same set of axes: y   x  1 and y  x  2 1. 2. Write down the coordinates of the point(s) of intersection for the two graphs. ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

125

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E. Sketch the graphs of the following on the same set of axes: 1. y   x 2 and y  x  2 2. Write down the coordinates of the point(s) of intersection for the two ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

126

Sketch of a parabola using a 5 point method. 1.

x –intercepts at y = 0 :

y- intercept at x = 0

Turning point: for y  ax 2  c the y- intercept and the turning point are the same.

Get two further points using substitution:

Example: A. Sketch the graph of y  x 2  4

Method: 2

x – intercepts at y = 0: x2  4  0 ( x  2)( x  2)  0 x = 2 or x = -2

y – intercept at x = 0: y in = -4

Turning point: (0; -4)

Substitution:

qx  = x 2-4

-5

-2

-4

At x = 1 thus y = (1)2 – 4 = -3 plot point ( 1; -3) At x = -1 thus y = (-1)2 –4 = -3 plot point ( -1; -3) 8

B. Sketch the graph of y  2 x 2  18

x – intercepts at y = 0: 2 x 2  18  0 2( x 2  9)  0 2( x  3)( x  3)  0 x = 3 or x = -3

-4

-2

-4

-6

-8

-10

y – intercept at x = 0. y-int = -18

-12

-14

-16

Turning point : ( 0; -18)

-18

-20

-2

By substitution: at x = 2 y = 2(2)2 – 18 =8 – 18 =- 10. plot point (2; -10) 2 at x = -2 y = 2(-2) –18 = 8 – 18 = -10 plot point ( -2 ; -10)

qx  = 2x 2-18

Grade 10 Core Mathematics

GOMATH WORKBOOKS

127

Exercise 11.3: Use a point by point method to sketch: y  x2  9

-14

-12

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

y  x 2  9

Grade 10 Core Mathematics

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128

-12

-10

-8

-6

-4

-2

-1

-2

-3

-4

y  2x  8 1.3 ___________________________________________________________________ 2

-12

-10

-8

-6

-4

-2

2 -1

-2

-3

-4

-5

-6

-7

Grade 10 Core Mathematics

GOMATH WORKBOOKS

129 y  3x 2  12

1.4

-12

-10

-8

-6

-4

-2

2 -1

-2

-3

-4

-5

-6

-7

Grade 10 Core Mathematics

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130

Finding Equations of Parabolas:

Given the turning point and any other point Method: Use the general form of the equation : y  ax2  q Example :

-2

2 -8

y  ax 2  q

y  ax 2  8

sub Minimum value of - 8 for q

0  a (2) 2  8

sub point (2;0) for x and y

0  4a  8

solve for a and sub back into equation.

4a  8 a2 y  2x 2  8

Given the two x – intercepts and any other point. Method: Use the general form of the equation: y  a( x  x1 )(x  x2 ) NB x1 and x 2 refer to the roots or x – intercepts Example (1;3) -2

y  a( x  x1 )(x  x2 ) y  a ( x  2)( x  2) sub the x – intercepts changing their signs when you do so. 3  a (1  2)(1  2) sub the other point in for the x and y values 3  a(1)(3) 3  3a solve for a and sub back into equation a  1 y  1( x  2)( x  2) y  x2  4

Grade 10 Core Mathematics

12.

GOMATH WORKBOOKS

131

Hyperbola Graphs:

Sketching: Use a table method (on the calculator or on paper) Example: Sketch the graph of y  y

1. X Y

6 x

-3 -2

-2 -3

-1 -6

1 6

2 3

3 2

f x  =

6 x

-10

-5

-2

-4

-6

-8

  

Asymptotes are are : x  0 and y  0 Domain: x  (;  ) x  0 . y  (;  ) y  0 . Range:

Grade 10 Core Mathematics

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132

Exercise 12.1: Complete the tables and use the values to sketch the graphs of: 1.

y

X Y

-3 2

6 x

-2 3

-1

3 -2

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

y

2. X Y

8 x

-8 -1

-4

-2

-1

8 1

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Grade 10 Core Mathematics

y

X Y

-8 1

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133

8 x

-4

-2

-1

8 -1

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

y

X Y

-4 -1

4 x

-2

-1

1 4

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Grade 10 Core Mathematics

y

X Y

-4

GOMATH WORKBOOKS

134

4 x

-2 2

-1

4 -1

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Shifts with hyperbola graphs: y

k q x

The q- value gives the vertical shift of the graph. It also gives the new horizontal asymptote. Example: y

  

4 2 x

Asymptotes are : x  0 and y  2 Domain: x  (;  ) x  0 . y  (;  ) y  2 . Range:

Grade 10 Core Mathematics

135

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Exercise 12.2: Write down the : Asymptotes, Domain & Range of each of the following: 1. y 

8 3 x

y

10 6 x

9 6 x

Grade 10 Core Mathematics

13.

GOMATH WORKBOOKS

136

Exponential Graphs:

General formula :

a>0,a1,x  y>0

y = ax

A. To sketch the graph y = ax (a) x  and y > 0 y = ax lies above the x – axis in quadrants 1 and 2. b) (i) If a > 1 , then as x increases , y increases. (ii) If 0 < a < 1 , then as x increases , y decreases. (i)

y = 2x [Where ‘a’ is a whole number]

X Y

-2

-1

1 4

1 2

(ii)

y = ( 12 )x. [Where ‘a’ is a fraction]

X Y

-2 4

-1 2

0 1

1 2

e.g. y = 2 x e.g. y = ( 12 )x.

