BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS 2016 MATHEMATICS GRADE 12 PAPER 2

Transcription

1 BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS 2016 MATHEMATICS GRADE 12 PAPER 2 Examiner: Mr J Ruiz Mesa Ttal marks: 150 Date: 25 July 2016 Instructins: Mderatr: Ms A Smith Time: 3hrs This questin paper cnsists f TEN questins. Answer ALL the questins in the space prvided n the questin paper. Clearly shw ALL the calculatins, diagrams, graphs, et cetera yu have used in determining the answers. An apprved calculatr (nn-prgrammable and nn-graphical) may be used, unless stated therwise. If necessary, answers shuld be runded ff t ONE decimal place, unless stated therwise. Diagrams are NOT necessarily drawn t scale. An infrmatin sheet is included. It is in yur wn interest t write legibly and t present the wrk neatly. Please ensure yur calculatr is in DEGREE mde. DO YOUR BEST!! Max Mark Actual Mark Max Mark Actual Mark Stats. Q1 SECTION A Ttal SECTION B Ttal Analytical Ge. Q2+Q6 Trignmetry Q3+Q4+Q7 Cmment Cmment Euclidean Ge. Q5+Q8+Q9+Q10

2 BLANK PAGE This page has been left blank intentinally Page 2 f 22

3 SECTION A Questin 1 a) The cumulative frequency curve alngside shws the Mid-Year examinatin mark fr a grup f 200 pupils at a schl. The passing mark fr the exam was set at 30. 1) Determine the percentage f pupils wh passed the examinatin. (2) 2) A bx and whisker plt f the data is given belw. Using the diagram, r therwise, determine the values f m, n and p. (3) 0 m n p 100 Page 3 f 22

4 b) The scatter diagram belw displays the amunt f mney Tanner spends n cffee in relatin t the number f hurs he spends studying per week. 1) On the diagram, draw a line f best fit fr the data. (1) 2) Using yur line, estimate hw much Tanner will spend n cffee if he studied fr 6,5 hurs in a week. (1) 3) Culd yu estimate accurately the amunt Tanner wuld have spent n cffee if he had studied fr 1 hur in a week? Clearly explain the reasns fr yur answer. (2) Page 4 f 22

5 c) The table alngside shws infrmatin abut the year marks fr Mathematics and Physical Sciences fr a grup f 50 grade 11 students at the end f ) Jseph scred a Mathematics year mark f 65% and a Physical Sciences year mark f 63%. Assuming that the pupils were ranked frm highest t lwest accrding t their marks, in which subject was Jseph better ranked? Substantiate yur answer. (2) 2) The Mathematics mark was adjusted and 4% added t each mark. Write dwn the new Mean, Median, Standard Deviatin and Range fr Mathematics. (4) 3) Mr. Naid decided t adjust the Physical Sciences marks by subtracting 3% frm each f the tp 10% f the Science students. Determine the Mean, Median and Range fr Physical Sciences after the adjustment. (5) [20] Page 5 f 22

6 Questin 2 a) The diagram alngside shws line l, thrugh pints A and B, with equatin 3x 4y 24. The line intersects the x-axis at A and the y-axis at B. 1) Determine the crdinates f A and B. (2) 2) Determine the gradient f line l. (2) 3) Line m, thrugh pints B and C, is perpendicular t line l. Determine the equatin f the line m. (2) 4) Determine the angle f inclinatin f line m. (2) Page 6 f 22

7 5) Pint D lies n the line segment AC such that the area f ABD is 15 units 2. Determine the crdinates f D. (3) b) In the diagram alngside the equatin f the circle 2 with centre C is: x 2 y 1 r 2 2. The circle passes thrugh the pint A(0 ; -2). A tangent is drawn thrugh A. 1) Determine the value f r. Leave yur answer in surd frm. (2) 2) Determine the equatin f the tangent thrugh A. (2) Page 7 f 22 [15]

8 Questin 3 2 a) It is given that csx 1 and 3 0 x 180 f: sin 45 xcsx 60. Determine, withut the use f the calculatr, the value (4) Page 8 f 22

9 b) Simplify, withut using a calculatr: 2 cs 15 sin15 cs75 2 cs 15 sin15 cs15 tan15 (5) 1 cs2a c) Given the identity tan A sin 2A 1) Prve the identity. (4) 2) Hence, r therwise, find the general slutin t the equatin 1 cs2 sin 2 (5) [18] Page 9 f 22

10 Questin 4 a) The graph belw shws the graphs f f x sin bx and gx acs2x y 2 fr x 0 ; f x -1-2 g 1) Write dwn the values f a and b. (2) 2) Write dwn the perid f g. (1) 3) Fr what value(s) f x is 3 g x f x? (2) 4) In yur diagram, mark the segment(s) n the x-axis that will satisfy the inequality: g f x x 0 (2) [7] Page 10 f 22

