Grade 12 workbook paper 2 by GoMath Workbooks

Compiled by Chesley Nell

2 GOMATH WORKBOOKS

Grade 12 Core Mathematics

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

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Grade 12 Core Mathematics

GRADE 12 CORE MATHEMATICS Contents: Paper Two: Topic:

Pages:

Analytical Geometry

( 4 – 15)

Trigonometry

(16 - 41)

Data Handling

(42 – 65)

Circle Geometry

(66 – 81)

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Grade 12 Core Mathematics

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 1  x 2 ) (y 1  y 2 )  ;   2 2  Mid – point =  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 Straight Line: y  mx  c or y  y1  m( x  x1 )

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

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Grade 12 Core Mathematics 5.

The equation of a circle centre not the origin

(x  a) 2  (y  b) 2  r 2 Where (a ; b ) represents the co-ordinates of the centre and (x ; y ) a point on the circumference.

Examples: 1. Determine the equation of a circle centre (2 ; 5) and radius 4 units (x  a) 2  (y  b) 2  r 2 (x -2)2 + (y – 5)2 = 42 (x -2)2 + (y – 5)2 = 16 OR

x2 – 4x +4 + y2 – 5y + 9 = 0

2. Determine the centre and radius of x2 + 6x + y2 – 4y - 12 = 0. One must use the method of “completing the square” to calculate the centre and radius. The given equation must be converted to the correct form i.e. (x  a) 2  (y  b) 2  r 2 x2 + 6x + y2 – 4y - 12 = 0. x2 + 6x +( 3 )2+ y2 – 4y + (2)2 = 12 + 9 + 4 (x + 3)2 + (y – 2)2 = 25 Centre ( -3 ; 2) and radius = 5

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Grade 12 Core Mathematics Tangents to Circles. Remember: Tangents are perpendicular to the radius of any circle.

Essential to know the centre of the circle in order to work out the gradient of the radius. Examples. 1. Determine the equation of the tangent to the circle x - 2x + y + 4y = 5 at the point (-2;-1) Use completing the square to form two binomials: To find the centre x2 - 2x + (-1)2 + y2 + 4y + (2)2 = 5 + (-1)2 + ( 2)2 (x-1)2 + (y+2)2 - 1 - 4 = 5 (x-1)2 + ( y +2)2 = 10 centre is (1;-2) gradient of radius =

 2 1 1 2

1 3

Gradient of tangent = 3

(radius perpendicular to tangent)

Equation of tangent is y = 3x + c Substitute (-2;-1) thus -1 = 3(-2) + 6 c=5 equation of tangent is y = 3x + 5

7 Grade 12 Core Mathematics Exercis 1.1: A:

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Analytical Geometry. Equations of circles of the form: (x  a) 2  (y  b) 2  r 2

1. Determine the equation of the circle with: 1.1 centre (2 ; 3) and radius 5 units.

1.2

Centre (4; -5) and radius 10 units

1.3

Centre (-2 ;-3) passing through (-2; 4)

8 Grade 12 Core Mathematics

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2. Determine the centre and radius of each of the following circles: 2.1

(đ?&#x2018;Ľ â&#x2C6;&#x2019; 4)2 + (đ?&#x2018;Ľ + 7)2 = 64

2.2

đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 â&#x2C6;&#x2019; 6đ?&#x2018;Ś â&#x2C6;&#x2019; 27 = 0

2.3

đ?&#x2018;Ľ 2 + 4đ?&#x2018;Ľ + đ?&#x2018;Ś 2 â&#x2C6;&#x2019; 5 = 0

2.4

đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 4đ?&#x2018;Ľ + đ?&#x2018;Ś 2 + 2đ?&#x2018;Ś â&#x2C6;&#x2019; 20 = 0

9 Grade 12 Core Mathematics 2.5

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đ?&#x2018;Ľ 2 + 6đ?&#x2018;Ľ + đ?&#x2018;Ś 2 â&#x2C6;&#x2019; 4đ?&#x2018;Ś â&#x2C6;&#x2019; 12 = 0

2.6

đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 + 4đ?&#x2018;Ľ + 6đ?&#x2018;Ś â&#x2C6;&#x2019; 3 = 0

Find the equations of the tangents in the following: 3.1

to circle đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 = 5 at the point (3 ; 4).

3.2

to circle đ?&#x2018;Ľ 2 + đ?&#x2018;Ś 2 = 36 at the point (â&#x2C6;&#x2019;2 ; 3).

10 Grade 12 Core Mathematics 3.3

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To circle (đ?&#x2018;Ľ + 1)2 + đ?&#x2018;Ś 2 = 20 which is parallel to 2đ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;Ľ = 0

3.4

to (đ?&#x2018;Ľ â&#x2C6;&#x2019; 2)2 + (đ?&#x2018;Ś + 3)2 = 16 which is parallel to the y â&#x20AC;&#x201C; axis.

