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1 CORE MATHEMATICS PII Page 1 of 4 HILTON COLLEGE TRIAL EXAMINATION AUGUST 016 Time: 3 hours CORE MATHEMATICS PAPER 150 marks PLEASE READ THE FOLLOWING GENERAL INSTRUCTIONS CAREFULLY. 1. This question paper consists of 4 pages. There is also a separate yellow information sheet. Please check that your paper is complete.. Read the questions carefully. 3. This question paper consists of 13 questions. Answer all questions. 4. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator. 5. Round off your answers to one decimal digit where necessary, unless otherwise stated. 6. All necessary working details must be shown. 7. It is in your own interest to write legibly and to present your work neatly. 8. Please note that the diagrams are NOT necessarily drawn to scale. 9. Please ensure that your calculator is in DEGREE mode. Please do not turn over this page until you are asked to do so EXAMINATION NUMBER: M E M O I pledge that I have neither given nor received help with this assessment. Date: Signed:

2 CORE MATHEMATICS PII Page of 4 QUESTION 1 SECTION A (a) Give the equation of the circle shown below: () x y 4 a radius a centre (b) Give the equation of the line which has an angle of inclination of 45 o and and passes through the point 0; 3. () y x 3 a slope a intercept (c) The longer diagonal of a kite bisects the shorter diagonal at 90 o Use the above statement to find the equation of the longer diagonal. (4) mid pt of AB 3;5 gradient of AB c 3 c 4 1 y x4 3 a slope 1 3 gradient of longer diagonal m substitution ca a mid-pt m - ve reciprocal 8

3 CORE MATHEMATICS PII Page 3 of 4 QUESTION Consider the sketch of Guy cycling up a hill. The equation of the straight line representing the x road is y and the equation of the circle representing the rear wheel is 3 x 8x y 4y 1004 (a) Determine the angle the road makes with the horizontal. () m tan m link to gradient tan 30 ca (b) Assuming that units are in cm, determine the diameter of the rear wheel, in cm. (4) x x y y x y m completing the square x y a radius diameter is is 3 64 ca cm ca 6

4 CORE MATHEMATICS PII Page 4 of 4 QUESTION 3 Determine the possible value(s) of k if the point D3; A 9;3. k is a distance of 10 units from the point k k k 3 64 m solving for k k 38 k ll or 5 ca ca m distance formula a 5 QUESTION 4 (a) The Grade 1 Core Maths Marks for Term are depicted in the cumulative frequency curve shown below: Q 1 k Q 3 (i) How many matrics do Core Maths? (1) 78 a

5 CORE MATHEMATICS PII Page 5 of 4 (ii) Determine the inter-quartile range of the marks. Show by means of (3) dotted lines where you have read off any values you have used in your calculation. Q3 74 Q1 5 m a for showing on graph for both correct IQR 74 5 m for subtracting Q 1 from Q 3 (iii) What percentage of pupils achieved a distinction (80% or more)? () % 78 a m for % calculation (iv) Give a value for k if 40% of candidates achieved a mark of less than k? () 40% ~ 31 candidates k 56% ca a

6 CORE MATHEMATICS PII Page 6 of 4 (b) Ten Grade 11 boys achieved the following marks in Science and Mathematics in the June 016 Examinations.) Science Maths (x) (y) (i) Determine the equation of the line of best fit, the least squares regression line. Give both parameters to 3 decimal places. () y 0.886x a slope a intercept (ii) Use your answer to (b) (i) to predict the Maths mark for a boy who achieves a mark of 80% for Science. () 78.9% a m for substitution if shown (iii) Calculate, to decimal places, the correlation coefficient and comment on what it means for the relationship between the Science and Mathematics marks given. (3) r 0.71 a a a There is a reasonably strong, positive relationship between performance in Maths and performance in Science. 15

7 CORE MATHEMATICS PII Page 7 of 4 QUESTION 5 (a) In each case simply give the quadrant(s) in which must lie if: (i) cos 0 and tan 0 (1) (ii) IV a 3 3 sin and tan (1) 5 4 III a (b) Give the equations for each of the following graphs: (i) () y sin x a sin a amplitude of (ii) () y cos x 1 a cos a reflected and shifted (iii) () y 3tanx a tan a amplitude

