NYGH 2019-S3EOY-IM1 with ans

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2019 Secondary 3 INTEGRATED MATHEMATICS 1 2 hours Thursday 10 October 2019 0845 – 1045 READ THESE INSTRUCTIONS FIRST 1. Write your name, register number and class on all the work you hand in. 2. Answer all questions. 3. Write your answers and working on the separate writing paper provided, unless otherwise stated. 4. Write in dark blue or black ink on both sides of the paper. 5. You may use an HB pencil for any diagrams or graphs. 6. Do not use staples, paper clips, glue or correction tape/fluid. 7. Omission of essential working will result in loss of marks. 8. The use of an electronic calculator is expected, where appropriate. 9. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degree to one decimal place. For 𝜋, use either your calculator value or 3.142, unless the question requires the answer in terms of 𝜋. 10. At the end of the examination, fasten all your work securely together. 11. The number of marks is given in brackets [ ] at the end of each question or part question. 12. The total number of marks for this paper is 80. Setter: CHY This document consists of 8 printed pages, including this cover page. 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 r , where  is in radians Sector area 2 2 1 r , where  is in radians

3 1 (a) Given 𝐀 = ( 3 0 −2 1) and 𝐁 = (2 −5 3 1 ), find k and m if 𝐀2 − 𝑘𝐁 = (13 −10 𝑚 3 ). [4] (b) Ginger Tea, Jasmine Tea and Milk Tea are drinks sold by three Bubble Tea outlets. During Christmas, all proceeds from the three outlets are donated to charity. The number of cups of each type of drink sold during this year’s Christmas is shown in the table below. Number of cups sold Shop A Shop B Shop C Types of Drink Ginger Tea 70 120 30 Jasmine Tea 130 90 160 Milk Tea 50 160 50 (i) Calculate (1 1 1) ( 70 120 30 130 90 160 50 160 50 ). [1] (ii) Explain what your answer represents. [1] For the Christmas charity drive, each outlet charges a fixed price for all types of drink. Shop A sells its drinks at $3.50 per cup, Shop B at $4.50 pe r cup while Shop C charges $4 per cup. (iii) By multiplying your answer from (i) with another matrix, find the total amount of money that will be collected from all the three Bubble Tea outlets. [2] [ Turn over]

4 2 The diagram below shows a circle wi th centre O, and A, B, C and D are points on the circumference of the circle. EF is a tangent to the circle at point B. Angle AOB = 110°, angle CBD = 22° and angle BDC = 35°. (a) Find, giving reasons for each answer, (i) angle DAB, [2] (ii) angle DAO, [2] (iii) angle EBC, [1] (iv) angle ADB. [1] (b) A point P is such that angle BPC is 30°. Is P inside, outside or on the circumference of the circle? Justify your answer with a brief reason. [2] (c) Explain whether points A, O and C are collinear. [2] O A B C D E F 110° 35° 22°

5 3 The diagram shows a cube ABCDEFGH of sides 6 cm. Find (a) the length of AM, where M is the midpoint of CH, [2] (b) angle AMF. [2] 4 In the diagram, P, Q, R and S are four points on level ground. The bearing of Q from P is 320° and the bearing of R from Q is 055°. PQ = 120 m, PS = 250 m, QR = 170 m and angle PSR = 48°. (a) (i) Find angle PQR. [2] (ii) Hence, show that the length of PR is 199 m. [2] (b) Calculate, using the value of PR given in (a)(ii), (i) the acute angle PRS, [2] (ii) the area of triangle PRS. [2] (c) A tree stands at S. An eagle sits on top of the tree, looking at a mouse as the mouse runs along PR. Given that the greatest angle of dep ression from the eagle to the mouse is 3.6°, find the height of the tree. [3] A D B C H G E M  6 cm 6 cm 6 cm F P R N 170 m Q 120 m 48° S 250 m [ Turn over]