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Grade 12 Core Mathematics B. 1.

A C 0

The Points A(-4 ;3) ; B(-4 ; -4) ; C(6 ; 1) and D(6 ; 8) lie on a cartesian plane. Determine: 1.1

the length of AD.

1.2

the mid-point of DC ______________________________________________________________ ____________________________________________________________

1.3

The gradient of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

12 Grade 12 Core Mathematics 1.4

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the inclination of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.5

the equation of BC ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6

Prove that ABCD is a parallelogram. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Points A( 2 ; - 3 ) ; B( - 1 ; p ) and C ( 4 ; 3 ) are co-linear. Calculate the value of p. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

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Grade 12 Core Mathematics 3 y

A(4 ; 12)

D ( 8; 4)

B(1 ; 3) C4;2) 0

In the figure ABCD is a quadrilateral with, A(4 ; 12 ); B(1 ; 3); C(4 ; 2) and D(8 ;4). 3.1

Find the length of AB. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.2

Find the co-ordinates of the mid-point of AC. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.3

If E(p ; -3) is co-linear with A and B, find the value of p. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

14 Grade 12 Core Mathematics 3.4

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Determine the angle of inclination of the line through BC with the x- axis. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.5 Determine the size of BCˆ D .

______________________________________________________________ ______________________________________________________________ 3.6

Show that AB  BC.

15 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 5.

Locus of a Point ( Not in syllabus)

The locus of a point P is the set of all possible positions/locations of P under given conditions. The locus is expressed in terms of x and y in equation form, since the point P can vary as long as the conditions are upheld. Example: NB a sketch is extremely useful when finding loci 1.Find the relationship between x and y if the distance of P(x;y) from the origin must remain equal to its perpendicular distance from the line x = 2. P(x ; x=2 0;0

Note: The points P form a parabola thus the locus of P will be a Parabola P is tyhe Pt (x;y) M is the point( 2;y) Condition on P is that PM = OP PM = OP ( y - y ) + (2 - x) = y - 0) + ( x - 0) 4 - 4x + x = y + x y = -4x + 4 or x = y + 1 (it is the inverse of y = - x + 1 This is the relationship between the co-ord's of P that satisfy the given condition ( PM + OP always) and this is the locus of P. P (x ; y) 5

2. P is 5 units from B(2;3).

B(2 ; 3)

P is the point(x ;y) BP = 5 thus BP2 = 25 (y - 3)2 + ( x - 2 )2 = 25 (square to get root of root sign.) y2 -6y + 9 + x2 - 4x + 4 = 25 thus y2 - 6y + x2 - 4x = 12 is the equation of the locus required.

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Grade 12 Core Mathematics

TRIGONOMTRY FORMULA SHEET Compound Formula: cos(a-b) = cosacosb + sina sinb

REDUCTION FUNCTION

cos(a+b) = cosa cosb - sina sinb 90°

Quadrant 2

sin(a-b) = sinacosb - sinb cosa



180  

sin(a+b) = sina cosb + sinb cosa 180°

Double Angles

Quadrant 1  90  

 90  

S A   180   T C 360  

cos2x  cos 2 x  sin 2 x/2cos 2 x  1/1  2sin 2 x Quadrant 3 sin2x  2sinxcosx

0 360

Quadrant 3 270°

Complimentary Functions: sinx  cos(90   x) x =0 y=1 r=1 90°° Quadrant 2

SPECIAL ANGLES Quadrant 1

x = -1 y = 0 180° r=1

S A T C Quadrant 3

0° 360°

x =1 y=0 r=1

Quadrant 4

270° x =0 y = -1 r=1

Basic ratios:

sin x 

y r

cos x 

x r

tan x 

y x

FUNDAMENTAL IDENTITIES ` Quotient Identities: Squared Identities: cos 2 x  sin 2 x  1 sin x tan x  cos x

17 GOMATH WORKBOOKS

Grade 12 Core Mathematics Solution of Triangles: sine rule :

a sin Aˆ



b sin Bˆ



c sin Cˆ

Cosine Rule: 1. a 2  b 2  c 2  2bc cos Aˆ

b2  c2  a2 OR cos Aˆ  2bc

a2  c2  b2 2. b 2  a 2  c 2  2ac cos Bˆ OR cos Bˆ  2ac 3. c 2  a 2  b 2  2ab cos Cˆ

OR cos Cˆ 

a2  b2  c2 2ab

Area rule : AreaABC  12 ac sin Bˆ ; 12 ab sin Cˆ ; 12 bc sin Aˆ

GRAPHS: y  asinb(x  c)  d ; ETC. a  affects the amplitude of the graph and also inverts the graph when negative.

