ASRJC 9649 2025 Prelim P2

[Turn over FURTHER MATHEMATICS 9649/02 Paper 2 29 August 2025 (Friday), 8 – 11 am 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your name on the work you hand in. 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. DO NOT WRITE ON ANY BARCODES. Answer all 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 6 printed pages and 2 blank pages. Anderson Serangoon Junior College ANDERSON SERANGOON JUNIOR COLLEGE JC2 Preliminary Examination 2025 Higher 2

2 Anderson Serangoon Junior College Section A: Pure Mathematics [50 marks] 1 �e Fibonacci numbers is defined by the recurrence relation 21nnnUUU−− += where 2n≥ and 01 1UU= = . { }nR is a sequence of real numbers where 1 n n n UR U − = , n +∈  . Given that this sequence converges, prove algebraically that its limit can be expressed in the form 2 ab+ , where a and b are integers to be determined. [5] 2 (a) As an alternative to the Newton-Raphson method for numerical root-finding, Chebyshev’s method computes successive approximations using the following recurrence relation: [ ] 2 1 3 f () f () f () f( ) 2f( ) n nn nn n n x xxxx x x + ′′= −− ′ ′ , n +∈  . Determine the general form that f( x) must take such that Newton- Raphson method and the Chebyshev’s method will lead to the same subsequent approximation given any initial value in the domain of f. [2] �e equation ln 0xx = has a root 1α = . (b ) Using a common initial approximation of 1 5x = , perform successive iterations of both Newton- Raphson and Chebyshev’s methods, and comment on your observations. [4] (c) Explain why Chebyshev’s method fails with an initial approximation of 0.5. [2] 3 �e Himmelblau function is defined by ( ) ( ) ( ) 2222f , 11 7xy x y x y= +− + + − . (a) Find the unit vector in the direction of the steepest descent at ( )1, 2, 68 and state its directional derivative. [4] (b) Given that there is a saddle point near ( )3 ,0 ,f ( 3 ,0 ) , use quadratic approximation to find an estimate for the coordinates of the saddle point, showing clearly that it is indeed a saddle point. [6]

3 Anderson Serangoon Junior College [Turn over 4 In this question, you are to show all numerical computation workings. Linear algebra forms the mathematical foundation of Large Language Models (LLMs) such as ChatGPT. At their core, LLMs use matrices and vectors to predict the next word in a sentence through sophisticated mathematical processes. Words are represented as hig h-dimensional vectors (typically of hundreds or thousands of dimensions) that capture complex features such as grammar, semantic meaning, and contextual relationships. �ese are processed through multiple layers of matrix operations involving attention mechanisms and neural networks. For this problem, we will consider a simplified model where an LLM processes just 3 words from a sentence, with each word represented as a vector in 3-dimensional space. For the following matrix 123 014 137  = E , the word vectors are its columns, namely 1 0 1    , 2 1 3    and 3 4 7    . (a) Determine whether the word vectors are linearly independent. [2] �e column space of matrix E represents the embedding subspace that the model can understand and work with. Any linguistic concept or word meaning that lies outside this subspace cannot be properly processed by the model, as it lacks the representational capacity to encode such information. (b) Write down a basis for the column space of E. [1] During language processing, the model encounters a new word or concept represented by the vector ( ) T 6 9 18=v . To determine whether the model can understand and process this concept, we need to check if it lies within the model’s representational capacity. (c) By performing row operations, determine whether the word vector v can be represented using the model’s current embedding subspace and state its implication. [3] To speed up calculations, the model uses a linear transformation to compress 3-dimensional word vectors into 2 dimensions. �e transformation is defined by the matrix 10 1 02 1 −= A . (d) Determine the rank and the nullity of matrix AE. Obtain a basis of the null space of AE. [4]

4 Anderson Serangoon Junior College 5 (a) Sketch the locus of the complex number u satisfying 6Re( ) 0u < . [3] (b) A variable complex number v satisfies 3 3 1arg 12 v v π − =− + . Show that the locus of v consists of three disconnected arcs on a unit circle with centre at the origin, stating the total length of the arcs. [4] (c) (i) On an Argand diagram, sketch the locus of points representing the complex number z that satisfies the inequalities 4 3i 3z+− ≤ and 1 4arg( ) sin2 25 zππ −< ≤+ . [3] (ii) Determine the exact maximum value of 3z+ . [2] (iii) Find the range of values of arg( 3 3i)z−− within the principal range ( ,]ππ− . [3] (iv) A variable complex number w satisfies Im( ) 3w =− . Determine the exact value of θ, where 0 2 πθ<< , such that the minimum possible value of iewz θ− is 2 as z and w varies. [2] Section B: Probability and Statistics [50 marks] 6 A company intends to conduct a study on eco-friendly packaging for their products. �e first part of the study involves estimating the proportion of consumers who are willing to pay more for eco-friendly packaging. (a) Show that, regardless of the sample proportion, a 95% confidence interval for a population proportion calculated using a large sample of size n must have a width less than 2 n . [2] (b) �e company surveyed n people and obtained a 95% confidence interval of (0.273669, 0.426331). Find the value of n. [2] In the second part of the study, the company wants to evaluate the effectiveness of their new eco-friendly packaging design. �ey conducted a controlled experiment to measure the reduction in material waste achieved by the new eco-friendly design compared to their current standard packaging. �e company randomly selects 12 product packages and measures the material waste reduction for each. �e waste reduction values (in grams) are: 13.88 14.21 14.53 13.11 15.29 14.00 14.73 13.60 14.32 15.07 12.94 14.15 (c) Construct a 99% confidence interval for the population mean waste reduction, giving your answer correct to 2 decimal places. [3] (d) State what is meant by a ‘99% confidence interval’. [1]

5 Anderson Serangoon Junior College [Turn over 7 A bird conservationist collected data on the number of Anna’s Hummingbird, a species native to North America, foraging in three different habitats at three distinct times of the day as shown in