Grade 11 workbook paper 2 by GoMath Workbooks

Compiled by Chesley Nell

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Forward: Welcome to “ GOMATH WORKBOOKS”. This workbook is designed to be a 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 11 Core Mathematics

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GRADE 11 CORE MATHEMATICS CONTENTS:

Paper Two: Topic

Pages

Analytical Geometry

(4 – 29)

Trigonometry

(30 – 73)

Data Handling

(74 – 117)

Grade 12 Statistical Data

(118 – 129)

Volumes & Surface Area

(130 – 150)

Circle Geometry

(151 – 176)

Grade 11 Core Mathematics

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PAPER TWO 1.

ANALYTICAL GEOMETRY:

Analytical geometry - Studies the properties of geometric figures Algebraically. This is pursued by the means of examining significant points (co-ordinates) of these figures in a Cartesian Plane. Hence also referred to as Co-ordinate Geometry.

Formulae: 1. Length of a line:

A(2 ; 5)

B(-4 ; -3)

Length of AB = (x1  x 2 ) 2  (y 1  y 2 ) 2 = (6) 2  (8) 2 = 100 = 10 2.

Mid – Point of a line

 (x  x 2 ) (y 1  y 2 )  ; Mid – point =  1  2 2    A(2 ; 5) C (x ; y )

B(-4 ; -3)

Mid – Point AB = C (-1; 1) 3.

Gradient of Straight Line: Gradient is represented using the symbol ‘m’ [from y= mx+c] M=

y [ i.e the difference in y divided by the difference in x] x

Equation of a circle centre origin on a cartesian plane. Given by x2 + y2 = r2

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Exercises 1.1: Distance between points: 1.

Find the distance between the given pairs of points: 1.1 (2 ; 3) and (4 ; 5) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 1.2 (6 ; 1) and ( -6 ;6) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 1.3 (3 ; -7) and (-1 ; 3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 1.4 (-4 ; 3) and (0 ; 0) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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1.5 (-2 ; 1) and -4 ; -1) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6 (-3 ;-1) and (4 ; -6) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 2.

Given the coordinates of the vertices of ď &#x201E;ABC , in each case ( 2.2 to 2.5) Determine: A. the perimeter of the triangle. B. Whether the triangle is equilateral, isosceles or scalene. C. Whether or not the triangle has a right angle. 2.1

A(1 ; -3) ; B(7 ; 3); C(4 ; 6)

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2.2

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A(5 ;1) ; B(1 ; 3) ; C(1 ; -2)

A(-2 ; -3) ; B(-4 ; 1) ; C(4 ; 5)

A(0 ; 0) B( 3 ; 1) ; C( 3 ; -1)

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Show that: 3.1

A(-3 ; 2) , B(3 ;6), C(9 ;-2) and D(3 ; -6) are vertices of a parallelogram.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 3.2 (6 ;-4) , (5 ;3) (-2 ; 2) and (-1 ; -5) are vertices of a square. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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Mid-points of lines: 4.

Calculate the coordinates of the midpoints of the line joining the following points: 4.1 (-3 ;1) and (1 ; 5) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.2 (-2 ; 3) and (6 ; 3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.3 (4 ; -1) and (-1 ; 3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.4 (0 ;0 ) and (3 ; -8) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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4.5 ( 3;1) and (3 3;ď&#x20AC;1) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 5.

Determine the values of x and y if: 5.1 (-3 ; 2) is the mid-point of the line joining (-1 ; 5) and (x ; y). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 5.2

(-1 ; y) is the mid-point of the line joining (0 ; -2) and x ; 8)

(x ; y) is the centre of a circle on diameter AB where A(-2 ; -1) and B(-1 ; 9).

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5.4 (x ; 3) is the centre of a circle with diameter MN. M (5 ; -2) and N(-7 ; y) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Calculate the lengths of the medians of ď &#x201E;ABC in which the coordinates of the vertices are as follows: A(-3 ;1), B(-5 ; -3) and C(1 ; -5). (NB: a median is the line from a vertex drawn to the mid-point of the side opposite the vertex) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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The gradients and Inclinations of straight lines:

y y1  y 2  x x1  x 2 y y1  y 2  Inclination = Tan  = x x1  x 2 Gradient = slope =

NB: The inclination of a line is measured from the positive x –axes to the line in question.

Angle of Inclination 

Calculate the gradients of the lines joining the following points: 7.1 (-3 ; 2) and (1 ; 1) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.2 (4 ; 3 ) and (-1 ; 8) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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7.3 (-3 ; -5) and (1 : 3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 8.

Write down the gradients of the lines perpendicular to the lines in 7. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Calculate the inclinations of the line AB in each of the following cases. 9.1 A(-3 ; 2) and B(-5 ; 0) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 9.2

A(-2 ; 1 ) and B(1 ; -2)

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9.3

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A( 3 ; 1) and B((2 3 ; -2)

A(-1 ;2) and B(1 ; -1)

A(-5 ; 2) and B(3 ; -1)

Calculate the gradients of lines with inclinations of: 10.1 45º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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10.2

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60º

150º

110º

Calculate the gradients of the following lines and state whether they are Parallel Perpendicular Neither. 11.1

A(0 ; -1) , B(-4 ; -2) , C( -3 ; 1) and D ( 1 ; 2)

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11.2

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A(6 ; -10) , B(0 ; 4) , C( 6 ; 0) and D ( -4 ; -3)

A(-3 ; 5) , B(5 ; -1) , C( -2 ; -1) and D ( 1 ; 3)

A(-2 ; -4) , B(3 ; 1) , C( 5 ; -1) and D ( -2 ; -8)

Show that the following points are collinear: ( lie on the same line) 12.1 A(-2 ; -6) , B(2 ; -4) , C( 4 ; -3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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12.2

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A(-5 ; 5) , B(1 ; 1) , C( 4 ; -1)

Determine the equaition of a line where the gradients and a point on the line are given as follows: 1 13.1 ; (2;3) 2 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

13.2

3 ; (3;1) 2

 2; (1;3)

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14.

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Lines passing through the following points: 14.1 (-2 ; 4) and (2 ; 2) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 14.2

(-1 ; 1) and (1 ; 5)

(-3 ; -2) and (-1 ; -1)

(3 ; -3) and (3 ; -6)

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15.

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A line with the slope of 3 and intersecting the y – axes at 2 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

16.

parallel to y  3x  2 and passing through (3 ; 1) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

17.

Through (-2 -1) and perpendicular to 3 y  2 x  6 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

18.

Through (-1 ; 3 ) and an inclination of 120º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

19.

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A(-2 ; 1), B(3 ; 3) and C(6 ; -3) are the vertices of a triangle . Determine: 19.1 The coordinates of M, the mid-point of AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 19.2 the gradient of AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 19.3 the equation of the perpendicular bisector of AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 19.4 the equation of the median BM ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 19.5 the equation of the altitude from B to AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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______________________________________________________________ Circle centre the origin: 20.

Determine the equation of a circle with centre origin and: 20.1 radius = 3 cm ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 20.2 radius = 3 2 cm ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 20.3 Passing through point (-2 ; 3) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 20.4

Passing through point ( -4 ; -2)

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A(-3 ; 4) is a point on a circle with centre at the origin: 21.1 Determine the equation of the circle. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 21.2

Determine the coordinates of B if AB is a diameter.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 21.3 Show that the point C(0 ; 5) lies on the circle. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 21.4 Prove that đ??´đ??śĚ&#x201A; đ??ľ is a right angle. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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Tangents to circle centre origin: 22.

B( 3;4 )

O (0;0)

x C

In the figure O is the centre of the circle and the origin of the set of axes. ABC is a tangent to the circle at point B. OBď &#x17E; AB Determine: 22.1 The gradient of OB.

22.2

The equation of AC, the tangent. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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22.3 The equation of the circle centre O. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Exercise 1.2: QUESTION 1: The Points A(-8 ;3) ; B(-1 ; 1) ; C(1 ; -4) and D(-6 ; -2) lie on a cartesian plane. 4

A(-8 ; 3)

B(-1 ;1)

-10

-5

-2

D(-6 ; -2)

-4

C(1 ; -4)

-6

Determine: 1.1

the length of AD.

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1.2

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the mid-point of DC

1.3

The gradient of BC

1.4

The length of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.5

the inclination of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6

the equation of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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1.7

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The perimeter of ABCD

1.8

State what shape is represented in the diagram ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

QUESTION 2: 2.1

Show that A(-5 ; -3); B(-1 ; 0) and C(3 ; 3) lie on the same straight line.

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2.2

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P(13 ; t) , Q(7 ; 2) and R(4 ; 1) are points in a Cartesian plane. If P , Q and R are collinear, then determine the value of t. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

QUESTION 3: B(2 ; 3) A(-5 ;2)

D( t ; t-3 ) C(-3 ;-2)

Three points A 5;2 ; B(2 ; 3)and C  3;2 in a Cartesian plane are given. 3.1 Calculate the distance AB. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 3.2

Calculate the gradient of AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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3.3 3.4

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Calculate the value of t if the point D(t ; t- 3) is such that AC // BD. Calculate the mid-point of BC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.5

Determine, by calculation , whether the quadrilateral is a parallelogram. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.6

Give the equation of BC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.7

What is the size of the angle of inclination of BC with the positive x â&#x20AC;&#x201C; axes. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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3.8

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Calculate the size of ACˆ B .

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TRIGONOMETRY

In trigonometry there are 3 different approaches to finding answers without the use of a calculator.

2.1

Using Pythagoras:

A ratio will be supplied and a clue in the form of a restriction stating in which quadrants the angle can be situated. Combining the two clues locate the precise quadrant for the sketch. 3 sin x Example: If sin x  and x  [90 ;360 ] find the value of 5 cos x

y 2  r 2  x 2 ( pythagoras ) y 2  5 2  32 y 2  16

5 3

-4

y  4

X X 3 sin x 3 5 3 Answer:  5     4 cos x 5 4 4 5 Exercise 2.1: 8 1.1 If sin x   and 90 < x < 270 17 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ Find without the use of a calculator: 1.1.1 cosx ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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1.1.2

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sin2x + cos2 x

1.2 If cos A 

3 , and A  [180 ;360 ] Determine then value of: 5

sin A  cos A

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1.2.4

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sin A cos A

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Negative angles: Clockwise rotation of the terminal ray gives rise to negative angles.

Co – terminal Angles: These are angles which share the same terminal ray. Thus all ratios of co-terminal angles must be equal.

210

Eg.

-150º Co-terminal angles. 210º -150

cos(-150º) = cos 210º = cos(180º + 30º) = - cos 30º tan ( -150º) = tan 210º = tan(180º + 30º) = tan 30º When finding a ratio of a negative angle, convert it to the ratio of the positive , co-terminal angle eg. sin ( -261º) = sin 99º

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Angles Greater than 360º The terminal ray of these angles is rotated through one or more rotation(s). Again coterminal angles arise. 