2 4

1 2

1 4

1 x y    or y  0,5 2   These 2 graphs are mirror images of each other, the axis of symmetry being the yaxes (x = 0 ). The x- axes is a vertical asymptote, as y will never equal zero.

y = 0,5x

y = 2x y

1 x

Grade 10 Core Mathematics

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137

Exercise 13.1: Complete the tables and use the values to sketch the graphs of: y  2x -2

1. X Y

-1

0 1

1 4

2 4

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

y  2x  1 -2 -1

2. X Y

5 4

0 2

2 5

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Grade 10 Core Mathematics

y  2x  1 -2 -1

3. X Y



GOMATH WORKBOOKS

138

0 0

3 4

2 3

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

y  2.2 x -2 -1

4. X Y

0 2

1 2

2 8

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Grade 10 Core Mathematics

5. X Y

y  2.2 x -2 -1 

1 2

GOMATH WORKBOOKS

139

0 -2

8

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Functional Notation Graph Interpretation: Exercise 13.2: 1. 1.1

If f ( x)  4 x  3 find: f ( 4)

f (7)

Grade 10 Core Mathematics

140

GOMATH WORKBOOKS

f ( a  b) 1.3 ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2. If g ( x)  3x 2  x  6 . Find: g (2) 2.1 ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

g (3) 2.2 ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

g ( a  b) 2.3 ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 3. Sketch the graph of 2 y  4 x  8 using the dual intercept method. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

GOMATH WORKBOOKS

141 6

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Sketch the graphs of y  x  3 and 2 y  3x  6 on the same system of axes and:

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

Grade 10 Core Mathematics

4.1

GOMATH WORKBOOKS

142

write down the co-ordinates of the point of intersection. _______________________________________________________________

___________________________________________________________________ ____________________________________________________________________

Sketch the graph of y 

4.2

8 8 and y  on the same system of axes. x x

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

4.2.1 Write down the domain and range for the two graphs above. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

4.2.2

GOMATH WORKBOOKS

143

Write down the asymptotes of the two graphs above.

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

4.3.1

Write down the domain and range of the graphs above:

Grade 10 Core Mathematics

GOMATH WORKBOOKS

144

Sketch the graphs of y  2 x and y  2 x  2 on the same system of axes. ____________________________________________________________________ 4.4

-10

-8

-6

-4

-2

-1

-2

-3

-4

-5

4.4.1 Write down the asymptotes of the two graphs ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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145

Write down the gradients; x and y intercepts of the following graphs: 5.1

5.2 8 6 4

-2

5.4

y 8

-3

-2 -9

Grade 10 Core Mathematics

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146

6. Write down the domain and range; the mapping and whether the graph represents a function or a no-function for each of the following: 6.1

6.2

(2 ; -9)

6.3

6.4

-12

-4

Grade 10 Core Mathematics

6.5

GOMATH WORKBOOKS

147

6.6

-7

6.7 O (3; 9)

6.8

• 0

•(-4 ; -4)

-4

• 4

-9

Grade 10 Core Mathematics

GOï&#x201A;·MATH WORKBOOKS

148

7. Find the equations of the following graphs: 7.1

7.2 y (4 ; 2)

-4

7.3 (0;1 0

Grade 10 Core Mathematics

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149

7.4 y (0;9)

(2 ; 5)

0 -2 (2 ; -4)

Grade 10 Core Mathematics

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150

8. đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = 3đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 12 and â&#x201E;&#x17D;(đ?&#x2018;Ľ) = đ?&#x2018;&#x161;đ?&#x2018;Ľ + đ?&#x2018;? are depicted below.

-2

h f -6

-12

8.1 Find the equation of â&#x201E;&#x17D;. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 8.2 Write down the domain and range of đ?&#x2018;&#x201C;. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 8.3 For which value(s) of x is đ?&#x2018;&#x201C; decreasing. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

151

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Use algebraic methods to find the coordinates of đ?&#x2018; , the point of intersection of đ?&#x2018;&#x201C; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; â&#x201E;&#x17D;. ____________________________________________________________________ 8.4

Grade 10 Core Mathematics

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152

Further Graph Interpretation: Find a and c in the following: 1. đ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;Ľ + đ?&#x2018;?

đ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;Ľ + đ?&#x2018;?

đ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;Ľ 2 + đ?&#x2018;?

Grade 10 Core Mathematics

5. đ?&#x2018;Ś = đ?&#x2018;&#x17D; đ?&#x2018;Ľ

GOď&#x201A;ˇMATH WORKBOOKS

153 6.

đ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;Ľ + đ?&#x2018;?

đ?&#x2018;&#x17D; đ?&#x2018;Ľ

đ?&#x2018;&#x17D;

8. đ?&#x2018;Ś = + đ?&#x2018;? đ?&#x2018;Ľ

Grade 10 Core Mathematics đ?&#x2018;&#x17D;

9. đ?&#x2018;Ś = đ?&#x2018;Ľ + đ?&#x2018;?

GOď&#x201A;ˇMATH WORKBOOKS

154

10. đ?&#x2018;Ś = đ?&#x2018;&#x17D;đ?&#x2018;? đ?&#x2018;Ľ

11. The graphs of đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = â&#x2C6;&#x2019;đ?&#x2018;Ľ 2 + 4 and đ?&#x2018;&#x201D;(đ?&#x2018;Ľ) = 3đ?&#x2018;Ľ + 4 are sketched alongside. 11.1 Calculate the coordinates of A, B, C and D.

y g

_______________________________________________ _______________________________________________

_______________________________________________ _______________________________________________ _______________________________________________

_______________________________________________

B x

_______________________________________________ f

_______________________________________________ _______________________________________________ E

Grade 10 Core Mathematics

155

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11.2 Determine the coordinates of E. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

11.3 Calculate the length of FB. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 11.4 List an expression for đ?&#x2018;&#x201D;(đ?&#x2018;Ľ) â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ľ). ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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156

11.5 For which values of x is đ?&#x2018;&#x201D;(đ?&#x2018;Ľ) â&#x2030;Ľ đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ E

12. D

B x

A H g

The graphs of đ?&#x2018;&#x201C;(đ?&#x2018;Ľ) = đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 9 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018;&#x201D;(đ?&#x2018;Ľ) = 2đ?&#x2018;Ľ + 6 are sketched above. 12.1

Calculate the coordinates of A, B , C and D.

Grade 10 Core Mathematics

12.2

157

GOď&#x201A;ˇMATH WORKBOOKS

State the lengths of OC, AB and OD

Calculate the coordinates of E.

IF đ?&#x2018;&#x201A;đ?&#x2018;&#x2020; = 1 Find FT.

