1-1. What is the area of the region bounded by the graphs of the two functions $y = 5 - |x - 1|$ and $y = |x|$? (1) $8$ (2) $9$ (3) $15$ (4) $12$
1-2. A completely calm balloon loses 5 percent of its air per day. After several days, half of the initial air remains. How many days does it take? $(\log 19 = 1.287,\ \log 2 = 0.301)$ (1) $17$ (2) $18.5$ (3) $21.5$ (4) $25$
1-3. From the equation $\log(x+2) + \log(2x-1) = \log(4x+1)$, what is the value of $\log(5x+2)$ in base 4? (1) $0.5$ (2) $0.75$ (3) $1.25$ (4) $1.5$
1-4. The graph below shows the function $y = a + b\cos\!\left(\dfrac{\pi}{2}x\right)$, with period $(4,\ 0)$. What is $b$?
[Figure: Graph of a cosine-based function with maximum value $4$ and minimum value near $0$, symmetric about the y-axis] (1) $-2$ (2) $-1$ (3) $1$ (4) $2$
1-5. How many distinct real roots does the equation $2 = (x^2 - 2x)^2 - (x^2 - 2x)$ have? (1) $1$ (2) $2$ (3) $3$ (4) $4$
1-6. If $f(x) = x + |x|$ and $g(x) = |x+1| + 1$, then the range of $\left(\dfrac{f}{g}\right)(x)$ is: (1) $[0,1)$ (2) $[0,2)$ (3) $[0,+\infty)$ (4) $[1,+\infty)$
1-7. Which one of the following functions is one-to-one? (1) $f(x) = x + \sqrt{x}$ (2) $g(x) = x - \sqrt{x}$ (3) $h(x) = 2x + \dfrac{1}{x}$ (4) $p(x) = \dfrac{x}{x^2+1}$
1-8. What is the general solution of the trigonometric equation $\sin 2x \sin 4x + \sin^2 x = 1$? (1) $k\pi + \dfrac{\pi}{6}$ (2) $(2k+1)\dfrac{\pi}{6}$ (3) $k\pi - \dfrac{\pi}{6}$ (4) $\dfrac{k\pi}{6}$
1-9. What is $\cos^{-1}\!\left(\dfrac{1}{2}\cot\dfrac{11\pi}{3}\right)$? (1) $-\dfrac{\pi}{3}$ (2) $-\dfrac{\pi}{6}$ (3) $\dfrac{\pi}{3}$ (4) $\dfrac{5\pi}{6}$ %% Page 4
\textbf{1-1.} What is the area of the region bounded by the graphs of the two functions $y = 5 - |x - 1|$ and $y = |x|$?
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(1) $8$ \hfill (2) $9$ \hfill (3) $15$ \hfill (4) $12$
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\textbf{1-2.} A completely calm balloon loses 5 percent of its air per day. After several days, half of the initial air remains. How many days does it take? $(\log 19 = 1.287,\ \log 2 = 0.301)$
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(1) $17$ \hfill (2) $18.5$ \hfill (3) $21.5$ \hfill (4) $25$
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\textbf{1-3.} From the equation $\log(x+2) + \log(2x-1) = \log(4x+1)$, what is the value of $\log(5x+2)$ in base 4?
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(1) $0.5$ \hfill (2) $0.75$ \hfill (3) $1.25$ \hfill (4) $1.5$
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\textbf{1-4.} The graph below shows the function $y = a + b\cos\!\left(\dfrac{\pi}{2}x\right)$, with period $(4,\ 0)$. What is $b$?
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\textit{[Figure: Graph of a cosine-based function with maximum value $4$ and minimum value near $0$, symmetric about the y-axis]}
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(1) $-2$ \hfill (2) $-1$ \hfill (3) $1$ \hfill (4) $2$
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\textbf{1-5.} How many distinct real roots does the equation $2 = (x^2 - 2x)^2 - (x^2 - 2x)$ have?
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(1) $1$ \hfill (2) $2$ \hfill (3) $3$ \hfill (4) $4$
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\textbf{1-6.} If $f(x) = x + |x|$ and $g(x) = |x+1| + 1$, then the range of $\left(\dfrac{f}{g}\right)(x)$ is:
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(1) $[0,1)$ \hfill (2) $[0,2)$ \hfill (3) $[0,+\infty)$ \hfill (4) $[1,+\infty)$
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\textbf{1-7.} Which one of the following functions is one-to-one?
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(1) $f(x) = x + \sqrt{x}$ \hfill (2) $g(x) = x - \sqrt{x}$ \hfill (3) $h(x) = 2x + \dfrac{1}{x}$ \hfill (4) $p(x) = \dfrac{x}{x^2+1}$
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\textbf{1-8.} What is the general solution of the trigonometric equation $\sin 2x \sin 4x + \sin^2 x = 1$?
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(1) $k\pi + \dfrac{\pi}{6}$ \hfill (2) $(2k+1)\dfrac{\pi}{6}$ \hfill (3) $k\pi - \dfrac{\pi}{6}$ \hfill (4) $\dfrac{k\pi}{6}$
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\textbf{1-9.} What is $\cos^{-1}\!\left(\dfrac{1}{2}\cot\dfrac{11\pi}{3}\right)$?
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(1) $-\dfrac{\pi}{3}$ \hfill (2) $-\dfrac{\pi}{6}$ \hfill (3) $\dfrac{\pi}{3}$ \hfill (4) $\dfrac{5\pi}{6}$
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