Eg.

Are Co-terminal angles. All ratios of both are equal.  + 360º Thus when finding a ratio of an angle greater than 360º convert it to the ratio of  Where   [0º ; 360º] Likewise, if the terminal arm is rotated clockwise through 360º , we get:  Are Co-terminal angles. All ratios of both are equal.  - 360º If the terminal ray is rotated through more than one positive or negative revolution , then the angles formed ; [eg  + 2.360º ;  - 2. 360º ;  + 3. 360º] have equal ratios to those of  [0º ; 360º] e.g sin( + 720º) = sin( -720º) = sin ( + k.360º) = sin  cos( + 720º) = cos( -720º) = cos ( + k.360º) = sin  tan( + 720º) = tan( -720º) = tan ( + k.360º) = sin  cos 730º = cos(2.360º + 10º) = cos (720 +10º) = cos 10º sin 380º = sin(360º + 20º) = sin 20º tan 1030º = tan(3.360º - 50º) = -tan 50º

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2.2. Using reduction function: In these types of questions specific angle sizes are not given. The angles are given as variables and the different reduction functions are utilized. i.e

sin(180   x)   sin x sin(90   x)  cos x

Example:

tan(180   x) sin(360   x) cos(90   x) ( tan x)( sin x)(sin x) sin x     tan x    ( sin x)(cos x)(tan x) cos x sin(180  x) sin(90  x) tan(180  x) Exercise 2.2: 2.1

cos(180   x). sin(90   x). tan(180   x) cos(360   x). sin 270

2.2

sin(180   x) tan(360   x) sin(90   x) sin(180   x) cos(90   x) tan(360   x)

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sin(360   x) sin(90   x) cos(90   x)

2.3

tan(180   x)( sin x) cos(180   x)

2.4

Using Special Angles: Specific angle sizes given :

A. Special angles and their multiples: 1 sin 30   2 e.g. 1 sin 330    sin 30    2 B. Complimentary angles. e.g. sin 40  cos 50 sin 25 sin 25 cos 65  or 1 cos 65 sin 25 cos 65

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Exercise 2.3: 3.1 3sin30º tan45º cos30º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.2

sin30º cos30º tan60º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.3

4sin60º + tan45º + 2cos30º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.4

4sin 2 45º - 3 sin 2 30º

3.5

cos 2 0º +cos 2 30º + sin 2 45º ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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3.6

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cos 30º tan 2 45º + tan 2 30º + tan 0

3.7

1 2 sin 60   cos 30  tan 2 30   sin 45 tan 2 60  2 3 3

3.8

   cos 60 . cos 40 . tan 330    sin 210 sin 50 . sin 270

3.9

   sin 315 . cos 135  cos 210   tan 135 . sin 270

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3.10

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   sin 315 . cos 20 . sin 240    tan 135 . sin 80 cos 180

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2.4. Reduction Function: NB: All angles greater than 90º can be written as ratios of acute angles using then relevant reduction function. Example: represent sin 240 as a ratio of an acute angle. 3/sin 240   sin(180   60  )

 - sin60  Exercise 2.4: Represent the following as ratios of acute angles: 4.1 sin 230 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.2 cos150 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.3 tan125 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.4 cos230 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.5 tan320 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.6 sin145 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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2.5. Calculator Work: Exercise:2.5 Use a calculator to Find: 3 sin 120  5.1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

5.2

2 tan135

cos 240  ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

5.3

2 tan125 cos 150 

sin 139  ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 5.4

If x  25,7  and y  137,4  calculate the value of the following: sin 2 x cos 2 y 3 tan y ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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2.6

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Solution of trig equations:

It is important to note the restrictions given ,to arrive at the correct solution. Find a Key L ( reference angle) and use the relevant reduction function to supply the correct answer in the correct quadrant. [NB: The Key L must not be manipulated or changed ] Example:1 Solve for x where x  [0 ;360 ] if :

Ex 1.

sin x  0,5

2 sin( x  30  )  1

KeyL  30 

sin( x  30  )  0,5

x  180   30 

Key ( x  30  )  30 

x  210  OR x  360   30  x  330



Ex 2:

x  30   180   30  x  240  OR x  30   360   30  x  360 

Exercise 2.6: Use a calculator to determine the values of x for x  [0  ;360  ] [ 1 decimal place.] 6.1

2 sin x  0,545 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.2

2 cos x  0,147 3

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6.3

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3 tan x  6,605 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.4

1 cos x  0,245  0 2

6.5

tan x  8,213  0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

6.6

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3 sin x  0,369 2

6.7

 2 sin x  0,546  0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.8

2 tan x  8,442  0 3

Grade 11 Core Mathematics

6.9

GOMATH WORKBOOKS

4 sin x  3,208  0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.10

2 sin( x  20 )  1,636

6.11

2 cos( x  30  )  0,262 3

Grade 11 Core Mathematics

6.12

GOMATH WORKBOOKS

2 sin( x  25)  0,345 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.13

3 tan(x  75 )  6,147

6.14

 2 cos( x  15)  1,605 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

6.15

GOMATH WORKBOOKS

2 sin( x  30)  0,445  0 3 ______________________________________________________________

6.16

3 tan( x  54)  21,213 4 ______________________________________________________________

Grade 11 Core Mathematics

2.7.

GOMATH WORKBOOKS

General Solution in trig equations

The general solution of an equation are not in specific form. i.e they do not give the final vale of the given variable. They are solutions that satisfy the given equation in a general form and when the parameters for the variable are supplied then the specific values can be found. [They involve k.360º which simply refers to the number of extra revolutions required to obtain a specific answer in a given restriction] Find the general solution for x in the following: Example:

2 sin 2 x  1,630 sin 2 x  0,815

+ve 1/2

KeyL (2 x)  54,6  In Q1: 2x = 54,6 + k .360º x = 27,3º + k .180º In Q2 2x = (180º - 54,6º) + k.360º 2x = 125,4º + k.360º x = 62,7º + k.180º

General solutions:

If the specific solution is required then the period (Domain)must be stated: Example: Now find the value(s) for x when x  [0  ;360  ] If: k = 0 then x = 27,3º or 62,7º k = 1 then x = 207,3º or 242,7º Exercise 2.7: Find the general solution of the following: 7.1

k=  2

k=  1 k= 0

k- represents the number of revolutions required to obtain the answers that fall within the stated restriction

tan 2 x  2,6 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

7.2.1

GOMATH WORKBOOKS

2 cos x  0,66  0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.2.2 Find the value(s) of x above if x  [180 ;360  ] in 7.2.1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7.3.1

sin

1 x  0,825 2 ______________________________________________________________

Grade 11 Core Mathematics

7.3.2

GOMATH WORKBOOKS

Find the specific solutions for 7.3.1 if x  [270 ;270 ] ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7.4

Find the general solution for the following: tan 3x  4,302 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7.6.1

Find the general solution for the following: 2 cos(2 x  60 )  0,684 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

7.6.2 Find the specific solutions for 7.6.1 if x  [360 ;360 ] ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.7.1 Find the general solution for the following: 4 sin(3x  120 )  2,812 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.7.2 Find the specific solutions for 7.7.1 if x  [180 ;360 ] ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Exercise 2.8. Use the fundamental identities to simplify the following: 8.1

sin x tan x

1-sin2x

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

8.3

1-cos2x ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

8.4

sin 2 x  1  cos 2 x

Grade 11 Core Mathematics

8.5

GOMATH WORKBOOKS

cos x ) tan x ______________________________________________________________

sin x(sin x 

sin x cos 2 x  sin 3 x 2 ______________________________________________________________

Proving Fundamental Identities:

Example: Prove the following: sin 4 x  cos 4 x  sin 2 x  cos 2 x

LHS  sin 4 x  cos 4 x  (sin 2 x  cos 2 x)(sin 2 x  cos 2 x)  sin 2 x  cos 2 x  RHS

NB sin 2 x  cos 2 x  1

Exercise 2.9: 9.1

cos x 1 1   tan x cos 2 x sin x ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

9.2

(1  cos 2 x) 

1 2

GOMATH WORKBOOKS

 cos 2 x

tan x ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 9.3

1  1  cos x   sin x  sin x  tan x  ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

9.4

sin 3 x  cos 3 x  (sin x  cos x)(1  sin x cos x ) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

9.5

GOMATH WORKBOOKS

(sin x  cos x) 2  1  2 sin x cos x ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

9.6

. (1  sin 2 x)(1  tan 2 x)  1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

9.7

GOMATH WORKBOOKS

56 1 cos x

cos x(1  tan 2 x) 

Finding values trigonometric functions in terms a given of a variable : Example: 10

If cos 40 = p , then represent the following in terms of p

10.1

sin 50 = cos 40º = p

10.2

cos140 = -cos40º = -p

NB: MAKE A SKETCH

sin 2 40   1  cos 2 40 

10.3

sin 40 

sin 2 40   1  p 2 sin 40   1  p 2

10.4

tan 40 =

sin 40



cos 40



1 p p2

1 - p2

40 p

If the answer cannot be found using the diagram above THEN make use of the Reduction formulae to get back to the given clue i.e, 40 in this case. 180 ±  ; 360 -  ; 90 ± 

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Exercise 2.10: 10.5 If sin 20 = k , then represent the following in terms of k: 10.1

sin 160 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

10.2

cos70 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

10.3

cos 20 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

10.4

tan 20 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

10.5

If cos 40  a determine the following in terms of a: 10.5.1 tan 40 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

10.5.2 cos 220 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 10.5.3 sin 40  ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

2.11 11.1

GOMATH WORKBOOKS

Solution of Triangles: Area Rule:

Theorem: The area of a triangle is half the product of the lengths of two sides of the triangle and the sine of the included angle. Example: Calculate the area of ABC

A 8cm 40º B 5cm C 1 ( AB )( BC ) sin Bˆ 2 1 Area of ABC  (8)(5) sin 40  2 ABC 

ABC  12,86cm 2

Exercise 2.11: Calculate the area of: 11.1.1

ABC in which BC = 7cm; AC = 6cm and Cˆ  27,6 .

11.1.2

EFG in which EG = 29cm ; EF = 54cm and Eˆ  61,4

Grade 11 Core Mathematics

11.1.3

GOMATH WORKBOOKS

FGH in which Hˆ  61,4 ; GH = 9,5cm and FH = 2,3cm

11.2

Calculate the area of a parallelogram in which two adjacent sides measure 100mm and 120mm and the angle between them is 65º. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

11.3

If the area of XYZ is 3000m2 and x  80m and y  150m , calculate two possible sizes of Zˆ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

2.12

GOď&#x201A;ˇMATH WORKBOOKS

Sine Rule: a b c ď&#x20AC;˝ ď&#x20AC;˝ sin A sin B sin C

In any ď &#x201E;ABC :

Theorem:

Example:

15cm 40Âş B

8cm

Find AË&#x2020; and AC

8 AC ď&#x20AC;˝ sin 20 sin 120 8 sin 120 AC ď&#x20AC;˝ sin 20 AC ď&#x20AC;˝ 20,3cm

8 15 ď&#x20AC;˝ sin A sin 40 8 sin 40 sin A ď&#x20AC;˝ 15 AË&#x2020; ď&#x20AC;˝ 20 ď Ż

The Ambiguous Case: This comes into effect when the side opposite the given angle is smaller than the side adjacent to the given angle. One angle and two sides are given.

