Department of Mathematics

Transcription

1 Department of Mathematics TIME: 3 hours Setter: CF DATE: 06 August 2018 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER II Total marks: 150 Moderator: DAS Name of student: PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 20 pages and an Information Sheet of 2 pages (i ii). Please ensure that your question paper is complete. 2. Read all the questions carefully. 3. Answer all the questions on the question paper and hand this in at the end of the examination. Remember to write your name on the paper. 4. Diagrams are not necessarily drawn to scale. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 6. All necessary working details must be clearly shown 7. Round off your answers to one decimal digit where necessary, unless otherwise stated. 8. Ensure that your calculator is in DEGREE mode. 9. It is in your own interest to write legibly and to present your work neatly. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total

2 SECTION A QUESTION 1 In the diagram below, ABC is an isosceles triangle with A( 2;1 ) and B (4;9). AB BC and BC is parallel to the y-axis. Angle is indicated. (a) Show that the length of AB is 10 units. (2) (b) Hence determine the coordinate of C. (2) (c) Calculate the coordinates of K, the midpoint of AC. (2) 2

3 (d) Determine the equation of AC in the form y mx c. (3) (e) Calculate the size of. (3) (f) Calculate the area of ABC. (3) (g) Write down the coordinates of D if ABCD is a parallelogram. _ (2) [17] 3

4 QUESTION 2 The table below shows the results from a survey on expenditure of monthly data usage for 100 learners from a school in Grahamstown. Expenditure (in rand) Frequency Cumulative Frequency 0 x < x < x < x < x < 250 a 250 x < (a) Determine the value of a. (1) (b) Complete the cumulative frequency table. (2) (c) Draw an ogive for the data on the set of axes below. Clearly label your axes. (4) (d) What is the modal class for the data? (1) [8] 4

5 QUESTION 3 (a) Use the diagram to prove the theorem which states that the opposite angles of a cyclic quadrilateral are supplementary. (5) 5

6 (b) In the diagram, the vertices of ABD lie on the circle with centre O. Diameter AC and chord BD intersect at T. Point W lies on AB. OT BD. Â 1 = 30. Determine, giving reasons, the size of: (1) Ĉ (3) (2) Â 2 (3) 6

7 (3) ˆB 2 (3) (4) If it is further given that BW WA, prove that TBWO is a cyclic quadrilateral. (3) [17] 7

8 QUESTION 4 This question must be answered without the use of a calculator. (a) If tan 8 and 180 ; 360 determine, by using a sketch, the value of: (1) 8 sin cos sketch (4) (2) sin 2 (3) (b) Simplify and hence determine the numerical value of: cos(90 ).tan 210.sin170 sin(180 ).cos100 (5) 8

9 (c) Prove that: 1 cos 2x sin x tan x sin 2x cos x (6) 2 (d) Determine the general solution for: sin 2x cos x 0. (6) [24] 9

10 QUESTION 5 In the diagram below, the graph of f ( x) sin( x 30 ) is drawn for the interval x 30 ; 150. (a) On the same set of axes sketch the graph of ( for 30 ; 150 g x) cos 3x x. (3) (b) Write down the period of g. (1) (c) Estimate the x -coordinate of the points of intersection of f and g and hence write down the values of x for which f ( x) g( x) in the interval x [ 30 ;150 ]. (2) (d) Given h ( x) g( x) 2, write down the range of h. (2) [8] 10

11 SECTION B QUESTION 6 A Science teacher wants to create a model by which he can predict a learner s test result based on a previous test written on the same content. Test marks are given below as percentages. First test (x) Second test (y) (a) Determine the equation of the line of best fit in the form and B correct to 3 decimal places. y A Bx, giving A (3) (b) Determine the correlation coefficient of the data correct to 3 decimal places. (1) (c) Describe the correlation between the two tests. (2) (d) Use your equation in (a) to predict the test mark for a learner who attained 46% in the first test. (2) (e) Determine x and y correct to 3 decimal places and hence show that the point ( x ; y) lies on the line of best fit. (4) [12] 11

12 QUESTION 7 In the diagram, diameter AB of circle ACB with centre D is given. The coordinates of A and B are ( 1;8 ) and (5;0) respectively. C is a point on the x-axis. (a) Determine (1) the equation of the circle ACB. (5) (2) the coordinates of C. (3) 12

13 (3) the equation of the tangent to the circle at C. (4) 2 2 (b) Another circle with equation x 4x y 8y 29 0 is given. (1) Determine, showing all working, whether the two circles are concentric (HAVE THE SAME CENTRE). (4) (2) Determine, with reasons, whether this circle lies inside or outside circle ACB. (2) [18] 13

14 QUESTION 8 In the figure below, AT : TC 2 : 1 and AD // TE. ABC has D and E on BC. BD 10 cm and DC 15 cm. CE (a) Write down the numerical value of ED (1) (b) Show that D is the midpoint of BE. (2) 14

15 (c) If FD 2, 5 cm, calculate TE giving a reason for your answer. (2) (d) Calculate the value of: Area ΔADC Area ΔABD (2) (e) Caluculate the value of: Area ΔTEC Area ΔABC (3) [10] 15

16 QUESTION 9 In the diagram below, ABCD is a cyclic quadrilateral with AD CD. Chords AC and BD intersect at H. BA is extended to E such that ED // AC. Prove that: (a) DE is a tangent to circle ABCD at D. (4) 16

17 (b) EBD EDA (4) (c) ED 2 EB. EA (2) (d) 2 ED HD. BD. EA 2 (4) [14] 17

18 QUESTION 10 In the diagram, P, T and R are three points in the same horizontal plane. SR is a vertical tower of height h metres. The angle of elevation of S from T is. In addition, P RT ˆ, R T ˆP 30 and PT = 6 m. (a) Express h in terms of TR and. (2) (b) Express TP ˆ R in terms of. (2) (c) Show that h 3(1 3 tan ). (6) 18

19 QUESTION 11 [10] Most modern soccer balls are stiched together from 32 panels of waterproofed leather, using 12 regular pentagons and 20 regular hexagons. The surface area of each hexagonal panel is 52,68 cm 2. The distance from the vertex of each pentagon to the centre of the pentagon is 3,8 cm. Calculate the total amount of leather that is used to make a soccer ball. [6] 19

20 QUESTION 12 A cylinder with radius 4 units fits snugly into a right-angled triangular box, the cylinder just touching all three sides of the triangular box. If KP x, K J ˆL 90 and the hypotenuse of the triangle is 24 units, determine, with reasons, the value(s) of x. [6] Total: