YTSS 4NA Additional Mathematics with Answer Key Question Paper 2 2023

NAME: ( ) CLASS: YISHUN TOWN SECONDARY SCHOOL PRELIMINARY EXAMINATION 2023 SECONDARY 4 NORMAL ACADEMIC ADDITIONAL MATHEMATICS PAPER 2 (4051/02) DATE : 15 August 2023 DAY : Tuesday DURATION : 1 hour 45 minutes MARKS : 70 READ THESE INSTRUCTIONS FIRST Do not turn over the cover page until you are told to do so. Write your name, class and class index number in the spaces at the top of this page. 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 number of marks for this paper is 70. MARKS OBTAINED FULL 1 4 2 4 3 5 4 5 5 6 6 8 7 9 8 10 9 10 10 9 TOTAL 70 This question paper consists of 14 printed pages including this cover page.

2 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation , 2. TRIGONOMETRY Identities Formulae for ABC sin sin sin a b c A B C 2 2 2 2 cosa b c bc A   1 sin2 bc A 02  cbxax a acbbx 2 42  AA AA AA 22 22 22 cot1eccos tan1sec 1cossin          A AA AAAAA AAA BA BABA BABABA BABABA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin tantan1 tantantan sinsincoscoscos sincoscossinsin         

3 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) [Turn over 1 Integrate with respect to x (a) 22( 1)x  , [2] (b) 32 .2 xx x  [2]

4 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) 2 (a) Write down the period of 12 tan .2y  [1] (b) On the axes below, sketch the graph of 12 tan 2y  for 22     radians. [3] 2   3 2  2 y 2 3 2   2  

5 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) [Turn over 3 (a) Show that 2f ( ) 3 9 7x x x   can be written as 2f ( ) ( ) ,x a x b c   where a, b and c are constants to be found. [4] (b) Hence, explain why the function f(x) is always positive. [1]

6 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) 4 A potato field is infested by insects. The percentage of the field that is infested, q %, t days after pesticide is sprayed is modelled by the equation   3 d5 d 21 q t t   . Just before the pesticide is sprayed, 5% of the field was already infested with insects. Once the percentage of the field that is infested decreases to 1% or less, the potatoes will be harvested. Find the minimum number of days for the potatoes to be harvested. [5]

7 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) [Turn over 5 The equation of a curve is 26 25 xy x  for 2 5x . (a) Express d d y x in the form 2 (4 ) (2 5 ) ax bx x   , where a and b are integers to be determined. [3] (b) State the range of values of x for which y is a decreasing function. [3]

8 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) 6 A curve has the equation 3 2 5.y x x   The normal to the curve at x = 1 meets the y-axis at A and the x-axis at B. (a) Find the area of triangle AOB, where O is the origin. [6] (b) The tangent to the curve at x = 1 is parallel to the tangent at point Q. Find the coordinates of Q. [2]

9 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) [Turn over 7 (a) By substituting 3 2 ,   show that 3cos3 4cos 3cos   . [3] (b) Hence solve the equation 3 14cos 3cos 2 for 180 180oo    . [6]

10 2023 Preliminary Exam Secondary 4 Additional Mathematics NA (4051/02) 8 The diagram shows a container used to contain a chocolate product. It is made up of a cone fixed to the top of a right circular cylinder of radius r cm and height h cm. The slant height of the cone is 2r cm. (a) Given that the volume of the cylindrical part is 108 cm3, express h in terms of r. [1] (b) Show that the total surface area, A cm2, of the container is given by 2 723Ar r  . [2] 2r cm h cm r cm