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) $\dfrac{5\pi}{6}$
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110- The value of $\displaystyle\lim_{x \to \frac{3\pi}{4}} \dfrac{1-\tan^2 x}{\sqrt{1+\sin 2x}}$ is which of the following?
(1) $-2\sqrt{2}$ (2) $-\sqrt{2}$ (3) $\sqrt{2}$ (4) $2\sqrt{2}$

111- If $f(x) = \sqrt{x^2 - |x| + |x|}$, then $\displaystyle\lim_{h \to 0^+} \dfrac{f(1+h)-f(1)}{h}$ is which of the following?
(1) $\dfrac{1}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{3}{2}$ (4) $\dfrac{5}{2}$

112- Point $M(x,2)$ lies on the curve $y=2$. It is a variable point. The line segment connecting point $M$ to the origin, makes an angle $\alpha$ with the positive $x$-axis. The rate of change of $\alpha(x)$ with respect to $x$, at the moment $x=4$, is which of the following?
(1) $-0/2$ (2) $-0/1$ (3) $0/05$ (4) $0/15$

113- For natural numbers $n \geq n_0$, the sequence $\left\{\dfrac{2n^2+1}{n^2+2n}\right\}$ converges to its limit point, with a distance less than $0.04$. The smallest value of $n_0$ is which of the following?
(1) $96$ (2) $97$ (3) $98$ (4) $99$

114- The sequence $\left\{\left(1+\dfrac{1}{n^2}\right)^n\right\}$ converges to which number?
(1) $\sqrt{e}$ (2) $\dfrac{1}{2}e$ (3) $1$ (4) $\dfrac{1}{e}$

115- The function with the rule $f(x) = \begin{cases} \dfrac{x-[x]}{x^2-x-6} & ; \ x \neq 2 \\ a & ; \ x = 2 \end{cases}$, for which value of $a$, is continuous on the interval $[2,3]$?
(1) $\dfrac{1}{11}$ (2) $\dfrac{1}{9}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{1}{6}$

116- The number of discontinuous points of the graph of the function $f(x) = \dfrac{3-\sqrt{x+4}}{1+\sqrt[3]{x+1}} + \dfrac{1}{x+5}$ is which of the following?
(1) zero (2) $1$ (3) $2$ (4) $3$

117- A line is tangent to the graph of the function $y = x^3 - 2x^2 + 3x$ at the point $x = 2$ and passes through it. The slope of this line is which of the following?
(1) $-\dfrac{2}{3}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{5}{3}$
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118- The line perpendicular to the graph of $f(x) = \dfrac{\cos 2x}{2 - \sin x}$, at the point of tangency with the $y$-axis, cuts the $x$-axis at which length?
(1) $0/1$ (2) $0/2$ (3) $0/3$ (4) $0/5$

119- From the relation $y^2 + xy^2 + x = 7$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1,2)$ is which of the following?
(1) $\dfrac{3}{4}$ (2) $\dfrac{4}{5}$ (3) $\dfrac{6}{5}$ (4) $\dfrac{3}{2}$

120- The function $f : \mathbb{R} \to \mathbb{R}$ is twice differentiable. For every real number $x$, the function $g(x) = f(4 - x^2)$ is defined. If $f^{-1}(1) = -5$ and $f^{-1}(1) = -1$, and $f''(1) = -1$, what is the value of $g''(\sqrt{3})$?
(1) $-3$ (2) $-2$ (3) $2$ (4) $3$

122- The figure below shows the graph of $y = \dfrac{x^2 + ax^2}{x^2 + bx + 1}$. What is the value of the relative minimum of the function?
[Figure: Graph of the function with a local minimum visible, axes labeled $x$ and $y$]
(1) $4/5$
(2) $6$
(3) $6/25$
(4) $6/75$

123- What is the mean value (average) of the function $f(x) = \dfrac{2x-1}{\sqrt{x}}$ on the interval $[1, 4]$?
(1) $\dfrac{17}{9}$ (2) $\dfrac{7}{3}$ (3) $\dfrac{22}{9}$ (4) $\dfrac{8}{3}$

124- If $F(x) = x\displaystyle\int_{2}^{x^2} \dfrac{dx}{\sqrt[3]{x^2-1}}$, and $F'(\sqrt{2})$ is given, what is its value?
(1) $3$ (2) $4$ (3) $4/5$ (4) $6$
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128. In triangle $ABC$, the side lengths are $BC = 9$, $AC = 8$, and $AB = 2$. The angle bisectors of angle $A$ intersect side $BC$ at $M$ and $N$. What is the length of $MN$?
(1) $4/2$ (2) $4/5$ (3) $4/8$ (4) $5/1$

