Author Topic: 4U Maths Question Thread (Read 665308 times) Tweet Share

beau77bro · « **Reply #1410 on:** September 15, 2017, 10:55:30 am »

Quote from: justwannawish on September 14, 2017, 01:18:49 pm

Hi!

I was wondering what are the best textbooks for 4U? I've heard that while cambridge is good for the harder questions, it isn't really realistic for the types of questions in the HSC? (Any recs for 3U as well? Is Fitzpatrick 'better' than Cambridge?)

Thank you

I'm not sure if you are a year 12 or 11, if you are a year 12 then this isn't as helpful. but if u are going into year 12 i hope this helps u pick and choose.
3U:
Terry Lee - is very good for harder level questions, a lot are HSC esk, some are just ridiculous. overall one of the top recommends personally. Explanations for the exercises are usually not great, but the worked solutions are pretty good.
Cambridge - very good explanations (quite wordy) and a lot of questions to practice with. Some are just not HSC type Questions and are just tedious, strange, or just why? but they aren't irrelevant and all the questions are very good for drawing out some kind of understanding and for testing yourself (i personally think there are too many and some are unnecessary to do)
Fitzpatrick - is very well written and nicely set out, it's not as hard as the TL and Cambridge but it has some really good, tough questions (i don't use it that much but it's good). the explanations are very good and its a pretty book ngl
AtarNotes - is something i recommend for revision (topic specific which helps when u only need to study for a couple of topics for an exam)--> to me there aren't enough questions to practice on or learn stuff. but when it comes time for exams the questions are challenging, they cover almost all types of questions you can be asked and the solutions are so comprehensive that it's super easy to fix any holes.
Papers - are ultimately the best for revision holistically. HSC, or trial you are gonna be challenged and get some solid practice. if you do 30 papers you will undoubtably be exponentially better (but u need to stick with it through hard Qs).
NOTE: YOU WILL NEED TO REVISE CONSISTENTLY TO KEEP ON TOP OF THINGS SO WORK YOU DID DOESN'T GO TO WASTE

4U:
Terry Lee - is again amazing, and very very challenging. (some topics are better than others, so in 4u ull probably have to do a couple textbooks to work out the best)
Cambridge - i honestly don't use it much, not very appealing and the questions are weird (good challenge exercises and diagnostic tests). explanations are good but it's just not as good as others.
Sydney Grammar Notes - ARE EASILY THE BEST RESOURCE overall - explanations are amazing, and as rui said it's the closest to Cambridge 3U. There are lots of Questions, lots of chapters and they go from easy to medium to hard (i forgot what they are called) and if you had to do just one this is the one (in my opinion because it covers all topics pretty well). still i would say use this in conjunction with terry lee. we used these notes from my teacher throughout - but i think if u can do this for the whole course.
Fitzpatrick - is good, I don't use it much because it's a bit repetitive (for what i did), is definitely useful and has good questions. better for starting out than challenging urself, but it asks a lot of questions which is good.
Atarnotes - i haven't started doing, but from reading it they are good challenging questions for revision. if they are as good as the 3u ones then it is very worthwhile
Trials - are again just the best thing for revision and practice but u have to do a lot of them to cover everything since the course is massive. and u need to spend time working things out (dont be afraid to use the solutions just use them well dont copy)

For harder ext 1 I recommend a book by kinny lewis. like just lots of Questions and really good solutions. They are hard, but they cover most things which is amazing. still should do everything under the sun for this topic since its expected to be like 30% of the course by NESA.

yea so there you go. I'm sure there are more out there, but that's my top picks - can't go too wrong with any but it's good to use a couple in my opinion.
HOPE THIS HELPS.

statues · « **Reply #1411 on:** September 15, 2017, 11:14:12 pm »

Hi -
I need a quick explanation of HSC 2016 q 15 b ii
So I can work to the answer without the limits of the integral (d and b) - I don't understand why they're there. I've read the suggested solutions posted by Rui. But I still don't get it. In other questions where I'm integrating to get an expression for time, I just integrate, solve for any constant and then substitute a point such as x in to get time. Why is it that in this question a definite integral is required?

