Korean SAT Exam Question with

University of Oxford Mathematician Dr Tom Crawford works through his solution to a killer question from the 2022 Korean Suneung CSAT exam with some help from @pkmath12345.
Use my link gauthmath.onelink.me/SUq5/d58... to download Gauthmath and don't forget to use code 4SBW8X to get a 1 month discount now!
Check out PK's channel here: / drpkmath
The question is taken from the Korean SAT exam taken in November 2022. This was apparently a 'killer' question which hardly any students were able to solve within the time limit. The exam is taken at the end of high school as part of the college admissions process in the Republic of Korea (South Korea).
PK begins by explaining a little about the exam before sharing the question. We are asked to work out the value of a cubic function at the point x=2, but in order to determine the coefficients of the cubic we must solve a series of equations involving exponential functions, trigonometric functions, derivatives and composite functions.
The full question is as follows:
A cubic function f(x) with positive leading coefficient, the function g(x) = exp(sin(pi*x)) - 1, and the composite function h(x) = g(f(x)) on the domain of all real numbers, satisfy the following conditions:
1) Function h(x) has the maximum value of 0 when x is 0.
2) In the open interval (0, 3), there are 7 distinct real solutions to h(x) = 1.
3). f(3)= 1/2, f’(3)=0, and f(2)= p/q.
What is p+q? (where p and q are coprime integers).
This video is sponsored by Gauthmath.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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[@TomRocksMaths]

Use my link gauthmath.onelink.me/SUq5/d58... to download Gauthmath and don't forget to use code 4SBW8X to get a 1 month discount now!

[@pkmath12345]

Thanks Dr. Tom for our collaboration. Just wanted to point out one thing. h(x) has local maximum of 0 when x is 0. Even though Dr. Tom and I took it as a local maximum when solving this question, I missed saying that when reading the question for Dr. Tom. Hope everyone enjoys👍👍👍

[@geochum]

Okay, that's what I was contemplating when I first read it. The way it was written on the board implied h(x) has a global maximum of 0 at x=0. And if that was the case, condition (2) would cause a contradiction. Glad you cleared things up.

[@david1610]

@DrPKMath what I noticed immediately when I started to watch this was how "boomy" your voice is in that classroom. Perhaps consider a local mike or recording in a room that has kinder acoustics!

[@zakolache4490]

In terms of test taking strategy, this definitetly one of those to skip & come back to later, if possible. Hell of a question

[@zingery4746]

Me: Doesnt understand a drop of math Also me: spent 30 mins of my Sunday watching two people find out the answer to the question a teenager would take normally

[@Min-ou8ti]

This feels nostalgic to me as I've been through these kinds of questions back in my high school years

[@sunggyulee9020]

Wow, thats insane. This is a question from a standardized test, meaning there is a time limit. I took the same algebraic approach as Tom and it took me a while. However, that approach takes too much time for the real test. This means that Korean examiners expect students to take the graph analysis approach, which requires quite a deep understanding of functions. I doubt that even the average engineer student in the us would be able to answer this that quickly.

[@Min-ou8ti]

Well tbf no average Korean high school student(myself included unfortunately...) could answer this reliably fast enough during a real SAT. This problem is truly one of the cream on the top, only meant to be solved by the top 1~5% who enter uni here that would be Korean equivalent(in terms of entrance difficulty at least) of Cambridge, Oxford etc Gotta distinguish those monsters from the rest somehow...

[@ThePhysicsMathsWizard]

Hello @Tom Rocks Maths & @DrPKMath great to see you working together on this one. As DrPKMath said at the start, it was indeed a 'killer question' especially for that level, but its ok for such a competitive exam. I like both methods :)

[@tiernanomahoney3925]

Hi, high school student from Ireland 🇮🇪 here. I solved this got 22/9, 22+9=31. Like Dr PK, I used a more graphical approach. Overall, problem is well-made.

[@jacobdaniels3246]

I'm American and I was able to solve this, but this is very difficult compared to anything we study in school. It's not even required to know calculus before college here.

[@drvanhelsingz5133]

The high school kid who does this in 10 should skip university and just start their own tech company. 😂

[@okanatalay3227]

In the second part, after you cleaned the board, it should be written 27a+9b+d=1/2, not =0. The way you solved the problem was an inspiring one. Looking forward to seeing your new questions and solutions. 👍

[@jonathanbrewer7072]

Thank you, Tom

[@leonkayombo4316]

Intelligible solutions. Good sense of smile, too

[@dd6240]

If my Oxford interview question for maths in 2 weeks looks like this I’m leaving the call😂

[@crazy4hitman755]

Looool same

[@chiragsharma5050]

It isn't hard like korian sat

[@zawaarudo1995]

How did it go?

[@dd6240]

@@zawaarudo1995 got an offer 💪🏾

[@choibeomgyuloml794]

@@dd6240 congrats! how hard would you say the interview was? similar to MAT or was it closer to STEP?

