DHS Prelims 2025 H2 Math P2 Questions

© DHS 2025 This document consists of 8 printed pages. Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9758/02 Paper 2 19 September 2025 3 hours Additional Materials : Printed Answer Booklet List of Formulae and Results (MF27) 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 on both sides of the paper. 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 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.

2 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 Section A: Pure Mathematics [40 marks] 1 A curve C has equation 9 .1yx x= + + (a) Using differentiation, find the range of values of x such that C concaves upwards. [2] (b) Sketch C. [3] 2 (a) U se the substitution ext = to show that ( ) ( ) 21 1e tan e d tan d .xx x t tt−− =∫∫ [2] (b) Hence find the exact value of ( ) 1 21 0 e tan e d .xx x− ∫ [5] 3 A curve C has equation 33 50x y xy+− = where x > 0. (a) Find d d y x in terms of x and y. [2] (b) Find the coordinates of the point on C at which the normal is parallel to the y-axis. [3] (c) Determine the nature of the stationary point. [2] 4 Relative to the origin O, the points A, B, and C have position vectors a, b and +3a b respectively. The point D lies on AB such that AD kAB= , where 0 1.k<< (a) Find a vector equation of the line OD in terms of k, a and b. [2] (b) The point E is the midpoint of BC. Find the value of k if O, D and E are collinear. [4] It is given that 1, 2 and 3 2 31.= = −=a b ab (c) By considering the scalar product (3 2 ) (3 2 ),−−abab  find the numerical value of ab and hence determine the angle between a and b. [4] (d) Give a geometrical interpretation of | |.ab [1] O A E C a b B D

3 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 [Turn over 5 A heated metal block at o160 C is left to cool in a laboratory with an ambient temperature of o30 C. The temperature, T in o C , of the metal block t minutes after it is left in the laboratory is model as a function (t).T The rate of cooling is proportional to the difference between the object’s temperature and the ambient temperature, with a positive constant of proportionality k. (a) Write down a differential equation for the situation and find the expression of ()Tt in terms k. [3] An experiment to study the cooling of the heated metal block starts at 10:00 am. However, a protective thermal casing is used to delay the cooling of the metal block for 5 minutes, resulting in the temperature of the metal block remaining constant at o160 C during this time. At 10:05 am, the casing is removed and cooling begins immediately. At 10:15 am, the temperature of the metal block is measured to be o100 C . (b) In order to use the solution obtained in part (a) to model the cooling process now, the time t needs to be replaced with ,ta+ where a is a constant. State the value of a. [1] (c) Due to the protective thermal casing, the temperature of the metal block o() CpTt , t minutes after 10:00 am can be modelled by the following piecewise function ( ) 160 for 0 , 5 ( ) for 5. p t Tt Tt a t  <≤= +≥ Using your answers from parts (a) and (b), show that 0.0619( ) 30 177e . tTt a −+=+ [3] (d) Find the time it takes for the metal block to cool to o60 C. [1] (e) Sketch the graph of ()pTt against t for 0.t≥ [2]

4 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 Section B: Probability and Statistics [60 marks] 6 The events A, B and C are such that P( ) 0.8A = , P( ) 0.2B = and P( ) 0.6C = . It is also known that P( ) P( )AB B∩= and P( ) 0.1.BC∩= (a) Find exactly the maximum and minimum possible values of (i) P( ' ),ABC∩∩ [3] ( ii) ( )P' .A B CA C∩∩ ∪ [2] (b) It is given further that A and C are independent, find the value of P( ' )ABC∩∩ . [2] 7 Twelve students from three CCAs organised a n overseas learning trip to Shanghai. It comprises five students from Tennis, four students from Bowling and three students from Softball. The available seats for the flight for them to choose from are shown on the diagram as follows: (a) In how many different ways can the students be seated if there is a particular student who must be seated nearest to the emergency exit? [2] (b) Find the number of ways they can be seated if two particular students do not want to be seated on any of the aisle seats. [3] (c) Find the number of ways they can be seated if students of the same CCA must be seated together either front and back or left and right, and cannot be separated by an aisle. For example, the three students from Softball can be seated at Row 56 Seat G, H and Row 57 Seat G, but not Row 56 Seat H, I and Row 57 Seat G. [3] Available Seat Occupied Seat Legend: Aisle Aisle Emergency exit Row 56 Row 57 C D E F G I Row 55 H

5 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 [Turn over 8 The bag contains 6 red discs, 6 green discs, 6 yellow discs and 6 blue discs. The discs are indistinguishable aside from their colour. 4 discs are taken one after another from the bag without replacement. The random variable X is the number of different colours obtained among the 4 discs. (a) Show that ( ) 465P2 . 1771X = = [3] (b) Hence determine the probability distribution of X. [3] Joe plays a game that comprises 8 rounds of drawing 4 discs from the bag. For each round, he wins $10 if he obtains at most 2 discs of different colours, otherwise he loses $5. After each round, all the discs are returned to the bag before the next round. (c) Find the probability that Joe wins some money at the end of the game. [2] (d) Explain, with working, whether it is worthwhile for Joe to play this game to win money. [2]

6 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 2 9 (a) The following scatter diagrams with 5 data points each have product moment correlation coefficients 0.7, 0, 1,− not necessarily in the given order. State the product moment correlation coefficients for the scatter diagrams I, II and III. [1] Scatter Diagram I Scatter Diagram II Scatter Diagram III (b) A private developer hires workers for renovation works for its townhouse projects. The renovation works required for each townhouse in a certain project are identical. The private developer wishes to investigate how the number of working days taken to complete the renovation works for a townhouse for a certain project, y, varies with the number of workers hired to renovate a townhouse , x. The relevant data collected by the private developer is given in the following table. x 3 5 8 10 13 15 18 20 y 92 50 33 25 21 19 17 16 (i) Draw a scatter diagram for these values. Use your diagram to comment on whether a linear model would be appropriate to model the