If đ?&#x2018;&#x201A;đ?&#x2018;&#x192; = â&#x2C6;&#x2019;4 Find GH.

Calculate the length of AD.

Grade 10 Core Mathematics

14.

158

GOMATH WORKBOOKS

Financial Maths:

No business can exist without the information given by figures. Borrowing, using and making money is the heart of the commercial world thus the principle of interest and interest rate calculations are extremely important. This leads into an examination of the principles involved in assessing the value of money over time and how this Information can be utilized in the evaluation of alternate financial decisions. Remember that the financial decision area is a minefield in the real world, full of tax implications, depreciation allowances, investment and capital allowances etc. The basic principles in financial decision making are established through the concept of interest and present value: –

Definition of interest: Interest is the price paid for the use of borrowed money Interest is paid by the user of the money to the supplier of it. It is calculated as a fraction of the amount borrowed or saved over a certain period of time. This fraction is also known as interest rate and is expressed as a percentage per year (per annum).

Simple interest is computed on the principle for the entire term of the loan and is thus due at the end of term. I = Prt I is the interest paid or earned P is the principle or Present value r is the interest rate per annum t is the time or term of loan

A  P(1  in) A  ( Amount) New Price per time period P  Original Price(Cost)(Principle) r i  Interest rate used for growth : i  100 n  the relevanttime period.

P

A (1  in)

Grade 10 Core Mathematics

GOMATH WORKBOOKS

159

Simple Interest ( Growth) Simple Interest:(Growth) NB interest is calculated on the original investment for each time period. i.e. The interest earned is a constant. Formula:

A  P (1  ni) Where:

A  FutureValue (Future amount.) P  present Value (original amount.) r ) 100 n  Number of interest periods.

i  interest level(i 

(When using i use the decimal of the interest rate quoted (or use

r for i) 100

Example 1: A Principle of R300,00 is invested in a Bank at 10% Simple Interest for a number of years; Complete the table below: Principle Interest period(yrs) Simple Interest (%) Interest

300 1

300 2

300 3

300 4

300 5

300 6

300 N

120

150

180

300 100in

Amount

330

360

390

420

450

480

200 

300 100in

Example 2:Invest R300,00 @ 15% per annum( per year) Year 0

R300

Simple Interest p.a

Year 1

R345

Year 2

R390

Year 3

R435

Year 4

R480

Year 5

R525

The value of the investment increases at a constant value of R45 per year. The relevant graph for simple growth is a straight line:

Grade 10 Core Mathematics

160

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Exercise 14.1: Question 1: Peter deposits R8500,00 for 8 years into a savings account. 1.1 If the money earns a simple interest rate of 12% per annum. How much will he have at the end of the investment period. ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 1.2

If he had received an interest rate of 20% , how much more would he have earned in interest?

Grade 10 Core Mathematics

161

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Question 2: John invests R20 000,00 into a capital venture that will earn him a simple interest of 10% for the first 5 years and then 15% for the last 5 years. What is his investment worth at the end of the 10 years? ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ Question 3: 3.1

Mrs Vezi invests R10 000,00 at 12,5% simple interest per annum.

How much interest will she earn after 5 years?

How much interest will she earn after 10 years?

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3.3

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What is the final amount in her account after 15 years if she does not have any withdrawals?

Calculate the Principle if James received an amount of R7500,00 earned in simple interest at a rate of 10% over 5 years.

Calculate the Principle if James received an amount of R37500,00 earned in simple interest at a rate of 15% over 10 years.

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Question 5: A loan of R100 is made at 10% simple interest p.a. 5.1

A line graph is used to display the amount at the end of each year. Draw this graph on the grid below. y

200 190 180 170

A m o u n t

160 150 140 130 120 110 100

Years

5.2

What do you notice about the increase every year? How is this displayed in the graph? ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5.3

If R100 is borrowed at 5% p.a. Simple Interest for 4 years display this on the same graph and show clearly how the two graphs differ from one another. ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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5.4

If R100 is borrowed at 15% p.a simple interest for 4 years explain without drawing the graph how the graph would differ from the other two (2). ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ COMPOUND INTEREST Compound interest arises when, in a transaction over an extended period of time, interest due at the end of a payment period is not paid, but added to the principal. Interest also earns interest i.e. it is compounded. The amount due at the end of transaction period is referred to as the compounded amount or accrued principal. Interest periods can vary : daily, monthly, quarterly, half-yearly or yearly. FOR GRADE 10 PURPOSES INTEREST IS ONLY CALCULATED YEARLY

Formula: Compound Growth: A  P (1  i ) n

A = Amount or Future Value P = Principal or Initial value i rate of interest per annum i 

r 100

n = number of years invested Compound Interest(Growth): Interest is calculated on the new amount each time period. i.e. Earn interest on interest, thus interest increases each year.

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Example 1: Year 0

R300

Compound Interest p.a.

Year 1

R345

Year 2

R396,75

51,75

Year 3

R456.26

59,51

Year 4

R524,70

68,44

Year 5

R603,41

78,71

The amount increases at a constant interest rate however it is calculated on the increasing amount each time period. The relevant graph for compound growth is curve shaped.

Example 2: Vusi invests R1000,00 at 12% compound interest for 5 years: 2.1 Write down a formula for the compound interest to be calculated over n yrs at r % p.a. n

r  n Answer: A  P 1   or A  P1  i  

100 

2.2 Use the formula to calculate the final amount earned after the 5 years to the nearest cent. A  P 1  i n

Answer:

A  10001  0,125 A  R1762,34

2.3 How much interest was earned on his investment:

i  A P Answer: i  1762,34  1000

Ci  R762,34

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2.4 Calculate what he would have earned in simple interest for the same time period: Pr t 100 1000 12  5 Si  100 Si  R 600. Si 

Answer:

It can be deduced that it is more beneficial to invest money at compound interest:

If the number of interest periods are increased then it can be deduced that the interest earned would increase accordingly:

i.e Calculating interest: 1. annually 2. half- yearly 3. quarterly 4. monthly 5. daily.

