129. In triangle $ABC$, with $BC = A$, altitude $AH = h$, and angle $\hat{A} = A\circ^\circ$, the triangle can be constructed. What is the maximum value of $h$? (1) $f\sin f\circ^\circ$ (2) $f\cos f\circ^\circ$ (3) $f\tan f\circ^\circ$ (4) $f\cot f\circ^\circ$
130. In a trapezoid circumscribed about a circle, the length of the line segment connecting the midpoints of the two legs is 12 units. What is the perimeter of the trapezoid? (1) $36$ (2) $44$ (3) $46$ (4) $48$
132. Plane $P$ and line $d$ and point $A$ are given, and point $Q$ on plane $A$ and line $d$ are assumed. If we draw the line passing through point $A$ and parallel to plane $P$ and intersecting line $d$, in which case is this impossible? (1) $Q \cap P \neq \phi$, $d \| P$ (2) $Q \cap P \neq \phi$, $d \not\| P$ (3) $Q \cap P = \phi$, $d \| P$ (4) $Q \cap P = \phi$, $d \not\| P$
149. The graph has vertices $(1,2)$, $(2,4)$, $(4,\circ)$, $(\circ,4)$, $(1,5)$, $(3,6)$, $(3,7)$, and $(5,7)$. Among the real numbers, how many have degree 4? (1) $2$ (2) $2$ (3) $4$ (4) $5$
151- The remainder of dividing the natural number $N$ by $31$ is $26$. If we divide this number by $43$, the remainder that comes out of the quotient is $N$. What is the largest single-digit number of $N$? (1) $2$ (2) $4$ (3) $6$ (4) $7$
153- In the directed graph below, by drawing at least a few new edges, a graph with the properties of transitivity and anti-symmetry is obtained. What is the result? [Figure: A directed graph with vertices $a$, $b$, $c$, $d$, $e$ and directed edges between them] \begin{flushright} (1) $2$ (2) $4$ (3) $5$ (4) Not possible \end{flushright}
156- A particle starts from rest with constant acceleration $\vec{a} = \hat{i} + 2\hat{j}$ and begins to move. The position vector at $t = 4$ is which of the following? (Quantities are in SI units.) (1) $\vec{r} = 8\hat{i} + 16\hat{j}$ (2) $\vec{r} = 8\hat{i} + 12\hat{j}$ (3) $\vec{r} = 4\hat{i} + 12\hat{j}$ (4) $\vec{r} = 4\hat{i} + 16\hat{j}$
157- The velocity–time graph of two moving objects A and B, both moving along the x-axis, is shown below. During the time that object A moves in the positive x direction, how many meters is the displacement of object B? [Figure: Velocity-time graph with V(m/s) on vertical axis and t(s) on horizontal axis. Object B is a straight line starting from 16 m/s decreasing, object A starts from 0, goes to 18 on time axis, then decreases to $-8$ and continues to $-20$. The graph shows values 16, 18 on axes and $-8$, $-20$ marked.]
158- Two balls A and B with initial speeds of $30\ \dfrac{\text{m}}{\text{s}}$ according to the figure below are launched simultaneously. From the moment the two balls pass beside each other, how many times greater is the displacement of ball A compared to the displacement of ball B? (Neglect air resistance. $g = 10\ \dfrac{\text{m}}{\text{s}^2}$) [Figure: Ball B is launched downward from height 180 m with initial velocity $V_0$ downward, and ball A is launched upward from the ground with initial velocity $V_0$ upward. The height between them is labeled $180\ \text{m}$.]
159- The trajectories of two projectiles launched simultaneously from a slope with equal initial speeds are shown below. Neglecting air resistance, which one reaches the ground sooner? [Figure: Two projectile paths A and B launched from a slope at height h. Path A has a larger arc and path B has a smaller arc closer to the slope.]
