153- For which natural number values of $n$, is the expression $1 + 5^{3n+2} + 5^{6n+4}$ divisible by 31?
(1) Only odd values (2) Only even values (3) Only multiples of 5 (4) All values

156. A car is moving along a straight road at a speed of $108\,\dfrac{\text{km}}{\text{h}}$. The driver sees an obstacle and applies the brakes. At a distance of $165\,\text{m}$, with a constant deceleration of $3\,\dfrac{\text{m}}{\text{s}^2}$, the car stops and remains stationary. If $t_1$ is the reaction time of the driver and $t_2$ is the time during which the car is moving after braking, which of the following is $\dfrac{t_2}{t_1}$?
(1) $5$ (2) $10$ (3) $15$ (4) $20$

157. A bullet is fired in vacuum with no initial velocity from a height $h$ and travels a distance of 3 meters in the last second of its motion. If the distance it has traveled before that equals the distance it travels in that last second, how many meters is $h$? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$
(1) $20$ (2) $25$ (3) $75$ (4) $80$

158. The equation of a moving object's path in SI is $y = -\dfrac{1}{5}x^2 + 3x$. If the velocity component along the $x$-axis is constant and equal to $5\,\dfrac{\text{m}}{\text{s}}$, what is the speed of the object (in meters per second) at the moment it passes through point $M(5\text{m},\,10\text{m})$? (The object is at the origin at $t = 0$.)
(1) $5$ (2) $5\sqrt{2}$ (3) $10$ (4) $10\sqrt{2}$

159. From the top of a building 40 meters high, a ball is thrown with an initial speed $V_0$ in a direction that makes an angle of 45 degrees with the horizontal, upward and to the right. If the ball hits the ground at a point 120 meters from the base of the building, what is $V_0$ in meters per second? (Air resistance is negligible and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$.)
(1) $40$ (2) $30$ (3) $50$ (4) $60$

160. A bullet of mass $200\,\text{g}$ is fired horizontally from a height of $35$ meters above the ground, with an initial speed of $30\,\dfrac{\text{m}}{\text{s}}$, at an angle of $37°$ above the horizontal. It hits the ground after $t$ seconds. What is the impulse vector of the bullet during this time in SI?
$$\left(g = 10\,\dfrac{\text{m}}{\text{s}^2},\quad \sin 37° = 0.6,\quad \text{air resistance is negligible.}\right)$$
(1) $-2\hat{j}$ (2) $+2\hat{j}$ (3) $-10\hat{j}$ (4) $+10\hat{j}$
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162- Suppose a planet whose radius is half the radius of Earth and whose mass is $\dfrac{1}{4}$ the mass of Earth. The gravitational acceleration on the surface of this planet will be how many times the gravitational acceleration on the surface of Earth?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) $2$

165- According to the figure below, light enters perpendicularly through face AB of a prism with refractive index $n = 2$, and continues along one of the faces of the prism. What is the angle of deviation of this ray with respect to its initial direction in degrees?
[Figure: A prism with vertex angle B at top, base AC at bottom, with angles $70^\circ$ at A and $70^\circ$ at C, and a ray entering perpendicular to face AB]

• [(1)] $50$
• [(2)] $90$
• [(3)] $100$
• [(4)] $160$

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166- A person moves at a speed of 20 cm/s toward a flat mirror. The mirror is also moving at a speed of 20 cm/s toward the person. How many centimeters per second does the image move toward the person?
$80$ (1) $60$ (2) $40$ (3) $20$ (4)

167- The contact point of a rolling sphere on a flat mirror is at rest. We observe that the maximum possible displacement for the image in a 40 cm mirror is. If a body is placed in front of this mirror at a distance of 120 cm from the mirror, how many centimeters will the image be?
$180$ (1) $150$ (2) $90$ (3) $40$ (4)

168- An object moves at a constant speed toward a converging lens. If at a certain moment the object is at a distance $2f$ from the lens, and the position changes so that the object moves from distance $2f$ to distance $f$ from the lens, how does the image move? (f is the focal length of the lens.)

• [(1)] It moves away from the lens slowly.
• [(2)] It moves away from the lens quickly.
• [(3)] It moves toward the lens slowly.
• [(4)] It moves toward the lens quickly.

