3U HSC Questions by Topic 1990-2006 210507

HOW I CAN KICK BUTT IN THE HSC IN 3U MATHS!!! By : Khiet Hoang – World’s craziest teacher

WARNING This book may be considered by some as offensive, insensitive and politically incorrect ……..and if you do then go and tell someone who cares! ……..uhh and it also contains lots of speling and grematical erors……and the odd maths error! Footnote: (should really be at the bottom of the page but who cares…….) The author of this book has about 500808 different split personalities so sometimes he (she/it/they/we) uses the first person (often “I”) to refer to himself. Other times he uses third person such as “the author”. I believe he uses perpendicular pronouns too…..so if you are confused already before you even read the rest of this book then he has succeeded! I can tell you that one of the author’s favourite sayings to his students is “are you scarred for life yet?” and then laugh very loudly in an infectious manner! You’d have to be one of my students to understand the joke! The man is crazy!!!! Acknowledgements: I would like to thank my Mum, my Dad for bringing me into this world, my wife, my kids, my dog…… No seriously, I’d like to thank all of my students and ex-students at Kellyville High and at the various coaching institutions in Sydney. By teaching you I have learnt a great deal about how the NSW 3U syllabus should be taught. Thank you for putting up with me and my eccentricities. I’d also like to thank my colleagues and friends who have read this and gave feed back and support. This humble grass hopper owes you. Most of all I must thank my wife, Vanessa for being so understanding about me writing this during the wee hours of the mornings in my “spare” time……….

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Content Page

Why I wrote this book

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4

Formulas 2U

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7

Formulas 3U

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8

3U course overview

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9

3U syllabus content

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10

Most Important Procedure for HSC

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18

Chapter 1 - Other inequalities (1.4 E) – Prelim 3U

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19

Chapter 2 - Circle geometry (2.6–2.10) – Prelim 3U

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21

Chapter 3 - Further trigonometry (5.6–5.9) – Prelim 3U

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32

Chapter 4 - Angles between two lines (6.6) – Prelim 3U

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45

Chapter 5 - division of lines into given ratios (6.7 E) – Prelim 3U

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47

Chapter 6 - Parametric representation (9.6) – Prelim 3U

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49

Chapter 7 - Permutations and combinations (18.1) – Prelim 3U

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57

Chapter 8 - Polynomials (16.1–16.3) – Prelim 3U

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63

Chapter 9 - Methods of integration (11.5) – HSC 3U

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67

Chapter 10 - Primitive of sin2x and cos2x (13.6 E) – HSC 3U

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74

Chapter 11 - Equation dN/dt

= k(N-P) (14.2 E) – HSC 3U

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75

Chapter 12 – Vel and Acln as a fn of x (14.3 E) – HSC 3U

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79

Chapter 13 - Projectile motion (14.3 E) – HSC 3U

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84

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Chapter 14 - Simple harmonic motion (14.4) – HSC 3U

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99

Chapter 15 - Inverse functions

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109

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118

Chapter 17 - Binomial theorem Coeffs and terms(17.1–17.3) – HSC 3U -------------

121

Chapter 18 - Further probability (18.2) – HSC 3U

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122

Chapter 19 - Binom thrm Max/Min term/coeff HSC 3U

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129

Chapter 20 - Roots of a poly – Newton’s Method (16.4) – HSC 3U-----------------------

133

Chapter 21 - Roots of a poly – halving interval (16.4) – HSC 3U

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136

Chapter 22 – Harder 2U series and sequence and algebra HSC 3U---------------------

137

Chapter 23 - Harder 2U area and volume HSC 3U

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139

Chapter 24 – Harder 2U - motion and rates of change HSC 3U

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144

Chapter 25 - Harder 2U - Derivatives HSC 3U

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155

Chapter 26 - Harder 2U – Curve sketching HSC 3U

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158

Chapter 27 - Harder 2U- Maxima and Minima HSC 3U

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161

(15.1–15.5) – HSC 3U

Chapter 16 - Induction (7.4) – HSC 3U

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Why I wrote this book. This book was compiled by an experienced teacher who has taught this course from as early as 1989. He was NOT a terribly good student at 2U/3U/4U maths (….who also refuses to recognise the new course names as the names are the only things that changed. The syllabus content has NOT) and had to teach himself these courses again during 1990-1991 using this book below. This book was written with the 1982 syllabus in mind and is only good until 2010 because the syllabus is due to be changed then and who knows what will become of the new syllabus. If there are only cosmetic changes then this book will still be useful. Else, if you’re ever short on toilet paper then………. This book explains the whole 3U course in easy to understand language and it is intended to be used as a self help guide for students as well as teachers. 90% of 2U/3U and especially 4U is about following certain procedures to obtain the answers. This book is just that. You can easily get over 90% for 2, 3 and 4U maths just by following the procedures in this book alone. Students often find 3U/4U (and even 2U) hard. This is the case NOT because the content is hard. I can teach my 10 year old son (in 2007) some 4U maths concepts! Will he be able to remember it is another story?!!! Will he want to learn?!!!! Maths IS hard though for the following reasons: 1)

There is TOO MUCH to REMEMBER. From the formula perspective alone, can you REMEMBER about 15 pages of typed up formulas, let alone know which one to use? If you cannot remember, then do General maths because 90% of the formulas are given to you. Somewhere in this book you’ll find all the formulas you’ll need to know.

