MikaStars39/nano-eval huggingface dataset:
In the construction of diaphragm walls in dense sandy soil foundations, where the groundwater level is flush with the ground, the saturated unit weight of the sand $\gamma_{\mathrm{sat}} = 20.2 \, \mathbf{kN/m}^3$, internal friction angle $\varphi = 38°$, and cohesion $c = 0$, submerged slurry retaining walls are used. The slurry inside the trench is level with the ground, forming an impermeable slurry skin. In order to balance the active earth pressure of the foundation sand with the slurry pressure and maintain the stability of the trench walls, the specific gravity of the slurry should be at least is () g/cm³.

Options:
A. $$
1. 3 0 \mathrm{g / c m}^{3}
$$
B. $$
1. 8 7 \mathrm{g / c m}^{3}
$$
C. $$
1. 4 5 \mathrm{g / c m}^{3}
$$
D. $$
1. 2 4 \mathrm{g / c m}^{3}
$$
E. $$
1. 1 0 \mathrm{g / c m}^{3}
$$
F. $$
1. 2 8 \mathrm{g / c m}^{3}
$$
G. $$
1. 9 5 \mathrm{g / c m}^{3}
$$
H. $$
1. 7 2 \mathrm{g / c m}^{3}
$$
I. $$
1. 6 3 \mathrm{g / c m}^{3}
$$
J. $$
1. 5 0 \mathrm{g / c m}^{3}
$$
Answer: D
The type of a certain construction site is Category II, located in an 8-degree seismic intensity zone, with a basic design seismic acceleration of 0.3g. The design earthquake grouping is the first group. The site has a slope height of 18m, a slope horizontal length of 20m, and the minimum distance between the building and the edge of the slope is 15m. The natural vibration period of the building is 0.3s. Under the condition of frequent earthquakes, the seismic influence coefficient of this building structure is ().

Options:
A. 0.31
B. 0.37
C. 0.28
D. 0.32
E. 0.34
F. 0.36
G. 0.35
H. 0.27
I. 0.29
J. 0.33
Answer: E
Mr. Li, male, 59 years old. Has a 7-year history of hypertension and irregularly takes nifedipine tablets to lower blood pressure, but the control is poor: $150-160/105$ mmHg. A chest X-ray a year ago indicated: heart silhouette enlarged toward the left lower side, boot-shaped heart. This morning, following a trivial quarrel with a neighbor, the patient suddenly developed dyspnea, cough with expectoration of pink frothy sputum, pallor, cyanosis of the lips, profuse sweating, and restlessness. He was brought to the emergency department by ambulance. Physical examination indicated: blood pressure: $185/120$ mmHg, wet crackles and wheezing sounds throughout both lungs, heart rate: 120 beats per minute, regular rhythm, with a detectable early diastolic gallop rhythm. Based on the above medical history, what is the most likely diagnosis for this patient?

Options:
A. Grade III hypertension, congestive heart failure, acute renal failure
B. Grade I hypertension, coronary artery disease, acute pulmonary edema
C. Grade I hypertension, chronic liver disease, acute myocardial infarction
D. Grade II hypertension, cardiomyopathy, acute respiratory distress syndrome
E. Stage I hypertension, atrial fibrillation, acute brainstem stroke
F. Grade I hypertension, hypertensive heart disease, acute left heart failure
G. Grade II hypertension, chronic kidney disease, acute left heart failure
H. Stage II hypertension, ischemic heart disease, acute right heart failure
I. Stage II hypertension, hypertensive retinopathy, respiratory failure
J. Stage I hypertension, valvular heart disease, pulmonary embolism
Answer: F
For an equilateral triangle with side length $a$, point charges $q$, $-q$, as well as $-2q$ are placed at its three vertices. Find the electric potential at the centroid of this triangle. How much work must an external force do to move a point charge of $+Q$ from infinity to the centroid?