30ď&#x201A;°

30ď&#x201A;° 5

Ambiguous Case đ?&#x;&#x2019; đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x;&#x2018;đ?&#x;&#x17D;Â°

đ?&#x;&#x201C;

= đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x2018;¨Ě&#x201A;

đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x2018;¨ =

đ?&#x;&#x201C;đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x;&#x2018;đ?&#x;&#x17D;Â° đ?&#x;&#x2019;

Normal Case đ?&#x;&#x201C;

đ?&#x;&#x2019;

đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x;&#x2018;đ?&#x;&#x17D;Â°

= đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x2018;¨Ě&#x201A;

đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x2018;¨ =

đ?&#x;&#x2019;đ?&#x2019;&#x201D;đ?&#x2019;&#x160;đ?&#x2019;?đ?&#x;&#x2018;đ?&#x;&#x17D;Â° đ?&#x;&#x201C;

Ě&#x201A; = đ?&#x;&#x2018;đ?&#x;&#x2013;, đ?&#x;&#x2022;Â° đ?&#x2019;?đ?&#x2019;&#x201C; đ?&#x;?đ?&#x;&#x2019;đ?&#x;?, đ?&#x;&#x2018;Â° đ?&#x2018;¨

Ě&#x201A; = đ?&#x;?đ?&#x;&#x2018;, đ?&#x;&#x201D;Â° đ?&#x2019;?đ?&#x2019;&#x201C; đ?&#x;?đ?&#x;&#x201C;đ?&#x;&#x201D;, đ?&#x;&#x2019;Â°(reject) đ?&#x2018;¨

Ě&#x201A; = đ?&#x;?đ?&#x;?đ?&#x;?, đ?&#x;&#x2018;Â° đ?&#x2019;?đ?&#x2019;&#x201C; đ?&#x;&#x2013;, đ?&#x;&#x2022;Â° đ?&#x2018;Ş

Ě&#x201A; = đ?&#x;?đ?&#x;?đ?&#x;&#x201D;, đ?&#x;&#x2019;Â° đ?&#x2018;Ş

Two possible answers Ě&#x201A; Ě&#x201A; đ?&#x2019;&#x201A;đ?&#x2019;?đ?&#x2019;&#x2026; đ?&#x2018;Ş For đ?&#x2018;¨

Only one possible answer Ě&#x201A; Ě&#x201A; đ?&#x2019;&#x201A;đ?&#x2019;?đ?&#x2019;&#x2026; đ?&#x2018;Ş For đ?&#x2018;¨

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Exercise 2.12: Solve the following triangles: 12.1.1

ABC in which Aˆ  30 ; Bˆ  45 and a  2

PQR in which Pˆ  115 ; Qˆ  20  and q  15,3

Grade 11 Core Mathematics

GOMATH WORKBOOKS

12.1.3 XYZ in which Yˆ  64 ; Zˆ  21 and x  30

12.1.4

ABC in which ; Bˆ  50,1 ; Cˆ  72,3

and AC  5,34

Grade 11 Core Mathematics

12.2

GOď&#x201A;ˇMATH WORKBOOKS

In ď &#x201E;PQR , đ?&#x2018;&#x192;Ě&#x201A; = 38,2; đ?&#x2018;&#x201E;đ?&#x2018;&#x2026; = 5,2đ?&#x2018;?đ?&#x2018;&#x161; and PR = 7,4 cm. Solve the triangle. i.e. find all missing values. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

12.3 In ď &#x201E;KMS , KË&#x2020; ď&#x20AC;˝ 30,7 ď Ż ; SË&#x2020; ď&#x20AC;˝ 19,1ď Ż and KM = 4,2m. Calculate the length of KS. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

2.13

GOMATH WORKBOOKS

Cosine Rule: In any ABC :

Theorem:

a 2  b 2  c 2  2bc cos A b 2  a 2  c 2  2ac cos B c 2  a 2  b 2  2ab cos C

Example: Calculate the length of AB:

AB  BC  AC  2( BC )( AC ) cos 40 2

A 

AB 2  12 2  15 2  2(12)(15) cos 40  AB 2  93,22400048

15cm

AB  9,7cm

40º B

12cm

Exercise 2.13: Solve the following triangles: 13.1

ABC in which Aˆ  60 ; AB = 5cm and AC = 8cm ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

13.2

GOMATH WORKBOOKS

PQR in which Qˆ  135 ;QR = 3 2 and PQ = 1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

13.3

GHK in which GH = 8cm ; HK = 9cm and GK = 10cm ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

LMN in which LM = 7cm ; MN = 13cm and NL = 8cm

13.4

13.5

20 m

120

ABCD is a trapezium in which: AD = 20 m. BC = 30 m, DAˆ B  120 and ABˆ D  30 . AD // BC.

30 B

Show that : 13.5.1

BD = 20 3m

30 m

[Hint: use special angles)

Grade 11 Core Mathematics

GOMATH WORKBOOKS

13.5.2 DC = 10 3m ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 13.6

In the following sketch AC  BC  x and Cˆ  y  C y x

13.6.1 Show that the distance AB  x 2(1  cos y) and hence,

Grade 11 Core Mathematics

GOMATH WORKBOOKS

13.6.2 Calculate AB if x  150m and y  112

In the diagram QP = 10,28 cm PR = 5,73 cm and Qˆ  32  P

10,28cm 5,73cm 32 Q

Calculate Pˆ

Grade 11 Core Mathematics

GOMATH WORKBOOKS

14.2 In the diagram AB = 5cm , AC = 4cm and BC = 6cm A

4cm 5cm

C 6cm B

Calculate all the angles of ABC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

14.3 In the figure PQR is right agled at Q: RS = 10cm, Rˆ  40  and PSˆQ  68 P 1

68

40 10cm

Calculate: 14.3.1 PS ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 14.3.2

Grade 11 Core Mathematics

GOMATH WORKBOOKS

14.4 In the diagram A B & C lie on the same horizontal plane . HC is a vertical height equal to 100m.

100m  ACB=104,5

26,5

A 21,8 P B

Calculate: 14.4.1 AB

14.4.2 CP

Grade 11 Core Mathematics

GOMATH WORKBOOKS

14.5 In the figure B and D are points in the same horizontal plane as C, the foot of a A

vertical tower AC. AC = BD = x and BDˆ C  90  

C 

Show that: BC  x

1 sin 2 

 2 cos 

Grade 11 Core Mathematics

GOMATH WORKBOOKS

DATA HANDLING: Data Collection: 1. Understand data collection as a potential source of bias. 2. Choose a sample that is free from bias. 3. Understand the need for random sampling methods. 4. Develop appropriate and relevant survey questions. Choosing your source. 

must be appropriate



whole group of things called :



the reference group



Target population.



Researchers that need to find information to help them understand an area will do research under specific conditions, defined Carefully ahead of time.



Data involving peoples opinions collected by:



1. A population census is collected from every member of the target population.



2. Survey – a sample or selected part of the target population.

A census requires a lot of planning ,is time consuming and costly. A survey is a more convenient way of collecting data from a ample of the community. Chosen people must be representative of the community. Too many of a certain section of the community will produce a biased or unrepresentative outcome. Biased data - an imbalance towards a particular result in the data. Not a true reflection of target population. The group chosen for a survey is called the sample. NB: Sample must be large enough to be representative but small enough to be convenient tom work with. NB:

Free from bias: Need to generalize findings Representative sample are more accurate.

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Random Numbers: A number chosen by chance – each number must have an equal chance of being the result. Random number tables: Lists from 0 to 9 that have random selection. Simple random samples: Chosen from target population where every person in population has had an equal chance of being chosen. Stratified Random Samples: Where target population separated into groups according to relevant information can help to make sample selection more representative—less biased. Univariate Numerical Data: Univariate data involves working with single variables or frequencies of single variables. Know: mean ; median; mode; quartiles; range inter-quartile range and semi-quartile range; Notation used: Mean of a set of data : x 

Sum Total of values  number of the values

x n

SKILLS NEEDED: 1.

Summarise sets of data by calculating The 5 –number summary.

Calculate the variance and standard deviation of sets of data manually and using technology(calculator)

Representing data – box & whisker diagrams – Ogives – histograms – and frequency tables.

Compare sets of data.

Effectively communicate conclusions from analysed data.

Differentiate between symmetrical and skewed data.

Critically analyze data misrepresentations.

Represent data with 2 variables as a Scatter plot and suggest types of graphs that will fit the data.

Grade 11 Core Mathematics

NB: 1.

Mean The average of the scores.

Mode The most repeated score.

Median ( Middlemost) in a ordered set of data.

Range = Maximum Value minus Minimum Value.

GOď&#x201A;ˇMATH WORKBOOKS

Exercise 3.1: 1.

Class results for a test out of 30 are recorded in the table below:

10A 16 12 16 11 14 15 22 16 17 15 26 23 16 22 16 17 24 19 16 10B 20 19 14 10 14 9 8 13 14 30 27 23 24 28 17 29 20 16 14 18 10C 5 20 14 12 7 2 12 21 14 26 14 14 12 14 21 24 14 14 1.1

Calculate the mean for each class.

1.2

Calculate the mode for each class.

1.3

Calculate the median for each class. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4

Calculate the range for each class. ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

1.5

GOď&#x201A;ˇMATH WORKBOOKS

Calculate the lower quartile (Q1) for each class.

1.6

Calculate the upper quartile (Q3) for each class.

1.7

Calculate the inter-quartile range for each class.

1.8

Calculate the semi-quartile range for each class ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

The table below shows the masses ( in kg) of people who go to gym on a regular basis. Complete the table and hence calculate the estimated mean.

Class interval: kg’s

Frequency

10 – 19 20 – 19 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99 100 – 109

0 1 3 7 18 31 38 49 13 0

Midpoint class interval 14,5 24,5 34,5 44,5 54,5 64,5 74,5 84,5 94,5 104,5

Frequency x midpoint 0 24,5 103,5 311,5 981 1999,5 2831 4140,5 1228,5 0

Grade 11 Core Mathematics

GOMATH WORKBOOKS

Example: Using energy as an issue. Energy consumed per person in same sub-Saharan African countries. Country Angola Benin Cameroon Congo Congo-Dem Rep Cote d’voire Eritrea Ethiopia Gabon Ghana Kenya Mozambique Namibia Nigeria Senegal South Africa Sudan Tanzania Toga Zambia Zimbabwe 1.