Example: R1000 invested at compound interest for 5 years at 12% p.a. Principle

Interest

A  P (1 

r tm ) 100m

Time Period = 5yrs Yearly

Half-Yearly

Monthly

Daily

m =4

m = 12

m= 365,25

m= no of time periods p.a.

m =1

R1000

R1762,34

R1790,85

R1806,11

R1816,70 R1821,94

R762,34

R790,85

R806,11

R816,70

12%

Compound Interest

m =2

Quartely

Ci = A – P It can be deduced from the table above that the more time periods involved in the calculation of compound interest the better the return on investment: The majority of Banks and investment companies employ daily interest calculations for principles invested with them.

R821,94

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Exercise 14.2: 1. Charlie wants to invest R7000,00 : There are 2 choices for him: What would the better choice be? Show all your working. 1.1

Invest R 7000,00 at 12% simple interest per annum over 5 years.

Invest R7000,00 at 12% compound interest over 5 years.

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3. R20 000,00 is invested for 15 years at 12% p.a. Calculate the value of the investment if the interest is calculated: 3.1

At simple interest.

At compound interest.

Calculate a simple interest rate that would yield the same return.

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4.2

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Calculate a compound interest rate that would provide the same return.

simple interest.

compound interest.

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6. The Ndlovu family uses a loan of R7200,00 to buy furniture They repay the loan at the end of 3 years. How much would they have to repay if the interest is calculated as: 6.1

16% p.a. simple interest?

13% p.a. compound interest?

Compound interest A  P(1  i) n

End of 1st year

End of 2nd year

End of 3rd year

End of 4th year

End of 5th year

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8. Use a table to draw two graphs on the same system of axes showing the difference between simple and compound interest . Use a the system of axes below for your graphs.

y 2100 A m o u n t

2000 1900 1800 1700 1600

i n R a n d s

1500 1400 1300 1200 1100 1000 5

no of years

8.1

What kind of graph does the simple interest formula produce?

Find the gradient of the line.

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8.3

What is the difference in the value of A for simple and compound interest after 5 years?

Use the formula to calculate the difference in A, ( the accumulated value of the investment), after 10 years.

€

R 9.9837

NZ$

USD

0.7747 0.7157

98.1100

1.5253 1.9117

ZAR

0.1002 1

0.0776 0.0717

9.8270

0.1528 0.1915

EUR

1.2909 12.8880 1

GBP

1.3972 13.9492 1.0823 1

137.0793 2.1312 2.6710

JPY

0.0102 0.1018

0.0079 0.0073

0.0155 0.0195

AUD

0.6556 6.5453

0.5079 0.4692

64.3209

NZD

0.5231 5.2225

0.4052 0.3744

51.3213

0.7979 1

0.9239

126.6502 1.9690 2.4678

1.2533

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Examples : R1000 will buy 1.1

$100 (to nearest dollar)

1.2

£72(to nearest pound)

1.3

€78(to nearest Euro)

1.4

¥9827(to nearest Yen)

Exercise 14.3: 1.Copy and complete the following currency table:

CURRENCY

USD

ZAR

0.133 1

EUR 2

JPY

7,5

GBP

€

NZD

NZ$

0.0666

300

0.1428

0.933 7

150

O,05

AUD

1 0.333 0.01

0,0105 1

0,3335

Use the table above for questions 2 to 4. 2

You have R1000 to spend in each of the above countries. How much of the local money can be purchased in each of the countries. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 3

What will it cost you in Rands to purchase the following currencies:

3.1

$800.

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3.2 ¥2000. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 3.3 A$500. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 3.4 €2000. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 3.5 £5000. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ How many rands can be exchanged for the following: 3.6 $200 ____________________________________________________________________ ___________________________________________________________________ 3.7 ¥1500 ____________________________________________________________________ ___________________________________________________________________

3.8 A$150 __________________________________________________________________ ___________________________________________________________________

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3.9 €350. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

3.10

£750.

Cost of Hire Purchase: A hire purchase agreement is a short - term loan It is calculated using simple interest The lending company also adds insurance to the cost to cover the loan amount. Example: Andy buys a plasma TV set for R12000,00 He buys the TV using hire purchase agreement involving monthly payments over 3 yrs. The simple interest rate is 14% p.a. calculate Andy,s monthly installment if a monthly insurance premium of R10 is added . Solution:

A  P (1  ni)

A  12000[1  3(0,14)] A  R17040 Monthly installments are = R

17090  10 36

= R473.33 + R1 = R483 .33

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Exercise 14.4: 1.

A couple buy new furniture for their home. The furniture costs R120 000,00 They pay for it using a hire purchase loan. The interest rate is 12% p.a. over a 5year period. Calculate the monthly repayment if there is no insurance added.

Sihle buys a washing machine for R6500,00. He pays 15% of the value in cash as a deposit and uses a hire purchase agreement to pay the balance over 36 months. The interest rate is 10% p.a. Calculate his monthly installments if an insurance premium of R6,50 is added monthly.

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Mr Moosa sees an advert in the newspaper for a surround sound audio system. The advert is worded as follows: “Pay ONLY R299,99 R7500.

per month for 36 months!” The cash price is

Assume that no insurance premium is added. Calculate: 3.1

how much interest is paid over 3 years?

The interest rate per annum if the advertisement refers to a hire purchase agreement.

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Ms Bokapane buys a washing machine from a appliance store . She asks for a quote to buy the machine through a hire purchase contract over 1

1 2

years. The following quote was received by her:

QUOTATION: Purchase price

R3800,00

10% cash deposit

R 380,00

Balance owing

R3420,00

Finance charges(interest)

R 820,00

Insurance over 18 months

R 684,00

Total amount to be paid

R4924,80

Monthly payment

R 273,60

1.1

Why do you think Ms Bokopane has to pay insurance costs?

1.2

Calculate the interest rate per year. (Note: the interest charged is calculated on the balance owing i.e. R3420,00 .The insurance is not included in this calculation)

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Cost of Inflation Inflation is the continuous increase in the cost of goods and services over a period of time. The rate of inflation is given as percentage per annum. It is the average increase (as a percentage ) in the cost of goods and services from one year to the next. The compound increase formula is used in inflation problems. Example: A car costs R80 000,00, and the average rate of inflation is expected to be 8,05% p.a. over the next 5 years. 1. Calculate the expected price of the same model of car in 5 years time, based on the stated inflation rate. 2. How much would the same model car cost 5 years ago, based on the same rate of inflation? Solution :

A  P(1  i ) n A  P (1  i ) n A  80000(1  0,0805)

A  R117818,00

80000  P(1  0,0805) 5 5

P

80000

(1  0,0805) 5 P  R54320,80

Exercise 14.5: 1.

