NYGH 2023-S3EOY-IM2 with Ans

Class Register Number Name End-of-Year Examination 2023 Secondary 3 INTEGRATED MATHEMATICS 2 2 hours Monday 2 October 2023 0845 – 1045 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 5 printed pages and 1 blank page. Setter: TCL NANYANG GIRLS' HIGH SCHOOL [Turn over

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = a acbb 2 42 −− . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin (A  B) = sin A cos B  cos A sin B cos (A  B) = cos A cos B ∓ sin A sin B tan (A  B) = BA BA tantan1 tantan  

[ Turn over 3 1 Given that 6x3 – 16x2 + 10x – 4 = A(x – 1)2(2x – 1) + B(x + 3)(x – 1) + C for all values of x, find the values of A, B and C. [4] 2 Express 2 10 19 ( 3)(2 1) x xx + −+ in partial fractions. [4] 3 Simplify the expression ( )3 1 3 5 35 + + and leave your answer in the form 5ab+ . [3] 4 The population, P, of a new town at the beginning of the year 2015 was 50000. The population increases such that after a period of t years, the new population can be modelled by 0.0650000e tP= . (a) Find the population of the town at the beginning of 2023. [2] (b) How many years would it take for the population to reach 100 000? [3] (c) Is this model reliable in predicting the population after a long period of time? Explain your answer. [1] 5 The polynomial 82 23 −++ bxaxx , where a and b are constants, has a factor (x – 2) but leaves a remainder of 9 when divided by (x + 1). (a) Find the value of a and of b. [4] (b) Factorise 82 23 −++ bxaxx completely. [2] (c) Hence, with the values of a and b found in part (a), solve the equation 3216 4 8 2y ay by+ = − . [3] 6 (a) Solve the equation (i) 212 2 2 4x x x +− = + , [4] (ii) 33log 3 8 4logx x=− . [5] (b) If 1=x is a solution to the equation ( ) 22 2 9 2 2x p x p x++ − = − where p is a constant, find the value of p. [2]

4 7 (a) Find the range of values of k for which the line 5y kx=− intersects the curve 22 4 3y x x= + + at two distinct points. [4] (b) (i) Explain why the expression 22 4 3xx++ is always positive for all real values of x . [2] (ii) Hence, solve the inequality 2 2 2 4 3 03 7 6 xx xx ++ −− . [3] 8 The function f is given by f ( ) cos 2x a x b=+ for 0 180x , where a and b are positive integers. f(x) has an amplitude of 5 and a minimum value of – 3. (a) State the value of a and of b. [2] (b) State the period of f ( )x . [1] (c) Sketch the graph of f ( )yx= . [3] 9 (a) The graph of ln( 4)yx=+ undergoes 2 successive transformations. I A scaling parallel to the y-axis with a factor of 2. II A translation parallel to the x-axis by 3 units. Find the equation of the resulting graph. [2] (b) Sketch the graph of ln( 4)yx=+ , showing clearly if any, the intercept(s) with the axes and the asymptote(s). [2] (c) Explain how you solve the equation 1e4xx −=− using your sketch in part (b). Hence, determine the number of solutions to the equation.[4] 10 Given that sin m = and  is an acute angle, express, in terms of m, (a) tan  [2] (b) cos (−) [1] (c) cosec  [2] 11 Solve the equation (a) 24cos 4sin 1 0xx+ − = for 0 360x [4] (b) 2sin tan 0xx−= for 02 πx [4]

[ Turn over 5 12 (a) It is given that 12tan 5A=− and 3tan 4B=− , where A and B are in different quadrants and AB . Without using a calculator, find the value of cos( )AB+ . [4] (b) Given that sin( ) 2 sin( ) 3 PQ PQ + =− , evaluate tan tan Q P . [3] END OF PAPER

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1 2023 S3 IM2 EOY Marking Scheme No. 1 𝐴 = 3 𝐵 = −1 𝐶 = −4 2 1 𝑥 − 3 − 2 2𝑥 + 1 − 4 (2𝑥 + 1)2 3 6√5 − 9 4(a) 80800 (3 s.f.) 4(b) 11.6 years 4(c) As 𝑡 → ∞, 𝑃 → ∞ Thus, this model is not reliable in predicting the population after a long period of time. Or This model does not take into account deaths rate or falling birth rate, etc. 5 (a) 𝑎 = 5 𝑏 = −14 5(b) f(𝑥) = (𝑥 − 2)(2𝑥 + 1)(𝑥 + 4) 5(c) 𝑦 = 1, − 1 4 , −2 6(a)(i) 𝑥 = 2 6(a)(ii) 𝑥 = √3 𝑜𝑟 3√3 6(b) 𝑝 = 3 7(a) 𝑘 < −4 or 𝑘 > 12 7(b)(ii) − 2 3 < 𝑥 < 3 8(a) 𝑎 = 5 𝑏 = 2 8(b) Period = 180° 8(c)

2 9(a) 𝑦 = 2 ln(𝑥 + 1) 9(b) (c) 9(c) Add 𝑦 = 𝑥 − 1 Number of solutions = 2 10 (a) tan 𝜃 = 𝑚 √1 − 𝑚2 10 (b) cos(−𝜃) = √1 − 𝑚2 10 (c) cosec 𝜃 = 1 𝑚 11(a) 𝑥 = 210°, 330° 11(b) 𝑥 = 0, 𝜋, 2𝜋 or 𝑥 = 𝜋 3 , 5𝜋 3 12(a) 16 65 12(b) 1 5− 𝑥 = −4