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

109- The value of $\dfrac{1}{\sin 15°} - \dfrac{1}{\cos 15°}$ is which of the following?
(1) $2$ (2) $\sqrt{6}$ (3) $2\sqrt{2}$ (4) $2\sqrt{3}$

110- The general solution of the trigonometric equation $\cos 2x = \sin x \sin 3x$ is which of the following?
(1) $\dfrac{k\pi}{2} - \dfrac{\pi}{6}$ (2) $\dfrac{k\pi}{3} + \dfrac{\pi}{6}$ (3) $k\pi + \dfrac{\pi}{2}$ (4) $\dfrac{k\pi}{3}$

111- The limit of $\dfrac{\sqrt{\cos 3x} - \sqrt{\cos x}}{x^2}$ as $x \to 0$ is which of the following?
(1) $-2$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{2}$ (4) $2$

112- The derivative of $f(x) = \sin\!\left(\dfrac{\pi}{2} + \tan^{-1}\dfrac{x}{2}\right)$ at the point $x = 2\sqrt{3}$ is which of the following?
(1) $-\dfrac{1}{24}$ (2) $-\dfrac{1}{16}$ (3) $\dfrac{1}{8}$ (4) $\dfrac{1}{4}$

113- The sequence $\left\{\left[\dfrac{(-1)^n}{n}\right]\right\}$, $n = 1, 2, 3, \ldots$ is how?
(1) Converges to $-1$ (2) Converges to zero (3) Divergent -- bounded (4) Divergent

114- The function with the rule $f(x) = \begin{cases} |x| + [-x] & ; \ x \notin \mathbb{Z} \\ a & ; \ x \in \mathbb{Z} \end{cases}$, for which value of $a$, is continuous on the set of real numbers?
(Note: $[\ ]$ denotes the floor function.)
(1) $-1$ (2) $1$ (3) $0$ (4) Always discontinuous

115- The $x$-intercept of the oblique asymptote of $y = x\sqrt{\dfrac{3x-3}{x-1}}$ is which of the following?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{3}{4}$

116- The smallest positive root of the equation $x^2 - 3x + 1 = 0$ lies in which interval?
(1) $\left(0, \dfrac{1}{2}\right)$ (2) $\left(\dfrac{1}{2}, \dfrac{2}{3}\right)$ (3) $\left(\dfrac{1}{3}, \dfrac{2}{5}\right)$ (4) $\left(\dfrac{2}{5}, \dfrac{1}{2}\right)$
%% Page 5 Mathematics 120-A Page 4

117. If $\theta$ is the angle between the left and right tangents to the graph of $y = |\ln x|$ at the corner point, then $\tan\theta$ equals:
(1) $-1$ (2) $1$ (3) zero (4) $\infty$

118. If $f$ is differentiable at $x = 4$ and $\displaystyle\lim_{x \to 4} \frac{f(x) + 5}{x - 4} = \frac{-3}{2}$, then $\dfrac{f(2x)}{x}$ at $x = 2$ equals:
(1) $-\dfrac{1}{4}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{4}$ (4) $\dfrac{1}{2}$

119. The function $f(x) = x + \ln x$ is defined (given). The equation of the tangent line to the graph of $f^{-1}$ at the point where it meets the bisector of the first quadrant is:
(1) $y + 2x = 3$ (2) $2x - y = 1$ (3) $2x + y = 3$ (4) $2y - x = 1$

120. The $x$-intercept of the normal line to the curve $x^2 + y^2 = 3xy + 3$ at the point $(1, 2)$ is:
(1) $2$ (2) $3$ (3) $4$ (4) $5$

121. The volume of a sphere is increasing at a constant rate of $3$ cubic centimeters per second. At the moment when the radius of the sphere is $8$ centimeters, the surface area of the sphere increases how many square centimeters per second?
(1) $1/2$ (2) $1/25$ (3) $1/5$ (4) $1/6$