2)

People fear and RESIST any CHANGE in their lives ………..and hence they do NOT REMEMBER new definitions or concepts given to them. When a new theory or formula is given to students the first thing they often ask is why? Or say “That’s too hard. How am I going to remember that” (cast your mind back to the quadratic formula…..Oh sorry you don’t remember that one? ………my point exactly. For those who do remember, that’s the one with a, b and c in it. How about the distance formula from a point to a line……..what you didn’t know there was such a formula? ......... Uhh yeah there is and that one is not even 3U maths. It is 2U maths !!!!!).

3)

Students DON’T UNDERSTAND the CONCEPTS ……..and hence do NOT REMEMBER them. Unfortunately maths is often typically taught by teachers putting a few QUESTIONS on the board, showing how the QUESTIONS ARE TO BE SOLVED. They then ask the students to do a million (or thereabout) questions for homework which the students, of course, find difficult to

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do because they didn’t understand the CONCEPTS of the topic. Then the teacher puts some more questions on the board the next day, again showing the students how to solve those questions and again ask the students to do another million (or more) questions for homework ………..and the vicious cycle goes on…………meanwhile the students fall behind. And the text book also “teaches” the same way. They show how two or three questions are done and hardly show the concepts behind it, or if they do it is not explicitly shown. One of the text books I’ve seen shows a couple of questions with solutions in one chapter and then the concepts are shown in the NEXT chapter. I think the rationale there was to show “Oh do you see how hard it is for you to do the question this way but with this formula it is so much easier!!!” Well, why wasn’t it shown in the same chapter at the beginning of the chapter in the first place? 4)

Students PONDER upon the beauty of THE QUESTION. This is the biggest problem of all. This happens in not only maths but in all subjects. This is the main reason why HSC students do not do as well as they should in the HSC. What is the problem? Simply put, students, when sitting an exam, spend far too much time on questions they CAN’T do and very little or no time on questions they CAN do and hence run out of time. Many of my students often say “Oh, I didn’t have time to do the last question and it was the easiest question of the exam!!!” My response is always: “Well then why didn’t you do that question first?” which is often met with another of the student’s question: “Oh can you do that?” “Uhh……yeah you can….” I often say to my 4U maths students that you can suck at 4U maths but the idea is to make others suck at it worse than you. That is unfortunately how the UAI works. It simply ranks you in order from highest to lowest. The aim of any HSC exam is NOT to finish the paper but to finish AS MUCH AS YOU CAN. So if you ponder upon the beauty of the questions that you could not do and “accidentally” ignore questions that you can do, than you will suck worse then your HSC peers. The rule of thumb is if you read the question and you do not understand or have a mental blank you count to 3 and if you still do not know then go onto the next question. Don’t sit there and ponder upon its beauty. That is the real secret behind the HSC!!! There is an actual procedure on how to attack an HSC paper also somewhere in this book…… I’ll give you two actual real life everyday situations and one of my personal experiences where people ponder upon the beauty of the questions: Situation 1 2 DAY FM (and other radio stations) often has a contest where they ask the contestants 10 questions in a minute. You answer all 10 correctly you get $1000 otherwise you get $10 (or $100 depending on the radio station) per question. If the contestant does not know the answer they are allowed to say “pass” and the next question is asked. All questions are always asked until they answer all the questions correctly or run out of time.

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Not many contestants get $1000!!! Most often get $50 or even $60. A lot get $30 - $40. Some even get $10 or nothing!!! This is NOT because they do not know a lot of the answers. It’s simply because they ponder upon the beauty of one or two questions and before you know it, the minute is up. Instead of saying “pass” STRAIGHT AWAY when they do not know the answer, typical contestants say “Uhh can you read the question again?” or “Uhhh…… I read that in Cosmopolitan the other day…..what was his name again?......Uhh, what was the question again?” Of course I am exaggerating here a little but you get the idea. Situation 2 Students often claim that they study 3-4 hours some say even 6 hours a day after school and still their marks do not reflect their efforts!!! Why is this so? It’s because they ponder upon the beauty of the questions!!! This happens for ALL subjects not just for maths. When students come across a difficult question or concept or theme or theory or a certain place in their essay etc they often sit and ponder sometimes for hours (sometimes on just one question - I know because I myself have done it!!!!) because if they leave the question then it means in their mind that they have failed and so they must do that particular question before they can do the next one. It’s quite a funny psychological thing! The right thing to do is of course to move onto something else or ask your teacher or friend or tutor etc the next day so that you can carry on your study in a productive and effective manner. If you ponder upon the beauty of one or two questions then you will never finish the million or so question for maths that were set for homework. Remember the teacher asked you to do the million questions but they did not say that you have to do them in any order. That means that you finish as many as you can in the time set. This approach of “skipping” questions should be done not just for homework but for class work and particularly for exams (especially HSC exams). Make sure you come back to the ones you skipped and attempt them when you have done all the others that you can do. One of my own personal experiences. In writing this book the author pondered upon the beauty of this section alone ……..sitting there thinking and thinking some more and so far for the last 2 pages spent 3 hours pondering what to write. What he should have done is he should have stood up. Go and make a cup of coffee and write it the next day……But no, he must finish this section otherwise he cannot write the next section……..that funny psychological bs…remember? ………Sorry got to go….got other things I must write for this book…..the pages of formulas for example which is much easier to write then the pages above….... I think I made my point………..NEVER

