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The Tuition Enquiry From Hell

(11)
Tuition given in the topic of A-Maths Tuition Questions from the desk of at 10:20 pm (Singapore time)

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It’s midnight.

It’s that time of the year again.

A hardworking Miss Loi remained behind at The Temple to mark her student’s homework diligently, even as the last glow had long faded from the dying cinders of joss sticks in the area.

The silence was deafening, since the last noisy Jφss Sticks session ended hours ago, and no getai has ever been held to ruffle the eerie tranquility of Novena during certain times of the year.

A sudden cold chill swept through the air, as the sound of someone knocking on the Temple Gates unsettled the deathly stillness of this moonless night.

Verifying that she didn’t accidentally lower the environmentally-friendly 25° temperature of her aircon, Miss Loi went outside the room to find a tall, expressionless man at her doorway, dressed in traditional Chinese costume.

Wah! You just come from the Beijing Olympics Opening Ceremony or did NDP ended late? Can I help you Sir?

I need mathematics tuition.

replied the man in a soulless voice.

Which level are you? When are you available for tuition?

I take Imperial Examination. Only available 12-6am.

Errr … sorry we’re closed during those hours and … umm … how old are you?

Next week my 300th birthday. Must pass Imperial Exam. Must have tuition! MUST HAVE MISS LOI!

With that, the man raised his arms outwards and started hopping menacingly towards a terrified Miss Loi.

While retreating, Miss Loi quickly whipped out her ancient Nokia N73 to call a friend who happens to be moonlighting as a temple medium:

HELP! I’M BEING ATTACKED BY A 僵屍! HOW?!

After a brief moment, came this reply:

Each term in the expansion of represents each of the n hops that the 僵屍 makes.

Given that the fifth term is independent of x, you must write the coefficient of the middle term on a piece of yellow paper and paste it on his forehead in order to stop the 僵屍.

Luckily, a stack of bright kawaii yellow Post-It notes lay within reach on Miss Loi’s desk. Now all she needs is the value of that coefficient.

Having just taught a class on Binomial Theorem, she recalled that it’s a pretty straightforward topic with questions that should easily be done in 10-15 minutes to bag your 6-8 marks (provided you can recall the two formulae corresponding to generally two kinds of questions for this topic:

1. The ones that require you to expand an expression.
2. Or the kinds (like this question) that require you to find a specific term and/or its coefficient in an expansion, utilizing heavily the following formula:

In the expansion of (a + b)2, the (r+1)th term:

Tr+1 = nCr anrbr

*Note that the Binomial Coefficient nCr can be denoted by

So unless you’re attempting this question alone in the middle of the night, it should be easy for you to find the coefficient before the 僵屍 grabs Miss Loi by the neck and drags her to the netherworld to conduct maths tuition!

N.B. Some of you should find this question familiar 😉

Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

1. commented in tuition class

2008
Aug
10
Sun
11:26pm
2. krisandro commented in tuition class

2008
Aug
10
Sun
11:44pm

2

*scratch armpit*

*catch no balls*

3. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2008
Aug
11
Mon
12:27am

3

Cockroach: *holds breath* ... ... ... *exhales after 10 secs* *僵屍 resumes hopping*

*Strikes out Krisandro's name from the cast coz he didn't take AMaths*

4. |c3^sNoW commented in tuition class

2008
Aug
11
Mon
11:12am

4

wah miss loi is environmentally friendly! i like!
this sounds like my A lvl last time leh! forgot how to do liao. wahaha. sorry cant save u miss loi >.<

5. DK commented in tuition class

2008
Aug
11
Mon
11:52am

5

wah.... didn't know need to use Amath to fight 僵屍 nowadays. I sure die 1.

6. ignorantsoup commented in tuition class

2008
Aug
11
Mon
8:05pm

6

OMG. By the time you finish solving the question you kena sucked liao.

7. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2008
Aug
12
Tue
12:08am

7

Night has fallen ... time to hide in closet again ... *closes closet door quietly*

8. NAOMI! commented in tuition class

2008
Aug
12
Tue
4:19pm
9. Li-sa commented in tuition class

2008
Aug
12
Tue
9:42pm

9

Miss Loi is quietly looking at your workings from inside the closet ...

The 5th term is a constant. Let the (r+1)th term be

The fifth term
=

The good thing about Binomial questions is that there's not many formula you need to work with.

So to get the coefficient of the middle term in this question, you'll first need to find the value of n BEFORE you can even know WHICH is the middle term.

The only clue given is that the 5th term is independent of x ⇒ the power of x in T5 is zero.

So the key here calculate n by EQUATING the power of the x-component in T5 with zero.

From the calculations using the Tr+1 formula above, the x component of T5 is (x2)n-4x-4

Using a bit of indices magic, you can simplify the above into x2(n-4)-4

and you'll get your n as follows:

As the fifth term is a constant, power of x of this term = 0 = 2(n-4)-4, n = 6

Always, always pinch/slap/remind yourself that a binomial expansion of (a + b)n contains a total of n + 1 terms.

So since n is 6, the middle term of 0, 1, 2, 3, 4, 5, 6 is actually the 4th term T4, which also means that the value of r has to be 3!

N.B. There's no middle term if n happens to be an odd number 😉

Now coefficient of the middle term (or 4th term)
= Note that you don't have to bother with the x terms in the Tr+1 formula, since we're only interested in the coefficient.

*hurls a yellow Post-It note with "-160" on it and totally covered in paste at 僵屍's forehead and bullseye! 僵屍vanishes*

Exciting question but it seems that most visitors above this comment are just singing:
"wo~wo~wo~wo~Everybody was 僵屍 fighting~"

Ha, thinking of this yellow paper thingy, in my earlier secondary school days, the yellow paper was a warning that one has broke too many times of a certain school rule; break only 1 more time and get a demerit. Seems at that time everyone was really a 僵屍............................

10. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2008
Aug
13
Wed
12:40am

10

Thank you so much Li-sa for saving Miss Loi from the 僵屍!

But there's a big mistake in your comment: in the movies, sticking yellow papers to their foreheads only stop them temporarily, they most certainly don't VANISH!

Wah Naomi, you actually tried this question ... finished my E-Maths homework already?

Anyway, high chance that students will encounter such a 僵屍 Tr+1 in your exams too!

11. Li-sa commented in tuition class

2008
Aug
14
Thu
9:01am

11

I don't really care about that "mistake" because there are no actual reports of jiang shi anyway. Only spirits that possess humans and make them go crazy.

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开课了!!!

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* Please refer to the relevant Ten-Year Series for the questions of these suggested solutions.