Paper H Sec 3

1   Hwa Chong Institution Secondary 3 PAPER H 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 92. 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 (a) When 2f( ) 5xxx  is divided by xa , the remainder is 1. Find the possible values of a. [2] (b) Simplify    3 2 27 2 29 345 2 0 xx xxx x x     . [4] 2 (a) The line 31xy meets the curve 2 33 4xx y at points A and B. Find the distance AB. [5] (b) Find the range of values of c for which the line 2yx c does not intersect the circle 22 9xy . [4] 3 The graph of 32 27 9yx x x  is shown below. (i) Write down the equation of the straight line to be drawn on the same axes in order to solve 32 25 6 0xxx   graphically. [1] (ii) Using the graph given or otherwise, find the range of values of x such that 32 25 6 0xxx   . [2] Please Turn Over y x

4   4 A plate of hor fun provides 91 g of carbohydrates, 27 g of fats and 35 g of proteins. A cup of bubble tea provides 75 g of carbohydrates, 4 g of fats and 4 g of proteins. Each gram of carbohydrate, fat and protein provides 4 calories, 9 calories and 4 calories of energy respectively. (i) Write down a 23 matrix to represent the nutrients for a plate of hor fun and a cup of bubble tea. [1] (ii) Write down a 31 matrix to represent the calories that nutrients provide. [1] (iii) Using matrix operations, calculate the total energy provided by a plate of hor fun and a cup of bubble tea. [3] 5 The population growth of a bacterial sample is represented by  f2 xx  , with an initial population of one cell and a doubling time of 1 hour, where x is the number of hours after the start of the experiment. (i) Explain why 1f exists. [1] (ii) Find an expression for   1f x . [1] (iii) Comment on the graphs of both f( )x and 1f( )x if they were drawn on the same axes with the same scale used. [1] 6 (a) Solve the following sim ultaneous equations. 1 3322 8xy   1 33625 1xy    [ 4 ] (b) Solve the equation 22 5 1xx  . [3] (c) By considering the valid range of x, without solving the equation, explain why 34 2 3 2xx  has no real solution. [2] 7 The function f is defined as 2f: 2 4 7 , 1 4xx x x     . (i) Express f( )x in the form of 2()ax b c  , where a, b and c are constants. [2] (ii) Sketch the graph of f( )yx , indicating clearly the y-intercept, the turning point and the end points. [4] (iii) Solve f( ) 7x  algebraically. [4]

5   8 The triangle TBF lying on a horizontal plane are marked on the map below. It connects the three MRT stations: Tan Kah Kee Station (T), Botanic Gardens Station (B) and Farrer Road Station (F) respectively. It is given that 820 mTB , 970 mBF , and 56TBF  , (i) Calculate the distance TF. [2] A drone D is hovering vertically above B such that the angle of elevation of D from F, DFB , is 18. (ii) Calculate the height BD. [2] (iii) Without any calculations, state which angle is larger, DFB or DTB . [1] 9 (a) The graph of 1y x has undergone two successive transformations. The new graph of 3 1y x is obtained. Describe the transformations in order. [2] (b) The graph of a function is stretched along y-axis by a factor of 2. The new graph of  243 6xy is obtained. Find the equation of the original function. [1] Please Turn Over T B F 820 m 970 m 56o

6   10 The figure shows a straight line 23yx intersecting the vertical line x  at A and the y-axis at B. Point C has the coordinates (, 0 ) such that AB is perpendicular to BC. (i) Show that 6  . [3] (ii) If the coordinates of D are (18, 6), find the area of quadrilaterial ABCD. [3] (iii) Show that the equation of the perpendicular bisector of CD is 22 7yx  . [ 3 ] (iv) Does point A lie on the perpendicular bisector of CD? Justify your answer. [2] 11 (a) Find the value of 55 5log 75 2 log 1 log 3 . [2] (b) Given that 2logux , express  2log 2x x in term of u . [3] (c) Prove that log logn n aa bb , where a, b > 0. Hence, find the value of   2 3 2021 2 3 2021 32 22 2log 2 log 3 log 3 log 3 ... log 3  . [5] y = 2x + 3 x = α y x B O C (α, 0) A D (18, 6)

7   12 (a) Express the equation 2138 1xy   where 0y in the form ln 1ym x c  , where m and c are constants. [3] (b) The variables x and y are related by the equation 31 22ya x b x   where a and b are constants. The diagram shows the straight line graph obtained by plotting yx against 2x . The line passes through the points (3, –2) and (5, 6). Find the values of a and b. [4] Please Turn Over x2 (5, 6) (3, –2)

8   13 The diagram below shows three circles, C1, C2 and C3, centred at P, Q and R respectively. The three circles touch the x-axis at A, O and C respectively. Circle C2 of radius r units touches two smaller circles, C1 and C3 , of equal radii 4 units externally at S and T. It is also given that AO = OC = 12 units. M is the point of intersection of QO and PR. (i) Find the equation of circle C1. [2] (ii) Write down, in term of r, the length of (a) QR, [1] (b) QM. [1] (iii) Hence, show that r = 9. [2] (iv) Show that QPM , when rounded to 3 significant figures, is 0.395 radians. [1] (v) Hence, determine the area of the two shaded regions bounded by the three circles and the x-axis. [4] End of Paper y x R O C (12, 0) P A (–12, 0) Q C1 C2 C3 . M . S T .