b  affects the period or frequency c  shifts the graph horizontall

d  shifts the graph vertically

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Grade 12 Core Mathematics

2. Trigonometry: Compound Angles.

cos( A  B)  cos A cos B  sin A sin B cos( A  B)  cos A cos B  sin A sin B

Examples: 1. cos 2 x cos x  sin 2 x sin x

Reduce by using compound angle Formula cos(A+B)=cosAcosB-sinAsinB

 cos(2 x  x)  cos 3x 2. cos(3x  45 )  cos 3x.cos 45  sin 3x.sin 45

Use reduction function to reduce to acute angles

3. cos 290  . cos 320   sin110  . sin140   cos 70  . cos 40   sin 70  . sin40   cos(70   40  )  cos 30  1  2

sin( x  y)  sin x cos y  sin y cos x sin( x  y)  sin x cos y  sin y cos x

1. sin 2 x cos x  sin x cos 2 x

 sin(2 x  x)  sin 3x 2. sin(x  90)  sin x cos 90  sin90 cos x

 Sinx(0)  1(cos x )  cos x

Use special angles to obtain answer

19 GOMATH WORKBOOKS

Grade 12 Core Mathematics 3. sin 50 cos10  sin 10 cos 50

 sin(50  10)  sin 60 

3 2

4. Using compound angles to prove a statement.

3 sin( x  60)  sin( x  30)  cos x LHS  3[sin x cos 60  sin 60 cos x]  [sin x cos 30  sin 30 cos x] LHS  3[ 12 sin x  LHS 

3 2

cos x] 

sin x  32 cos x 

3 2

Use Expansion for sine Compound formula

sin x  12 cos x

LHS  cos x LHS  RHS . Exercise 2.1: 1. Use the compound angle formulae to simplify each expression to one term only: cos 3x cos 2x  sin 3x.sin 2x . 1.1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 1.2

sin 3x cos 2x  cos 3x.sin 2x . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.3

cos 5 x. cos 2 x  sin5 x sin2 x . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

20 Grade 12 Core Mathematics 1.4

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sin 3x.sin 2x  cos 3x. cos 2x . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.5

sin 2 x.sin x  cos 2x. cos x . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6

sin 2 x. cos x  cos 2x.sin x . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.7

sin50 . cos 10  cos 50 . sin10 . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.8

sin81 . cos 23  sin23 . cos 81 . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

21 Grade 12 Core Mathematics 1.9

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cos 18 . sin31  sin18 cos 31 . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2. Expand each of the following using compound angle formulae. 2.1 sin(x  20 ) . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 2.2

cos(2 x  10 ) . ______________________________________________________________

2.3

sin(a  2b) . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

22 Grade 12 Core Mathematics 2.4

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cos(a  2b) . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.5

sin(2a  20 ) . ______________________________________________________________

cos(a  30 ) . ______________________________________________________________

3. Evaluate the following without a calculator: sin40 . cos 20  cos 40 . sin20 3.1 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

23 Grade 12 Core Mathematics 3.2

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cos 40 . cos 20  sin40 . sin20 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.3

cos 100 cos 280  sin100 . sin280 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3.4

sin80 . sin40  sin10 . sin50 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4. Prove the following using compound angle formula: 4.1 sin(180   )   sin ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

24 Grade 12 Core Mathematics 4.2

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cos(360   )  cos ______________________________________________________________

cos(a  60 )  cos(a  60)  cos a ______________________________________________________________

5.3

sin(a  30 )  sin(a  30 )  cos a ______________________________________________________________

25 Grade 12 Core Mathematics 5.4

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cos( P  Q)  cos( P  Q)  2 cos P cos Q ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

5.5

sin5 A  sin3 A  2 sin4 A. cos A ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

6. Prove that: sin75  cos 105  sin15  cos 15  0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

26 Grade 12 Core Mathematics Double Angle Formulae. 1.

sin 2 x  2 sin x cos x i.e.

sin( x  x)  sin x cos x  sin x cos x  2 sin x cos x

cos 2 x  cos 2 x  sin 2 x OR  2 cos 2 x  1

OR  1  2 sin 2 x

cos 2 x  cos( x  x)  cos x cos x  sin x sin x  cos 2 x  sin 2 x OR i.e

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

tan 2 x 

2 tan x 1  tan 2 x

tan 2 x  tan( x  x) i.e

tan x  tan x 1  tan x tan x 2 tan x  1  tan 2 x 

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27 GOMATH WORKBOOKS

Grade 12 Core Mathematics

Proving Identities which include Double Angles. EXAMPLES: 1. Expand using appropriate sinx  sin2x  tanx formula 1  cosx  cos2x sin x  2 sin x cos x LHS  1  cos x  2 cos 2 x  1 sin x (1  2 cos x ) LHS  Factorise and cos x (1  2 cos x ) simplify sin x LHS  cos x LHS  tan x Simplify using LHS  RHS basic Identities 2. 8sinAcosA. cos2Acos4A  sin8A