PONDER UPON THE BEAUTY OF

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Most Important Procedure for HSC. Out of all of the procedures in this book this is by far the most important one. Here it is: 0) Take a few (10) deep breaths so you can relax and also get some brain food (oxygen) into the brain. 1) Read the whole paper in 2 minutes (you WILL be given at least 5 minutes reading time in the HSC regardless of how crappy your school is in NOT giving you reading time in the trials or in any test. I give reading time for EVERY single exam I give to my students from year 7 to year 12 even if it’s just for a minute or 10 seconds – for my one minute exam!). Of course you are not to read word for word question for question in the 2 minutes (you just don’t have time for that unless you are a very good speed reader). Instead you glance at the whole paper to see what you can do and plan to do those first. Trust me, this is the most productive 2 minutes of the exam. 2) Write down the formulas next to each question when you are asked to start writing since your mind is still fresh at the start of the exam. There’s nothing worse then you not remembering the formulas mid-way through the exam even though you know the topic. And don’t stress if you don’t remember at the start. You’ll dig it out later out of that brain of yours during the exam. The idea is not to stress your self but to remain calm…easier said then done. Remember it’s just a STUPID test (all tests are stupid believe me and you’ll unfortunately go through a lot of tests in your lifetime). It does not change the fact that you are a wonderful person! (pay me later for that one later!!!) 3) Do the exam in the order that you can skipping the ones that you can’t. If question 1 is the hardest question for you then skip it straight away and go to question 2. If question 2 is also not obvious to you then go to Q3. I often deliberately put question 1 as the hardest question in my exams to train my students to skip. Remember if you have read the question and for some reason it’s not obvious to you then don’t sit there and ponder upon its beauty. Go onto the next question straight away. Don’t be afraid to do this because even though you consciously leave the question your mind subconsciously works at the solution to try and find the answer. Your mind is a wonderful machine for that very reason. And I can prove it to you. All of us have problems of our own. But how often do we CONSCIOUSLY think about our problems during the course of the day. We still carry on with our busy lives but our brain continues to work on the problem and then somehow the solution presents itself!!! Just make sure though that you make a marking on the questions that you skipped so you can come back to do them later. 4) When you get to the end of the paper go back to the beginning of the exam and repeat steps 2 to 4 until you finish every question in the exam or when time is up, whichever is sooner. 5) Check the exam for silly mistakes or check whether you have answered what they asked for. They may have asked you to factorise for example and you gave them the roots instead! If you do finish the whole paper then don’t just sit there and show off “oh look at me I’m a smart ar…I finished the paper first before yous all!!!” Remember the idea of any HSC paper is NOT to finish the whole paper but to FINISH CORRECTLY AS MUCH AS YOU CAN.

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Formulas 2U

Refer to separate handout

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Formulas 3U

Refer to separate handout

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Syllabus content:

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3U Syllabus Preliminary Course Other inequalities (1.4 E) – Prelim 3U

 

Quadratic inequalities Fraction inequalities

Circle geometry (2.6–2.10) – Prelim 3U

Further trigonometry (sums and diffs, t formulae, IDs and eqs) (5.6–5.9) – Prelim 3U

Angles between two lines (6.6) – Prelim 3U Internal and external division of lines into given ratios (6.7 E) – Prelim 3U Parametric representation (9.6) – Prelim 3U

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Permutations and combinations (18.1) – Prelim 3U

Polynomials (16.1–16.3) – Prelim 3U

Harder applications of the Preliminary 2 Unit course – Prelim 3U

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HSC Course Methods of integration (11.5) – HSC 3U

Substitution procedure: Steps: 1 2 3 4 5

find du/dx change the limits if any get rid of x and dx and anything else along the way and substitute into the integrand.(the thing you’re trying to integrate integrate answer the question (if indefinite then change the variables back to x terms)

Normal Integration: Rearrange the integrals into the form of one of the 11 on the standard integral table. Area under curve and x-axis A=

 ydx

Area bounded by the curve and y-axis A=

 xdy

Area between two curves A =  topcurve  bottomcurve Volume around x-axis 2 V =   y dx

Volume around y-axis 2 V =   x dy

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Primitive of sin2x and cos2x (13.6 E) – HSC 3U sin 2 x 

1 1  cos 2 x 2 2

sin 2 x 

1 1  cos 2 x 2 2

because

because

cos 2 x  1  2 sin 2 x

cos 2 x  2 cos 2 x  1

Equation dN/dt = k(N-P) (14.2 E) – HSC 3U

dN  k ( N  P) dt

(1)

A solution to (1) is: N  P  Ae kt

(2)

Two parts to growth and decay: 1)

prove that (2) is a solution to (1) by differentiation wrt t and the fact that N  P  Ae kt and therefore Ae kt  N  P and kAe kt  k ( N  P )

2)

solve numerical questions

Velocity and acceleration as a function of x (14.3 E) – HSC 3U

x = displacement = distance from origin x = 0 dx

= velocity = dt  v use this to integrate if v is given in terms of t = acceleration =

dv d 2 x d 1 2 dv  2  ( v )v dt dt dx 2 dx

use this to integrate

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horizontal motion

vertical motion

=0

= Vcos

= -g 

x=

y=

Simple harmonic motion (14.4) – HSC 3U

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Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U

