western cape education department · web viewthe telematic teaching project is a collaborative...

Western Cape Education Department

TELEMATIC TEACHING PROJECT 2011

GRADE 11 MATHEMATICS RESOURCE MATERIAL

Mathematics Grade 11 Telematics Resource Materials 2011

Dear Grade 11 Learner

The Telematic teaching project is a collaborative initiative between the Western Cape Education department and University of Stellenbosch.

It is not meant to replace class room teaching or your teacher. It is meant as extra support in order to consolidate what your teacher does during normal classroom time.

It therefore implies that you are getting extra teaching and revision, presented by expert teachers who were specially appointed for this teaching intervention.

Mathematics is not a spectator “sport”. Success is determined by the measure of your participation and involvement beyond the teaching that you receive. It means that you must develop the necessary discipline and work ethics.

These lessons will also not cover the entire curriculum for Grade 11 but will focus on certain areasThis will be done on the basis of the revision question paper question papers included in this resource . The aim is to adequately prepare you for the final examination paper in mathematics.Please work out other papers as well with the help of your teacher.

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You are therefore encouraged to do your homework regularly and supplement this with a regular revision programme. In order for you to set up a regular study programme the following concept checklist will assist you to work systematically through the content: The papers included in this resource will assist to revise all these concepts and expose you to the type of questions that may be set in the examinationPAPER 1 PAPER 2

1. Algebra(40 marks)1.1 Algebraic fractions1.2 Exponents and surds1.3 Quadratic equations (factors and the formula)1.4 Quadratic inequalities1.5 Simultaneous equations

2. Number patterns (20 marks)2.1 General patterns (linear and quadratic)

3. Functions and graphs (50 marks)3.1 Line, Parabola; hyperbola; exponential and

their transformations

4. Financial mathematics (25 marks)4.1 Simple and compound interest4.2 Nominal and effective interest rates4.3 Depreciation (reducing balance and straight

line)

5. Linear programming (15 marks)5.1 Setting up constraints (inequalities)5.2 Sketch the feasible region5.3 Set up the objective function5.4 Investigate vertices of the feasible region

to optimise

1. Data handling (35 marks)1.1 Mean, median, mode, quartiles1.2 variance and standard deviation1.3 Five number summary and box and

whisker plots1.4 Cumulative frequencies and ogives1.5 Scatter plots and lines of best fit

2. Analytical geometry (35 marks)2.1 Distance formula, midpoint of a line

segment2.2 Gradient, inclination and equation of a line

3. Transformation geometry (20 marks)3.1 Basic transformations (translations,

reflections and rotations through 90° and 180°)

3.2 Enlargements

4. Trigonometry (50 marks)4.1 Basic definitions in a right angled triangle and on the Cartesian plane4.2 Reduction formulae4.3 Special angles4.4 Compound and double angle formulae4.5 Identities4.6 Equations (especiallygeneral solution)4.7 Solution of triangles and problems in 2D

(area; sine and cosine formulae)4.8 Trigonometric graphs

5. Measurement (10 marks)Surface area and volumes of right pyramids, cones and spheres

Good luck with you preparation for the final examination

Raymond SmithSenior Curriculum PlannerJuly 2011

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PAPER 1Time: 3 hours Marks: 150

Question 1

1.1 Solve for x in each of the following;1.1.1 7−5 x=x2

(5)

1.1.21+ x+1

x+2+ 2

x−1=0

(6)1.1.3 x2−6<−5 x (5)

1.2 Simplify each of the following

1.2.1

(3 x )−2

3 x−3(3)

1.2.2

x−1+ y−1

x−1 y− y−1 x (5)

1.2.3 √108 x12+√243 x12(3)

1.3 Solve for x : 23 x−6=√8 (3)

1.4 Solve for both x and y in the system of equations below.

xy+6=0 and x+3 y+3=0 (7)

[37]Question 2

2.1 Given the functions y= f ( x )=− 1

2( x+1 )2+2

and y=g ( x )=−2 x−6 :2.1.1 Write down the co-ordinates of the turning point of f

