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March 29, 2024, 02:57:38 am

Author Topic: 3U Maths Question Thread  (Read 1230296 times)  Share 

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Neutron

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #45 on: February 24, 2016, 02:41:06 pm »
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Hey! Okay so i don't seem to be able to find an easy way to do this (like I got an answer but the method was waaay too complicated I reckon but I got -8 degrees and 48 seconds and -24 degrees and 54 seconds??) D: I was wondering whether you guys could help me! thanks :D

Solve the following equation:
2cos2ϴ=1-3sin2ϴ   0≤ϴ≤360

Thank you!

Neutron

jakesilove

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #46 on: February 24, 2016, 06:49:37 pm »
+1
Hey! Okay so i don't seem to be able to find an easy way to do this (like I got an answer but the method was waaay too complicated I reckon but I got -8 degrees and 48 seconds and -24 degrees and 54 seconds??) D: I was wondering whether you guys could help me! thanks :D

Solve the following equation:
2cos2ϴ=1-3sin2ϴ   0≤ϴ≤360

Thank you!

Neutron

Hey Neutron!

I also can't find an easy way to answer that question! I get the same answers, although make sure that since the range is between 0 and 360, you add 360 degrees to your two answers! If anyone can think of an easy solution, I'd love you to post it!

Jake
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RuiAce

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #47 on: February 25, 2016, 12:24:09 am »
+1
Hey! Okay so i don't seem to be able to find an easy way to do this (like I got an answer but the method was waaay too complicated I reckon but I got -8 degrees and 48 seconds and -24 degrees and 54 seconds??) D: I was wondering whether you guys could help me! thanks :D

Solve the following equation:
2cos2ϴ=1-3sin2ϴ   0≤ϴ≤360

Thank you!

Neutron
The presence of the 2cos(2θ) is not able to eliminate the 1 as no matter what expansion of the double angle formula is used, a 2 will pop out instead of a 1. This makes expanding all double angles, unfortunately, a folly.

The neatest method here would be to attack 3sin(2θ)+2cos(2θ) with the auxiliary angle method. Yes, admittedly this isn't tidy either.

Phillorsm

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #48 on: February 29, 2016, 10:15:22 pm »
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Hey Jake, or whoever answers this...
So In your lectures you mentioned that with Induction you are allowed to skip steps in the proof to save time. I was just wanting to check with you if this is what you meant by that, and if it would get all the marks :)

P.S. Sorry if the image is a bit small, it wouldnt let me upload anything over 512kB.

RuiAce

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #49 on: February 29, 2016, 11:11:04 pm »
+1
Hey Jake, or whoever answers this...
So In your lectures you mentioned that with Induction you are allowed to skip steps in the proof to save time. I was just wanting to check with you if this is what you meant by that, and if it would get all the marks :)

P.S. Sorry if the image is a bit small, it wouldnt let me upload anything over 512kB.

This is interesting. I would ALWAYS restate the LHS and RHS before rearranging the equation by either substituting in the assumption or just algebra. Also, when you use the assumption you should always state "by assumption" or something along the lines of it e.g. by inductive hypothesis.

And your final statement, whilst it can be short, has to be there. Just say "Hence true by induction"

Edit: Also you skipped heaps of algebra. That's going to confuse the examiner.

Happy Physics Land

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #50 on: February 29, 2016, 11:20:19 pm »
+1
Hey Jake, or whoever answers this...
So In your lectures you mentioned that with Induction you are allowed to skip steps in the proof to save time. I was just wanting to check with you if this is what you meant by that, and if it would get all the marks :)

P.S. Sorry if the image is a bit small, it wouldnt let me upload anything over 512kB.

Yeah hmm if you recall at the end of Jake's induction lecture he said the best way to write your conclusion would be "since its true for n=1, assumed true for n=k and proven true for n=k+1, then by the principle of mathematical induction the statement is true" This will be the most secure way to write your conclusion, just in case you encounter an old-fashion marker who wants a complete formal style conclusion.
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jakesilove

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #51 on: February 29, 2016, 11:36:35 pm »
+1
Hey Jake, or whoever answers this...
So In your lectures you mentioned that with Induction you are allowed to skip steps in the proof to save time. I was just wanting to check with you if this is what you meant by that, and if it would get all the marks :)

P.S. Sorry if the image is a bit small, it wouldnt let me upload anything over 512kB.