11 Questin 5 a) In each f the fllwing cases mark the statement which is true abut the given diagram. 1) O is the centre f the circle. A, B and C are pints n the circumference. (1) ACB ˆ 2AOˆ B ACB ˆ AOˆ B AOB ˆ 2ACˆ B AOB ˆ 1 2 ACˆ B 2) O is the centre f the circle. A, B, C and D are pints n the circumference. (1) ADB ABC 360 DAB DCB 180 DAB DCB ABC BCD 3) O is the centre f the circle. AT and AN are tangents t the circle. (1) y 65 y x 90 x c 14cm x y 65 Page 11 f 22

12 Questin 5 (Cnt.) 4) O is the centre f the circle. A, B, C and D are pints n the circumference. PA and PC are tangents t the circle. (1) A w D z 24 B z x O y w 24 x 2z P x 2w C 5) O is the centre f the circle. A, B, C and D are pints n the circumference. A (1) D n 50 m n B m 50 m 42 C Page 12 f 22

13 b) In the diagram alngside AB is a tangent t the circle. O is the centre f the circle, OA and OC are radii and OA OC. Use the diagram t write a reasn fr each f the statements given belw. 1) s t 45 (1) 2) r 45 (1) 3) u 65 (1) 4) m 25 (1) 5) n 45 (1) c) Use the diagram alngside t determine the values f x, y, z and w. N reasns are required. 1) x = (1) 2) y = (1) 3) z = (2) 4) w = (1) Page 13 f 22 [15] SECTION A: [75]

14 SECTION B Questin 6 a) The diagram alngside shws three f the vertices f trapezium ABCD. A(-2 ; 1), C(5 ; 0) and D(3 ; 2) where AB CD. The furth pint B(p ; -4). 1) Shw that p = 3 (4) 2) If N(x ; y) is a pint n AB and NBCD is a parallelgram, determine the crdinates f N. (4) Page 14 f 22

15 b) In the diagram alngside, tw cncentric circles are given with centre P. Line intersect the y 7 2x 2 2 larger circle, x y 2x 10y 39, at A and B. The line is als a tangent t the smaller circle at H. 1) Determine the centre and radius f the larger circle. (5) 2) Determine the crdinates f H. (7) 3) If the pint H is H(5; -3), determine the equatin f the inner circle. (3) Page 15 f 22 [23]

16 QUESTION 7 a) Use the diagram alngside and cmplete the statement belw. (2) cs A b) In the diagram alngside A, B and C are three pints n a hrizntal straight line such that AB = 60m and BC = 30m. The angles f elevatin frm A, B and C respectively, t the tp f the twer, P, are. The ft f the twer Q is n the same, and hrizntal plane as A, B and C. PQ = h. Given that tan tan 13, tan 1 15 and 1) Write dwn the length f QA, QB and QC in terms f h. (3) 2) Shw that 2 h QCB ˆ cs using QBC. (2) 1200h Page 16 f 22

17 3) Using QAC write dwn the value f csqca ˆ. (3) 4) Hence, r therwise, determine the height f the clck twer (h). (4) [14] Page 17 f 22

18 QUESTION 8 a) In the diagram alngside, O is the centre f the circle. C and ÔD 130 A. ĈE 40 Determine the size f angle A. N reasns are required. (3) b) In the diagram alngside, O is the centre f the circle. AQ is a diameter, SA and AB are chrds such that AQ bisects SÔB. M and E are pints n SA and AB such that BM and SE intersect at O. AM = ME. Given that Q ÔB = x 1) Shw, giving reasns fr yur answers, that Ê 1 = x. (3) 2) Shw that AMOE is a cyclic quadrilateral. (4) Page 18 f 22

19 3) Shw that OM is the perpendicular bisectr f AS. (6) [16] Page 19 f 22

20 QUESTION 9 a) In the diagram alngside. O is the centre f the circle. TAN is a tangent at A. Prve the therem that states: The angle between the tangent and the chrd is equal t the angle in the alternate segment. i.e. prve NÂD AÊD (6) b) In the diagram alngside, PT and PQ are tangents t the circle at T and Q respectively. PK QA. K is a pint n AT and R is the intersectin f TQ and PK. A 1) Prve, giving reasns, that PTKQ is a cyclic quadrilateral. (6) Page 20 f 22

21 2) Prve, giving reasns, that PK bisects TKˆ Q. (6) [18] Page 21 f 22

22 Questin 10 The diagram alngside represents a painting (AB) n display at a gallery. Frm a viewpint (V) 3m away frm the wall the angle between the tp and the bttm f the painting is the same as the angle f elevatin t the bttm f the painting. If the tp f the painting is 4m frm the flr, determine hw far the bttm f the painting is frm the flr (BC). (4) [4] Ttal: 150 Page 22 f 22