Graad/ Grade 12 Augustus

Laaste toets/ Last Test Totaal/ Total : 55 punte/ marks

Eksaminator: Tyd/ Time : 1 uur/ hour

Moderator:

INSTRUKSIES INSTRUCTIONS

1. Beantwoord ALLE vrae.2. Sakrekenaars mag gebruik word.3. Dit is tot jou eie voordeel om netjies en

volledig te werk.4. Rond alle antwoorde af tot 2 desimale syfers.5. Trek ‘n lyn na elke vraag.6. Geniet die vraestel.

1. Answer ALL questions.2. Calculators may be used.3. It is to your advantage to work neatly and show

all workings.4. Round off all answers to two decimal places.5. Draw a line after each question.6. Enjoy the paper.

VRAAG 1/QUESTION 1

Op die eerste skooldag van elke maand word inligting oor die middaguur-temperatuur (in ° C) en die getal 500 ml-bottels water wat in pouse by die snoepie van ‘n sekere skool verkoop word, aangeteken. Die data word in die tabel hieronder getoon en op die spreidiagram voorgestel. Die kleinstekwadrate-regressielyn vir hierdie data is op die spreidiagram geskets.

On the first school day of each month information is recorded about the temperature at midday (in ° C) and the number of 500 ml bottles of water that were sold at the tuck shop of a certain school during the lunch break. The data is shown in the table below and represented on the scatter plot. The least squares regression line for this data is drawn on the scatter plot.

Middaguur-temperatuur (in ° C ¿Temperature at midday (in ° C ¿

18 21 19 26 32 35 36 40 38 30 25

Getal bottels water (500 ml)Number of bottles of water (500 ml)

12 15 13 31 46 51 57 70 63 53 23

1.1 Identifiseer ‘n uitskieter in die data.

Identify an outlier in the data. (1)

1.2 Bepaal die vergelyking van die kleinstekwadrate-regressielyn.Determine the equation of the least squares regression line. (2)

1.3 Skat die getal 500 ml-bottels water wat verkoop sal word indien die middaguur-temperatuur 28℃ is.Estimate the number of 500 ml bottles of water that will be sold if the temperature is 28 ℃ at midday. (2)

1.4 Verwys na die spreidiagram. Sou jy sê dat die verband tussen die middaguur-temperatuur en die getal 500 ml-bottels water wat verkoop word, swak of sterk is? Motiveer jou antwoord.Refer to the scatter plot. Would you say that the relation between the temperature at midday and the number of 500 ml bottles of water sold is weak or strong? Motivate your answer. (2)

1.5 Gee ‘n rede waarom die waargenome neiging vir hierdie data nie onbepaald kan aanhou nie.Give a reason why the observed trend for this data cannot continue indefinitely. (1) [8]

VRAAG 2/QUESTION 2

‘n Sirkel met C(3;-1) as middelpunt en ‘n radius van 10 eenhede is geskets. PTR is ‘n raaklyn aan hierdie sirkel by T. R(k;21), C en P is die hoekpunte van ‘n driehoek. TR = 20 eenhede.A circle having C(3;-1) as centre and a radius of 10 units is drawn. PTR is a tangent to this circle at T. R(k;21), C and P are the vertices of a triangle. TR = 20 units.

2.1 Gee ‘n rede waarom TC⊥TR .Give a reason why TC⊥TR . (1)

2.2 Bereken die lengte van RC. Laat jou antwoord in wortelvorm.Calculate the length of RC. Leave your answer in surd form. (2)

2.3 Bereken die waarde van k as R in die eerste kwadrant lê.Calculate the value of k if R lies in the first quadrant. (4)

2.4 Bepaal die vergelyking van die sirkel wat C as middelpunt het en deur T gaan. Skryf jou antwoord in die vorm ¿Determine the equation of the circle having centre C and passing through T. Write your answer in the form ¿ (2)

2.5 PS, ‘n raaklyn aan die sirkel by S, is ewewydig aan die x-as. Bepaal die vergelyking van PS.PS, a tangent to the circle at S, is parallel to the x-axis. Determine the equation of PS. (1)

2.6 Die vergelyking van PTR is 3 y−4 x=35The equation of PTR is 3 y−4 x=35

2.6.1 Bereken die koördinate van P. Calculate the coordinates of P.

2.6.2 Bereken, deur ‘n rede te gee, die lengte van PT. Calculate, giving a reason, the length of PT.

(2)

(2)

2.7 Beskou ‘n ander sirkel met vergelyking ¿ en middelpunt M.Consider another circle with equation ¿ and having centre M.

2.7.1 Skryf die koördinate van middelpunt M neer. Write down the coordinates of centre M.

2.7.2 Skryf die lengte van die radius van hierdie sirkel neer. Write down the length of the radius of this circle.

2.7.3 Bereken of die sirkel met middelpunt C en die sirkel met middelpunt M mekaar sny, raak of nie sny nie.

Calculate if the circle with centre C and the circle with centre M do intersect, touch or do not intersect.

(1)

(1)

(3) [19]VRAAG 3/QUESTION 3

3.1 Vyf seuns en vier meisies gaan fliek. Hulle sit almal langs mekaar in dieselfde ry.Five boys and four girls go to the movies. They are all seated next to each other in the same row.

3.1.1 Een seun en meisie is ‘n paartjie en wil langs mekaar op enige punt van die ry vriende sit. Op hoeveel verskillende maniere kan die hele groep sit?

One boy and girl are a couple and want to sit next to each other at any end of the row of friends. In how many different ways can the entire group be seated? (2)

3.1.2 Indien al die vriende willekeurig langs mekaar sit, bereken die waarskynlikheid dat al die meisies langs mekaar sal sit.If all the friends are seated randomly, calculate the probability that all the girls are seated next to each other. (3)

3.2 3.2.1 Hoeveel verskillende 4 syfergetalle (deur die syfers 0 tot 9 te gebruik) is moontlik as die laaste syfer ‘n onewe getal moet wees en die syfers mag herhaal?

How many different 4 digit numbers (by using 0 to 9)is possible if the last digit must be an uneven number and digits may be repeated.

(2)

3.2.2 Wat is die waarskynlikheid dat die getal met twee nulle sal eindig?What is the probability that the number will end with two zeros? (3) [10]

VRAAG 4/QUESTION 4

4.1 In ∆ PQR is PS ∥VT en QS :SR=2 :3. T is ‘n punt op PR sodat PT :TR=2 :7. Bereken QU :UT .In ∆ PQR , PS ∥VT and QS :SR=2 :3. T is a point on PR and PT :TR=2 :7. Calculate QU :UT .

(5)

4.2

Koordevierhoek EFGH is in die diagram hieronder geskets. Koord EH verleng en koord FG verleng ontmoet by K. M is ‘n punt op EF sodat MG ∥EK . Verder is KG = EF.In the diagram below, cyclic quadrilateral EFGH is drawn. Chord EH produced and chord FG produced meet at K. M is a point on EF such that MG ∥EK . Also KG = EF.

4.2.1 Bewys dat/Prove that:

(a) ∆ KGH∨¿∨∆ KEF(b) EF2=KE. GH(c) KG2=EM . KF

(4)(3)(3)

4.2.2 As gegee word dat KE = 20 eenhede, KF = 16 eenhede en GH = 4 eenhede, bereken die lengte van EM.

If it is given that KE = 20 units, KF = 16 units and GH = 4 units, calculate the length of EM. (3) [18]