NYGH 2025-S3EOY-IM1 with Ans

Class Register Number Name End-of-Year Examination 2025 Secondary 3 INTEGRATED MATHEMATICS 1 2 hours Tuesday 14 October 0845 – 1045 READ THESE INSTRUCTIONS FIRST 1. Write your name, register number and class in the spaces at the top of this page. 2. Write in dark blue or black ink. 3. You may use a 2B pencil for any diagrams or graphs. 4. Do not use staples, paper clips, glue or correction tape/ fluid. 5. Write your answers and working in the space provided. 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. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 21 printed pages and 1 blank page. NANYANG GIRLS' HIGH SCHOOL [Turn over For examiners’ use Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total / 80

2 Mathematical Formulae Quadratic Equation For the equation ax2 + bx + c = 0, 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 2 4 2 b b acx a −  −=

3 [ Turn over 1 The axes below show a point with coordinates (1, 1). Sketch, on the same axes, the graphs of (a) 1y x= , [1] (b) 3yx= , [1] (c) 2yx= . [1] 2 Variables x and y are related by the equation 2y ax bx=+ , where a and b are constants. A straight line graph can be drawn to represent this equation. State which variables should be plotted on each axis and explain how the values of a and b can be calculated. [4] x O y . (1, 1)

4 3 In the diagram, A, B, C and D are four points on a circle. ACE is a straight line. Angle 40BAC= and angle 134BCE= . (a) Find angle BDC, giving a reason for your answer. [1] (b) Explain why AC is not a diameter of the circle. [2] C B D A E 40 134

5 [ Turn over 4 The variables x and y are such that when values of 1 y are plotted against 1 x , a straight line is obtained. Two points 52, 2   and 96, 2   lie on the straight line. Find an expression for y in terms of x. [4]

6 5 In the diagram, A, B, C and D are four points on a circle, centre O. AT and CT are tangents to the circle. AO and BC are parallel. Angle 30OAD= and angle 40OCD= . (a) Find angle ADC. [1] (b) Find, giving reasons for each answer, (i) angle CAT, [1] (ii) obtuse angle AOC, [1] T C D O B A 40° 30°

7 [ Turn over 5 (b) (iii) angle OAB, [2] (iv) angle ATC. [2]

8 6 The graph of 3211 4322y x x x= + − + is drawn on the grid. (a) By drawing a tangent, find the gradient of the curve at ( )3, 6− . [3]

9 [ Turn over 6 (b) By drawing a suitable straight line on the graph, solve the equation 3211 5 0.22x x x+ − = [3] (c) (i) Draw the line 2yx= on the graph. [1] (ii) The gradient of a point on the curve is 2. Its x-coordinate is p. Given that 0p , find the value of p. [2] (d) The equation 3211 4 5 022x x x+ − + = has only one solution. Explain how this can be seen from your graph. [2]

10 7 A parallelogram ABCD has vertices ( )3, 0A − , ( )4, 3B and ( )7, 7C . (a) Find the coordinates of D. [2] (b) Find the equation of AD. [2] (c) Find the area of the parallelogram ABCD. [2]