Springfield Mathmatics 4Exp Amaths Paper 1 Prelim 2021

[Turn Over SPRINGFIELD SECONDARY SCHOOL Preliminary Examination 2021 STUDENT NAME CLASS S - INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 Secondary 4 Express 25 Aug 2021 2 hours 15 minutes Candidates answer on the question paper No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams, graphs or rough working. Do not use staples, paper clips, glue or correction fluid. Answer all the 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. You are reminded of the need for clear presentation in your answers. 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 90. For Examiner’s Use Total /90 Do not turn over this question paper until you are told to do so. ____________________________________________________________________________ This question paper consists of 16 printed pages.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42 Binomial Expansion  nrrnnnnn b...bar n...banbanaba         221 21 where nis a positive integer and   ! 11 !! ! r rn...nn rnr n r n    2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  cosec 2 A = 1 + cot 2 A BABABA sincoscossin)sin(  BABABA sinsincoscos)cos(  BA BABA tantan1 tantan)tan(   AAA cossin22sin  AAAAA 2222 sin211cos2sincos2cos  A AA 2tan1 tan22tan  Formulae forABC Abc Abccba C c B b A a sin2 1 triangleof Area cos2 sinsinsin 222   

3 [Turn Over 1 Solve the simultaneous equations. 2 9 5 3 2 yx x y     [5]

4 2 The equation of a curve is 1 2sin 2y x  . (a) State the minimum and maximum values of y. [2] (b) Sketch the graph of 1 2sin 2y x  for 0 360x   . [3]

5 [Turn Over 3 (a) Explain why there are no odd powers of x in the expansion of 102kx x     , where k is a constant. [3] (b) Given k = 3, find the coefficient of 2x in the expansion of 102kx x     . [3]

6 4 The function f is defined by 2 1f ( ) 3 xx x   . (a) Determine the range of values of x for which f is a decreasing function. [4] (b) A point T moves along the curve y = f(x) in a way that the y-coordinate of T is decreasing at a rate of 0.1 units per second. Find the rate of increase of the x-coordinate of T when x = 2. [2]

7 [Turn Over 5 The population, P of a certain micro-organism, present at t hours after initially being observed is given by the formula 160 220ktP e  , where k is a constant. It took 4 hours to reach a population of 620. (a) Calculate the population of the micro-organisms after 24 hours. [4] (b) Explain why the formula is not suitable to calculate the long-term population of the micro-organisms. [2]

8 6 For a particular curve 2 2 sin 3cos3d y x xdx   . The curve passes through the point , 42P     and the gradient of the curve at P is 4. Find the equation of the curve. [6]

9 [Turn Over 7 A portion of a suspension bridge has a curved cable hanging between two pillars which can be modelled by a quadratic function. In the model, x m is the horizontal distance from the left pillar and y m is the height of the cable above the bridge. Vertical supporting wires are spread out in 10 equal intervals of 0.5 m apart. At its lowest point, the cable is situated 2.5 m above the bridge. (a) Write down the co-ordinates of the lowest point. [1] (b) Form a quadratic equation in the form 2( )y a x b c   to model this situation. Hence, find the value of a, b and c. [4] O x y 0.5m 15m 15m 2.5m bridge bridge

10 8 Given that 12 4sin and tan13 3A B  , where A and B are in the same quadrant. Evaluate (a) tan( )A B , [3] (b) sin 2B, [2] (c) cos( )A B . [2]