169- We want to make a sphere from a metal with density $\dfrac{g}{\text{cm}^3} \times 6$, with a radius of $5\,\text{cm}$. How many kilograms will this sphere be?
$1.57$ (1) $2.36$ (2) $3.14$ (3) $4.71$ (4)

170- Two spheres of the same metal type A and B, with outer radius $20\,\text{cm}$ and inner radius $20\,\text{cm}$, and the inner cavity radius of B is $10\,\text{cm}$. If we heat both spheres by the same amount of temperature, and the change in volume of sphere A is $\Delta V_A$ and the change in volume of metal in sphere B is $\Delta V_B$, which of the following is the ratio $\dfrac{\Delta V_A}{\Delta V_B}$?
$\dfrac{\gamma}{\lambda}$ (1) $1$ (2) $2$ (3) $\dfrac{\lambda}{\gamma}$ (4)

171- A container holds $1000$ g of water and $200$ g of ice at zero degrees Celsius. In thermal equilibrium, a piece of special metal is heated to $400\,\dfrac{\text{J}}{\text{kg}\cdot\text{K}}$ and placed inside the container at $250$ degrees Celsius. At minimum, how many grams of metal must be placed so that some ice remains in the container?
$$\left( L_f = 336000\,\frac{\text{J}}{\text{kg}},\quad c_{\text{water}} = 4200\,\frac{\text{J}}{\text{kg}\cdot\text{K}} \text{ and heat loss is negligible.} \right)$$
$375$ (1) $672$ (2) $860$ (3) $950$ (4)
\begin{flushright} Calculation Space \end{flushright}
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172. In the figure below, the piston mass is one kilogram, the mass of the gas on it is 4 kilograms, and the gas temperature inside the container is 27 degrees Celsius. If we want to bring the gas to rest at 87 degrees Celsius by gradually adding weight on top of the piston, how many kilograms of weight should be added so that the piston does not move?
[Figure: A container with a piston, gas trapped below the piston]
(Piston base area is $5\,\text{cm}^2$, atmospheric pressure is $10^5$ pascals, and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$.)

• [(1)] 2
• [(2)] 3
• [(3)] 6
• [(4)] 7

173. In the figure below, valve $R$ is closed and the temperature of the trapped air in the tube is raised from 39 degrees Celsius. By how many degrees should the temperature be increased so that the height difference of the mercury columns in the two tubes increases by 2 centimeters? (The atmospheric pressure at the mercury level is 78 centimeters of mercury, and the diameters of the two tubes are equal. Neglect the volume of mercury.)
[Figure: A U-tube manometer with valve R at the top, mercury columns with a height difference of $\Delta$cm indicated]

• [(1)] 22
• [(2)] 100
• [(3)] 211
• [(4)] 384

174. A container with a fixed volume of 14 liters contains a mixture of 6 grams of hydrogen gas and 112 grams of nitrogen gas at 27 degrees Celsius. What is the total pressure of the mixed gases in atmospheres?
$$\left(M_{N_2} = 28\,\frac{\text{g}}{\text{mol}},\quad M_{H_2} = 2\,\frac{\text{g}}{\text{mol}},\quad 1\,\text{atm} = 10^5\,\text{pa},\quad R = 8\,\frac{\text{J}}{\text{mol.K}}\right)$$
(1) $6$ (2) $8$ (3) $9$ (4) $12$

175. 10 grams of hydrogen gas at constant pressure is heated from $27^\circ\text{C}$ to $127^\circ\text{C}$. How many kilojoules of work is done by the gas in this process?
$$\left(R = 8\,\frac{\text{J}}{\text{mol.K}}\right)$$
(1) $2$ (2) $4$ (3) $6$ (4) $8$

Space for calculations
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176. A fixed amount of an ideal monatomic gas follows the cycle shown in the figure below. If the gas absorbs $1500\,\text{J}$ of heat during process $ab$, how many joules of internal energy does it gain during process $ca$?
\begin{minipage}{0.35\textwidth} [Figure: P-V diagram showing a cycle with points a, b, c. The x-axis is V with values $V_1$ and $4V_1$, the y-axis is P with values $P_1$ and $2P_1$. Point a is at $(V_1, P_1)$, point b is at $(4V_1, P_1)$, and point c is at $(V_1, 2P_1)$.] \end{minipage} \begin{minipage}{0.55\textwidth}