[@rayhanalam9651]

I just love the fact how you are the exact opposite of how I imagined an Oxford Professor. Take love from Bangladesh.

[@benknox9743]

I’m excited

[@pekkakontkanen]

Tom rocks 👍

[@eyupaydn8871]

Good day, teacher. I would appreciate it if you explain how we make the transition from exponential numbers to radical numbers.

[@joshwylie1772]

Don’t know how I would handle that in an exam

[@darcash1738]

Dang I figured it had more complex techniques involved and followed it down the wrong rabbit hole

[@arkapravorajkonwar8020]

i too got stuck between 7 or 8 , thinking after a lot of time i just gave up and thought either one is correct ; i got answers 86 and 31.

[@soyokou.2810]

I wonder if you could use a depressed cubic to use fewer conditions

[@davidc4408]

As a maths graduate from UK, this is certainly something you do at first year level and not really high school. High school teen maybe able to do with further maths A level

[@kassidydavis5565]

I absolutely love watching mathematicians doing math on a chalkboard 😍❤️ it makes my love for nerds grow exponentially it’s so soothing 😂🤣

[@hivan04]

i’m so happy i did a level maths 😭😂

[@user-ht4oq2cx2g]

Korean college admission math problems have essentially become complex calculus puzzles, because the curriculum being solely focused on calculus. In the past, subjects like matrices and complex planes were taught, but many people gave up on mathematics, leading to their removal from the curriculum. As a result, the difficulty of the remaining calculus increased. Therefore, as a joke, people sometimes refer to 'Cubic Function Research Institute' (implying that while other mathematical research institutes deal with problems of modern mathematics, Korean research institutes study cubic functions for college entrance exams)

[@aidanhancock2117]

..... i suck at maths, despite excessive revision and i think this pretty much confirms its one of those things you either instinctively understand, or suck at.

[@mujtabaalam5907]

This is not a typical math problem

[@DarkBoo007]

Even if I knew how to do this problem, I would've skipped it. The complexity and length in just a short amount of time... It'll take me around 25 minutes to complete this problem :(

[@yeeunpark8291]

I’m Korean !

[@Rodhern]

Lol, I am four minutes in and already lost track of what is going on, 1) h(x) has maximum 0, 2) there are real solutions to h(x)=1. Uhm, 1 is larger than 0. Going to watch some more now. Edit: For those looking to the comments, like I did an hour or so ago, the maximum in question is a LOCAL maximum. So, in short, in the following information "on the domain of all real numbers, h(x) has the maximum 0", the important part is the comma, i.e. the maximum value is NOT for "all real numbers". Still a very enjoyable video, thanks Tom. Edit2: The fact that it is a local maximum has since been addressed by DrPKMath in another comment.

[@thatfamousscientist9786]

Interesting...

[@clairep8225]

This seems awfully complicated for an exam to get *into* university 😳😳

[@0FFICERPROBLEM]

yeah... insane

[@Simonpeter316]

As a Korean high-school student, this is how it always has been. Haha

[@mujtabaalam5907]

You don't need to answer it, it's mainly there as a time sink / differentiator for the 99.99th percentile

[@adityasingh3963]

Not as hard as questions in JEE Advanced exam. (Exam to get into best engeneering University in India.)

[@mooshiros7053]

@@adityasingh3963 JEE advanced is kind of unique though because you have to do well on the JEE main first to even be able to take it. The CSAT is taken by every Korean high schooler, no matter what school they want to go to.

[@benknox9743]

I study Scottish advanced higher mathematics could you potentially do a sqa exam that’d be awesome

[@sammurphy1103]

That would be indeed.

[@Jackjanes24]

Why is the ratio between the two points 1:2? Am I being stupid?

[@xXspacecowboy2011Xx]

At about 24:40 shouldn’t it be sin(piZ)=ln(2) instead of log(2)? You clearly know what you’re doing…but I get confused easily!

[@JO06]

Hes taking logx=lnx its just notation

[@user-us9kp7nu1e]

Try solving jee advanced maths questions

[@katkit4978]

I'm writing as a student who solved the problem shown in the video in the actual college entrance exam. When you first encounter that type of problem, everyone has a hard time solving it, and most of them try an algebraic approach. However, if a composite function comes out on the Korean SAT math test, I would have been able to solve it more simply because I knew that I would always solve it using a graph. Moreover, Korean academies train students by making many problems very similar to Korean SAT math, so students can solve those high-level problems in three to four minutes. (In fact, the problem shown in the video is not a difficult problem compared to the problems made at the academy!)