Energy use per person 323 164 280 141 204 151 108 6 659 206 243 278 98 585 138 272 205 312 57 309 430

Ranking 18 8 15 6 9 7 4 1 21 11 12 14 3 20 5 13 10 17 2 16 19

(a) Which has least residential energy per person? Why so low. Answer: Ethiopia has smallest residential sector energy per person. A poor country that uses a lot of wood as fuel source. (b) Which has highest per person? Is it extreme? Why? Answer: Gabon has the highest residential sector energy use per person. The mean per person in Gabon is not an extreme value as there is only a moderate gap between it and its closest data value ( Nigeria: 585)

What is the range for the data set? Answer: Range is 659 -6 = 653.

What is the size of the data set? Answer: n = 21

(a) Ranked data set – get the median. What does this mean? Answer: Median = 206 When the values are ordered from smallest to largest, the median is the middlemost value. (b) How many countries above the median and how many below? Answer: 21 data values. Middle value is 11th = 206. 10 countries lie above and 10 below.

Grade 11 Core Mathematics

GOMATH WORKBOOKS

(c) What percentage of the data lies below the median? Answer: Of data values 50% lie above and 50% lie below the median. 5.

(a) Find Q1 : What % below? Answer: Q1 = 141. 25% of the data lies below 1st quartile. (b) Find Q3: What % above? Answer: Q3 = 309. 25% of data lies above 3rd quartile. 6. Calculate the inter-quartile range. What does this mean? Answer: Inter-quartile range : 309 – 141 = 168. It is the middle 50% of the data. 7.

5 number summary Answer: Minimum Value 6 Q1 = 141 Median = 206 Q3 = 309 Maximum Value =

659

Key Ideas: 1. Extreme Data value – lies far away from rest of data (maybe errors or a genuine observation) 2. Range: Max – Min. 3. Median: Middlemost score: divides data set into 2 halves. 4. Median of Ist half called First quartile (Q1): lower quartile. Median of 2nd half called Q3 or third quartile: upper quartile. 5. Inter-quartile Range : Q3 – Q1 Measures the spread of middle 50% of data. If large then there is a high level of variation in the data. If Small then data is widely spread. 6. 5 number summary – Min; Q1; Median; Q3; Max. 25% of data lies below Q1 thus 75% above 75 % of data lies below Q3 thus 25% above.

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Representing 5-number summary: Median

Box and whisker diagram:

Highest Value

Lowest Value

659

Q1 141 100

309

206 200

600

300

Ogive curve drawn from the Box & Whisker Diagram:

100

0 100

200

300

700

600

659

141 100

206 200

309 300

600

Key Ideas: 1.

Quartiles in 5 - Number Summary gives an indication of frequency of data.

Ogive or Cummulative Frequency Graph of 5- Number Summary gives a rough idea of how the data values in a set are distributed.

Slope will be steeper where quartile range closest together indicating a higher concentration of data values. Slope thus a measure of the spread of data. Steep slope closer with small range. A gradual slope shows spread out data with a greater range.

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MEAN, VARIANCE AND STANDARD DEVIATION USED TO SUMMARIZE DATA SET. Learners in Portia’s class record the number of hours out of 24 that they switch lights on at home . Portia wants to compare her use of electricity to others. She wants to calculate the average or mean electricity use of lights among the learners taking part in the exercise. First Column Second Column Third Column Fourth Column Data Value (Data value – mean) (Data value – mean)² Portia 7 1 1 Nomfundiso 7 1 1 Unathi 8 2 4 Busiswe 4 -2 4 Pearl 5 -1 1 Sonwabo 6 0 0 Zuzeka 2 -4 16 Bulelani 9 3 9 Total 48 0 36 mean 

48 6 6

36  4,5 8 sum of squared Variance  n-1 36 Variance  7 Variance  5,14

Squared deviation =

Std Deviation  5,14 Std Deviation  2,27

Key Ideas: 

If one adds up all the differences between the data values and the means the answer will be zero (0) because the negative values will ‘:cancel out” the positive ones. For this reason, the square of the differences is used for the average.



The variance is defined as the sum of the squared differences between each data value and the sample mean, this sum then divided by one less than the number of data values in the sum.



The standard deviation is defined as the square root of the variance.



The square root of 5,44 is 2,27 Because the square root has been taken the variance has been reduced to a measurement that has the same units as the data. If the data is symmetrical, a SD of 2,27 suggests that roughly two-thirds of the data values should fall between 2,27 units below the mean and 2,27 units above the mean.

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Measures of Central Tendency:

First Column Northern and Western European Countries Poland Ireland Netherlands United Kingdom Germany Denmark Norway Sweden Belgium Iceland Total

Second Column Third Column Household energy (Data value – mean) use per person(kgoe) 503 647 654 716 774 822 854 903 937 2114 8924

Fourth Column (Data value – mean)²

-389,4 -245,4 -238,4 -176,4 -118,4 -70,4 -38,4 10,6 44,6 1221,6 0

151632,36 60221,16 56834,54 31116,96 14018,56 4956,16 1474,56 112,36 1989,16 1492306,56 1814662,4

mean  892,4 1814662,4 Variance   201629,1556 9 StdDev  449,031 5 number summary;

503;654;798;903;2114

100

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2114

503 654

798

903

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Key Ideas:     



M.O.C.T.describe the middle or typical value in a data set. M.O.D. measure the spread of data. Outlier values: data values that are very different from others- i.e. they lie far from the median. Outlier values have a large effect on calculations of mean and std deviation. Causing statistics to change as outliers are included or omitted form data sets. Generally speaking if data is symmetrically distributed the mean will be a 2 good measure of location and ± of the data will often lie between 1 (one) 3 std deviation above and below the mean. If skewed the median becomes a better measure of central tendency and the inter-quartile range a better measure of spread as they are not affected by outliers as mean and std dev are.

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Grouped Data: Frequency tables and histograms. Examine represented data after grouping data values into convenient intervals. Representations will include : 1.

Histograms

Frequency polygons

Ogive curves.

For the above we first need a frequency table. Choose a class interval to group the data values. Example: Class Interval

Class width

0 – 100 100 – 200 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800 800 – 900

100 100 100 100 100 100 100 100 100

Class Mark (midpoint of interval) 0 50 150 250 350 450 550 650 750 850

1300 – 1400

100

1350

2500 – 1600

100

2550

Total NB: Class intervals 0 – 100 include up to 99. 100 is in the next set 100 – 200.

Frequency 0 0 1 3 6 3 2 2 1 1 0 1 0 1 0 21

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

Histograms and frequency tables give a quick picture of the spread of data.



Skewed data – median a better measure of middle data.



For symmetrical data use the mean as a measure. HISTOGRAM 8

Median (419)

F r e q

Mean (587,62)

u e n c y

200

400

600

800

1000

1200

1400

Per capita energy use (kgoe) for Sub-Saharan Africab coun tries (n = 21)

   

Data is skewed to the right Median is 419 Mean is 587,62 The median is a better measure of middle or centre of the data. FREQUENCY POLYGON 8

F r e q

u e n c y

200

400

600

800

1000

1200

Per capita energy use (kgoe) for Sub-Saharan Africab coun tries (n = 21)

1400

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CUMULATIVE FREQUENCY TABLE: Class Interval 0 – 100 100 – 200 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800 800 – 900 1300 – 1400 2500 – 1600 Total

Frequency 0 0 1 3 6 3 2 2 1 1 0 1 0 1 0 21

Cumulative Frequency

CUMULATIVE FREQUENCY GRAPH - OGIVE CURVE:

200

400

600

800

2600 1200 1000 1400 Per capita energy use (kgoe) for Sub-Saharan Africab coun tries (n = 21)

0 1 4 10 13 15 17 18 19 19 20 20 21 21 198

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The Normal distribution curve: In general for a large sample the graph will be bell shaped. The reason for this is that a few scores will be high and a few low. The majority of scores will be in the central region. The results are thus said to be normally distributed. If we have a normal distribution then:  The mean , median and mode will be the same.  It is symmetric and has an equal number of scores on either side of the mean.  The majority of scores lie within 3 std dev’s from the mean.  Two thirds (67%) of sample lie within one std dev of mean.  95% lies within two std dev’s of mean.  99% of sample lies within three std dev’s of mean. Symetrical and Skewed Data A curve is symmetrical when one half is a mirror image of the other. NB: Not all corves are symmetrical. If extremely high or low scores are added to distribution the the mean tends to shift towards these scores and the curve becomes skewed. 

If the greater number of scores are massed on the right and a few are much lower scores than most others then the distribution is negatively skewed ( Skewed to the left)



If the greater number of scores are massed on the left and a few are much higher scores than most others then the distribution is positively skewed ( Skewed to the right) The relative position of the mean and median gives an idea whether the distribution is symmetrical, negative of positively skewed. (mean – median)= 0 Symmetric (mean – median) > 0 Positively skewed (skewed to right) (mean – median) < 0 negatively skewed. ( skewed to left)

   

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Example: 1.

The following marks were recorded for a maths class: 28 53 75 63

45 75 63

36 58 75 63

36 60 78 67

36 60 81 68

38 60 83 68

45 71 84 69

42 71 84 76

45 75 90 79

1.1 1.2 1.3 1.4

Do a stem and leaf diagram for the data Find the median, mode and mean for the data Find the lower and upper quartile Calculate: 1.4.1 the interquartile range 1.4.2 the semi-interquartile range 1.4.3 the range for the class 1.5 Write down the maximum and minimum scores. 1.6 Do a box and whisker diagram using the five-number summary

Answer: Stem 2 3 4 5 6 7 8 9

Leaf 8 6668 2555 38 0003337889 115555689 1344 0

Mode = 75 ; Mean = 62.9 ; Number = 35 Interquartile range = 30 ; Semi- interquartile ; range = 15 Range = 62 Standard Deviation = 16.6 Lowest = 28 ; Q1 = 45 ; Median = 67 ; Q3 = 75 ; Highest = 90

28 45 0

67 50

75 80

100

x  Q2  62.9  67  4.1  0 Data is negatively skewed i.e. skewed to the left. The marks are concentrated to the right of the median and spread out to the left of median.

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Exercise 3.2: 1. The number of points scored by four (4) Formula One racing drivers over a number of races is in the table below A B C D

1 1 1 2

1 2 1 2

1 6 2 2

2 8 2 4

6 8 4 4

6 8 4 6

8 8 6 6

8 8 6 8

1.1

Calculate the mean for each of the drivers.

8 8 8 8

8 10 8 10

10 10 10 10

10 10 10

10 -

List the Five Number Summary for each driver.

1.3

Calculate the difference between the mean and median for each driver.