Options:
A. $$
{\frac{q Q} {2 \pi\varepsilon_{0} a}}
$$
B. $$
- {\frac{2 q Q} {4 \pi\varepsilon_{0} a}}
$$
C. $$
- {\frac{q Q} {2 \pi\varepsilon_{0} a}}
$$
D. $$
- {\frac{q Q} {4 \pi\varepsilon_{0} a}}
$$
E. $$
- {\frac{\sqrt{3} Q} {2 \pi\varepsilon_{0} a}}
$$
F. $$
- {\frac{\sqrt{3} q Q} {2 \pi\varepsilon_{0} a}}
$$
G. $$
{\frac{2 q Q}{4 \pi \varepsilon_{0} a}}
$$
H. $$
{\frac{\sqrt{3} q Q} {4 \pi\varepsilon_{0} a}}
$$
I. $$
{\frac{q Q}{4 \pi\varepsilon_{0} a}}
$$
J. $$
{\frac{\sqrt{3} q Q}{2 \pi\varepsilon_{0} a}}
$$
Answer: F
The moment of inertia of a flywheel is $J$, and at $t=0$, its angular velocity is $\omega_{0}$. Thereafter, the flywheel undergoes a braking process. The magnitude of the resistance torque $M$ is proportional to the square of the angular velocity $\omega$, with a proportionality coefficient $K > 0$.Find the time required from the start of braking to $\omega=\omega_{0} / 3$.

Options:
A. $$
\frac{2 J} {K \omega}
$$
B. $$
\frac{6 J}{K \omega}
$$
C. $$
\frac{J}{2 K \omega}
$$
D. $$
\frac{3 J}{K \omega^{2}}
$$
E. $$
\frac{J}{K \omega_{0}}
$$
F. $$
\frac{4 J}{K \omega_{0}}
$$
G. $$
\frac{J}{K \omega^{2}}
$$
H. $$
\frac{3 J}{K \omega_{0}}
$$
I. $$
\frac{5 J}{K \omega^{2}}
$$
J. $$
\frac{4 J}{K \omega}
$$
Answer: A
An object undergoes simple harmonic motion along the $x$-axis, with an amplitude of $0.12 \ \mathrm{m}$ and a period of 2.0 s. At $t=0$, the object is located at 0.06 m and is moving in the positive direction of the $x$-axis.
What are the displacement from the equilibrium position, velocity, and acceleration of the object at $t=0.5 \ \mathrm{s}$?

Options:
A. $$
\approx0. 1 2 \, ( \mathrm{m} )
$$
$$
\approx-0. 1 8 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 9 0 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
B. $$
\approx0. 0 \, ( \mathrm{m} )
$$
$$
\approx-0. 1 5 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 8 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
C. $$
\approx0. 0 8 \, ( \mathrm{m} )
$$
$$
\approx-0. 2 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-1. 0 1 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
D. $$
\approx0. 0 9 \, ( \mathrm{m} )
$$
$$
\approx-0. 1 0 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 8 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
E. $$
\approx0. 1 \, ( \mathrm{m} )
$$
$$
\approx-0. 1 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 9 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
F. $$
\approx0. 1 1 \, ( \mathrm{m} )
$$
$$
\approx-0. 0 5 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-1. 0 3 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
G. $$
\approx0. 2 \, ( \mathrm{m} )
$$
$$
\approx-0. 1 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
H. $$
\approx0. 0 7 \, ( \mathrm{m} )
$$
$$
\approx-0. 2 2 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 9 8 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
I. $$
\approx0. 1 5 \, ( \mathrm{m} )
$$
$$
\approx-0. 2 0 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-1. 1 0 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
J. $$
\approx0. 0 5 \, ( \mathrm{m} )
$$
$$
\approx-0. 2 5 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
\approx-0. 7 9 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
Answer: E
The density of the polyethylene terephthalate sample, measured using a density gradient tube, is $\rho\!=\! 1.40 \mathrm{g/cm}^{3}$. Calculate the result of mass crystallinity$f_{\mathrm{c}}^{w}$ and volume crystallinity of the sample$f_{\mathrm{c}}^{V}$.

Options:
A. $$
f_{\mathrm{c}}^{V} \!=\! 6 0. 4 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 7. 2 \, \%
$$
B. $$
f_{\mathrm{c}}^{V} \!=\! 4 9. 5 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 6 0. 0 \, \%
$$
C. $$
f_{\mathrm{c}}^{V} \!=\! 5 1. 2 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 0. 8 \, \%
$$
D. $$
f_{\mathrm{c}}^{V} \!=\! 5 3. 8 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 5. 9 \, \%
$$
E. $$
f_{\mathrm{c}}^{V} \!=\! 5 5. 9 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 6 2. 1 \, \%
$$
F. $$
f_{\mathrm{c}}^{V} \!=\! 4 8. 0 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 3. 7 \, \%
$$
G. $$
f_{\mathrm{c}}^{V} \!=\! 5 6. 2 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 4. 5 \, \%
$$
H. $$
f_{\mathrm{c}}^{V} \!=\! 4 7. 8 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 8. 4 \, \%
$$
I. $$
f_{\mathrm{c}}^{V} \!=\! 5 8. 7 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 5 1. 3 \, \%
$$
J. $$
f_{\mathrm{c}}^{V} \!=\! 5 2. 0 \, \%
$$
$$
f_{\mathrm{c}}^{w} \!=\! 4 9. 7 \, \%
$$
Answer: D
A beam of argon ion laser with a wavelength of $0.5145 \mu\mathrm{m}$ and an output power of 3W is uniformly projected onto a white screen with an area of $0.2 \mathrm{cm}^{2}$. What is the illuminance on the screen? If the reflection coefficient of the screen is 0.8, what is its luminance? How many photons does the screen receive per minute?