6 5 Solutions to this question by accurate drawing will not be accepted. The diagram below (not drawn to scale) shows a quadrilateral ABCD. AB meets the y-axis at 10, and BC meets the x-axis at –4. Points A and C lie on the y-axis and x-axis respectively. (a) BC is parallel to the line y – 2x + 9 = 0. Find the equation of BC. [2] (b) The length of AB is 2√2 units. Show that the coordinates of B are (2, 12). [3] (c) The point D (–3, p) is equidistant from points B and C. Find the value of p. [4] (d) Calculate the area of quadrilateral ABCD. [2] 6 Write down a possible equation for each of the following graphs. [3] x y O (1, 1) (1, –2) (a) (b) x y O x y O (c) y B A 10 C D x –4 O

7 7 Answer the whole of this question on the insert provided. The diagram below shows part of the graph of 𝑦 = − 2 5 𝑥2 + 1 𝑥 + 3. 𝑦 = − 2 5 𝑥2 + 1 𝑥 + 3 [ Turn over] (a) By drawing a tangent, find the gradient of the graph at the point where x = –2. [2] (b) Use your graph to solve the inequality − 4 5 𝑥2 + 2 𝑥 + 2 ≥ −5 in the range of −3 ≤ 𝑥 ≤ 3.5. [3] (c) State the range of values of k for which the equation − 2 5 𝑥2 + 1 𝑥 + 3 = 𝑘 has less than two solutions. [1] (d) (i) On the same axes, draw the graph of 2𝑦 + 𝑥 = 2. [1] (ii) Write down the x-coordinates of the points at which the two graphs intersect. [2] (iii) Find the equation, in the form 4𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, which is satisfied by the values of x found in d(ii), given that a, b, and c are integers. [2]

8 8 The diagram shows a sector ABCDEF with an inscribed circle BDFG, centre O. ABC and AFE are tangents to the circle at points B and F respectively. The arc CDE touches the circle at D. The radius of the circle BDFG is 7 cm. (a) The arc length of the arc BGF is 14𝜋 3 cm. Show that angle BAF = 𝜋 3 radians. [3] (b) Calculate the length of OA. [2] (c) Find the area of the shaded region. Leave your answer in terms of 𝜋. [3] 9 The variables x and y are related by the equation 𝑥𝑦 = 𝑥+𝑝𝑦 𝑞 . When the graph of 𝑥 𝑦 against x is drawn, a straight line is obtained which has a gradient of 5 and passes through the point (4, –12). (a) Find the value of p and of q. [4] (b) Find the value of x when 3y = x. [2] (c) The equation 2𝑥 + 𝑎𝑥𝑦 − 4𝑏𝑦 = 0 is plotted on the same axes of 𝑥 𝑦 against x. It has an infinite number of solutions when solved simultaneously with 𝑥𝑦 = 𝑥+𝑝𝑦 𝑞 . Find the value of a and of b. [3] 10 A circle C1 of equation x2 + y2 + 16x – 30y = 0 meets a second circle C2 at points (0, 0) and (7, 7). Given that the radii of C1 and C2 are equal, find (a) the radius of C1, [2] (b) the equation of C2. [3] END OF PAPER A B C F D E G O

9 Answers to IM1 1(a) k = –2, m = –2 (b)(i) (250 370 240) (ii) 250 represents the total number of drinks sold by A, 370 represents the total number of drinks sold by B, and 240 represents the total number of drinks sold by C. (iii) $3500 2(a)(i) 57° (ii) 22° (iii) 35° (iv) 55° (b) 𝑃 is outside the circumference of the circle. P cannot be on the circumference. According to the property “angles in the same segment”, if P is on the circumference of the circle, angle BPC will be equal to angle BDC, which is 35°. P cannot be inside the circumference, because BPC is less than 35o. Since angle BPC is less than 35°, P must be outside the circle. (c) ∠𝐶𝐷𝐴