Maths notes for 4038 and 4016 paper


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Maths notes for 4038 and 4016 paper

  1. 1. Notes for Additional and Elementary Mathematics and Supplementary Questions<br />
  2. 2. Dear user of this note,<br />This note is to supplement what you will receive in your 4 years of secondary school education. These are my experience of learning maths and my tips for you. I definitely tell you that by spotting ques is for you to make sure a pass in exam only. However, I do have to remind you if you have the ability to ace it, please do your best and not spotting ques. (Sounds like a paradox ah). I wish you all the best in your O levels after using this note. I hope it brushes up your fundamental skills of mathematics argumentations and presentation. Also a reminder, this paper is not the usual O level style as my ques are half difficult and half easy. You might not be able to answer all ques as o level is usually 70-80% easy and 20-30% difficult only.<br />Best wishes,<br />Creator of this note<br />Fabian<br />
  3. 3. AM (4038) 2009 O LvlStatistics<br />27.8% of AM paper is on differentiation and integration which mean that it will determine a quarter of your marks. You would therefore have to be very good in it. In addition, Trigo will decide another 18.4% of your marks. This means that you need this 3 chapter to almost pass your AM which I presume is easier for you guys to focus on.<br />TRIGO: 18.4%<br />DIFF & INT: 27.8%<br />EXP & LOG & MODULUS & POLYNOMIAL & DISCRIMINANT: 12.8%<br />BIONOMIAL: 5%<br />CIRCLE: 5.6%<br />
  4. 4. AM (4038) 2009 O LvlStatistics<br />LINEAR LAW: 5%<br />THIS 11/18 chapters give you 74.6 % of your AM paper hence these Chapters are what you should focus on when you have no time during an exam. Although it varies yearly, but I believe it will still give you a min of 62% even if it changes a lot. This will allow you to pass with more than what you have now. However, if you do have the time to finish studying all, please do so. Use Allen’s Tips although I don’t really trust what he said but I have to admit it is still 60% useful.<br />Usually the most commonly asked question by anyone would be how do I predict questions. This can be done by reviewing all the chapters that you have study. <br />After doing paper 1, those chapter that did not come out will come out for paper 2<br />
  5. 5. Other than trigo which will come out more than 1 ques, the rest of the chapters will be tested for1 ques.<br />1 ques on maxima and minima, 1 ques on area under graph, 1 ques on kinematics and 1 ques on linear law will definitely be tested.<br />Also, for quad and inequality, usually they will ask find alpha3-beta3 or alpha4-beta4<br />(Statistics of 4038 paper quoted from pass with distinction by topics)<br />AM (4038) 2009 O LvlStatistics<br />
  6. 6. Quad Func & Inequalities<br />If the roots of the quad Eqn 2x2-5x-1=0 are α and β, find the value of (β3+(1/ α3))( α3+(1/β3)).<br />A quad func y=f(x) attains its min value 11/3 at x=4/3 and y=5 when x=2. Find an expression, in terms of x, for y.<br />Show that, if x is real, the values of the expression (x2-3x+1)/(2x2-3x+2) can only lie btw -1 and 5/7 inclusive.<br />
  7. 7. Logarithm, Indices and Surds<br />Rationalise (4+2√5)/(1+√2+√5)<br />Find the Exact Value of x if 27x=(√(3√27))<br />Given that ((9n+2 – 3n+2)/25 )= 2a3b, where a and b are Z, find the value of a and express b in terms of n.<br />Exact form of (lnx)lnx=x<br />If a˃b˃1 and , find the value of <br /> without the use of calculator.<br />
  8. 8. Trigonometry(Factor Formula, Basic identities, Double angle, Integration, Sketch)<br />For all Trigonometry identities question, Start solving from the more complicated side to the less complicated side. If you are stucked halfway, please also try solving from the other side. Then, merge the answer as one by making one side writing the steps from the bottom.<br />A trigonometry proving may be a mixture of usually 1-2 type of trigonometry equations.<br />If they asked you to solve questions for eg, given that sinx=(3/5) and 180˂x˂270, give the value of cosx. This ques you have to draw triangle to ans. What I would suggest is that u don’t use the Cartesian plane method if u will confuse with the angle quadrant. <br />
