1-1. If $\alpha = \sqrt[3]{3\sqrt{3}-4}$ and $\beta = \sqrt[3]{3\sqrt{3}+4}$, then the expression $(\alpha^2 + \beta^2 - \alpha\beta)(\alpha^2 + \beta^2 + \alpha\beta)$ equals which of the following? (1) $6$ (2) $8$ (3) $6\sqrt{2}$ (4) $7\sqrt{2}$
1-2. For which values of $m$, the parabola $y = (m-2)x^2 - 2(m+1)x + 12$ cuts the $x$-axis at two points with negative lengths (i.e., both $x$-intercepts are negative)? (1) $m > 2$ (2) $-1 < m < 2$ (3) every value of $m$ (4) no value of $m$
1-3. The graphs of two functions $f(x) = 3^{ax+b}$ and $g(x) = \left(\frac{1}{q}\right)^x$ intersect at a point with $x$-coordinate $-1$. If $f(2) = \frac{1}{3}$, what is $f^{-1}(27)$? (1) $-2$ (2) $-2$ (3) $1$ (4) $2$
1-4. The figure below shows part of the graph of the function $y = a - 2\cos\!\left(bx + \dfrac{\pi}{2}\right)$. What is $a + b$? [Figure: Graph of a cosine function; the curve passes through $y=1$ on the $y$-axis, with marked $x$-values at $\dfrac{\pi}{18}$ and $\dfrac{13\pi}{18}$] (1) $\dfrac{1}{2}$ (2) $1$ (3) $\dfrac{3}{2}$ (4) $2$
1-5. If the expression $a - 10 + 14x - 4x^2 + ax^3 + 4x^4$ is divisible by the three terms $x^2 - 2x + 1$, what is $a$? (1) $1$ (2) $2$ (3) $3$ (4) $4$
1-6. If the solution set of the inequality $\sqrt{3x+4} > 2|x-1| - x$ is the interval $(a, b)$, what is the length of this interval? (1) $\dfrac{5}{2}$ (2) $2$ (3) $\dfrac{7}{2}$ (4) $4$
1-7. The domain of the function $f(x) = \sqrt{1 - \log(x^2 - 3x)}$ in interval form is which of the following? (1) $[-2, \circ) \cup (3, 5]$ (2) $[-2, \circ) \cup (3, 5)$ (3) $[-2, 3)$ (4) $(\circ, 5]$ %% Page 4 Mathematics 120-C Page 3
\textbf{1-1.} If $\alpha = \sqrt[3]{3\sqrt{3}-4}$ and $\beta = \sqrt[3]{3\sqrt{3}+4}$, then the expression $(\alpha^2 + \beta^2 - \alpha\beta)(\alpha^2 + \beta^2 + \alpha\beta)$ equals which of the following?
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(1) $6$ \hfill (2) $8$ \hfill (3) $6\sqrt{2}$ \hfill (4) $7\sqrt{2}$
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\textbf{1-2.} For which values of $m$, the parabola $y = (m-2)x^2 - 2(m+1)x + 12$ cuts the $x$-axis at two points with negative lengths (i.e., both $x$-intercepts are negative)?
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(1) $m > 2$ \hfill (2) $-1 < m < 2$ \hfill (3) every value of $m$ \hfill (4) no value of $m$
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\textbf{1-3.} The graphs of two functions $f(x) = 3^{ax+b}$ and $g(x) = \left(\frac{1}{q}\right)^x$ intersect at a point with $x$-coordinate $-1$. If $f(2) = \frac{1}{3}$, what is $f^{-1}(27)$?
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(1) $-2$ \hfill (2) $-2$ \hfill (3) $1$ \hfill (4) $2$
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\textbf{1-4.} The figure below shows part of the graph of the function $y = a - 2\cos\!\left(bx + \dfrac{\pi}{2}\right)$. What is $a + b$?
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\textit{[Figure: Graph of a cosine function; the curve passes through $y=1$ on the $y$-axis, with marked $x$-values at $\dfrac{\pi}{18}$ and $\dfrac{13\pi}{18}$]}
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(1) $\dfrac{1}{2}$ \hfill (2) $1$ \hfill (3) $\dfrac{3}{2}$ \hfill (4) $2$
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\textbf{1-5.} If the expression $a - 10 + 14x - 4x^2 + ax^3 + 4x^4$ is divisible by the three terms $x^2 - 2x + 1$, what is $a$?
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(1) $1$ \hfill (2) $2$ \hfill (3) $3$ \hfill (4) $4$
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\textbf{1-6.} If the solution set of the inequality $\sqrt{3x+4} > 2|x-1| - x$ is the interval $(a, b)$, what is the length of this interval?
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(1) $\dfrac{5}{2}$ \hfill (2) $2$ \hfill (3) $\dfrac{7}{2}$ \hfill (4) $4$
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\textbf{1-7.} The domain of the function $f(x) = \sqrt{1 - \log(x^2 - 3x)}$ in interval form is which of the following?
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(1) $[-2, \circ) \cup (3, 5]$ \hfill (2) $[-2, \circ) \cup (3, 5)$ \hfill (3) $[-2, 3)$ \hfill (4) $(\circ, 5]$
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\textbf{Mathematics \hfill 120-C \hfill Page 3}
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