RHS  sin 8 A

continue expanding using double angle formulae

RHS  2 sin 4 A cos 4 A RHS  4 sin 2 A cos 2 A cos 4 A RHS  8 sin A cos A cos 2 A cos 4 A RHS  LHS. 2. cos4A  cos2A 1  sin4A  sin2A tanA 2 cos 2 2 A  cos 2 A  1 look at the numerator and denomenato r searately 2 sin 2 A cos 2 A  sin 2 A (2 cos 2 A  1)(cos 2 A  1) LHS  and use the specific expansion needed. sin 2 A(2 cos 2 A  1) cos 2 A  1 LHS  If cos functions needed use only cos double L expansion etc. sin 2 A LHS 

2 cos 2 A 2 sin A cos A cos A LHS  sin A 1 LHS  tan A LHS  RHS LHS 

NB constants can be written in form of cos 2 x  sin 2 x 2  2sin 2 x  2cos 2 x etc.

28 Grade 12 Core Mathematics 3. cos3x  4cos 3 x  3cosx

LHS  cos(2 x  x) LHS  cos 2 x cos x  sin 2 x sin x LHS  cos x(2 cos 2 x  1)  2 sin x cos x sin x LHS  2 cos 3 x  cos x  2 sin 2 x cos x LHS  2 cos 3 x  cos x  2(1  cos 2 x) cos x LHS  2 cos 3 x  cos x  2 cos x  2 cos 3 x LHS  4 cos 3 x  3 cos x LHS  RHS 4. cos3x  1  4sin 2 x cosx cos(2 x  x) LHS  cos x cos 2 x cos x  sin 2 x sin x LHS  cos x cos 2 x cos x  2 sin 2 x cos x LHS  cos x LHS  cos 2 x  2 sin 2 x

LHS  1  2 sin 2 x  2 sin 2 x LHS  1  4 sin 2 x LHS  RHS .

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29 GOMATH WORKBOOKS

Grade 12 Core Mathematics 5. 1 sina  cosA  tan2A  cos2A cosA  sinA 1 sin 2 A LHS   cos 2 A cos 2 A 1  sin 2 A LHS  cos 2 A sin 2 A  2 sin A cos A  cos 2 A LHS  cos 2 A  sin 2 A (sin A  cos A)(sin A  cos A) LHS  (cos A  sin A)(cos A  sin A) sin a  cos A LHS  cos A  sin A LHS  RHS .

Exercise 2.2:

Rewrite 1 as sin2x + cos2x

Proving Identities.

2 cos x sin x  sin 2 x 1 sin 2 x

30 Grade 12 Core Mathematics 2.

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1 sin 2 x sin x  cos x   cos 2 x cos 2 x cos x  sin x

sin x  sin 2 x  tan x 1  cos x  cos 2 x

31 Grade 12 Core Mathematics 4.

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sin 3x  sin x  2 sin x 1  cos 2 x

sin 2 x  tan x 1  cos 2 x

32 Grade 12 Core Mathematics 6.

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sin 4 x  sin 2 x. cos 2 x  1  cos x 1  cos x ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

sin 2 x  tan x  tan x cos 2 x

33 Grade 12 Core Mathematics 8.

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sin 2 x  cos x cos x  sin x  cos 2 x sin x  1

1  cos 2 x tan 2 x  cos 2 x tan x

34 GOMATH WORKBOOKS

Grade 12 Core Mathematics General Solution of Trigonometric Equations:

If an equation is solved in general terms specific answers are not given unless asked for: The variable “k” is used to refer to the number of revolutions that are utilized to find the specific solution: Examples: 1.

Find the general solution for x in the following:

2 sin 2 x  1,630 sin 2 x  0,815 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:

2. 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

3. (basic) cos x  0,5

x  {-180  ;180  }

key  30  x  150   360k or Or x  30  180k

x  210   360k

If k  1 then x  150  If k  0 then x  150 

35 Grade 12 Core Mathematics 4.

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(Difficult)

