RI KI Y6 Prelim P2 2019

This document consists of 4 printed pages [Turn over] RAFFLES INSTITUTION 2019 Year 6 Preliminary Exam Higher 2 Knowledge and Inquiry 9759/02 Paper 2 24 September 2019 2 hours Additional Materials: Answer Paper READ THESE INSTRUCTIONS FIRST Do not turn this sheet over until you are told to do so. Write your name and CT group on all the work you hand in. Write in dark blue or black ink. Do not use staples, paper clips, highlighters, glue or correction fluid/tape. Section A Answer Question 1. Section B Answer any two questions. At the end of the examination, fasten your answers to each section separately. You will be asked to submit your answer to each section separately. The number of marks is given in brackets [ ] at the end of each question or part question.

2 RI 2019 KI Y6 PRELIM P2 Section A You must answer question one. 1 Why on earth is mathematics beautiful? Does it matter? Looking at two contrasting solutions to a mathematical problem, I am struck by how different they are, but also by how I am instantly drawn to one over another. Not because one is explained better than the other (both are described well, in fact), but because one is shorter. The longer one doesn’t quite get to the heart of the matter; it is a bit cluttered with unnecessary distractions, although it does eventually answer the question. The other uses a different approach, which captures the essence of the ideas – it helps the reader to understand why this piece of mathematics works this way, not just that it does. For a mathematician, the “why” is critical, and the best mathematical solutions are the ones that answer the question simply and elegantly. It is no different from Science adopting the law of parsimony – that we choose the simpler solution of two competing hypotheses that make the same predictions. After all, all assumptions introduce possibilities for error; if an assumption does not improve the accuracy of a theory, its only effect is to increase the probability that the overall theory is wrong. If it matters to Science, then, it should, all the more, matter to Mathematics. Besides, there is something beautiful about a beautiful proof. Euler’s equation eiπ + 1 = 0, for example, demonstrates this very well. In addition to containing three of the basic arithmetic operations once each (addition, multiplication, exponentiation), it also knits together five fundamental mathematical constants: i) the number 0, ii) the number 1, iii) the number π (pi), which is used everywhere in Euclidean geometry, iv) the number e, the base of natural algorithms and a figure that is widely used in mathematical analysis, and v) the number i, the imaginary unit of complex numbers, which is crucial in providing insig ht into algebra and calculus. All that information, in such an elegant and powerful package. Some might argue that we should not concern ourselves with beauty in mathe