NTU Entrance Test

February 4, 2008 at 7:37 pm (MaTh, Science..)

Nah,, berhubung aku gak ikut NTU Entrance test nanti.. :cry:

[karena gak diijinkan abangku tercinta,, dan karena Jakarta lagi musim banjir :mrgreen: ]

Kali ini aku mau coba telaah 1 soal NTU (sampel sih..)

[You can find out the sample entrance test in here]

Aku cuma terkejut karena soal ini memuat bilangan kompleks,, dimana sluruh anak SMA di Indonesia pasti gak pernah temui…(apakah ini berarti tingkat pendidikan SMA di Singapore lebih baik??)

The problem::

Use the relationship e^{i\theta}=\cos{\theta}+i\sin{\theta} to express \cos{5\theta} in terms of \cos{\theta}. Hence show that x=\cos{\frac{\pi}{10}} is a root of the equation 16x^4-20x^2+5=0

Solution (sorry if I’m mistaken..):

The relationship would be like this: e^{ip}=\cos{p}+i\sin{p}

So,, if we put p=n\theta \rightarrow e^{in\theta}=\cos{n\theta}+i\sin{n\theta}

The full relationship will be like this::

e^{in\theta}= e^{(i\theta)n}=(\cos{\theta}+i\sin{\theta})^n=\cos{n\theta}+i\sin{n\theta}

Taking n=5 we have:

(\cos{\theta}+i\sin{\theta})^5 =\cos{5\theta}+i\sin{5\theta}

The binomial Newton’s theorem says: (x+y)^n= \sum_{k=0}^n C_k^n x^k \cdot y^{n-k}

So,, we can expand (\cos{\theta}+i\sin{\theta})^5=\sum_{k=0}^5 C_k^5 (\cos{\theta})^k \cdot (i\sin{\theta})^{5-k} (C=combination)

Generally i is defined as i=\sqrt{-1} \rightarrow i^2=-1;i^3=-i;i^4=1;i^5=i

(\cos{\theta}+i\sin{\theta})^5= i\sin^5{\theta}+ 5\sin^4{\theta}\cdot \cos{\theta} - 10i\sin^3{\theta}\cdot \cos^2{\theta}-10\sin^2{\theta}\cdot\cos^3{\theta}+i\sin{\theta}\cdot\cos^4{\theta}+\cos^5{\theta}

Thus,,

\cos{5\theta}+i\sin{5\theta}= i\sin^5{\theta}+ 5\sin^4{\theta}\cdot \cos{\theta} - 10i\sin^3{\theta}\cdot \cos^2{\theta}-10\sin^2{\theta}\cdot\cos^3{\theta}+i\sin{\theta}\cdot\cos^4{\theta}+\cos^5{\theta}

 

That’s why::

\cos{5\theta}=\cos^5{\theta} +5\sin^4{\theta}\cdot \cos{\theta}-10\sin^2{theta}\cdot \cos^3{\theta}

\sin{5\theta}=\sin^5{\theta}+ i\sin{\theta}\cdot \cos^4{\theta} -10i\sin^3{\theta}\cdot \cos^2{\theta}

\sin^2{x}=1-\cos^2{x}

\cos{5\theta}

=\cos^5{\theta} +5\sin^4{\theta}\cdot \cos{\theta}-10\sin^2{\theta}\cdot \cos^3{\theta}

=\cos^5{\theta} +5(1-\cos^2{\theta})^2\cdot \cos{\theta}-10(1-\cos^2{\theta}\cdot \cos^3{\theta}

=\cos^5{\theta}+5(1-2\cos^2{\theta}+cos^4{\theta})\cdot \cos{\theta}-10\cos^3{\theta}+10\cos^5{\theta}

=\cos^5{\theta}+5\cos{\theta} -10\cos^3{\theta}+5\cos^5{\theta} -10\cos^3{\theta}+10\cos^5{\theta}

