37. By placing the digit 3 between two digits of the three-digit number $(a \circ a)$, a new number $a$ is formed. What is the maximum number of values that $a$ can take?
(1) zero (2) $1$ (3) $2$ (4) $3$

39. What is the minimum number of ordered pairs to be selected from the natural numbers such that, when selected in order, at least two of the selected pairs have the same sum of first components and the same sum of second components, with each being a multiple of 5?
(1) $13$ (2) $14$ (3) $25$ (4) $26$

40. The sum of degrees of vertices of graph $G$ is $48$. If the minimum degree of a vertex is at least one, what is $\Delta(\overline{G}) + q(\overline{G})$?
(1) $17$ (2) $11$ (3) $19$ (4) $13$
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41. In the decay process ${}^{11}_{6}\mathrm{C} \longrightarrow {}^{11}_{5}\mathrm{B} + \mathrm{x}$, what is $\mathrm{x}$?
(1) proton (2) $\beta^+$ (3) $\beta^-$ (4) neutron

43. The length of a steel suspension bridge was 900 meters in the coldest year and reached 900.9 meters in the hottest year of that year. The difference between the highest and lowest temperatures of the bridge in that year is how many degrees Celsius?
($\alpha = 1.25 \times 10^{-5}\,\mathrm{K}^{-1}$)
(1) 70 (2) 80 (3) 90 (4) 100

44. In which process performed on a gas, is the work positive and the internal energy of the gas decreases?
(1) isochoric compression (2) isobaric compression (3) isobaric expansion (4) adiabatic expansion

49. Assume that satellites orbit the Earth in circular orbits. Which of the following statements is correct?

1. [(1)] The orbital speed of a satellite around the Earth is proportional to the square root of the satellite's distance from the center of the Earth.
2. [(2)] The square of the orbital period of a satellite around the Earth is proportional to the cube of the satellite's distance from the center of the Earth.
3. [(3)] The acceleration of a satellite is proportional to the square root of the satellite's distance from the center of the Earth.
4. [(4)] The weight of a satellite is inversely proportional to the square root of the satellite's distance from the center of the Earth.

53 -- A string of length $60\,\text{cm}$ is fixed at both ends and vibrates, forming $3$ antinodes along its length. If $300\,\text{Hz}$ is a harmonic frequency of this string, how many Hz is the fundamental frequency and what is the harmonic number of $300\,\text{Hz}$?

• [(1)] $500$ and $300$
• [(2)] $120$ and $300$
• [(3)] $120$ and $100$
• [(4)] $500$ and $100$

54 -- If the distance from a sound source is halved and at the same time the power of the sound source is doubled, how does the sound intensity level change? $\left(\log 2 \approx 0.3\right)$

• [(1)] becomes 8 times.
• [(2)] becomes 9 times.
• [(3)] increases by 4 decibels.
• [(4)] increases by 9 decibels.

55 -- The length of a simple pendulum is changed to $17\,\text{cm}$, and its period increases by $12.5\%$. What was the period (before the change) in seconds? $\left(g = \dfrac{\pi^2\,\text{m}}{\text{s}^2}\right)$

• [(1)] $1.2$
• [(2)] $1.4$
• [(3)] $1.6$
• [(4)] $1.8$

56 -- The position--time equation of simple harmonic motion in SI is $x = A\cos 5\pi t$. If the average speed over the time interval $t_1 = 0\,\text{s}$ to $t_2 = 0.2\,\text{s}$ is $1.5\,\dfrac{\text{m}}{\text{s}}$, the amplitude of oscillation is how many centimeters?

• [(1)] $1.5$
• [(2)] $3$
• [(3)] $4.5$
• [(4)] $6$

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57. According to the figure, a string fixed between two supports oscillates at its first harmonic with frequency $f$. If the distance between the two supports is $50\,\text{cm}$ and the transverse wave speed in it is $25\,\dfrac{\text{m}}{\text{s}}$, how many milliseconds does it take for each particle of the string to complete one oscillation?
[Figure: string fixed at both ends vibrating in first harmonic mode]

• [(1)] 25
• [(2)] 2
• [(3)] 5
• [(4)] 4

58. In the hydrogen atom, an electron absorbs a photon with energy $12.75\,\text{eV}$ and moves from orbit $n^0$ to orbit $n$. The values of $n^0$ and $n$ are respectively ($E_R = 13.6\,\text{eV}$):

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

59. In a photoelectric device, the work function of metal $Z$ is $4\,\text{eV}$. We perform two experiments with this device. In the second experiment, we use half the wavelength of the incident light, and the maximum kinetic energy of the photoelectrons is 6 times that of the first experiment. What is the wavelength of the light used in the first experiment in nanometers? ($c = 3\times10^{8}\,\dfrac{\text{m}}{\text{s}}$ and $h = 4\times10^{-15}\,\text{eV·s}$)

• [(1)] 180
• [(2)] 240
• [(3)] 360
• [(4)] 480

60. What is the enrichment process for a sample of uranium?

• [(1)] Converting as much uranium-235 to uranium-238 as possible
• [(2)] Converting as much uranium-238 to uranium-235 as possible
• [(3)] Increasing the percentage of uranium-238 isotopes
• [(4)] Increasing the percentage of uranium-235 isotopes

61. By decreasing the electric charge of a capacitor, if we reduce its energy by $\dfrac{3}{4}$, the initial potential of it becomes what fraction of its initial potential?

