So I took a look at some questions from a Korean
university entrance exam and I was pretty shocked by how hard they were.
This is the CSAT and I guess it's the South Korean version of the American SAT
which you may be more familiar with. But I think this Korean exam
ranks as much more difficult and is right up there with some of the world's toughest exams.
It is taken by Korean high school students and the score determines which university they can
go to. It is very competitive because many in Korea believe that a university's ranking makes
a huge difference to one's career prospects. To get into those top unis you might only be able to
get one or two questions wrong on the whole exam but that becomes seemingly impossible when you
realize that the exam covers five different topics on one day and takes over eight hours to complete.
I think one of the most difficult sections is mathematics. It covers calculus;
with limits, derivatives, integration, trig calculus, even integration by parts.
Plenty of conic sections, vectors, dot products, 3D geometry, probability and statistics.
And students would get three to four minutes to solve each question.
For question 15 we are told that two functions: y equals a to the x,
and y equals root three, meet at the point A. There's also this point B which is at four zero
and we're asked what is the product of all the values of the little a such that the straight
lines AB and OA are perpendicular to each other. Now if you ask me that sounds pretty tricky.
I heard the notorious questions hide near the end so let's check out question 30 on this 2020 paper.
What it says is that we have these two functions: one is a log and one is an exponential, and they
meet at only one point when little a can be said to be f(t). It's a function of this variable t.
Then we're asked to find the square of f prime when t is equal to one-third.
Have a think about what you might want to do here, i'll come back at the end
and show my version of a solution but first let's look at some other sections.
There's also Korean, English, History, Science and some other optional papers. The english test
starts off with a listening section and for this the whole country goes silent, no planes take off
or land in South Korea for that 30 minute period, so that shows how seriously this exam is taken.
For this recent question they had to listen to what seems like a youtube intro and identify the
purpose of the paragraph but later in the english section the questions get a lot more abstract with
language that even a native english speaker would find intense. There's a question about an African
proverb "till the lions have their historians, tales of hunting will always glorify the hunter".
And there's this paragraph here asking us about the fact that "people don't usually
think of touch as a temporal phenomenon but it is every bit as time-based as it is spatial".
The physics paper contains all the usual suspects but seems mostly based on conceptual understanding
rather than heavy calculations. Understanding an eclipse,
how to determine the wavelength of a laser, and orbital mechanics.
Here is just a glance at chemistry, biology, and geology and astrophysics.
500 000 students take the exam each year but around 20% of those are people taking it again
because they weren't happy with their previous score. Those students may keep studying for the
same exam year after year until they can get into a good university and move on with their
life. South Korea is an extremely well-educated country, their performance as well as enrollment
and graduation rates are ranked among the top few in the world. They have around 80 percent of high
school graduates going on to university so it may come as no surprise that there is also a booming
private coaching industry centered around passing this exam. They call them hagwons or cram schools.
So let's take a look back at that math problem. It seems you first have to be smart enough to figure
out that if our two functions would look like this for the exponential and this for the log that if
they touch at only one point it would be something like this, where the functions are tangent and
do not intersect. So we could set our function 1 equal to function 2 to find that point where they
touch. That will give us this star equation here but that doesn't immediately take us any further
instead we need a bit of a brain wave that we can also set the derivatives, that is dy/dx, equal to
each other because at this point, since they're tangent, the slopes of each line would be equal.
So we can differentiate each side and set them equal. Noticing that our right hand side hasn't
changed during this process we can replace it with the definition we had for it earlier
giving us a version without any a and just t's and x's, which can simplify down to give us that
all of this is equal to one and i'll call that square because we're also going to use that later.
This might seem like another dead end but we have to remember that we're being asked to find f prime
of t which I prefer to think of as da/dt and how would we get a da/dt? Well we could go back up to
the equation that we started with and if we were to differentiate both sides of this with respect
to t, this little a in here due to implicit differentiation would give us a da/dt term.
So that's how we can get it. I differentiate both sides of it with respect to t, remembering that
a is a function of t, a bit of product rule a bit of chain rule and you'll end up with this.
Now we're getting good at replacing this term in here which crops up again. I'm going to replace it
with the result of that initial differentiation so i'll end up with this line here.
Now really we want to get da/dt by itself since that's what we're interested in. We can get rid of
some of these terms by multiplying both sides by x minus t over t cubed. And also remembering that
earlier we found that all this was merely equal to one because that crops up right here we'll set
that equal to one, that simplifies things down quite a bit. A little bit more rearranging and
we get something for da/dt which is the same as what they're calling f dash of t and because they
wanted f dash of t at t is equal to one third, we can put in one third for our value of t,
giving minus eight. And then they want all of that squared so that would be sixty-four.
Perhaps the calculus and algebra involved here is not that advanced but the way that
it's arranged and the fact you need to keep going back to results you found
earlier and somehow knowing how to get your way out of these seemingly dead ends does make this
problem really difficult. Certainly a stretch to be able to do this in just three or four minutes
and remember for this exam just thinking that you can skip questions that are too hard
is not really a feasible option if you want to get those top scores
because getting just one or two wrong on the whole thing can actually make the difference.
If you want to rank at the top I think you're expected to be able to do all of them.
Best of luck to all students studying for this exam or a similar one. Thanks for watching this
video and you can check out others in the playlist to see how other countries might do things.
Thanks to my Patreon supporters for making this video possible
and a special shout out to today's Patron Cat of The Day, Maeve.