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1.4

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Draw a Box and Whisker plot for each driver.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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1.5

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Discus each drivers distribution of scores in terms of the spread about the median and mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6

Compare the performance results for each driver by using the information obtained above. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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The following set of data records the number of chocolates sold by a convenience store over a period of 44 days.

2.1

Draw a Stem and Leaf Plot to organize the data.

List the Five Number Summary. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.3

Draw a Box and Whisker Plot. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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2.4

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Draw the cumulative frequency graph (Ogive Curve) using the Box and Whisker plot as a starting point. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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% Frequency

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The following table (grouped frequency distribution) shows the mark obtained by 220 learners in a Science exam. 1-10 2 3.1

11-20 6

21-30 11

31-40 22

41-50 39

51-60 59

61-70 45

71-80 20

81-90 91-100 11 5

Complete the cumulative frequency table below for this data:

Marks

Class midpoint

Frequency

1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 Total

5.5 15.5 25.5 35.5 45.5 55.5 65.5 75.5 85.5 95.5

2 6 11 22

Cumulative Frequency 2 8 19 41

3.2 On a set of axes draw the cumulative frequency graph ( Ogive Curve) for the data.

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3.3

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Determine the lower quartile (Q1).

3.4

Determine the median.

3.5

Determine the upper quartile (Q3). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

The following is list of heights of learners in a class. Heights are in centimeters(cm) 152 149 151 4.1

153 155 142

147 153 183

151 167 168

138 180 150

181 132 145

159 157 145

Determine: 4.1.1 median ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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4.1.2

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arithmetic mean

4.1.3

standard deviation

4.1.4

first and third quartiles.

Draw a box-and-whisker diagram for the data set.

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Complete the table below:

Interval 0  x  10 10  x  20 20  x  30 30  x  40 40  x  50 50  x  60 5.1

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Class midpoint 4,5 24,5 44,5 54,5

Frequency 7 11 22 25 10 0

Cumulative frequency 7 18 65 75

Draw a histogram of the information.

5.2

Determine the mean and standard deviation.

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5.3

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Draw a graph to determine whether the median is likely to be closer to 30 or to 40.

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6.1

100

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The ages to nearest year of 27 members of a Cricket Club are : 17 21 23 19 27 18 20 21 28 18 21 24 30 25 19 22 27 35 27 22 20 30 27 21 23

31 18

Organize the ages using a stem & leaf diagram. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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6.2

101

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Using 5 classes, and starting at 16, construct a frequency table and histogram for the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.3

Use the histogram to construct a frequency polygon on the same set of axes.

6.4

Describe the shape of the frequency polygon. Use the shape to predict the relation ship between the median and mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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6.5 6.5.1

102

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Find: Median ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.5.2

An estimate of the mean using grouped data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6.5.3

Mean – median ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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103

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The following table shows the prices , correct to nearest rand, of second-hand VW Golf cars for sale in Car Finder . a magazine listing second-hand cars for sale. Prices in Rands 0 - 19 999 20 000 – 39 999 40 000 – 59 999 60 000 – 79 999 80 000 – 99 999 100 000 – 119 999 120 000 – 139 999 140 000 – 159 999

7.1

Frequency 8 19 45 25 23 9 11 1

Find estimates of: 7.1.1 The mean and standard deviation of the prices. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7.1.2

The median price.

Calculate mean – median.

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104

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7.2.2. Use the answer to predict whether the distribution is symmetrical, positively skewed or negatively skewed. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7.2.3 Draw a histogram to illustrate the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.2.4

Describe the shape of the histogram. Does the answer confirm the prediction in 7.2.2.

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105

Bivariate Data and Scatter Plots: Relationships between two variables can cause problems in investigation and interpretation. 

Scatter plots can be utilized to show the relationship between 2 sets of information.



If the cloud of data produced is gently sloping before rising sharply the relationship may represent an exponential function.



If the cloud has a steep positive slope and then decreases rapidly and becomes steeply negative it could be represented by a quadratic function. For linear relationships:



.If correlation is positive the cloud slopes to the right.



If correlation is negative the cloud slopes to the left.



No correlation points scattered all over.

The following representations numbered A to H are all scatter plots. A.

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A. B. C. D. E. F. G. H.

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106

Parabola Function (with a < 0 ; steeply positive then steeply negative) Exponential Function. No Relationship or correlation Linear Function (positive correlation) Linear Function (negative correlation) Linear Function (No correlation) Quadratic Function. (with a > 0; Steeply negative going to steeply positive) Linear Function (No correlation)

Exercise 3.3: 1.

Botanists compared the use of growth hormones and the length of growth in plants:

Look at the table below and answer the questions that follow: Amount of growth hormones in ml 0,5 1 2 1,5 2 2,5 3 2,5 3 1,5 1.1

Plant growth in cm 1,5 2 2,5 3,5 4 5 8 7 6 3

Draw a scatter plot of the data.

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1.2

107

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Why do we represent the amount of growth hormone on the x â&#x20AC;&#x201C; axis?

Discuss the correlation between the variables.

1.4

Suggest whether a linear, quadratic or exponential function would best fit the data? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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108

The following marks were recorded for a maths class:

64 88 75 2.1

80 75 67 80

75 62 72 65

74 65 74 68

72 55 50 79

66 73 64 89

53 84 75 72

82 90 80

80 78 80

Do a stem and leaf diagram for the data

2.2

Find the median, mode and mean for the data

2.3

Find the lower and upper quartile ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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2.4

109

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Calculate:

2.4.1 the interquartile range.

2.4.2

the semi-interquartile range.

2.4.3

the range for the class.

2.4.4 Write down the maximum and minimum scores.

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2.5

110

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Do a box and whisker diagram using the five-number summary (L;Q1;M; Q3;H). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.6

Standard Deviation. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.7

What % of scores lie within 1 standard deviation from the mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.8

What % of scores lie within 2 standard deviations of the mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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111

The following marks were recorded for a maths class: 53 68 70 76

3.1

60 75 55 65

45 70 67 74

50 67 64 67

62 50 60 75

75 73 70 54

85 90 82 67

76 80 85 68

69 67 64

Do a stem and leaf diagram for the data

3.2

Find the median, mode and mean for the data ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.3

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3.4

112

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Calculate:

3.4.1 the interquartile range.

3.4.2

the semi-interquartile range.

3.4.3

the range for the class.

3.5

Write down the maximum and minimum scores.

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3.6

113

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3.7

3.8

3.9

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114

The following marks were recorded for a maths class: 23 48 30 46

4.1

30 45 25 45

31 50 35 44

52 47 24 53

42 20 30 45

15 43 40 54

45 60 52 35

36 40 75 28

29 37 34

Do a stem and leaf diagram for the data

4.2

4.3

Find the lower and upper quartile

54 38 45

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4.4

115

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Calculate:

4.4.1 the interquartile range.

4.4.2

the semi-interquartile range.

4.4.3

the range for the class.

4.5

Write down the maximum and minimum scores.

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4.6

116

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4.7

4.8

4.9

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117

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Misrepresenting Data: 

Misrepresenting data can happen both intentionally and unintentionally. It is always important to find out the source of the data to be able to critically evaluate its graphs and tables.



In reports, the actual data is not usually given in detail. It is up to us to read the graphs critically and, if we feel the graph is a misrepresentation, to question why.



Changing the information on the axes of the graphs ( or simply leaving any axes labeling out altogether) is a common used technique for distorting information.



The kinds of changes that can be made to information given on sets of axes:



Adjusting the scales on one or both of the axes.



Varying the scale on an axis



Changing the scale around on one of the axes.

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118

OPTIONAL TO COMPLETE IN GRADE 11 IF TIME AVAILABLE:

GRADE 12 STATISTICAL DATA.

MODIFYING BOX AND WHISKER DIAGRAMS. Box and Whisker diagrams: The inter-quartile range gives the spread of the middle 50% of data values and is not affected by the extremes. Outliers are values that stand apart from the rest of the values. An outlier is a value that is more than 1,5 times the interquartile range from the nearest quartile. E.G. If Q1 = 46 and Q3 = 60 then the IQR = 14. IQR x 1,5 = 14 x 1,5 = 21 Q1 – 21 =46 – 21 = 25. : Any value less than 25 will be an outlier Q3 + 21 = 60 + 21 = 81 : Any value greater than 81 will be an outlier. Outliers greatly affect the mean but have no more affect on the median or mode than any other value. THE BOX AND WHISKER DIAGRAM CAN BE MODIFIED TO MAKE IT MORE DESCRIPTIVE BY EXCLUDING OUTLIERS. If Q1 = 20 and Q3 = 34 then the IQR = 14. IQR x 1,5 = 14 x 1,5 = 21 Q1 – 21 =20 – 21 = -1. : Any value less than -1 will be an outlier Q3 + 21 = 34+ 21 = 55 : Any value greater than 55 will be an outlier.

Original Box & Whisker

80 20

Modified Box & Whisker 12

55 20

34 30

100

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119

STANDARD DEVIATION: From a given table of data: 67; 70; 71; 71; 73; 74; 75; 75; 75; 77; 78; 78; 78; 78; 79; 80; 81; 82; 82; 83; 86; 86; 87; 91

67 70 71 73 74 75 77 78 79 80 81 82 83 86 87 91 x  78,2

1 1 2 1 1 3 1 4 1 1 1 2 1 2 1 1

n  24

F x1 67 70 142 73 74 225 77 312 79 80 81 164 83 172 87 91

xx

( x  x )2

11,2 8,2 7,2 5,2 4,2 3,2 1,2 O,2 -0,8 -1,8 -2,8 -3,8 -4,8 -7,8 -8,8 -12,8

125,44 67,24 51,84 27,04 17,64 10,24 1,44 0,04 0,64 3,24 7,84 14.44 23,04 60,84 77,44 163,84

 (x  x ) 1

STD DEV

s

 ( x  x1 ) 2 n 1

 

 (x  x ) 1

= 652,24

 5,8

s  5.9

Standard Deviation is a Measure of Dispersion about the mean: It measures how far each data item is from the mean and takes into account all data items. If the differences of the scores above the mean are added to the differences below the mean the answer eill be zero. The differences are recorded as positives whether the score is above or below the mean. Variance is defined as OR

 (x  x ) 1

n 1

, when working with a sample of a population.

when working with a population n Variance is called the standard deviation and is considered the best measure of dispersion. The symbol “  ” is used to denote Standard Deviation when referring to a population AND “s” when referring to a sample of a population. A small standard deviation indicates that the data items are clustered around the mean. While a large standard deviation indicates that the items are more spread out.

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120

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STATISTICAL DATA & THE CALCULATOR Mean & Std Deviation using a calculator The Casio f(x) 82ES and STD Dev Key MODE  2: STAT THEN 1 –VAR To enter Data into table:

key (data list) into each row.