Options:
A. 60.12lx,50.23lx,7.58 $\times10^{20}$
B. 58.36lx,44.89lx,7.91 $\times10^{20}$
C. 63.77lx,48.54lx,7.79 $\times10^{20}$
D. 60.78lx,48.44lx,7.67 $\times10^{20}$
E. 62.58lx,51.32lx,7.69 $\times10^{20}$
F. 59.43lx,47.91lx,7.83 $\times10^{20}$
G. 65.22lx,50.67lx,7.53 $\times10^{20}$
H. 62.29lx,49.83lx,7.76 $\times10^{20}$
I. 64.15lx,52.18lx,7.65 $\times10^{20}$
J. 61.95lx,46.27lx,7.82 $\times10^{20}$
Answer: H
Given that the resistance of a certain photoresistor is $550 \Omega$ under an illumination of $500 \mathrm{lx}$, and $450 \Omega$ under an illumination of $700 \mathrm{lx}$, what is the resistance of this photoresistor under illuminations of 550 lx and 600 lx?

Options:
A. 521.2Ω,491.5Ω
B. 518.8Ω,493.3Ω
C. 522.3Ω,489.5Ω
D. 519.4Ω,492.0Ω
E. 517.1Ω,494.0Ω
F. 523.8Ω,487.7Ω
G. 515.0Ω,495.0Ω
H. 520.0Ω,490.0Ω
I. 516.2Ω,496.1Ω
J. 525.5Ω,488.0Ω
Answer: B
Given that the $\mathrm{RL2048DKQ} (26 \times 13 \, \mu\mathrm{m}^{2})$ has a light sensitivity of $4.5 \, \mathrm{V} / \mu\mathrm{J} \cdot \mathrm{cm}^{-2}$ in the $250 \mathrm{\sim} 350 \, \mathrm{nm}$ band, the RL2048DK is used as a photodetector with a 16-bit A/D data acquisition card (reference voltage of 2.5V, full-scale input voltage of 5V) to detect the measured spectrum. A line with an amplitude of 12500 is detected at the center position of pixel 1042. Assume the spectrometer has been calibrated with two known spectral lines: one at 260nm with a center position at pixel 450, and the other at 340nm with a center position at pixel 1550. What is the spectral radiant energy of this spectrum?

Options:
A. $$
8. 4 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
B. $$
4. 2 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
C. $$
7. 1 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
D. $$
5. 9 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
E. $$
1. 8 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
F. $$
5. 1 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
G. $$
2. 7 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
H. $$
6. 3 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
I. $$
9. 6 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
J. $$
3. 5 \times1 0^{-9} \ ( \ \mu\ \mathrm{J} )
$$
Answer: C
The spacing between the mirrors of a He-Ne laser is 0.2m. If the gain curve width is $1.5 \times 10^{9} \mathrm{Hz}$, then how many possible longitudinal modes could be induced?

Options:
A. 9
B. 10
C. 2
D. 6
E. 13
F. 7
G. 12
H. 5
I. 3
J. 8
Answer: I
The laser rangefinder tested two distances to two targets in the same direction, $L_{1}=3.5 \text{ km}, \; \; L_{2}=$ 5 km, with the ratio of the pulse amplitudes measured twice being 10:1. What is the average atmospheric attenuation coefficient in this direction?

Options:
A. 4.00dB/km
B. 3.75dB/km
C. 5.00dB/km
D. 5.75dB/km
E. 2.00dB/km
F. 3.25dB/km
G. 6.50dB/km
H. 6.00dB/km
I. 2.50dB/km
J. 4.34dB/km
Answer: J
A certain pressurized water reactor uses $\mathrm{U O_{2}}$ as fuel, with an enrichment of 2.43% (by mass) and a density of $10000 \, \text{kg/m}^3$. When the neutron energy is $0.0253 \, \text{eV}$, what are the macroscopic absorption cross-section and macroscopic fission cross-section of $\mathrm{U O_{2}}$?

