A Level Maths Resource Site

Ok, I have been doing past year papers and I always drop so many marks on stupid errors.

I am not joking when I say this.. but I could do a pastpaper and purposely aim not to make any careless mistakes, yet my work will still be riddled with them!

I am so worried this will pull down my grades in exams. Any advice on how to deal with it?

Student X

At times 1+1 just equals to 3 when you are under immense stress, so I usually advise my students not to inspect a recently completed problem for too long (and simply move on) , because chances are, you might not be able to identify the simplest of errors when you are so invested in solving the question.

Come back to it when the entire paper is dealt with; in that sense when you look at the same problem once again, your state of mind is of a refreshed one. Hence a greater likelihood of you being able to detect your faults in a heartbeat. Just don't suffer a heart attack and get down to fixing things in a calmly manner.

Peace.

Best Regards,

Mr Koh

On what intervals of t is the curve described by the given parametric equation concave up? Concave down?

x=t^2  ;      y=t^(3) + 3t

Student  X

The signage of the second order derivative( ie positive,negative or zero) would help to discover the nature of concavity for the curve at various intervals.

dy/dx =dy/dt * dt/dx = (3t^2 + 3)/ (2t)

(d^2 y)/(dx^2)= d/dx (dy/dx) = d/dt (dy/dx) * ( dt/dx)

                   =  [2t*(6t)-2(3t^2 + 3)]/(4t^2)  * 1/(2t)

                   =  (6t^2 - 6)/(8t^3) =  (3t^2 - 3)/(4t^3)     

(Note that since dy/dx is actually a function in t, implicit differentiation must be employed when seeking the expression for (d^2 y)/(dx^2) .)

When the curve is concave downwards,  (d^2 y)/(dx^2)<0 =====> (3t^2 -3)/(4t^3) < 0

                                                                                           [(t-1)(t+1)]/ (t^3) < 0

                                                                           ie  t < -1   or    0 < t < 1  (shown)   

When the curve is concave downwards,  (d^2 y)/(dx^2)   =====> (3t^2 - 3)/(4t^3) > 0

                                                                                          [(t-1)(t+1)]/ (t^3) > 0

                                                                           ie  -1 < t < 0   or    t > 1  (shown) 

Hope this helps. Peace.          

Best Regards,

Mr Koh

Student X

Your answer isn't wrong, it just requires further manipulation. Kindly see the following:

Hope this helps. Peace.

Best Regards,

Mr Koh

Hi Mr Koh,

                  I encountered this particular question in my revision package:

                    A solution was provided for it:

                  However, I don't understand the part which I have underlined-why do we need to subtract 

                  145 (T10)?

 Student  X         

Perhaps this slightly different manner of looking at things might help:

Change in height of the tree at the end of the 10th year is 145cm.

Height of the tree at the end of the 10th year is 775cm.

Then

height of the tree at the end of the 11th year is 775+ 145r    (r denotes the common ratio of the GP)

and

height of the tree at the end of the 12th year is 775+ 145r + 145r ^2,    so on and so forth.

Continuing things in this way,

At the end of the infinite year of growth, height of tree

= 775+ (145r + 145r ^2 +145r^3 +.................) -----------------------(1)

Noting that the terms in brackets for (1) belongs to the sum of infinity of a GP,            

(1) becomes  775 + 145r / (1-r) which is also = 2000

Migrating the value 775 to the RHS of the above equation gives 145r / (1-r) = 1225

Therefore 145r = 1225 -1225r =====> r= 245/274 (shown)

Hope this helps. Peace.

Best Regards,

Mr Koh

Could you please explain why when forming differential equations where the rate of change is negative i.e temperature loss, depreciating value,

you would have to write things as dy/dt= -ky

because ' k is a positive constant'- this is repeatedly emphasized in books.

why can't you just do this: dy/dt= ky and simply state that k is negative?

Student X

You are not wrong in your assertion; that is if the problem has cited specific conditions for you to further solve for the value of the unknown constant.

However, usually the minus sign is employed (ie letting k to be positive) because during instances when you are merely able to obtain the general solution, important exponential decay factors which contribute to stabilization of values in the long run (for example e^-kt) become more prominent/identifiable due to the minus sign showing up.

Hope this helps. Peace.

Best Regards,

Mr Koh

I have a short question to ask:

If  a, b and c are vectors,  and  a•b = c•b , is it absolutely correct so say a=c by comparison?

Student X

No. That is incorrect. Let me give you one simple counter-example:

[ Taking ( x, y, z) = x i + y j + z k  in general ]

Consider  a = ( 1, 0, 1)       b = (2, 1, 1)     c = ( 1, 1, 0) ,

While  a•b = c•b = 3,  clearly   a ≠ c.

Hope this clarifies. Peace.

Best Regards,

Mr Koh

The O level results will be released pretty soon and I doubt I can pass my exams save for my E-maths paper because I faced quite a few problems in the past year, for example being assigned lousy lecturers, having to put up with annoying construction noises beside my residence, shifting to a new home etc. I do not wish to retake my O levels as a second round of examination fees would be rather costly, especially for science-related subjects. On top of this, I would be required to officially take up lessons at a recognised private institution, which would in turn add to my financial burden, not to mention it could be a complete waste of time if the teaching over there is ineffective.