To calculate the MEAN : key SHIFT -1 THEN No 5 : VAR : 2: x To calculate Std Dev: key SHIFT : 1 then 5 : VAR then 3 xn [OR 4 : xn  1] Mean & Standard Deviation on a Frequency Table using a Casio: Key MODE 2 : STAT then 1: 1 – VAR. Set a table up: Key SHIFT SETUP Scroll down to new screen: then key 3:STAT ; 1 : ON. Enter Data into column 1 and frequency into column 2 When complete key [AC] For the MEAN: key Shift 1 then 5:VAR then 2 : x . For Std Dev: key Shift 1 then 5 : VAR 3: xn [OR 4 : xn  1] Example: Speed in Kph 5060708090100110120-

Midpoint of Interval X 55 65 75 85 95 105 115 125

No of cars ‘f’ 20 27 25 54 21 15 8 5

mean  82,48 StdDEv  17,5 n  175 175 Median lies in interval 80 – 89 thus Median = 84,5  87,5 2

Total No fX 1100 1755 1875 4590 1995 1575 920 625

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SYMMETRIC & SKEWED DATA:

a) Equal spread either side of the median in a box & Whisker diagram portrays a symmetrical spread. b) If data values are spread out more on one side than the other of the median then the data is said to be skewed.

xM 0

SYMMETRICAL

NORMAL DISTRIBURION

Q2 DIAGRAM SKEWED TO LEFT

X X M 0 NEGATIVELY SKEWED DISTRIBUTION

DIAGRAM SKEWED TO RIGHT

 X

 X M 0 POSITIVELY SKEWED DAT

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Using Standard Deviation to reach conclusions: Provided that a sample is reasonably large and the data is not too skewed ( that is , it does not have some very large or very small values), it is possible to make the following approximate statements. 

About 66% of the individual observations will lie within one standard

deviation of the mean. 

For most sets of data, about 95% of the individual observations will lie within

2 standard deviations of the mean. 

Almost all of the data will lie within 3 standard deviations of the mean.

Exercise: 4.1 1. The following marks were recorded for a maths class:

45 75 84

54 53 75 63

46 58 75 92

44 81 78 67

22 60 60 68

28 54 37 68

37 71 56 69

56 71 25 76

45 44 90 98

1.1 Do a stem and leaf diagram for the data.

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1.3 Find the lower and upper quartile.

1.4 Calculate: 1.4.1

the interquartile range.

1.4.2

the semi-interquartile range.

the range for the class.

______________________________________________________________ ______________________________________________________________ 1.4.4

Write down the maximum and minimum scores.

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124

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1..4.5 Do a box and whisker diagram using the five-number summary (L;Q1;M; Q3;H) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4.6

Standard Deviation

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4.7

What % of scores lie within 1 standard deviation from the mean.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4.8

What % of scores lie within 2 standard deviations of the mean.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4.9

State whether the data is negatively or positively skewed and give a reason for your decision.

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125

The following table represents the maths scores for the entire grade 11 maths group at Northwood School. The data is grouped due to the size of group.

Class 0 to 9 10 to 19 20 to 29 30 to 39 40 to 49 50 to 59 60 to 69 70 to 79 80 to 89 90 to 99 100 to 109 Totals 2.1 2.2

Frequency(f) 15 10 17 40 35 22 20 20 15 5 1 200

Mid-points(X) 4.5 9.5 14.5 19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5

fX 67.5

Complete the last column of the table i.e (fX) Find the modal class ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.3

Find the median class ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.4

Find the interval where Q1 and Q3 lie. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.5

Calculate the estimated mean. ď&#x192;Ľ fX NB estimated mean = n ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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2.6

126

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Use the grouped data to display the data on a histogram ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.7

Draw the relevant frequency polygon on the histogram.

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127

Complete the table and calculate the variance and the standard deviation.

40; 50; 65; 65; 70; 75; 75; 75; 75; 78; 78; 78; 78; 78; 79; 80; 81; 81; 82; 82; 82; 86; 88; 90

40 50 65 70 75 78 79 80 81 82 86 88 90 x  75,5

1 1 2 1 4 5 1 1 2 3 1 1 1

n  24

F x1 40 50 130 70 300 390 79 80 162 246 86 88 90

xx

( x  x )2

35,5 25,5

1260.25 650.25

-12,5 -14,5

156.25 210.25

 (x  x ) 1

STD DEV

s s

 ( x  x1 ) 2 n 1

 

 ( x  x1 ) 2 n

 

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128

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Complete the table

Height (h) in cm 135  h < 140 140  h < 145 145  h < 150 150  h < 155 155  h < 160 160  h < 165 165  h < 170 170  h < 175 175  h < 180 4.1

Mid points

Frequency

137,5

Cumulative Frequency 2

142,5

147,5

Coordinates (140 ; 2)

Calculate the estimated mean.

4.2

Draw a histogram of the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.3

Draw a frequency polygon on the histogram.

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4.4

129

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State the modal group, median height ,upper and lower quartiles for the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.5

Sketch the Ogive Curve for the data.

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Volume and Surface Area of 3-D Shapes Formulae: Volumes of Rectangular Prisms:

V  (area of base)  height 1. Square Base: V  (side of base)2  height (length) V  length  breadth  height (length 2. Rectangular Base : 1 3. Trapezium Base : V  sum of parallel sides   height (length) 2 1 4. Triangular Base : V  (base  height)  length 2 Volume of cylinders: V  (area of base)  height V   r 2h Volume of a Cone: 1 V   r 2h 3 Volume of Pyramid: 1 V  (area of base)  height 3 Volume of Sphere: V 

4  r3 3

Surface Areas of Shapes: Hint Draw a net diagram of the shape: Net Diagrams of 3 D shapes: Rectangular Prisms:

Volume  side1 side2  side3 Volume = area of base X height l Net: Rectangular Prisms: b

hxl

b h

hxl

hxb

b hxl

hxl Surface area = 2(h x l) + 2( h x b) + 2(l x b)

hxb

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Net : Square base prism

Surface area = 4(h x s) + 2( s x s) s Net of a Cylinder

sxh

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sx s

sxh

sx s

2r

Surface Area = area of circles plus area of reactangle Surface Area = r 2  r 2  2rh Surface Area = 2r 2  2rh

Cone: Surface Area = r h 2  r 2

Sh  h 2  r 2

Or Surface area =

1 circumference x slant height 2

NB Slant height =

h2  r 2 Circumference = 2  r

Net Diagram of a cone:

Slant height

Arc length Formulae for Surface Area of Shapes. PRISMS: Rectangular bases:

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S ď&#x20AC;˝ 2lb ď&#x20AC;Ť 4hb Triangular Bases: 1. S ď&#x20AC;˝ bh ď&#x20AC;Ť bl ď&#x20AC;Ť 2l b 2 ď&#x20AC;Ť h 2

ď&#x192;&#x17E; If triangle is isosceles.

Slant height

perpendicular height

Length Base

1.2 S ď&#x20AC;˝ bh ď&#x20AC;Ť 3bl ď&#x192;&#x17E; If equilateral.

perpendicular height

Base Length

Base

Cylinders: S ď&#x20AC;˝ 2ď ° r 2 ď&#x20AC;Ť 2ď ° rh S ď&#x20AC;˝ 2(ď ° r 2 ď&#x20AC;Ť ď ° rh)

Square base pyramid: 1 slant height ď&#x20AC;˝ ( base) 2 ď&#x20AC;Ť (height ) 2 2 1

đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;&#x;đ?&#x2018;&#x201C;đ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x2019; đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; = (đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019; )2 + 4(2 đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018; đ?&#x2018;&#x2019; Ă&#x2014; đ?&#x2018; â&#x201E;&#x17D;) Side refers to the length of the base of the pyramid) Base refers to the length of the base of the triangular side of the pyramid OTHER 3- D SHAPES Cone:

CurvedSurface ď&#x20AC;˝ ď ° r h 2 ď&#x20AC;Ť r 2 or 1 CurvedSurface ď&#x20AC;˝ circumference ď&#x201A;´ slant height 2 Sphere: S ď&#x20AC;˝ 4ď ° r 2

Grade 11 Core Mathematics

133

Pictures of Different 3-Dimensional Shapes:

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134

Surface area and volume of cylinder with section removed: Calculate the surface area of the shape: Calculate the ratio of arc to circle 

Arc Leave  as a symbol Circumference

Area of Arc(sector) = Use the ratio above and multiply  r2 ( area of full circle) Area of flat sections = l  b (length x breadth) they are rectangles. Area of curved surface = Arc  height Total Surface area = 2( area of arc) + 2( area of rectangular flat sections + area of curved surface Find the volume of the shape. Volume of shape = Area of arc x height of cylinder. Surface area and volume of cone with section removed: Calculate the ratio of arc to circle

Arc Leave  as a symbol Circumference

Area of Arc(sector) = Use the ratio above and multiply  r2 ( area of full circle) Calculate the Slant Height as follows ( Pythagoras) Slant height = h 2  r 2 1 Area of curved surface = ( area of curved sector) x slant height 2 1 Area of straight side =  base  height 2 Total area = area of sector( arc) + area ofn curved side + 2( area of straight side) Volume of Cone =

1 (Area of Arc) x height 3

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135

Volumes of prisms & the effect of the factor--k Exercise 5.1: 1. Calculate the volume and surface area of the following closed prisms:

Prism P Q R S T 52 47 43 39 36 Length (mm) 20 18 17 15 14 Breadth (mm) 85 77 70 64 58 Height (mm) Volume Surface Area Determine the following ratios correct to 2 decimals. VolumeP VolumeQ 2.1 2.2 VolumeR VolumeQ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.3

VolumeR VolumeS

2.4

VolumeS VolumeT

SurfaceAre aP SurfaceAre aQ 3.2 SurfaceAreaQ SurfaceAre aR ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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3.3

136

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SurfaceAre aR SurfaceAre aS 3.4 SurfaceAre aS SurfaceAre aT ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Are the volumes of the prisms approximately in proportion? Give reasons for your answers. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

5.1 How much smaller in volume is prism T than prism P? Give the scale factor (not the change in volume). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 5.2 Are the surface areas of the prisms in proportion? Give reasons for pour answers. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 5.3 How much smaller in surface area is prism T than prism P? Give the scale factor (not the change in area). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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137

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Determine the scale factor used : 6.1 to reduce the dimensions of the prisms. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 6.2

To enlarge the dimensions of the prisms. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

7. What is “The Golden Ratio” ? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 7.1

Determine which ratio of the faces comes closest to this ratio. NB: You must choose a ratio greater than 1. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

8.1 Reduce each of the dimensions of prism P by a factor of

1 , then calculate the 2

volume and surface area of the new prism X. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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138

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8.2 How much smaller in volume and in surface area is this new prism X? Give the scale factor in each case. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 9.1

Using the answers to question 8 , estimate the volume and surface area of prism Y, where each dimension of prism P has been enlarged by a factor of 2. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

9.2

Calculate the volume and surface area of prism Y, and compare your answers to your answers to question 9.1

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139

Examples: h b l Dimensions of the prism is length; breadth and height. i.e. l ; b ; h Volume of prism = l  b  h [ k is a factor of 2] Prism l (cm) b (cm) H (cm)

A B C D E F G H

4 8 4 4 8 8 4 8

3 3 6 3 6 3 6 6

2 2 2 4 2 4 4 4

Volume (cm3)

Vxk

24 48 48 48 96 96 96 192

V Vx2 Vx2 Vx2 Vx4 Vx4 Vx4 Vx8

No of Factor Sides Doubled 0 k 1 k 1 k 1 K2 2 2 K 2 k2 2 3 k 3

It is noticed : When 1 dimension is doubled then the volume is doubled as well When 2 dimensions were doubled then the volume is 4 times the original. When all 3 dimensions are doubled the volume is 8 times the original. This holds for any factor value. i.e. k = 3 the volumes increase accordingly : 1 trebled thus volume trebled 2 trebled thus volume is 9 times original 3 trebled thus volume is 27 times the original. Factors affecting the volumes are:

Volume of prism = l  b  h [ k is a factor of 3] Prism l (cm) b (cm) h (cm) A B C D E F G H

4 4 4 12 12 12 4 12

3 3 9 3 9 3 9 9

2 6 2 2 2 6 6 6

Volume (cm3) 24 72 72 72 216 216 216 648

Vxk Factor V Vx3 Vx3 Vx3 Vx9 Vx9 Vx9 V x 27

k k k k2 k2 k2 k3

Grade 11 Core Mathematics

10.