Options:
A. $$ 4 0. 9~m^{-1} ,~ 3 1.5~m^{-1} $$
B. $$ 4 0. 5~m^{-1} ,~ 2 8.5~m^{-1} $$
C. $$ 4 5. 1~m^{-1} ,~ 3 5.0~m^{-1} $$
D. $$ 4 3. 9~m^{-1} ,~ 3 1.8~m^{-1} $$
E. $$ 4 4. 3~m^{-1} ,~ 3 2.5~m^{-1} $$
F. $$ 4 1. 7~m^{-1} ,~ 2 9.0~m^{-1} $$
G. $$
4 3. 2~m^{-1} ,~ 3 2.0~m^{-1}
$$
H. $$ 4 2. 5~m^{-1} ,~ 3 3.0~m^{-1} $$
I. $$ 4 6. 8~m^{-1} ,~ 3 6.0~m^{-1} $$
J. $$ 4 7. 6~m^{-1} ,~ 3 4.5~m^{-1} $$
Answer: G
For a cylindrical stack with a radius of $R$ and a height of $H$, the savings in the reflector layer are $\delta_{\mathrm{r}}$ and $\delta_{\mathrm{H}}$, respectively. What is the ratio of the maximum value to the average value of the equivalent bare stack's thermal neutron flux density, i.e., the coefficient of non-uniformity of the thermal neutron flux density?

Options:
A. $$
\frac{\pi} {8}^{3} ( \frac{a} {a+2 \delta_{x}} ) ( \frac{b} {b+2 \delta_{y}} ) ( \frac{c} {c+2 \delta_{z}} )
$$
B. $$
\frac{\pi} {8}^{2} ( \frac{u} {u+2 \delta_{m}} ) ( \frac{v} {v+2 \delta_{n}} ) ( \frac{w} {w+2 \delta_{o}} )
$$
C. $$
\frac{\pi} {7}^{3} ( \frac{r} {r+2 \delta_{j}} ) ( \frac{s} {s+2 \delta_{k}} ) ( \frac{t} {t+2 \delta_{l}} )
$$
D. $$
\frac{\pi} {9}^{3} ( \frac{j} {j+2 \delta_{g}} ) ( \frac{k} {k+2 \delta_{h}} ) ( \frac{l} {l+2 \delta_{i}} )
$$
E. $$
\frac{\pi} {4}^{3} ( \frac{d} {d+2 \delta_{a}} ) ( \frac{e} {e+2 \delta_{b}} ) ( \frac{f} {f+2 \delta_{c}} )
$$
F. $$
\frac{\pi} {5}^{3} ( \frac{g} {g+2 \delta_{d}} ) ( \frac{h} {h+2 \delta_{e}} ) ( \frac{i} {i+2 \delta_{f}} )
$$
G. $$
\frac{\pi} {11}^{3} ( \frac{s} {s+2 \delta_{s}} ) ( \frac{t} {t+2 \delta_{t}} ) ( \frac{u} {u+2 \delta_{u}} )
$$
H. $$
\frac{\pi} {10}^{3} ( \frac{p} {p+2 \delta_{p}} ) ( \frac{q} {q+2 \delta_{q}} ) ( \frac{r} {r+2 \delta_{r}} )
$$
I. $$
\frac{\pi} {6}^{2} ( \frac{x} {x+2 \delta_{m}} ) ( \frac{y} {y+2 \delta_{n}} ) ( \frac{z} {z+2 \delta_{p}} )
$$
J. $$
\frac{\pi} {2}^{3} ( \frac{v} {v+2 \delta_{v}} ) ( \frac{w} {w+2 \delta_{w}} ) ( \frac{x} {x+2 \delta_{x}} )
$$
Answer: A
The pressure of $N_{2}$ is 101 MPa, and the specific enthalpy is 129 kJ/kg. Determine the dryness fraction of the two-phase mixture.

Options:
A. 0.8535
B. 0.6335
C. 0.5435
D. 0.6635
E. 0.7135
F. 0.6735
G. 0.6235
H. 0.5935
I. 0.6935
J. 0.7635
Answer: D
The refrigerant in Lai's refrigeration system is R22, with a cooling capacity of 80 kW. It operates using a vapor compression cycle with an evaporation temperature of $-10^{\circ} C$ and a condensation temperature of $42^{\circ} C$. Determine the vapor fraction of the mixture at the evaporator inlet, expressed in terms of both mass fraction and volume fraction.