2 cos 2 x ď&#x20AC;Ť cos x ď&#x20AC;Ť 2 ď&#x20AC;˝ 0 4 cos 2 x ď&#x20AC; 2 ď&#x20AC;Ť cos x ď&#x20AC;Ť 2 ď&#x20AC;˝ 0 cos x(4 cos x ď&#x20AC;Ť 1) ď&#x20AC;˝ 0

cos x ď&#x20AC;˝ 0 x ď&#x20AC;˝ 90 ď&#x20AC;Ť 180k ď Ż

4 cos x ď&#x20AC;˝ ď&#x20AC;1 cos x ď&#x20AC;˝ ď&#x20AC;0,25 keyď&#x192;? ď&#x20AC;˝ 75,5ď Ż x ď&#x20AC;˝ ď&#x201A;ą104,5ď Ż ď&#x20AC;Ť 360k

TRIG EQUATIONS INVOLVING COMPLIMENTARY ANGLES. Examples: 1. Cos x = sin40 = cos[90-40] convert sine to cosine by using a complimentary function) = cos 50 The Key Angle is now 50 and answers lie in quadrants 1 and 4 x = 50 + 360k OR x = 310 + 360k 2.

sin(2 x ď&#x20AC;Ť 5ď Ż ) ď&#x20AC;˝ cos(ď&#x20AC;20ď Ż ď&#x20AC;Ť x) = sin[90-(-20 + x)] Use the complimentary angle = sin (110 - x) {1/2}

Use (110 - x) for the KEY ANGLE

2 x ď&#x20AC;Ť 5ď Ż ď&#x20AC;˝ 110 ď&#x20AC; x ď&#x20AC;Ť 360k 3x ď&#x20AC;˝ 105 ď&#x20AC;Ť 360k x = 35 + k.120 OR

2 x ď&#x20AC;Ť 5ď Ż ď&#x20AC;˝ 180 ď&#x20AC; (110 ď&#x20AC; x) ď&#x20AC;Ť 360k

2 x ď&#x20AC;Ť 5ď Ż = 70Â° + x + 360k đ?&#x2019;&#x2122; = 65Â° + 360k

36 Grade 12 Core Mathematics

GOMATH WORKBOOKS

Exercise 2.3: A. Find the general solution for the following: 1.

sin x  0,235 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

3 cos x  1,2066 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

tan 2 x  4,302 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2 tan 3x  2,3648 3 ______________________________________________________________

37 Grade 12 Core Mathematics

GOMATH WORKBOOKS

B: Find the specific solutions of the above equations if x  [360 ;360  ] ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

38 GOMATH WORKBOOKS

Grade 12 Core Mathematics Exercise 2.4: Example: 1. Solve for 2 sin x  3 cos x  0 , where x  0 ;360





2 sin x  3 cos x 2 sin x 3 cos x  cos x cos x 2 tan x  3

1 to convert to cos x a tan function

multiply by

3 2 Key L = 56,3 tan x  

x  123,7  / 303,7 

Find the general solution of : 4sin2x – 3 = 0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Solve for x for x[-180;90] when cos2x – 7cosxtanx = 4 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

39 Grade 12 Core Mathematics 3.

GOMATH WORKBOOKS

Solve for x if 4sinxcosx = 1 and x[0;360] ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Find the general solution of 4cos2x – 6cosx + 5 = 0 ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

40 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 5.

2 Solve for : 2 sin x ď&#x20AC; sin x ď&#x20AC; 1 ď&#x20AC;˝ 0

ď &#x203A;

where x ď&#x192;&#x17D; 0ď Ż ;360ď Ż

ď ?

2 Solve for: 2 sin x ď&#x20AC;Ť 5 cos x ď&#x20AC;˝ 4

ď &#x203A;

where x ď&#x192;&#x17D; 0ď Ż ;360ď Ż

ď ?

Find the general solution of đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ľ(đ?&#x2018;Ľ + 20Â°) = đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x2018;Ľ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

41 Grade 12 Core Mathematics 8.

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Find the general solution of cos(2đ?&#x2018;Ľ + 23Â°) = sin(4đ?&#x2018;Ľ + 25Â°) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _____________________________________________________________

42 GOMATH WORKBOOKS

Grade 12 Core Mathematics

Data Handling

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

43 GOMATH WORKBOOKS

Grade 12 Core Mathematics 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

xx 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

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

( x  x )2 125,44 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 will 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.

44 GOMATH WORKBOOKS

Grade 12 Core Mathematics

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. 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

45 GOMATH WORKBOOKS

Grade 12 Core Mathematics 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 DATA

46 GOMATH WORKBOOKS

Grade 12 Core Mathematics

Stem & Leaf diagrams; standard deviation an Ogive curves. Example: 1. The following marks were recorded for a math 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 Do a stem and leaf diagram for the data 1.2 Find the median, mode and mean for the data 1.3 Find the lower and upper quartile 1.4 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.

47 GOMATH WORKBOOKS

Grade 12 Core Mathematics 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 3.1: 1.