Hey!!

I definitely agree with HPL: You need to write out the full concluding remark. However, in regards to the algebra I would recommend doing a few more steps. Definitely expand brackets/collect like terms etc. Basically, what I was suggesting is that if you can clearly see how the LHS is going to equal the RHS, do some algebra steps and then just pretend you've quickly collected like terms and simplified. Only skip steps if you can see exactly how it's going to work out: as a general rule, try to do each step of algebra!

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amandali

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #52 on: March 05, 2016, 09:20:43 am »
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how to sketch y=inverse cos(x^2)

jakesilove

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #53 on: March 05, 2016, 12:06:06 pm »
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how to sketch y=inverse cos(x^2)

Hey Amandali!

The first thing to remember with inverse trigonometric graphs is their domain and range. We know that inverse cos has a domain of



and a range of



Then, all I would recommend is plotting points! See what happens when you sub in x=-1, x=-0.5, x=0, x=0.5, x=1. This will give you a general idea of the graph, so that you can sketch the final graph!



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Re: 98 in 3U Maths: Ask me Anything!
« Reply #54 on: March 06, 2016, 03:00:39 pm »
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how to solve  lim    tan3x/tan2y
                       x->0

jakesilove

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #55 on: March 06, 2016, 06:16:38 pm »
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how to solve  lim    tan3x/tan2y
                       x->0

Hey! I assume you actually meant 2x, instead of 2y?

In that case, we know that the limit of tan(a)/a as a approaches zero is equal to one. Therefore, the limit of tan(ax) as x approaches zero is equal to ax!

For tan(3x), as x approaches zero, the value will approach 3x. For tan(2x), as x approaches zero the value will approach 2x. Therefore, we now have 3x/2x=3/2, which is our answer!
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Re: 98 in 3U Maths: Ask me Anything!
« Reply #56 on: March 15, 2016, 09:12:53 am »
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Hello Im also a year 12 student doing my maths extension I, and I have encountered similar issues beforehand as you, because binomial is very tedious and the sigma notations always scare me because I am unfamiliar with it. So I began doing questions first from maths in focus which provides easy questions on binomial theorem and then moved on to do more exercises in the cambridge book. It is very challenging however beneficial to do those development and extension questions as well because it is likely that your teacher will confront you with a similar style question.

