1-1. If $f(x) = 3 - e^x$, and $g(x) = \sqrt{x f^{-1}(x)}$, what is the domain of $g$? (1) $[0, 2]$ (2) $[0, 3)$ (3) $[2, 3)$ (4) $[1, 3)$
1-2. For which values of $a$ does the second-degree equation $a = 0$, i.e., $x^2 - 2(a-2)x + 14 - a = 0$, have two positive roots? (1) $-2 < a < 2$ (2) $2 < a < 5$ (3) $2 < a < 14$ (4) $5 < a < 14$
1-3. The function $f(x) = a + \log_2(bx - 4)$ passes through the two points $(6, 2)$ and $(10, 12)$. What is $a$? (1) $3$ (2) $4$ (3) $5$ (4) $6$
1-4. The figure shows part of the graph of the function $y = \dfrac{1}{2} + 2\cos mx$. At the point $x = \dfrac{16\pi}{3}$, what is the value of the function?
[Figure: Graph of a cosine-type function with period $4\pi$ shown on the $x$-axis] (1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) zero
1-5. The graphs of the two functions $y = 3^x + \dfrac{4}{3}$ and $y = \left(\dfrac{\sqrt{3}}{3}\right)^{2x}$ intersect at point A. What is the distance of point A from the point $(1, -1)$? (1) $1$ (2) $\sqrt{2}$ (3) $2$ (4) $\sqrt{5}$
1-6. For which value of $m$ is the sum of both roots of the second-degree equation $0 = 2x^2 - (m+1)x + \dfrac{1}{8}$ equal to $2$? (1) $3$ (2) $4$ (3) $5$ (4) $6$
1-7. If $f(x) = \dfrac{1+x^2}{1-x^2}$ and $g(x) = \sqrt{x - x^2}$, what is the domain of $g \circ f$? (1) $[0, 1)$ (2) $\{0\}$ (3) $(-1, 1)$ (4) $\mathbb{R} - \{1, -1\}$
1-8. What is $\sin\!\left(\dfrac{\pi}{2} + \cos^{-1}\!\left(-\dfrac{\sqrt{3}}{2}\right)\right)$? (1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) zero %% Page 4
\textbf{1-1.} If $f(x) = 3 - e^x$, and $g(x) = \sqrt{x f^{-1}(x)}$, what is the domain of $g$?
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(1) $[0, 2]$ \hfill (2) $[0, 3)$ \hfill (3) $[2, 3)$ \hfill (4) $[1, 3)$
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\textbf{1-2.} For which values of $a$ does the second-degree equation $a = 0$, i.e., $x^2 - 2(a-2)x + 14 - a = 0$, have two positive roots?
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(1) $-2 < a < 2$ \hfill (2) $2 < a < 5$ \hfill (3) $2 < a < 14$ \hfill (4) $5 < a < 14$
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\textbf{1-3.} The function $f(x) = a + \log_2(bx - 4)$ passes through the two points $(6, 2)$ and $(10, 12)$. What is $a$?
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(1) $3$ \hfill (2) $4$ \hfill (3) $5$ \hfill (4) $6$
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\textbf{1-4.} The figure shows part of the graph of the function $y = \dfrac{1}{2} + 2\cos mx$. At the point $x = \dfrac{16\pi}{3}$, what is the value of the function?
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\textit{[Figure: Graph of a cosine-type function with period $4\pi$ shown on the $x$-axis]}
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(1) $-\dfrac{1}{2}$ \hfill (2) $\dfrac{1}{2}$ \hfill (3) $1$ \hfill (4) zero
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\textbf{1-5.} The graphs of the two functions $y = 3^x + \dfrac{4}{3}$ and $y = \left(\dfrac{\sqrt{3}}{3}\right)^{2x}$ intersect at point A. What is the distance of point A from the point $(1, -1)$?
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(1) $1$ \hfill (2) $\sqrt{2}$ \hfill (3) $2$ \hfill (4) $\sqrt{5}$
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\textbf{1-6.} For which value of $m$ is the sum of both roots of the second-degree equation $0 = 2x^2 - (m+1)x + \dfrac{1}{8}$ equal to $2$?
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(1) $3$ \hfill (2) $4$ \hfill (3) $5$ \hfill (4) $6$
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\textbf{1-7.} If $f(x) = \dfrac{1+x^2}{1-x^2}$ and $g(x) = \sqrt{x - x^2}$, what is the domain of $g \circ f$?
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(1) $[0, 1)$ \hfill (2) $\{0\}$ \hfill (3) $(-1, 1)$ \hfill (4) $\mathbb{R} - \{1, -1\}$
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\textbf{1-8.} What is $\sin\!\left(\dfrac{\pi}{2} + \cos^{-1}\!\left(-\dfrac{\sqrt{3}}{2}\right)\right)$?
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(1) $-\dfrac{1}{2}$ \hfill (2) $\dfrac{1}{2}$ \hfill (3) $1$ \hfill (4) zero
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