NYGH 2024-S3EOY-IM1 with Ans

Class Register Number Name End-of-Year Examination 2024 Secondary 3 INTEGRATED MATHEMATICS 1 2 hours Thursday 10 October 0845 – 1045 Additional Material: NIL 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 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 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 19 printed pages and 1 blank page. Setter: OLH NANYANG GIRLS' HIGH SCHOOL [Turn over For examiners’ use Q1 / 4 Q2 / 10 Q3 / 6 Q4 / 8 Q5 / 9 Q6 / 8 Q7 / 8 Q8 / 9 Q9 / 11 Q10 / 7 Total / 80

2 Mathematical Formulae Quadratic Equation For the equation ax2 + bx + c = 0, 2 4 2 b b acx a −  −= 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 (a) On the diagram below, sketch the graph of y = (6 – x) (x + 2), indicating the intercept(s) and turning point clearly. [2] (b) The diagram below shows the graph of ny kx= . Write down the value of n and the corresponding value of k. [2] x y O (1, –2) x y O

4 2 The diagram shows a circle ABCDE, with centre O. PDR and RA are tangents to the circle. Angle BAD = 54°, angle ODC = 62° and angle ADO = 28°. (a) Giving reasons for each answer, find (i) angle BOD, [1] (ii) angle BED, [1] (iii) angle BCD, [1] (iv) angle RPC, [2] (v) angle ARD, [2] (vi) angle ABD. [1] (b) Explain clearly if OARD is a cyclic quadrilateral. [2] 28° 62° 54° R C O D B P E A

5 [ [Turn Over Continuation of working space for Question 2.

6 3 (a) Given that tan ABD m =− , find, in terms of m, the value of (i) sin ABD , [2] (ii) cos CBD . [1] (b) When a graph of y x is plotted against y, the resulting line has a gradient of 3 and an intercept on the y x - axis of −1. Express y in terms of x. [3] A B C D

7 [ [Turn Over 4 It is known that x and y are related by an equation of the form y = hx2 – x + k, where h and k are constants. A straight-line graph is obtained when y + x is plotted against x2, for x  0. (i) Use the graph to estimate (a) the value of h and of k, [3] (b) the value of y when x = 2. [2] (ii) Explain how the above graph may be used to solve the equation 2( 1) 1x h k+ = − . Hence, find the value of x that satisfies this equation. [3] y + x x2 6 4 2 0 –2 –4 1 2 3 4 5 6 7 8

8 Continuation of working space for Question 4.

9 [ [Turn Over 5 (a) ABCD is a quadrilateral with coordinates A(3, 4), B(b, b + 1), C(6, 1) and D(–3, –2). (i) Find the value of b, if the area of ABCD is 30 square units. [2] (ii) Find the coordinates of M, if ACMD is a rectangle. [2] (iii) Given that K is a point on AC produced such that AK : CK is 3 : 2, find the coordinates of K. [2]

10 5 (b) The horizontal and vertical distances between two points P and N are 9 units and 5 units respectively, where P is on the left of and lower than N. Write down the gradient of the line PN and hence find the acute angle the line PN makes with the x–axis. [3]