=16\cos^5{\theta}-20\cos^3{\theta}+5\cos{\theta}

Or we can rewrite like this:

\cos{5x}=\cos{x }(16\cos^4{x}-20\cos^2{x}+5)

By putting x=\frac{\pi}{10} we have

\cos{\frac{\pi}{2}}=\cos{\frac{\pi}{10}}(16\cos^4{\frac{\pi}{10}}-20\cos^2{\frac{\pi}{10}}+5)=0

As \cos{\frac{\pi}{10}}\not = 0 we may conlude that x=\frac{\pi}{10} is the root of 16x^4-20x^2+5=0

Thanks for your attention (どうも　ありがとう　ございます。)

9 Comments

  1. rocky said,

    March 12, 2008 at 11:55 pm

    hello!!
    i was just serching about ntu entrance test on yahoo..
    and i found this page..

    i think u r also a foreigner student who take ntu entrance test..
    did u take the test?

    oh.. sorry..
    i’m alos a foreigher student and i’m going to take the test
    if u dun mind.. may i share the imfomation about the test?
    i know it must be weird for u..
    but i have no idea about the test so…

    if i was rude or disturb you i’m so sorry..

    Reply

  2. mharteens said,

    March 13, 2008 at 2:02 pm

    Hi Rocky..
    sorry for writing some sentence in Bahasa Indonesia…
    I’ve written that I didn’t take the test (I wrote this in Bahasa)
    coz I’ve been awarded Monbukagakusho Scholarship Rocky..

    Yes,, of course we may share the information about the test..
    thanks for your attention.. :smile:

    Reply

  3. rocky said,

    March 13, 2008 at 9:44 pm

    Oh thank u for the answer!!!
    actually i didn’t expect u would answer back for me..
    thank u very much!!

    wow !! so u already entered ntu..huh???
    it’s cool man!!
    i mean i’m waiting soooooooo long for entering ntu.
    it’s almost 1 year..huhu..
    but the meantime i didn’t prepare for the test..

    anw finally and fortunately..i got a chane to take the test..
    my test is coming soon..on april
    so i’m worried about my test..

    anyway thanks a lot man!!^^
    have a gd life in ntu and s’pore^^

    Reply

  4. mharteens said,

    March 14, 2008 at 1:00 am

    wow..there’s a misunderstanding ..^^
    I didn’t take the test,, coz I’ve chosen to study in Japan..
    I’ve got the MEXT Scholarship..

    Good luck for your coming NTU entrance test.. :smile:
    がんばって。。

    Reply

  5. ardianto said,

    April 13, 2008 at 9:06 pm

    Keren Tin…
    Bisa nomor 1 di google pakai keyword yang menjual
    Coba ketik “NTU Entrance Test” di Google..
    Blog kau nomer 1…

    Reply

  6. mharteens said,

    April 13, 2008 at 10:15 pm

    ah..masak sih??
    wah..jadi terharu.. :mrgreen:
    yg aku tulis gak salah kan!?malu bener kalo salah.. :grin:
    hmm..kalo diliat dari statistic sih,,
    ini peringkat ketiga..dengan 94 views..

    thanks 4 the info Ardianto..

    Reply

  7. ardianto said,

    April 21, 2008 at 10:15 pm

    がんばって。。

    Ganbatte kan bacanya? Artinya apa Tin?

    Jawabannya bener kok… :)

    Reply

  8. akbar said,

    October 17, 2008 at 10:46 pm

    bang martin jago banget nih….hohohoho
    NTU NTU…..bikin orang makin gendeng…
    *keluh kesah orang yg ditolak

    Reply

  9. kevin said,

    February 16, 2009 at 2:08 pm

    Teen, keren banget dpt sholar di jpn. Gimana dptinnya? kasi tau dong prosesnya. Aku mo coba, tp denger2 testnya lbh kaco dr ntu, tul ga?

    Reply

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