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Thread: Cambridge HSC MX1 Textbook Marathon/Q&A

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    Bump


    "verify the result on the third column of the Pascal triangle." Does anyone have any idea on this? Or is my answer shown above good enough ?




    • Binomial Coefficients in Pascal's Triangle: http://www.mathwords.com/b/binomial_...nts_pascal.htm
    • Pascal's Triangle: http://www.mathwords.com/p/pascal's_triangle.htm .

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by InteGrand View Post




    • Binomial Coefficients in Pascal's Triangle: http://www.mathwords.com/b/binomial_...nts_pascal.htm
    • Pascal's Triangle: http://www.mathwords.com/p/pascal's_triangle.htm .



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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Im aware that Pascal's triangle is:

    1
    1 2 1
    1 3 3 1
    1 4 6 4 1
    1 5 10 10 5 1
    1 6 15 20 15 6 1

    I'm also aware there is no 3rd column in the first row.
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post



    Yes, k = 2.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    And what is the purpose of n?

    If you look at Pascal's Triangle, the 3rd column values are: 1,3,6,10,15...........I struggle to see any link between that equation and those values.
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    And what is the purpose of n?

    If you look at Pascal's Triangle, the 3rd column values are: 1,3,6,10,15...........I struggle to see any link between that equation and those values.
    Pascal's Triangle:

    1
    1 1
    1 2 1 (n = 2)
    1 3 3 1 (n = 3)
    1 4 6 4 1 (n = 4)
    1 5 10 10 5 1 (n = 5)
    etc.

    The rule was that nC2 + n+1C2 = n2.

    In terms of the blue numbers above, the sum of consecutive ones will be n2, where n is therow of the upper one out of the two.

    E.g. 6 + 10 = 16 = 42 in the n = 4 row.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    OK I see it now.

    Out of curiosity Integrand, I was keen to know what was your educational background? Your quite advanced in terms of knowledge and I think it's fair to say your the most knowledgeable on the forum. (I think others would agree)
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    I'm not the most knowledgeable.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by InteGrand View Post
    I'm not the most knowledgeable.
    he seems knowledgeable in maths and physics and not so much chemistry as he couldn't really answer my chem questions

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    OK I see it now.

    Out of curiosity Integrand, I was keen to know what was your educational background? Your quite advanced in terms of knowledge and I think it's fair to say your the most knowledgeable on the forum. (I think others would agree)
    He probably did Uni maths and physics

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by si2136 View Post
    He probably did Uni maths and physics
    Yeah that's what I thought lol, I wish there was a chemistry version of him

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by eyeseeyou View Post
    Yeah that's what I thought lol, I wish there was a chemistry version of him
    Chem isn't that useful outside of hs compared to physics or maths, as there are a lot of pathways that go with them (all types of Engineering, Maths, Physics, Computer Science)

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread


    I'm having a hard time finding a systematic way of doing parts (c), (d) and (e). Is it necessary to check every number and eliminate the doubles or is there a easier way to do it? Thanks in advance.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    I don't want to look up the answer in the back of the book. Want to discuss it on an open forum.









    Here is my working out:





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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    I don't want to look up the answer in the back of the book. Want to discuss it on an open forum.









    Here is my working out:





    When we expand a 10th power using binomial expansion, there'll be 11 terms. So the middle term is the sixth term (as there will be five terms either side of this, for a total of 11 terms).

    We don't need to expand everything to find just the sixth (or any particular) term. We can write down the general "kth term" using the binomial theorem, and then substituting k = 5 gives us the term we want (reason to sub. in 5 is that k goes from 0 to 10, not from 1 to 11).

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by Blitz_N7 View Post

    I'm having a hard time finding a systematic way of doing parts (c), (d) and (e). Is it necessary to check every number and eliminate the doubles or is there a easier way to do it? Thanks in advance.
    Divisible by 5 - consider the last digit first. It has to be 5 (because 0 is not an option). Then arrange everything else

    Divisible by 3 - This is trickier. The SUM of the two numbers has to be divisible by 3.