160- In the figure below, if the pulley and strings are massless and frictionless, the tension $T$ is how many Newtons? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$ [Figure: A fixed pulley attached to ceiling with a string over it; tension $T$ shown at top; two masses $4\,\text{kg}$ and $1\,\text{kg}$ hanging on either side]
161- Two objects A and B with equal initial velocities are thrown horizontally onto a horizontal surface. If the mass of object A is half the mass of object B, and the coefficient of friction of B is 2 times the coefficient of friction of A, find the distance that object A travels until it stops. How many times the distance that object B travels until it stops? $$\frac{1}{2} \;(4) \qquad\qquad \frac{\sqrt{2}}{2} \;(3) \qquad\qquad 1 \;(2) \qquad\qquad 2 \;(1)$$
162- In the figure below, a body is under horizontal force $F_1$ and is at the threshold of motion, and under horizontal force $F_2$ it moves with constant velocity. If the friction forces in these two cases are $f_1$ and $f_2$ respectively, which statement is correct? $(\mu_s > \mu_k)$ [Figure: A block on a rough surface with horizontal force $F$ applied]
163- On a curved road, the maximum allowed speed is $54\,\dfrac{\text{km}}{\text{h}}$. If the transverse slope of the road makes an angle of $37°$ with the horizontal, what is the radius of curvature of this curve in meters? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2},\ \sin 37° = 0.6\right.$ and friction on the road surface is negligible.) $$35 \;(1) \qquad\qquad 40 \;(2) \qquad\qquad 50 \;(3) \qquad\qquad 60 \;(4)$$ %% Page 12 Physics120-CPage 11
165. In the figure below, if body $AB$ rotates $\text{A}$ degrees about point $A$, and on the same surface a mirror of size $25$ degrees is also placed, we rotate the mirror clockwise by $25$ degrees; by how many degrees does the angle between the body and its image in the mirror change? [Figure: A mirror on a surface with angle $30°$ marked, and body $AB$]
166. An object is placed at a distance of $15\,\text{cm}$ in front of a concave mirror with radius $40\,\text{cm}$, and the image length is $4\,\text{cm}$. How many centimeters should the object be moved so that the image also has a length of $4\,\text{cm}$? (1) $5$ (2) $15$ (3) $25$ (4) $55$
167. In a converging lens, the distance between the object and the image is $20\,\text{cm}$ and the magnification is $0.5$. If the object is placed $20\,\text{cm}$ from the lens, by how much will the magnification change? (1) $0.2$ (2) $0.4$ (3) $0.6$ (4) $0.8$
168. A ray of light $SI$ hits face $AB$ of a prism and refracts such that the refractive index of the prism with respect to air is $\dfrac{4}{3}$. The entry into the prism: $(\sin 53° = 0.8)$ [Figure: A triangular prism with angles $53°$, $53°$, and $37°$ at vertices $A$, $S$, and $C$ respectively, with ray $I$ hitting face $AB$ and point $B$ at top]
It returns along the original path.
It exits from face $BC$ into air.
It undergoes total internal reflection on face $BC$.
It exits tangentially from face $BC$ of the prism.
\begin{flushright} Calculation Space \end{flushright} %% Page 13 Physics120-CPage 12
169- The efficiency of a heat engine (Carnot) is 30\%. If the temperature of the hot source is 4 times the temperature of the cold sink, what is the temperature of the cold sink in degrees Celsius? (1) $28$ (2) $35.5$ (3) $45.5$ (4) $91$
170- If the $(P-T)$ diagram of 5 moles of ideal gas A with volume 10 liters and $n$ moles of ideal gas B with volume 16 liters is as shown below, what is $n$? \begin{minipage}{0.45\textwidth} [Figure: P-T diagram showing two lines from origin, line (B) with steeper slope and line (A) with lesser slope, with $P_1$ and $\frac{4}{7}P_1$ marked on the pressure axis] \end{minipage} \begin{minipage}{0.45\textwidth}
171- A gas inside a container at constant pressure $2\times10^5\ \text{Pa}$ is cooled and its volume goes from $6\ \text{lit}$ to $2\ \text{lit}$. If in this process $2800\ \text{J}$ of heat is released, by how many joules does the internal energy of the gas decrease? (1) $1200$ (2) $1800$ (3) $2000$ (4) $2600$
172- In a container that is thermally insulated, there is a piece of ice at zero degrees Celsius. If we add 800 grams of water at 50 degrees Celsius to the container, then bring it to thermal equilibrium, 100 grams of ice remains in the container. How many grams was the initial piece of ice? $$\left(C_{\text{water}} = 4200\ \frac{\text{J}}{\text{kg}\cdot\text{K}} \text{ and } L_f = 336000\ \frac{\text{J}}{\text{kg}}\right) \text{ (only heat exchange between water and ice)}$$ (1) $300$ (2) $400$ (3) $500$ (4) $600$