• [(1)] $1500$
• [(2)] $1800$
• [(3)] $2100$
• [(4)] $2400$

\end{minipage}

177. Inside a uniform electric field, a point charge $q = +2\,\mu\text{C}$ moves from point A to point B. The electric force does work on this charge. If the work done during this transfer equals $J = +5\times10^{-5}\,\text{J}$, the change in electric potential energy of the charge is how many joules, and $V_B - V_A$ equals how many volts?

• [(1)] $-\Delta\times10^{-5}$ and $-25$
• [(2)] $-\Delta\times10^{-5}$ and $+25$
• [(3)] $+\Delta\times10^{-5}$ and $-25$
• [(4)] $+\Delta\times10^{-5}$ and $+25$

178. Four charged particles are placed at the vertices of a square. The electric forces on the charged particle $q_2$ are zero. What is $\dfrac{Q}{q}$?
\begin{minipage}{0.4\textwidth} [Figure: Square with charges at vertices: $q_1 = q$ (top-left), $q_2 = Q$ (top-right), $q_3 = q$ (bottom-right), $q_4 = -\dfrac{1}{2}Q$ (bottom-left).] \end{minipage} \begin{minipage}{0.5\textwidth}

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

\end{minipage}

179. In the figure shown, what is the equivalent capacitance between points A and B in picofarads?
\begin{minipage}{0.45\textwidth} [Figure: A network of capacitors between points A and B, with capacitors labeled $C_5$, $C_4$, $C_3$, $C_2$, $C_1$, $C_6$, $C_7$ with values $5\,\text{pF}$, $4\,\text{pF}$, $3\,\text{pF}$, $2\,\text{pF}$, $1\,\text{pF}$, $6\,\text{pF}$, $7\,\text{pF}$ respectively, and $C_A = 4\,\text{pF}$, $C_F = 4\,\text{pF}$.] \end{minipage} \begin{minipage}{0.45\textwidth}

• [(1)] $12$
• [(2)] $8$
• [(3)] $6$
• [(4)] $4$

\end{minipage}
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180. In the circuits below, the capacitors are given measured values and have electrical charge. If the plates of all capacitors in each circuit are connected together, which capacitor's charge decreases?
[Figure: Two circuits shown. Circuit (1): $C_1 = 2C$ with charge $2q$ in parallel with $C_2 = C$ with charge $q$. Circuit (2): $C_3 = 2C$ with charge $q$ in parallel with $C_4 = C$ with charge $q$.]

• [(1)] $C_1$ and $C_2$
• [(2)] $C_4$ and $C_3$
• [(3)] $C_3$
• [(4)] $C_4$

181. In the circuit shown, what is the current passing through battery $\varepsilon_2$, in amperes? (Both batteries are ideal.)
[Figure: Circuit with $\varepsilon_1 = 12\,\text{V}$, $r_1 = 0$, $\varepsilon_2 = 16\,\text{V}$, $r_2 = 0$, and resistors $2\,\Omega$, $4\,\Omega$, $3\,\Omega$, $6\,\Omega$, $8\,\Omega$ connected in a network.]

• [(1)] $0.5$
• [(2)] $1.5$
• [(3)] $2$
• [(4)] $2.5$

182. Two copper and aluminum wires of equal length, at a certain temperature, have equal electrical resistance. If the densities of copper and aluminum are respectively $9\,\dfrac{\text{g}}{\text{cm}^3}$ and $2.7\,\dfrac{\text{g}}{\text{cm}^3}$, and the specific resistance of copper is $\dfrac{1}{2}$ times the specific resistance of aluminum, how many times is the mass of the aluminum wire compared to the copper wire?

• [(1)] $\dfrac{3}{5}$
• [(2)] $\dfrac{4}{5}$
• [(3)] $\dfrac{5}{4}$
• [(4)] $\dfrac{5}{3}$

Workspace
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