Paper E Sec 3 IP

1 Name: _______________________________ ( ) Class: ________ HWA CHONG INSTITUTION Paper E INTEGRATED MATHEMATICS Level : Secondary Three Duration : 2 hours 30 minutes Do not open this booklet until you are told to do so. INSTRUCTIONS TO CANDIDATES Write your name, class and index number on all the work you hand in. 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. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. You are expected to use an electronic calculator to evaluate explicit numerical expressions. The use of a graphing calculator is not permitted. You are reminded of the need for clear presentation in your answers. Omission of essential working will result in loss of marks. This question paper consists 7 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c   2 4 2 b b acx a    2. TRIGONOMETRY Mensuration Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Formulae for ABC sin sin sin a b c A B C 2 2 2 c 2 cosa b bc A   1 = sin2 ab C

3 1 (a) Simplify    38 1 1 2.2 3 2 1 1 2 x xx x x      [3 ] (b ) Find the values of x satisfying both ineq ualities 1 4 5 3x    and   4 5 18.xx   [4 ] 2 (a) Find the value of x , given that 2 2 3 28. 2 x x   [3] (b) Using a suitable substitution, or otherwise, solve for x if   1 24 7 2 5. x x  [4 ] 3 S olve each of the following equations . (a ) 5log 2log 5 3 xx [3] (b ) 9 4 2 5xx    [4 ] 4 (a) Express 22 6 13xx i n the form   2 ,a x b c where a , b and c are constants. [2] (b) Given that 22f ( ) 2x x x k kx    , find the range of values of k fo r which f ( ) 0x  for all real values of x . [4] 5 The equation of a circle 1C is 225 20 125 50 5 .y y x x    Find (i) the coordinates of the cent re of the circle 1C and its radius, [3] (ii) the equation of another circle 2C which has the same centre as the circle 1C and passes through the point  5, 7 . [2] 6 The cubic polynomial f ( x ) has roots 3, 3 and k . (i) Given that the coefficient of 3x is 2, and that f ( x ) has a remainder of 8 when divided by x + 1, find the value of k . [3 ] (ii) Find the remainder when f ( x ) is divided by x – 5. [2 ] [Turn Over S3Math10A

4 7 (a) Given 54 11  B , find P if 2213. 13  PB [2] (b) The table below shows the price of the items for sale in two mini-markets in Ponggol. Product Price in mini-market A Price in mini-market B Eggs $3.30 per tray of 6 $3.20 per tray of 6 Meat $1.90 per 100 g $2.00 per 100 g Milk $3.40 per litre $3.45 per litre Mr Tan’s shopping list consists of 2 trays of eggs, 500 g of meat and 3 litres of milk. If 3.30 3.20 1.90 2.00 3.40 3.45    C , determine matrix Q such that the amount spent in each mini-market can be calculated using matrices C and Q. Find the amount spent in each mini-market and hence conclude which mini-market to shop for the items. [3] 8 Functions f and g are defined by f : 2 7xx  and 3g : , 2 2 xxx x  . Solve the equations (i) 1g ( ) f(3)x  , [3] (ii) f( ) 3xx  . [3] 9 The roots of the quadratic equation 2x2 + 7x + 14 = 0 are  and  . (i) Write down the values of  and  . [1] (ii) Find the quadratic equation whose roots are 2  and 2  . [3]

5 10 The figure below shows the graph of sin for 0 2 ,y a bx c x     where a, b and c are integers. State the values of a, b and c. [3] 11 The figure below, not drawn to scale, shows a circle with centre O and radius 8 m. Two tangents, TA and TB, have length l m each. P is a point on the arc AB. (i) Write AOT in terms of l. [1] (ii) Show that the area, S m2, of the shaded region is given by 18 64 tan 8 lSl    . [2] (iii) Find the perimeter of the shaded region ATBP in terms of l. [2] (iv) The figure above lies on horizontal ground such that a pole of length 12 m stands vertically at O. Find the angle of elevation, in degrees, of the top of the pole from T when 8l . [2] [Turn Over l l A T B O 8 8 P 0 1 4 x y –2 2π

6 12 (a) Given that 1 tan 31 tan x x   , express tan x in exact form. [2] (b) Find all angles which satisfy the equation (i) 4sin 2 1, 0 180 ,xx    [2] (ii) 23cos 4sin cos , 0 2 .x x x x    [3] 13 In the diagram P, Q, R and S are points on level ground. P, S and R lie on a straight line and P is due north of Q. Given that 6SQ km and 10RS  km, 38QPS   and 56PSQ   , calculate (i) QR, [2] (ii) ,SQR [2] (iii) the bearing of S from R, [1] (iv) the shortest distance from S to QR. [2] 14 (i) Given that f( ) lnxx and  g( ) 2ln 3xx , describe the transformation of the graph of f( )x to the graph of g( )x in two successive steps. [2] (ii) Sketch the graph of  2ln 3yx , showing clearly the asymptote and the x-intercept. [2] (iii) Find the equation of the straight line to be drawn and explain how the graphical solution of the equation   12 2e 3 e x x   could be obtained. [2] R S Q P 10 km 6 km 56  38  North

7 15 Solution to this question by accurate drawing will not be accepted. ABC is an isosceles triangle with  6,6A , B lies on the x-axis and AB = AC . Given that the equation of BC is 24yx and M is the midpoint of BC, find (i) the coordinates of B, C and M, [5] (ii) the equation of the perpendicular bisector of BC. [2] (iii) Given that coordinates of D are  10, 2 , determine the name of this quadrilateral ABDC . [2] (iv) Find the area of quadrilateral ABDC. [2] 16 Answer the whole of this question on a sheet of graph paper. Two positive variables x and y are related by the equation 2 ebxya , where a and b are unknown constants. The table shows some experimental values of the two variables. x 0.5 1 1.5 2 2.5 3 3.5 4 y 2.87 3.69 4.73 6.08 7.80 10.6 12.9 16.5 (i) Using a scale of 4 cm to 1 unit on both axes, plot ln y against x. [3] (ii) Use your graph to estimate the values of a and b. [3] (iii) Use your graph to estimate the value of y when x = 1.8. [1] End of Paper