122. For the graph of $f(x) = \cos^2 x - 2\cos x$; $x \in [0, 2\pi]$, at which base point is the inflection point and local minimum?
(1) $\left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ (2) $\left(\pi, \dfrac{4\pi}{3}\right)$ (3) $\left(\dfrac{2\pi}{3}, \pi\right)$ (4) $\left(\dfrac{4\pi}{3}, \dfrac{3\pi}{2}\right)$

123. The area bounded by the curve $y = \sqrt{1 - \cos 2x}$ and the $x$-axis over one period is:
(1) $2$ (2) $2\sqrt{2}$ (3) $3$ (4) $3\sqrt{2}$

124. The value of $\displaystyle\int_0^4 |1 - \sqrt{x}|\, dx$ is:
(1) $\dfrac{4}{3}$ (2) $\dfrac{5}{3}$ (3) $2$ (4) $3$

125. The area of a regular octagon inscribed in a circle of radius $2$ is:
(1) $8\sqrt{2}$ (2) $8(\sqrt{2}-1)$ (3) $4(1+\sqrt{2})$ (4) $4(2+\sqrt{2})$
%% Page 6

126. In a rectangle with side lengths 3 and 4 units, from each vertex, a perpendicular is drawn to the other diagonal of this rectangle. What is the area of the resulting parallelogram?
[Figure: Rectangle with diagonals and perpendiculars drawn]

• [(1)] $5/25$
• [(2)] $5/75$
• [(3)] $6$
• [(4)] $7/5$

127. The region bounded by a $2\times 5$ rectangle and a semicircle with diameter 3 units rotates around line $\Delta$. The volume of the resulting solid is how many times $\pi$?
[Figure: Rectangle with semicircle, dimensions 5 and 2 shown, axis $\Delta$]

• [(1)] $15$
• [(2)] $15/5$
• [(3)] $16/5$
• [(4)] $17$

128. Quadrilateral $ABCD$ is inscribed in a circle. If $AB$ is the farthest chord and $BC$ is the closest chord to the center of this circle, which relationship between the angles cannot hold?

• [(1)] $\hat{D} > \hat{C}$
• [(2)] $\hat{B} > \hat{C}$
• [(3)] $\hat{A} > \hat{B}$
• [(4)] $\hat{B} > \hat{D}$

129. In an isosceles triangle, if the altitude is 8 units and the radius of the inscribed circle is 3 units, what is the length of the base of this triangle?

• [(1)] $15$
• [(2)] $12$
• [(3)] $14$
• [(4)] $16$

134. The distance from point $(1,3,2)$ to the line of intersection of the plane $2x - y - z = 4$ with the plane $xOy$ is:
(1) $2$ (2) $\sqrt{6}$ (3) $3$ (4) $\sqrt{10}$

135. For which value of $a$, the angle between the tangent line to the circle $x^2 + y^2 - 2x + y = 1$ and the line $3x + 2y = a$ at their intersection point is $90°$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

137. Matrix $A = \begin{bmatrix} 5 & 2 & -1 \\ 4 & 3 & -2 \\ 1 & 6 & 7 \end{bmatrix}$ is written as the sum of a symmetric matrix and a skew-symmetric matrix. The determinant of the symmetric matrix is:
(1) $16$ (2) $18$ (3) $22$ (4) $24$

138. If one unit is added to all entries of the second column of matrix $A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & a & 7 \\ 3 & b & 6 \end{bmatrix}$, what number is added to the value of the original determinant of the matrix?
(1) $-3$ (2) $-2$ (3) $3$ (4) $6$
%% Page 8

139. If $\begin{bmatrix} \cos 15° & \sin 15° \\ -\sin 15° & \cos 15° \end{bmatrix}^n = -I$, what is the smallest natural number $n$?
(1) $6$ (2) $12$ (3) $18$ (4) $24$

140. Three planes with the matrix equation $\begin{bmatrix} 1 & 3 & -1 \\ 3 & -2 & 3 \\ 5 & 4 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 7 \\ 3 \\ 9 \end{bmatrix}$ are given. The common intersection of these two planes is which of the following?
(1) Parallel (2) Coincident (3) Divergent (4) Pass through one point

141. We represent the statistical data with stem-and-leaf plot as shown below. We show the variance of the data inside the box. Which is it?

