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.

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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: https://www.seh.ox.ac.uk/people/tom-c...

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