Many thanks

RuiAce · « **Reply #1412 on:** September 15, 2017, 11:16:57 pm »

Quote from: statues on September 15, 2017, 11:14:12 pm

Hi -
I need a quick explanation of HSC 2016 q 15 b ii
So I can work to the answer without the limits of the integral (d and b) - I don't understand why they're there. I've read the suggested solutions posted by Rui. But I still don't get it. In other questions where I'm integrating to get an expression for time, I just integrate, solve for any constant and then substitute a point such as x in to get time. Why is it that in this question a definite integral is required?

Many thanks

In the future, please provide a link or screenshot to the question.

Are these my solutions that I posted ages ago last year you're referring to?

statues · « **Reply #1413 on:** September 15, 2017, 11:24:38 pm »

Quote from: RuiAce on September 15, 2017, 11:16:57 pm

In the future, please provide a link or screenshot to the question.

Are these my solutions that I posted ages ago last year you're referring to?

Yup - the one here: https://atarnotes.com/forum/index.php?topic=168150.0

RuiAce · « **Reply #1414 on:** September 15, 2017, 11:36:13 pm »

Quote from: statues on September 15, 2017, 11:24:38 pm

Yup - the one here: https://atarnotes.com/forum/index.php?topic=168150.0

$\text{Firstly, a remark that you are right}\\ \text{in that it }\textbf{can}\text{ be done via only indefinite integrals.}$
$\text{However, definite integrals speed things up}\\ \text{An example would be to use the binomial expansion}\\ (1+x)^n = \binom{n}{0}+\binom{n}{1}x+\dots + \binom{n}{n}x^n\\ \text{to find an expression for }\\ \binom{n}{0}x+\binom{n}{1}\frac{x^2}{2}+\dots + \binom{n}{n}\frac{x^{n+1}}{n+1}$
$\text{One way to do this is to perform an indefinite integral}\\ \text{before simply substituting in }x=0\text{ to find the constant of integration}\\ \text{However, the following argument requires considerably less writing:}$
$\begin{align*}(1+t)^n &= \binom{n}{0} + \binom{n}{1}t +\dots + \binom{n}{n}t^n\\ \int_0^x (1+t)^n \, dt &= \int_0^x \left[\binom{n}{0}+\binom{n}{1}t+\dots+\binom{n}{n}t^n\right]dt\\ \left[\frac{(1+t)^{n+1}}{n+1}\right]_0^x&=\left[\binom{n}{0}t+\binom{n}{1}\frac{t^2}{2}+\dots + \binom{n}{n}\frac{t^{n+1}}{n+1}\right]_0^x \\ \frac{(1+x)^{n+1}-1}{n+1} &=\binom{n}{0}x+\binom{n}{1}\frac{x^2}{2}+\dots + \binom{n}{n}\frac{x^{n+1}}{n+1} \end{align*}$
$\text{A remark that you }\textit{should've}\text{ been presented such a method}\\ \text{when handling questions on resisted motion.}\\ \text{This is the method of separation of variables.}\\ \text{It is also covered in my book.}$
________________________________________
$\text{Essentially, a definitely integral is }\textbf{faster}\text{ than an indefinite one}\\ \text{in that we substitute our point }x=0\text{ in }\textbf{straight away.}\\ \text{Basically, killing two birds with one stone.}$
$\text{The answer provided last year could be redone by the}\\ \text{separation of variables method for clarity.}\\ \text{The computations would then be as thus:}$
$\begin{align*}\frac{dt}{dx}&= -\frac{1}{\mu \sqrt2}\sqrt{\frac{b}{b-x}}\\ dt &= -\frac{1}{\mu \sqrt2}\sqrt{\frac{b}{b-x}}\,dx\end{align*}$
$\text{When }t=0, x=b\\ \text{When }t\text{ is equal to, say, }T\\ \text{i.e. when }t=T, x=d$
$\therefore \int_0^T t\,dt =-\frac{1}{\mu \sqrt2} \int_b^d\sqrt{\frac{b}{b-x}}\,dx \\ \text{Upon evaluating the LHS, we arrive at what the solutions stated:}\\ T = -\frac{1}{\mu \sqrt2} \int_b^d\sqrt{\frac{b}{b-x}}\,dx$
$\text{And of course, after the substitution, the desired integral is found.}$
Whilst it probably isn't required out of a 3U student, in general a 4U student is expected to be able to handle applications of definite integrals in the place of indefinite integrals as well.

statues · « **Reply #1415 on:** September 15, 2017, 11:46:40 pm »

Thank you so much Rui!
Do you know what section it is in, in your MX2 notes, I can't seem to find it.