  9. 9. Trigonometry(Factor Formula, Basic identities, Double angle, Integration, Sketch)<br />5<br />3<br />From this right angle triangle, using pythygoras theorem, we know that adjacent is 4. Cos a is therefore (4/5).second issue is whether there is or is no negative sign to the value. This is very easy and can be done by checking quadrant ASTC. The third quad is only tan positive hence the ans is –(4/5). The only hardest ques is when they ask for sin(x/2), cos(x/2) and tan(x/2). Firstly, use the double angle eqn. Then, sub in the value of the trigonometry function involving x only. Leave the (x/2) trigofunc aside as unknowns. Then, u may form quad eqn and solve. Check your quad to reject one of the answer(+ or -). The tips here is you can type your calculator but not presenting decimal in your answer script. Then, how would u know?? Rmb √2=1.41…, √3=1.71…., √5=2.23…., (1/√2)=0.707…., (√3/2)=0.866…..<br />x<br />
  10. 10. Trigonometry(Factor Formula, Basic identities, Double angle, Integration, Sketch)<br />Prove 2cosec22Ө-cosec2 Ө=-2cot2 Өcosec2 Ө<br />Prove cos 3 Ө +sin3 Ө=(1+2sin2 Ө)(cos Ө-sin Ө)<br />Given cos2 Ө=(1/9), calculate without a calculator, the values of sin Ө<br />Prove cos4 ӨcosӨ+sin Өsin2 Ө=cos3 Өcos2 Ө<br />Prove (2+sin2 Ө)cos Ө+(1+cos2 Ө)sin Ө=2(1+sin2 Ө)cos Ө<br />Find angles btw 00-3600 inclusive for 2 tan Ө=3sin Ө<br />Prove sin4 Ө-cos4 Ө=sin2 Ө-cos2 Ө<br />Prove cosec Ө+cot Ө=cot(Ө/2)<br />Prove sin2 Ө-tan Өcos2 Ө=tan Ө<br />Sketch 1+ from 0≤x≤π and state the range of y<br />Prove sin3 Ө=3sin Ө-4sin3 Ө and Evaluate , giving your answers in exact form.<br />Exact value of principal value of <br />Find the exact value of o<br />
  11. 11. Linear Law<br />Usually students would think linear law is a hard chapter as how the hell would I know I drawn the correct graph or not. Don’t worry, let me teach/ Rewind you the linear law “Cheating method”. And you will think linear law is an easy chapter to ace<br />Use the Eqn given to form your own eqn. How do you know your eqn is correct or not. Firstly, if the axes are given to you, compare with your answer and make sure you have those axes. For eg, the question say a graph of lgx against x2, then your Y axis is lgx and X axis is x2. Also, you must make sure your eqn is in a visible form of Y=mX+C, where your Y cannot have a constant and can only have variable x and y, for eg, lny, ex. Your X must come with a constant which is your m and can contain variable x and y. For eg, xlga, my, lnalnx. Your C can only have constant and no x and y. For eg, lga, lgb, g, q, y, lna, ea.<br />Using your eqn and the table of value given to you as x and y, change the value into the value for the axis. If eqn is lnx=alny+lnb, than your X axis value is lny which Y axis is lnx and just press Calculator.<br />After pressing the calculator, you will have a set of value. Now, don’t plot and perform cheating method on a piece of paper not to be submitted like question paper. Just write values approximately here to save time. Firstly, sub the first set of value into the eqn and make C the subject. For eg, when lny=2 when lnx=1 then put it as 1=a2+lnb and change it to lnb=2a-1. Then sub in 2nd set of value and make C the subject then solve the 2 eqn simultaneously. You will be able to get your grad and y-intercept. However, since y-inter is lnb, you have to know what is b by applying elnb. This is for cases with ln and lg.<br />
  12. 12. 4. Now, you plot all the point on the graph paper and start drawing the line for the y-inter value you had fine. However, don’t plot it as a point. Use the ruler and y-inter to look for line of best fit where most or all points lie on the line. Remember to label your line and write the scale use no matter it is stated or not in the ques.<br />5. Then, the ques will ask you to find the unknown a and b usually. Then you say from the graph, Y-inter=…therefore lnb=…, and for grad remember your big triangle. Then, check your value found previously and with your graph. The value must be within ±0.2 only. If not, redraw.<br />Linear Law<br />
  13. 13. Modulus<br />Always sketch a modulus graph inside outwards. This means that sketch the graph from the most basic eqn to the modulus or any other requirements. Range are the range of value of y while the range os x value are known as domain.<br />Sketch the graph of y=|2+|2+|x2+4x+3||| from -5˂x˂4. State the range of the graph. Using a additional str line, solve the eqn2-x˃|2+|2+|x2+4x+3|||.<br />Sketch the graph of y=2+|2-x| from -5˂x˂4. State the range of the graph. Using a additional str line, solve the eqn 2-x˃2+|2-x|<br />
