2022 AHS AMath Prelims P1

Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over NAME: _______________________________ ( ) CLASS: 4 ( ) 9o ADDITIONAL MATHEMATICS 4049 / 01 Paper 1 29 August 2022 Candidates answer on the Question Paper. 2 hours 15 mins No Additional Materials are 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 18 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 Cabsin 2 1

3 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 1 A polynomial, P( )x , is 3 22 6 27x ax x   , where a is a constant. The quadratic expression 2 9x bx  is a factor of P( )x , where b is a constant. (a) Find the remaining factor of P( )x . [2] (b) Show that the equation P( ) 0x  has only one real root. [3]

4 Secondary Four Additional Mathematics Preliminary Examinations 2022 2 Find the remainder when the polynomial   2 12f 5 121x x  is divided by 1 3 2 5 x     . [4] 3 Solve the equation   74 7 log 1log 3 log 16 2 xx   . [5]

5 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 4 Carbon14 has a half life of 5,73 0 years , which means that it takes 5,730 years for the original mass of Carbon14 to be reduced by 50%. To determine the age of a plant or animal fossil, scientists determine the amount of Carbon14 in the specimen as the Carbon14 undergoes radioactive decay. The amount of Carbon14 in a piece of fossilised bone is given by 0 e ktM M  , where k is a constant, M is the mass of Carbon14 in the specimen, 0M is the initial mass of Carbon14 and t is measured in years. (i) Determine the value of k. [2] (ii) Determine the percentage of Carbon14 that has decayed in the specimen if the fossil is estimated to be about 900 years old. [2] (iii) Express the rate of change of percentage of Carbon14 in terms of t. [2]

6 Secondary Four Additional Mathematics Preliminary Examinations 2022 5 Express    2 2 10 3 2 1 x x x    in partial fractions. [5]

7 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 6 The function f is defined as  f cos 2 1x x  for 0 2 πx  . The function g is defined as  g 2 tanx x for 0 2 πx  . (i) State the period of f. [1] (ii) On the same axes, sketch, for 0 2 πx  , the graphs of f and of g. [3] (iii) Hence, state the number of solutions for which cos 2 2 tan 1 0x x   [1] 0 x y 1 2 3

8 Secondary Four Additional Mathematics Preliminary Examinations 2022 7 Given that   4sin 5A B  and   12cos 13A B  where  A B and  A B are acute angles, (i) what can be deduced about the size of A relevant to B? [1] (ii) find, without using the additional formulae, the value of (a)  tan A B , [2] (b)  tan A B , [2] (iii) Hence, using the answers in (ii), prove that 63tan 2 16A . [3]

9 Secondary Four Additional Mathematics Preliminary Examinations 2022 [Turn over 8 The diagram shows a quadrilateral PQRS where the coordinates of P, Q and R are  6,1 ,  2,2 and  1,k respectively. The line QS is the perpendicular bisector of PR, and PQ QR . The equation of the line PS is 4 23y x   . The line QW makes an acute angle  with the xaxis and 3tan 5  . (a) Find (i) the value of k, [3] y x P Q R S W O

10 Secondary Four Additional Mathematics Preliminary Examinations 2022 (ii) the coordinates of S. [4] (b) Explain why the gradient of QW is 3 5 . [1]