RuiAce · « **Reply #1416 on:** September 15, 2017, 11:48:10 pm »

Quote from: statues on September 15, 2017, 11:46:40 pm

Thank you so much Rui!
Do you know what section it is in, in your MX2 notes, I can't seem to find it.

Should be under 6.3.2 Vertical Resisted Motion

Shlomo314 · « **Reply #1417 on:** September 17, 2017, 03:35:03 pm »

Could someone help me out with this integration using the t method please?

Integrate tanx/(1+cosx)

cheers

RuiAce · « **Reply #1418 on:** September 17, 2017, 04:03:15 pm »

Quote from: Shlomo314 on September 17, 2017, 03:35:03 pm

Could someone help me out with this integration using the t method please?

Integrate tanx/(1+cosx)

cheers

$t = \tan \frac{x}{2} \implies x = 2\tan^{-1}t \implies dx = \frac2{1+t^2}dt\\ \text{and }\tan x = \frac{2t}{1-t^2}, \cos t = \frac{1-t^2}{1+t^2}$
$\begin{align*}\int \frac{\tan x}{1+\cos x}dx &= \int \frac{\frac{2t}{1-t^2}}{1 + \frac{1-t^2}{1+t^2}} \cdot \frac{2\, dt}{1+t^2}\\ &= \int \frac{\frac{4t}{1-t^2}}{(1+t^2 + 1-t^2)}dt\\ &=\int \frac{2t}{1-t^2}dt\\ &= -\int \left(\frac{1}{t-1}+\frac{1}{t+1}\right)dt \tag{partial fractions}\\ &= -\ln |t-1| - \ln |t+1| + C\\ &= -\ln \left| \tan \frac{x}{2}-1\right|-\ln \left| \tan \frac{x}{2}+1\right|+C\end{align*}$
$\text{Though in reality, the u-substitution }u=\cos x\\ \text{would've been a little neater}$
Subject to minor computation error

Shlomo314 · « **Reply #1419 on:** September 19, 2017, 06:55:46 pm »

Hi guys, can someone help me on this question please?
Find the number of arrangements that can be made using the letters of the word PARTICULAR without altering the relative positions of vowels and consonants

I don't understand the part where it says the 'relative positions of vowels and consonants.

Cheers Atar team

RuiAce · « **Reply #1420 on:** September 19, 2017, 07:19:46 pm »

Quote from: Shlomo314 on September 19, 2017, 06:55:46 pm

Hi guys, can someone help me on this question please?
Find the number of arrangements that can be made using the letters of the word PARTICULAR without altering the relative positions of vowels and consonants

I don't understand the part where it says the 'relative positions of vowels and consonants.

Cheers Atar team

$\text{Denoting }C\text{ as a consonant and }V\text{ as vowel}\\ \textbf{we must preserve this ordering of consonants and vowels.}\\ CVCCVCVCVC$
$\text{The easiest way to do this is to fix the consonants}\\ \text{and vowels in this position}\\ \text{then only arrange the consonants and vowels}\\ \textbf{among themselves.}$
____________________________
$\text{The consonants are }PRTCLR\\ \text{They have }\frac{6!}{2!}\text{ arrangements}$
$\text{The vowels are }AIUA\\ \text{They have }\frac{4!}{2!}\text{ arrangements}$
$\text{So since we aren't considering any separate cases}\\ \text{the answer is }\frac{6!}{2!}\times \frac{4!}{2!}$

atangirala1 · « **Reply #1421 on:** September 20, 2017, 05:47:34 pm »

Hey RuiAce ,
Obviously I want to join ur atar notes forum but I'm currently unable to do so.
May u plz guide me and let me join ur Ex2 math science group

RuiAce · « **Reply #1422 on:** September 20, 2017, 05:54:33 pm »

Quote from: atangirala1 on September 20, 2017, 05:47:34 pm

Hey RuiAce ,
Obviously I want to join ur atar notes forum but I'm currently unable to do so.
May u plz guide me and let me join ur Ex2 math science group

What group is this?

justwannawish · « **Reply #1423 on:** September 20, 2017, 09:03:07 pm »

