PAGE 21Wednesday, 29 June 2011 PAGE 25Wednesday, 29 June 2011

Winter School Xtra: Mathematics Grade 12 RevisionNumber and Number RelationshipsQuestion 1Consider the following sequence:

1.1. Write down the next two terms of the sequence 3 5 7 9; ; ; ; ...............2 4 8 16

. (2)

1.2. Determine the nth term of the sequence in simplifi ed form. (5)

Question 2By using appropriate formulae and without using a calculator, calculate the value of the following:

2.1 (4)

2.2 (2)

Question 3In the fi gure below, a stack of cans is shown. There are 30 cans in the fi rst layer, 29 cans in the second layer (lying on top of the fi rst layer), 28 cans in the third layer. This pattern of stacking continues.

Determine the maximum number of cans that can be stacked in this way. (4)

Question 4Consider the sequence: The sequence has a constant second diff erence of 1

4.1. Determine the value of a and b. (4)

4.2. Determine the nth term of the sequence. (5)

4.3. Hence, prove that the sum of any two consecutive numbers in the sequence equals a square number. (4)

Question 5A motor car costing R200 000 depreciated at a rate of 8% per annum on the reducing balance method. Calculate how long it took for the car to depreciate to a value of R90 000 under these conditions. (4)

Question 6Mpho starts a fi ve year savings plan. At the beginning of the month he deposits R2000 into the account and makes a further deposit of R2000 at the end of that month. He then continues to make month end payments of R2000 into the account for the fi ve year period (starting from his fi rst deposit). The interest rate is 6% per annum compounded monthly.

6.1. Calculate the future value of his investment at the end of the fi ve year period. (4)

6.2. Due to fi nancial diffi culty, Mpho misses the last two payments of R2000. What will the value of his investment now be at the end of the fi ve year period? (4)

Learn Xtra Winter School MathematicsBroadcasting 9:00 Monday 4 July 2011

Question 1

TABLE below shows the area, the population, and the gross domestic product (GDP) per province in South Africa during 2007/2008.

TABLE: Area, population and GDP per province during 2007/2008

PROVINCE AREA (in km2) POPULATION GDP (in millions of rand)

Western Cape 129 370 4 839 800 199 412

Eastern Cape 169 580 6 906 200 112 908

KwaZulu-Natal 92 100 10 014 500 2312 616

Northern Cape 361 830 1 102 200 30 087

Free State 129 480 2 965 600 75 827

North West 116 320 3 394 200 87 127

Gauteng 17 010 9 688 100 462 044

Mpumalanga 79 490 3 536 300 94 450

Limpopo 123 910 5 402 900 93 188

1.1. According to the Agricultural Research Council, 80% of South Africa’s land surface area is used for farming. However, only 11% of the farming land is suitable for the planting of crops (arable land). 3,2 million hectares of the farming land in the Free State is suitable for the planting of crops (arable land).

(a) Calculate the total area (in km2 ) of land that is used for farming in South Africa. (4)

(b) Calculate the percentage of land in South Africa suitable for planting crops (arable land) that is found in the Free State.

1 hectare (1 ha) = 0,01 km2 (5)

Question 2The following information about the Free State was given in the 2007/2008 South African Yearbook:

Capital: BloemfonteinHome languages: Sesotho: 64,4% Afrikaans: 11,9% IsiXhosa: 9,1%Population: 2 965 600 (mid-year population estimates in 2007)Area: 129 480 km2Percentage of total area of South Africa: 10,6%Gross domestic product (GDP) in 2004 (latest fi gure available): R75 827 millionPercentage of South Africa’s GDP in 2004: 5,5%

Winter School Xtra: Mathematics Winter School Programme Grade 12 revision

2.1. Calculate the number of people in the Free State whose home languages were NOT Sesotho, Afrikaans or isiXhosa during the period 2007/2008. (4)

2.2. If a person is randomly selected from the Free State, determine the probability that the home language of the person is NOT Afrikaans or isiXhosa. (3)

2.3. Surveys have shown that 60% of the inhabitants of the Free State are employable. This means that the workforce is 60% of the total population of the Free State.