    Or you can play it smart. Because the numbers are forcibly 2 digits, then:
    From 0-100 there are 33 numbers divisible by 3
    From 0-9 there are 3 numbers divisible by 3
    Then do subtraction

    Divisible by 6 - I would've done this using set theory. Divisible by 6 just means both even and divisible by 3 simultaneously

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by InteGrand View Post
    When we expand a 10th power using binomial expansion, there'll be 11 terms. So the middle term is the sixth term (as there will be five terms either side of this, for a total of 11 terms).

    We don't need to expand everything to find just the sixth (or any particular) term. We can write down the general "kth term" using the binomial theorem, and then substituting k = 5 gives us the term we want (reason to sub. in 5 is that k goes from 0 to 10, not from 1 to 11).
    Yep I know there are 11 terms if it is of the form







    I personally think its good to do it both ways 1) the long way and 2) the shorter way for learning purposes.

    Food for thought, what happens if the term was to the power of 11 ?

    i.e ? Would we then look at the 6th term AND 7th term as the middle term?
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    Yep I know there are 11 terms if it is of the form







    I personally think its good to do it both ways 1) the long way and 2) the shorter way for learning purposes.

    Food for thought, what happens if the term was to the power of 11 ?

    i.e ? Would we then look at the 6th term AND 7th term as the middle term?
    Unnecessarily expanding feels like a waste of time and does not teach much at all.


    They wouldn't ask you for a middle term that way, they will either make it plural (to which therefore yes) or just not ask it.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by leehuan View Post
    Unnecessarily expanding feels like a waste of time and does not teach much at all.


    They wouldn't ask you for a middle term that way, they will either make it plural (to which therefore yes) or just not ask it.
    OK Cool, what I did notice with those questions were they were asking them to even powers .
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    In Question 13 (b) they ask for the 2 middle terms eg (a+3b)^5
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by leehuan View Post
    Unnecessarily expanding feels like a waste of time and does not teach much at all.


    They wouldn't ask you for a middle term that way, they will either make it plural (to which therefore yes) or just not ask it.
    Alot of the introductory problems with Binomials they want to get you to write the whole expansion out by using Binomial Theorem or Pascal's Triangle. Once you develop better understanding and knowledge, obviously you look at quicker methods.
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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by davidgoes4wce View Post
    In Question 13 (b) they ask for the 2 middle terms eg (a+3b)^5
    If they ask for two then use what you said about sixth and seventh term for the higher power. But it's a bit more unlikely to be asked in the exam.

    Quote Originally Posted by davidgoes4wce View Post
    Alot of the introductory problems with Binomials they want to get you to write the whole expansion out by using Binomial Theorem or Pascal's Triangle. Once you develop better understanding and knowledge, obviously you look at quicker methods.
    Do you really need to do more than 5 questions that make you write out a full binomial expansion to see what's going on?

    Plus Pascal's triangle isn't really necessary. It helps to visualise things such as Pascal's identity but other than that so long as you know you can replace the terms with \binom{n}{k} nothing to worry about

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    On the topic of binomial stuff


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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Quote Originally Posted by seanieg89 View Post
    On the topic of binomial stuff

    I think this is how to do that one







    Edit: WOAH THERE. Start from (1+x)0, not (1+x)m otherwise the series is wrong

    Result is not affected as there is no term in xm in the expansion of (1+x)k, where 0 ≤ k ≤ m-1
    Last edited by leehuan; 5 Jul 2016 at 4:59 PM.

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    Re: Year 12 Mathematics 3 Unit Cambridge Question & Answer Thread

    Yr 12 Cambridge Ex 10 A Q 23.

    I spent more than 30 minutes and still couldn't come to the answer for this one. I had a few attempts at drawing the diagram but still couldn't come up with it.

    Q23. A rectangular field is 60 metres long and 30 metres wide. A cow wanders randomly around the field. Find the probability that the cow is:

    (a) More than 10 metres from the edge of the field

    (b) not more than 10 metres from the corner of the field


    Answers are:



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