• [(1)] $\dfrac{1}{4}$
• [(2)] $\dfrac{3}{4}$
• [(3)] $\dfrac{7}{16}$
• [(4)] $\dfrac{9}{16}$

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62. A charge $q = 2\,\text{nC}$ moves along the direction of a uniform electric field from point A to point B, and the electric potential energy decreases by $2\,\text{mJ}$. The potential is $V_B = V_A$ increases by a few volts, and the direction of motion compared to the direction of the electric field is:
(1) $-10^5$ and opposite to the field direction (2) $+10^5$ and opposite to the field direction
(3) $+10^5$ and in the field direction (4) $-10^5$ and in the field direction

63. In the figure below, three positive point charges at the vertices of an equilateral triangle are fixed. The net electric field at point M (center of the triangle) is $E$. If we remove charge $q_2$, how large does the net electric field at point M become?
[Figure: Equilateral triangle with vertices labeled $q_2 = 2\,\mu\text{C}$ (top right), $q_1 = 1\,\mu\text{C}$ (left), $q_3 = 3\,\mu\text{C}$ (bottom), and center point M]
(1) $\sqrt{5}$
(2) $2\sqrt{5}$
(3) $\dfrac{3}{2}$
(4) $\dfrac{2}{3}$

64. In the figure below, the potential difference across a $6\,\Omega$ resistor and an $8\,\Omega$ resistor are equal. The current intensity passing through the $8\,\Omega$ resistor is how many amperes?
[Figure: Circuit with resistors $15\,\Omega$, $R$, $8\,\Omega$, $6\,\Omega$, $12\,\Omega$, $6\,\Omega$, battery $16\,\text{V}$ with internal resistance $r = 2\,\Omega$]
(1) $0.2$
(2) $0.3$
(3) $0.4$
(4) $0.5$
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65. In the circuit below, an ideal ammeter reads $0.25$ A and an ideal voltmeter reads $5$ V. What is $R_1$?
[Figure: Circuit with $10\,\Omega$ resistor, $R_2$, voltmeter (V), ammeter (A), $R_1$, battery $12\,\text{V}$, internal resistance $r$, and $6\,\Omega$ resistor]

• [(1)] $12$
• [(2)] $16$
• [(3)] $18$
• [(4)] $24$

66. In the figure below, if we close the switch K, the electric potential difference across the two terminals of the battery becomes how many times?
[Figure: Circuit with resistors $R$, $R$, $R$, switch $K$, internal resistance $r$, battery]

• [(1)] $\dfrac{4}{5}$
• [(2)] $\dfrac{5}{6}$
• [(3)] $\dfrac{14}{15}$
• [(4)] $\dfrac{15}{16}$

67. According to the figure, three loops with equal currents of $0.5\,\text{A}$, each with radius $15\,\text{cm}$, are placed such that the plane of each loop is perpendicular to the other two loops. The magnitude of the magnetic field at point O (the center of the loops) is how many tesla? $\left(\mu_0 = 4\pi \times 10^{-7}\,\dfrac{\text{T.m}}{\text{A}}\right)$
[Figure: Three circular loops arranged mutually perpendicular to each other, with center point O]

• [(1)] $2\sqrt{3} \times 10^{-6}$
• [(2)] $2\sqrt{2} \times 10^{-6}$
• [(3)] $4 \times 10^{-6}$
• [(4)] $2 \times 10^{-6}$

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68. An electron passes through an environment that contains a uniform magnetic field and a uniform electric field. If the magnitude and direction of the electron's velocity remain constant along this path, which statement is correct?

1. [(1)] Both fields are parallel to the path of the electron and in opposite directions.
2. [(2)] Both fields are perpendicular to the path of the electron and in opposite directions.
3. [(3)] The magnetic field is necessarily perpendicular to the path of the electron, but the electric field may not be perpendicular to this path.
4. [(4)] The electric field is necessarily perpendicular to the path of the electron, but the magnetic field may not be perpendicular to this path.

69. An ideal solenoid without a core, with length $15.7$ cm, has $1000$ turns. If the cross-sectional area of each turn is $8\,\text{cm}^2$, what is its inductance in millihenries? $\left(\mu_0 = 4\pi \times 10^{-7}\,\dfrac{\text{T.m}}{\text{A}}\right)$
(1) $6.4$ (2) $64$ (3) $1.6$ (4) $16$

70. A wire in the shape of a circular loop with radius $10\,\text{cm}$ is placed on a horizontal surface. A uniform magnetic field perpendicular to the surface, with a tilt angle of $30°$ from the plane of the loop, decreases from $6000$ gauss to zero in $15.7$ milliseconds. What is the average induced EMF in the loop in volts?
(1) $0.6\sqrt{3}$ (2) $0.6$ (3) $1.2\sqrt{3}$ (4) $1.2$