(a) Identify any TWO possible reasons why 40% of the inhabitants are not employable. (2)

(b) According to the Labour Force Survey of March 2007, the offi cial unemployment rate in the Free State was 26,4% of the workforce.

Calculate the number of unemployed people in the Free State at the time of this survey. (5)

Question 3

3.1. Ronwyn and Bronwyn are twins. They plan to celebrate their 21st birthday by having a big party. Ronwyn has decided that she wants a round cake, while Bronwyn has decided to have a ring cake, as shown in the pictures below.

The dimensions of each cylindrical cake is as follows:

The following formulae (using π= 3,14) may be used: Volume of a cylinder = π x (radius)2 x height Volume of a cylindrical ring = π x (R2 – r2 ) x height where R = outer radius and r = inner radius

Total outer surface area of an open cylinder = π x (radius)2 + 2x π x radius x height

3.1.1. Using the volume of each cake, determine which of the two cakes is better value for money if the costs of the two cakes are the same. Give a reason for your answer, showing ALL your calculations. (10)

3.1.2. Ronwyn decides that her round cake will be a fruit cake. The cake will be covered with marzipan icing on the top of the cake as well as around the sides. Determine the total outer surface area of the cake that the marzipan icing will cover. (6)

3.2 .The twins can choose from the following two options for the catering for their party:

OPTION 1: R120 per head, which includes the payment for the venue, but excludes the 14% value-added tax (VAT).

OPTION 2: R3 200 for the hire of the venue and then R80 per head for catering, which includes the 14% VAT.

Analyse the two options and determine which ONE would be the cheaper option if 100 people in total will attend the party. Show ALL calculations. (5)

Question 4Thandi washes her dishes by hand three times daily in two identical cylindrical basins. She uses one basin for washing the dishes and the other for rinsing the dishes. Each basin has a radius of 30 cm and a depth of 40 cm, as shown in the diagram below.

Thandi is considering buying a dishwasher that she will use to wash the dishes daily.

4.1. Calculate the volume of one cylindrical basin in cm3. Volume of a cylindrical basin = x(radius)2 x height , using = 3,14 (2)

4.2. Thandi fi lls each basin to half its capacity whenever she washes or rinses the dishes. Calculate how much water (in litres) she will use daily to wash and rinse the dishes by hand. (1 000 cm3 = 1 liter) (5)

4.3. A manufacturer of a dishwasher claims that their dishwasher uses nine times less water in comparison to washing the same number of dishes by hand.

4.3.1. How much water would this dishwasher use to wash Thandi’s dishes daily? (2)

4.3.2. Is the claim of the manufacturer realistic? Justify your answer by giving a reason(s). (3)

18

2

12

k

k

−

=

∑

1

2

12

k

k

−

=

∞∑

3 ; ;10 ; ; 21; ........a b

0 ,0 ≥≥ yx

Winter School Xtra: Maths Literacy Grade 12 RevisionSolving Problems in ContextWinter School Xtra: Maths Literacy

Winter School Programme Grade 12 revision

9:00

Learn Xtra Winter School Maths LiteracyBroadcasting 13:30 Monday 4 July 201113:30

Question 7Simphiwe takes out a twenty year loan of R100 000. She repays the loan by means of equal monthly payments starting three months after the granting of the loan. The interest rate is 18% per annum compounded monthly.

7.1. Calculate the amount owing two months after the loan was taken out by Simphiwe. (2)

7.2. Calculate the monthly repayments. (4)

Question 8A clothing company manufactures white shirts and grey trousers for schools.

• A minimum of 200 shirts must be manufactured daily.• In total, not more than 600 pieces of clothing can be manufactured daily.• It takes 50 machine minutes to manufacture a shirt and 100 machine

minutes to manufacture a pair of trousers.• There are at most 45 000 machine minutes available per day.

Let the number of white shirts manufactured in a day be x.

Let the number of pairs of grey trousers manufactured in a day be y.