8 CORE MATHEMATICS PII Page 8 of 4 (c) If sin 0 p then determine the following in terms of p: (i) cos 0 () cos 0 1sin 0 1 p a m identity / diagram (ii) sin 00 () sin 0 p (iii) cos 50 () sin 0 m reduction p a (iv) cos140 () cos 40 1sin 0 m double angle formula a p 1 m a reduction

9 CORE MATHEMATICS PII Page 9 of 4 (d) Consider the diagram with lengths and angles as marked: (i) Determine, to one decimal place if necessary, the length of AC. () 6 sin 30 m AC 6 AC 1 sin 30 cm correct method ca (ii) Hence, or otherwise, determine, to one decimal place, the length of AD. (3) AD cos AD 1.5 cm ca m cos rule a 1

10 CORE MATHEMATICS PII Page 10 of 4 QUESTION 6 (a) Complete the following statements: (i) If a line cuts two sides of a triangle in the same proportions then. (1) It is parallel to the third side a (ii) The exterior angle of a cyclic quadrilateral is. (1) Equal to the opposite interior angle a (iii) If two triangles have corresponding sides in the same proportion then. (1) They are equiangular a OR They are similar a (b) In the diagram below CP is a tangent to the circle at C. AB CD. AE = ED. PCD ˆ 40 and BDC ˆ 60

11 CORE MATHEMATICS PII Page 11 of 4 Calculate, giving reasons, the sizes of: (i) ˆB () a 40 tan - chord theorem (ii) ˆB 1 (1) a with reason 60 alt. ' s on lines a (iii) Ê () a a 10 opp. ' s of CQ (iv) ˆD 1 () a a 30 sum of ' s of isos (v) Â 1 () a a ˆD 0 opp. s of CQ 1 Â 100 sum s of 1

12 CORE MATHEMATICS PII Page 1 of 4 QUESTION 7 (a) In the diagram below, FE DC and DE BC. AF = 5 cm and FD = cm. Calculate, with reasons, the length of DB to one decimal place. (5) a a AF AE 5 prop. int. theorem FD EC m AD AE 5 but prop. int. theorem DB EC 7 5 a DB DB.8cm ca seocnd use of prop. int. theorem

13 CORE MATHEMATICS PII Page 13 of 4 (b) In the diagram below A is the centre of the circle. CD = 6 cm and EF = 1 cm. Calculate, giving reasons, the length of the radius. Hint: let the radius be x. (3) CE 3 cm from centre x 1 3 x Pythag. x 5 ca wj wj 8 TOTAL FOR SECTION A: 75 MARKS

14 CORE MATHEMATICS PII Page 14 of 4 QUESTION 8 Consider the diagram: SECTION B (a) Determine to 1 decimal place, giving reasons where necessary. (5) 1 tan 1tan 11 3 m a ca opp s of CQ ca reason (b) If it is further given that BC is a diameter of the circle then find the equation of the circle. (4) centre is r x y 15 4 ; ; a ca m a 9

15 CORE MATHEMATICS PII Page 15 of 4 QUESTION 9 Consider the diagram below, showing the circle with equation x y x y Determine, to 1 decimal place, the length of the tangent (AB) to the circle from the point A8; 1, to the point of contact B, giving reasons where necessary. x y r 5 and r 5, AB AO r pythagoras radius tangent but AO AB a AB 10 units ca a m finding centre and radius a m with reason 6

16 CORE MATHEMATICS PII Page 16 of 4 QUESTION 10 (a) John was asked to prove the tan-chord theorem, viz. that the angle between a tangent and a chord is equal to any angle subtended by that chord in the alternate segment. He knows that he needs to prove that A ˆ ˆ D and he has remembered that he needs to construct the diameter and to join FC. He has made the constructions correctly with dotted lines but now he is stuck! Complete the proof for John. (6) let Aˆ x Aˆ Aˆ 90 radius tangent Aˆ 90x but Cˆ Cˆ 90 in semi circle Fˆ x s of but Dˆ x s in same segment Aˆ Dˆ a m wj wj wj wj

