NYGH 2023-S3EOY-IM1 with ans

Class Register Number Name End-of-Year Examination 2023 Secondary 3 INTEGRATED MATHEMATICS 1 2 hours Thursday 5 October 0845 – 1045 Additional Material: Insert READ THESE INSTRUCTIONS FIRST 1. Write your name, register number and class on all the work you hand in. 2. Write in dark blue or black ink. 3. You may use an HB pencil for any diagrams or graphs. 4. Do not use staples, paper clips, glue or correction tape/ fluid. 5. Write your answers and working on the separate writing paper provided, unless otherwise stated. 6. Answer all questions. 7. Omission of essential working will result in loss of marks. 8. The use of an approved scientific 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. This document consists of 9 printed pages and 1 blank page. Setter: LJW NANYANG GIRLS' HIGH SCHOOL [Turn over

2 Mathematical Formulae Mensuration Area of triangle ABC = 1 2 𝑎𝑏 sin 𝐶 Arc length = 𝑟𝜃, where 𝜃 is in radians Sector area = 1 2 𝑟2𝜃, where 𝜃 is in radians Trigonometry 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴

3 [ [Turn Over 1 Answer the whole of this question on the insert provided. (a) On the diagram above, sketch the graph of 2 4yx= . [2] (b) The diagram below shows the graph of ny kx= . Write down a possible value of n and the corresponding value of k. [2] x y O (–1, 2) (1, 2) y x O (–2, 1)

4 2 The diagram shows a circle ABCDE, centre O. TAP and TBQ are tangents to the circle. Angle AOB = 130° and angle BEC = 30°. (a) Giving reasons for each answer, find (i) angle BDC, [1] (ii) angle AEB, [1] (iii) angle ABC, [1] (iv) angle TAB, [1] (v) angle ATB. [1] (b) Explain why a semicircle, with OT as the diameter, passes through A. [2] A B C D T O P Q 30° E 130°

5 [ [Turn Over 3 (a) EFGH is a quadrilateral with coordinates E(2, 1), F(9, –5), G(7, 4) and H ( )3, 5 . (i) Find the area of EFGH. [2] (ii) Given that P divides EG internally such that EP : PG is 2 : 3, find the coordinates of P. [2] (b) A is the point (1, 1), B is the point (1, 3 ) and O is the origin. (i) Explain why the line AB is parallel to the y-axis. [1] (ii) Given that OATB is a parallelogram, find the coordinates of T. [2] (iii) Find the angle AOB. [2] 4 The variables x and y are related in such a way that when a graph of yx− against 2xy is drawn, the resulting straight line passes through the points (2, 7) and (5, 1). (a) Express y in terms of x. [4] (b) Find the values of x when y = 3. [3] (c) If the variables x and y are plotted on another graph of 22y x y+ against x, find the gradient of the new straight line. [1] 5 The equation of a circle, C1, is 22 6 10 15 0x y x y+ − − + = . (a) Find the radius and the coordinates of the centre of C1. [3] (b) Determine whether the x-axis is a tangent to C1. [2] The equation of the tangent to a second circle, C2, at point A is 3 4 9yx=− + . B is (5, 13) and AB is a diameter of C2. (c) Show that the coordinates of A are ( 3, 7)− . [4] (d) Find the equation of C2. [3]

6 6 In the diagram, A, B and C represent three points on level ground. AB = 170 m and BC = 90 m. Angle ABC = 120°. The bearing of C from B is 110°. Calculate (a) AC, [3] (b) the area of triangle ABC, [2] (c) the bearing of A from C. [4] A tower, TB, of height 30 m, stands vertically at B. (d) A man is at a point on AC. Calculate the greatest angle of depression of the man from the top of the tower. [3] B A C 170 m 90 m North 120°

7 [ [Turn Over 7 (a) Given that A 12 34 = , B x y = , and that A2B 11 23 = , find the value of x and of y. [4] (b) A coffee company sets up two coffee stalls, X and Y, with both serving three types of coffee. The table shows the number of cups of each type of coffee served during a day and the selling price of each cup of coffee. Americano Cappuccino Latte Stall X 32 20 65 Stall Y 48 35 50 Selling price (per cup) $4.50 $5.00 $6.00 (i) Given that A 32 20 65 48 35 50 = and B 4.5 5 6  = , evaluate the product AB. [1] (ii) Explain what the elements of AB represent. [2] A cup of Americano is made from 18 grams of coffee beans. A cup of Cappuccino is made from 16 grams of coffee beans. A cup of Latte is made from 20 grams of coffee beans. (iii) Stall X operates 5 days a week and Stall Y operates 7 days a week. Write down two matrices P and Q such that PAQ will give the total mass of coffee beans used by the coffee company in a week. [2] (iv) Evaluate this product PAQ. [2]

8 8 Answer the whole of this question on the insert provided. The diagram below shows the graph of 223 60 2xxy x −−= for 3 10x . (a) The equation 223 60 2 1.1xx x −− = has only one solution. Explain how this can be seen from your graph. [2] (b) The gradient of the tangent to the curve at point P is 0.4. Use your graph to find the x–coordinate of point P. [2] (c) The equation 2223 60 2 2x x x x− − =− + can be solved by drawing a suitable straight line on the same axes. (i) Find the equation of this straight line. [1] (ii) By drawing this straight line, solve the equation 2223 60 2 2x x x x− − =− + . [2] – – – –

9 [ [Turn Over 8 cm 5 cm D B O CA 9 In the diagram, ABC is a semicircle with centre O and radius 5 cm. D lies on AC such that CBD is a sector with centre C and radius 8 cm. (a) Show that the angle BOA is 1.287 radians, correct to 4 significant figures. [3] (b) Find the perimeter of the shaded region. [3] (c) Find the area of the shaded region. [4] END OF PAPER

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