CJC Prelims 2025 H2 Math P2 Questions

9758/02/J2PRELIM/2025 [Turn Over CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS 9758/02 Paper 2 17 Sep 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 8 printed pages.

2 9758/02/J2PRELIM/2025 Section A: Pure Mathematics [40 marks] 1 The diagram shows the graphs of ( )fyx= and ( )f'yx= . The graph of ( )fyx= has turning points at ( )3, 4− and ( )3, 4 , crosses the x -axis at ( )2, 0− and ( )2, 0 , and the equations of asymptotes are 1x=− , 1x= and 2y= . The graph of ( )f'yx= crosses the x -axis at ( )3, 0 , and the equations of asymptotes are 1x= and 0y= . On separate diagrams, sketch the graphs of (a) ( )fyx= , [2] (b) ( ) 1 ,f 'y x= [3] (c) ( )f 1,yx= −− [3] labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable. 2 Functions f and g are defined by ( ) 2 4f: , , 4 4 x xx x → ∈≠ −  , 1g : ln 1 , , 0x xx x →+ ∈>  . (a) Sketch the graph of ( )fyx= , stating the equations of any asymptotes, the coordinates of the points where it crosses the axes and the coordinates of the turning points, if any. [2] (b) Show that gf exists. Hence find the rule, domain and range of gf. [4] (c) If the domain of f is further restricted to xk< , state the largest value of k for which the function 1f − exists. [1] (d) Using the restricted domain found in part (c), find 1f − in a similar form. [3] y x O y =2 (−2, 0) (2, 0) (3, 4) (−3, 4) x = −1 x = 1 x y O y =0 (3, 0) x = 1

3 9758/02/J2PRELIM/2025 [Turn Over 3 Fig. 1 shows a net of a hexagonal pyramid folded from a star-shaped cardboard of equal edge length cma . The net consists of a hexagon with equal sides of cmx and six isosceles triangles with base cmx and side cma . The net is folded to form a right pyramid with a hexagonal base of edge length cmx and vertical height cmh , as shown in Fig. 2. The hexagonal base is made up of six equilateral triangles of side length cmx . The volume of a right hexagonal pyramid with base edge cmx and height cmh is given by 23 2V xh= . (a) Show that the volume of the hexagonal pyramid, V satisfies the expression given by ( ) 2 24 63 4V ax x= − [2] (b) Find, in terms of a, the maximum possible volume of the hexagonal pyramid. You need not show that this value is a maximum. [4] (c) Find, in terms of a, the total surface area of the hexagonal pyramid when the volume is a maximum. [4] a x Fig. 1 h x Fig. 2

4 9758/02/J2PRELIM/2025 4 One of the roots of the equation 4322 14 33 26 0z z z zp− + − += , where p is a constant is 3i+ . (a) Based on the above information only, a student claims that the equation has a root 3i− . State, with a reason, why the student’s claim may not be true. [1] (b) Show that 10p= . [2] For the rest of this question, do not use a calculator. (c) Find the roots of the equation 4322 14 33 26 10 0zzzz− + − += and mark them clearly on a single labelled Argand diagram. [7] (d) The points of the Argand diagram in part (c) form the vertices of a quadrilateral. Identify the type of quadrilateral and determine its area. [2] Section B: Probability and Statistics [60 marks] 5 The basketball club in a college has 5 centers, 8 forwards and 7 guards. W ith the National School Games approaching, the coach wishes to find out the opinions of members of the club about the training programme. He gives a questionnaire to all the members of the club and receives replies from everyone. (a) Explain whether the 20 members form a sample or a population. [1] The coach then decides to select teams to play in the matches for National School Games. A basketball team to play in a match consists of 1 center, 2 forwards and 2 guards. (b) Explain an advantage for choosing a random sample in each category of members for the match. [1] (c) How many different teams can be formed? [1] In the club, one particular forward is the classmate of one particular guard. Both classmates are injured and cannot participate in a particular match. The coach decides that one of the remaining guards can play either as a guard or as a forward. (d) How many different teams can now be formed? [3]

5 9758/02/J2PRELIM/2025 [Turn Over 6 A group of 100 students are asked if they are student leaders, in a sports CCA, or studying in a science faculty. The number of students who are student leaders is 25, the number of students who are in a sports CCA is 30 and the number of students studying in a science faculty is 40. There are 15 student leaders who are in a sports CCA. The number of students who are student leaders, in a sports CCA and studying in a science faculty is x. The number of students who are in a sports CCA and studying in a science faculty but not a student leader is y. One of the students is chosen at random. A is the event that the student is a student leader. B is the event that the student is in a sports CCA. C is the event that the student is studying in a science faculty. It is given that A and C are independent. (a) Complete the Venn diagram below to represent all the above information. You are allowed to give expressions in terms of x and y. [3] It is further given that B and C are independent. (b) Find y in terms of x. Hence, find the greatest and least possible values of y. [4] A B C x y

6 9758/02/J2PRELIM/2025 7 Happie, the owner of a store selling novelty items, is organizing a publicity stunt for his store. Using a stock of ultra-rare Lubaba dolls that he acquired, he sets up a game where a player opens “mystery boxes” to try to find a Lubaba doll hidden inside one of the identical boxes used for the game. The rules of the game are: • The player pays an initial $5 to start the game to open one of n boxes. • If the opened box is empty, the player can pay an additional $3 to open a second box. • If the second box is empty, the player can pay $2.50 again to open a third box. • If the third box is still empty, the player loses the game. The doll that is not won w ill then be donated away and the game is reset for the next player. As the Lubaba dolls are considered rare collectors’ items, it can be assumed that in every game, the players will keep opening the boxes until they win the doll and that they have the means to pay for the maximum of 3 allowable tries at opening the boxes. Let X be the random variable denoting the number of empty boxes a player has opened in a game. (a) Show that ( )P2 1X n= = . [1] (b) Find the probability distribution of X, leaving your answers in terms of n. [2] (c) If 10n= , find ( )