Korean CSAT Mock Exam September #21

Calculus Level 4

Two functions $f(x) = \sin(kx)+2$ and $g(x) = 3\cos{12x}$ are defined at a closed interval $[-2\pi,2\pi]$ . Let $a$ be the $y$ value(s) of the point(s) where $f(x)$ and $g(x)$ intersect.

How many natural numbers $k$ satisfies $\{ x|f(x)=a \} \subset \{ x|g(x)=a \}$ ?

BONUS: How many natural numbers $k$ that satisfies $\{ x|g(x)=a \} \subset \{ x|f(x)=a \}$ instead?

This is a problem from 2020.

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1 solution

Inquisitor Math
Nov 10, 2020

The following is my approach for this problem: Let’s focus on one period of each function. (Denote those as $f_1, g_1$ .) It is easy to observe that an even number of $g_1$ should ‘fit into’ $f_1$ . (This is because both functions f,g are periodic and symmetric.) Thus $\dfrac{T_g}{T_f} = \dfrac{12}k$ is an even number. This is equivalent to saying that $k$ divides 6. Therefore, $k=1,2,3,6$ .

(If you solved this differently, plz share!)

Korean CSAT Mock Exam September #21

This is a problem from 2020.

1 solution

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