(2)

2.1.2 Calculate the roots of the equation f ( x )=0 (4)2.1.3 Write down the equation of the axis of symmetry of f . (1)

2.1.4 Sketch the graphs of y=f ( x ) and y=g ( x ) on the same system of axes (4)

2.1.5 Determine the values of x for which f ( x )≥g ( x ) (4)

2.1.6 Describe in words the difference between shape of y=f ( x ) and y=2 f (x )(2)

2.2 Given h( x )=( 1

4 )x−1

−2

2.2.1 Write down the equation of the asymptote of h (1)4.2 Determine the coordinates of the intercepts of h with the x and y axes (6)

4

D

C

B

A

O >x

^y


4.3 Write down the equation of the reflection of h( x )=( 1

4 )x−1

−2 in the y axis. (2)

[30]Question 3

3.1 Sketched below are the graphs of p ( x )= 3

x−1−2

and q ( x )=x−1

3.1.1 Calculate the co-ordinates of A and B. (4)

3.1.2 Write down the equation of the horizontal

asymptote of p ( x ) (2)

3.1.3 Write down the domain of p ( x ) (2)

3.1.4 Show that p (−2 )=q (−2 ) and state the significance of this fact to the sketch. (3)

3.1.5 Determine the co-ordinates of D if CD=4 where CD is perpendicular to the x -axis. (5)

3.2 The graphs below are the functions p( x )=a .bx−c and s( x )=bx−c

x

y

-2 -1 0 1 2

-2

-1

1

2

s

p

3.2.1 What is the value of c (2)

3.2.2 If the curve of s( x )passes through the point (1 ;1 ) , find the value of b. (3)

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3.2.3 If p( x ) cuts the y−axis at y=−2

3 , find the value of a. (3)3.2.4 Determine the equation of r if p( x ) is shifted 2 units to the right to create r ( x ) (2)

[26]

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Question 4

4.1 The diagram shows a sequence of patterns. Each one is made by surrounding the previous pattern (shade black) by squares that are shaded grey.

4.1.1 On the square grid on your diagram sheet, draw the next pattern (2)

4.1.2 There are two sequences formed. The first, is the number of squares added to each new pattern (the grey squares), the second is the total number of squares making up the pattern. Write down the first six terms of each sequence (6)

4.1.3 Find a formula for the n−th term in each sequence. (4)

4.2 Consider the following patterns of diamond shapes:

4.2.1 How many diamonds() are there in the next pattern? (2)4.2..2 How many diamonds are there in the nth pattern? (4)4.2.3 Which pattern has 960 diamonds? (4)

[22]

Question 5

5.1 Nthabi is running a small business. She has just bought equipment for R 500 000

5.1.1 She decides to depreciate the equipment at 20% p.a. on the straight line basis. When will she write the equipment off? (2)

5.1.2 Nthabi changes her mind and depreciates the equipment at 25% p.a. on the reducing balance. Calculate the value of the equipment after 5 years. Give your answer correct to the nearest Rand. (4)

5.2 After just 2 years, a laptop computer is one third it’s original value. Assuming reducing balance depreciation, what was the annual rate of depreciation? (5)

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5.3 Byron deposits R2500 into a bank account and makes no withdrawals for 8 years. At the end of the fifth year he deposits an additional R1200. If the interest rate for the first 4 years is 8% p.a compounded quarterly and 9,5% p.a compounded semi-annually for the remaining four years, what will have accrued in the account at the end of the eighth year. (7)

5.4 Which is the better investment offer: 10,28% p.a. compounded daily (use 365 days in a year) or 10,3% p.a. compounded monthly? (6)

[24Question 6

The sketch below represents the feasible region ABCDEF of a linear programming problem.