17 CORE MATHEMATICS PII Page 17 of 4 (b) Consider the diagram below showing parallelogram ABCD with diagonals AC and BD drawn, intersecting at G. EF BD, CF = 3 cm and FD = 5 cm (i) Determine the ratio CE EA giving reasons. (4) CE CF 3 prop. int. theorem EG FD 5 let CE 3p then EG 5p m GA 8 p diagonals of parm. bi sect one another CE 3p 3 EA 13p 13 a wj a (ii) Determine Area CEF Area ABCD giving reasons. (4) CEF CGD AAA a areas of similar figures are in the ratio CEF 3 9 equal to the square of the ratios of wj CGD 8 64 their sides 1 diagonals bisect area of parm and but CGD parm. ABCD wj 4 CGD DAG equal base and height CEF 9 a ABCD 56

18 CORE MATHEMATICS PII Page 18 of 4 (c) In the diagram below, AB is a diameter of circle ABC with centre O. Chord BC is produced to D. OD AB and OD cuts AC at E. Prove, giving reasons: (i) That AOCD is a cyclic quadrilateral. (4) Cˆ Cˆ 90 in semi circle Cˆ 90 s on straight line Cˆ Oˆ a wj wj AOCD is a cyclic quadrilateral converse s in same segment wj (i) C ˆ ˆ D1 (3) Dˆ Aˆ s in same segment wj 1 1 but Aˆ Cˆ isos. radii Cˆ Dˆ a wj

19 CORE MATHEMATICS PII Page 19 of 4 (iii) OCE ODC (4) Cˆ Dˆ proved a 1 Oˆ is common 1 3 Eˆ Cˆ Cˆ third of OCE ODC AAA a wj wj (iv) OE.OD OC OE OD OC s OC OE. OD OC () a wj 7

20 CORE MATHEMATICS PII Page 0 of 4 QUESTION 11 The marks of a class of 3 boys have a mean of 73% and a standard deviation of 10. Angus and Gavin have marks of 79% and 67% respectively. Calculate, to decimal places, the standard deviation of the remaining boys if Angus and Gavin leave the class. Variance 1 i1 3 i1 initially 100 x x 300 i Suppose Angus and Gavin are number and 3 then now, x x 36 and x3 x 36 m Since x is unchanged by their leaving : x x i a so the new variance a and the new standard deviation ca 1 a 5

21 CORE MATHEMATICS PII Page 1 of 4 QUESTION 1 (a) Determine the general solution of the following equations: (i) 1 sin cos 0cos sin 0 (4) 1 sin 0 m key angle 30 m k or k with k Z for both k or k a a (ii) sin cos3 (5) sin sin 903 a m with k Z k or k for both.590k or 45180k ca ca m

22 CORE MATHEMATICS PII Page of 4 (b) In the diagram below, AB is a straight line 1000m long. P represents an object moving along AB. DC is a vertical tower with C, A and B points in the same horizontal plane. The angles of elevation of D from A and B are 0 and respectively. (i) Find the length of AC rounded to decimal places. () 154 tan 0 a AC 154 AC tan 0 m ca (ii) Find the value of to the nearest degree. (6). cos BC AC AB AC AB BAC BC cos 30 BC m ca 154 now tan m trig ratio a tan 13.0 ca ˆ m cos rule a

23 CORE MATHEMATICS PII Page 3 of 4 (iii) Let be the angle of elevation of D from P. (5) Determine the maximum value of to one decimal place. The angle will be greatest when P is closest to C which will happen when CP is perpendicular to AB m When this happens: CP sin m using 90 o triangle CP 43.11sin 30 CP m a now 154 tan m 1 tan ca

24 CORE MATHEMATICS PII Page 4 of 4 QUESTION 13 A point P moves in the plane in such a way that its distance from the point A ;3 is always equal to its distance from the line y 1. Two possible positions for P are shown and labelled as P1 and P to aid your understanding. By letting the point P be the point P xy, ; determine, in standard form, the equation of the function on which P moves while satisfying the condition of being equidistant A ;3 and the line y 1. from the point x y y x y y x x y y y y x x y ca x y x3 ca 4 m distance formula m squaring a m equating distances TOTAL FOR SECTION B: 75 MARKS 6 TOTAL FOR PAPER: 150 MARKS