MikaStars39/nano-eval huggingface dataset:
The surface area of the metal is $A$. Work $\Phi$ is required to move a nitrogen atom from the metal surface to infinity. Helium atoms do not interact with each other; they can move freely on the two-dimensional metal surface. If the volume of helium gas is $V$, the pressure is $p$, and the temperature is $T$, and it is in thermal equilibrium with the metal, what is the average number of helium atoms adsorbed on a unit metal surface area? Provide your answer using the quantities given in the problem and fundamental constants.

Options:
A. $$
\frac{h} {\sqrt{k \pi m}} \dot{p} T^{-3 / 2} \mathrm{e x p} \Big( \frac{\Phi} {2 k T} \Big)
$$
B. $$
\frac{h k^2} {2 \pi m} \dot{p} \mathrm{e x p} \Big( \frac{\Phi} {k T^2} \Big)
$$
C. $$
\frac{\sqrt{2 \pi m k}} {h k} \mathrm{e x p} \Big( \frac{k T} {\Phi} \Big) p T^{-1}
$$
D. $$
\frac{h} {k \sqrt{m}} \dot{p} T^{-1 / 2} \mathrm{e x p} \Big( \frac{2 \Phi} {k T} \Big)
$$
E. $$
\frac{k} {h \sqrt{2 \pi m k}} \dot{p} T^{1 / 2} \mathrm{e x p} \Big( \frac{-\Phi} {k T} \Big)
$$
F. $$
\frac{\pi m k} {h} \dot{p} T^{-2} \mathrm{e x p} \Big( \frac{\Phi} {k} \Big)
$$
G. $$
\frac{k T} {h \sqrt{2 \pi m}} \dot{p} \mathrm{e x p} \Big( \frac{\Phi^2} {k T} \Big)
$$
H. $$
\frac{\Phi} {h \sqrt{2 \pi k m}} \dot{p} T^{-1} \mathrm{e x p} \Big( \frac{k} {T} \Big)
$$
I. $$
\frac{h k} {2 \pi m k^2} \dot{p} \mathrm{e x p} \Big( \frac{\Phi} {k^2 T^2} \Big)
$$
J. $$
\frac{h} {k \sqrt{2 \pi m k}} p T^{-3 / 2} \mathrm{e x p} \Big( \frac{\Phi} {k T} \Big)
$$
Answer: J
In an equilateral triangle, each vertex has a particle with a spin quantum number of 1/2. The interaction Hamiltonian between them is given by
$$
H=\frac{\lambda} {3} ( \sigma_{1} \cdot\sigma_{2}+\sigma_{2} \cdot\sigma_{3}+\sigma_{3} \cdot\sigma_{1} )
$$
where $\sigma$ represents the Pauli operators and $\lambda$ is the interaction constant. Try to determine the energy level $E$ and derive the partition function $Z$ .

Options:
A. $$
E=\frac{\lambda}{6}[5S(S-1)-9]
$$
$$
Z= 2\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 6\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 8\mathrm{sinh}\Big( \frac{\lambda}{kT}\Big)
$$
B. $$
E=\frac{\lambda}{4}[4S(S+2)-10]
$$
$$
Z= 5\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 5\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 10\mathrm{cosh}\Big( \frac{\lambda}{kT}\Big)
$$
C. $$
E=\frac{\lambda}{6}[3S(S+1)-9]
$$
$$
Z= 2\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 2\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 4\mathrm{sinh}\Big( \frac{\lambda}{kT}\Big)
$$
D. $$
E=\frac{\lambda}{5}[3S(S+1)-7]
$$
$$
Z= 6\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 6\mathrm{exp}\Big( \frac{\lambda}{2kT}\Big)= 12\mathrm{cosh}\Big( \frac{\lambda}{2kT}\Big)
$$
E. $$
E=\frac{\lambda}{6}[4S(S+1)-8]
$$
$$
Z= 6\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 8\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 14\mathrm{cosh}\Big( \frac{\lambda}{kT}\Big)
$$
F. $$
E=\frac{\lambda}{7}[4S(S+3)-11]
$$
$$
Z= 7\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 7\mathrm{exp}\Big( \frac{\lambda}{2kT}\Big)= 14\mathrm{cosh}\Big( \frac{\lambda}{2kT}\Big)
$$
G. $$
E=\frac{\lambda}{8}[4S(S+1)-12]
$$
$$
Z= 3\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 5\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 8\mathrm{sinh}\Big( \frac{\lambda}{kT}\Big)
$$
H. $$
E=\frac{\lambda}{6}[4S(S+1)-9]
$$
$$
Z= 4\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 4\mathrm{exp}\Big( \frac{\lambda}{kT}\Big)= 8\mathrm{cosh}\Big( \frac{\lambda}{kT}\Big)
$$
I. $$
E=\frac{\lambda}{9}[2S(S+1)-8]
$$
$$
Z= 3\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 3\mathrm{exp}\Big( \frac{\lambda}{2kT}\Big)= 6\mathrm{cosh}\Big( \frac{\lambda}{2kT}\Big)
$$
J. $$
E=\frac{\lambda}{8}[3S(S+1)-10]
$$
$$
Z= 4\mathrm{exp}\Big(-\frac{\lambda}{kT}\Big)+ 4\mathrm{exp}\Big( \frac{\lambda}{2kT}\Big)= 8\mathrm{cosh}\Big( \frac{\lambda}{2kT}\Big)
$$
Answer: H
Consider a glass in which some constituent atoms can occupy two slightly different positions, resulting in an energy level deviation. For the $i$th atom, this energy level difference is △ and $- \Delta_i$.
(1) If all △ are equal, denoted as A, what is the contribution to the heat capacity ? (excluding Debye heat capacity);
(2) If △ is randomly distributed around $\Delta_{0} > 0$, what is the low-temperature behavior of the heat capacity, specifically the behavior when $k T \ll \Delta_{0}$ (integrals that do not depend on physical variables can be retained).

Options:
A. $$
3 N k \Big( {\frac{\Delta} {k T}} \Big)^{4} \cdot{\frac{1} {( e^{2\Delta/ k T}+e^{-2\Delta/ k T} )^{4}}}
$$
$$
3 k \! \int\! \left( \frac{\Delta} {k T} \right)^{4} \! e^{-4 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
B. $$
5 N k \Big( {\frac{\Delta} {k T}} \Big)^{5} \cdot{\frac{1} {( e^{\Delta/ 2k T}+e^{-\Delta/ 2k T} )^{5}}}
$$
$$
5 k \! \int\! \left( \frac{\Delta} {k T} \right)^{5} \! e^{-5 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
C. $$
8 N k \Big( {\frac{\Delta} {k T}} \Big)^{2} \cdot{\frac{1} {( e^{\Delta/ k T}+e^{-\Delta/ k T} )^{2.5}}}
$$
$$
8 k \! \int\! \left( \frac{\Delta} {k T} \right)^{2.5} \! e^{-2.5 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
D. $$
4 N k \Big( {\frac{\Delta} {k T}} \Big)^{2} \cdot{\frac{1} {( e^{\Delta/ k T}+e^{-\Delta/ k T} )^{2}}}
$$
$$
4 k \! \int\! \left( \frac{\Delta} {k T} \right)^{2} \! e^{-2 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
E. $$
10 N k \Big( {\frac{\Delta} {k T}} \Big)^{9} \cdot{\frac{1} {( e^{\Delta/ 2k T}+e^{-\Delta/ 2k T} )^{9}}}
$$
$$
10 k \! \int\! \left( \frac{\Delta} {k T} \right)^{9} \! e^{-9 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
F. $$
N k \Big( {\frac{\Delta} {2k T}} \Big)^{6} \cdot{\frac{1} {( e^{\Delta/ 2k T}+e^{-\Delta/ 2k T} )^{6}}}
$$
$$
k \! \int\! \left( \frac{\Delta} {2k T} \right)^{6} \! e^{-6 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
G. $$
6 N k \Big( {\frac{\Delta} {2k T}} \Big)^{3} \cdot{\frac{1} {( e^{\Delta/ 2k T}+e^{-\Delta/ 2k T} )^{3}}}
$$
$$
6 k \! \int\! \left( \frac{\Delta} {2k T} \right)^{3} \! e^{-3 \Delta/ 2k T} \rho( \Delta) \mathrm{d} \Delta
$$
H. $$
2 N k \Big( {\frac{\Delta} {k T}} \Big) \cdot{\frac{1} {( e^{\Delta/ k T}+e^{-\Delta/ k T} )}}
$$
$$
2 k \! \int\! \left( \frac{\Delta} {k T} \right) \! e^{-\Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
I. $$
7 N k \Big( {\frac{\Delta} {2k T}} \Big)^{7} \cdot{\frac{1} {( e^{2\Delta/ k T}+e^{-2\Delta/ k T} )^{7}}}
$$
$$
7 k \! \int\! \left( \frac{\Delta} {2k T} \right)^{7} \! e^{-7 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
J. $$
9 N k \Big( {\frac{\Delta} {k T}} \Big)^{8} \cdot{\frac{1} {( e^{\Delta/ k T}+e^{-\Delta/ k T} )^{8}}}
$$
$$
9 k \! \int\! \left( \frac{\Delta} {k T} \right)^{8} \! e^{-8 \Delta/ k T} \rho( \Delta) \mathrm{d} \Delta
$$
Answer: D
Potassium chloride $(^{39} \mathrm{K}^{35} \mathrm{Cl})$ is a colorless and transparent ionic crystal with a density of $\rho = 2.0 \mathrm{g/cm}^{3}$: (1) calculate the nearest neighbor distance between $\mathrm{K^{+}}$ and $\mathrm{Cl^{-}}$ in this solid; (2) based on the given optical properties, derive the lower limit of the energy gap between the valence band and the conduction band; (3) provide the relationship between the specific heat capacity $c$ of solid KCl and the temperature $T$;

