120. In a trapezoid, a line segment connecting the midpoints of the legs divides the area in the ratio $3$ to $5$. What is the ratio of the parallel sides of the trapezoid?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{3}$ (3) $\dfrac{2}{5}$ (4) $\dfrac{3}{5}$
121. In triangle $ABC$, $M$ is the midpoint of $BC$. The bisectors of angles $AMB$ and $AMC$ intersect the two sides of the triangle at $P$ and $Q$ respectively. Point $O$ is the intersection of $AM$ and $PQ$. What does $OM$ equal?
(1) $\dfrac{1}{4}BC$ (2) $AQ$ (3) $OA$ (4) $OP$

121-A Mathematics Page 3

122. In quadrilateral $ABCD$, the two non-adjacent sides and the two diagonals are each equal to a side of a rhombus. Which of the following conclusions about the quadrilateral is correct?

1. [(1)] The two non-adjacent sides are equal to each other.
2. [(2)] The two diagonals are perpendicular to each other.
3. [(3)] The two sides include the vertices of the rhombus, and are equal.
4. [(4)] The two non-adjacent sides are parallel.

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123. Point $A$ and line $d$ and plane $P$ are given. In a plane drawing passing through point $A$, parallel to line $d$ and perpendicular to plane $P$, in which case is the number of answers infinite?
(1) $d \cap p = d$ (2) $d \cap p \neq \phi$ (3) $d \parallel p$ (4) $d \perp p$

124. In a cube, the plane passing through one edge and the midpoint of another edge divides the cube into two unequal pieces. What is the ratio of the volumes of these two pieces?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{3}$ (3) $\dfrac{1}{\sqrt{5}}$ (4) $\dfrac{1}{\sqrt{3}}$

128. Point $A$ is on plane $d$ and lines $d$ and $d'$ intersect. In drawing an isosceles triangle with vertex $A$, where the other two vertices lie on each of the two lines, which geometric transformation is used?
(1) Translation (2) Reflection (3) Homothety (4) Rotation

129. In the figure of the rhombus, what is the length of the larger diagonal?
\begin{minipage}{0.45\textwidth} [Figure: Rhombus with sides labeled 1, 2, 2, 3, 4] \end{minipage} \begin{minipage}{0.45\textwidth} (1) $2\sqrt{10}$
(2) $2\sqrt{11}$
(3) $4\sqrt{3}$
(4) $5\sqrt{2}$ \end{minipage}
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137- If $A = \{1,2,\{1,2\},\{1,\{1,2\}\},\{2\}\}$ and $B = \{\{1\},\{1,2\}\}$, the number of subsets of $A \cap B'$ is which of the following?
(1) $4$ (2) $16$ (3) $32$ (4) $33$

144. The unit price of two types of goods is 220 and 140 tomans respectively. In how many ways can one purchase both types of goods with a total of 19000 tomans?
(1) $10$ (2) $11$ (3) $12$ (4) $13$

146. A simple regular $(-4)$-regular graph with 6 vertices has how many edges of length 4?
(1) $9$ (2) $10$ (3) $12$ (4) $15$

149. From the set of numbers $\{5, 8, 11, \ldots, 65, 68, 71\}$ which forms an arithmetic sequence, at least how many members must be selected so that we are certain that at least two numbers exist in this subset whose sum is $82$?
(1) $11$ (2) $12$ (3) $13$ (4) $14$

150. For which values of $x$, the set $(1 - x + 1, 2x - 1)$ has exactly 3 elements?
(1) $\varnothing$ (2) $\{2\}$ (3) $2 < x < 2.5$ (4) $1.5 < x < 2$

151. Points $A\begin{vmatrix}3\\5\end{vmatrix}$ and $B\begin{vmatrix}9\\11\end{vmatrix}$ are in a plane with different coordinate axes, and points $M$ and $N$ always move along the two axes. What is the minimum length of the broken line $AMNB$?
[Figure: Coordinate plane with points B and A in first quadrant, M on y-axis, N on x-axis]
(1) $18$
(2) $19$
(3) $20$
(4) $21$

152. The proposition $(p \Rightarrow q)$, with which of the following propositions below, is also equivalent?
(1) ${\sim}p \vee q$ (2) $p \vee {\sim}q$ (3) ${\sim}p \wedge q$ (4) $p \wedge {\sim}q$

153. The surjective proposition $\forall x \in \mathbb{N}, \exists y \in \mathbb{N}; P(x,y)$, with which expression of $P(x,y)$ is correct?
(1) $y - x = 6$ (2) $x - y = 6$ (3) $x + y = 6$ (4) $xy = 6$