Quote from: beau77bro on September 15, 2017, 10:55:30 am

I'm not sure if you are a year 12 or 11, if you are a year 12 then this isn't as helpful. but if u are going into year 12 i hope this helps u pick and choose.
3U:
Terry Lee - is very good for harder level questions, a lot are HSC esk, some are just ridiculous. overall one of the top recommends personally. Explanations for the exercises are usually not great, but the worked solutions are pretty good.
Cambridge - very good explanations (quite wordy) and a lot of questions to practice with. Some are just not HSC type Questions and are just tedious, strange, or just why? but they aren't irrelevant and all the questions are very good for drawing out some kind of understanding and for testing yourself (i personally think there are too many and some are unnecessary to do)
Fitzpatrick - is very well written and nicely set out, it's not as hard as the TL and Cambridge but it has some really good, tough questions (i don't use it that much but it's good). the explanations are very good and its a pretty book ngl
AtarNotes - is something i recommend for revision (topic specific which helps when u only need to study for a couple of topics for an exam)--> to me there aren't enough questions to practice on or learn stuff. but when it comes time for exams the questions are challenging, they cover almost all types of questions you can be asked and the solutions are so comprehensive that it's super easy to fix any holes.
Papers - are ultimately the best for revision holistically. HSC, or trial you are gonna be challenged and get some solid practice. if you do 30 papers you will undoubtably be exponentially better (but u need to stick with it through hard Qs).
NOTE: YOU WILL NEED TO REVISE CONSISTENTLY TO KEEP ON TOP OF THINGS SO WORK YOU DID DOESN'T GO TO WASTE

4U:
Terry Lee - is again amazing, and very very challenging. (some topics are better than others, so in 4u ull probably have to do a couple textbooks to work out the best)
Cambridge - i honestly don't use it much, not very appealing and the questions are weird (good challenge exercises and diagnostic tests). explanations are good but it's just not as good as others.
Sydney Grammar Notes - ARE EASILY THE BEST RESOURCE overall - explanations are amazing, and as rui said it's the closest to Cambridge 3U. There are lots of Questions, lots of chapters and they go from easy to medium to hard (i forgot what they are called) and if you had to do just one this is the one (in my opinion because it covers all topics pretty well). still i would say use this in conjunction with terry lee. we used these notes from my teacher throughout - but i think if u can do this for the whole course.
Fitzpatrick - is good, I don't use it much because it's a bit repetitive (for what i did), is definitely useful and has good questions. better for starting out than challenging urself, but it asks a lot of questions which is good.
Atarnotes - i haven't started doing, but from reading it they are good challenging questions for revision. if they are as good as the 3u ones then it is very worthwhile
Trials - are again just the best thing for revision and practice but u have to do a lot of them to cover everything since the course is massive. and u need to spend time working things out (dont be afraid to use the solutions just use them well dont copy)

For harder ext 1 I recommend a book by kinny lewis. like just lots of Questions and really good solutions. They are hard, but they cover most things which is amazing. still should do everything under the sun for this topic since its expected to be like 30% of the course by NESA.

yea so there you go. I'm sure there are more out there, but that's my top picks - can't go too wrong with any but it's good to use a couple in my opinion.
HOPE THIS HELPS.

I think I'll go with Fitzpatrick and Terry Lee for the moment. Thank you for your advice

Quote from: RuiAce on September 14, 2017, 10:24:43 pm

There is no such thing as best textbooks to rely on HSC questions/preparation, especially given that the ultimate focus is always past papers.

For some decently worthwhile mentions, if you're using the small Cambridge book that is only ever so decent. As for question types, they are either too easy or too hard.

On the other hand, things like Terry Lee's textbook are generally recommended for selective schools, due to the difficulty of the questions.

4U Fitzpatrick is decent, provided the newer version is chosen. The old version is quite useless.

What's generally regarded as a standout is the Sydney Grammar textbook, as it is most similar to the 3U Cambridge textbook. This textbook, however, must be purchased from the school. It was never published (presumably due to the new syllabus things), and comes as only a PDF.

Thank you for your advice. Is the Sydney Grammar book available for students who don't go there?

RuiAce · « **Reply #1424 on:** September 20, 2017, 09:12:56 pm »

Quote from: justwannawish on September 20, 2017, 09:03:07 pm

Thank you for your advice. Is the Sydney Grammar book available for students who don't go there?

Yes.

But like said, you have to order through your school.

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