The positive integers, starting at 1, are grouped into sets containing integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.
(i) Write down the expressions, in terms of for
(a) the number of integers in the set,
(b) the first integer in the set,
(c) the last integer in the set.
(ii) Given that the integer occurs in the set, find the integer value of .
(iii) The sum of all the integers in the set is denoted by , and the sum of all the integers in all of the first sets is denoted by . Show that may be expressed as .
“An Inclusive Society, A Stronger Singapore” Budget Plan 2012-2016: fulfilled promises, financial burden and new directions
In 2012, the Singapore government introduced a 5-year budget plan aiming to make Singapore a fair, stronger and inclusive society. The main features of this ambitious project were:
Restructuring to sustain growth
Less dependency on foreign labour
More grants and support for SMEs
Building a fair and inclusive society
Improving the social mobility of lower income families through improved financial and educational subsidy and/or grant schemes
Helping seniors live long and well by offering housing purchase schemes, credit incentives for hiring aged workers, increased retirement savings and more affordable healthcare
Supporting those with disabilities and special needs through subsidies for employing caretakers, providing educational and skills support facilities and building elder-friendly infrastructure
Sharing the fruits of Singapore’s economic growth by giving out GST vouchers in cash, Medisave and U-save
(taken from http://www.singaporebudget.gov.sg/budget_2012/download/FY2012_Budget_in_Brief.pdf)
It is not difficult to understand why the government initiated such a plan. In the words of PM Lee “We all have something to contribute,” (channelnewsasia.com, 31 Oct 2015) hence maximising the potential of our human resource: young, old, poor, rich, abled and disabled, and keeping them healthy and happy is a strategic move to support Singapore’s future economic growth.
It is now 2017 and the government has indeed delivered most of what was promised in the budget plan. To highlight a few of these achievements, our neighbourhoods and transport systems are now more elder-friendly, we receive our GST cash vouchers as promised, we have two spanking-new hospitals in the West and are currently in the midst of integrating our healthcare groups to streamline operations and make healthcare more affordable, we built a $25million Enabling Village to help those with disabilities and special needs and we have decreased the local to foreign worker dependency ratio. However no success story is without sacrifice and just who or what were the sacrificial lambs in this budget plan? Moreover, which areas should the government focus on in the next one?
Tax payers are once again not spared from funding Singapore’s progressive aspirations and why should they? The money is redistributed and re-invested to make Singapore a fairer and stronger society. However tax rates have gone up again for YA2017 for the rich and upper middle classes, from a range of 17-20% to a range of 18-22% and with the addition of two new income tiers (www.iras.gov.sg). The question is, is this really fair? I heard that little “Yes!” go up in your head. Truth be told, it depends on how the rich make their money. For those who own successful local businesses, the new tax hikes are going to hurt and might even discourage further business growth or expansion. For those who make their riches through stocks and shares, guess what? These are not declared when filing income tax! Perhaps we should work towards minimising the number of those who profit from loopholes in the system if we truly wish to build a fairer – also towards the rich— and more inclusive society.
Finally, what new directions might we find in the AY2017 budget? The Committee on the Future Economy (CFE) of Singapore recently released a report highlighting key growth sectors and proposed seven strategies to sustain Singapore’s economy in the coming years.
Key growth sectors: finance, hub services, logistics, urban solutions, healthcare, the digital economy and advanced manufacturing
Deepen and diversify international connections
Acquire and utilise deep skills
Strengthen enterprise capabilities to innovate and scale up
Build strong digital capabilities
Develop a vibrant and connected city of opportunity
Develop and implement Industry Transformation Maps (ITMs)
Partner each other to enable innovation and growth
(taken from https://www.gov.sg/microsites/future-economy/the-cfe-report)
One can therefore expect more money in the AY2017 budget to be channelled into re-skilling the workforce, encouraging innovation and improving digital infrastructure and technology i.e. building a Smart Nation. Support for the elderly, disabled and those with special needs might take on a minor role in the budget this year because it was the focus of AY2012-16’s budget plan.