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Induction (7.4) – HSC 3U In the beginning there was a statement! Eg: “Mr, Hoang is gay!” This statement would then have to be proved by the process of induction. So what is induction then? It’s just s number of steps to follow: Step 1 Prove statement is true for FIRST TERM (normally for n = 1) . A lot of students and even teachers normally write “prove n =1” for this step which of course is wrong! …..and it has a conclusion at the end of the step: “therefore the statement is true for the first term (normally n =1)” Step 2 Part a) ASSUME statement is true for some term n = k So substitute n = k into the statement and label that as equation (1) Part b) PROVE that statement is true for the NEXT TERM (normally but NOT always for n = k + 1) This is the hardest part of induction where you prove the statement with n = k + 1 (or what ever the next term happens to be) substituted into the statement. This steps requires to be a master at algebra but you can skip this step if you don’t know or have a mental blank and still get marks for the other steps. Just don’t sit there and ponder upon its beauty! Step 3 Conclude with a standard line: “From step 1 the statement is true for the first term n = 1 (normally). It follows from repeated applications of step 2 that the statement is true for the NEXT TERM n = 2,3,4,….. Hence statement is true for all positive integer n. (replace the underlined with the initial condition of n that goes with the statement.

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Binomial theorem (17.1–17.3) – HSC 3U

Further probability (18.2) – HSC 3U

Iterative methods for numerical estimation of the roots of a poly eqn (16.4) – HSC 3U  Newton’s method  Halving the interval method Harder applications of HSC 2 unit topics, including 10.5 E, 13.4 E, 14.1 E– HSC 3U

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Most Important Procedure for HSC. Out of all of the procedures in this book this is by far the most important one. Here it is: 0) Take a few (10) deep breathes so you can relax and also get some brain food (oxygen) into the brain. 1) Read the whole paper in 2 minutes (you WILL be given at least 5 minutes reading time in the HSC regardless of how crappy your school is in NOT giving you reading time in the trials or in any test. I give reading time for EVERY single exam I give to my students from year 7 to year 12 even if it’s just for a minute or 10 seconds – for my one minute exam!). Of course you are not to read word for word question for question in the 2 minutes (you just don’t have time for that unless you are a very good speed reader). Instead you glance at the whole paper to see what you can do and plan to do those first. Trust me this is the most productive 2 minutes of the exam. 2) Write down the formulas next to each question when you are asked to start writing since your mind is still fresh at the start of the exam. There’s nothing worse then you not remembering the formulas mid-way through the exam even though you know the topic. And don’t stress if you don’t remember at the start. You’ll dig it out later out of that brain of yours during the exam. The idea is not to stress your self but to remain calm…easier said then done. Remember it’s just a STUPID test (all tests are stupid believe me and you’ll unfortunately go through a lot of tests in your lifetime). It does not change the fact that you are a wonderful person! (pay me later for that one later!!!) 3) Do the exam in the order that you can skipping the ones that you can’t. If question 1 is the hardest question for you then skip it straight away and go to question 2. If question 2 is also not obvious to you then go to Q3. I often deliberately put question 1 as the hardest question in my exams to train my students to skip. Remember if you have read the question and for some reason it’s not obvious to you then don’t sit there and ponder upon its beauty. Go onto the next question straight away. Don’t be afraid to do this because even though you consciously leave the question your mind subconsciously works at the solution to try and find the answer. Your mind is a wonderful machine for that very reason. And I can prove it to you. All of us have problems of our own. But how often do we CONSCIOUSLY think about our problems during the course of the day. We still carry on with our busy lives but our brain continues to work on the problem and then somehow the solution presents itself!!! Just make sure though that you make a marking on the questions that you skipped so you can come back to do them later. 4) When you get to the end of the paper go back to the beginning of the exam and repeat steps 2 to 4 until you finish every question in the exam or when time is up, whichever is sooner. 5) Check the exam for silly mistakes or check whether you have answered what they asked for. They may have asked you to factorise for example and you gave them the roots instead! If you do finish the whole paper then don’t just sit there and show off “oh look at me I’m a smart ar…I finished the paper first before yous all!!!” Remember the idea of any HSC paper is NOT to finish the whole paper but to FINISH AS MUCH AS YOU CAN.

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Other inequalities (1.4 E) – Prelim 3U year Question 1990

marks 2

comment Q1 /Q1

1991

Q1 /Q2

1992

Q1 /Q3

1993

Q1 /Q4

1994 1995

Q1 /Q5

1996

3

1997 1998

Q1 /Q6

Q7 /Q7

1999 2000

3

Q1 /Q8

2001

3

Q4 /Q9

2002 2003

3

Q1/ Q10

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Other inequalities (1.4 E) – Prelim 3U 2004

2

Q1/ Q11

3 2005

2006 2007 2008 2009 2010

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New syllabus expected

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Q1 /Q12

Circle geometry (2.6–2.10) – Prelim 3U year Question 1990

marks

1991

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comment Q3 /Q13

Q2 /Q14

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Circle geometry (2.6–2.10) – Prelim 3U 1992

Q3 /Q15

1993

Q4 /Q16

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Circle geometry (2.6–2.10) – Prelim 3U 1994

Q2 /Q17

1995

Q6 /Q18

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Circle geometry (2.6–2.10) – Prelim 3U 1996

Q2 /Q19

1997

Q2 /Q20

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Circle geometry (2.6–2.10) – Prelim 3U 1998

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Q4 /Q21

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Circle geometry (2.6–2.10) – Prelim 3U 1999

Q3 /Q22

2000

Q5 /Q23

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Circle geometry (2.6–2.10) – Prelim 3U 2001