Options:
A. $$
2. 9 \, \AA
$$
$$
1. 5 0 \times1 0^{-1 8} \mathrm{J}
$$
$$
\beta T^{2}
$$
B. $$
5. 0 \, \AA
$$
$$
2. 0 0 \times1 0^{-1 9} \mathrm{J}
$$
$$
\beta T^{3}
$$
C. $$
3. 3 \, \AA
$$
$$
1. 2 0 \times1 0^{-1 8} \mathrm{J}
$$
$$
\alpha T^{2}
$$
D. $$
3. 6 \, \AA
$$
$$
1. 1 0 \times1 0^{-1 8} \mathrm{J}
$$
$$
\delta T^{2}
$$
E. $$
4. 5 \, \AA
$$
$$
1. 9 0 \times1 0^{-1 9} \mathrm{J}
$$
$$
\gamma T^{3}
$$
F. $$
4. 0 \, \AA
$$
$$
1. 7 0 \times1 0^{-1 9} \mathrm{J}
$$
$$
\gamma T^{2}
$$
G. $$
4. 2 \, \AA
$$
$$
1. 8 0 \times1 0^{-1 8} \mathrm{J}
$$
$$
\alpha T^{2}
$$
H. $$
3. 7 \, \AA
$$
$$
1. 0 0 \times1 0^{-1 9} \mathrm{J}
$$
$$
\alpha T^{4}
$$
I. $$
3. 4 \, \AA
$$
$$
3.1 eV
$$
$$
c = \alpha T^{3}
$$
J. $$
5. 1 \, \AA
$$
$$
1. 4 0 \times1 0^{-1 9} \mathrm{J}
$$
$$
\delta T^{3}
$$
Answer: I
In leveling measurement, point A is the backsight, and point B is the foresight. The elevation of point A is $H_{A}$, with a backsight reading of $a$ and a foresight reading of $b$. Then the elevation of point B is ( ).

Options:
A. $$
H_{A}+b-a
$$
B. $$
H_{A}/a-b
$$
C. $$
H_{A}-a+b
$$
D. $$
H_{A}/b-a
$$
E. $$
H_{A}*a/b
$$
F. $$
H_{A}-b-a
$$
G. $$
H_{A}*b/a
$$
H. $$
H_{A}-a-b
$$
I. $$
H_{A}+b+a
$$
J. $$
H_{A}+a-b
$$
Answer: J
A simply supported steel beam with a rectangular cross-section is subjected to a concentrated force $P$ at mid-span. When the concentrated force increases to $2P$, and the beam's width is doubled while all other conditions remain unchanged, what is the maximum bending stress at mid-span in comparison to the original?

Options:
A. 0.5x
B. 0.75x
C. 4x
D. 2x
E. 1x
F. 5x
G. 1.5x
H. 3x
I. 1.25x
J. 2.5x
Answer: E
The steady-state current between parallel plates with a distance of $0.4 \mathrm{cm}$ and a voltage of $8 \mathrm{kV}$ is $3.8 \times 10^{-8} \mathrm{~A}$. Keeping the electric field unchanged, if the distance between the plates is reduced to $0.1 \mathrm{cm}$, the current decreases to $3.8 \times 10^{-9} \mathrm{~A}$. What is $\alpha$? How many electrons are emitted per second from the cathode due to external ionizing factors?

Options:
A. $$
7.499,\ 2. 3 3 9 9 \times1 0^{8 }
$$
B. $$
6.785,\ 1. 5 6 4 0 \times1 0^{1 0}
$$
C. $$
5.876,\ 3. 2 2 9 6 \times1 0^{8 }
$$
D. $$
9.012,\ 0. 8 4 2 1 \times1 0^{1 1}
$$
E. $$
6.432,\ 2. 1 9 0 3 \times1 0^{1 0}
$$
F. $$
5.992,\ 9. 9 8 1 5 \times1 0^{9 }
$$
G. $$
4.231,\ 1. 4 7 5 8 \times1 0^{1 2}
$$
H. $$
8.674,\ 1. 2 0 3 0 \times1 0^{1 0}
$$
I. $$
7.675,\ 1. 1 0 2 5 \times1 0^{1 0}
$$
J. $$
8.111,\ 1. 7 5 0 1 \times1 0^{1 0}
$$
Answer: I
Assume that the mean free path of electrons in the gas is $\lambda$. Also, assume that at $x=0$ in a uniform electric field, there are $n_{0}$ electrons moving in the direction of the electric field. What is the number of these $n_{0}$ electrons that collide in each interval of $x$ as $0 \sim \lambda\,, \, \, \, \lambda\sim 2 \lambda\,, \, \, \, 2 \lambda\sim 3 \lambda\,, \, \, \, 3 \lambda\sim 4 \lambda$, etc., expressed as percentages?

Options:
A. $$
5 9. 0\ \% n_{0},~
2 7. 2\ \% n_{0},~
7. 8 \%\ n_{0},~
4. 2 \%\ n_{0}
$$
B. $$
6 5. 7\ \% n_{0},~
2 8. 1\ \% n_{0},~
9. 2 \%\ n_{0},~
2. 9 \%\ n_{0}
$$
C. $$
7 1. 6\ \% n_{0},~
2 2. 9\ \% n_{0},~
9. 4 \%\ n_{0},~
2. 3 \%\ n_{0}
$$
D. $$
7 0. 4\ \% n_{0},~
2 3. 6\ \% n_{0},~
8. 0 \%\ n_{0},~
3. 9 \%\ n_{0}
$$
E. $$
7 2. 5\ \% n_{0},~
2 0. 7\ \% n_{0},~
8. 3 \%\ n_{0},~
3. 4 \%\ n_{0}
$$
F. $$
6 3. 2\ \% n_{0},~
2 3. 3\ \% n_{0},~
8. 6 \%\ n_{0},~
3. 1 \%\ n_{0}
$$
G. $$
6 0. 3\ \% n_{0},~
2 6. 0\ \% n_{0},~
9. 1 \%\ n_{0},~
2. 8 \%\ n_{0}
$$
H. $$
6 8. 9\ \% n_{0},~
2 1. 4\ \% n_{0},~
7. 5 \%\ n_{0},~
4. 0 \%\ n_{0}
$$
I. $$
5 4. 1\ \% n_{0},~
2 5. 8\ \% n_{0},~
9. 7 \%\ n_{0},~
2. 4 \%\ n_{0}
$$
J. $$
6 7. 8\ \% n_{0},~
2 4. 5\ \% n_{0},~
8. 1 \%\ n_{0},~
3. 6 \%\ n_{0}
$$
Answer: F
To ensure a steel rod is always $5 \, \mathrm{cm}$ longer than a copper rod at any temperature, what are their lengths $l_{01}$ and $l_{02}$ at $0 \, \mathbb{°C}$ ?
( It is known that the linear expansion coefficients for the steel rod and the copper rod are $\alpha_{1} \,=\, 1.2 \times 10^{-5} \, \mathrm{K}^{-1}$ and $\alpha_{2} \,=\, 1.6 \times 10^{-5} \, \mathrm{K}^{-1}$, respectively. )

Options:
A. $$
\, 2 2 \, \ \mathrm{c m} ,1 7 \, \ \mathrm{c m}
$$
B. $$
\, 2 1 \, \ \mathrm{c m} ,1 6 \, \ \mathrm{c m}
$$
C. $$
\, 2 0 \, \ \mathrm{c m} ,1 5 \, \ \mathrm{c m}
$$
D. $$
\, 1 5 \, \ \mathrm{c m} ,1 0 \, \ \mathrm{c m}
$$
E. $$
\, 1 9 \, \ \mathrm{c m} ,1 4 \, \ \mathrm{c m}
$$
F. $$
\, 2 4 \, \ \mathrm{c m} ,1 9 \, \ \mathrm{c m}
$$
G. $$
\, 1 8 \, \ \mathrm{c m} ,1 3 \, \ \mathrm{c m}
$$
H. $$
\, 2 5 \, \ \mathrm{c m} ,2 0 \, \ \mathrm{c m}
$$
I. $$
\, 2 3 \, \ \mathrm{c m} ,1 8 \, \ \mathrm{c m}
$$
J. $$
\, 1 7 \, \ \mathrm{c m} ,1 2 \, \ \mathrm{c m}
$$
Answer: C
Fill a balloon with hydrogen gas under standard conditions. The volume of this balloon can change with variations in external pressure. Upon completion of inflation, the volume of the balloon is $566 m^{3}$, and the volume of the balloon skin can be neglected. If the weight of the balloon skin is 12.8 kg and the atmospheric temperature at a certain altitude remains at 0 $\mathbf{°C}$, how much additional weight the balloon can carry at this altitude without descending ?