Nonetheless, there is so only so much money can do in helping a country to survive. As pointed out by PM Lee, Singaporeans need to practice active citizenship. This means making the effort to keep oneself fit and healthy, using the incentives and support systems in place to reskill or retrain oneself to meet with changing labour demands, revising educational content to ensure our students are prepared for the future by the time they graduate and managing ones expectations and demands in times of economic difficulty. Policies, systems and budget plans will change but the brunt of it can be softened by adopting the right attitude and mentality. Thus when the AY2017 budget comes out, hold your tongue for a moment, understand the changes and safety nets in place and learn to embrace them. If not, a corner of Hong Lim Park beckons you.
All solutions here are SUGGESTED. Casey will hold no liability for any errors. Comments are entirely personal opinions.
Question 21 is a flawed question. When unpolarised light goes through a polarizer, the I is halved while the A is reduced by a factor of root 2. But based on the information Cambridge provides, the answer is C.
Note to all: Casey will not respond to most of the comments as he is busy. You may contact him by SMS at +65 9474 5005 if you have a burning question.
Feel free to explain the answers, if you are confident. Many thanks.
This post is going out to all students that are taking their A levels GP 2016 this year: the listed topics are just some of the common questions that schools have set in this prelim. You may want to take note of it and prepare for these issues just in case. Being prepared beats being caught off guard right?
S&T with Artificial Intelligence/ robotics
S&T with eco-cities/green cities
Sports (take note it is usually cross-topical). It could be sports with arts; sports with S&T; Sports with gender; Sports with war etc.
Government collaboration especially in areas like war and terrorism; migrant issues and how does government interference infringes on privacy and rights of individuals
Arts and Culture (significance of museums/historical places etc).
Ageism + SG society
P.S Do take note that S&T questions are usually quite narrow; while arts and culture questions tend to be broad-based. Prepare accordingly to the question types and requirements! Good luck to all who are mugging now.
All solutions here are SUGGESTED. KS will hold no liability for any errors. Comments are entirely personal opinions.
As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks. As these workings and answers are rushed out asap, please pardon me for my mistakes and let me know if there is any typo. Many thanks.
I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare. I’ll try my best to attend to the questions as there is H2 Math Paper 2 coming up and I have a lot of prepare.
Side note: I think the paper is tedious, but definitely manageable. Hard ones, could have been q3, q7a, 10, 11ib. So if you did your tutorials and past year papers well, with proper time management and no careless, 70 is manageable. To get an A, you need to fight for that 30 marks which really test you on your comprehension skills and precisions. And these questions should distinct the students who deserve an A.
Comments/Explanations: Firstly, it is fine to give all answers in 3SF since the question did not mention anything about exact. I just happen to recognise such functions. Besides that, it can be easily done with the Graphic Calculator.
First, observe that turning point for is at and . This means that undergoes 3 transformations in the following order. Scale of factor parallel to y-axis, then a translation in units in the positive y-direction, followed by a translation in units in the positive x-direction.
Since we know the is simply after undergoing the above transformation, we can conclude that
. Since .
Coordinates of intersection
Coordinates of turning point
Comments/ Explanations: A rather tough question to be honest. Students must be able to relate the given and figure out what the question is asking. Students comprehension skills are put to test here.
Comments/ Explanations: Proving the first part is not that easy as students just got to just try and figure something out, rather than staring for link. Use the GC to solve the next part, but be careful to copy the correct . The last part tests the students’ understanding on summation of GP and using of .
Comments/ Explanations: Students could have evaluated the vector before doing the cross or dot product. Knowing that its a unit vector is a cue to students that the modulus is 1. This is also a fairly easy question
Since is true, and is true is true, by mathematical induction, is true for all positive integers n.
Comments/Explanations: This is a standard tutorial question. Students should know that they are not required to memorise summation of or , though they are handy here. Thus, being a bit more observant, they should realise they can easily use (i) to resolve the sum. There should be no need to simplify the answer in (iii) since it is only 3 marks.
Since coefficients of given equation are all real, is also a root, given that is a root.
By comparing coefficients,
(rejected since )
Comments/ Explanation: First part of (a) is manageable, while the long division can be tedious. Students can also use the quadratic formula (b) is quite easy so long as students know their basics of roots well.
This value is in both the domains of and so it should satisfy.
We have that is not . Thus, the inverse does not exist.
Comments/ Explanation: The first part is your typical tutorial question. Picking the correct is a bit hard though, so students need to give it a good thought. The second part, as point out by Mr. Wee WS, is similar to an Olympiad Question on functional equations. To prove that inverse does not exists, there are other ways too. So feel free to try.
