Grade 12 Mathematical Literacy Preliminary Exam Paper 1

Grade 12 Mathematical Literacy Preliminary Exam Paper 1 Memorandum

© e-classroom www.e-classroom.co.za

Sutherland is a small Karoo town in the Northern Cape, famous for its warm hospitality, snowy winters, starlit nights and dramatic landscapes. Fascinating history, unique natural assets and numerous outdoor activities make Sutherland a popular destination.

QUESTION 1

1.1 Income tax on individuals, is the largest source of income in the National Budget. The table below shows the assessed individual taxpayers by province.

Province Number of Taxpayers

Taxable Income (R million)

Average Annual Taxable Income

(Rand)

Eastern Cape Fee State Gauteng KwaZulu – Natal Limpopo Mpumalanga North West Northern Cape Western Cape Unknown

440 406 258 998

1 847 903 707 335 261 252 301 871 276 937 111 572 801 441 166 857

85 408 46 167

502 977 147 225 51 431 64 031 50 967 20 942

182 771 29 634

193 930 178 252 272 188 208 140 196 864 212 114 184 038 187 699 228 053 177 601

TOTAL 5 174 572 1 181 553 228 338

(www.treasury.gov.za)

http://www.e-classroom.co.za/



1.1.1 Round the number of taxpayers in the Northern Cape to the nearest 100 000. (2) 111 572 ≈ 100 000 TL1 Data

1.1.2 Round the number of taxpayers in Gauteng to the nearest million. (2) 1 847 903 ≈ 2 000 000 TL 1 Data

1.1.3 Using the rounded values from questions 1.1.2 and 1.1.3, give the number of taxpayers in the Northern Cape as a ratio of the number of taxpayers in Gauteng. (2) 100 000 : 2 000 000ca = 1:20(simplify) TL1 Data

1.1.4 Write down the taxable income of the taxpayers in the Northern Cape in words. (3) Twenty billion nine hundred and forty-two million rand TL1 Fin

1.1.5 Show how the value of R187 699 for the average taxable income for the Northern Cape was obtained. (3) 20 942 000 000a ÷m 111 572a = R187 699 (given) TL1 Fin

1.1.6 What is the average monthly taxable income of a taxpayer living in the Northern Cape? (3) R187 699a ÷ 12ma = R15 641,58 2dp TL1 Fin

1.1.7 If the number of registered taxpayers in South Africa was 12,4 million, then what percentage of these actually paid income tax during that financial year? (3) 𝟓𝟓 𝟏𝟏𝟏𝟏𝟏𝟏 𝟓𝟓𝟏𝟏𝟓𝟓𝒂𝒂𝟏𝟏𝟓𝟓 𝟏𝟏𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒𝟒𝟒𝒂𝒂

× 𝟏𝟏𝟒𝟒𝟒𝟒 = 41,7%ca TL1 Fin




Mrs Hultzer and her family went to Sterland to do some stargazing while they were in Sutherland. Jurg, the owner, had bought the property just outside Sutherland as an investment in 2015 and organises a stargazing tour each evening.

1.2 Sterland has an FNB Business Account, which charges the following service fees:

FNB Business Account: Pay-as-you-use pricing option

Transaction Service Fee

Monthly account fee R200

Cash deposit fee at FNB branch

Minimum fee per deposit R30,00

Value of deposit Deposit fee

R0 – R4 999.99 R8,40 + R2,55 per R100 or part thereof

R5 000 – R14 999.99 R8,40 + R1,49 per R100 or part thereof

1.2.1 What does this account cost the business on a monthly basis? (2) R200 TL1 Fin

1.2.2 How do you know if the R30,00 minimum fee must be paid? (2) Calculate the deposit fee. If the calculated fee is less than R30,00 then the R30,00 must be paid. TL4 Fin

1.2.3 Calculate the total cost to this business when Jurg deposits R11 250,00 at an FNB branch. (3) R8,40 + 1,49a x 113a = R8,40 + 168,37 = R176,77ca TL3 Fin




1.2.4 The FNB Business account pays 2,4% pa interest. The interest is compounded monthly.

(a) Calculate the monthly interest rate. (2) 2,4% ÷ 12ma = 0,2%a TL1 Fin

(b) Calculate how much interest Jurg will earn on the R11 250,00 if he leaves itin the account for 2 months. (5) R11 250 x 100,2%ca = R11 272,50 R11 272,50ca x 100,2% = R11 295,05ca R11 295,05 – R11 250ma = R45,05ca TL2 Fin

1.3 Sterland made a profit of R210 000 during the past year. A graph representing the total cost and income, is shown below:

1.3.1 Determine Sterland’s fixed costs for the year and indicate on the graph where you

determined this value. (2)

