2019 CCHY MYE AMath 4047 P2 sol

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 1 of 20 Mid-Year Examination (2019) Secondary 4 Express / 5 Normal (Academic) Candidate Solutions Name Register No Class Additional Mathematics Paper 2 4047 / 2 Date : 14th May 2019 Duration : 2 hours 30 minutes Additional Materials : Graph Paper READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. 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 marks for this paper is 100. Setter : Ang Wee Hoon Fyn This paper consists of 20 printed pages, INCLUDING the cover page. For examiner’s use / 100 CHUNG CHENG HIGH SCHOOL YISHUN 义 顺 [Turn over

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 2 of 20 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation a 2x + bx + c = 0, x = 2 4 2 b b ac a    Binomial Expansion nba = na + 1 n    1na b + 2 n    2na 2b +    +    r n rna  rb +    + nb , where n is a positive integer and    r n =  !! ! rnr n  = 1 1 ! n n n r r     2. Trigonometry Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin ( A  B ) = sin A cos B  cos A sin B cos ( A  B ) = cosA cos B  sin A sin B tan ( A  B ) = BA BA tantan1 tantan   sin 2A = 2 sin A cos A cos 2A = cos2 A - sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A tan 2A = 2 2 tan 1 tan A A Formulae for ABC sin a A = sin b B = sin c C a2 = b2 + c2 – 2bc cos A area of ABC = 1 2 ab sin C

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 3 of 20 1. The table below shows the experimental values of two variables x and y. x 0.5 1.0 1.5 2.0 2.5 3.0 y 1.20 1.00 0.86 0.74 0.66 0.59 It is known that x and y are related by an equation of the form ay x b  , where a and b are constants. (i) Using a scale of 4cm to 1 unit on the x axis, and 4cm to 0.5units on the 1 y axis, draw a straight line graph. [3] (ii) Use your graph to estimate the value of a and of b. [2] (iii) Find the value of x where 10 9y [2] From another set of experimental data, it is found that x and y are related by the equation 7 28 25xy y  . (iv) By drawing a suitable straight line on your graph, illustrate the second situation. Hence find the value of x which is consistent in the two experiments. [3]

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 4 of 20 2. (a) (i) Show that 522  xx is always positive for all real values of x. [2] 2 22 5 ( 1) 1 5x x x      2( 1) 4x   2 2 ( 1) 0 ( 1) 4 4 x x      2 2 5x x   is always positive. OR 2 24 ( 2) 4(1)(5)b ac    16 Since 2 4 0b ac  and 0a , therefore 522  xx is always positive. (ii) Hence, find the largest integer value of k for 152 73 2 2   xx kxx to be true for all real values of x. [4] 2 2 2 2 3 7 2 5 02 5 2 5 x kx x x x x x x         2 2 2 ( 2) 2 02 5 x k x x x      Since 2 2 5 0,x x   22 ( 2) 2 0x k x    2 4 0b ac  2 2 2 ( 2) 4(2)(2) 0 4 4 16 0 4 12 0 ( 6)( 2) 0 6 2 k k k k k k k k                 Largest integer value of k = 1 (b) Simplify 2 3 2(1 ) 1 45 3 9 3 3 27 x x x x         . [3] 2 3 2 2 2 3 2(1 ) 1 2 2 3 3 45 3 9 3 5 3 3 3 3 3 27 3 3 x x x x x x x x                 1 1 5(3 ) 3 3 13 5 3 13 3 14 x x x x x             

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 5 of 20 3. The roots of a quadratic equation22 2 1 0x x   are  and . (i) show that 3 3 2 2 8   , [4] Sum of roots    Product of roots  2 2 1   1 2     3 3 2 2 2 2 ( )( ) ( ) 3 31 1 2 11 2 1 2                                   3 3 3 3 3 3 2 2 2 2 ( ) 12 2 1 2 1 1 8 8 ( )shown                  (ii) find the equation whose roots are3 2  and 3 2  . [2] Sum of roots 3 3 2 2    8 Product of roots 3 3 2 2              3 3 4 4 1 2 32       Hence the quadratic equation is 2 8 32 0x x  

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 6 of 20 4. A student learns a new topic from a teacher. After learning the topic for t days, the percentage, P %, of the topic that the student remembers can be modelled by20e80  ktP , where k is a constant. (i) Explain with clear working steps why the student is able to remember 100% of what he learnt from the teacher on the day it was taught. [1] 80 20ktP e   When 0t 080 20P e  80 20 100    Therefore, the student can only remember 100% of what he has learnt only on the day it was taught. (ii) Find the value of k if the student can only remember 40% of what he had learnt from the teacher a week ago. [2] When 7, 40t P  7 7 7 7 40 80 20 80 20 1 4 1ln ln 4 17 ln 4 0.19804 0.198 (3 . ) k k k k e e e e k k k s f             

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 7 of 20 (iii) How many days would have elapsed when a student is only able to retain at most half of what he learnt from the teacher? [2] When 50, 0.19804P k  0.19804 0.19804 0.19804 50 80 20 3 8 3ln ln 8 30.19804 ln8 4.95268 5 t t t e e e t t t            It takes the students 5 days to retain half of what he has learnt. (iv) Lester claimed that in the long run, a student will not be able to remember that he learnt from the teacher entirely. Do you agree with Lester? Support your decision with clear working steps. [1] 80 20ktP e   After a long time, t gets very large, kte approaches 0 80(0) 20P   20 Hence the students will not forget the entire topic.

CCHY Mid-Year Examination (2019) Additional Mathematics Sec 4E/5N Page 8 of 20 5. (i) In the answer space below, sketch the graph of 1 21 3y x    for 0x . [1] (ii) On the same diagram, sketch the graph 1 21 3y x  for 0x . [1] (iii) Calculate the coordinates of the point of intersection of your graphs. [2] 1 1 2 2 1 1 2 2 1 3 1 3 1 1 3 2 x x x x x y           Hence the point of intersection is 1, 2 (iv) Determine, with explanation, whether the tangents to the graphs at the point of intersection are perpendicular. [3] 1 2 3 2 1 3 3 2 1 3 2 y x dy xdx when x dy dx        1 2 1 2 1 3 3 2 1 3 2 y x dy xdx when x dy dx       3 3 9 2 2 4       Since the product of the two gradients is not –1