Q3 /Q24

2002

Q2 /Q25

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Circle geometry (2.6–2.10) – Prelim 3U 2003

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Q4 /Q26

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Circle geometry (2.6–2.10) – Prelim 3U 2004

Q2 /Q27

Q6 /Q28

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Circle geometry (2.6–2.10) – Prelim 3U 2005

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Q5 /Q29

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Circle geometry (2.6–2.10) – Prelim 3U 2006

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Q3 /Q30

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U year Question 1990

marks 2

comment Q2 /Q31a

2 Q31b 1991

Q6 / Q32

1992

Q2 /Q33

Q3 /Q34 3D

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1993

Q2 /Q35

Q4 /Q36

1994

Q1 /Q37

Q5 /Q38

Q7 /Q39 3D

1995

1

Q1 /Q40

1996

3

Q4 /Q41

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1996

Q4 /Q42

3D TRIG

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1997

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Q3 /Q43

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1997

Q4 /Q44

1998

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Q1/Q45

Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1998

Q2/Q46

1999

Q2/Q47

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 1999

Q6/Q48

3D TRIG

2000

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Q2/Q49

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2000

Q3/Q50

3D TRIG

2001

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2001

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2002

2003

1

Q1/Q53

2

Q2/Q54

2

Q1/Q55

Q2/Q56

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Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2003

Q7/Q57 3D TRIG

2004

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- 43 -

K Hoang 2007

Q2/Q58

Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2004

Q3/Q59 3D TRIG

2005

325618808.doc

Q4/Q60

- 44 -

K Hoang 2007

Further trigonometry (sums and diffs, t formulae, IDs and eqs, 3D trig) (5.6–5.9) – Prelim 3U 2006

2

2

2007 2008 2009 2010

325618808.doc

New syllabus expected

- 45 -

K Hoang 2007

Q1/Q61

Angles between two lines (6.6) – Prelim 3U year 1990

Question

marks

comment Q1/Q62

1 3 1991 1992

Q1/Q63

1993 1994

Q1/Q64

325618808.doc

- 46 -

K Hoang 2007

1995 1996

Angles between two lines (6.6) – Prelim 3U Q1/Q65

1997 1998

2

Q1/Q66

1999 2000

2

Q1/Q67

2001 2002 2003 2004

2005 2006 2007 2008 2009 2010

325618808.doc

Q1/Q68

New syllabus expected

- 47 -

K Hoang 2007

Internal and external division of lines into given ratios (6.7 E) – Prelim 3U year 1990 1991

Question

marks

comment Q1/Q69

1992 1993 1994

Q1/Q70

1995 1996

Q1/Q71

1997 1998 1999

Q1/Q72

2000 2001

Q1/Q73

2002

Q1/Q74

325618808.doc

- 48 -

K Hoang 2007

2003 2004

Internal and external division of lines into given ratios (6.7 E) – Prelim 3U Q1/Q75

