$$(3x-1)(x+1)+(3x-1)(x-2)=0$$ What is the sum of the real numbers $x$ that satisfy the equation? A) $\frac{2}{3}$ B) $\frac{3}{4}$ C) $\frac{3}{5}$ D) $\frac{5}{6}$ E) $\frac{7}{6}$
$$(2x-1)\left(4x^{2}-1\right)<0$$ Which of the following open intervals is the solution set of the inequality in real numbers? A) $\left(-\infty, \frac{-1}{2}\right)$ B) $\left(\frac{-1}{2}, 0\right)$ C) $\left(\frac{-1}{2}, \frac{1}{2}\right)$ D) $\left(\frac{1}{4}, \frac{1}{2}\right)$ E) $\left(\frac{1}{2}, \infty\right)$
$$f(x) = \sqrt{2-|x+3|}$$ Which of the following is the domain interval of the function? A) $3 \leq x \leq 5$ B) $-1 \leq x \leq 5$ C) $-3 \leq x \leq 4$ D) $-3 \leq x \leq 0$ E) $-5 \leq x \leq -1$
A function defined from real numbers to a subset $K$ of real numbers $$f(x) = \begin{cases} -x+8, & \text{if } x < 3 \\ x+2, & \text{if } x \geq 3 \end{cases}$$ Given that the function is surjective, which of the following is the set $K$? A) $[3, \infty)$ B) $[5, \infty)$ C) $[3,5]$ D) $(-\infty, 5)$ E) $(-\infty, 3)$
For given positive real numbers $a$, $c$ and negative real number $b$, $$a^{2}b > abc + c^{2}$$ Given that the inequality is satisfied, which of the following is necessarily true? A) $a = |b|$ B) $a = c$ C) $c > |b|$ D) $a < c$ E) $c < a$
Binary operations $*$, $\oplus$, $\odot$ defined on the set of rational numbers I. $a * b = a - b$ II. $a \oplus b = a + b + ab$ III. $a \odot b = \frac{a+b}{5}$ are defined as follows. Accordingly, which of these operations satisfy the associative property? A) Only I B) Only II C) Only III D) I and II E) II and III
$$P(x) = 2x^{3} - (m+1)x^{2} - nx + 3m - 1$$ Given that the polynomial is completely divisible by $x^{2} - x$, what is $m - n$? A) $\frac{-1}{3}$ B) $\frac{-1}{2}$ C) $\frac{3}{2}$ D) $2$ E) $3$
Which of the following is the domain of the function $f$ whose graph is given above? A) $[-3,0) \cup [4,7)$ B) $(-3,0) \cup (3,7]$ C) $[-3,2] \cup (3,7)$ D) $(-3,3) \cup (3,7]$ E) $[-3,2) \cup (4,7]$
The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as $$f(x) = \begin{cases} 2\sin x, & \text{if } \sin x \geq 0 \\ 0, & \text{if } \sin x < 0 \end{cases}$$ Accordingly, which of the following is the image of the open interval $(-\pi, \pi)$ under $f$? A) $[-2,2]$ B) $(-1,2)$ C) $[0,1]$ D) $(0,2)$ E) $[0,2]$
The function $f(x) = mx - 1 + \frac{1}{x}$ is given. Accordingly, what is the smallest value of $m$ that satisfies the property $f(x) \geq 0$ for all $x > 0$? A) $\frac{1}{2}$ B) $\frac{1}{3}$ C) $\frac{1}{4}$ D) $\frac{1}{5}$ E) $\frac{1}{6}$
Let $P(x)$ be a third-degree polynomial function such that $$P(-4) = P(-3) = P(5) = 0, \quad P(0) = 2$$ Given this, what is $P(1)$? A) $\frac{7}{3}$ B) $\frac{8}{3}$ C) $\frac{7}{4}$ D) $\frac{9}{4}$ E) $\frac{8}{5}$
The parabola $f(x)$ and the line $d$ are shown in the Cartesian coordinate plane above. Accordingly, which of the following systems of inequalities has the shaded region as its solution set? A) $\left.\begin{array}{l} y - x^{2} + 2x \leq 0 \\ y - x + 2 \geq 0 \end{array}\right\}$ B) $\left.\begin{array}{l} y - x^{2} + 2x \geq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$ C) $\left.\begin{array}{l} y - x^{2} + 4x \leq 0 \\ 2y - x + 2 \leq 0 \end{array}\right\}$ D) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 4 \leq 0 \end{array}\right\}$ E) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$