Comments/Explanation: First part of the question is rather straightforward and students just need to be really meticulous. The next part tests students on their understand of a unit vector, or distance between planes. Students MUST leave the final answers in cartesian form. The last part is simply testing their knowledge of lines and planes. I will say that system of linear equations is tested in part 1 already.
I’m sure most of you are already done with your prelims or nearly done… In fact, some of you have already gotten back your GP papers. How did you fare? I’m sure most of your teachers would have told you that Prelim results are usually quite an accurate indication of your final A level grade (except maybe a grade jump or two if your school is pitching it at a high standard).
Whatever grade that you have received is a really a time and reality check for you. You have 6 weeks left to the final exams! How are you going to spend your time revising considering u are left with this time frame?
Talk to me if you need any help and advice. Most importantly, pls do not burn out or break down.
This is a special post going out to all of you. Considering that your A levels are drawing near and you are already in the midst of your prelims, what are some topics that you are still struggling with? What are some essays that you would like to see a post on?
The team will do their best to meet your requests! Just comment on the section below the post 🙂
Traditions are important because it provides people with a way to govern themselves in a world where moral standards are now often being compromised. Most traditions have moral and ethical messages embedded within them, and these serve as guidelines as to how people should govern themselves and, therefore, gain a degree of control over their lives. Before the invention of today’s modern laws, people used traditions as their guiding principles and while some traditions were abandoned, most of them are recycled and reused as the basis or foundations for some of the existing laws today. At times, traditions could have such a powerful and pervading influence, so much so that they become unspoken laws, laws of logic that the masses hold to. Examples of traditions that we hold dear would be the Chinese New Year festival, where it teaches the value of sharing and family unity and importance.
Traditions are also important to keep us rooted in our own self identity especially in this globalized world where the spread of Americanization serves to dilute our own culture and values. The unique practices that people still and continually adhere to, serve as a reminder of our origins and give us a glimpse into the lives of our ancestors. It is only through undergoing the activities that our ancestors have undergone, that we are able to understand our personal histories and be aware of the cultural identities that we each possess, particularly, in Singapore where it also serves as a binding force for a heterogeneous society, where a plethora of different cultures and traditions abound.
However, retaining a sense of tradition may not be important as it may lead to segregation and divide people especially in a multi-racial and multi-religious country such as Singapore. While having a sense of tradition unites people together, it only applies to people who belong to a particular group who share the same traditions. This inadvertently creates an “us” versus “them” kind of mind set within the different parties and it would serve to heighten the sense of suspicion and fear. Also, given that Singapore is increasingly globalized and greater influx of people are coming to this city to work and travel, it does not make sense to retain traditions. Traditions in this case no longer serve any purpose and they should be re-adapted to fit the contexts of the present day.
Finally, traditions may not even hold any role in today’s world where people’s lives are getting more and more fast-paced- simply because people do not have the time to uphold these traditions. For example, it has been observed that the annual Chinese Qing Ming festival is in the danger of being virtually extinct. The number of families that turn up on the day of Qing Ming has been decreasing most likely because of tight and busy working schedules which incidentally, has been cited as one of the common reasons by Singaporean Chinese. In a world where stress is placed on individuals to succeed and to attain a degree of sustainable income to support themselves, such traditions serve to hinder their progress and may make them lose out on the race, therefore, traditions may not always be important.
“Ever justifiable”: set a criteria for such a question. Possible criteria that you can set: 1) law implemented is unlawful/ unreasonable e.g. when the punishments outweigh the crime 2) breaking the law suffices as the last hope/action and it actually benefits society as a whole.
Law: A set of rules/legislations that follow a social contract theory.
Assumption: Other peaceful means of handling the problems are not viable, resulting in individuals having to break the law.
It may be counter productive and may result in the country breaking down into lawlessness.
It may invalidate the people efforts in breaking the law in the long run. The law exists for a reason. It exists to make people feel safe in the face of possibility of lawlessness.
It may subsequently open up a myriad of “valid reasons” for breaking the law, making other laws redundant in a sense.
Provided the cause is a noble one and breaking the law is used as the last resort and this action benefits society as a whole. One should also consider if there is a high chance of success of breaking law in order to preserve the rights of individuals.
P.S These discussion points are raised during class by students themselves. If you have any other additional points or to contest any of these points, just type them in the comment section.