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140

Complete the following table:

h b l Dimensions of the prism is length; breadth and height. i.e. l ; b ; h Volume of prism = l  b  h Factor k =______. Prism

l (cm)

b (cm)

h (cm)

Volume (cm3) 72

Vxk V

Factor k

Vx2

Volume of prism = l  b  h Factor k =______. Prism

l (cm)

b (cm)

h (cm)

Volume (cm3) 24

Vxk Factor V Vx3

Vx Vx

216

Vx 9

Vx 216

Vx Vx

Grade 11 Core Mathematics

11.

141

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A cold - drink can measures approximately 65 mm in diameter and 75mm in height. 11.1 Calculate the volume of the can ( in mm3 and cm3). ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 11.2

The writing on the can says that it contains 200 ml of liquid. How much air space is there in the can? ( 1ml ď&#x20AC;˝ 1cm 3 )

What is the height of the liquid in the can?

Grade 11 Core Mathematics

12.1

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142

Calculate the total surface area of the can (in cm2 ) , assuming that the can is a closed cylinder. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

12.2

If the metal to make the can costs 0,25 cents (4 đ?&#x2018;&#x153;đ?&#x2018;&#x201C; đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ą) per square centimeter, calculate the cost of making each can. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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143

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Exercise 5.2: Use the formulae provided at the beginning of this section to do this exercise. Find the volumes and Surface areas of the following solids: 5.2.1

đ??śđ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2019; đ?&#x2018;¤đ?&#x2018;&#x2013;đ?&#x2018;Ąâ&#x201E;&#x17D; â&#x2C6;ś đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;˘đ?&#x2018; = 4đ?&#x2018;?đ?&#x2018;&#x161; & â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x2013;đ?&#x2018;&#x201D;â&#x201E;&#x17D;đ?&#x2018;Ą = 11đ?&#x2018;?đ?&#x2018;&#x161;

5.2.2

đ?&#x2018;&#x2020;đ?&#x2018;?â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2019; đ?&#x2018;¤đ?&#x2018;&#x2013;đ?&#x2018;Ąâ&#x201E;&#x17D; â&#x2C6;ś đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;˘đ?&#x2018; = 8đ?&#x2018;?đ?&#x2018;&#x161;

Grade 11 Core Mathematics

5.2.3

144

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𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑟𝑖𝑠𝑚: ℎ𝑒𝑖𝑔ℎ𝑡 = 10𝑐𝑚 ; 𝑙𝑒𝑛𝑔𝑡ℎ = 25𝑐𝑚; ℎ𝑒𝑖𝑔ℎ𝑡 = 5𝑐𝑚

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 11 Core Mathematics

145

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Apothem From Wikipedia, the free encyclopedia

Apothem of a hexagon The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent and have the same length. For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane parallel to the base), the apothem is the height of a trapezoidal lateral face. [1] For a triangle (necessarily equilateral), the apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's centers, since an equilateral triangle's centers coincide as a consequence of the definition.

Grade 11 Core Mathematics

146

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𝑆𝑞𝑢𝑎𝑟𝑒 𝑏𝑎𝑠𝑒 𝑝𝑦𝑟𝑎𝑚𝑖𝑑: 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 = 71,2 𝑚𝑒𝑡𝑟𝑒𝑠 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑏𝑎𝑠𝑒 = 233,5 𝑚𝑒𝑡𝑟𝑒𝑠 𝑇ℎ𝑒 𝒂𝒑𝒐𝒕𝒉𝒆𝒎 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑙𝑖𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 𝑡𝑜 𝑖𝑡𝑠 𝑐𝑒𝑛𝑡𝑟𝑒 .

5.2.4

Grade 11 Core Mathematics

5.2.5

147

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𝑆ℎ𝑎𝑝𝑒 𝑖𝑠 𝑎 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑎𝑛𝑑 𝑎 𝑐𝑜𝑛𝑒: 𝑟𝑎𝑑𝑖𝑢𝑠 = 5𝑐𝑚 ; 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 = 20𝑐𝑚 √89 𝑐𝑚 𝑎𝑛𝑑 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 ℎ𝑒𝑖𝑔ℎ𝑡 =

Grade 11 Core Mathematics

5.2.6

148

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𝑆ℎ𝑎𝑝𝑒 𝑖𝑠 𝑎𝑐𝑜𝑛𝑒 𝑜𝑛 𝑎 ℎ𝑒𝑚𝑖𝑠𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒: 𝑟𝑎𝑑𝑖𝑢𝑠 = 3,5𝑐𝑚 𝑎𝑛𝑑 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑛𝑒 = 10𝑐𝑚

Grade 11 Core Mathematics

5.2.7

149

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𝑆ℎ𝑎𝑝𝑒 𝑖𝑠 ℎ𝑎𝑙𝑓 𝑎 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟: 𝐿𝑒𝑛𝑔𝑡ℎ 𝑖𝑠 20 𝑐𝑚 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑏𝑎𝑠𝑒 = 6 𝑐𝑚

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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150

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5.2.8

Grade 11 Core Mathematics

151

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8. CIRCLE GEOMETRY Introduction: Circle Geometry is the study of shapes and angles formed within circles.

Shapes involved: Quadrilaterals Parallogram Rectangle Square Rombus Kite Trapezium

Triangles: Equilateral Isosceles Right Angled Scalene

Parallel lines and angles formed Corresponding Angles Alternate Angles Co-interior Angles

Geometry Theorems:

Grade 11 Core Mathematics

152

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Chord Theorems: Theorem 1: (proof required) The line segment joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. (reason: Mid-pt chord theorem)

Given: Circle centre O and C, the midpoint of AB. RTP: OC  AB Proof: Join AO and OB In ∆AOC & BOC AC = CB ( C is midpt) OC is common AO = OB (radii) ∆AOC ≡∆BOC ( SSS) C1 = C2 But C1 + C2 = 180 ( L’s on a str line) C1 = C2 = 90 OC  AB 2.

Converse of 1. (proof required) The perpendicular from the centre of a circle to a chord bisects the chord. ( Reason: Mid pt chord theorem)

Given:

Circle centre O and OC  AB

R.T.P.

AC = CB

Proof:

Join AO and OB In ∆AOC & BOC AC = CB ( C is midpt) OC is common C1 = C2 (both 90) ∆AOC ≡∆BOC ( RHS) AC = CB

Theorem 3: (proof required)

Grade 11 Core Mathematics

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153

The angle which an arc of a circle subtends at the centre of the circle is twice the angle it subtends at any point on the circle. ( Reason: Angle at centre) C C

A O O 1

O 1

1 2

B A A

Fig 1 fig 2 fig 3 Given: Circle centre O , đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ľ the angle at the centre subtended by AB and đ??´đ??śĚ&#x201A; đ??ľ the at the circumference. RTP: đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ľ = 2đ??´đ??śĚ&#x201A; đ??ľ Proof: Join CO and produce to D AO = OC (Radii) đ??´Ě&#x201A; = đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A; (Lâ&#x20AC;&#x2122;s opp = sides) đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = đ??´Ě&#x201A; + đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A; (ext L triangle) ď &#x153; đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ľ = 2đ??´đ??śĚ&#x201A; đ??ľ Similarly đ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??ľđ??śĚ&#x201A; đ?&#x2018;&#x201A; Thus in figure 1 & 2 đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ + đ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A; + 2đ??ľđ??śĚ&#x201A; đ?&#x2018;&#x201A; = 2(đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A; + 2đ??ľđ??śĚ&#x201A; đ?&#x2018;&#x201A;) = 2đ??´đ??śĚ&#x201A; đ??ľ Thus in fig3. đ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ â&#x2C6;&#x2019; đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??ľđ??śĚ&#x201A; đ?&#x2018;&#x201A; â&#x2C6;&#x2019; 2đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A; = 2(2đ??ľđ??śĚ&#x201A; đ?&#x2018;&#x201A; â&#x2C6;&#x2019; đ??´đ??śĚ&#x201A; đ?&#x2018;&#x201A;) = 2đ??´đ??śĚ&#x201A; đ??ľ

Grade 11 Core Mathematics

154

GOď&#x201A;ˇMATH WORKBOOKS

Theorem 4: (proof not required) The angle subtended at the circle by a diameter is a right angle. (Reason: Angle in a semi-circle) C

Given: Circle centre O and AB the diameter. đ??´đ??śĚ&#x201A; đ??ľ is an angle in the semi-circle. RTP: đ??´đ??śĚ&#x201A; đ??ľ = 90Â° Proof: đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ľ = 2đ??´đ??śĚ&#x201A; đ??ľ ( L at centre = 2 L at circumference) đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ľ = 180Â° ( St line) 2đ??´đ??śĚ&#x201A; đ??ľ = 180Â° đ??´đ??śĚ&#x201A; đ??ľ = 90Â° Theorem 5: (proof required) Angles in the same segment are equal: (Reason: ď&#x192;? in same segt) B

D A

Given: Circle centre O and đ??´đ??ľĚ&#x201A; đ??ˇ and đ??´đ??śĚ&#x201A; đ??ˇ angles in the same segment. RTP: đ??ľĚ&#x201A; = đ??śĚ&#x201A; Proof: Join AO and OD đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??ľĚ&#x201A; ( L at centre = 2 L at circle) đ??´đ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??śĚ&#x201A; ( L at centre = 2 L at circle) 2đ??ľĚ&#x201A; = 2đ??śĚ&#x201A; đ??ľĚ&#x201A; = đ??śĚ&#x201A;