2005

2006 2007 2008 2009 2010

325618808.doc

Q1/Q76

New syllabus expected

- 49 -

K Hoang 2007

Parametric representation (9.6) – Prelim 3U year 1990

Question

marks

1991

325618808.doc

comment Q5/Q77

Q5/Q78

- 50 -

K Hoang 2007

Parametric representation (9.6) – Prelim 3U 1992

325618808.doc

Q5/Q79

- 51 -

K Hoang 2007

Parametric representation (9.6) – Prelim 3U 1993

Q7/Q80

1994

Q3/Q81

325618808.doc

- 52 -

K Hoang 2007

Parametric representation (9.6) – Prelim 3U 1995

325618808.doc

Q3/82

- 53 -

K Hoang 2007

1996 1997 1998 1999

325618808.doc

Parametric representation (9.6) – Prelim 3U Q4/83

- 54 -

K Hoang 2007

2000 2001

Parametric representation (9.6) – Prelim 3U Q6/84

2002

325618808.doc

2

- 55 -

K Hoang 2007

Q1/85

Parametric representation (9.6) – Prelim 3U 2003

Q1/86

2004

Q4/97

2005

Q4/88

325618808.doc

- 56 -

K Hoang 2007

Parametric representation (9.6) – Prelim 3U 2006

2007 2010

325618808.doc

Q2/89

New syllabus expected

- 57 -

K Hoang 2007

Permutations and combinations (18.1) – Prelim 3U year 1990

Question

marks

comment Q4/90

1991

Q4/91

1992

Q6/92

1993

Q1/93

325618808.doc

- 58 -

K Hoang 2007

Permutations and combinations (18.1) – Prelim 3U 1994

This should be a series and sequence question. NOT permutation and combination

325618808.doc

- 59 -

K Hoang 2007

Q6/94

Permutations and combinations (18.1) – Prelim 3U 1993

Q4/95

1994

Q7/96

325618808.doc

- 60 -

K Hoang 2007

Permutations and combinations (18.1) – Prelim 3U 1995

Q3/97

1996

Q1/98

1997 1998 1999

Q2/99

325618808.doc

- 61 -

K Hoang 2007

Permutations and combinations (18.1) – Prelim 3U 2000

Q2/100

Q6/101

2001

Q2/102

2002

Q3/103

2003

Q3/104

Q4/105

325618808.doc

- 62 -

K Hoang 2007

Permutations and combinations (18.1) – Prelim 3U 2004

Q2/106

2005 2006

Q3/107

2007 2008 2009 2010

325618808.doc

New syllabus expected

- 63 -

K Hoang 2007

Polynomials (16.1–16.3) – Prelim 3U year 1990

Question

marks 2

comment Q2/108

1991

Q1/109

1992

Q6/110

1993

Q2/111

Q3/112a

325618808.doc

- 64 -

K Hoang 2007

Polynomials (16.1–16.3) – Prelim 3U 1994

Q4/112b

1995

Q2/112c

1996

112j

1997

Q2/112d

1998

Q1/112e

Q2/112f 1999

325618808.doc

Q1/112h

- 65 -

K Hoang 2007

Polynomials (16.1–16.3) – Prelim 3U 2000

2

Q1/113

Q4/114

2001

2

Q1/115

2002

2

Q2/116

Q4/117

2003

325618808.doc

Q4/118

- 66 -

K Hoang 2007

Polynomials (16.1–16.3) – Prelim 3U 2004

Q3/119

2005 2006

Q4/120

2007 2008 2009 2010

325618808.doc

New syllabus expected

- 67 -

K Hoang 2007

HSC Course Methods of integration (11.5) – HSC 3U year Question 1990

marks comment 2 Q1/121a

2

Q121b

1991

Q1/122

1992

Q1/123

Q4/124

325618808.doc

- 68 -

K Hoang 2007

Methods of integration (11.5) – HSC 3U 1993

Q1/125

1994

Q1/126

Q3/127 1995

Q1/128 2

1996

325618808.doc

3

- 69 -

K Hoang 2007

Q1/129

Methods of integration (11.5) – HSC 3U 1996

Q2/130

1997

Q1/131 3

4

Q4/132 Q7/133

1998

325618808.doc

- 70 -

K Hoang 2007

Methods of integration (11.5) – HSC 3U 1999

2

Q1/134

Q3/135

Q4/136

2000

3

Q1/137

Q2/138

325618808.doc

- 71 -

K Hoang 2007

Methods of integration (11.5) – HSC 3U 2001

2

Q1/139

3

Q2/140 1

2002

2

3

325618808.doc

- 72 -

K Hoang 2007

Q1/141

Methods of integration (11.5) – HSC 3U 2003

3

Q1/142

2004

2

Q1/143

3

2005

1

Q1/144

3

Q2/145

325618808.doc

- 73 -

K Hoang 2007

Methods of integration (11.5) – HSC 3U 2005

Q3/146

Q4/147 2006

2007 2008 2009 2010

325618808.doc

2

New syllabus expected

- 74 -

K Hoang 2007

Q1/148

Primitive of sin2x and cos2x (13.6 E) – HSC 3U year Question 1990

marks

comment Q1b/149

1991

Q3/150

1996

Q3/151

1998

Q1/152

2001

3

Q2/153

2002

3

Q2/154

2003

2

Q5/155

2004

2

Q3/156

2006

2

Q3/157

2010 325618808.doc

New syllabus expected - 75 -

K Hoang 2007

Equation dN/dt = k(N-P) (14.2 E) – HSC 3U year Question 1990

marks

1991 1992

325618808.doc

comment Q6/158

Q5/159

- 76 -

K Hoang 2007

Equation dN/dt = k(N-P) (14.2 E) – HSC 3U 1993

Q6/160

1994 1995 1996

Q5/161

1997 1998 1999 325618808.doc

- 77 -

K Hoang 2007

2000 2001 2002

Equation dN/dt = k(N-P) (14.2 E) – HSC 3U Q3/162

2003

325618808.doc

Q5/163

- 78 -

K Hoang 2007

2004 2005

Equation dN/dt = k(N-P) (14.2 E) – HSC 3U Q2/164

2006

2007 2008 2009 2010

325618808.doc

2

New syllabus expected

- 79 -

K Hoang 2007

Q5/165

Velocity and acceleration as a function of x (14.3 E) – HSC 3U year Question 1990 1991 1992 1993 1994 1995 1996

325618808.doc

- 80 -

marks

comment

Q166

K Hoang 2007

1996 1997

325618808.doc

Velocity and acceleration as a function of x (14.3 E) – HSC 3U Q5/167

- 81 -

K Hoang 2007

Velocity and acceleration as a function of x (14.3 E) – HSC 3U 1998

Q6/168

1999 2000 2001

Q7/169

325618808.doc

- 82 -

K Hoang 2007

2002 2003

Velocity and acceleration as a function of x (14.3 E) – HSC 3U Q6/170

2004

325618808.doc

Q5/171

- 83 -

K Hoang 2007

2005 2006

2007 2008 2009 2010

325618808.doc

Velocity and acceleration as a function of x (14.3 E) – HSC 3U Q4/172

New syllabus expected

- 84 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U year Question 1990

marks

2 2 2

325618808.doc

- 85 -

K Hoang 2007

comment Q2/173

Projectile motion (14.3 E) – HSC 3U 1991

325618808.doc

Q6/174

- 86 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 1992

325618808.doc

Q7/175

- 87 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 1993

325618808.doc

Q7/176

- 88 -

K Hoang 2007

1994 1995

325618808.doc

Projectile motion (14.3 E) – HSC 3U Q7/177

- 89 -

K Hoang 2007

1996 1997

325618808.doc

Projectile motion (14.3 E) – HSC 3U Q7/178

- 90 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 1998

325618808.doc

Q6/179

- 91 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 1999

325618808.doc

Q7/180

- 92 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2000

325618808.doc

Q7/181

- 93 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2000

325618808.doc

Q7 /182 CONT’

