NYGH 2018-S3EOY-IM1

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2018 Secondary Three INTEGRATED MATHEMATICS 1 2 hours Thursday 4 October 2018 0845 – 1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Answer questions 1 – 11 before attempting the Bonus Question. 2. Write your answers and working on the separate writing paper provided. 3. Write your name, register number and class on each separate sheet of paper that you use and fasten the separate sheets together with the string provided. Do not staple your answer sheets together. 4. Omission of essential steps will result in loss of marks. 5. 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 approved scientific calculator is expected, where appropriate. 4. You are reminded of the need for clear presentation in your answers. Setter: CL This document consists of 10 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 Area of triangle ABC = 2 1 ab sin C 2. MENSURATION Arc length θr= , where θ is in radians Sector area θ2 2 1 r= , where θ is in radians

3 [ Turn over 1. Boyle’s Law states that at constant temperature for a fixed mass, the absolute pressure, P, is inversely proportional to the volume, V, of a gas. A sample of a gas has a volume of 0.005 m 3 at a pressure of 7600 Pascal (Pa). (i) Find an equation connecting P and V. [2] (ii) What will the volume of the gas be if the pressure is increased to 18600 Pa? [1] (iii) Sketch a graph to represent the relationship between P and V. [1] 2. (a) Sketch each of the graphs of y = axn separately, where (i) a < 0, n = 2, (ii) a > 0, n = ̶ 2. [2] (b) Write down a possible equation for each of the following graphs. (i) (ii) [2] 3. A small circle of centre P, radius 4 cm, touches a large circle of centre Q, radius 9 cm, at the point R. The circles are lying on a straight line ABCD. The point E is the foot of the perpendicular from P to QC. Find the perimeter of the shaded region. [5] E R C P Q BA D O 2 y x y x 2 O 4 cm 9 cm

4 4. Variables x and y are known to be related by an equation of the form 𝑥𝑥 = 𝑎𝑥3 + 𝑏𝑥2, where a and b are constants and 𝑥 ≠ 0 . When the graph of 𝑦 𝑥2 against 1 𝑥 is plotted, a straight line passing through the points (0.1, 0.9) and (0.5, 1.5) is formed. Estimate (i) the value of a and of b, [3] (ii) the value(s) of x when y = 2. [3] If a new line is formed when 𝑦 𝑥 is plotted against x, (iii) find the intercept on the x-axis. [2] 5. The diagram shows a triangle ABC where A has coordinates (2, 8) and AC is parallel to 5y + 8x + 4 = 0. AB meets the y-axis at D (0, 4) and AC meets the x-axis at F. (i) Find the equation of the line AC. [2] (ii) Determine if DF is perpendicular to AB. [3] (iii) Given that AB:BD = 3:2, find the coordinates of B. [2] (iv) If the equation of BC is 4y + x + 20 = 0, find the coordinates of C. [3] (v) Find the area of quadrilateral BCFD. [2] y x O A(2, 8) B C D (0, 4) F 4y + x + 20 = 0

5 [ Turn over 6. An outbreak of a contagious disease has hit a local secondary school. The number of lower and upper secondary boys and girls in the school and the percentage of lower and upper secondary students who are currently healthy, ill or carriers of the disease are given in the table below. Boys Girls Healthy Ill Carriers Lower Secondary 120 160 25% 30% 45% Upper Secondary 140 120 35% 25% 40% (i) P is a 2 × 2 matrix representing the number of boys and girls in the lower and upper secondary sections respectively. Q is a 2 × 3 matrix representing the percentage of lower and upper secondary students who are currently healthy, ill or carriers of the disease. Evaluate PQ and describe what the elements of matrix PQ represent. [2] (ii) As a result of the outbreak, the school sent all the students for a check -up. The students rece ived treatment if they were ill or carriers. The check -up and/or treatment for students who are healthy, ill and carriers cost $10, $25 and $15 respectively. Write down a 3 × 1 matrix R such that the produc t PQR gives the total medical cost the school incurred for the boys and for the girls respectively. Find the total medical cost the school incurred for the boys and for the girls respectively. [3] (iii) Given that the School Health Services decided to give the female students a discount of 30% of the medical cost, using matrix multiplication , find the total medical costs incurred by the school for all the students as a result of the discount. [2] 7. At a cross-country run, the first-aid post F is located 650 m north of an ambulance A . A milo van M is located 450 m away from the first-aid post such that the bearing of M from A is 319°. (i) Find the two possible values of angle AMF. [3] (ii) A drone D is hovering at a point vertically above the ambulance. A student on duty at the first -aid post spotted the drone at an angle of elevation of 5°. She walked south towards t he ambulance and reached a point N . The angle of elevation of the drone from the student at N is 10°. Find the distance the student walked towards N. [3]

6 8. Answer the whole of this question on the INSERT provided. The graph of 𝑥 = 1 5 𝑥3 + 2 𝑥 is partially shown on the INSERT. (a) By drawing a tangent, find the gradient of the graph at the point where x = 0.5. [3] (b) Find the x-coordinate of a point on the curve at which the gradient is 5. [2] (c) By drawing suitable straight lines on the graph, find (i) the range of values of x for which 1 5 𝑥3 + 2 𝑥 ≤ 6 − 𝑥, [2] (ii) the solutions to the equation 𝑥4 − 5𝑥2 − 15𝑥 + 10 = 0. [3] x y 10 0 20 5 5 1 4 2 3 15

7 [ Turn over 25° 40° B H F G E O D A C J 9. ABCD is a tangent to the smaller circle BGHE. The circles BCJE and BGHE intersect at B and E. JC is the diameter of the larger circle BCJE, with centre O. EFGC and JHFB are straight lines that intersect at F. It is given that angle BEC = 40° and angle EJB = 25°. (a) Find (i) angle BJC, [1] (ii) angle EBA, [1] (iii) angle JCE, [2] (iv) angle BFC. [1] (b) State the geometrical property used in (a)(i). [1] (c) Given that minor arc JE = 3 cm, find the length of minor arc EB, explaining your reasons clearly. [2]

8 10. The diagram below shows circle C 1 which passes through the points (1, 5) and (− 6, 6) and has its centre lying on the line 𝑥 + 𝑥 = −1. (i) Show that the centre of circle C1 is (−3, 2). [4] (ii) Hence, find the equation of circle C1. [2] The line y = 7 is a tangent to circles C 1, C2 and C3. Circles C2 and C3 have a radius of 3 and touch the y-axis at (0, 4). (iii) Write down the coordinates of the centre of C 2 and of C3. [2] (iv) Explain why it is possible to draw a circle C 4 passing through the centres of the three circles. Find the centre of C4. [2] C1 C2 C3 y = 7 x + y = -1 (0, 4) (-6, 6) (1, 5) 𝑥 + 𝑥 = −1

9 [ Turn over 4.92 km 4.5 km 4.84 km 2 km 3.55 km Q B P CA 11. Three cell towers A , B