Stem	\multicolumn{4}{l}{Leaf}
2	5	6	7	9
3	1	3	4	5 6
4	$\circ$	1	2	4

(1) $9/25$ (2) $9/75$ (3) $10/15$ (4) $10/85$

142. A population with mean $12$ and variance $12/6$, and another population with mean $24$ and variance $7/2$, form a new combined population. If the two populations have equal size, what is the standard deviation of the new population?
(1) $7/9$ (2) $3$ (3) $3/1$ (4) $3/2$

143. In the sequence $\{U_n\}$, with initial conditions $U_1 = U_2 = 1$ and recurrence $U_{n+1} = U_n + U_{n-1}$, using induction, the expression $(U_n^2 - U_{n+1} \times U_{n-1})$ equals which number?
(1) $-1$ (2) $1$ (3) $(-1)^n$ (4) $(-1)^{n+1}$

144. In a bag there are $7$ white beads, $5$ black beads, and $3$ green beads. We draw beads from the bag until we are sure we have at least $4$ white beads or $3$ black beads or $2$ green beads. What is the minimum number of beads drawn?
(1) $6$ (2) $7$ (3) $8$ (4) $9$

147- We toss two coins and one die together. What is the probability that both heads ("ro") appear on the coins and 6 appears on the die?
(1) $\dfrac{3}{8}$ (2) $\dfrac{5}{8}$ (3) $\dfrac{5}{12}$ (4) $\dfrac{7}{12}$

148- Two numbers are randomly chosen between 2 and 5. What is the probability that both numbers are between $0.3$ and $0.5$?
(1) $0.2$ (2) $0.25$ (3) $0.3$ (4) $0.35$

154- Six numbered balls are randomly placed in 3 boxes. What is the probability that no box remains empty?
(1) $\dfrac{5}{14}$ (2) $\dfrac{5}{12}$ (3) $\dfrac{3}{7}$ (4) $\dfrac{7}{12}$

155- A sample space consists of 5 outcomes $a, b, c, d, e$. If $P(a) = \dfrac{1}{4}$ and $P(\{a,b,c\}) = \dfrac{2}{3}$, then $P(\{b,c,e\} \mid \{a,b,c\})$ equals what?
(1) $\dfrac{5}{8}$ (2) $\dfrac{5}{12}$ (3) $\dfrac{5}{8}$ (4) $\dfrac{3}{4}$
\rule{\textwidth}{0.4pt} Calculation Space
%% Page 10 Physics 120-A Page 9

161- According to the figure below, a horizontal force $F$ is applied to a body. What is the minimum value of $F$ (in terms of the body's weight) so that the body remains stationary on the inclined surface? ($\sin 53^\circ = 0.8$, $g = 10\dfrac{m}{s^2}$)
[Figure: A block of mass $m$ on an inclined plane at $53^\circ$ with a horizontal force $F$ applied, $\mu_s = 1$]
(1) $\dfrac{1}{2}$ (2) $\dfrac{3}{5}$ (3) $\dfrac{4}{5}$ (4) $1$

163- In the figure below, $M = 2000\,\text{kg}$ and $m = 2400\,\text{kg}$. If the system is released from rest, the acceleration of mass $M$ is approximately how many $\dfrac{m}{s^2}$ and in which direction? ($g = 10\dfrac{m}{s^2}$, and the mass of the rope and pulleys are neglected.)
[Figure: A pulley system with mass $M$ hanging on one side and mass $m$ on the other side]

• [(1)] 1.5 and upward
• [(2)] 3 and upward
• [(3)] 1.5 and downward
• [(4)] 3 and downward

164- The magnitude of motion (momentum) of a body with mass 2 kg is $6\,\dfrac{\text{kg}\cdot\text{m}}{\text{s}}$. What is the kinetic energy of the body in joules?
(1) $3$ (2) $6$ (3) $9$ (4) $12$