8.1. Write down the constraints, in terms of x and y, to represent the above information. (You may assume: ) (3)

8.2. Use graph paper to represent the constraints graphically. (3)

8.3. Clearly indicate the feasible region by shading it. (1)

8.4. If the profi t is R30 for a shirt and R40 for a pair of trousers, write down the equation indicating the profi t in terms of x and y. (2)

8.5. Using a search line and your graph, determine the number of shirts and pairs of trousers that will yield a maximum daily profi t. (2)

PAGE 22 Wednesday, 29 June 2011 PAGE 25Wednesday, 29 June 2011

Learn Xtra Winter School Schedule 11-15 July 2011

Mon 11 July Tue 12 July Wed 13 July thurs 14 July Fri 15 July

09:00 Maths P1: Functions and graphs English FAL: Language and editing Physical Sciences P1: Electricity and magnetism Maths P2: Trigonometry Physical Sciences P2: Rates and

chemical equilibrium

13:30 Maths Literacy: Mixed questions Accounting: Financial statements Life Sciences P1: Genetics Geography: Mapwork Life Sciences P2: Reproduction

17:30 Maths P1: Number and number relationships English FAL: Visual literacy Physical Sciences P1: Mechanics Maths P2: Trigonometry Physical Sciences P2: Redox and

chemical industries

20:00 Maths Literacy: Problems in context Accounting: Manufacturing accounts Life Sciences P1: Evolution Geography: Geomorphology Life Sciences P2: Life processes

Mindset Learn: DStv Channel 319 and TopLearn: TopTV Channel 319

Winter School Xtra: Mathematics Grade 12 Revision Mathematics paper 1: CalculusQuestion 1

Diff erentiate f from fi rst principles :

1.1. f(x) = x1 (4)

1.2. f(x) = -2x + 3 (5)

Question 2Use the rules of diff erentiation to determine

2.1. d yd x

if y = (2- 5x)² (2)

2.2. d yd x

if y = x2 – 2 x 3

1 (2)

Question 3The graph below represents the functions f and g with f(x)= ax3 + cx – 2 and g(x) = x – 2. A and (– 1; 0) are the x-intercepts of f. The graphs of f and g intersect at A and C.

3.1. Determine the coordinates of A. (1)

3.2. Show by calculation that a = 1 and c = – 3. (4)

3.3. Determine the coordinates of B, a turning point of f. (3)

3.4. Show that the line BC is parallel to the x-axis. (7)

3.5. Find the x-coordinate of the point of infl ection of f. (2)

3.6. Write down the values of k for which f(x)=k will have only ONE root. (3)

3.7. Write down the values of x for which f ’ x<0 (1)

Question 4A tourist travels in a car over a mountainous pass during his trip. The height

above sea level of the car, after t minutes, is given as s(t) = 5 t ³ - 65 t ² + 200 t + 100 metres. The journey lasts 8 minutes.

4.1. How high is the car above sea level when it starts its journey on the mountainous pass? (2)

4.2. Calculate the car’s rate of change of height above sea level with respect to time, 4 minutes after starting the journey on the mountainous pass. (3)

4.3. Interpret your answer to QUESTION 4.2. (2)4.4. How many minutes after the journey has started will the rate of change of

height with respect to time be a minimum? (3)

Learn Xtra Winter School MathematicsBroadcasting 17:30 Monday 4 July 2011

Winter School Xtra: Mathematics Winter School Programme Grade 12 revision

Question 1

A bus tyre has a diameter of 120 cm. The ratio of the diameter of a bus tyre to the diameter of a minibus tyre is 12:7.

Calculate the distance travelled by the minibus (rounded off to the nearest km) if the minibus’s tyre rotated 1 862 times during the journey.

The following formulae may be used: Circumference = 2 × π × r where r = radius and using $ = 3,14

Number of rotations = Distance travelled

Circumference of tyre (6)

Question 2Mosima’s LCD TV screen is a new slim model that is only 39,7 mm thick. The rectangular screen is 45 cm high and 60 cm wide. The TV stands on a round base with a diameter of 20 cm,that is 2 cm thick and is held up by a swivel that is 5 cm high, as shown in the diagram below.

Determine the volume (in cm3) of the rectangular box that the TV will be delivered in if there is an allowance of 5 mm for all measurements to package the TV, as shown in the side view above.

Given the formula: Volume = length × breadth × height (5)

Question 3An aquarium is a place where collections of fi sh and other aquatic animals are displayed. The fi sh are kept in open rectangular glass tanks. A water pump is used to circulate and refresh the water in the tanks.

