120-C Page 2

125. In an equilateral triangle, two squares are constructed on two of its sides. The area of the shaded triangles is how many times the area of the original triangle?
[Figure: Equilateral triangle with squares on two sides and shaded regions]
(1) $\dfrac{\sqrt{3}}{2}$ (2) $\dfrac{2\sqrt{3}}{3}$ [6pt] (3) $1$ (4) $\sqrt{3}$

126. In triangle $ABC$, segment $DE$ is parallel to side $BC$ and $AD = \dfrac{4}{5}DB$. The area of triangle $EBC$ is how many times the area of triangle $EBD$?
[Figure: Triangle ABC with point D on AB and point E on AC, DE parallel to BC]
(1) $2$ (2) $2.25$
(3) $2.5$ (4) $2.75$

128- In a regular trapezoid, one of the base angles is $60°$ and the base lengths are $6$ and $10$ units. The area of the four triangles formed by the intersection of the internal diagonals of this trapezoid is how much?
(1) $8$ (2) $10$ (3) $14$ (4) $16$

129- In triangle $ABC$, $AC = 6$, $BM = 5$, and $M$ is the midpoint of $AC$. The bisectors of angles $AMB$ and $CMB$ intersect the other two sides of this triangle at $P$ and $Q$ respectively. What is the length $PQ$?
(1) $3.25$ (2) $3.5$ (3) $3.75$ (4) $4$
[Figure: Circle with inscribed quadrilateral $ADBC$, point $M$ inside, with diagonals drawn]

130- In the figure, $\hat{A}_1 = \hat{A}_2$, $AD \cdot BC$ is obtained. Which of the following is equal to $AD \cdot BC$?
(1) $DM \cdot AC$ (2) $BM \cdot AC$
(3) $AB \cdot CD$ (4) $BD \cdot BM$

132- Two skew lines $d$ and $d'$ and a point $A$ on line $d$ are given. We want to draw a line through point $A$ that passes through both lines $d$ and $d'$ and is perpendicular to both $d$ and $d'$. How many solutions are there?
(1) No solution (2) Always one solution (3) More than one solution (4) One solution or no solution

136- In the ellipse $5x^2 - 16y + 18x - 16y^2 = 5$, the sum of the focal radii of every point on it from its two foci is:
(1) $4\sqrt{2}$ (2) $6$ (3) $4\sqrt{3}$ (4) $8$

149- The degrees of the vertices of a simple and connected graph with vertices $4, 3, 1, a, b, c$ are given. If $p$ is the number of vertices of the graph, $q$ is the number of edges of the graph, and $q = \dfrac{3}{2}p$, then the number of elements of the set $\{a, b, c\}$ is which of the following?
$$1 \ (1) \hspace{2cm} 2 \ (2) \hspace{2cm} 3 \ (3) \hspace{2cm} 4 \ (4)$$

153- The number of equivalence relations on the set $\{a,b,c,d\}$ that include $(a,b)$ is which of the following?
$$3 \ (1) \hspace{2cm} 4 \ (2) \hspace{2cm} 5 \ (3) \hspace{2cm} 6 \ (4)$$

156. The sum of vectors $\vec{A} = 9\hat{i} + 12\hat{j}$ and $\vec{B}$ is a vector in the positive y-direction and has the same magnitude as vector $\vec{A}$. What is the magnitude of $\vec{A} - \vec{B}$?
(1) $9$ (2) $9\sqrt{2}$ (3) $9\sqrt{7}$ (4) $9\sqrt{5}$

157. A particle moves on a plane and its position vector in SI units is given by $\vec{r} = (6t)\hat{i} + (-t^2 + 4t)\hat{j}$. At the moment $t = 1\ \text{s}$, the velocity vector makes what angle (in degrees) with the positive x-axis?
(1) $30$ (2) $0$ (3) $60$ (4) $90$

158. The position–time graph of a particle moving along the x-axis is shown in the figure below, in the form of a parabola. If the distance traveled by the particle in the time interval $t = 3\ \text{s}$ to $t = 9\ \text{s}$ is 12 meters, what is the displacement of the particle in this interval?

[Figure: A parabolic x(m) vs t(s) graph. The curve rises to a maximum of 48 m, then decreases, crossing the t-axis at approximately $t = 6$ s.]
(1) zero (2) $3$ (3) $6$ (4) $12$

159. Two balls A and B are thrown from height $h$ at the same time from the same point, with speeds $V_A = 32\ \dfrac{\text{m}}{\text{s}}$ and $V_B = 22\ \dfrac{\text{m}}{\text{s}}$, both thrown vertically upward. At the moment ball B reaches its highest point, what is the distance between the two balls?
(Air resistance is neglected. $g = 10\ \dfrac{\text{m}}{\text{s}^2}$.)
(1) $7.5$ (2) $22$ (3) $32$ (4) $46.5$