Options:
A. 759.1kg
B. 762.3kg
C. 758.9kg
D. 770.5kg
E. 764.4kg
F. 660.8kg
G. 655.7kg
H. 868.8kg
I. 691.9kg
J. 967.2kg
Answer: F
A clean solid surface exposed to an ideal gas with a molecular mass of $m$, molecular number density of $n$, and temperature $T$ absorbs gas molecules at a certain rate [measured in molecules $\left. / (\mathbf{s} \cdot \mathbf{m}^{2}) \right]$. If the probability of absorption for molecules striking the surface, with a normal velocity component less than $\boldsymbol{v}_{r}$, is zero, and for those greater than $\boldsymbol{v}_{r}$ is one, give the expression for the absorption rate. ( i.e. number of molecules absorbed per unit area of the surface per unit time )

Options:
A. $$
n \sqrt{\frac{2 k T} {\pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {k T} \Bigr) \
$$
B. $$
n \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}} {2 k T} \Bigr) \
$$
C. $$
2 n \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {2 k T} \Bigr) \
$$
D. $$
n \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \frac{1}{\exp\Bigl(\frac{m v_{\mathrm{r}}^{2}} {2 k T} \Bigr)} \
$$
E. $$
\frac{n}{2} \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {2 k T} \Bigr) \
$$
F. $$
n \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {2 k T} \Bigr) \
$$
G. $$
n \sqrt{\frac{k T} {2 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {k T} \Bigr) \
$$
H. $$
n \sqrt{\frac{k T} {4 \pi m}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {2 k T} \Bigr) \
$$
I. $$
n \sqrt{\frac{k T} {\pi m}} \ \cdot\ \exp\Bigl(-\frac{2 m v_{\mathrm{r}}^{2}} {k T} \Bigr) \
$$
J. $$
n \cdot \sqrt{\frac{k T} {2 \pi}} \ \cdot\ \exp\Bigl(-\frac{m v_{\mathrm{r}}^{2}} {2 m T k} \Bigr) \
$$
Answer: F
In a certain particle accelerator, particles are accelerated in a container with a pressure of $1.3 \times 10^{-4} \, \mathrm{Pa}$ and a temperature of 273 $\mathbf{K}$. If the molecular diameter of the gas is $2.0 \times 10^{-10} \mathrm{~m}$, what is the mean free path of the gas molecules?

Options:
A. $$
\overline{{{{\lambda}}}}=3. 5 \times1 0^{2} \mathrm{~ m}
$$
B. $$
\overline{{{{\lambda}}}}=4. 8 \times1 0^{3} \mathrm{~ m}
$$
C. $$
\overline{{{{\lambda}}}}=7. 9 \times1 0^{4} \mathrm{~ m}
$$
D. $$
\overline{{{{\lambda}}}}=9. 7 \times1 0^{2} \mathrm{~ m}
$$
E. $$
\overline{{{{\lambda}}}}=2. 3 \times1 0^{1} \mathrm{~ m}
$$
F. $$
\overline{{{{\lambda}}}}=5. 1 \times1 0^{4} \mathrm{~ m}
$$
G. $$
\overline{{{{\lambda}}}}=8. 4 \times1 0^{3} \mathrm{~ m}
$$
H. $$
\overline{{{{\lambda}}}}=2. 6 \times1 0^{4} \mathrm{~ m}
$$
I. $$
\overline{{{{\lambda}}}}=1. 6 \times1 0^{2} \mathrm{~ m}
$$
J. $$
\overline{{{{\lambda}}}}=6. 2 \times1 0^{1} \mathrm{~ m}
$$
Answer: I
A rubber band with a free length of $l_{0}$, a cross-sectional area of $S$, and an elastic modulus of $E$, is tied to an iron wire ring using cotton thread. When the iron wire ring is slowly pulled out from a soap solution, a layer of soap film forms on the iron wire ring. After heating a pin and using it to pierce the portion of the soap film surrounded by the rubber band, the rubber band is stretched into a circle with a length of $l$. What is the surface tension coefficient $\sigma$ of the soap film?

Options:
A. $$
\frac{\pi E S^{2}} {l_{0}} \cdot\frac{( \, l \,-\, l_{0} \, )} {l}
$$
B. $$
\frac{\pi S E} {l} \cdot\frac{( \, l_{0} \,-\, l \, )} {l_{0}}
$$
C. $$
\frac{\pi E S} {l} \cdot\frac{( \, l \,-\, l_{0} \, )} {l_{0}^{2}}
$$
D. $$
\frac{\pi S E} {l_{0}} \cdot\frac{( \, l \,-\, l_{0} \, )} {l}
$$
E. $$
\frac{2 S E} {l_{0}} \cdot\frac{( \, l \,-\, l_{0} \, )} {l^{2}}
$$
F. $$
\frac{2 \pi S E} {l_{0}} \cdot\frac{( \, l \,-\, l_{0} \, )} {l}
$$
G. $$
\frac{\pi S} {l_{0}^{2}} \cdot\frac{( \, l \,-\, l_{0} \, )} {E}
$$
H. $$
\frac{S E} {l} \cdot\frac{( \, l \,-\, l_{0} \, )} {\pi}
$$
I. $$
\frac{S E l_{0}} {\pi} \cdot\frac{( \, l_{0} \,-\, l \, )} {l}
$$
J. $$
\frac{\pi S} {l_{0}} \cdot\frac{( \, l \,-\, l_{0} \, )} {E l^{2}}
$$
Answer: D
Two cylindrical adiabatic calorimeters with the same cross-sectional area contain ice with a height of $h=0.25 \ \mathrm{m}$ in one and water of the same height at a temperature of $t_{1}=10 \ \mathrm{°C}$ in the other. The water is now poured onto the ice, and the total height of the ice and water interface is immediately marked. After a period of time, thermal equilibrium is reached, and the interface has risen by $\Delta h = 0.005 \ \mathrm{m}$. What is the initial temperature $t_{0}$ of the ice?
( Given that the density, specific heat capacity, and latent heat of fusion of the ice are $\rho_{0}=0.9 \times 10^{3} \ \mathrm{kg/m^{3}}$, $c_{0}=2100 \ \mathrm{J/(kg \cdot K)}$, and $\lambda=3.4 \times 10^{5} \ \mathrm{J/kg}$, respectively, and the specific heat capacity of water is $c_{1}=4200 \ \mathrm{J/(kg \cdot K)}$. )

Options:
A. $$
-4 2. 3 \mathrm{~} \mathrm{°C}
$$
B. $$
-6 3. 7 \mathrm{~} \mathrm{°C}
$$
C. $$
-3 9. 9 \mathrm{~} \mathrm{°C}
$$
D. $$
-4 5. 0 \mathrm{~} \mathrm{°C}
$$
E. $$
\approx-5 4. 6 \mathrm{~} \mathrm{°C}
$$
F. $$
-5 7. 4 \mathrm{~} \mathrm{°C}
$$
G. $$
-6 0. 2 \mathrm{~} \mathrm{°C}
$$
H. $$
-4 9. 8 \mathrm{~} \mathrm{°C}
$$
I. $$
-3 8. 1 \mathrm{~} \mathrm{°C}
$$
J. $$
-5 1. 5 \mathrm{~} \mathrm{°C}
$$
Answer: E
Calculate the Fourier transform of the following signal at $N$ uniformly sampled points over the interval $\left[ 0, 2 \pi \right]$: $a \cos(\omega_{0} n) R_{N}(n)$.

Options:
A. $$
0 \,, 1 \,, \cdots, N-1
$$
B. $$
N \,, N-1 \,, \cdots, 0
$$
C. $$
-N \,, -N+1 \,, \cdots, -1
$$
D. $$
-\pi \,, 0 \,, \cdots, \pi
$$
E. $$
0 \,, \pi \,, \cdots, 2\pi
$$
F. $$
e \,, 2e \,, \cdots, Ne
$$
G. $$
1 \,, 1 \,, \cdots, 1
$$
H. $$
1 \,, 2 \,, \cdots, N
$$
I. $$
0.5 \,, 1.5 \,, \cdots, N-0.5
$$
J. $$
0 \,, 0 \,, \cdots, 0
$$
Answer: A
Assume the displacement of the spring oscillator system is the superposition of two vibrations with different frequencies and amplitudes, i.e.,
$$
x(t) = \sin \omega t + \frac{1}{2} \sin 2\omega t
$$
Question: At what time is the displacement greatest? At what time is the velocity greatest?