129. In the figure below, $AD$ is tangent to the circle with center $O$, and $OH$ is perpendicular to $AC$. If $\widehat{DBC} = 2\widehat{DAC}$, how many times is angle $\widehat{COH}$ equal to angle $\widehat{DAC}$?
\begin{minipage}{0.45\textwidth} [Figure: Circle with center $O$, tangent line $AD$, points $B$, $C$, $H$ marked] \end{minipage} \begin{minipage}{0.45\textwidth} \begin{flushright} (1) $2.5$
(2) $3$
(3) $3.5$
(4) $4$ \end{flushright} \end{minipage}

130. Two circles with radii $4$ and $8$ are internally tangent at point $A$. A chord $BC$ of the large circle is tangent to the small circle, and the line through the center of the small circle parallel to the radical axis passes through point $P$. What is $PB \times PC$?
(1) $24$ (2) $32$ (3) $36$ (4) $48$
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131- The reflection of line $\Delta$ across the line $y = -x$ is line $\Delta'$. The equation of line $\Delta$ is $2y + x = 6$. With respect to line $x = -x$, the equation of line $\Delta'$ is which of the following?
(1) $y + 2x = -6$ (2) $y + 2x = 2$ (3) $y + 2x = -2$ (4) $y - 2x = \Lambda$

133- Vector $\mathbf{a}$ makes an angle of $60°$ with each of the axes $Ox$ and $Oy$, and makes a right angle with the axis $z$. This vector is perpendicular to which plane? Which equation does the plane satisfy?
(1) $x - \sqrt{2}y + z = 0$ (2) $2x + 2y + \sqrt{2}z = 0$
(4) $x + y - \sqrt{2}z = 0$ (3) $x + y + \sqrt{2}z = 0$

134- If $\mathbf{a} = (2, -3, 1)$ and $\mathbf{b} = (1, 2, -4)$, the volume of the parallelepiped constructed on vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} \times \mathbf{b}$ is which of the following?
(1) $225$ (2) $220$ (3) $245$ (4) $250$

135- The shortest distance between the two lines $\dfrac{x-1}{3} = -y + 4 = \dfrac{z}{5}$ and $\begin{cases} x = 2 \\ y = 5 \end{cases}$ is which of the following?
(1) $\dfrac{3}{\sqrt{10}}$ (2) $\dfrac{4}{\sqrt{10}}$ (3) $\sqrt{10}$ (4) $2\sqrt{5}$

136- For which value of $a$, the asymptote $x = \dfrac{21}{8}$ of the conic $2y^2 - 12y + ax + 8 = 0$ holds?
(1) $12$ and $3$ (2) $16$ and $3$ (3) $12$ and $5$ (4) $16$ and $5$

137- For which value of $a$, the distance between the foci of the hyperbola $3x^2 + 4y^2 + 16y + a = 0$ equals $2$?
(1) $2$ (2) $4$ (3) $6$ (4) $8$

138- If $A = \begin{bmatrix} 1 & 3 & 6 & 24 \\ \frac{1}{2} & 1 & 2 & 8 \end{bmatrix}$, $B = \begin{bmatrix} \frac{1}{6} & \frac{1}{2} & 1 & 4 \\ \frac{1}{24} & \frac{1}{8} & \frac{1}{4} & 1 \end{bmatrix}$, and $C = \begin{bmatrix} A \\ B \end{bmatrix}$, the sum of the main diagonal entries of matrix $C^T$ is which of the following?
(1) $16$ (2) $18$ (3) $20$ (4) $24$

139- The values of $x$ from the relation $\begin{vmatrix} 0 & x-3 & x-2 \\ x+3 & 0 & -4 \\ x+2 & 6 & 0 \end{vmatrix} = 0$ are which of the following?
(1) $-1, -6$ (2) $-1, 6$ (3) $1, -6$ (4) $1, 6$
\hrule
Calculation Space
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140-- Three pages with matrix equations. If $\begin{vmatrix} a & -1 & 3 \\ b & 2 & 4 \\ c & -2 & 1 \end{vmatrix} = 5$, then the three pages intersect. With which of the following lengths do they intersect?
$$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 4 \\ 3 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$$
(1) $-\dfrac{1}{3}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{1}{2}$

142-- In the following frequency table, what is the standard deviation using the quick method?