- 94 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2001

325618808.doc

Q4/183

- 95 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2002

325618808.doc

Q6184

- 96 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2003

325618808.doc

Q7/185

- 97 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2004

325618808.doc

Q6/186

- 98 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2005

325618808.doc

Q6/187

- 99 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2006

325618808.doc

Q6/188

- 100 -

K Hoang 2007

Projectile motion (14.3 E) – HSC 3U 2006

2008 2009 2010

325618808.doc

Q6 CONT’

New syllabus expected

- 101 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U year Question 1990

marks

Comment Q3/190

1991

Q4/191

1992

Q2/192

325618808.doc

- 102 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 1993

Q3/193

1994 1995

325618808.doc

- 103 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 1996

325618808.doc

Q6/194

- 104 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 1997

Q2/195

1998

Q3/196

325618808.doc

- 105 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 1999

Q6/197

2000

Q4/198

2001

Q4/199

325618808.doc

- 106 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 2002

Q4/200

2003

Q3/201

325618808.doc

- 107 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 2004

325618808.doc

Q7/202

- 108 -

K Hoang 2007

Simple harmonic motion (14.4) – HSC 3U 2005

Q5/203

2006

Q4/204

2007 2008 2009 2010

325618808.doc

New syllabus expected

- 109 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U year Question 1990

marks

comment Q4/205

1991

Q5/206

1992

Q3/207

1993

Q3/208

325618808.doc

- 110 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 1994

Q6/209

1995

Q4/210

325618808.doc

- 111 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 1996

3

Q3/211

Q7/212

325618808.doc

- 112 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 1997

Q6/213

1998

Q4/214

325618808.doc

- 113 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 1999

Q5/215

2000

Q3/216

325618808.doc

- 114 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 2000

Q5/217

2001 2002

2

Q1/218

Q7/219

2003

325618808.doc

Q2/220

- 115 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 2003

325618808.doc

Q5/221

- 116 -

K Hoang 2007

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 2004

Q5/222

2005

325618808.doc

2

- 117 -

K Hoang 2007

Q1/223

Inverse functions and inverse trigonometric functions (15.1–15.5) – HSC 3U 2005

Q7/224

2006

Q2/225

Q5/226 2009 2010 325618808.doc

New syllabus expected - 118 -

K Hoang 2007

Induction (7.4) – HSC 3U year Question 1990

marks

comment Q227

1991

Q4/228

1992

Q4/229

1993

Q5/230

1994

Q3/231

1995

325618808.doc

- 119 -

K Hoang 2007

1996 1997

Induction (7.4) – HSC 3U Q5/232

1998

Q3/233

1999

Q5/234

2000

Q4/235

2001

325618808.doc

3

- 120 -

K Hoang 2007

Q6/236

Induction (7.4) – HSC 3U 2002

Q5/237

2003

3

Q3/238

2004

Q4/239

2005

Q4/240

2006

Q5/241

2007 2010 325618808.doc

New syllabus expected - 121 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U coefficients and terms year Question 1990 1995

marks

comment Q3/242

1996

Q7/243

2000

Q2/244

2001

Q2/245

2002 2003

2

2004 2005

2006 2007 2010

325618808.doc

Q2/246

Q2/247

New syllabus expected

- 122 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability year Question 1990

marks

comment Q6/248

1991

Q2/249

1992

Q1/250

1993 1994

Q3/251

325618808.doc

- 123 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 1995

325618808.doc

Q252

- 124 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 1996

Q5/253

1997

Q3/254

325618808.doc

- 125 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 1998

Q5/255

1999

Q3/256

2000

325618808.doc

- 126 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 2001

Q5/257

2002

Q4/258

325618808.doc

- 127 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 2003

Q3/259

2004

Q4/260

2005

Q6/261

325618808.doc

- 128 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U binomial probability 2006

2007 2008 2009 2010

325618808.doc

Q6/262

New syllabus expected

- 129 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U Identities and max term/coefficients year Question 1990

marks

comment Q6/263

1991 1992

Q6/264

1993 1994 1995 1996 1997

Q7/265

325618808.doc

- 130 -

K Hoang 2007

Binomial theorem (17.1–17.3) – HSC 3U Identities and max term/coefficients 1998

Q7/266

1999

Q7/267

2000

325618808.doc

- 131 -

K Hoang 2007

2001 2002

Binomial theorem (17.1–17.3) – HSC 3U Identities and max term/coefficients Q7/268

2003 2004

325618808.doc

Q7/269

- 132 -

K Hoang 2007

2005 2006

2007 2008 2009 2010

325618808.doc

Binomial theorem (17.1–17.3) – HSC 3U Identities and max term/coefficients Q2/270

