2025 CJC JC2 H2 Math Prelim Paper 2 answer key

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  fy x and  f 'y x . The graph of  fy x has turning point s 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 'y x crosses the x -axis at  3, 0 , and the equations of asymptotes are 1x and 0y . On separate diagrams, sketch the graphs of (a)  fy x , [2] (b)   1 ,f 'y x [3] (c)  f 1 ,y x  [3] labelling clearly the equation(s) of any asymptote(s), coordinates of any axial intercept(s) and turning point(s) where applicable. y x O y =2 (2, 0) (, 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 (a) Solution: Examiners Feedback: Well attempted. Most students were able to achieve full credit. Others were still able to get the portion of the graph, 0x correct. y x O y =2 (2, 0) (, 0) (3, 4) (3, 4) x = 1 x = 1 x y O y =0 (3, 0) x = 1 y x O y =2 (, 0) (3, 4) x = 1

4 9758/02/J2PRELIM/2025 (b) Examiners Feedback: Poorly attempted. Students were able to identify x-intercept (1,0) and vertical asymptote, 3x but unable to sketch the curve. Others were able to go on and identify that there is a maximum point in the curve   1 ,f 'y x but indicated this point in the 1st quadrant instead of the 4th quadrant. x y O y =0 (3, 0) x = 1 y x O x = 3 (1, 0)

5 9758/02/J2PRELIM/2025 [Turn Over (c)       f translate 1 unit in positive -axis direc tion f 1 reflect in -axis f 1 y x x y x x y x        Examiners Feedback: Poorly attempted. Students made several careless mistakes including labeling of coordinates of the axial intercepts and turning points and asymptotes. Students are advised to sketch an intermediate graph if they are not confident in sketching the final graph directly. This will allow them to secure maximum marks in the exam. y x O y =2 (2, 0) (, 0) (3, 4) (3, 4) x = 1 x = 1 y x O y = 2 (1, 0) (3, 0) (4, 4) (2, 4) x = 0 x = 2 (1, 0)

6 9758/02/J2PRELIM/2025 2 Functions f and g are defined by   2 4f : , , 4 4 x x x x      , 1g : ln 1 , , 0x x x x         . (a) Sketch the graph of  fy x , 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] Examiners Feedback: Majority of the candidates were able to sketch the graph well. However, many candidates could not get full credit due to lapses in the labelling of the features of the graph. Common mistakes observed include:  Missing vertical asymptotes.  Not labelling the horizontal asymptote when the question asked to state.  Not giving the y-intercept in coordinates form when required by the question. Candidates should review:  The techniques of curve sketching which includes the features such as asymptotes, intercepts and turning points.  Read the question carefully to identify any requirements from the question (e.g. coordinates form, state equation of asymptotes etc). (b) Show that gf exists. Hence find the rule, domain and range of gf. [4] For gf to exist, f gR D , fR 0,  gD 0,  Since f gR D , gf exists (shown) x y y = 0 x = 4 y = f (x) O

7 9758/02/J2PRELIM/2025 [Turn Over Hence, find the rule,         2 2 2 gf 4g 4 1ln 1 4 4 4ln 1 4 x x x x                                          domain    gf fD D , 4 4,     OR  gf fD D 4 \   and range of gf. Method : Sketch  gfy x for given domain  gfR 0,  Method : Mapping        f g f f gfD , 4 4, R 0, R 0,            Examiners Feedback: This part of the question was not very well attempted. Many candidates missed out certain components (did not find rule/domain/range) in this part of the question. Common mistakes observed include:  Unable to establish the range of f and domain of g.  Incorrect condition for gf to exist, e.g. g fD R , g fR D , f gD R etc  Incorrect gfD such as gf gD D , or writing as    , 4 4,   .  Incorrect method to find gfR such as using gf fR R or gf gR R without mentioning about the restricted domain. Note: Marks were not awarded even though the answer is correct here. Candidates should review: The condition for existence of a composite function, as well as to find the rule, domain and range of the composite function (including the use of the restricted domain to find the range). (c) If the domain of f is further restricted to x k , state the largest value of k for which the function 1f  exists. [1] Largest value of k is 4. Examiners Feedback: This part of the question was well done. Most candidates were able to recognise that the largest value of k occurs at the turning point. However, some candidates were confused by the vertical asymptote and gave 3k  instead. x y O 4 y = gf (x) x y y = 0 y = g (x) O

8 9758/02/J2PRELIM/2025 (d) Using the restricted domain found in part (c), find 1f  in a similar form. [3]       2 2 2 4 4 4 2 24 rej 4 . 4 r 4 4 o y x x x y y y x x y x             1 ffD R (0,    1f : , , 0 24 x x x x    Examiners Feedback: This part of the question was well done. Most candidates were able to let  fy x and make x the subject to find the rule. Common mistakes observed include:  Not including the  when taking square roots to both sides of the equation.  Not knowing or reject the wrong expression.  Used the restricted domain of f as the domain of the inverse.  Not writing in similar form when the question requires the inverse in the similar form. Candidates should review: The concept of finding the rule and domain of an inverse function, especially for questions that require the decision to reject one of the expression. (Decision is based on domain of f)

9 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 s