The marks , out of 150, for 30 learners were as follows:

100

109

122

118

124

127

105

112

128

107

114

115

121

135

111

117

120

130

123

141

107

113

116

119

121

131

129

139

1.1

Organise the marks using a stem & leaf diagram. ____________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

48 Grade 12 Core Mathematics

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1.2 Draw a Box & Whisker diagram to illustrate the dispersion of the marks. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 1.3

Determine the mean for the above data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.4

Determine the standard deviation, ‘s’ (correct to 1 decimal place ) ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.5

What percentage of calls lie within one standard deviation of the mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

1.6

What can the teacher conclude about these marks. ______________________________________________________________ ______________________________________________________________ _____________________________________________________________ ______________________________________________________________

49 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 2.

The following marks were recorded for a maths class:

23 48 30 46 2.1

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

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 ______________________________________________________________ ______________________________________________________________

29 37 34

50 Grade 12 Core Mathematics 2.4

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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.

2.5

Write down the maximum and minimum scores.

51 Grade 12 Core Mathematics

GOď&#x201A;ˇMATH WORKBOOKS

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

2.7 Standard Deviation. 2.7.1

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

2.7.2

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

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

52 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 3.

The following marks were recorded for a math class: 80 75 67 80

64 88 75 3.1

75 62 72 65

74 65 74 68

72 55 50 79

66 73 64 89

53 84 75 72

82 90 80 90

Do a stem and leaf diagram for the data

3.2

Find the median, mode and mean for the data

80 78 80

53 Grade 12 Core Mathematics 3.3

GOď&#x201A;ˇMATH WORKBOOKS

Find the lower and upper quartile

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 3.4

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.

______________________________________________________________ ______________________________________________________________

54 Grade 12 Core Mathematics 3.6

GOď&#x201A;ˇMATH WORKBOOKS

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

3.7

Standard Deviation. 3.7.1 What % of scores lie within 1 standard deviation from the mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 3.7.2

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

55 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 4. Girls 64 Boys 75 4.1

The following marks for a class of Girls and Boys were recorded : 90 85 57 72

95 92 62 75

84 90 85 88

82 75 80 79

76 83 64 89

80 64 75 72

88 75 95 80

Do a back to back stem and leaf diagram for the data

4.2

Find the median, mode and mean for both sets of data

70 72 80 81

56 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 4.3

Find the lower and upper quartile of each set of data

4.4

Calculate: 4.4.1

the interquartile ranges for: 4.4.1.1

girls

4.4.1.2

boys

4.4.2

the semi-interquartile ranges for: 4.4.2.1 girls

57 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 4.4.2.2 boys

4.4.3

the ranges for: 4.4.3.1 girls ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.4.3.2

boys

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.5

Write down the maximum and minimum scores of each set of data ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.6 Do separate box and whisker diagrams for the girls and the boys ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _____________________________________________________________

58 Grade 12 Core Mathematics 4.7

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Standard Deviation : 4.7.1 the girls. ______________________________________________________________ ______________________________________________________________

4.7.2 the boys ______________________________________________________________ ______________________________________________________________

4.8

What % of scores lie within 1 standard deviation from the mean for: 4.8.1 girls ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.8.2 boys ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

4.9

What % of scores lie within 2 standard deviations of the mean for: 4.9.1 girls ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 4.9.2 boys ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

59 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics

5. 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

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

Mid-points(X) 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5 104.5

5.1

Complete the last column of the table i.e (fX)

5.2

Find the modal class

fX 67.5

Find the median class ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

5.4

Find the interval where Q1 and Q3 lie.

______________________________________________________________ 5.5

Calculate the estimated mean. NB estimated mean =

ď&#x192;Ľ fX n

60 Grade 12 Core Mathematics 5.6

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

5.7 Draw the relevant frequency polygon on the histogram.

61 GOMATH WORKBOOKS

Grade 12 Core Mathematics STANDARD DEVIATION: 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

xx 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

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

( x  x )2 125,44 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.

62 GOMATH WORKBOOKS

Grade 12 Core Mathematics 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.

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

63 GOMATH WORKBOOKS

Grade 12 Core Mathematics Exercise 3.2:

1. 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 35,5 25,5

( x  x )2

-12,5 -14,5

156.25 210.25

1260.25 650.25

 (x  x ) 1

STD DEV

s

 ( x  x1 ) 2 n 1

 

 ( x  x1 ) 2 n

s   ______________________________________________________________

64 GOMATH WORKBOOKS

Grade 12 Core Mathematics 2.