Grade 11 Core Mathematics

GOMATH WORKBOOKS

155

Theorem 6: (proof not required)_ Equal angles at the centre stand on equal chords: A B A

O O

O C D B

Theorem 7: (proof not required) Equal angles are subtended by equal chords: (Reason: equal ,s; equal chord) F

E A D

Theorem 8: (proof not required) If two chords of a circle are equal , then they are equidistant from the centre: (Reason :equal chords; equidistant) D

A E

Grade 11 Core Mathematics

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156

Theorem 9: (proof not required) Two chords of a circle are equal (or of two different circles) if they subtend supplementary angles at the circumference. i.e. if đ?&#x2018;Ľ + đ?&#x2018;Ś = 180Â° then AB = DE C y

A D E x B

Cyclic Quadrilaterals: (quads within a circle) Theorem 10: (proof required) The opposite angles of a cyclic quad are supplementary: (Reason: opp ď&#x192;?,s cyclic quad) A

Given: Cyclic quadrilateral ABCD with circle centre O. Ě&#x201A; = 180Â° RTP: đ??´Ě&#x201A; + đ??śĚ&#x201A; = 180Â° & đ??ľĚ&#x201A; + đ??ˇ Proof: Join BO and DO đ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??śĚ&#x201A; ( L at centre = 2 L at circle) đ?&#x2018;&#x2026;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122;đ?&#x2018;&#x2019;đ?&#x2018;Ľđ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 2đ??´Ě&#x201A; ( L at centre = 2 L at circle) đ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ + đ?&#x2018;&#x2026;đ?&#x2018;&#x2019;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122;đ?&#x2018;&#x2019;đ?&#x2018;Ľđ??ľđ?&#x2018;&#x201A;Ě&#x201A;đ??ˇ = 360Â° ( Lâ&#x20AC;&#x2122;s at a point) 2đ??śĚ&#x201A; + 2đ??´Ě&#x201A; =360Â° đ??śĚ&#x201A; + đ??´Ě&#x201A; = 180Â° Ě&#x201A; = 180Â° Similarly it can be proved that đ??ľĚ&#x201A; + đ??ˇ

Grade 11 Core Mathematics

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157

Theorem 11: (proof required) The exterior angle of a cyclic quad is equal to the interior opposite angle. (Reason: ext ď&#x192;? cyclic quad) A

C E

Given:

Cyclic quad ABCD with BC produced to E.

RTP: đ??ˇđ??śĚ&#x201A; đ??¸ = đ??´Ě&#x201A; Proof: đ??ˇđ??śĚ&#x201A; đ??¸ + đ??ˇđ??śĚ&#x201A; đ??ľ = 180Â° (Lâ&#x20AC;&#x2122;s on a st line) đ??´Ě&#x201A; + đ??ˇđ??śĚ&#x201A; đ??ľ = 180Â° (opp Lâ&#x20AC;&#x2122;s of a cyclic quad) đ??ˇđ??śĚ&#x201A; đ??¸ = đ??´Ě&#x201A;

Tangents to Circles: Theorem 12. (proof not required) A radius( or Diameter) is always perpendicular to a tangent at the point of tangency. (Reason: rad ď &#x17E; Tan)

Grade 11 Core Mathematics

158

GOď&#x201A;ˇMATH WORKBOOKS

Theorem 13: (proof not required) Tangents drawn from the same point to a circle are equal in length: (Reason: tan from same point)E

Theorem 14. (proof required) The angle formed between a tangent and a chord is equal to the angle in the alternate segment (Reason: Alt Segt Thm)

D E O

Given: Tangent AB touching circle centre O at C. Chord CE Ě&#x201A; & đ??¸đ??śĚ&#x201A; đ??´ = đ??šĚ&#x201A; RTP: đ??¸đ??śĚ&#x201A; đ??ľ = đ??ˇ Proof: Draw diameter COG and join DG. Ě&#x201A;2 = đ?&#x2018;Ś (i) Let đ??şĚ&#x201A; = đ?&#x2018;Ľ đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??ś Ě&#x201A; đ??şđ??¸ đ??ś = 90Â° ( L in a semi-circle) đ?&#x2018;Ľ + đ?&#x2018;Ś = 90Â° ( L sum â&#x2C6;&#x2020;) đ?&#x2018;Ś + đ??¸đ??śĚ&#x201A; đ??ľ = 90Â° ( Rad ď &#x17E; Tan) đ?&#x2018;Ľ = đ??¸đ??śĚ&#x201A; đ??ľ Ě&#x201A; ( Lâ&#x20AC;&#x2122;s in same segt) BUT đ?&#x2018;Ľ = đ??ˇ Ě&#x201A; đ??¸đ??śĚ&#x201A; đ??ľ = đ??ˇ Ě&#x201A; + đ??šĚ&#x201A; = 180Â° ( opp Lâ&#x20AC;&#x2122;s cyclic quad) (i) đ??ˇ đ??¸đ??śĚ&#x201A; đ??ľ + đ??¸đ??śĚ&#x201A; đ??´ = 180Â° (Lâ&#x20AC;&#x2122;s on st line) Ě&#x201A; (proved in i) But đ??¸đ??śĚ&#x201A; đ??ľ = đ??ˇ đ??¸đ??śĚ&#x201A; đ??´ = đ??šĚ&#x201A;

Grade 11 Core Mathematics

GOï&#x201A;·MATH WORKBOOKS

159 GEOMETRIC RYDERS (PROBLEMS)

O is the centre of the circle. AB = 60mm ; OM = 40mm Calculate the radius of the circle and the Length of CD if ON = 30mm

N C O

M A

CD = 80 mm; AB = 60mm and AB // CD If the radius is 50mm find the distance between the chords

B A

CD // AB C

Grade 11 Core Mathematics

GOMATH WORKBOOKS

160

Prove CD = 2AB NB: A and B are centres of the circles.

M is the mid-point of AB, O is the centre. Prove AMC  BMC

_M _C

Grade 11 Core Mathematics

GOMATH WORKBOOKS

161

Find the sizes of x and y in each case. 5.

6. A

A x O

110

70 C

x B

x A B

Grade 11 Core Mathematics

162

GOMATH WORKBOOKS

Prove that Bˆ1  Aˆ  90 O

1 B

1 C

Grade 11 Core Mathematics

GOMATH WORKBOOKS

163

Find the values x ; y and z in 11 to 16. 11.

12. A

D D

x z

60

2 1

20 B

x C

BOˆ C  130 and DOˆ C  60 ______________________________________________________________

Grade 11 Core Mathematics

GOMATH WORKBOOKS

164

13. F x A 2

1 20

4 B

85 D

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Grade 11 Core Mathematics E

14. 100

GOMATH WORKBOOKS

165

15. O is the centre of circle ABCD D

A z

O x

B E

120 D

y C

Grade 11 Core Mathematics

166

GOMATH WORKBOOKS

16. DE is a tangent to circle ABC. A z

O y

x 62 D

A 2 1

Grade 11 Core Mathematics

167

GOMATH WORKBOOKS

17. EF is tangent to circle ABCD and BC = CD . Find 5 angles equal to Cˆ1 , (giving reasons.)

18.

D A E

Grade 11 Core Mathematics

168

GOMATH WORKBOOKS

Chord AB is parallel to chord CD. Cˆ1  Cˆ 2

Prove that BC = DE

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

19.

D 1

Grade 11 Core Mathematics

169

GOMATH WORKBOOKS

Chord AD equals chord AC and Sˆ1  Sˆ3 Prove: 19.1 Eˆ  Bˆ 19.2 AE  AB

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 20.

Grade 11 Core Mathematics

170

AB and CD are two chords which intersect at T. AK  CD and DL  AB .

A T L

GOMATH WORKBOOKS

Prove: 20.1 AKLD is a cyclic quad 20.2 KL // CB. 20.3 If AK and Dl produced cut CB at M and N respectively, prove AMND is a cyclic quad.

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 21.

Grade 11 Core Mathematics

P 1 2

F 1 2

171

PA and PC are tangents to the circle at A and C. AD //PC and PD cuts the circle at B. CB is produced to meet AP at F. AB, AC and DC are joined.

A 4 3

4 3 2 B 1

2 3

GOMATH WORKBOOKS

1 2 D

Prove: 21.1 AC is the bisector of PAˆ D . 21.2 Bˆ1  Bˆ 3 . 21.3 AP = AC. 21.4 APˆ C  ABˆ D . 21.5 Aˆ 4  Pˆ2

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Grade 11 Core Mathematics

172 FEˆ C  ADˆ C (i.e.Eˆ 3  Dˆ 1 2 )

B 3 1 2

2 A

GOMATH WORKBOOKS

2 E1 3 6 4 5

Prove: 22.1 CEBF is a cyclic quad. 22.2 CGAF ARE concyclic. 22.3 AC bisects BCˆ G

1 2 D

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Grade 11 Core Mathematics

23.

173

AB is a diameter. ADP and BCP are straight lines. PQTR is a straight line.

1 2

D 1 2

GOMATH WORKBOOKS

4 1 T 3 2

1 Q 2 6 3 4 5

1 2

Prove: 23.1 DQCP is a cyclic quadrilateral. 23.2 If Qˆ 5  TBˆ C , then PT  AB 23.3 DATQ is a cyclic quad.

1 2 C

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Grade 11 Core Mathematics

174

GOMATH WORKBOOKS

24. AOD and EOB are diameters. AF  EB Prove : 24.1 EFHD is cyclic. 24.2 BAˆ D  DHˆ C . 24.3 Cˆ 3  Aˆ1 .

A 1 E

2 3

3 1

2 O F 1

24.4 2

2 2 1 H 1 2 D

EB bisects ABˆ C (i.e Bˆ 3  Bˆ1 2 )

1 1

3 C

Grade 11 Core Mathematics

GOMATH WORKBOOKS

175

25. T

In the figure BOD is a diameter of the circle with centre O. BA and BC are chords of the circle. BA produced and CD produced meet in T and AD produced and BC produced meet in S. Prove:

25.1 25.2 25.3

D O

ATSC is a cyclic quad

ADˆ B  ATˆS OA is a tangent to circle ATSC.

C S

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Grade 11 Core Mathematics

GOMATH WORKBOOKS

176

26. D 3 2

1 T 2 1

3 2 4

1 B W

In the figure above, TD is a tangent to circle ABCD at D. AD // BC, AB and DC produced meet at W. TBS is a straight line. If WBˆ T  CBˆ D , Prove that: 26.1 BWTD is a cyclic quadrilateral. 26.2 TBS is a tangent to the circle at ABCD. 26.3 TW // BC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________