Options:
A. $$
{\frac{\pi} {2}}, \frac{\pi} {3}
$$
B. $$
0, {\frac{\pi} {3}}
$$
C. $$
{\frac{\pi} {4}}, 0
$$
D. $$
{\frac{\pi} {3}}, \frac{\pi}{2}
$$
E. $$
{\frac{\pi} {3}}, {\frac{\pi} {6}}
$$
F. $$
{\frac{\pi} {6}}, 0
$$
G. $$
{\frac{\pi} {3}}, \pi
$$
H. $$
{\frac{\pi} {2}}, 0
$$
I. $$
{\frac{\pi} {3}}, {\frac{\pi}{4}}
$$
J. $$
{\frac{\pi} {3}}, 0
$$
Answer: J
String length $L \ =\ 31.4\ \mathrm{cm}$, linear density $\rho_{\mathrm{L}} \;=\; 0.1 \; \mathrm{g/cm}$ with both ends fixed. There is a standing wave on the string in the form of $y=2 \sin(x/5) \cos(3t)$, where both $x$ and $y$ are in cm, and $t$ is in seconds. Find: the phase velocity, frequency, and wave number; the displacement amplitude and velocity amplitude at $x = \frac{L}{2}$ and $x = \frac{L}{4}$; the energy density at the aforementioned two points; and the total energy of the string's vibration are( ).

Options:
A. c = 16 cm/s, 0.47 Hz, 0.19; at x = $\frac{L}{2}$, 1.5 cm, 3 cm/s; at x = $\frac{L}{4}$, 2 cm, 5 cm/s;
$$
0,3.1 \times 10^{-6} \mathrm{~ J};
$$
$$
3.1 \times 10^{-6} \mathrm{~ J};
$$
B. c = 13 cm/s, 0.49 Hz, 0.28; at x = $\frac{L}{2}$, 2.5 cm, 4 cm/s; at x = $\frac{L}{4}$, 0.5 cm, 7 cm/s;
$$
0,2.9 \times 10^{-6} \mathrm{~ J};
$$
$$
2.9 \times 10^{-6} \mathrm{~ J};
$$
C. c = 14 cm/s, 0.50 Hz, 0.22; at x = $\frac{L}{2}$, 0.5 cm, 1 cm/s; at x = $\frac{L}{4}$, 1.5 cm, 2 cm/s;
$$
0,2.5 \times 10^{-6} \mathrm{~ J};
$$
$$
2.5 \times 10^{-6} \mathrm{~ J};
$$
D. c = 17 cm/s, 0.46 Hz, 0.12; at x = $\frac{L}{2}$, 0 cm, 0 cm/s; at x = $\frac{L}{4}$, 1 cm, 1 cm/s;
$$
0,3.3 \times 10^{-6} \mathrm{~ J};
$$
$$
3.3 \times 10^{-6} \mathrm{~ J};
$$
E. c = 20 cm/s, 0.48 Hz, 0.3; at x = $\frac{L}{2}$, 1 cm, 2 cm/s; at x = $\frac{L}{4}$, 3 cm, 9 cm/s;
$$
0,
3.2 \times 10^{-6} \mathrm{~ J};
$$
$$
3.2 \times 10^{-6} \mathrm{~ J};
$$
F. c = 12 cm/s, 0.52 Hz, 0.1; at x = $\frac{L}{2}$, 1 cm, 0 cm/s; at x = $\frac{L}{4}$, 0, 5 cm/s;
$$
0,3.0 \times 10^{-6} \mathrm{~ J};
$$
$$
3.0 \times 10^{-6} \mathrm{~ J};
$$
G. c = 10 cm/s, 0.45 Hz, 0.25; at x = $\frac{L}{2}$, 2 cm, 4 cm/s; at x = $\frac{L}{4}$, 1 cm, 3 cm/s;
$$
0,1.9 \times 10^{-6} \mathrm{~ J};
$$
$$
1.9 \times 10^{-6} \mathrm{~ J};
$$
H. c = 11 cm/s, 0.51 Hz, 0.15; at x = $\frac{L}{2}$, 3 cm, 5 cm/s; at x = $\frac{L}{4}$, 1 cm, 8 cm/s;
$$
0,2.7 \times 10^{-6} \mathrm{~ J};
$$
$$
2.7 \times 10^{-6} \mathrm{~ J};
$$
I. c = 18 cm/s, 0.50 Hz, 0.14; at x = $\frac{L}{2}$, 4 cm, 7 cm/s; at x = $\frac{L}{4}$, 1.2 cm, 4 cm/s;
$$
0,2.6 \times 10^{-6} \mathrm{~ J};
$$
$$
2.6 \times 10^{-6} \mathrm{~ J};
$$
J. c = 15 cm/s, 0.48 Hz, 0.2; at x = $\frac{L}{2}$, 0, 0; at x = $\frac{L}{4}$, 2 cm, 6 cm/s;
$$
0,2.8 \times 10^{-6} \mathrm{~ J};
$$
$$
2.8 \times 10^{-6} \mathrm{~ J};
$$
Answer: J
The diaphragm of the receiver is a circular steel sheet with rigidly fixed edges, with a diameter of $4 \; \mathrm{cm}$ and a thickness of $0.02 \; \mathrm{cm}$. When it undergoes centrally symmetrical vibration, the restoring force of the diaphragm's vibration is entirely generated by its own stiffness. If the diaphragm's fundamental frequency vibration is entirely produced by the restoring force of tension and the fundamental frequency value equals the former, what should the tension of the diaphragm be?

Options:
A. $$
1 2 3 2 \ \mathrm{Hz}; 6 3 7 5 \ \mathrm{N/m}.
$$
B. $$
1 2 3 2 \ \mathrm{Hz}; 2 0 0 0 \ \mathrm{N/m}.
$$
C. $$
1 3 5 0 \ \mathrm{Hz}; 6 3 7 5 \ \mathrm{N/m}.
$$
D. $$
1 2 3 2 \ \mathrm{Hz}; 3 0 0 0 \ \mathrm{N/m}.
$$
E. $$
1 0 0 0 \ \mathrm{Hz}; 6 3 7 5 \ \mathrm{N/m}.
$$
F. $$
1 2 3 2 \ \mathrm{Hz}; 1 0 0 0 \ \mathrm{N/m}.
$$
G. $$
1 5 0 0 \ \mathrm{Hz}; 6 3 7 5 \ \mathrm{N/m}.
$$
H. $$
2 0 0 0 \ \mathrm{Hz}; 4 3 7 5 \ \mathrm{N/m}.
$$
I. $$
1 2 3 2 \ \mathrm{Hz}; 5 0 0 0 \ \mathrm{N/m}.
$$
J. $$
1 2 3 2 \ \mathrm{Hz}; 8 5 0 0 \ \mathrm{N/m}.
$$
Answer: A
The resonant tube can also be used to measure standard atmospheric pressure and the speed of sound in air at room temperature (20°C). This involves a tube partially filled with water (placed vertically), where the length of the air column can be altered by raising or lowering the water surface. It is known that when the water level is 0.344 m below the reference mark, a tuning fork with a frequency of $f_{1}=200 \mathrm{~Hz}$ causes the tube to resonate. When the water level is 0.136 m below the reference mark, a second tuning fork with a frequency of $f_{2}=400$ Hz causes the air column to resonate again. What is the speed of sound in the air?

Options:
A. $$
3 5 0. 6 \mathrm{~ m / s}.
$$
B. $$
2 9 0. 3 \mathrm{~ m / s}.
$$
C. $$
3 2 5. 0 \mathrm{~ m / s}.
$$
D. $$
2 8 0. 5 \mathrm{~ m / s}.
$$
E. $$
3 1 5. 7 \mathrm{~ m / s}.
$$
F. $$
4 0 0. 0 \mathrm{~ m / s}.
$$
G. $$
3 9 9. 1 \mathrm{~ m / s}.
$$
H. $$
3 3 2. 8 \mathrm{~ m / s}.
$$
I. $$
3 6 0. 2 \mathrm{~ m / s}.
$$
J. $$
3 8 5. 9 \mathrm{~ m / s}.
$$
Answer: H
A plane sound wave with a frequency of 500 Hz and a sound pressure level of 60 dB (reference sound pressure $0.0002 \mu\mathrm{bar}$) is incident perpendicularly on the interface between air and another medium (whose characteristic impedance is 830 Rayl). Determine: the effective value of the sound pressure of the reflected wave; the effective value of the sound pressure of the transmitted wave; how far from the interface does the sound pressure amplitude in the standing wave field equal the sound pressure amplitude of the incident wave?