160. A ball is thrown vertically upward from the ground with initial speed $V_0$ and reaches a height of 80 meters. If we throw the ball with the same initial speed at the appropriate angle for maximum range, what is the maximum range of the ball?
($\text{Air resistance is negligible and } g = 10\ \dfrac{\text{m}}{\text{s}^2}$.)
(1) $80$ (2) $80\sqrt{2}$ (3) $160$ (4) $160\sqrt{2}$
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161- According to the figure below, force $\mathbf{F}$ is applied to mass $m_1$ and the system begins to move with constant acceleration. The coefficient of lateral (kinetic) friction between each of the two masses and the horizontal surface is $\mu_k$. In the same situation, if force $\mathbf{F}$ is applied, the coefficient of lateral friction for each of the two masses with the horizontal surface is halved. How many times will the force applied to both masses together become?
[Figure: Force F applied horizontally to mass $m_1$ which is in contact with mass $m_2$, both on a rough surface]

• [(1)] $1$ (2) $2$
• [(3)] $\dfrac{1}{2}$ (4) $\dfrac{1}{4}$

162- If $m$, $V$, and $P$ are, respectively, the mass, speed, and momentum of an object, which relation represents the kinetic energy of that object?
$$\text{(1)}\ \frac{m \cdot V}{2P} \qquad \text{(2)}\ \frac{PV}{2m} \qquad \text{(3)}\ \frac{P^2}{2m} \qquad \text{(4)}\ \frac{mP^2}{2}$$

163- In the figure below, an appropriate force $F$ parallel to the inclined surface is applied to a body of mass $m = 20\ \text{kg}$ so that the body moves down the incline at constant speed. How much work does force $F$ do in the time that the body travels $2$ meters down the incline?
$$\left(g = 10\ \frac{\text{m}}{\text{s}^2},\ \sin 37^\circ = 0.6\right)$$
[Figure: Mass $m$ on an inclined plane at $37^\circ$ with $\mu_k = 0.25$, force $F$ directed up the incline]

• [(1)] $-260$ (2) $-160$
• [(3)] $+160$ (4) $+260$

166. The temperature of a metal disk is increased by 250 degrees Celsius, and as a result its surface area increases by one percent. What is the linear thermal expansion coefficient of the metal in SI units?
\[ (1)\ 2\times10^{-5} \qquad (2)\ 4\times10^{-5} \qquad (3)\ 2\times10^{-6} \qquad (4)\ 4\times10^{-6} \]

167. A piece of ice at $-20°C$ is placed inside 250 grams of water at $20°C$. After thermal equilibrium is established, 50 grams of ice remains unmelted. What was the initial mass of the ice?
\[ \left(C_{\text{water}} = 4.2\ \frac{\text{J}}{\text{g.K}},\quad C_{\text{ice}} = 2.1\ \frac{\text{J}}{\text{g.K}},\quad L_f = 336\ \frac{\text{J}}{\text{g}},\quad \text{heat exchange is only between water and ice.}\right) \]
\[ (1)\ 50 \qquad (2)\ 100 \qquad (3)\ 250 \qquad (4)\ 300 \]

168. The temperature of 2 moles of an ideal gas increases from $30°C$ to $80°C$ at constant pressure. How many joules of work is done on the gas in this process?
\[ \left(R = 8.3\ \frac{\text{J}}{\text{mol.K}}\right) \]
\[ (1)\ 415 \qquad (2)\ -415 \qquad (3)\ 830 \qquad (4)\ -830 \]

169. According to the figure below, a certain amount of ideal gas undergoes a process from state $i$ to state $f$. Which of the following statements about this process is correct?
[Figure: P-V diagram showing a curved path from point $i$ at $(V_1, P_1)$ to point $f$ at $(3V_1, 0.4P_1)$]

1. The process is isothermal.
2. The process is adiabatic.
3. The gas has absorbed heat.
4. The work done on the gas is positive.

170. The coefficient of performance of a refrigerator is 4. This refrigerator converts 2 kilograms of water at $10°C$ into ice at $-8°C$. How many kilojoules of heat does the refrigerator release to the environment in this process?
\[ \left(L_f = 336\ \frac{\text{kJ}}{\text{kg}},\quad C_{\text{water}} = 4200\ \frac{\text{J}}{\text{kg.°C}},\quad C_{\text{ice}} = 2C_{\text{ice}}\right) \]
\[ (1)\ 433 \qquad (2)\ 493 \qquad (3)\ 867 \qquad (4)\ 987 \]

171. A ball with diameter 20 cm is placed between a flat mirror and a concave mirror, and its shadow and penumbra are formed on the concave mirror. The radius of the light source is 4 cm, and the distance from the center of the light source to the center of the ball is 30 cm, and the line connecting these two centers is perpendicular to the flat mirror. If the diameter of the ball's shadow is 40 cm, what is the width of the penumbra in centimeters?
\[ (1)\ 4 \qquad (2)\ 5 \qquad (3)\ 8 \qquad (4)\ 10 \]

172. A concave mirror has an object placed on its principal axis, and the image length is $\dfrac{1}{4}$ of the object length. If we move the object to the location of the image, how many times the object length will the new image length be?
\[ (1)\ 16 \qquad (2)\ 8 \qquad (3)\ 4 \qquad (4)\ 1 \]
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