The average rate of inflation over the last 10 years was 7,3% p.a. The current price of a 2,5kg packet of white sugar is R10,25. 1.1

Calculate the expected price of sugar in 10 years time if the rate of inflation continues at the same level.

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1.2

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How much did the 2,5kg packet cost 10yrs ago (the rate of inflation continues at the same level.)

A kettle costs R180,00. Determine the expected cost of a similar kettle in 5 years time , based on an inflation rate of 18% p.a.

A block of 500grams margarine costs R16,19 . Determine the cost in 10 years time if the price is expected to rise by 9% p.a. as a result of inflation.

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The current annual fees for a Bachelor of Commerse degree at UKZN are R20500,00. Determine the expected cost of studying the same degree in years time if the fees are to increase by 9% p.a. as a result of inflation. Give the answer correct to the nearest rand.

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TIME LINES USED FINANCIAL CALCULATIONS Time lines are useful when dealing with complicated problems, such as changes in the interest rate during an investment period or when several deposits or withdrawals are made from a savings account. It helps to summarise the information and give a visual representation of the data in an ordered manner. Example: R7000 is deposited into a savings account , and 4 years later another R5000 is added to the savings. Calculate the value of the savings at the end of 7 years if the interest rate is 12% p.a. for the first 3 years and then increased to 13,5% for the remaining period. Solution: T0

R7000

R5000 12% p.a.

13,5 % p.a.

T6 etc indicates the time period of the investment.

Balance after 3 years :

Balance after 4 years :

Balance after 7 years:

A  7000(1  0,12) 3 A  R9834,496

A  9834,496(1  0,135)  5000 A  R16162,152 A  16162,152(1  0,135) 3 A  R 23631,26(nearest cent)

n (years) Interest rates

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Example 2: In order to save for her sons University fees , Mrs Gumede deposits R8000 into a savings account at the end of January when her son is 10 years old. The rate of interest is 14% p.a. compounded annually. When her son is 18yrs old he starts a University course which has a duration of 3 years. The first fees are R9000 , payable at the end of January . The fees increase by 10% each year. Calculate: 1.

the second and third years fees.

the balance in the account after the first years fees have been paid.

the balance in the account after the second years fees are paid.

how much additional cash will be needed to cover the third years fees?

R8000

T10

(R9000)

n (years)

14% p.a.compounded annually NB the bracket around R9000 indicates a withdrawal. 2.

Second years fees ( fees at T9):

9000  (0.1)(9000)  R9900 Third years fees ( fees at T10) 9900  (0.1)(9900)  R10890 3.

Balance at T8: 8000(1  0.,14) 8  9000  R22820,69  9000  R13820,69 (to nearest cent)

Balance at T9:

13820,69(1  0,14)  9900  R5855,59 (to nearest cent) 5.

Money in savings account at T10:

5855,59( I  0,14)  R6675,37 Additional cash required: 10890  6675,37  R 4214,63 (to nearest cent)

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Exercise 14.6: 1. Jack deposits R1000 into a savings account . One year later he adds R2000 to the savings. At the end of the second year he deposits R4000 into the same account, and finally he adds R8000 to the savings account at the end of the 3rd year. Calculate the amount (A) in Jacks account at the end of the 4th year if the interest is calculated at 11,5% compounded annually. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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2. R6500 is deposited into a savings account , and 3 years later R7400 is added to the savings. At the end of 5 years , R5800 is withdrawn from the account. How much money will be in the account at the end of 10 years if the interest rate is 11% p.a.? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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3. R21000 is invested where the interest rate is 7,5% p.a for the 1st 3 years. The rate is then then increased to 8,25% p.a. for the next 4 years. Calculate the value of the investment at the end of the 7 years. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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4. Mrs Smith invests R8000 for in a savings account when her two sons are 7 and 10 years old. She pays each of them R15000 in the year they turn 21. 4.1

Calculate how much money is in the savings account after she has paid her younger son. The interest rate is 14% p.a. compounded annually.

4.2

Is this fair on the sons? Explain your answer.

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5. Mr Ndlovu places R52000 in the bond market as a fixed saving for 12 years. The interest paid during first 5 years is 10,5% p.a. It is then increased to 12 % p.a. for the next 3 years, and then finally increased to 14% for the last 4 years. In each case the interest compounded annually. Calculate how much Mr Ndlovu will have in his savings account at the end of the 12 year period. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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15 Probability Theory: The language of probability:       

Probability tells us how likely something is to happen. We often use the word chance for probability. The thing you want to happen is called an event. An experiment etc that you are carrying out is called a trial. The result of a trial is called an outcome. some outcomes always happen . We say they are certain to happen. Some outcomes never happen. We say they are impossible. Some outcomes are not certain, but they are not impossible either. They may or may not happen. These probabilities are greater than 0, but less than 1.



A probability of 1 means that there is a 50-50 chance or an even chance 2

of the outcome occurring. We can write probabilities as fractions, decimals or percentages. To compare probabilities we compare the sizes of the fractions, decimals or percentages. Probabilities can be placed on a probability line as shown below: Fractions

1 2

Decimals Percentages

0 0%

0,5 50%

1 100%

Impossible

Even chance

Certain

Example Activity 1: Task: Rank the following activities from lowest to highest probability: Event Probability

Answer: Event Probability %

1 3 5

60%

2 0,25

3 78%

2 0,25

3 78%

25%

78%

5 30%

1 3

33 13 %

Answer: 0,25 ; 30%; 1 ; 3 ; 78% at events 2; 5; 4; 1; 3 3

4 1 3

30%

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Example Activity 2: All the learners in Grade 10 at Northwood were asked what they wanted to do when they left school. Different people worked out the results of the survey and gave the results in different ways. Results: Do nothing Get a job

0,15 60%

Go to University

1 25

Go to Technikon

12% 9 100

Don’t know a)

Arrange the results in order of probability of popularity of options, starting from least popular choice.