R30 000 (a on graph) TL1 Fin

R-

R20.00

R40.00

R60.00

R80.00

R100.00

R120.00

R140.00

0 200 400 600 800 1000

Amou

nt (i

n th

ousa

nds)

Number of people

Total Costs compared to Total Income for 2018

Total Income

Total CostA

a




1.3.2 How much does Jurg charge per person for the stargazing? Round your answer to

the nearest ten rand. (3)

R80 000a ÷ 600ma = R133,33… ≈ R130 per personca (can use any point off the Total Income graph) TL2 Fin

1.3.3 If the Total cost for 500 people is R40 000, calculate the cost price per person for an

evening of stargazing. (4)

R40 000 – R30 000 ma= R10 000ca R10 000 ÷500ma = R20 per personca TL2 Fin

1.3.4 Calculate the percentage profit per person that attends a star-gazing tour. (3) 𝑹𝑹𝟏𝟏𝑹𝑹𝟒𝟒−𝑹𝑹𝟓𝟓𝟒𝟒𝒄𝒄𝒂𝒂

𝑹𝑹𝟓𝟓𝟒𝟒𝒄𝒄𝒂𝒂× 𝟏𝟏𝟒𝟒𝟒𝟒 = 550%ca TL2 Fin

[ca from 1.3.3: (130-80)/80 x 100 = 50/80 x 100 ≈ 62,5%]

1.3.5 Explain what the point A on the graph represents. (2)

The break-even point: After about 260 people have been stargazing he will start to make a profit. TL4 Fin

1.3.6 Use the graph to estimate the profit Jurg would make if he only had 800 people on

tours for 2018. (3)

Total Income = R105 000 Total costs = R45 000 Profit = R105 000 – R45 000 = R60 000ca TL2 Fin




1.4 Jurg needs to buy a new telescope and sees the following advert in a newsletter he subscribes to.

1.4.1 The telescope was $799 and is on sale for

$699. Calculate what percentage discount is

being offered. (3) 𝟔𝟔𝟔𝟔𝟔𝟔−𝟏𝟏𝟔𝟔𝟔𝟔𝒂𝒂

𝟏𝟏𝟔𝟔𝟔𝟔𝒂𝒂× 𝟏𝟏𝟒𝟒𝟒𝟒 = - 12,5% = 12,5% discount

TL2 Fin

1.4.2 Use the exchange rate shown below to calculate how many rands Jurg would have

to pay for the telescope before tax and import duties? (2)

699 ÷ 0,068ma = R10 279,41a TL1 Fin

1.4.3 The American inflation rate is currently 2,5%. If Jurg decides to wait for the sales

next year before he purchases the telescope, what can he expect to pay? (3)

R10 279,41 xm 102,5%a = R10 536, 40ca OR 699 xm 102,5%a = $716,48ca TL1 Fin

[57]




QUESTION 2

Situated just outside Sutherland is the Southern African Large Telescope (SALT), which is the largest single optical telescope in the Southern Hemisphere and among the largest in the world.

SALT can detect the light from faint or distant objects in the universe, a billion times too faint to be seen with the unaided eye – as faint as a candle flame would appear at the distance of the moon. [Source:www.salt.ac.za]

2.1 The Sterland telescopes that Jurg has have a primary mirror of 22cm. How many times smaller is the mirror in Jurg’s telescope than the primary mirror in SALT? (3) 11m = 1 100cm conv 1 100 ÷ 22ma = 50 Jurg’s telescope has a mirror which is 50 times smallerca TL1 Meas

2.2 The total telescope cost, including its instruments and operations, was approximately US$32 million, of which South Africa contributed a third and the international partners provided the balance. SALT was built at less than a fifth of the cost of a conventional 10-metre telescope.

2.2.1 Write the cost of $32 million using only digits. (2) $32 000 000a TL1 Fin

2.2.2 Calculate the approximate cost of a conventional 10-metre telescope, in dollars.(2)

32 000 000 x 5ma = $160 000 000 ca TL1 Fin


http://www.salt.ac.za/



2.2.3 South Africa paid a third of the cost and is entitled to one third of the observation time. In April observation started from sunset at 7pm and continued until sunrise at 4am. Calculate how many hours of observation time South Africa would get each evening. (3) 19h00 to 24h00 = 5 hours 00h00 to 04h00 = 4 hours 9 hours of observationa ÷ 3ma = 3 hours each eveningca TL1 Meas

2.2.4 Edge sensors, which detect changes in relative positions of the mirror segments, provide data on mirror misalignment approximately every 60 seconds. Use the time determined in Question 2.2.3 to calculate how many data alerts South Africa would get in one week. (4) 1 alert every 60seconds = 1 alert every minuteconv 3 hours = 180 minutes conv 180 alerts x 7ma = 1 260 alerts in one weekca TL1 Meas

