NYGH 2017-S3EOY-IM2 with ans

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2017 Secondary Three INTEGRATED MATHEMATICS 2 2 hours Tuesday 10 Oct 2017 0845 – 1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Write your name, register number and class on all the work you hand in. 2. Answer questions 1 - 12 before attempting question 13 (Bonus Question). 3. Write your answers and working on the separate writing paper provided. 4. Write in dark blue or black pen 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 fluid. 7. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. 8. You are reminded of the need for clear presentation in your answers. INFORMATION FOR CANDIDATES 1. At the end of the examination, fasten all your work securely together. 2. The number of marks is given in brackets [ ] at the end of each question or part question. 3. The total number of marks for this paper is 80. Setter: GT This document consists of 8 printed pages. NANYANG GIRLS' HIGH SCHOOL Nanyang Girls' High School

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = a ac b b 2 42 − ± − . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A Formulae for ∆ABC C c B b A a sin sin sin= = a2 = b2 + c2 − 2bc cos A ∆ = 2 1 ab sin C 3. MENSURATION Arc length θr= , where θ is in radians Sector area θ2 2 1 r= , where θ is in radians Nanyang Girls' High School

3 1 By rationalising the denominators, simplify 3 3 2 3 2 3 1 −+ − + . [3] 2 Given that 6 4 3 22 − > − −mxx x for all real values of x, find the range of values of m. [3] 3 The diagram shows the layout plan of a garden. ABC is an arc of a circle, centre O and radius 12 m. ADC is an arc of a circle, centre E and radius 18 m. π18 13= ∠AOC radians and π9 4= ∠AEC radians. (a) The segment ADC is a fish pond. Calculate the area of this region. [3] (b) The shaded region ABCD is allocated for growing plants. Calculate the perimeter of this region. [2] [Turn over E D B C A O 12 m 18 m π 9 4 π18 13 Nanyang Girls' High School

4 4 Sound is measured in decibels according to the formula      = 0 lg10 P Pd , where d represents the noise rating in decibels, P represents the intensity of the sound and P0 represents the intensity of the weakest sound that the human ear can hear. A thunderclap has a noise rating of 120 decibels and a vacuum cleaner has a noise rating of 80 decibels. Calculate the ratio of the intensity of a thunderclap to that of a vacuum cleaner. [3] 5 The roots of the equation 0 6 22 = − +x x are α and β. (i) Show that 12 103 − =α α . [2] (ii) Find a quadratic equation whose roots are 2α β and 2β α . [5] 6 (a) Sketch the graph of ( ) 2ln − =x y , labelling the asymptote(s) and intercept(s) clearly. Find the equation of the straight line that needs to be added onto the same axes so as to solve the equation e xx 2e3 −= . It is not necessary to draw the line on your sketch. [4] (b ) Sketch the graph of 12sin −    = xy for 0 ≤ x ≤ 2π. [3] 7 (a) Solve the equation ( ) ( ) xxx 3 3 3 2 321 3 − =− . [4] (b) Solve the equation ( ) 1 2ln1ln ln − − = +x x e . [3] (c) Given that m=4lg , express each of the following in terms of m. (i) 3 400lg , [2] (ii) 10 log 16 1 . [3] Nanyang Girls' High School

5 8 Answer the whole of this question on the INSERT provided. The diagram shows the graph of y = f(x) for 2 5− ≤ ≤ −x . On separate axes, sketch the graphs of (a) ( ) 4 f+ =x y , [2] (b) ( )x yf2 3= , [2] (c) ( )x y1f−= , [2] labelling clearly the images of the points A and B as A’ and B’ respectively under the respective transformations. 9 (a ) Given that A and B are in different quadrants, p A− =cos , p B2tan = , 90° ≤ A ≤ 360°, 90° ≤ B ≤ 360° and p is a positive integer. Express the following in terms of p. (i) tan A, [2] (ii) cos B. [2] (b) Solve the equation xx 2cos10 9sin3 = + for ° ≤ ≤ °360 0x . [4] [Turn over y x B (−2, 0) A (−5, 4) ∙ ∙ y = f(x) Nanyang Girls' High School

6 10 (a) The function f is defined by 3 8 2 : f2 + −x x x , for 4 0≤ ≤x . (i) By completing the square, express 3 8 22 + −x x in the form ( ) c b x a+ + 2 . [2] (ii) Find the range of f. [2] (iii) Does f have an inverse? Explain your answer. [1] (iv) The function g is defined by 3 8 2 : g2 + −x x x , for x ≥ k. State the smallest value of k for which the function g−1 exists. [1] (b) A function h is defined as 1 3 2: h −x xx  for 3 1≠x . Find the values of x which map onto themselves under the function h. [3] 11 (a) The function ( ) 410 f 2 3 − + + =bx ax x x where a and b are constants, is exactly divisible by 2−x and leaves a remainder of − 21 when divided by 1−x . (i) Find the values of a and b. [4] (ii) Hence, factorise ( )xf completely and solve the equation ( ) 0 f=x . [3] (b) Express ( )( ) 2 2 1 2 1 3 2 + + − − x x x x as partial fractions. [4] Nanyang Girls' High School

7 12 Figure I shows the start time of dawn and end time of dusk in Helsinki, Finland, for the 12 months of the whole year (January to December). For example, on the marked day in March, as shown by the dotted line, the start time of dawn is 06 00, there is sunshine from 06 00 to 17 45, the end time of dusk is at 17 45 and there is darkness after 17 45. (Source: https://www.gaisma.com/en/location/helsinki.html) Figure I It is given that the start time of dawn is modelled by the function 56cos41 +    = th π , while the end time of dusk is modelled by the function     − = 6cos5 192 th π . 1h and 2h are the number of hours after midnight and t is the time in months from the start of the year. (i) State the amplitude of 2h . [1] (ii) It is believed that the best period of the year for tourists to visit Helsinki is when the days are extremely long, that is, the difference in start time of dawn and end time of dusk is at least 14 hours. Use Figure I to find the best period of the year to visit Helsinki. [2] (iii) It is also known that the start time of dawn in Beijing, China, is modelled by the function     + = 6cos2 3 2 11 3 th π . Calculate the values of t when dawn starts at 06 15 in Beijing, China. [3] [Turn over Bonus Question End time of dusk Start time of dawn Darkness Darkness Sunshine h (hours) t (months) 06 00 17 45 Nanyang Girls' High School

8 13 Given that ( )( ) abb a ccab log4 1log log= , prove that b a= . [3] END OF PAPER Nanyang Girls' High School

1 Sec 3 IM2 EOY Answers only Qn Solution 1 43 3 11+ 2 1 7< < −m 3a Area of pond sf) (3 m 7 . 662 3b Total perimeter sf) (3 m 4 . 52 4 The ratio is 10 000 : 1. 5i ( ) 12 10 0 6 4 12 0 6 2 6 2 0 6 2 0 6 2 3 3 3 23 2 − = = − − + = − − + = − + = − + α α α αα α αα α α α α α Or ( ) ( ) 12 10 2 6 2 6 2 6 2 6 62 6 2 2 3 3 23 2 2 − = − − = − = − = − = − − −= − − − − = α α αα α α α α α α α α α α α β 5ii 0 3 28 18 06 1 9 14 equation New 2 2 = − + = − + x x x x 6a 1 3+ =x y 6b Graph of 12sin − =xy Nanyang Girls' High School

2 7a 1=x 7b 1=x 7ci 3 2 3 1 +m 7cii m2 1 − 8a 8b 8c