The probabilities add up to 1 . Explain why this is so.

Answer: Do Nothing

0,15

Get a Job

60%

Go To University

1 25

15 100 60 100 4 100

Go to Technikon

12%

12 100

Don’t Know

9 100

Go to University; don’t know; go to Technikon; do nothing; get a job.

15  60  4  12  9 100  1 100 100

All grade 10 surveyed all gave answers so 100% or value of 1.

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Exercise 15.1: 1.

There is a hot-drink vending machine in an office block. A survey found that the probability that an office worker , buying a drink from this machine, would choose: Soup

Coffee

Tea

Hot Chocolate

1 20 2 5 3 10 1 4

Which drink bought from the machine is: Most likely to be chosen? Least likely to be chosen? Arrange the drinks in order of popularity. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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Listing Outcomes: When a coin is tossed into the air there are two possible outcomes: Heads or Tails. When a dice is thrown, there are 6 possible outcomes: 1; 2; 3; 4; 5; 6. For any event , you can usually list all the possible outcomes. Outcomes which give the “event” you are interested in are called favourable outcomes for that event. e.g. Consider “getting an even number” when you throw a dice. 

The activity is to “throw the dice”



All the possible outcomes are : 1, 2, 3, 4, 5, 6



The event you are interested in is “ getting an even number”



The favourable outcomes are 2. 4 and 6.

e.g. A coin is flipped and a dice is thrown at the same time, and you want to get a head and an even number.  

Coin Outcomes  

The activity is “flip” the coin and toss a dice” All the possible outcomes can be worked out in a table as follow.

1 H, 1 T,1

H T

Dice Outcomes 2 3 H, 2 H, 3 T, 2 T, 3

4 H, 4 T, 4

5 H, 5 T, 5

The event you are interested in is “ a head and an even number” The favourable outcomes are (H, 2), (H, 4) and (H, 6).

Example Activity 3: Look at each of the activities in turn:  List all the possible outcomes for the activity.  List all the favourable outcomes for each named event. 3.1

Activity: throw a dice. Event 1:

get a 1

Event 2:

get a multiple of 3.

Event 3: get a prime number. 

The possible outcomes are 1, 2, 3, 4, 5, 6. Event

1. get a 1 2. Get a multiple of 3 3. Get a prime number

Favourable outcomes 1 3;6 2; 3; 5

6 H, 6 T, 6

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193

Activity: take a coin from a purse containing a 5c coin, a 10c coin, a 20c coin, a 50c coin, a R1 coin, a R2 coin and a R5 coin. Event 4: get a 20c piece.

Event 5: get a silver piece

Event 6: get a coin worth less than 50c. 

The Possible outcomes are : Get a 5c ; 10c; 20c; 50c; R1; R2; R5.

Event 4. Get a 20c coin 5. Get a silver coin 6. Get a coin worth less than 50c 3.3

Favourable Outcomes 20c R1; R2; R5 5c; 10c; 50c

Activity: take a letter from the word PROBABILITY Event 7: get a T

Event 8: get a vowel

Event 9: get a consonant. 

The possible outcomes are : P; R ; O ; B ; A ; B ; I ;L; I ; T; Y. Event

7. Get a T 8. Get a vowel 9. Get a consonant

Favourable Outcomes T O; A; I P;R;B;L;T;Y

Exercise 15.2: 2.1 In my left-hand pocket I have a 50c coin, a 20c coin and a 10c coin. In my right-hand pocket I have R1 coin, a R2 coin and a R5 coin. 2.1.1

I choose one coin at random from either pocket. List all the possible outcomes.

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I choose all outcomes that satisfy the condition : the total must be greater than R2,50.

2.2 I roll dice: 2.2.1

List all the possible outcomes ( that is , possible pairs of numbers)

2.2.2

To get my score I must multiply the value shown on the dice by the value shown on the other dice. List all outcomes that satisfy the condition: the score must be a multiple of 6.

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Sample Spaces: Suppose you have taken the four picture cards of the suit of hearts from a pack. You shuffle them and draw a card at random. You could describe what you have done as a statistical experiment or trial, since there are no laws determining which card you will draw. The fact that you draw an ace, say , is an outcome. We can represent all possible Outcomes in set notation, S  A; K ; Q; J . The possible outcomes can also be represented in a venn diagram.

A; K;Q and J are called elements of the sets.

Venn Diagram

This set is referred to as the probability space or sample space

Example Activity 4: Consider the rolling of an ordinary cubical dice, where the number on the upper face, when it comes to rest, is used. 4.1 Describe the experiment or trial involved. 4.2 Give an example of an outcome in this case. 4.3 Write down the sample space of the experiment. 4.4 Represent the sample space be means of a venn diagram.

Answers: 4.1 4.2 4.3

Rolling the dice. 4

S  1;2;3;4;5;6

4.4

1 4

2 3 5 6

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Exercise 15.3: 3.1

Consider the tossing of a coin: 3.1.1 Describe a trial

Give an example of an outcome.

3.1.3

Write down the sample space, S of the experiment.

Represent the sample space by means of a venn diagram.

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3.2

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The diagram below shows a four sided spinner with sides labeled 1, 2, 3and 4 respectively. 3.2.1 Describe a trial using such a spinner.

___________________________________________________________

4 3

___________________________________________________________ 3.2.2

Give an example of an outcome.

3.2.3

Write down the sample space, S for the experiment.

Give a venn diagram for S.

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Events: A bag contains 3 blue and 5 white counters; The blue counters are labled b1; b2 and b3. While the white counters are labled w1; w2; w3…….w5. Suppose the trial consists of drawing out a counter, after the bag has been given a good shake, and then replacing the counter. The sample space here is: S = {b1; b2; b3; w1; w2; w3;w4 ;w5.} The number of possible outcomes in the sample space, S , is denoted by n(S). Here n(S) = 8 The drawing of a blue counter , no matter what its lable is, can be represented by the set B ={ b1; b2; b3} A few of the outcomes or elements of S are involved in the situation described by B. We call the drawing of blue counter ( no matter the lable) an event. The number of outcomes in event B is the number of elements in B. This number is denoted by n(B), which equals 3. Another event related to drawing counters from the bag, would be that of drawing a white counter. We can represent this event by the set : W = { w1; w2; w3;w4 ;w5} n(W) = 5 The sets B and W are subsets of S. The sample , S, is the universal set. Definition: An event is a set which consists of one or more of the elements of the sample space.