Options:
A. Answer: 0.0066 $\mathrm{Pa}$; 0.027 $\mathrm{Pa}$; 17.2 $\mathrm{cm}$
B. 0.0082 $\mathrm{Pa}$; 0.029 $\mathrm{Pa}$; 18.5 $\mathrm{cm}$
C. 0.0058 $\mathrm{Pa}$; 0.026 $\mathrm{Pa}$; 20.3 $\mathrm{cm}$
D. 0.0064 $\mathrm{Pa}$; 0.030 $\mathrm{Pa}$; 21.8 $\mathrm{cm}$
E. 0.0069 $\mathrm{Pa}$; 0.028 $\mathrm{Pa}$; 15.2 $\mathrm{cm}$
F. 0.0055 $\mathrm{Pa}$; 0.024 $\mathrm{Pa}$; 16.1 $\mathrm{cm}$
G. 0.0061 $\mathrm{Pa}$; 0.031 $\mathrm{Pa}$; 19.0 $\mathrm{cm}$
H. 0.0070 $\mathrm{Pa}$; 0.025 $\mathrm{Pa}$; 15.9 $\mathrm{cm}$
I. 0.0078 $\mathrm{Pa}$; 0.023 $\mathrm{Pa}$; 19.5 $\mathrm{cm}$
J. 0.0075 $\mathrm{Pa}$; 0.022 $\mathrm{Pa}$; 14.7 $\mathrm{cm}$
Answer: A
A hemisphere with a radius of $a$ and a piston with a radius of $a$ are each mounted on an infinitely large rigid barrier, thus radiating sound waves only to one side of the barrier. The maximum amplitude of the oscillation velocity of both sound sources is $u_{0}$, and they have the same vibration frequency, satisfying $ka \ll 1$, where $k$ is the wave number. Question: At a distance where $r \gg a$, what is the ratio of sound pressure on the axis of the piston source to the sound pressure on the axis of the hemisphere source? What is the ratio of the total power radiated by the two sources?

Options:
A. $$
{\frac{1} {5}} \, ;{\frac{1} {3}}
$$
B. $$
{\frac{5} {8}} \, ;{\frac{1} {2}}
$$
C. $$
{\frac{4} {9}} \, ;{\frac{3} {4}}
$$
D. $$
{\frac{1} {2}} \, ;{\frac{1} {4}}
$$
E. $$
{\frac{3} {6}} \, ;{\frac{7} {8}}
$$
F. $$
{\frac{3} {7}} \, ;{\frac{4} {5}}
$$
G. $$
{\frac{2} {3}} \, ;{\frac{3} {5}}
$$
H. $$
{\frac{2} {5}} \, ;{\frac{5} {6}}
$$
I. $$
{\frac{3} {4}} \, ;{\frac{1} {3}}
$$
J. $$
{\frac{1} {3}} \, ;{\frac{2} {5}}
$$
Answer: D
A loudspeaker: Total mass of the diaphragm (including the voice coil) $m = 10$ g; diaphragm diameter $\boldsymbol{a} = 0.1$ m; sound power $s = 2000\ \text{N/m}$; mechanical resistance $R_{\mathrm{m}} = 1\ \text{N} \cdot \text{s/m}$; voice coil inductance $L_{0} = 0.2\ \text{mH}$; voice coil resistance $R_{0} = 5\ \Omega$; voice coil wire length $l = 5\ \text{m}$; magnetic gap field strength $B = 0.9\ \text{T}$. If the diaphragm is flat-mount on an infinitely large baffle and can be considered as a piston vibration, try to find:
Electromechanical coupling coefficient $K_{\mathrm{m}}^{2} \Big(=\frac{L_{\mathrm{m}}}{L_{\mathrm{m}}+L_{0}} \Big)$;
Effective electromechanical coupling coefficient $K_{\mathrm{m (\text{off})}}^{2} \Big( \approx 1 - \left( \frac{\omega_{0}}{\omega_{\mathrm{r}}} \right)^{2} \approx 1 - \left( \frac{\omega_{\mathrm{n}}}{\omega_{\mathrm{m}}} \right)^{2} \Big)$;
Mechanical $Q$ value $Q_{\mathbf{M}}$.

Options:
A. $$
0. 9 8 ; 0. 9 8 ; 4. 7
$$
B. $$
0. 5 9 ; 0. 8 1 ; 4. 5
$$
C. $$
1. 0 2 ; 0. 5 3 ; 4. 0
$$
D. $$
1. 1 6 ; 0. 3 4 ; 4. 8
$$
E. $$
0. 3 7 ; 0. 6 9 ; 5. 2
$$
F. $$
0. 8 4 ; 1. 0 5 ; 2. 1
$$
G. $$
1. 2 3 ; 0. 6 1 ; 2. 9
$$
H. $$
0. 7 1 ; 0. 9 4 ; 1. 6
$$
I. $$
0. 6 5 ; 0. 7 2 ; 3. 5
$$
J. $$
0. 5 4 ; 0. 7 8 ; 3. 8
$$
Answer: A
The electric field is perpendicular to the length of the length-extending chip made of PZT-4, with a density and compliance constants of $p=7.5×10^{3}$ kg/m$^{3}$, $s^E_{11}=12.3×10^{-12}$ m$^{2}$/N, $s^D_{33}=7.9×10^{-12}$ m$^{2}$/N. If the fundamental resonant frequency is 30 kHz, find the length $l$ of the chip.

Options:
A. l=8.16cm
B. l=4.88cm
C. l=5.25cm
D. l=5.48cm
E. l=7.04cm
F. l=3.94cm
G. l=6.72cm
H. l=4.12cm
I. l=6.30cm
J. l=7.85cm
Answer: D
In a certain sea area during summer, the measured temperature gradient is -0.5°C/m, and the salinity s=35‰ is independent of depth. The sea surface temperature is 25°C. Find the sound speed gradient at a water depth of 40m.

Options:
A. -2.25/s
B. -2.30/s
C. -2.02/s
D. -1.95/s
E. -1.85/s
F. -1.60/s
G. -2.05/s
H. -1.90/s
I. -1.75/s
J. -2.15/s
Answer: C
The speed of sound in the sea surface is 1500 m/s, with a 60 m thick isothermal layer at the surface, and below it is a steep negative temperature gradient layer. Assuming a submarine is located just below the isothermal layer, find the minimum horizontal distance at which it cannot be detected by surface sonar.

Options:
A. 4.01km
B. 3.67km
C. 2.83km
D. 4.15km
E. 3.79km
F. 3.32km
G. 4.56km
H. 3.45km
I. 2.48km
J. 2.93km
Answer: F
A submarine is navigating in a convergence zone with a gain of 15 dB, emitting a line spectrum at a sound source level of 160 dB at 500 Hz. Another submarine, 30 miles away, is monitoring with an omnidirectional hydrophone. Assume the background noise is equivalent to deep-sea noise under sea state 3 conditions. If the receiving bandwidth of the second submarine is 100 Hz, with a detection probability of 50% and a false alarm probability of 0.20%, how much time should be observed in one pass using incoherent processing?

Options:
A. 2.8s
B. 4.1s
C. 0.9s
D. 1.4s
E. 2.3s
F. 3.2s
G. 3.8s
H. 1.5s
I. 4.5s
J. 1.7s
Answer: D
Consider a "bilinear" underwater acoustic channel with a channel axis depth of 700m and a sound speed of 1450m/s at the axis. The sound speed gradient is ∓0.2/s. The horizontal distance between the transmitter and receiver is 20km. If the pulse width of the transmitted signal is 3.0ms, calculate the total duration of the received waveform.

Options:
A. 455.6ms
B. 436.5ms
C. 458.2ms
D. 450.1ms
E. 439.3ms
F. 442.7ms
G. 441.9ms
H. 430.8ms
I. 446.9ms
J. 448.4ms
Answer: I
A jet aircraft is flying in the air, emitting a sound power level of 170 dB. Assuming the directivity index is -10 dB in the direction forming a 120° angle with the jet's direction, find the sound pressure level at a distance of 2 miles from the aircraft in that direction.

Options:
A. 104dB
B. 88dB
C. 110dB
D. 95dB
E. 69dB
F. 91dB
G. 85dB
H. 72dB
I. 115dB
J. 78dB
Answer: E
The maximum sound pressure level that the human ear can tolerate is 10 $^{4}$ μbar, while at the other extreme, for certain frequencies, the human ear can perceive a sound pressure of 10 $^{-4}$ μbar. Try calculating the difference between these two extreme sound pressure levels using 20 μPa as the reference sound pressure.

Options:
A. 158dB
B. 162dB
C. 152dB
D. 168dB
E. 160dB
F. 155dB
G. 175dB
H. 150dB
I. 165dB
J. 170dB
Answer: E
An electroacoustic transducer, with a resonant frequency of 10 kHz and an equivalent mechanical quality factor $Q_{\mathrm{m}}$ of 10, is used to receive pulse sound. How long must the pulse duration で be at minimum to allow the vibration signal amplitude to basically reach stability?