2.3 The temperature forecast for Sutherland is provided below.

SUN 4/28

Actual Temp 28° /-1°C

MON 4/29

Actual Temp 27° /6°C

TUE 4/30

Actual Temp 23° /3C°

WED 5/1


THU 5/2


FRI 5/3


SAT 5/4


[Source: www.accuweather.com]

2.3.1 Convert the minimum temperature for Monday the 29th of April into degrees Fahrenheit with the following formula: (3)

(°F) = (°C) × 95+ 32

oF = 6a x 𝟔𝟔𝟓𝟓 + 32m = 42,8oF a TL1 Meas

2.3.2 What is the chance, as a percentage, that the day the Hultzer family visited SALT that week, the maximum temperature was 23oC? (3) 𝑹𝑹𝐚𝐚𝟏𝟏𝐚𝐚

× 𝟏𝟏𝟒𝟒𝟒𝟒 = 42,9%ca TL1 Prob




2.3.3 SALT’s primary mirror edge sensor system can keep the mirror segments well aligned for a week for a temperature range of up to 20 degrees Celsius. Calculate the temperature range during the week shown in the forecast and decide if the mirror segments would stay aligned. (3) Temperature range = 28 – (-2)a = 30oCca The mirror segments would NOT stay aligned.ca TL2 Data

2.3.4 SALT has a hexagonal primary mirror array of 11 metres across, made up of 91 individual hexagonal mirrors each with a flat diameter of 1,2m and edges of 0,7m. Each mirror weighs approximately 100kg.

2.3.5 Use calculations to show that there are only nine individual hexagonal mirrors across the flat diameter of the primary mirror. (2) 11 ÷ 1,2mma = 9,166..a ≈ 9 hexagons. TL1 Meas

2.3.6 A scientist from the United Kingdom claims that the total weight of all the individual, hexagonal mirrors making up the primary mirror is approximately 1 433 stone. Use calculations to verify his claim.

You may use the following conversions: 1 stone = 14 pounds 1 pound = 0,45359 kg (6) 100 x 91ma = 9 100 kga

1 pound = 0,45359 kg ? pounds = 9 100 kg m 9 100 ÷ 0,45359 = 20 062,17 pounds ca

1 stone = 14 pounds ? stone = 20 062,17 pounds m 20 062,17 ÷ 14 = 1 433 stone .: He is correctca TL2 Meas




2.3.7 Calculate the area of each hexagonal mirror. Hint: Each mirror can be divided into 6 identical triangles each with a height of 0,6m. (4)

Area of one triangle = 𝟒𝟒,𝟏𝟏 × 𝟒𝟒,𝟔𝟔𝒎𝒎𝒂𝒂𝟓𝟓

= 0,21m2 a Area of each mirror = 0,21 x 6ma = 1,26 m2 ca TL3 Meas

2.3.8 The mirrors are resurfaced with a 5cm thick aluminium layer once a year. Calculate the volume of melted aluminium required to resurface each mirror. (3) Volume = 1,26 xm 0,05conv = 0,063 m3ca TL2 Meas

2.3.9 How many litres of melted aluminium is poured onto each mirror in order to resurface it if 1m3 = 1 kilolitre? (2) 0,063 m3 = 0,063 klca = 63 litresconv TL1 Meas

[40]




QUESTION 3

3.1 A scale drawing of SALT is shown

below. Study the diagram and then

answer the questions that follow.

3.1.1 What type of scale is

provided on the diagram?

(2)

A bar scale a TL1 Map

3.1.2 Use the scale which has

been provided to show that

the numerical scale (ratio scale) of the drawing is approximately 1:500 (3)

2,3 cma : 10metres [CHECK AFTER PRINTING] 2,3 cm : 1 000cm conv

1 : 454,5 ≈ 500m TL2 Map

3.1.3 Use the scale to calculate the diameter of the dome as indicated by the points A

and B. (4)

1 : 500 9 cm a : ? m [MEASURE AFTER PRINTING] 9 x 500 = 4 500cm ca = 4,5m conv/appropriate units TL2 Map

3.1.4 The drawing above is a cut-away Southern elevation of SALT.

(a) What elevation would show the full length of the 34-m high mirror segment

alignment (CCAS) tower? (2)

Northern elevation a (or Eastern elevation) TL1 Map

A B




(b) How long would the 34m high tower be drawn on the elevation from (a) if the

same scale of 1:500 was used? (3)

1 : 500? cm : 3 400cmconv3 400 ÷ 500m = 6,8cm ca TL2 Map

Refer to the map below to answer the questions which follow.

