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6.1 In the figure below AB is a tangent at A to the circle with centre O. D and E are points on the circle Prove the theorem that states that DAB = E. (5) 6.2 In the figure PT is a tangent to the circle at T and TR is a chord. PR cuts the circle at Q. S is a point on TR so that PTSQ is a cyclic quadrilateral. PS, TQ and SQ are joined. Let S, = x O 6.2.1 Prove that PS is a tangent to circle QSR at S. (5) 6.2.2 Prove that PT - PS. (6) [16]QUESTION 3 ABCD is a cyclic quadrilateral and CB is extended to E. DA = 155mm and DC = 136mm. ABE =810 and CE - 193 mm. AD | CE. 155 mm D 136 him 193 mm 3.1 Give a reason why D = 81". (1) 3.2 Calculate the length of AC. (3) 3.3 Calculate the size of DAC. (3) 3.4 Calculate the area of AACE (4 ) QUESTION 4 The organiser of the regional school soccer league decided to have a trophy made to present to the winning team of the season. He had it made in the shape of a cylinder with a sphere on top of it. The radius of the cylinder is 5 cm and its height is 20 cm. The total volume of metal used to make this trophy is 2000 cm'. V = -arth V = arch 4 SOCCER 3 TROPHY 20 cm Calculate the radius of the sphere.QUESTION 2 2.1 Jane bought a laptop for R9 600. Calculate the book value of the laptop after 3 years, if it depreciates at 20% p.a. on the reducing balance method. (2) 2.2 Sandra invests R20 000 in a savings account that pays interest at the rate of 12% p.a., compounded monthly. How much will she have accumulated in this account after 4 years? (3) 2.3 Thuli has two sons: Bongani and Sbu. They share the same birthday. On the day that Bongani had his 12" birthday and Sbu his 15" birthday, she opened an account for each of them, and invested some money in each account. The total amount that she invested in the two accounts was R 150 000. Both accounts earn interest at 9.6% p.a., compounded quarterly. She divided the R 150 000 between the two accounts in such a way that each of the boys will receive the same amount of money on his 25" birthday. How much money does she invest for Bongani? (5) [10] QUESTION 3 ABCD is a cyclic quadrilateral and CB is extended to E. DA = 155mm and DC = 136mm. ABE = 810 and CE = 193 mm. AD | CE. 155 mm D E 136 Mm 193 mm 3.1 Give a reason why D = 81". (1) 3.2 Calculate the length of AC. (3) 3.3 Calculate the size of DAC. (3) 3.4 Calculate the area of AACE (4) [111QUESTION 1 1.1 For two events A and B, it is given that P(A) = 0,4 and P(B) = 0,3. Calculate P(A or B) if: 1.1.1 A and B are mutually exclusive. (2) 1.1.2 A and B are independent. (3 ) 1.2 A survey was carried out among 100 learners about which movies they have watched recently. The results are given below: 43 watched Braven (B) 41 watched Hereditary (H) 50 watched The Kissing Booth (K) 6 watched all 3 movies 7 watched Braven and The Kissing Booth but not Hereditary 18 watched Hereditary and The Kissing Booth . 15 watched only Hereditary The above information is represented in the Venn diagram below. B 15 K 1.2.1 Write down the values of a, b, c and d in the Venn diagram above. (4) 1.2.2 Calculate the probability that a learner selected at random from this group has not watched Braven or Hereditary or The Kissing Booth. (3) 1.2.3 Calculate the probability that a learner selected at random from this group has watched at least 2 of these movies. (3) [15]QUESTION 5 5.1 Given / (x) = 2cos 3.r. 5.1.1 Use the grid on DIAGRAM SHEET 2 and draw a sketch graph of f for the interval .re [-90 ; 909]. Clearly show all intercepts with the axes and turning points of the graph. (3) 5.1.2 Write down the period of f. (1) 5.1.3 Write down the range of f. (1) 5.2 In the diagram below the graphs of g(x) =sin(x + a) and h(x)=btance are drawn in the interval re [-180 ; 1809]. g and h intersect at D and E. The coordinates of Dare (-1350 ; -1) and the coordinates of E are approximately (-22.30 ; 0,4). 450 5.2.1 Write down the values of a. b and c. (3) 5.2.2 Write down the equations of the asymptotes of / in the interval [-180 : 1809]. (2) 5.2.3 Solve for x if g (x)-h(x) 20 in the interval [-180 : 09]. (4) 5.2.4 Describe the transformation of g to m if m(x)= cos.x-1. (2) [16]QUESTION 4 4.1 The sketch below shows APQR with PQ = 32 mm. QR = 60 mm and P= 50. 50 R 32 mm 60 mm Q Calculate the size of R . (3) 4.2 ADEF is shown in the sketch below. G is a point on DF and EG is drawn. DG = DE =>. FG = #x and EG = V3x. 3.x G D E 4.2.1 Calculate the size of D. (4) 4.2.2 Calculate the area of AGEF in terms of .x, in its simplest form. (5) [12]QUESTION 3 3.1 Two bags. A and B. are filled with coloured balls. It is equally likely that Jade will choose bag A or bag B. Bag A is filled with 5 green balls (G) and 3 yellow balls (Y). Bag B is filled with 2 green balls (G) and 6 yellow balls (Y). Jade randomly selects a bag and draws a ball from it. 3.1.1 Draw a tree diagram to represent the above information. Show the probabilities associated with each branch as well as the possible outcomes. (3) 3.1.2 Write down the probability that Jade will select bag A and draw a green ball from it. (1) 3.1.3 Calculate the probability of Jade choosing a yellow ball. (3) 3.2 A and B are two events in a sample space. P(not A) = 0,48 and P(B) = 0.32. 3.2.1 Determine P(A). (1) 3.2.2 Determine P(A and B) if A and B are independent events. (2) 3.3 During the Olympic Games in 2016, complaints about a certain hotel fell into 3 main categories: Reception (R). Food (F) and Accommodation (A). A total of 173 complaints were received about this hotel. There were: 1 15 complaints about the Reception. 55 complaints about the Food. 63 complaints about the Accommodation 23 complaints about the Reception and the Food but not the Accommodation. 12 complaints about the Reception and the Accommodation but not the Food. 17 complaints about the Food and the Accommodation but not the Reception. Let the number of complaints about all 3 categories be .r. 3.3.1 Draw a Venn Diagram to represent the information above. (3) 3.3.2 Determine the value of .x. (2) 3.3.3 Determine the probability that a complaint. selected at random from those received, was about: (a) The Accommodation only. (1) (b) At least two of the categories. (2) [18]QUESTION 2 The amount of data, in megabytes, that a student used browsing the internet each day was recorded. The information is given in the frequency table below. Amount of data used (in megabytes) Number of days 100