$x$	27	29	31	33	35
$f$	7	15	13	11	9

(1) $2.6$ (2) $2.7$ (3) $2.8$ (4) $2.9$

146-- In how many ways can the set $\{a, b, c, d, e, f, g\}$ be partitioned into two three-element sets and one single-element set such that $\{a\}$ is missing?
(1) $45$ (2) $50$ (3) $56$ (4) $60$
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147- Urn A contains 5 beads with odd digit numbers and Urn B contains 4 beads with non-zero even digit numbers. One bead is drawn from each urn. With which probability is the product of the two numbers greater than 10?
(1) $0.6$ (2) $0.65$ (3) $0.7$ (4) $0.75$

148- Three people are working on decoding a message. Their probabilities of success are $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{1}{2}$ respectively. What is the probability that at least one of them succeeds?
(1) $\dfrac{19}{24}$ (2) $\dfrac{5}{6}$ (3) $\dfrac{11}{12}$ (4) $\dfrac{23}{24}$

150- The five-digit number $N = \overline{a7\!4\!6b}$ is a multiple of 36. What is the largest remainder when $N$ is divided by 11?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

151- When natural number $A$ is divided by 23, the remainder is 5, and when $A$ is divided by 17, the remainder is 9. What is the largest remainder when the three-digit number $A$ is divided by 12?
(1) zero (2) $2$ (3) $6$ (4) $7$

153- How many non-negative integer solutions does the inequality $x + y + z \leq 5$ have?
(1) $50$ (2) $54$ (3) $56$ (4) $60$

154- A fair coin is tossed repeatedly. What is the probability that the number 4 appears before the number 6?
(1) $\dfrac{1}{3}$ (2) $\dfrac{1}{2}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$

155- Given $P(X = x) = \dfrac{\dbinom{5}{x}\dbinom{4}{r-x}}{a}$\,; $x = 0, 1, 2, 3$, for which value of $a$ is this a probability function?
(1) $48$ (2) $56$ (3) $64$ (4) $84$
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179- A stone is thrown vertically upward from height $h$ with initial velocity $V_0$ and hits the ground after $4$ seconds. If in the last second of its motion it travels $\dfrac{h}{2}$, how many meters is $h$? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$

• [(1)] $60$
• [(2)] $90$
• [(3)] $120$
• [(4)] $180$

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180. The equation of motion of a body in SI is $x = 2t^3 - 6t^2 + 6t$ . In the time interval zero to 2 seconds, which statement is correct?

1. [(1)] Average acceleration is equal to zero.
2. [(2)] The direction of motion changes once.
3. [(3)] The motion is first decelerating and then accelerating.
4. [(4)] The motion is initially in the direction of the $x$-axis and then opposite to the $x$-axis.

181. The position–time graph of a moving object is shown below as a parabola. In the time interval 0 to $8\,\text{s}$, what are the magnitude of average velocity and average speed in SI?
[Figure: A parabolic position-time graph with $x(\text{m})$ on the vertical axis and $t(\text{s})$ on the horizontal axis. The curve reaches a maximum of 24 m, passes through 16 m, and returns to 0 at $t = 8\,\text{s}$.]

1. [(1)] 1 and zero
2. [(2)] 2 and zero
3. [(3)] 1 and 1
4. [(4)] 2 and 2

182. Balls A and B are at distance $d$ from each other, and are simultaneously thrown horizontally from the ground surface. Ball A is thrown with speed $30\,\dfrac{\text{m}}{\text{s}}$ in the vertical direction, and ball B is thrown with speed $V_\circ$ at an angle of $45°$ to the horizontal. If both balls collide at a point, how many meters is $d$? (Neglect air resistance, $g = 10\,\dfrac{\text{m}}{\text{s}^2}$)
$$30 \quad (1) \qquad 45 \quad (2) \qquad 60 \quad (3) \qquad 90 \quad (4)$$

183. A projectile of mass $m$ with initial speed $V_\circ$ is launched at angle $\alpha$ to the horizontal and reaches the ground after $2t$ seconds. What is the magnitude of the change in momentum of the projectile during the first $t$ seconds of motion? (Neglect air resistance.)
$$2mgt \quad (1) \qquad mgt \quad (2) \qquad \dfrac{mv_\circ}{2} \quad (3) \qquad \dfrac{2mv_\circ}{2} \quad (4)$$
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