[@chiragsharma5050]

Rip to all Asians in my country condition is same fuck education system

[@qwerty-bs4xk]

부엉이 ㅎㅇ

[@henrymarkson3758]

This is a great question. It requires little more than tools available to high school calculus students. This should be the last question in a final year high school exam to differentiate the good students from the truly talented students who have thought deeply about functions and calculus. Comparing the 2 methods in this video. If this exam allowed the use of technology (a graphics calculator) I would vote for PK, because the graphical exploration of the behaviour of functions is a legitimate approach and part of the art of maths. If the exam did not allow the use of technology, I would vote for Tom's classical approach, it's much more intuitive. Therefore, great problem and the result: Tom 1 - PK 1

[@user-en6qh7zx6f]

It is not allowed to use any tools And I had tried that question and it took 20m😢 I solved it in different way using the shape of graph I think condition2 is the most important

[@gyuro7988]

No tools and it's a 100 min test for 30 questions so average 3.3 min each question. Students are probably expected to know how to solve it using the graphical approach as it's faster, and thats means they need to know the graphs in their head. Tom's approach is too slow and won't be enough time to solve it in time (competitive unis require a near perfect score in maths but to be fair the incorrect rate for this question was 95%)

[@dave_thebrave]

the sat exam questions for when nashville said "i did not have sex."

[@meowpurrr]

is this comparable to a british a level or gcse?

[@mouldyvinegar5665]

Not comparable at all, both A-Level and especially GCSE are much easier. The closest parallels would be something you get for a university admission test, such as the MAT.

[@meowpurrr]

@@mouldyvinegar5665oh ok, the mat isn't that difficult tho only contains like half of a2 level maths, is nothing compared the step 3

[@davidc4408]

​@@mouldyvinegar5665further math A level would cover that

[@oraz.]

If it's a timed exam I would just skip this, I'm sorry.

[@iteerrex8166]

Are you kidding me? Some high school kid did this in 10 mins? Or they were allowed 10 mins?

[@g76agi]

Considering their school culture i wouldnt be surprised if they did this problem in 4th grade

[@prima2178]

student who took the test here. That question was the last question of the test for the "Calculus" Branch. Compared to prior last test questions in the past years, this question was the easiest. For me it took about 5~7minutes. I know a friend of mine (A math wiz) who took about 3~4 minutes. For normal people who isn't friendly with math, this might look hard, but compared to the preparations that korean high schoolers undertake for this exam, this problem was relatively easy.

[@iteerrex8166]

@@prima2178 I had no idea of Korean curriculum, I was comparing to USA high schools, which are awful. Thankfully my dad was teaching me math at home, so I was ahead of school by 5 years.

[@warlord1981nl]

My answer to this question is: "No."

[@xiemo6704]

How often do you go goblin mode tom?

[@whatever__]

Take the jee main or jee advance math challenge ,

[@m.moonsie]

NO

[@steelbrotherhoodof2359]

a multiplication for a sine function so it can cross the zero on the y axxis. need to deliver also a zero out of a negative number. -1 × ( 1 +( 1 / -1))=0, or suchs wich is -y x ( 1 +( 1 ÷ y)) = + 1 for a sine function, (sin x (y ÷ ( 1 + (1/y))))×enlargedmentX delivers -1. so the zero axis can be crossed safetly without erros. although you can not use your calcutor you need to write your own math program. wich allow the zero calculation. although x and multplication sign X or a dot can be easily confused.

[@steelbrotherhoodof2359]

and then THAT formula, with a sine function is, the whole thing times sine.

[@steelbrotherhoodof2359]

where: y=-5 declare y a negative value of 5 for y=1 to 10 log it 10 times pset (x,y),10 plot a point on the y axx next y next in line a basic code for the former formula wich has no name yet. other then maybe "near value in sequence formula" wich sounds rubbish. or. "a not so random multiplication of self" formula.

[@steelbrotherhoodof2359]

a program can be written by raising for the comouter the y axcis above zero with a 100 or so. so it does not cross the zero line. while for the user display. it is compensated too apropiate data. so for the comouter it does not cross the zero but for the observer it does. wich is just display something else.

[@steelbrotherhoodof2359]

so the machine sees the value 100 in its calculatiom and the user sees zero as the x axx.

[@steelbrotherhoodof2359]

xxx

[@sachamoser9783]

Congagr5z

[@emilwandel]

I couldn't follow the second approach. To little explanation. Leaving to much to the viewer. I like graphical approaches, but I couldn't follow the prof. He was jumping from graph to graph and condition to condition. Clearly Tom is a more structured teacher who suits me more. I could follow him easily but it was a lengthy solution, which will always bring me to the end. I like learning a method which always brings me to the solutions but also think about variant solutions which are faster when you have the right insights and practice with these functions. I don't see the need for high schoolers to do these convoluted questions, what does it show that you practiced maths. It doesn't seem creative and rather methodical. Sad times maths should be more creative in school to keep the fun.

[@georgesheffield1580]

Murikan kids cant make change ,even alot of collage students cant . Wonder why , teachers and curriculum are bothe incompetant .

[@studentstudent5044]

Is the Kotean SAT only acceptable in Korea?

[@Simonpeter316]

As I know, there are some colleges that except the Korean SAT in the US and Canada. Maybe some other countries' colleges would except it too