Let $A = \{1,2,3,4\}$ and $B = \{-2,-1,0\}$. For any element $(a,b)$ taken from the Cartesian product set $A \times B$, what is the probability that the sum $a + b$ equals zero? A) $\frac{1}{4}$ B) $\frac{1}{5}$ C) $\frac{1}{6}$ D) $\frac{1}{7}$ E) $\frac{2}{7}$
$$3\sin x - 4\cos x = 0$$ Given this, what is the value of $|\cos 2x|$? A) $\frac{3}{4}$ B) $\frac{3}{5}$ C) $\frac{4}{5}$ D) $\frac{7}{25}$ E) $\frac{9}{25}$
$$\frac{(\sin x - \cos x)^{2}}{\cos x} + 2\sin x$$ Which of the following is this expression equal to? A) $\frac{1}{\cos x}$ B) $\frac{1}{\sin x}$ C) $1$ D) $\arcsin x$ E) $\arccos x$
$$\frac{\tan 60^{\circ}}{\sin 20^{\circ}} - \frac{1}{\cos 20^{\circ}}$$ Which of the following is this expression equal to? A) 4 B) 2 C) 1 D) $\frac{\sqrt{3}}{2}$ E) $\frac{1}{2}$
$$\frac{1+\cos 40^{\circ}}{\cos 55^{\circ} \cdot \cos 35^{\circ}}$$ Which of the following is this expression equal to? A) $\cos 20^{\circ}$ B) $2\cos 20^{\circ}$ C) $4\cos 20^{\circ}$ D) $\cos 40^{\circ}$ E) $2\cos 40^{\circ}$
In the complex number plane $$|z-1| = |z+2|$$ Which of the following does this equation represent? A) The line $x = 1$ B) The line $x = \frac{-1}{2}$ C) The line $x = 2$ D) The circle $(x-1)^{2} + y^{2} = 1$ E) The circle $x^{2} + (y+2)^{2} = 1$
Let $\bar{z}$ denote the conjugate of $z$. For the complex number $z = 2 + i$, $$\frac{z}{\bar{z}-1}$$ Which of the following is this expression equal to? A) $\frac{1}{2} + \frac{3}{2}i$ B) $\frac{2}{3} - \frac{3}{2}i$ C) $1 + 3i$ D) $2 - 3i$ E) $3 + i$
$$z = 1 + i\sqrt{3}$$ Which of the following is this complex number equal to? A) $2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$ B) $2\left(\cos\frac{\pi}{6} - i\sin\frac{\pi}{6}\right)$ C) $2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ D) $4\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ E) $4\left(\cos\frac{\pi}{3} - i\sin\frac{\pi}{3}\right)$
Let $b$ and $c$ be real numbers. One root of the polynomial $P(x) = x^{2} + bx + c$ is the complex number $3 - 2i$. Accordingly, what is $P(-1)$? A) 5 B) 10 C) 20 D) 25 E) 30
$$\log_{3} 5 = a$$ Given this, what is the value of $\log_{5} 15$? A) $\frac{a}{a+1}$ B) $\frac{a+1}{a}$ C) $\frac{a}{a+3}$ D) $\frac{a+3}{a}$ E) $\frac{4a}{3}$
$$\frac{1}{\log_{2} 6} + \frac{1}{\log_{3} 6}$$ Which of the following is this expression equal to? A) $\frac{1}{3}$ B) $1$ C) $2$ D) $\log_{6} 2$ E) $\log_{6} 3$
For positive real numbers $a$, $b$, $c$ different from 1, $$\log_{a} b = \frac{1}{2}, \quad \log_{a} c = 3$$ Given this, what is the value of the expression $\log_{b}\left(\frac{b^{2}}{c\sqrt{a}}\right)$? A) $\frac{3}{2}$ B) $\frac{5}{2}$ C) $\frac{5}{3}$ D) $-6$ E) $-5$
The sequences $\{a_{n}\}$ and $\{b_{n}\}$ are defined as follows. $$a_{n} = \begin{cases} 0, & \text{if } n \equiv 0 \pmod{3} \\ n, & \text{if } n \equiv 1 \pmod{3} \\ -n, & \text{if } n \equiv 2 \pmod{3} \end{cases}, \quad b_{n} = \sum_{k=0}^{n} a_{k}$$ Accordingly, what is $b_{4}$? A) $-2$ B) $-1$ C) $0$ D) $2$ E) $3$
The angle formed by the lines $d_{1}$ and $d_{2}$ given above measures $30^{\circ}$. First, a perpendicular $A_{1}B_{1}$ is drawn from point $A_{1}$ on line $d_{1}$ to line $d_{2}$. Then, a perpendicular $B_{1}A_{2}$ is drawn from point $B_{1}$ to line $d_{1}$, and a perpendicular $A_{2}B_{2}$ is drawn from the foot of the perpendicular $A_{2}$ to line $d_{2}$, and this process continues. Given that $|A_{1}B_{1}| = 12$ cm, what is the sum of the lengths of all perpendiculars drawn to line $d_{2}$ in this manner, $|A_{1}B_{1}| + |A_{2}B_{2}| + |A_{3}B_{3}| + \cdots$, in cm? A) 32 B) 36 C) 38 D) 40 E) 48