[Source: www.google.com/maps/place/Sutherland,+6920]

3.1.5 The Hultzer family followed the R354 in a Northerly direction to get from

Maatjiesfontein to Sutherland. They had booked to stay at Perlman House. Was

Perlman House on the left-hand side or right-hand side of the road as they

approached? (2)

On the LEFT of the road.a TL1 Maps

3.1.6 If a crow flies directly from Sterland to SALT, what direction will it be flying? (2)

North Easta TL1 Maps




3.1.7 Write a clear set of directions for the Hultzer family to get from Sterland to SALT.

(3)

Leave Sterland and turn NORTH onto the R354. After a short distance turn RIGHT (East) onto the R356. After a few kilometres turn LEFT (North) onto the D127. SALT will be a few kilometres along the road. direction roads clear TL2 Maps

[21]




QUESTION 4

The South African Astronomical Observatory (SAAO) was founded in Cape Town in early 1820.

By the mid-20th century, light and air pollution in Cape Town was making it difficult to perform

astronomical work. Late in 1972 all the major South African telescopes were relocated to a

hilltop close to Sutherland where there were dark skies and clear weather evenly spread

throughout the year. In 2000 SALT was built at the same site. There are now 15 telescopes

from different international facilities hosted at the site in Sutherland.

[Source: www.saao.ac.za]

Telescope Country Mirror Diameter

1,9m SAAO Telescope South Africa 1,9𝑚𝑚




Alan Cousins Telescope South Africa 0,75𝑚𝑚

BiSON United Kingdom 1𝑚𝑚

IRSF Japan 1,4𝑚𝑚

MONET Germany 1,2𝑚𝑚

YSTAR Korea 0,5𝑚𝑚

SALT Germany, India, New Zealand, Poland, South Africa, United Kingdom and USA

11𝑚𝑚

SuperWASP United Kingdom 1𝑚𝑚

KELT USA 1𝑚𝑚

SUMI-hut Japan 1𝑚𝑚

Solaris Poland 0,5𝑚𝑚

LCOGT USA 1,0𝑚𝑚




4.1 For how many years did the SAAO have its major telescopes in Cape Town? (2)

Early 1820 – Late 1972 = 152a + 1 = 153 yearsa TL1 Data

4.2 Which country has the most telescopes? How many does it have? (2)

South Africaa 6 (or 5𝟏𝟏𝑹𝑹) a TL1 Data

4.3 Calculate the total percentage of the telescopes that belong to Japan. (3) 𝟓𝟓𝒂𝒂𝟏𝟏𝟓𝟓𝒂𝒂

x 100 = 13%ca TL1 Data

4.4 If a telescope was chosen at random on the Sutherland site, what is the probability, as a

fraction, that it will be owned by either Japan or Korea? (3) 𝑹𝑹𝒂𝒂𝟏𝟏𝟓𝟓𝒂𝒂

= 𝟏𝟏𝟓𝟓 simplify TL1 Prob

4.5 Consider the mirror diameters of the telescopes in the table and then answer the questions

that follow.

4.5.1 Arrange the telescopes in ascending order of their mirror diameter. (3)

Telescope Diameter (m)

0,5m SAAO Telescope 0.5 YSTAR 0.5 Solaris 0.5 0,75m SAAO Telescope 0.75 Alan Cousins Telescope 0.75 1,0m SAAO Telescope 1 BiSON 1 SuperWASP 1 KELT 1 SUMI-hut 1 LCOGT 1 MONET 1.2 IRSF 1.4 1,9m SAAO Telescope 1.9 SALT 11

smallest largest all correct TL1 Data




4.5.2 Calculate the mean, median and mode of the mirror diameters. (6)

Mean = 24,5a ÷ 15ma = 1,63m 2dp Median = 1m a Mode = 1m a TL1 Data

4.5.3 Draw a box-and-whisker plot on the grid below to represent the mirror diameters. (5)

min (0,5) Q1 (0,75) median (1) Q3 (1,2) Max (11m) TL2 Data

4.6 A group of astronomers recorded the number of significant sightings together with the

temperature over a fortnight. Their results are plotted on the graph below.

4.6.1 What type of graph did they use? (2)

Scatter plota TL1 Data

4.6.2 Draw the line of best fit on the graph above. (2)

(See graph) ma TL2 Data

Y a

0

2

4

6

8

10

12

-4 -2 0 2 4 6 8 10 12 14

Num

ber o

f sig

nific

ant s

ight

ings

Temperature (oC)

Significant sightings relative to the Temperature

Y a




4.6.3 Describe the correlation of their findings. (2)

It is a NEGATIVE correlation. The warmer the temperature the less significant sightings they see. TL1 Data

4.6.4 Use the graph to predict the number of significant sightings they will have if the

temperature is predicted to be – 2oC. Indicate on the graph, using the letter ‘Y’,

where you took your reading. (2)

9 significant sightingsca TL1 Data

[32]


Documents

Grade 12 Mathematical Literacy Preliminary Exam Paper 1