RI Promos Solns 2025

RAFFLES INSTITUTION 2025 Year 5 H2 Mathematics Promotion Examination Questions and Solutions with comments ______________________ 2025 Yr 5 H2 Math Promotion Examination Solution with Comments General Comments • Wrong Exam Index Numbers A small number of students wrote the wrong Exam Index Number on their scripts, though fewer than during the Timed Practice. For example, 260227 was mistakenly written as 262027, which belongs to another student. 跚跛跜距 Please double-check your Index Number before writing it down in future assessments. • Writing Outside the Designated Boxes Quite a significant number of scripts had solutions written outside the boxes provided for each part of the question. 굓굔굕굖 For the actual A-Level exams, your scripts will be digitally scanned, and any writing outside the boxes may not be captured. 跚跛跜距 Always keep your work within the provided boundaries. • Writing Solutions in the Wrong Question Space More students than in the Timed Practice wrote solutions for one question in the space meant for another question. 跚跛跜距 Do not do this — different markers may be assigned to different questions, and your answer may be missed if it is in the wrong section. • Using Additional Pages and not using the correct Additional Page. A significant number of students who used Additional Pages did not indicate this on the original allocated space. There are also a few students wrote solutions on the provided school foolscap paper and submitted together with the answer booklet 跚跛跜距 When you use an Additional Page, always write “See Additional Page” clearly in the allocated space, or the marker may miss your continued work. 跚跛跜距 The provided school foolscap paper is meant for rough work. Any work that needs to be submitted for marking should be inside the Answer Booklet or the Additional Page. • Multiple Solutions for the Same Question Some students wrote two or three different solutions for the same question without cancelling the undesired ones. 跚跛跜距 You must cancel any solution that you do not want to be marked. The marker will only mark 1 solution. • Presentation and Legibility Most scripts were well presented, with clear and logical mathematical arguments. 踀踁 However, some scripts were difficult to mark because:  handwriting was hard to read  writing was too small  ink was too light  Reminder: Markers can only award marks for work that is legible. Please ensure your handwriting is neat, written at a reasonable size, and clearly visible.

2 ______________________ 2025 Yr 5 H2 Math Promotion Examination Solution with Comments 1 Fig. 1 A tablet is dissolving in water and is modelled as a cylinder, as shown in Fig. 1. At t seconds after being dropped into the water, the radius of the tablet is x mm, and its thickness is 1 3xmm. The circular cross-sectional area of the tablet is decreasing at a constant rate of 0.5 mm² per second. When x = 5, find the exact value of (a) d d x t , [2] (b) the rate of change of the volume of the tablet. [2] (a) [2] Let A be the cross-sectional surface area at time t. Then, 2A x . d d2d d A x xt t  When x= 5, d0.5 2 (5)d d 0.05 1or d 20 x t x t        Most students used the chain rule instead of differentiating implicitly wrt t. Both methods are accepted but it is encouraged to do implicit differentiation for this question. When chain rule was used, there were quite a few careless mistakes. There were also quite a number of bad presentation where 5x was substituted at the last step just to get the answer. Common mistakes seen: 1) When students used this chain rule: d d d d d dx A t t x A  , they did not take the reciprocal of d d A x when substituting for d d x A. 2) Writing 1 20 as 1 20 . 3) Did not consider “decreasing rate for A” and omitted the sign , i.e used d 0.5d A t  instead of d 0.5d A t  4) Failed to read the question carefully and did not give the “exact value” as required.

3 ______________________ 2025 Yr 5 H2 Math Promotion Examination Solution with Comments 5) Did not know the formula for the area of a circle, used 2A x instead of 2A x (b) [2] Let V be the volume of the tablet at time t. Then, 2 3 2 1 1 3 3 d d d d V x x x V x xt t          Thus,  2 5 d 0.05 5 1.25d x V t         The rate of change of the volume of the tablet when x = 5 is 31.25mm per second. This part was done better than part (a) as the chain rule was applied in a more direct manner. Students should also remember to answer the question by writing a short statement and giving the correct units. Common mistakes: 1) 31 3V x 2d d V t x (differentiated wrt to x but on the LHS treated it as d d V t ) 2) Differentiating 31 3V x  wrongly. ( 2d d 3V xx  )

4 ______________________ 2025 Yr 5 H2 Math Promotion Examination Solution with Comments 2 Do not use a calculator in answering this question. Showing your working, find the complex numbers z and w which satisfy the simultaneous equations  * 2i 5 7i, 4 i 3i. z w w z       [5] [5]  * 2i 5 7i ----(1) 4 i 3i ----(2) z w w z       From (2),   * * 4 i 3i 4 i 3i ----(3) w z z w           * * * Sub 3 into 1 , 2i 4 i 3i 5 7i 2 8i 6 11 7i 2 8i 11 7i w w w w w w                         Let i . 2 8i i i 11 7i 2 8i 2i 8 i 11 7i 3 8 8 i 11 7i w x y x y x y x x y y x y x y x y                        Comparing Re and Im parts, 8 8 7 11 3 67 673 8 11 8 7 8 7 1, 1 x x xx y x y y x x y                   Sub 1 i into 3 , 4 i 1 i 3i 4 5i 1 3i 3 2i 1 i, 3 2i w z w z                  There were a few methods to find the complex numbers. Some students let iw a b  and iz c d  , substituted them into the given equations, compared real and imaginary components to get 4 equations to solve for , , ,a b c d. This was accepted. There were quite a number of careless mistakes made when they were solving the equations simultaneously. As the use of calculator is not allowed for this question, students must show clear working when they are solving simultaneous equations. Common mistakes: 1) Make w the subject first and write: 5 (7 2 )i * 5 (7 2 )i w z w z       Take note that z is also a complex number with real and imaginary components. And so, Im( ) 7 2w z  . It should be * 5 (7 2 *)iw z   2) Using GC to solve for unknowns even when the question does not allow the use of calculator. Similar Qns:  Promo Topical Revision Exercise Page 10 Qn 5, 6(a)(i)  RI 2024 Promo Qn 7(b)

5 ______________________ 2025 Yr 5 H2 Math Promotion Examination Solution with Comments 3 The point A on the Argand diagram below represents the complex number 1w with modulus r and argument . It is given that the complex numbers 2w and 3w satisfy the equations 2 1w w and 3 2iw w . Let B, C and D represent the complex numbers 2w, 3w and 2 3w w respectively. (a) On the copy of the Argand diagram in the Printed Answer Book, plot the points B, C and D, indicating clearly the modulus and argument of 2w and 3w. [3] (b) State, in radians, the angle BOC. [1] (c) By considering the quadrilateral OBDC, find 2 3w w in terms of r and 2 3arg ( )w w in terms of . [2] (a) [3] Most students were able to locate the points in the correct quadrant, but were not able to find the argument. Students need to know that arguments are measured from the positive Re axis, and points above the real axis has positive argument and points below the real axis has negative argument. While the question did not give a specific point, students could arbitrary let point A be (say 1+i) and calculate out the argume