8

6

4

2

5 10>x

F

E

DC

B

A

6.1 Two of the constraints are 1≤x≤6 and 6 y+5 x≤60 . Write down the inequalities that represent the other constraints. (5)

6.2 The point P(5 ; p ) lies in the feasible region and the point Q (q ;3 ) is not in the feasible region. Write down one possible value of p and one possible value of q . (2)

6.3 Find the values of x and y for which the objective function O1=8 x+4 y has a minimum value. (2)

6.4 Write down the minimum value of O1 (1)

6.5 Find the values of x and y for which O2=3 y+4 x is maximised. (2)

6.6 Write down the maximum value of O2 (1)

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[13]

9

y

x

L( -5 ; -2 )

M( -1 ; -6 )

K( 5 ; 4 )

A( 4 ; 3 )

y

x

C( -2 ; 5 )

D( 0 ; 1 )

B( 2 ; 7 )


PAPER 2

Paper 2Time: 3 hours Marks: 150

Question 1

1.1 In the diagram below, L(-5; -2), M(-1; -6) and K(5; 4) are the vertices of KLM in a Cartesian plane.

Determine:

1.1.1 Q, the midpoint of MK (3)

1.1.2 the gradient of LM (3)

1.1.3 the inclination of LM (3)

1.1.4 The length of LM (3)

1.1.5 the equation of the line parallel to LM passing through N. (3)

1.1.6 Show that the line in question 1.5 passes through the point QP the midpoint of KL(4)

1.1.7 Show that LM = 2PQ (4)

1.2 In the diagram below, A(4 ; 3), B (2 ; 7), C (-2 ; 5) and D (0 ; 1) are four points in a Cartesian plane.

1.2.1 Show that AC BD (5)

1.2.2 Show that AC bisects BD (4)

1.3.3 State, giving a reason, which type of quadrilateral ABCD is.

(2)

[31]

10

8

6

4

2

-5 5

y

x

C'

B'

A'

CB

A

6

4

2

5

y

x

R

M

T

P


Question 2

2.1 The diagram alongside shows Δ ABC with its transformation Δ A/ B/ C/.

2.1.1 Write down the coordinates of A and A/ (2)

2.1.2 Describe the above transformation. (2)

2.1.3 If Δ A//B//C// is the rotation of Δ ABC through 180º , sketch Δ A//B//C// (2)

2.2 The diagram below shows quadrilateral PRMT with P(3 ; 5).

2.2.1 Sketch, using a scale factor of 4, the enlargement of PRMT through the origin. Label this diagramKLMN (4)

2.2.2 Write down the coordinates of K and N on the sketch. (2)

2.2.3 Determine the area of PRMT (4)

2.2.4 Determine PRTM : KLMN (2)

2.3 If H(3 ; 5) is rotated through the origin in a clockwise direction through an angle of 90o, write down the coordinates of H/, the image of H. (2)

2.4 The point P(x;y) is first rotated through 1800 , then reflected in the line y = –x, and then translated 4 units up and 3 units left, in that order. Write down the general formula for the combination of these transformations. (4)

2.5 Δ ABC has vertices A(2 ; 6), B(– 3 ; 4) and C(1 ; – 8 ) and its transformation

Δ A'B'C ' has vertices A'(1 ; 3), B'(−3

2; 2)

and C'( 12

; −4). Describe the

transformation in words . (3)[22]

11

120

81,8

8cm

5cm3cm

D

C

B

A


Question 3

3.1 If cosθ=− 2

√13 and 180 °≤θ≤360 ° , use a sketch to determine the value oftanθ (3)

3.2 If x=87 ,6 ° and y=240 ,2 ° , use a calculator to evaluate the following expression

correct to two decimal places

sin ycos x

+3 tan2 x(2)

3.3 Determine the solution to the following equation for 1800 ≤ θ ≤ 3600. Give answer(s) correct to two decimal places 3−tan θ=2,4 (2)

3.4 Simplify the following expressions and show ALL the calculations without using a calculator

3.4.1

cos (1800−x) sin ( x−900)−1tan2 (5400+ x ) sin (900+x ) cos (−x ) (6)

