2022 ZHSS AMath Prelims P2

[Turn over ZHONGHUA SECONDARY SCHOOL PRELIMINARY EXAMINATION 2022 SECONDARY 4 EXPRESS/ 5 NORMAL (ACADEMIC) Candidate’s Name Class Register Number ADDITIONAL MATHEMATICS 4049/02 PAPER 2 13 September 2022 2 hour 15 minutes Candidates answer on the Question Paper READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, glue or correction fluid. 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. 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. ___________________________________________________________________ This question paper consists of 21 printed pages (including this cover page) For Examiner’s Use 90

2 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax a acbbx 2 42 −−= Binomial expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21)( , where n is a positive integer and ! )1()1( )!(! ! r rnnn rnr n r n +−−=−=       2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= Abcsin2 1=

3 [Turn over 1 (a) By using suitable substitution or otherwise, solve the equation ( )e 3 e 5ex x x− −= , giving your answer correct to 2 decimal places. [4]

4 [Turn over (b) Solve 75xx+ = − . [3] 2 Given that tan p = and that  is acute, find in terms of p, (a) ( )sin − [1] (b) tan 2   − [1]

5 [Turn over 3 (a) Express 23 36 106xx− − − in the form 2()a x b c++ . [2] (b) Hence, sketch the graph of 23 36 106y x x=− − − on the axes below. Indicate clearly the coordinates of the points where the graph crosses the y-axis and the turning point on the curve. [3] y x

6 [Turn over 4 (a) Given that 2 sin cos xy x += , show that 2 d 1 2sin d cos yx xx += . [3] (b) Hence, evaluate 4 0 2 2 2sin dcos x xx  +  in exact form. [4]

7 [Turn over (c) Find the y-coordinate of the maximum turning point of 2 sin cos xy x += when 02 x  . [6]

8 [Turn over 5 The diagram shows a straight coastline ABC. A turtle, at point X, at sea, swam 35 m in a straight line to point B, where John was throwing food to feed the turtles. The turtle then turned 90 and swam 50 m in a straight line to point Y at sea. Angle 90XBY =  , angle XBA = and XA and YC are perpendicular to ABC. (a) Show that 35cos 50sinAC =+ . [2] Y A B C X 35 m 50 m

9 [Turn over (b) Express AC in the form of ( )cosR − , where 0R and 0 90   . [4] (c) John’s father claims that the coastline AC is 70 m. Without measuring, John said that it was incorrect. Explain, with clear workings, how John arrived at this conclusion. [1]

10 [Turn over 6 (i) Write down the general term in the binomial expansion of 12 3 px x  − , where p is a constant. [1] (ii) Write down the power of x in this general term. [1] (iii) The constant term in the binomial expansion of 12 3 px x  − is 27500− . Find the value of p. [3]