New syllabus expected

- 133 -

K Hoang 2007

Iterative methods for roots of a poly eqn (16.4) – HSC 3U Newton’s method year Question 1990

marks

comment Q3/271

1991

Q3/272

1992

Q2/273

1993

Q5/274

1994

Q4/275

1995 1996

325618808.doc

- 134 -

K Hoang 2007

1997 1998

Iterative methods for roots of a poly eqn (16.4) – HSC 3U Newton’s method Q5/276

1999

Q2/277

2000

Q4/278

325618808.doc

- 135 -

K Hoang 2007

Iterative methods for roots of a poly eqn (16.4) – HSC 3U Newton’s method 2001

Q3/279

2002

Q3/280

2003

Q4/281

2004 2005 2006

Q3/282

2007 2008 2009 2010

325618808.doc

New syllabus expected

- 136 -

K Hoang 2007

Iterative methods for roots of a poly eqn (16.4) – HSC 3U halving the interval method year Question 1990 1991 1995

marks

comment Q2/283

1996

Q2/284

1997 1998 2005

Q3/285

2006 2007 2008 2009 2010

325618808.doc

New syllabus expected

- 137 -

K Hoang 2007

Harder applications of HSC 2 unit topics– HSC 3U – OTHER eg algebra + SERIES year Question 1990 1995

marks

Comment

2

Q1/286

1996 1997

Q1/287

1997

Q6/288

325618808.doc

- 138 -

K Hoang 2007

Harder applications of HSC 2 unit topics– HSC 3U – OTHER eg algebra + SERIES 1998

1

Q1/289

Q2/290

Q5/291 1999

1

Q4/292

2000 2001

1

Q1/293

2002

2

Q2/294

2003 2004 2005 2006 2007 2008 2009 2010

325618808.doc

New syllabus expected

- 139 -

K Hoang 2007

Harder applications of HSC 2 unit topics,– HSC 3U areas and volumes year Question 1990 1991 1992 1993 1994 1995

marks

comment

Q2/295

Q5/296

325618808.doc

- 140 -

K Hoang 2007

Harder applications of HSC 2 unit topics,– HSC 3U areas and volumes 1996

Q3/297

1997 1998

325618808.doc

- 141 -

K Hoang 2007

Harder applications of HSC 2 unit topics,– HSC 3U areas and volumes 1999

325618808.doc

Q3/298

- 142 -

K Hoang 2007

2000 2001

Harder applications of HSC 2 unit topics,– HSC 3U areas and volumes Q5/299

2002

325618808.doc

Q6/300

- 143 -

K Hoang 2007

2003 2004 2005

2006 2007 2008 2009 2010

325618808.doc

Harder applications of HSC 2 unit topics,– HSC 3U areas and volumes Q5/301

New syllabus expected

- 144 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION year Question 1990 1991

marks

comment Q3/302

1992 1994

325618808.doc

Q4/303

- 145 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 1994

325618808.doc

Q5/304

- 146 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 1995

325618808.doc

Q6/305

- 147 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 1996

325618808.doc

Q3/306

- 148 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 1997

325618808.doc

Q4/307

- 149 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 1998

Q4/308

1999

325618808.doc

- 150 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 2000

325618808.doc

Q7/309

- 151 -

K Hoang 2007

2001 2002

325618808.doc

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION Q5/310

- 152 -

K Hoang 2007

2003 2004

325618808.doc

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION Q3/311

- 153 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 2005

325618808.doc

Q7/312

- 154 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U rates of change, MOTION 2006

2008 2009 2010 325618808.doc

Q5/312

New syllabus expected - 155 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U derivatives year Question 1990

marks

1991

comment Q5/313

Q2/314a

314c 1992

Q1/315

1993

Q1/316

1997

2

1997 1998

325618808.doc

Q1/317

Q1/318a

- 156 -

K Hoang 2007

1998 1999

Harder applications of HSC 2 unit topics, HSC 3U derivatives

2000

2

Q1/318b

2

Q1/318c

Q3/318d

2000 2001

2

Q1/318e

Q2/318f

2002

2

Q1/319

2002 2003

2

Q2/320

2004

2

Q2/321

325618808.doc

- 157 -

K Hoang 2007

Harder applications of HSC 2 unit topics, HSC 3U derivatives 2005

2

Q2/322

Q3/323

2006

2008 2009 2010

325618808.doc

2

New syllabus expected

- 158 -

K Hoang 2007

Q1/324

Harder applications of HSC 2 unit topics– HSC 3U- curve sketching year Question 1990 1991

marks

Q325

1992

325618808.doc

comment

Q7/326

- 159 -

K Hoang 2007

Harder applications of HSC 2 unit topics– HSC 3U- curve sketching 1993

Q2/327

Q5/328

1994

325618808.doc

Q2/329

- 160 -

K Hoang 2007

Harder applications of HSC 2 unit topics– HSC 3U- curve sketching 1994

Q7/330

1995 1999

Q1/331

2000 2001 2002

Q5/332

2003 2006 2007 2008 2009 2010 325618808.doc

New syllabus expected - 161 -

K Hoang 2007

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima year Question 1990

marks

1991

325618808.doc

comment Q7/333

Q7/334

- 162 -

K Hoang 2007

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima 1992

325618808.doc

Q4/335

- 163 -

K Hoang 2007

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima 1993

Q6/336

1994 325618808.doc

- 164 -

K Hoang 2007

1995 1996 1997 1998 1999 2000

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima

Q6/337

2001

325618808.doc

- 165 -

K Hoang 2007

2002 2003

325618808.doc

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima Q6/338

- 166 -

K Hoang 2007

2004 2005

325618808.doc

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima Q3/339

- 167 -

K Hoang 2007

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima 2006

Q7/340

A  r 2 (  sin  cos  )

325618808.doc

- 168 -

K Hoang 2007

Harder applications of HSC 2 unit topics HSC 3U – maxima & minima 2006

2008 2009 2010

325618808.doc

Q7 CONT’

New syllabus expected

- 169 -

K Hoang 2007