At first, I wanted to study electrical engineering; however since my Mathematics and Physics skills are below average and in addition to that I am a slow learner ( I believe my understanding of certain electrical principles in Physics is sketchy at best) . Thus, I don't think I would be able to do well in that discipline. Which in turn places a big question mark on my future career intentions- what should I do? If I may be honest, I am not even sure what I am exactly good at in particular at the current moment.

I would like to ask if you know of a counselor or a professional career adviser whom I can talk to, because right now, I am really stuck in no man's land. I have not a single clue with regards to how I should proceed from here.

Student X

Firstly, congratulations in drafting a full set of excuses to "account" for your current predicament. Lousy lecturers? How about taking the initiative to seek alternate avenues of help instead of " sitting and suffering in silence"? Annoying construction noises near your house you say? Heard of something called the library?

Secondly, you do not wish to retake your O levels, which pretty much severs all possible routes which could have given you access to higher education. Put it simply, studies are out, so declaring you are "unsure of certain electrical principles in your physics" or whatever other artsy fartsy dreamy personal academic inadequacies in that long reflective paragraph of yours becomes totally pointless wouldn't you agree?

As long as that defeatist streak remains untamed within you, no amount of advice from a well-meaning professional counselor is going to make a difference, because you will simply turn once again to putting the blame squarely on every cat, dog and circumstance; anything or anyone but yourself. By the way, let's be clear on one thing: given your current state, you should be worrying about landing a decent job. A career is a luxurious contemplation way beyond you.

Take a cold shower, harden your resolve to make a real change in your attitude towards life. That will be task number one. A very, very urgent task.

Peace.

Best Regards,

Mr Koh

A discussion thread online (involving mostly Singaporean youths) about which universities are well-recognised in the world eventually trained its focus on the National University of Singapore (NUS).

Reproduced below is my initial response to someone declaring NUS to be ranked amongst the top internationally:

"NUS likes to imagine it is amongst the big boys on the international stage of varsity education, unfortunately imagination and actual reality are miles apart.

It embraces the statistical metrics which cast its academics in good light, on the other hand becoming extremely defensive against those which provide cutting yet valid criticisms. This is the National University of Singapore for you.

That said, NUS is still very much a decent place to get your degree.

Peace. "

A forumner therefore further queried:

"Would NUS still be the best option for a Singaporean student not looking to study overseas though? I've always wanted to enter NUS due to its fairly impressive place in the worldwide rankings, but what you said has got me thinking a bit. "

I replied :

"If you are looking to study locally, NUS should definitely be on your list .( I won't say it is the best option in the absolute sense though, because this is a situation of "to each his own"). After all Singapore is so tiny, the choices available are extremely limited. Needless to say, you can also consider NTU and SMU, as well as the newly up and running SUTD.

Interestingly though, NTU is seen in a less favorable light compared to its peer institutions NUS and SMU, or at least this is the trend I have observed amongst my charges who apply for local universities over the past 5, 6 years. Most of them would rather settle for a course they bluntly dislike in NUS/SMU, than pursue something they are crazily passionate about in NTU. Some food for thought here.

Above all, while the education system in Singapore is no doubt evolving and self-renewing, the learning atmosphere here is still rather stifling IMHO. So if you can afford to uproot, fly far, far away to the States or the UK to secure your Bachelor's. A different country, a different schooling experience altogether will help you mature faster.

Good luck. Peace. "

Mr Koh (18 December 2012)

I am pretty much stuck with this SLE question, as I am not exactly sure how to use my graphic calculator to solve for x, y and  z:

x+y+z  = 2a   

2x-y+z = 3a-2b

x-2z = 2b-a     

where a and b are unknown constants.

Student X

There is no need to use the graphic calculator for this. The problem is uniquely designed-how so?

x+y+z  = 2a   -----------------(1)

2x-y+z = 3a-2b  -------------(2)

x-2z = 2b-a    ----------------(3)

(1)+(2)+(3): you get 4x  = 4a =======> x = a;

with this, it isn't hard to discover that  y =b     and   z = a-b   (shown)

Hope this helps. Peace. 

Best Regards,

Mr Koh

Do you think that secondary Maths teachers in the UK would be able to solve the International Mathematical Olympiad questions?

How do they compare to say, STEP or the Cambridge Tripos?

Student X

I can't comment on the state of affairs within the UK since I clearly don't reside there, but here in Singapore, probably not.

Conquering IMO problems requires a set of skills vastly different from those used in preparing for the A levels/ mainstream Cambridge examinations. The latter can be typically aced with rigorous exposure to run of the mill questions, so in that sense practice does indeed make things perfect.

However for the former, you need imagination, and the ability to hypothesize/reason in unconventional ways to produce elegant, efficient answers. Stuff mainstream secondary school teachers in Singapore are definitely not trained for.

Not to mention you need to possess a certain flair to see through the extremely fine cracks and disassemble the context accordingly. That can't be nurtured IMHO; either you have it, or you don't.

My two cents worth. Peace.

Best Regards,

Mr Koh