Here are several tips that I think has helped me a lot with binomial:
1. It is very important to remember that the binomial co-efficients always starts with nC0, not nC1
2. When proving binomial identities, a lot of those can be related back to pascal's triangle. So if you are confused about whats the significance of the proof or what you are trying to prove, write down the first few rows of pascal's triangle to give you a better observation of what exactly you are trying to prove
3. DO NOT BE SCARE OF SIGMA NOTATIONS! A lot of my friends instantly give up as soon as they seen sigma notation because its such a weird representation of a series of numbers. Sometimes we dont think of sigma notation as normal maths, but rather, some "alien language". But its VERY CRUCIAL TO REMEMBER that SIGMA NOTATIONS ARE OUR FRIENDS. It is just A SERIES, nothing more, just A SERIES OF NUMBERS. It is helpful for us because instead of having to tediously look a long, boring chain of numbers, a simple sigma notation essentially summarise it for us in simple expressions. On the bottom of sigma notation there is r= some number or k = some number, this just means that for the expression next to the sigma sign, the initial variable is what r or k represents. E.g. for 3^r, r = 0, that means we start with 3^0. On the top of the sigma sign there is usually "n", which is indicative that the series terminates at r = n, whatever that n value maybe. E.g. for 3^r, we terminate at 3^n.
4. It is beneficial sometimes when solving binomial questions to expand the binomial out. If it is too long an expansion, just write out the first 3-4 terms and the last 3 terms. This helps us to find patterns that can help us to solve the question.
5. In Binomial questions associated with integration or differentiation, we almost always find a value for x (i.e. let x = something) to make our solution look more similar to what the question requires for us to prove/find. A sneaky tip is that HSC examiners would usually write the question in a way that students will let x = 0 or 1.
6. When we are proving an identity in binomial theorem, its not always compulsory to start with the side thats more complicated. This is counter-intuitive to what we have always been learning because we are always used to solving something thats looks more intimidating because there is a higher chance that we can somehow manipulate it to make it look more neat/tidy, and resemble the other side of the equation. In binomial theorem, this is not always the case. For example, consider the proof for "Sum of nCr from r=0 to r=n) = 2^n". Logically, we would begin with the left hand side because it is more complicated and we would hope for a neat result to come out in the end. However, if we begin with the right hand side it will be much easier because RHS = 2^n = (1+1)^n = sum of (nCr x 1^r) from r=0 to r=n. Since 1^r is always 1, we can effectively prove that 2^n = sum of nCr from r=0 to r=n.
7. It almost always helpful that when you are stuck on a binomial proof question to go back to the basics of expanding (1+x)^n, or remembering that (1+x)^n = the sum of (nCr x x^r ) from r= 0 to r=n.
8. When finding the constant term that involves expanding two binomials, expand both and select one term from each binomial expansion that will cancel each other's variable out when multiplied together, leaving us with just a number.
9. Transformations of (1+x)^n will always change the position of the greatest co-efficient in the expansion. (1+x)^n will have its greatest co-efficient at the centre, (1+3x)^n will have its greatest co-efficient shifted to the right and (1+5x)^n will have its greatest co-efficient shifted even further to the right. Adversely, (3+x)^n will have its greatest co-efficient shifted to the left and (5+x)^n will have its greatest co-efficient shifted even more to the left and so on.

These are all just some of my tips that l found very helpful to know. Im not sure how much this will help you but yeah good luck in everything this year!

Best Regards

Happy Physics Land

Thanks so much, this is a great overview and simplification of binomial
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Re: 3U Maths Question Thread
« Reply #57 on: March 19, 2016, 08:39:11 pm »
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Hi, just wondering,
we have only just done inverse trigonometry but there was a question in our test that we had not learnt how to attempt. We had to find the integral of the inverse of sin. How would you attempt this question?

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Re: 3U Maths Question Thread
« Reply #58 on: March 19, 2016, 10:16:02 pm »
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Hi, just wondering,
we have only just done inverse trigonometry but there was a question in our test that we had not learnt how to attempt. We had to find the integral of the inverse of sin. How would you attempt this question?

Hey atarz!

If possible can you please supply me with the actual question? If the question only asks for the integral of sin-1(x), then I have provided a solution below. It is quite infrequent for school exams to ask you such questions, especially when 4 unit candidates can solve it relatively easily using integration by parts. I cant see easier ways of doing this except for using Integration by parts which definitely isnt a 3 unit concept. You can try doing this question also with areas (i.e. the area under sin-1x). I'm suspecting that there may be a part beforehand in the question that may help with integrating inverse of sine using 3u methods?

Anyways, I have posted my solution below through integration by parts:



Best Regards
Happy Physics Land
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Re: 3U Maths Question Thread
« Reply #59 on: March 20, 2016, 03:17:50 am »
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Hey atarz!

If possible can you please supply me with the actual question? If the question only asks for the integral of sin-1(x), then I have provided a solution below. It is quite infrequent for school exams to ask you such questions, especially when 4 unit candidates can solve it relatively easily using integration by parts. I cant see easier ways of doing this except for using Integration by parts which definitely isnt a 3 unit concept. You can try doing this question also with areas (i.e. the area under sin-1x). I'm suspecting that there may be a part beforehand in the question that may help with integrating inverse of sine using 3u methods?

Anyways, I have posted my solution below through integration by parts:

(Image removed from quote.)

Best Regards
Happy Physics Land

Nice! Another method for integration by parts would be by a substitution at the start, it simplifies things considerably. But yeah, I don't think this can be done without some application of integration by parts, not that I can see right now anyway! Perhaps I am spoiled by the method  ;)