Options:
A. $\tau$ = 4 ms
B. $\tau$ = 0.5 ms
C. $\tau$ = 5 ms
D. $\tau$ = 1.5 ms
E. $\tau$ = 7 ms
F. $\tau$ = 2 ms
G. $\tau$ = 1 ms
H. $\tau$ = 8 ms
I. $\tau$ = 6 ms
J. $\tau$ = 3 ms
Answer: G
At what decibel level (reference 1 μPa) are sound waves no longer applicable within the scope of linear acoustics? What is the ratio of particle velocity to sound speed for a plane sound wave with a sound pressure level of 230 dB (reference 1 μPa) passing through a water medium?

Options:
A. 0.18m/s,1.0x10 $^{-5}$
B. 0.22m/s,1.3x10 $^{-6}$
C. 0.19m/s,1.6x10 $^{-4}$
D. 0.25m/s,1.5x10 $^{-4}$
E. 0.31m/s,1.8x10 $^{-3}$
F. 0.30m/s,1.7x10 $^{-3}$
G. 0.29m/s,1.5x10 $^{-5}$
H. 0.27m/s,1.9x10 $^{-4}$
I. 0.15m/s,1.2x10 $^{-5}$
J. 0.21m/s,1.4x10 $^{-4}$
Answer: J
If the concentrations of ADP and Pi inside the cell are $3mM$ and $1mM$ respectively, and the standard free energy change for ATP hydrolysis is $\Delta G^{\circ}=-7.3 \mathrm{kcal} \cdot \mathrm{mol}^{-1}$, try to calculate the concentration of ATP at equilibrium (temperature is 37°C). However, under these conditions, if the concentration of ATP is $10mM$, what would be the $\Delta G_{T\cdot P}$ of the reaction $\mathbf{ATP} \rightarrow \mathbf{ADP} + \mathbf{Pi}$?

Options:
A. $$
- 5 1. 2 9 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
B. $$
- 5 3. 6 6 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
C. $$
- 4 9. 8 5 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
D. $$
- 4 2. 1 0 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
E. $$
- 6 2. 3 4 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
F. $$
- 5 5. 4 0 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
G. $$
- 5 8. 7 8 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
H. $$
- 6 0. 1 9 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
I. $$
- 5 0. 9 4 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
J. $$
- 4 7. 3 2 \mathbf{k J} \cdot\mathrm{m o l}^{-1}
$$
Answer: A
Add $10 \text{mCi}^{14}\text{C}$-labeled inulin (Inula Flower Pollen $\mathbf{C}_{6} \mathbf{H}_{10} \mathbf{O}_{5}$) to a $15 \text{ml}$ yeast suspension. Then centrifuge the suspension and carefully remove the supernatant. The deposited yeast pellet occupies a volume of $0.2 \text{ml}$ and contains 10,000 counts per minute, with a counting efficiency of 25%. Calculate the proportion of the interstitial space in the yeast pellet is(). (Assume the inulin cannot pass through yeast cells and is not adsorbed on the cell surface.)

Options:
A. $$
2 5. 6 \%
$$
B. $$
1 2. 3 \%
$$
C. $$
2 8. 4 \%
$$
D. $$
2 0. 5 \%
$$
E. $$
1 0. 2 \%
$$
F. $$
1 5. 8 \%
$$
G. $$
1 1. 1 \%
$$
H. $$
1 8. 7 \%
$$
I. $$
1 3. 4 \%
$$
J. $$
1 7. 9 \%
$$
Answer: I
Determine the activity of $\beta$-galactosidase in the cell-free extract of $Penicillium Chrysogeum$. The extract ($0.25ml$) is incubated with $3 \times 10^{-3} M$ p-nitrophenyl-$\beta$-galactoside and buffer, with a total volume of $1.0ml$. At regular intervals, remove $0.1 ml$ and add it to $2.9ml$ of $0.1N NaOH$ solution. The concentration of liberated p-nitrophenol can be determined by the absorbance at $400nm$, using a blank consisting of p-nitrophenyl-$\beta$-galactoside in buffer plus $NaOH$. (The $\epsilon_{m}$ value for p-nitrophenol in $0.1N NaOH$ is 18,300). The cell-free extract contains $5mg$ of protein per ml. In a $1cm$ path length cuvette, the absorbance of the NaOH solution is:
Incubation Time (minutes) A400nm
2 0.09
4 0.18
6 0.27
Calculate the specific activity of $\beta$-galactosidase in the cell-free extract.

Options:
A. 0. 1 0 2 units / mg protein
B. 0.067 units/mg protein
C. 0. 1 5 9 units/mg protein
D. 0.059 units/mg protein
E. 0. 0 2 8 units / mg protein
F. 0.094 units/mg protein
G. 0.041 units/mg protein
H. 0. 1 2 3 units / mg protein
I. 0. 0 3 6 units / mg protein
J. 0. 0.084 units/mg protein
Answer: D
The $K_{m} \!=\! 2.9 \times 10^{-4} M$ and $K_{i} \!=\! 2 \times 10^{-5} M$ for a certain enzyme. When $\left[ S \right]= 1.5 \times 10^{-3} M$, what concentration of a competitive inhibitor will result in 75% inhibition?

Options:
A. $$
9. 4 \times 1 0^{-8} M
$$
B. $$
8. 3 \times 1 0^{-3} M
$$
C. $$
3. 7 \times1 0^{-4} M
$$
D. $$
2. 6 \times 1 0^{-5} M
$$
E. $$
2. 1 \times 1 0^{-4} M
$$
F. $$
4. 5 \times 1 0^{-5} M
$$
G. $$
1. 5 \times 1 0^{-7} M
$$
H. $$
5. 9 \times 1 0^{-3} M
$$
I. $$
7. 2 \times 1 0^{-6} M
$$
J. $$
6. 8 \times 1 0^{-4} M
$$
Answer: C
Yeast cells are grown in a synthetic medium using $^{35}SO_{4}$ (specific radioactivity $4.78 \times 10^{7}$ counts per minute $/ \mu$mol) as the sulfur source. After several days of growth, the cells are harvested and lysed to prepare an extract. Then, 500 ml of non-radioactive reduced glutathione is added to the 50 ml extract. Glutathione is then separated again from the mixture. The specific radioactivity of this compound is $6.97 \times 10^{6}$ counts per minute $/ \mu$mol. Try to calculate the concentration of glutathione in the extract.

Options:
A. 0.95mg/ml
B. 1.90mg/ml
C. 2.00mg/ml
D. 1.25mg/ml
E. 1.71mg/ml
F. 0.60mg/ml
G. 1.50mg/ml
H. 1.45mg/ml
I. 2.34mg/ml
J. 0.85mg/ml
Answer: E
Glutamate Dehydrogenase Catalyzed Reaction:
Glutamate $+ \mathbf{NAD}^{+} + \mathbf{H}_{2}O ⇄ \alpha$-ketoglutarate $+ NADH$
$$+ \mathbf{NH}_{4}^{+} + \mathbf{H}^{+}$$
The initial concentrations of glutamate and $NAD^{+}$ are $20 \, \text{mM}$ and $1 \, \text{mM}$ respectively. If the reaction takes place in a phosphate buffer at $pH 7$. The concentration of $NADH$ is measured by spectrophotometry. $A_{340 \text{nm}}$ (in a absorption cup with a thickness of 1 cm) increases to 0.561. Calculate the equilibrium constant for the above reaction.

Options:
A. $$
9. 0 1 \times1 0^{-9} \; M
$$
B. $$
4. 7 8 \times1 0^{-5} \; M
$$
C. $$
8. 9 3 \times1 0^{-7} \; M
$$
D. $$
1. 9 9 \times1 0^{-1 2} \; M
$$
E. $$
7. 6 4 \times1 0^{-1 1} \; M
$$
F. $$
2. 3 1 \times1 0^{-1 4} \; M
$$
G. $$
6. 4 7 \times1 0^{-8} \; M
$$
H. $$
3. 5 2 \times1 0^{-6} \; M
$$
I. $$
4. 0 5 \times1 0^{-1 5} \; M
$$
J. $$
5. 8 7 \times1 0^{-1 3} \; M
$$
Answer: I
If a spaceship travels at a constant speed of $v=\sqrt{0.999} c$, how much day's worth of food and other supplies should be stocked for a round trip journey to the Centaurus constellation, which is 4 light-years away from Earth, and back ? ( accurate to one decimal place )

Options:
A. 31.4 days
B. 32.7 days
C. 28.1 days
D. 30.5 days
E. 29.2 days
F. 27.9 days
G. 30.8 days
H. 28.6 days
I. 33.8 days
J. 26.5 days
Answer: E
When a light source approaches the earth at a speed of $v_{1}$, a stationary observer A on the earth sees the light source emitting green light with a wavelength of $\lambda_{1}=5000 \AA$. Meanwhile, another observer B, moving relative to Earth at a speed of $v_{2}$ (along the same line of motion as the light source), sees the light as red with a wavelength of $\lambda_{2}=6000 \AA$. When the light source moves away from the earth at the same speed $v_{1}$, observer A also sees red light with a wavelength of $\lambda_{2}=6000 \AA$.
What is the wavelength $\lambda_{0}$ of the light emitted by the light source ?