An open-top fi sh tank has the following dimensions: length = 2,5 m; breadth = 2 m; height = 1,5 m

Sketch of a fi sh tank Fish in an aquarium

3.1. Determine the volume of the fi sh tank in kilolitres if 1 m3 = 1 k�, where volume = length x breadth x height. (3)

3.2. Determine the total surface area (in m2) of glass used for the open-top fi sh tank, where:

surface area = (l x b) + 2 x (l x h) + 2 x (h x b) and (4)

l = length, b = breadth and h = height.

3.3. Calculate the cost of 20 m2 of special glass for the fi sh tank @ R480,00 per m2. (3)

3.4. The water pump costs R3 999,00. The suppliers gave the aquarium a 15% discount.

Calculate how much the aquarium paid for the pump. (3)

3.5. The tank is fi lled with 6 900 of water at a rate of 2 300 of water per hour. Calculate the time taken to fi ll the tank. (2)

Question 4Gerrie van Niekerk is a primary school learner who lives in Krugersdorp. He lives on the corner of Wishart Street and 5th Street.

Refer to the map of part of Krugersdorp, Gauteng (on the right), and use it to answer the questions that follow.

Winter School Xtra: Maths Literacy Grade 12 RevisionShape and SpaceWinter School Xtra: Maths Literacy

Winter School Programme Grade 12 revision

17:30

Learn Xtra Winter School Maths LiteracyBroadcasting 20:00 Monday 4 July 201120:00

4.1 Give a grid reference for the Jays Shopping Centre where Gerrie and his mother do their weekly grocery shopping. (1)

4.2 Gerrie’s grandmother lives with them and goes to the hospital for her medication once a month. What is the relative position of Krugersdorp Central Hospital with respect to Gerrie’s home? (1)

4.3 Gerrie’s father drives from Jays Shopping Centre to the petrol station to buy petrol for his car. Describe his route if the exit from Jays Shopping Centre is in 4th Street. (3)

5.4 Gerrie walks from home to Paardekraal Primary School by: • Crossing 5th Street and walking in an easterly direction along

Wishart Street• Turning right and walking in a southerly direction along 4th Street • Turning left and walking in an easterly direction along Onderste Street • Turning right, and walking in a southerly direction along 3rd Street

The entrance to the school is on the corner of 3rd Street and Pretoria Street. The distance on a map with a scale 1:11 000 is 11cm. Calculate the actual distance. Gerrie walks to school. Give your answer in kilometres. (4)

Learn Xtra Winter School Schedule 4-8 July 2011

Mon 4 July Tue 5 July Wed 6 July

09:00Maths P1: Number

and number relationships

English FAL: Visual literacy

Physical Sciences P1: Mechanics

13:30Maths Literacy:

Problems in context

Accounting: Cash � ow, statements, and interpretation

of ratios

Life Sciences P1: DNA and RNA

17:30 Maths P1: CalculusEnglish FAL:

Comprehension and summarising

Physical Sciences P1: Waves, sound

and light

20:00 Maths Literacy: Space and shape

Accounting: Financial

statements

Life Sciences P1: Evolution

thurs 7 July Fri 8 July

09:00

Maths P2: Transformations and co-ordinate

geometry

Physical Sciences P2: Organic chemistry

13:30 Geography: Geomorphology

Life Sciences P2: Life processes

17:30 Maths P2: Trigonometry

Physical Sciences P2: Rates

and chemical equilibrium

20:00 Geography: Climatology

Life Sciences P2: Environmental

studies

Mindset Learn: DStv Channel 319 and TopLearn: TopTV Channel 319

B C

2

3

HELP DE

SK

Are you in Grade 10, 11 or 12 and struggling with a particular Maths, Science, Life Science or Maths Lit question?

Why not ask our panel of expert teachers? Submit your question to the Help Desk and get a response within 48 hours...free! Submit your questions to the Help Desk via:

i. Web: www.learnxtra.co.za

ii. MXit: learnxtrahelpdesk

iii. Facebook: www.facebook.com/learnxtra

iv. Email: helpdesk@learnxtra.co.za

v. Phone: 086 105 8262

l l