Union & Intersection: In a shoe box there are cards numbered from1 to 10.

S  1;2;3;4;5;6;7;8;9;10

Consider the events:  Drawing a factor of 6: X = { 1 ;2 ;3 ; 6}  Drawing a factor of 9: Y = {1; 3 ;9} The union of these sets is all the elements in X or Y. X or Y ={ 2;6;1;3;9} The intersection of these sets all the elements that are both in X and in Y NB: 1. For an event to occur, it is not necessary for all the outcomes of the event to occur. If any one of the outcomes is the result of a trial , the event is said to have occurred. B thus occurs if any blue counter is drawn. It is not necessary to draw all three blue counters before we can say that B has occurred. 2. The sets W and B are to be disjoint since they have no elements which belong to both sets.

Example: 1. A trial for this question is the rolling of a normal dice. Write down the sample space, S.

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Describe the event, E , where an even number is obtained on rolling the dice, in terms of a set. Use set notation to describe the event, O , of obtaining an odd number. Describe the event, F of obtaining a factor of 6, in set notation. Write down: n(S) n(E) n(O) Represent S and the sets, E, O and F, in a single venn diagram. Answers: 1.1 1.2 1.3 1.4 1.5.1 1.5.2 1.5.3 1.5.4

S  1;2;3;4;5;6 E  2;4;6 O  1;3;5 F  1;2;3;6 n( S )  6 n( E )  3 n(O)  3 n( F )  4

1.6

1 3

2 4

5 F

Exercise15. 4: 4.1

The trial to be considered in this instance is the spinning of a coin.

4.1.1 Represent the sample space, S, in set notation. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 4.1.2 Write down a set for the event, H, of obtaining a head. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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4.1.3 Write down a set for the event, t , of obtaining a tail. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

4.1.4 What is the value of each of the following: 4.1.4.1 n(S ) ____________________________ 4.1.4.2 n(H ) ____________________________

4.1.4.3 n(T ) ____________________________ 4.1.5 Represent S, H and T in a single venn diagram. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

4.2

Max has a bag containing three(3) red , four(4) blue and six(6) green marbles. The bag is shaken and a marble is withdrawn. The colour of the marble is noted and the marble is replaced.

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4.2.1 Write down a sample space , S, for this experiment. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

4.2.2 Express the event of drawing a green marble, G, in terms of a set. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 4.2.3 Write down a set, B, or R, to represent the event of drawing either a red or a blue marble. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 4.2.4 Write down: 4.1.4.1 n(S ) _____________________________ 4.1.4.2 n(G ) _____________________________ 4.1.4.3 n(BorR) _____________________________

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Calculating Probabilities:   

Outcomes which have an equal chance of happening are called equally likely outcomes. For example: when you throw a fair dice , each outcome in a sample space S  1;2;3;4;5;6 is likely to occur. When all the outcomes of an activity are equally likely , you can calculate the probability of an event happening by using the following definition: P(E) =

number of favourable possibilities Total number of possible outcomes

n( E ) n( S )

For example: when you throw a fair dice the possible outcomes are S  1;2;3;4;5;6 i.e the total number of possible outcomes: n( S )  6 Event 1: Get a 4: The only possible outcome is a 4 i.e. E = {4} i.e. number of favourable outcomes : n( E )  1 Probability of getting a 4 = P(4) =

n( E ) 1  n( S ) 6

Event 2: Get a number more than 2: Favourable outcomes : E  3;4;5;6 Number of favourable outcomes : n( E )  4 Probability of getting more than 2 = P(more than 2) =

n( E ) 4 2   n( S ) 6 3

Event 3: Get a number more than 6: Favourable outcomes written as the “empty set.” E = { } For example: Number of favourable outcomes : n( E )  0 Probability of getting more than 6 = P(more than 6) =

n( E ) 0  0 n( S ) 6

Event 4: Get a number less than 10: Favourable outcomes: E  1;2;3;4;5;6 Number of favourable outcomes: n( E )  6 Probability of getting a number less than 10 = P(less than 10) =

n( E ) 6  1 n( S ) 6

i.e. It is certain that you will get a number less than 10 when you roll a dice.

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Probability can be either found by theory or experiment. The theory method relies on logical thought; the experimental method relies on the results of repetition of many, many trials..

Example: Need to find the probability of getting a head when tossing an unbiased coin. Theory method: Unbiased means that both events (head or tail) are equally likely to occur. There are two (2) possible outcomes: S  H ; T  There is one favourable outcome: E  H  So , the probability of getting a head is 1 or 0,5. 2

Experimental method: Suppose you toss an unbiased coin 100 times, and count the number of heads you obtain. Suppose you get 49 heads. The relative frequency of getting a head is therefore 49 or 0,49. Only after 100

many,many trials does the relative frequency give a good value for the probability.

Mutually exclusive and complimentary events. Mutually Exclusive Events 

The events A and B exclude each other.



If A happens B cannot happen.



If B happens A cannot happen.



Both events cannot happen at any one trial.



The sets of outcomes for the events are disjoint( Have no elements in common)



For two mutually exclusive events A, B: P(A and B) =0

P(A and B) =

n( AorB) n( A)  n( B) = n( S ) n( S ) n( A) n( B)  = = P(A) + P(B) n( S ) n( S )

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Complimentary events: For any two(2) complimentary events R and Q. >

n( R )  n(Q)  n( S ) .

P( R)  P(Q) 

n( R) n(Q)  n( S ) n( S ) n( R)  n(Q) n( S ) 1 = = n( S ) n( S )



It is useful to write this in the form P(Q) = 1- P(R).