Options:
A. $$
5 5 0 0 \AA
$$
B. $$
5 2 0 0 \AA
$$
C. $$
6 0 0 0 \AA
$$
D. $$
4 9 0 0 \AA
$$
E. $$
5 3 0 0 \AA
$$
F. $$
5 4 7 7 \AA
$$
G. $$
5 8 9 0 \AA
$$
H. $$
5 3 7 6 \AA
$$
I. $$
5 6 0 0 \AA
$$
J. $$
5 1 3 4 \AA
$$
Answer: F
An electron with a rest mass of $m$ and a charge of $e$ moves in a plane perpendicular to a uniform magnetic field. Ignoring radiation, the electron's orbit is a circle with a radius of $R$. Given that the electron's energy $E \gg mc^{2}$, derive the expression for the magnetic induction $B$. If $R=30\, \mathrm{m}$ and $E=2.5 \times 10^{9} \, \mathrm{eV}$, what is the numerical value of $B$ ?

Options:
A. $$
B=0. 3 8 \mathrm{T}.
$$
B. $$
B=0. 0 9 \mathrm{T}.
$$
C. $$
B=0. 2 8 \mathrm{T}
$$
D. $$
B=0. 5 0 \mathrm{T}.
$$
E. $$
B=0. 6 2 \mathrm{T}.
$$
F. $$
B=0. 2 2 \mathrm{T}.
$$
G. $$
B=0. 1 8 \mathrm{T}.
$$
H. $$
B=0. 1 2 \mathrm{T}.
$$
I. $$
B=0. 4 5 \mathrm{T}.
$$
J. $$
B=0. 3 0 \mathrm{T}.
$$
Answer: C
A collision between an electron and a positron, both with a rest mass of $9.11 \times 10^{-31} \mathrm{kg}$, results in the particles disappearing and producing electromagnetic radiation. If both particles are at rest before annihilation, what is the energy released as radiation ?

Options:
A. $$
1. 0 2 \mathrm{M e V}
$$
B. $$
4. 3 7 \mathrm{M e V}
$$
C. $$
2. 4 8 \mathrm{M e V}
$$
D. $$
6. 7 4 \mathrm{M e V}
$$
E. $$
5. 1 3 \mathrm{M e V}
$$
F. $$
9. 9 0 \mathrm{M e V}
$$
G. $$
0. 0 5 \mathrm{M e V}
$$
H. $$
2. 0 1 \mathrm{M e V}
$$
I. $$
3. 5 6 \mathrm{M e V}
$$
J. $$
1. 8 9 \mathrm{M e V}
$$
Answer: A
A $\pi^{0}$ meson with rest mass $m$ moving at velocity $\boldsymbol{v}$ decays into two $\gamma$ photons. Assume that in the rest frame of the $\pi^{0}$ meson, the distribution of the $\gamma$ photons by direction of emission is isotropic. With respect to the experimental reference frame, what is the probability that one of the $\gamma$ photons disperses at an angle to the direction of motion of the $\pi^{0}$ meson ?

Options:
A. $$
{\frac{1} {8}} {\frac{1-\beta^{7}} {( 1-\beta\operatorname{t a n} \theta)^{6}}} \operatorname{s i n} \theta\mathrm{d} \theta
$$
B. $$
{\frac{1} {4}} {\frac{1-\beta^{3}} {( 1-\beta\operatorname{s i n} \theta)^{4}}} \operatorname{t a n} \theta\mathrm{d} \theta
$$
C. $$
{\frac{1} {3}} {\frac{1-\beta^{4}} {( 1-\beta\operatorname{s i n} \theta)^{3}}} \operatorname{c o s} \theta\mathrm{d} \theta
$$
D. $$
{\frac{1} {9}} {\frac{1-\beta^{8}} {( 1-\beta\operatorname{c o s} \theta)^{9}}} \operatorname{c o s} \theta\mathrm{d} \theta
$$
E. $$
{\frac{1} {2}} {\frac{1-\beta^{2}} {( 1-\beta\operatorname{c o s} \theta)^{2}}} \operatorname{s i n} \theta\mathrm{d} \theta
$$
F. $$
{\frac{1} {7}} {\frac{1-\beta^{6}} {( 1-\beta\operatorname{s i n} \theta)^{7}}} \operatorname{t a n} \theta\mathrm{d} \theta
$$
G. $$
{\frac{1} {2}} {\frac{1-\beta^{2}} {( 1-\beta\operatorname{c o s} \theta)^{3}}} \operatorname{c o s} \theta\mathrm{d} \theta
$$
H. $$
{\frac{1} {0}} {\frac{1-\beta^{9}} {( 1-\beta\operatorname{s i n} \theta)^{5}}} \operatorname{t a n} \theta\mathrm{d} \theta
$$
I. $$
{\frac{1} {5}} {\frac{1-\beta^{2}} {( 1-\beta\operatorname{t a n} \theta)^{2}}} \operatorname{c o s} \theta\mathrm{d} \theta
$$
J. $$
{\frac{1} {6}} {\frac{1-\beta^{5}} {( 1-\beta\operatorname{c o s} \theta)^{3}}} \operatorname{s i n} \theta\mathrm{d} \theta
$$
Answer: E
Consider the photoreaction of the $\pi$ meson
$$
\gamma+\mathrm{p} \to \mathrm{p}+\pi^{0}
$$
The rest mass energy of the proton is $938 \mathrm{MeV}$, and the rest mass energy of the $\pi$ meson is $135 \mathrm{MeV}$. If the initial proton is stationary relative to the laboratory, what is the threshold energy of the gamma ray in the laboratory reference frame for this reaction to occur ?

Options:
A. $$
1 4 4. 7 \mathrm{M e V}
$$
B. $$
1 3 8. 9 \mathrm{M e V}
$$
C. $$
1 4 2. 8 \mathrm{M e V}
$$
D. $$
1 4 8. 3 \mathrm{M e V}
$$
E. $$
1 3 5. 6 \mathrm{M e V}
$$
F. $$
1 4 9. 0 \mathrm{M e V}
$$
G. $$
1 4 1. 3 \mathrm{M e V}
$$
H. $$
1 5 0. 2 \mathrm{M e V}
$$
I. $$
1 4 0. 5 \mathrm{M e V}
$$
J. $$
1 5 2. 1 \mathrm{M e V}
$$
Answer: A
The mass of a muon is approximately $100 \mathrm{MeV}/c^{2}$, and its rest lifetime is $2 \mu s$. Assuming the Earth's magnetic field is strong enough to keep it moving in orbit, how much energy at least does a muon need to complete a trip around the Earth?

Options:
A. $$
2. 9 \times1 0^{9} ( \mathrm{M e V} )
$$
B. $$
7. 8 \times1 0^{8} ( \mathrm{M e V} )
$$
C. $$
3. 2 \times1 0^{5} ( \mathrm{M e V} )
$$
D. $$
5. 5 \times1 0^{7} ( \mathrm{M e V} )
$$
E. $$
6. 7 \times1 0^{6} ( \mathrm{M e V} )
$$
F. $$
4. 1 \times1 0^{5} ( \mathrm{M e V} )
$$
G. $$
8. 6 \times1 0^{7} ( \mathrm{M e V} )
$$
H. $$
9. 4 \times1 0^{4} ( \mathrm{M e V} )
$$
I. $$
1. 0 \times1 0^{6} ( \mathrm{M e V} )
$$
J. $$
3. 3 \times1 0^{8} ( \mathrm{M e V} )
$$
Answer: E
Two photons with energies $E$ and $\varepsilon$ collide head-on. The speed of the center of momentum frame relative to the laboratory frame is $\beta = \frac{E - \varepsilon}{E + \varepsilon}$. If the collision of the two photons produces an electron-positron pair and $\varepsilon = 1 \text{ eV}$, what is the minimum value of $E$?

Options:
A. $$
2 5 0 \mathrm{G e V}
$$
B. $$
2 2 0 \mathrm{G e V}
$$
C. $$
3 0 1 \mathrm{G e V}
$$
D. $$
3 4 5 \mathrm{G e V}
$$
E. $$
2 8 0 \mathrm{G e V}
$$
F. $$
3 7 2 \mathrm{G e V}
$$
G. $$
2 6 1 \mathrm{G e V}
$$
H. $$
1 5 9 \mathrm{G e V}
$$
I. $$
2 9 8 \mathrm{G e V}
$$
J. $$
1 9 6 \mathrm{G e V}
$$
Answer: G
In the rest frame of a $\pi$ meson $(m_{\pi} = 140 \mathrm{MeV})$ decaying into $\mu^{-}$ and $\mathbf{v}$, what is the maximum momentum of the emitted $\mu^{-}$?