3.4.2sin 63° . cos2 135° . tan 315 °sin 240 ° . tan150 ° . cos27 ° (7)

3.5 Determine the general solution of the equation: 4 sin2 x−3=0 . (6)

3.6 Given the functions: f ( x )=cos2x and g ( x )=sin ( x+300 )

3.6.1 Solve the equation cos2 x=sin ( x+300 ) for x∈ [−1800 ;1800 ] (6)

3.6.2 Sketch graphs of f ( x )=cos2x and g ( x )=sin ( x+300 ) on the same system of

axes for x∈ [−1800 ;1800 ] . Show the co-ordinates of all points of intersection with

the axes, all turning points and all points at which f ( x )=g ( x ) (8)3.6.3 State the range of f if the graph of f undergoes a positive, vertical shift of 1 unit. (2)3.6.4 Write down the new equation of g if it is shifted 60º horizontally to the left. (2)

3.7 In the diagram below, ABCD is a quadrilateral with dimensions as shown in the sketch

Calculate:

3.7.1 AC (3)

3.7.2 D̂ correct to 1 decimal place. (3)

3.7.3 The area of ΔACD (2)

[52]

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Question 4

4.1 A time-capsule in which mementoes will be saved consists of a cylindrical body with a with a hemisphere on top. The diameter of the cylinder is 30 cm and the height of the cylinder is 150 cm. Calculate the total volume of the time-capsule.

(5)

4.2 A tent manufacturer makes a tent in the shape of a pyramid with a square base. The square ground sheet of the tent is attached to the tent. If slant height, h, is 350 cm and a side of the base is 200 cm , calculate the total surface area of the tent.

(5)[10]

13

30 cm

150 cm

200 cm

h


Question 5

5.1 Below are the percentage scores that 15 learners obtained in a Physical Science Examination

72 57 63 81 60 51 96 6678 54 39 69 90 30 39

5.1.1 Determine the median for the above data? (2)5.1.2 Determine the upper and lower quartiles. (4)5.1.3 Draw a box and whisker diagram for the data. (3)5.1.4 Determine the inter quartile range (1)5.1.5 Use a calculator to determine the standard deviation for this data (3)5.1.6 How many learners fall within one standard deviation from the mean (2)

5.2 Below are box and whisker plots that depict the results of a pollution survey in a Western Cape River conducted from 2005 to 2007. The survey measured to pollution (total dissolved solids) in milligrams per litre of water.

5.2.1 Determine the five number summary for 2006. (2)5.2.2 Which year had the greatest range of results? Explain your answer. (2)5.2.3 Using the data, comment on the pollution levels in the river over the three years of

monitoring. (3)

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500 1000 1500 2000 25000

2007

2006

2005

TOTAL DISSOLVED SOLIDS (mg/l)


5.3 The traffic department investigated where it would be most appropriate to in install speed cameras. As part of their investigation a survey was done of the different speeds of vehicles on a stretch of a national road. The following table shows the results of the survey:

SPEED(in km/h)

FREQUENCY(Of vehicles)

CUMULATIVE FREQUENCY

40 < d ¿ 60 49

60 < d ¿ 80 92

80 < d ¿ 100 134

100 < d ¿ 120 158

120 < d ¿ 140 49

140 < d ¿ 160 17

160 < d ¿ 180 1

5.3.1 How many vehicles were observed in the survey? (1)5.3.2 Complete the cumulative frequency column. (2)5.3.3 Represent the information in the table by drawing an ogive. (4)5.3.4 Use your graph to determine the median speed. Indicate on your graph

using the letter T where you would read off your answer. (3)

5.4 The number of learners attending university from this school has grown over the last 7 years. Their records show the following;

Year of Matric Number of learners that went to University2001 12002 22003 22004 52005 142006 252007 42

5.4.1 Draw a scatter plot to represent the data. (3)

5.4.2 Which function is the most appropriate to represent the line of best fit for this data?(2)[37]

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