NYGH 2015-S3EOY-IM1

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2015 Secondary Three INTEGRATED MATHEMATICS 1 2 hours Monday 12 October 2015 0845 – 1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Answer questions 1 – 10 before attempting the Bonus Question. 2. Answer the whole of question 8 on the graph paper provided. 3. Write your answers and working on the separate writing paper provided. 4. Write your name, register number and class on each separate sheet of paper that you use and fasten the sheets together with the string provided. Do not staple your answer sheets together. 5. Omission of essential workings will result in loss of marks. 6. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES 1. The number of marks is given in brackets [ ] at the end of each question or part question. 2. The total number of marks for this paper is 80. 3. The use of an electronic calculator is expected, where appropriate. 4. You are reminded of the need for clear presentation in your answers. Setter: CMF This document consists of 8 printed pages. NANYANG GIRLS' HIGH SCHOOL [ Turn over

2 Mathematical Formulae 1. TRIGONOMETRY Formulae for ∆ABC C c B b A a sinsinsin == a2 = b2 + c2 − 2bc cos A ∆ = 2 1 ab sin C

3 1 Given that 17 15cos −=A and °<<° 1800 A , find, without using a calculator, the value of (i) Asin , [1] (ii) )180(tan A−° , [2] (iii) )90(cos °−A . [2] 2 The following equations (i) xb x ay −= , (ii) xaby += 2 , may be represented by a straight line in the form cmXY += . a, b, m and c are constants and X and Y are each functions of x and/or y. Copy the following table and insert in it an expression each for Y, X, m and c. [4] Y X m c xb x ay −= xy xaby += 2 [Turn over

4 3 In the diagram, ∆ABD and ∆ACE are right-angled triangles. BCD and AED are straight lines. Given that AB = 12 cm, AC = 38 cm, AE = 16 cm and °=∠ 30ADC , (a) find the exact value of CED∠sin , [2] (b) show that ∆CED is isosceles, [2] (c) find the radius of the circle through A, B and D. [2] 4 (a) On different axes, sketch the graph of (i) 3 2 1 xy −= , [2] (ii) nxay = , where a > 0 and n = −2. [2] (b) Write down a possible equation for this graph, where y-intercept = 2 and asymptote is 1=y . [2] 5 The variables x and y, where 0>x and 1≥y , are related in such a way that when xy −2 is plotted against y , a straight line graph is obtained. Given that the line passes through the points (3, 5) and (1, 1), (i) express y in terms of x, [4] (ii) find the value of y when 9=x . [1] 12 A D C B 16 30o E 38 y = 1 x 0 y 2

5 6 A circle, with centre C, passes through the points P (−5, 0) and Q (7, −12). The point C lies on the line 133 −= xy . (i) Find the equation of the perpendicular bisector of PQ. [4] (ii) Find the coordinates of C. [2] (iii) Find the equation of the circle. [2] 7 The diagram shows a circle ABCDE with centre O. XY is a tangent to the circle at A and CDX is a straight line. °=∠ 52BAO , °=∠ 18ADE , °=∠ 94BCD and °=∠ 68EDX . (a) Stating your reasons clearly, find (i) AOE∠ , [2] (ii) XAE∠ . [2] (b) (i) Show that BA is parallel to CX. [2] (ii) Hence, or otherwise, find AXD∠ . [2] (c) Is AODX a cyclic quadrilateral? Explain your answer clearly. [2] [Turn over 94o A B C D E O Y 68o 18o X 52o

6 8 Answer the whole of this question on a sheet of graph paper. The variables x and y are connected by the equation xxy 352 +−= . Some corresponding values of x and y are given in the table below. x 0.25 0.5 1 1.5 2 3 4 5 6 y 7.5 2 0 0 0.5 2 3.75 5.6 7.5 (a) Using a scale of 2 cm to represent 1 unit on each axis, draw a horizontal x-axis for 625.0 ≤≤ x and a vertical y-axis for 81 ≤≤− y . On your axes, plot the points given in the table and join them with a smooth curve. [3] (b) By drawing a tangent, find the gradient of the curve at (2, 0.5). [2] (c) Use your graph to find the solutions to the equation 02 3 2 9 =+− xx . [2] (d) Find the equation, in the form 07 2 =++ CBxx , which is satisfied by the x-coordinates of the points where the line xy 2 34−= intersects the curve xxy 352 +−= . [2] (e) (i) On the same axes, draw the line xy −= 3 for 30 ≤≤ x . [1] (ii) State the value of c such that cxxx +−=+− 352 has exactly 1 solution for 625.0 ≤≤ x . [1]

7 C B North 123 m A 42o 120 m 9 Three markers, A, B and C, are placed on a horizontal field, where B is due north of A. AC = 120 m, BC = 123 m and °=∠ 42ACB . (a) Calculate (i) AB, [2] (ii) the bearing of C from A. [3] (b) A man standing at A is flying a drone. The drone, D, is 65 m vertically above B. Calculate the angle of depression of an object at C from the drone. [2] (c) The man starts to walk from A to C. Calculate his distance from B when his angle of elevation of D is the greatest. [2] (d) The marker at B is relocated along the path BC to a point E such that AE = AB. Calculate the area of ∆AEC, giving your answer to 3 significant figures. [3] [Turn over

8 10 Solutions to this question by accurate drawing will not be accepted. The diagram below shows part of the curve )4)(2( +−= xxy with its minimum point at T. The curve cuts the x-axis at R and S and intersects the line 10+−= xy at P and Q. (i) Find the coordinates of S, R and T. [3] (ii) Find the coordinates of P and Q. [4] (iii) Find the area of quadrilateral PQRS. [2] The point X is such that STRX is a kite and X lies on the line PQ. (iv) Find the coordinates of X. [2] (v) Find the value of ofarea ofarea RXP RQX ∆ ∆ . [2] Bonus Question 11 A region A is bounded by the lines 5,5 ±=±= yx , 102 += xy and 102 −= xy . The largest circle with centre at (0, 0) that can be fitted into region A has an area of πk square units. Find the value of k. [3] End of paper Q 10+−= xy 0 )4)(2( +−= xxy y x P S R T X  