  14. 14. EM (4016) O Level Statistics<br />Usually the most commonly asked question by anyone would be how do I predict questions. This can be done by reviewing all the chapters that you have study. <br />Paper 2 will definitely have 1 Ques of Graph Plotting with Finding of grad. The exact value of Gradient can be know by differentiating the eqn and sub the x value of that point to find as learnt in AM(4038). Then, by using the drawing method(legal method), compare both answer.If it differs by more than 0.2, you need to redraw the grad line. <br />Also, paper 2 will have 1 whole ques of word problem that will lead you to a ques that says, for eg, reduce the eqn to………., and find……. If it say round off to 2sf , it means you have to use general formula to find your answer. If not stated, it is most likely solved by factorise. This type of ques is a give away hence it is necessary to score full marks. <br />What people usually get wrong is due to errors in sign which can be prevented with the use of bracket and reading the ques well. <br />After doing paper 1, those chapter that did not come out will come out for paper 2 with these 2 ques.<br />
  15. 15. Matrix ques will asked you to explain what a particular matrix means which u have to use proper sentences to explain each row by column.<br />Other than foundation chapters which would have more than 1 ques, the other chapters taught to you will most likely be tested in 1 ques only.<br />The hardest ques for bearing would be the ambiguous case for sin angle as the value of sin 0 to 90 = sin 180 to 91( the inverse of angle)<br />Mensuration would most likely test on pyramid, sphere, frustum and it will likely be linked to congruency and similarity<br />(Analysed by Fabian from 2009-2011 using PRSS Papers and 4016 examinations)<br />EM (4016) O Level Statistics<br />
  16. 16. Word prob<br /> <br />1. The diagram shows the design of a company symbol. <br />It Consist of 3 circles.<br />The smallest circle has center O and rad 2x cm.<br />The largest circle has center O and rad 2y cm.<br />The third circle touches both the other 2 circle as shown.<br />The 3 region forms are coloured Green, yellow and blue as shown.<br />(a)Explain why the rad of third circle is (x+y)cm<br />(b)Write down, in terms of π,x and y, expression for the area of the region that is coloured<br />(i)yellow<br />(ii)blue<br />(c)The area of the blue region is twice the area of yellow region. Write down an eqn involving x and y and simplify toY2-6xy+5x2=0<br />(d) (i) Factorise Y2-6xy+5x2<br /> (ii) Solve Y2-6xy+5x2=0, express y in terms of x<br />(e)Calculate the fraction of the design that is yellow<br />(f)Find the probability that 1 of 3 arrows will land on the blue region.<br />(g)Find the ratio of blue:Green:yellow<br />.<br />
  17. 17. Word prob are never hard as ppl will juz be confuse if they made the wrong eqn since the start or the reduce to…..ques. just remember this analogy: If your numerator is amount of rice and denominator is the no of people, if your people is smaller there will be less people and more food is resulted. Then the reversed also. This is just like an eqn if the denominator is smaller, the answer is larger. Even if you cannot answer some parts of the ques, there must at least be 1 part you can do.(solve the eqnques)<br />Word prob<br />
  18. 18. LCM AND HCF<br />There is a tip for this.<br />Three lights flash every 12 minutes, 18 minutes and 28 minutes respectively. Given that they first flashed together at 9.36 am, at what time will they flash together for the third time?<br />12=22X3<br />18=32X2<br />28=22X7<br /> <br />LCM=22X32X7. Why??<br />This is because you have to remember to calculate LCM is to take all the possible base of the no give and choosing the highest power of each base from the no given to you. You can see the comparison at the top.<br />HCF=2<br />This is because it is common between all the no. This ques are tricky as you have to consider the units for the time given for every ques.<br /> <br />
  19. 19. LCM AND HCF<br />If ques say to find the smallest no of k if 12k is a perfect cube /square no?<br />For cube, the idea is to make all the power of all bases to be multiple of 3 with the shortest and smallest no.<br />K=2X32<br />For square, the idea is to make all the power of all bases to be multiple of 2 with the shortest and smallest no.<br />K=3<br />If ques ask the LCM of 12 and K is 84 find the smallest K,<br />84=22X7x3 then K is 7<br />Why??<br />This is because you have to minus off the stuff that 84 and 12 have in common. And the remaining is the ans. We know the method to select LCM but what is they have same base with diff power.?? We only choose the one with highest base.<br />If ques ask the LCM of 12 and K is 24 find the smallest K,<br />24=23X3 then K is 23. This is because the only possibility that why the LCM have 23 is because the other number have.<br />
  20. 20. Thank You<br />