Complete the table below:

Height (h) in cm 135  h < 140

Mid points 137,5

Frequency 2

Cumulative Frequency 2

140  h < 145

142,5

145  h < 150

147,5

150  h < 155

155  h < 160

160  h < 165

165  h < 170

170  h < 175

175  h < 180

2.1

Coordinates (140 ; 2)

Calculate the estimated mean. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2.2

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

65 Grade 12 Core Mathematics 2.3

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Draw a frequency polygon on the histogram

2.4

State the modal group, median height ,upper and lower quartiles for the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 2.5

Sketch the Ogive Curve for the data. ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

66 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics

4.Circle Geometry: Ratio / Proportion and Similar Triangles: Proportion Theorem: Theorem: A line parallel to one side of a triangle divides the other two sides (internally or externally) in the same proportion. A

H K

â&#x2020;&#x2019;

Given: â&#x2C6;&#x2020;ABC , D on AB and E on AC ( or AB and AC produced in either direction) DE | | BC. Required to prove:

đ??´đ??ˇ đ??ˇđ??ľ

đ??´đ??¸ đ??¸đ??ś

Proof: Construction: Draw DK and EH the altitudes for bases AD and AE. Draw DC and BE. đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??´đ??ˇđ??¸ đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??ľđ??ˇđ??¸

đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??´đ??ˇđ??¸ đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??śđ??¸đ??ˇ

1 đ??´đ??ˇ.đ??¸đ??ť 2 1 đ??ˇđ??ľ.đ??¸đ??ť 2

đ??´đ??ˇ

1 đ??´đ??¸.đ??ˇđ??ž 2 1 đ??¸đ??ś.đ??ˇđ??ž 2

đ??´đ??¸

đ??ˇđ??ľ

đ??¸đ??ś

( đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; â&#x201E;&#x17D;đ?&#x2018;Ą đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; đ??´đ??ˇ)

( đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; â&#x201E;&#x17D;đ?&#x2018;Ą đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; đ??ˇđ??ž)

But đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??ľđ??ˇđ??¸ đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??śđ??¸đ??ˇ ( đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; đ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018; đ?&#x2018;&#x2019; đ??ˇđ??¸ ; đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x161;đ?&#x2018;&#x2019; â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x2013;đ?&#x2018;&#x201D;â&#x201E;&#x17D;đ?&#x2018;Ą; đ??ˇđ??¸ || đ??ľđ??ś)

â&#x2C6;´

đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??´đ??ˇđ??¸

â&#x2C6;´

đ??´đ??ˇ

đ??´đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D; đ?&#x2018;&#x153;đ?&#x2018;&#x201C; â&#x2C6;&#x2020;đ??ľđ??ˇđ??¸

đ??ˇđ??ľ

đ??´đ??¸ đ??¸đ??ś

67 GOMATH WORKBOOKS

Grade 12 Core Mathematics Similar Triangle Theorem:

Theorem: If two triangles are equiangular, then the corresponding sides are in proportion, and then the triangles are said to be similar. A  D 

̂ ; 𝐵̂ = 𝐸̂ 𝑎𝑛𝑑 𝐶̂ = 𝐹̂ Given: ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 with 𝐴̂ = 𝐷 Required to Prove:

𝐷𝐸

𝐴𝐵

𝐸𝐹

𝐵𝐶

𝐷𝐹 𝐴𝐶

Proof: In ∆𝐴𝑃𝑄 𝑎𝑛𝑑 ∆𝐷𝐸𝐹 𝐴𝑃 = 𝐷𝐸 (𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛) ̂ ( 𝐺𝑖𝑣𝑒𝑛) 𝐴̂ = 𝐷 𝐴𝑄 = 𝐷𝐹 ( 𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛) ∆𝐴𝑃𝑄 ≡ ∆𝐷𝐸𝐹 ( 𝑆𝐴𝑆) 𝑃̂1 = 𝐸̂ 𝑃̂1 = 𝐵̂

(𝐸̂ = 𝐵̂ ) (𝐶𝑂𝑅𝑅𝐸𝑆𝑃𝑂𝑁𝐷𝐼𝑁𝐺  ′𝑠 𝑒𝑞𝑢𝑎𝑙)

𝑃𝑄||𝐵𝐶 𝐴𝑃

𝐴𝑄

∴ 𝐴𝐵 = 𝐴𝐶

(𝑃𝑄||𝐵𝐶 𝑖𝑛 ∆𝐴𝐵𝐶)

𝐵𝑢𝑡 𝐴𝑃 = 𝐷𝐸 𝑎𝑛𝑑 𝐴𝑄 = 𝐷𝐹 𝐷𝐸

𝐷𝐹

∴ 𝐴𝐵 = 𝐴𝐶

68 GOMATH WORKBOOKS

Grade 12 Core Mathematics Exercise 4.1: Mixed examples on proportion and similarity: 1.

Calculate as requested. A

1.1

1.2

x+ 1

Find x in 1.1

Find BC in 1.2

1.3 8

D 12

1.4 D

x B



y 6

69 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 2.