Q can also be written R or “not R “

Example: A bag contains three(3) red balls, five (5) white balls, two(2) green balls and four(4) blue balls. 1. Calculate the probability that a red ball will be drawn from the bag. 2.

Calculate the probability of that a ball which is not red will be drawn.

Solution: Let R be the event that a red ball is drawn:

n( R ) 3  n( S ) 14

P(R) =

R and R are complimentary events. 3 11  14 14 P ( R )  P ( B )  P (W )  P (G )

P ( R )  1  P ( R )  1 

Note: Alternatively



4 5 2 11    14 14 14 14

Exercise 15.5: [Use venn diagrams when necessary:] 5.1

250 tickets were sold for a raffle. Stephanie bought 10 tickets. What is the probability that Stephanie: ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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5.1.1 wins the prize? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 5.1.2 Does not win the prize? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 5.2

There are 250 cars in the car park. 165 are Volkswagen. What is the probability that the first car to leave the car park is: 5.2.1 a Volkswagen? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 5.2.2 not a Volkswagen ? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 5.3

Jon has 14 loose socks in a drawer. Four of these socks are black and three are white. Calculate the probability that the first sock taken at random from the drawer is. 5.3.1 black. 5.3.2 not black. 5.3.2. white 5.3.4 not white 5.3.5 black or white 5.3.6 not black nor white.

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5.4

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A packet contains 20 fruit – flavoured sweets. There are four(4) pineappleflavoured, five(5) melon-flavoured, two(2) lemon-flavoured, three(3) bananaflavoured and six(6) strawberry-flavoured sweets.

Probability Trees: Tree diagrams are used to count the various ways in which outcomes happen. Example: A bag has 6 red beads and 4 blue beads in it. First one bead is drawn and then a second without the first been put back. Calculate the probability that : 1. 2. 3.

the first bead drawn is red both beads are blue. one bead is red and the other is blue in any order.

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Solution:

5 9

4 9

6 10

6 9

Blue 3 9

1.1

Red Blue

Red

4 10

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1 3 4 15

Red

4 15

Blue

2 15

There are 6 red beads and 10 beads in total: P(first red) = 6  3 10

1.2

The probability of getting one blue bead first and blue bead second has to be calculated. The probabilities along the path therefore have to be multiplied. P(BB) = 4  3  2 10

[NB: there is no replacement so there are 3 blue beads left after the Ist blue bead is drawn and 9 beads in total] 1.3

Either red then blue or blue then red. The probabilities for the separate cases are thus added: P( RBorBR) 

4 4 8   15 15 15

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Exercise 15.6: 1. A coin and a dice are thrown together. 1.1 Draw a tree diagram to show all the possible outcomes. 1.2 What is the probability of obtaining: 1.2.1 A head on a coin and 1 on the dice? 1.2.2 A tail on the coin and an even number on the dice? ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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2. A bag has 6 red beads and 4 blue beads in it. First one bead is drawn and then a second without the first being put back. Calculate the probability that: 2.1 the first bead drawn is red. 2.2 both beads are blue. 2.3 one bead is red and the other blue in any order. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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Further Questions: Worksheet 1: [Independent Events] 1. Two coins are flipped together. Using H for heads and T for tails , list all the pairs of combinations from flipping the coins. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 2. Sandwiches are available in chicken(C), beef(B) and ham(H). They can have fillers of Salad(S) ; chilli(CH) and coleslaw(CO) Using Letter codes list all pairs of combinations of sandwiches. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 3. Two die are thrown together. The table shows all the combinations of possible outcomes. Find the probability of the following scores happening. a. 12 b. 2 c. 10 d. 7 e. 5

Dice Two

Dice One 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 __________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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4. Two spinners , each with 4 coloured regions (red; blue; green & yellow) are spun one after another. What is the probability of the following outcomes? a) Two colours the same. b) Red followed by red. c) Two different colours. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5. Two coins are tossed up at the same time . What is the probability of the following outcomes? a) One head and one tail b) Two heads c) Both coins land the same way up. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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6. 3 Balls, one yellow, one black and one white are in a cloth bag. One ball is removed from the bag , then another without replacing the first ball. a) What is the probability that a black ball is selected followed by a yellow ball b) On a new selection of two out of three what is the chance that a black ball is left in the bag?

7. A standard pack of cards consists of four suits of 13 cards- diamonds(red ), Hearts (red), Clubs (black)and spades(black). If two cards are chosen randomly, a) What are the chances that the second card will be the same suit as the first. b) What is the probability that both cards will be aces? c) What is the probability that both cards will be black. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Worksheet 2: [Mutually Exclusive Events] 1. A cloth bag contains a mix of black, white and red balls. Selecting one ball at random, the probability of getting a black ball is 0.2, while the probability of getting a white ball is 0.5. a) Selecting one ball at random, what is the possibility of selecting a red ball? b) If in total there are 5 white balls in the bag, how many black ball are there? c) If there are in total 3 red balls in the bag , how many balls are there all together? ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2. A paper bag contains 5 red sweets. 7 green sweets and 8 blue sweets. If a sweet is selected at random from the bag, find the probability that the sweet is: a) Red or blue b) Not green c) Green or red d) Not blue ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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3. A tall metal box contains beads of many colours . The probability of selecting a bead of a particular colour is as follows: White 0.2 Red 0.3 Black 0.4. What is the probability that a bead taken from the bos will be: a) Black or Red b) Not White or Black c) A different colour to white, Black or Red. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4. A pack of cards consists of different numbers of white , black or red cards. If the probability of choosing a white or red card is 0.6 and the probability of choosing a white or black card is 0.7, what is the probability of choosing each of the coloured cards individually? ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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5. In a herd of 30 cattle there are 8 coloured black, 12 coloured white and 7 without horns. Find the probability that: a) A cow is coloured white or black b) A cow has horns. c) A cow is neither white nor black. ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 6. Scientists examining climate classify winters as mild , normal, hard or severe. The probability that a winter will be mild or normal is 0.4 The probability that a winter will be severe is 0.1 a) What is the probability of having a hard winter? b) If the probability that a winter will be mild is three times the probability that it is normal, what is the probability that it will be mild or severe? ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________