154. Which set is a minimal vertex cover for the graph below?
[Figure: Graph with vertices $a, b, c, d, e, f, g$ and edges connecting them]
(1) $\{a, c, e, g\}$
(2) $\{a, d, e, g\}$
(3) $\{a, b, d, e\}$
(4) $\{a, d, e, f\}$

155. The number of Latin squares compatible with the Latin square $\begin{pmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}$ is which of the following?
(1) $2$ (2) $3$ (3) $4$ (4) $6$
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165. According to the figure below, three balls are thrown from the top of a building similarly, from one point with equal speeds. If $W_1$, $W_2$, and $W_3$ are the weights of the balls at the moment of landing on the ground, which relationship is correct?
[Figure: Three projectiles launched from a building rooftop at different angles]

• [(1)] $W_1 = W_2 = W_3$
• [(2)] $W_2 > W_1 > W_3$
• [(3)] $W_2 < W_3 < W_1$
• [(4)] $W_2 = W_3 > W_1$

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Physics 121-A Page 11

166. If the impulse of a bullet in SI units increases from 20 to 22, by what percentage does the kinetic energy of the bullet increase?
(1) $15$ (2) $17$ (3) $21$ (4) $42$

167. A 2 kg object slides down a frictionless inclined surface and hits a spring at point B with speed $2\,\dfrac{\text{m}}{\text{s}}$. It passes point A and hits the spring at point B. If the maximum compression of the spring is x and the energy stored in the spring at that point is 10 joules, how many centimeters is x? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$
[Figure: An inclined plane with a 2 kg object sliding down. Point A is near the top, point B is where the object meets the spring. The spring is at the bottom. Height $h$ and horizontal distance $r_\circ\,\text{cm}$ are marked. The incline angle is $r_\circ^\circ$.]
(1) $10$
(2) $20$
(3) $30$
(4) $40$

168. According to the figure below, a light ray enters from medium (1) (transparent) into other transparent media. If the speed of light in medium (1) is 25\% less than the speed of light in medium (2), and the speed of light in medium (4) is 40\% greater than the speed of light in medium (3), how many times is the refractive index of medium (2) equal to the refractive index of medium (3)?
$$(\sin 53^\circ = 0.8,\quad \sin 45^\circ = 0.7)$$
[Figure: A light ray passing through four media $n_1$, $n_2$, $n_3$, $n_4$ with angles of incidence $53^\circ$ and $45^\circ$ shown, along with a normal line.]
(1) $\dfrac{4}{3}$
(2) $\dfrac{6}{5}$
(3) $\dfrac{3}{4}$
(4) $\dfrac{5}{6}$
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169. In the figure below, the SI unit vector diagram includes red and green light rays entering a transparent liquid from air. Which of the figures below correctly shows the refraction path of light?
[Figure: Main diagram showing incident rays S and N with interface, liquid medium, and ray I. Four answer choices (1), (2), (3), (4) each showing different refraction configurations with red (فرمز) and green (سبز) rays]

170. The figure below shows a snapshot of a sinusoidal wave at a moment in time, showing displacement along the x-axis of a rope being pulled. The linear mass density of the rope is $\frac{\text{kg}}{\text{m}}$ and it moves. If the tension in the rope is $80\,\text{N}$ and the linear mass density (mass per unit length) is $0.2\,\frac{\text{kg}}{\text{m}}$, how many centimeters does each particle of the rope travel in $0.05\,\text{s}$?
[Figure: Graph with y(cm) axis showing sinusoidal wave, amplitude 2 cm, x(cm) axis extending to 15 cm]

• [(1)] $2$
• [(2)] $4$
• [(3)] $8$
• [(4)] $16$

171. The linear mass density (mass per unit length) of a wire used in a musical instrument is $4\times10^{-3}\,\frac{\text{kg}}{\text{m}}$, and this wire is stretched between two points with a tension of $250\,\text{N}$. If the resonant frequency produced is $312.5\,\text{Hz}$, the wavelength of the wave created in it is how many meters?
$$1/25 \quad (4) \qquad 0/80 \quad (3) \qquad 0/75 \quad (2) \qquad 0/50 \quad (1)$$
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172. A simple pendulum with length $80\,\text{cm}$ has a small damping. The pendulum is oscillating. How should we change the length of the pendulum so that its period of oscillation becomes half?

• [(1)] Decrease by 60 cm. (2) Increase by 60 cm.
• [(3)] Decrease by 20 cm. (4) Increase by 20 cm.