In the figure GF//EA. BA is a tangent at B.

E G

Prove: 2.1 â&#x2C6;&#x2020;đ??´đ??ľđ??ś ď źď źď ź â&#x2C6;&#x2020;ADB . 2.2 đ??´đ??ľ 2 = đ??´đ??ˇ. đ??´đ??ś Ě&#x201A;2 . 2.4 đ??¸Ě&#x201A;2 = đ??ˇ 2.5 â&#x2C6;&#x2020;đ??¸đ??śđ??´|||â&#x2C6;&#x2020;đ??ˇđ??¸đ??´. 2.6 đ??´đ??¸ 2 = đ??´đ??ˇ. đ??´đ??ś. 2.7 đ??´đ??¸ = đ??´đ??ľ

1 2

x 1 2 3

C 2

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

70 Grade 12 Core Mathematics

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71 GOMATH WORKBOOKS

Grade 12 Core Mathematics 3.

In the diagram , the chords Ad and BC of circle ABCD are produced to meet at F . E is a point on AF such that EC is a tangent to the circle at C and BD//CE.

A 1

Prove with reasons: 3.1 BC = DC

2 3

1 2

3.2

BAF /// DCF B

3.3

BA DE  AF EF

3.4

ECD /// EAC CE 2 

3.5

2 5

2 3 4

AE.BC .EF CF

72 Grade 12 Core Mathematics

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73 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics 4.

4.1

In the diagram AC is a tangent, FE //AC. Prove with reasons: â&#x2C6;&#x2020;đ??ľđ??¸đ??šď ź ď ź ď ź â&#x2C6;&#x2020;đ??ľđ??šđ??ˇ

4.2

BE 2 ď&#x20AC;˝ BD.BF

D F

1 1

A ______________________________________________________________ B C

74 GOMATH WORKBOOKS

Grade 12 Core Mathematics 5.

In the diagram KL // PM and NM = LM

K Q

Prove with reasons: NR 1 5.1  NK 2 5.2

NQL /// PML

5.3

QL.PL = 2NM2

4 4

2 3

1 2

75 GOMATH WORKBOOKS

Grade 12 Core Mathematics 6.

O is the centre ; BP = OB = AO . PT is a tangent and EP  AP. E

Prove with reasons: 6.1 TEPB is a cyclic Quad. 6.2 6.3

ATB  APE . TP = PE.

O B

6.4

ATPEPB.

6.5

2BP2 = BT.BE

76 Grade 12 Core Mathematics

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______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _____________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ _____________________________________________________________

77 GOï&#x201A;·MATH WORKBOOKS

Grade 12 Core Mathematics 7

MN is a tangent at R and PQRS is a cyclic quadrilateral. QT // SR and PS // QR Prove with reasons:

Q 2 1

7.1

QR bisects PQT. P

7.2

MN //PT S M

78 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics đ??ľđ??ś = đ??śđ??ˇ đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??şđ??ľ is a tangent to the larger circle. Hint: đ??żđ?&#x2018;&#x2019;đ?&#x2018;Ą đ??śđ??¸Ě&#x201A; đ??š = đ??ľĚ&#x201A;2 = đ??¸Ě&#x201A; = đ?&#x2018;Ľ

Prove with reasons: 8.1

ď &#x201E;CDF /// ď &#x201E;CED

8.2 8.3 8.4

CDË&#x2020; E ď&#x20AC;˝ CGË&#x2020; B

ď &#x201E;CGB /// ď &#x201E;CDE EC.GB = DE . CD

1 3 1 2 2

3 x

F 1

79 Grade 12 Core Mathematics

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80 GOď&#x201A;ˇMATH WORKBOOKS

Grade 12 Core Mathematics

9.1 AB is a tangent to the smaller circle and BC is a common chord. Ě&#x201A;1 = đ?&#x2018;Ľ đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??ˇ Ě&#x201A;2 = đ?&#x2018;Ś đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą: đ?&#x2018;&#x2122;đ?&#x2018;&#x2019;đ?&#x2018;Ą đ??ˇ Prove with reasons: B 9.1 AC bisects BCË&#x2020; D 2 9.2

ď &#x201E;ABC /// ď &#x201E;DEC

9.3

BC.DC ď&#x20AC;˝ AC.EC

9.4

ADË&#x2020; C ď&#x20AC;˝ BAË&#x2020; C ď&#x20AC;Ť ACË&#x2020; D

9.5

ď &#x201E;ABE /// ď &#x201E;ACB

9.6

AC 2 ď&#x20AC;˝ AE. AC

A 2 1

1 2

E 1 1 2

81 Grade 12 Core Mathematics

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