2022 AHS AMath Prelims P2

Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over NAME: _______________________________ ( ) CLASS: 4 ( ) 9o ADDITIONAL MATHEMATICS 4049 / 02 Paper 2 30 August 2022 Candidates answer on the Question Paper. 2 hours 15 mins One sheet of graph paper is required. READ THESE INSTRUCTIONS FIRST Write your name, index number and class in the space at the top of this page. Write in dark blue or black pen. You may use a 2B pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters and 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. 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 number of marks for this paper is 90. ------------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Question Marks Question Marks Table of Penalties 1 8 2 9 Clarity/Logic 3 10 Accuracy/Precision 4 11 Units 5 12 Total: 6 13 7 Parent’s Name & Signature: Date: This document consists of 16 printed pages. S4 ANGLICAN HIGH SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATIONS 2022 90

2 Secondary Four Additional Mathematics Preliminary Examinations 2022 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  Binomial expansion nrrnnnnn bbar nbanbanaba                  ......21)( 221 , where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n       2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  AAec 22 cot1cos  BABABA sincoscossin)sin(   cos cos cos sin sinA B A B A B     tan tantan 1 tan tan A BA B A B    AAA cossin22sin  AAAAA 2222 sin211cos2sincos2cos  A AA 2tan1 tan22tan   Formulae for ABC C c B b A a sinsinsin  Abccba cos2222  Area of Cab sin 2 1

3 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 1 Determine the range of values of k such that the curve 2 23 3y x k k    intersects the line 3y kx at two distinct points. [4] 2 Find the range of values of x where the curve 2 2 6 4 12 9 xy x x    is below the xaxis. [4]

4 Secondary Four Additional Mathematics Preliminary Examinations 2022 3 The curve 3 23 16 6y x x   intersects the line 10 12y x  . Find the xcoordinates of the points of intersection. [5] 4 Solve the equation     22 1 2 12 2 2 15x x    . [5]

5 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 5 (a) By considering the general term in the binomial expansion of 10 3 1 5 x x      , explain why there are only three terms of non-negative powers of x in this expansion. [4] (b) In the expansion of   5 2 1 4 nx x   , in ascending powers of x, the sum of the coefficients of the term in x and the term in x2 is 8218 . Find the value of n. [4]

6 Secondary Four Additional Mathematics Preliminary Examinations 2022 6 A formula for calculating the number of living cells, y, of a host body per hour, t, is given by  e e by a t  , where a and b are constants. When y is plotted against lnt , a straight line graph is obtained and it passes through the points  0,3 and 2,7 . Find the value of a and of b. [5] 7 Prove the following identities. (a) 2 1 1 2 1 cos 1 cos sin      . [2]

7 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over (b) cos sin cosecsin 2 cos 2 1       . [2] (c) By using (a) and (b), show that there is no solution to the equation 1 1 cos sin 01 cos 1 cos sin 2 cos 2 1            for 0 2 π  . [2]

8 Secondary Four Additional Mathematics Preliminary Examinations 2022 8 The diagram shows a triangle ABC in a semi-circle, centre O and radius 4 cm. It is given that 2cmAX  , 2ABC   and XY is parallel to BC. (a) Show that the perimeter of the figure XBCY, P cm, is given by the equation 10cos 2 6sin 2 6P     . [3] (b) Express P in the form  cos 2 6R    , where 0R and 0 90    . [3] X B A C O Y

9 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over (c) Find the maximum value of P and the corresponding value of  . [3]

10 Secondary Four Additional Mathematics Preliminary Examinations 2022 9 A circle 1C is given by the equation 2 2 10 6 9 0x y x y     . Another circle 2C with centre  7, 2Q  touches the circle 1C externally as shown in the diagram below. (i) Find the coordinates of the centre P and the radius of circle 1C . [3] (ii) Show that the equation of circle 2C is    2 27 2 64x y    . [2] P Q C1 C2