Options:
A. $$
\ 3 5 \mathrm{M e V} / c ;
$$
B. $$
\ 2 0 \mathrm{M e V} / c ;
$$
C. $$
\ 4 0 \mathrm{M e V} / c ;
$$
D. $$
\ 5 0 \mathrm{M e V} / c ;
$$
E. $$
\ 3 0 \mathrm{M e V} / c
$$
F. $$
\ 1 0 \mathrm{M e V} / c ;
$$
G. $$
\ 2 5 \mathrm{M e V} / c ;
$$
H. $$
\ 4 5 \mathrm{M e V} / c ;
$$
I. $$
\ 1 5 \mathrm{M e V} / c ;
$$
J. $$
\ 5 5 \mathrm{M e V} / c ;
$$
Answer: E
When protons strike a thin target, neutral mesons are produced, and each meson decays into two $\gamma$ photons. The photon emitted forward has an energy of 96 MeV, and the photon emitted backward has an energy of $48 \mathrm{MeV}$. What is the $\beta$ value of the meson ?

Options:
A. $$
{\frac{1} {6}}
$$
B. $$
{\frac{2} {3}}
$$
C. $$
{\frac{3} {4}}
$$
D. $$
{\frac{2} {5}}
$$
E. $$
{\frac{1} {8}}
$$
F. $$
{\frac{1} {5}}
$$
G. $$
{\frac{1} {7}}
$$
H. $$
{\frac{1} {2}}
$$
I. $$
{\frac{1} {3}}
$$
J. $$
{\frac{1} {4}}
$$
Answer: I
An electron is moving in a one-dimensional harmonic oscillator potential well, $\omega = 10^{5} \mathrm{rad/s}$, with an amplitude of $x_{0} = 10^{-8} \mathrm{cm}$. What is the energy radiated per cycle ?

Options:
A. $$
\approx6. 6 \times1 0^{-9} \mathrm{e V}
$$
B. $$
\approx1. 0 \times1 0^{-9} \mathrm{e V}
$$
C. $$
\approx9. 9 \times1 0^{-9} \mathrm{e V}
$$
D. $$
\approx4. 4 \times1 0^{-9} \mathrm{e V}
$$
E. $$
\approx7. 7 \times1 0^{-9} \mathrm{e V}
$$
F. $$
\approx8. 8 \times1 0^{-9} \mathrm{e V}
$$
G. $$
\approx1. 1 \times1 0^{-9} \mathrm{e V}
$$
H. $$
\approx3. 3 \times1 0^{-9} \mathrm{e V}
$$
I. $$
\approx2. 2 \times1 0^{-9} \mathrm{e V}
$$
J. $$
\approx5. 5 \times1 0^{-9} \mathrm{e V}
$$
Answer: G
We have the geodesic equation for the two-dimensional spacetime $\mathrm{d} s^{2}=-( 1+x )^{2} \mathrm{d} t^{2}+\mathrm{d} x^{2}$. Give the solution to the equation.

Options:
A. $$
\begin{aligned} {{}} & {{} {{}\pm\sin\left(\frac{C_{2}\pm\tau}{e}\right)}} \\ {{}} & {{} {{}+\log\left|\frac{1+x}{1-x}\right|}} \\ \end{aligned}
$$
B. $$
\begin{aligned} {{}} & {{} {{}\sinh\left(\frac{x}{C_{2}\pm\tau}\right)}} \\ {{}} & {{} {{}+\arctan\left(\frac{x}{1+x}\right)}} \\ \end{aligned}
$$
C. $$
\begin{aligned} {{}} & {{} {{}\pm\arctan\left(\frac{1+x}{1-x}\right)}} \\ {{}} & {{} {{}+C_{1}+\tau+\sqrt{2 e}}}\\ \end{aligned}
$$
D. $$
\begin{aligned} {{}} & {{} {{} \cosh^{-1}\left(\frac{C_{3}\pm\tau}{x}\right)}} \\ {{}} & {{} {{}+2\sqrt{1-\left(\frac{x}{C_{3}}\right)}}} \\ \end{aligned}
$$
E. $$
\begin{aligned} {{}} & {{} { x = \pm\sqrt{e^{2}-( C_{1} \pm\tau)^{2}}-1}} \\ {{}} & {{} { t = \mathrm{a r c t a n h} \left( \frac{C_{1} \pm\tau} {e} \right)+C_{2}}} \\ \end{aligned}
$$
F. $$
\begin{aligned} {{}} & {{} {{}\pm\arccos\left(\frac{C_1+\tau}{x}\right)}} \\ {{}} & {{} {{}+e^{x^{2}-1}}} \\ \end{aligned}
$$
G. $$
\begin{aligned} {{}} & {{} {{}\log\left(\frac{1+x^{2}}{1-x^{2}}\right)}} \\ {{}} & {{} {{}-\tanh^{-1}\left(\frac{C_{4}\pm\tau}{x}\right)}} \\ \end{aligned}
$$
H. $$
\begin{aligned} {{}} & {{} {{} \arcsin\left(\frac{2}{1+x^{2}}\right)}} \\ {{}} & {{} {{}+C_{4}+\arctanh\left(\tau\right)}} \\ \end{aligned}
$$
I. $$
\begin{aligned} {{}} & {{} {{}\frac{1}{2}\log\left|\frac{1+x+1}{1-x-1}\right|}} \\ {{}} & {{} {{}\pm\arctan\left(\frac{2\tau}{2}\right)+C_{3}}} \\ \end{aligned}
$$
J. $$
\begin{aligned} {{}} & {{} {{}\frac{1}{\sqrt{x^{2}-1}}}} \\ {{}} & {{} {{}+C_{3}+\log\left(\tau e\right)}} \\ \end{aligned}
$$
Answer: E
Given that 695xy155 is a multiple of 99, find $x, \mathbf{y}$.

Options:
A. x=0,y=8
B. x=4,y=7
C. x=1,y=4
D. x=8,y=0
E. x=6,y=9
F. x=9,y=6
G. x=3,y=2
H. x=2,y=3
I. x=5,y=1
J. x=7,y=5
Answer: H
Find the remainder when $\left( 1 2 3 7 1^{5 6}+3 4 \right)^{2 8}$ is divided by 111.

Options:
A. 94
B. 29
C. 57
D. 45
E. 52
F. 14
G. 38
H. 63
I. 70
J. 81
Answer: I
Let $p$ and $q$ be two coprime positive integers.
What is
$$
\sum_{0 < x < \frac{q} {2}} \left[ \frac{p} {q} x \right] + \sum_{0 < y < \frac{q} {2}} \left[ \frac{q} {p} y \right]
?$$

Options:
A. $$
\frac{p+1} {2} \cdot\frac{q-1} {3}
$$
B. $$
\frac{p-1} {3} \cdot\frac{q-1} {4}
$$
C. $$
\frac{p+3} {2} \cdot\frac{q-2} {4}
$$
D. $$
\frac{p-2} {3} \cdot\frac{q+3} {2}
$$
E. $$
\frac{p-1} {3} \cdot\frac{q-2} {3}
$$
F. $$
\frac{p-1} {2} \cdot\frac{q-1} {2}
$$
G. $$
\frac{p+1} {4} \cdot\frac{q-1} {2}
$$
H. $$
\frac{p+2} {3} \cdot\frac{q-2} {2}
$$
I. $$
\frac{p-1} {3} \cdot\frac{q+1} {2}
$$
J. $$
\frac{p-2} {2} \cdot\frac{q+2} {3}
$$
Answer: F
Two cyclists set off towards each other simultaneously from points $A$ and $B$. The location where they first meet is $70 \ \mathrm{km}$ from point A. After meeting, they continue to ride at the same speed. Upon reaching points $B$ and $A$, respectively, they turn around and head back to their starting points along the original route. The location where they meet for the second time is $90 \ \mathrm{km}$ from point $B$. Find the distance between points A and $B$.

Options:
A. 170km
B. 140km
C. 110km
D. 120km
E. 130km
F. 150km
G. 190km
H. 160km
I. 180km
J. 200km
Answer: D
What is $\begin{aligned}\sqrt[3]{\sqrt[3]{2}-1}\end{aligned}$?

Options:
A. $\sqrt[3]{\frac18}-\sqrt[3]{\frac27}+\sqrt[3]{\frac47}$
B. $\sqrt[3]{\frac12}-\sqrt[3]{\frac23}+\sqrt[3]{\frac43}$
C. $\sqrt[3]{\frac15}-\sqrt[3]{\frac56}+\sqrt[3]{\frac47}$
D. $\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$
E. $\sqrt[3]{\frac17}-\sqrt[3]{\frac38}+\sqrt[3]{\frac58}$
F. $\sqrt[3]{\frac13}-\sqrt[3]{\frac14}+\sqrt[3]{\frac34}$
G. $\sqrt[3]{\frac16}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$
H. $\sqrt[3]{\frac13}-\sqrt[3]{\frac23}+\sqrt[3]{\frac45}$
I. $\sqrt[3]{\frac{1}{5}}-\sqrt[3]{\frac25}+\sqrt[3]{\frac45}$
J. $\sqrt[3]{\frac14}-\sqrt[3]{\frac25}+\sqrt[3]{\frac45}$
Answer: D