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1 Hwa Chong Institution Secondary 3 PAPER i CANDIDATE NAME CLASS REGISTER NUMBER Integrated Mathematics 2 hours 30 minutes Additional materials: Writing papers READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work that you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Answer all questions. 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. The use of an approved scientific calculator is expected, where appropriate. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100. This document consists of 8 printed pages.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c  2 4 2 bb a cx a   Mensuration Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians 2. TRIGONOMETRY Formulae for ABC sin sin sin abc A BC 22 2 c2c o sab b c A 1=s i n2 ab C

3 1 Solve the simultaneous equations 384 231 0 . xy xy     [4] 2 (a) Simplify 432 1 6 2 4 , 6  expressing your answer in the form of 6,k where k is a constant . [3] (b) Solve the equation    21 285 5 25 .xx x  [ 4 ] 3 (i) Factorise 32 128x  completely. [2] (ii) Hence or otherwise, solve the equation 32 128 4x x  . [3] 4 (a) Find the range of values of x for which   32 63 2xx x   . [3] (b) Given that the line 2yb a x is a tangent to the curve 22452 0 0 ,xy  show that 22 51 6 .ba [ 4 ] 5 (a) Solve the equation 54x x . [ 3 ] ( b ) Solve the equation 2 3log 2 log 3 5xx   . [5] 6 (i) On the same axes, sketch the graphs of yx and 27 ,yx  indicating clearly the x-intercepts and y- i n t e r c e p t s . [ 4 ] (ii) Hence, or otherwise, s olve the inequality 27x x   . [ 4 ] [Turn over

4 7 (a) The points    3, 0 , 9,18AB and  13,10C lie on a quadratic curve f( )yx and B is the maximum point on the curve. Find the equation of the curve in the form 2 ,ya x b x c  where a, b and c are constants. [3] (b) Peter used 20 metres of fence to enclose two flower beds. One flower bed is in the shape of an equilateral triangle of side x metres. The other is in the shape of a circle of radius r metres. (i) Express r in terms of x and  . [ 1 ] (ii) Show that the total area, A 2m of the two flower beds is given by  2232 0 3 .4 x xA    [ 3 ] (iii) Hence, by completing the square, find the minimum total area of the two flower beds. [3] 8  3, 3P  and 0, 2T lie on a circle with centre  1,4Q  . (i) Given that PS is the diameter of the circle, find the coordinates of S . [2] (ii) Find the equation of the circle with centre Q in the general form. [3] (iii) 2,3R lies outside of the circle. By calculating the gradient of RS and ,PS show that RS is a tangent to the circle in (i). [3] (iv) Find the area of the quadrilateral PTRS. [2]

5 9 (i) G i v e n t h a t f( ) 2 xx e and g5 2 xx e , describe 2 successive transformations that will map the graph of  f x to the graph of  g x . [ 2 ] (ii) Sketch the graph of 5 2 xye , indicating clearly the asymptote, the values of the x-intercept and y- i n t e r c e p t . [ 4 ] (iii) In order to solve the equation 6ln 4 x x  , a graph of a suitable straight line is drawn on the same axes as the graph of 52 xye . Find the equation o f t h e s t r a i g h t l i n e . [ 3 ] 10 (a) Given that 23 43  A and 21 35  B , find 3 AB . [2] (b) E’Lite shop sells air-conditioning unit, instant water heater and humidifier. The number of air-conditioning unit, instant water heater and humidifier sold in 2020 and 2021 is given in the following table. (i) Write down a 23 matrix C which represents the number of air-conditioning unit, instant water heater and humidifier sold in 2021 and 2020 respectively. [ 1 ] (ii) Given that the profit made fro m selling one air-conditioning unit is $240, one instant water heater unit is $95 and one humidifier unit $110, write down a column matrix D that represents the profit made from selling each u n i t . [ 1 ] (iii) Hence, find matrix E where .EC D [ 2 ] (iv) Explain what matrix E represents. [1] [Turn over Type Year air-conditioning instant water heater humidifier 2021 290 240 200 2020 235 362 117

6 11 The function 32f( ) 7 4xa x xb x  where a and b are constants, is divisible by 2x and leaves a remainder of 185 when divided by 3x . (i) Show that 6a and 12b . [ 3 ] (ii) Use the value of a and b from (i), by using long division, find the remainder when f( )x is divided by 223x  . [ 3 ] 12 ABCD is a square of sides 6 cm. Q is the mid- point of DC. An arc BRP is drawn with centre Q and P lies on DC produced. (i) Show that the PQB = 2 . 0 3 . [ 2 ] (ii) Calculate the perimeter of CPRBC. [3] (iii) Calculate the area of the shaded region. [3]

7 13 T h e v a r i a b l e s x and y are connected by the equation 223 0 ,xa x y where a is a constant. (i) Express y x in terms of x and a . [ 1 ] (ii) When 1y x   is plotted against ,x a straight line is obtained which passes through 2, 2 and 5, .h Find the value of a and of h. [ 4 ] [Turn over 1y x

8 14 In the diagram, the points A, B and C are on level ground. The bearing of C from A is 217 and 75BAC  . Given that 60 mAB and 72 mAC , find (i) the length ,BC [ 2 ] (ii) the angle ABC , [ 2 ] (iii) the bearing of C from B. [ 2 ]