$$\left|\begin{array}{rrr} 2 & -3 & 2 \\ 1 & 2 & 0 \\ 2 & 3 & 0 \end{array}\right|$$ What is the value of this determinant? A) $-1$ B) $-2$ C) $-3$ D) $-4$ E) $-6$
$$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$ Given that $A^{t}$ is the transpose of the matrix and $A^{-1}$ is its inverse matrix, which of the following is the product $A^{t} \cdot A^{-1}$? A) $\begin{bmatrix} \frac{5}{2} & -3 \\ \frac{9}{2} & -5 \end{bmatrix}$ B) $\begin{bmatrix} \frac{3}{2} & -2 \\ 1 & 3 \end{bmatrix}$ C) $\begin{bmatrix} -2 & \frac{-9}{2} \\ 3 & \frac{5}{2} \end{bmatrix}$ D) $\begin{bmatrix} \frac{9}{2} & 3 \\ \frac{-5}{2} & -1 \end{bmatrix}$ E) $\begin{bmatrix} -3 & -1 \\ \frac{5}{2} & -2 \end{bmatrix}$
$$\begin{array}{r} 2x + 2y - z = 1 \\ x + y + z = 2 \\ y - z = 1 \end{array}$$ In the solution of the system of equations above, what is $x$? A) $-3$ B) $-2$ C) $-1$ D) $0$ E) $3$
For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, $$f'(x) = 2x^{2} - 1, \quad f(2) = 4$$ Given this, what is the value of the limit $\displaystyle\lim_{x \rightarrow 2} \frac{f(x)-4}{x-2}$? A) 3 B) 4 C) 5 D) 6 E) 7
The graph of the function $f: \mathbb{R}\setminus\{-1\} \rightarrow \mathbb{R}\setminus\{2\}$ is shown in the figure above. Accordingly, $$\lim_{x \rightarrow -\infty} f(x) + \lim_{x \rightarrow 0} f(x)$$ What is the sum of these limits? A) $-2$ B) $-1$ C) $0$ D) $1$ E) $3$
For the function $f(x) = 2x^{3} - ax^{2} + 3$, what should $a$ be so that the equation of the tangent line to the curve at some point is $y = 4$? A) $-3$ B) $-1$ C) $0$ D) $1$ E) $3$
$$f(x) = x^{4} - 5x^{2} + 4$$ What is the maximum value of the function on the interval $\left[\frac{-1}{2}, \frac{1}{2}\right]$? A) 8 B) 6 C) 4 D) 2 E) 0
$$f''(x) = 6x - 2, \quad f'(0) = 4, \quad f(0) = 1$$ For the function $f$ that satisfies these conditions, what is the value of $f(1)$? A) 4 B) 5 C) 6 D) 7 E) 8
The tangent line drawn from a point $A(x, y)$ on the parabola $y^{2} = 4x$ has slope 1. Accordingly, what is $x + y$, the sum of the coordinates of point $A$? A) 1 B) 2 C) 3 D) 4 E) 5
A workplace consisting of a corridor, kitchen, and study room has the model shown above as rectangle ABCD, and the perimeter of this rectangle is 72 meters. For the kitchen in this workplace to have the largest area, what should $x$ be in meters? A) 1 B) 2 C) 3 D) 4 E) 5
If the line tangent to the parabola $y = x^{2} + bx + c$ at the point $x = 2$ is $y = x$, what is the sum $b + c$? A) $-2$ B) $-1$ C) $0$ D) $1$ E) $2$
What is the area in square units of the (finite) region bounded by the curve $y = x^{3}$ and the line $y = x$? A) $\frac{1}{2}$ B) $\frac{3}{2}$ C) $1$ D) $\frac{1}{3}$ E) $\frac{2}{3}$
For the function $f$ whose graph is given above, $$\int_{1}^{3} \frac{x \cdot f'(x) - f(x)}{x^{2}}\, dx$$ What is the value of the integral? A) $\frac{7}{2}$ B) $\frac{3}{2}$ C) $\frac{2}{3}$ D) $\frac{1}{3}$ E) $\frac{5}{4}$
$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$ What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$? A) 2 B) 4 C) 6 D) 8 E) 10