Options:
A. $$Fraction_{mass} = 29\%, Fraction_{volume} = 8.987\%$$
B. $$Fraction_{mass} = 33\%, Fraction_{volume} = 12.890\%$$
C. $$Fraction_{mass} = 32\%, Fraction_{volume} = 11.456\%$$
D. $$Fraction_{mass} = 26\%, Fraction_{volume} = 10.432\%$$
E. $$Fraction_{mass} = 27\%, Fraction_{volume} = 9.124\%$$
F. $$Fraction_{mass} = 28\%, Fraction_{volume} = 10.567\%$$
G. $$Fraction_{mass} = 34\%, Fraction_{volume} = 13.643\%$$
H. $$Fraction_{mass} = 30\%, Fraction_{volume} = 9.735\%$$
I. $$Fraction_{mass} = 31\%, Fraction_{volume} = 10.210\%$$
J. $$Fraction_{mass} = 25\%, Fraction_{volume} = 8.123\%$$
Answer: H
The operating temperatures of the nitrogen single-stage throttle liquefier are set at 290K and 71.9K. Assume the nitrogen gas is isothermally and reversibly compressed to 10.1MPa, with its low pressure corresponding to the saturated vapor pressure of liquid nitrogen at 71.9K (0.05MPa). If the heat exchanger is ideal and the system is perfectly insulated, determine the liquefaction rate and the degree of thermodynamic perfection of the liquefaction system.

Options:
A. $$0.046,0.093$$
B. $$0.051,0.087$$
C. $$0.049,0.088$$
D. $$0.037,0.084$$
E. $$0.058,0.085$$
F. $$0.060,0.092$$
G. $$0.071,0.094$$
H. $$0.043,0.081$$
I. $$0.032,0.075$$
J. $$0.055,0.089$$
Answer: C
When R22 is discharged from the compressor, the pressure is 2.0 MPa and the temperature is $90^{\circ}C$. Determine the specific volume, specific enthalpy, and specific entropy of this refrigerant at this time.(The saturation temperature corresponding to a saturation pressure of 2.0 MPa for R22 is 52°C.)

Options:
A. $$0.013246m^{3}/kg,452kJ/kg,1.77kJ/ (kg\times K )$$
B. $$0.014789m^{3}/kg,456kJ/kg,1.82kJ/ (kg\times K )$$
C. $$0.015678m^{3}/kg,460kJ/kg,1.80kJ/ (kg\times K )$$
D. $$0.011234m^{3}/kg,454kJ/kg,1.76kJ/ (kg\times K )$$
E. $$0.014286m^{3}/kg,455kJ/kg,1.79kJ/ (kg\times K )$$
F. $$0.012345m^{3}/kg,450kJ/kg,1.75kJ/ (kg\times K )$$
G. $$0.014500m^{3}/kg,457kJ/kg,1.78kJ/ (kg\times K )$$
H. $$0.015900m^{3}/kg,459kJ/kg,1.83kJ/ (kg\times K )$$
I. $$0.013567m^{3}/kg,453kJ/kg,1.74kJ/ (kg\times K )$$
J. $$0.016379m^{3}/kg,458kJ/kg,1.81kJ/ (kg\times K )$$
Answer: E
Consider the scenario of water flow through a sluice gate in an open channel. If the channel bottom is horizontal, the water is regarded as an incompressible homogeneous fluid, and viscous effects can be neglected. Let the water depths far upstream and far downstream of the gate be $h_{1}$ and $h_{2}$, respectively, where the flow is uniform. When water flows steadily under the influence of gravity, an external force must be applied to the gate to hold it in place. The force $\mathbf{R}$ required per unit width of the gate is ( ) .

Options:
A. $$
\frac{\rho g ( h_{1}-h_{2} )^{3}} {2 \left( h_{1}-h_{2} \right)}
$$
B. $$
\frac{2 \rho g ( h_{1}-h_{2} )^{3}} {2 \left( h_{1}+h_{2} \right)}
$$
C. $$
\frac{\rho g ( h_{1}-h_{2} )^{3}} {2 \left( h_{1}+h_{2} \right)}
$$
D. $$
\frac{\rho g ( h_{1}-h_{2} )^{2}} {2 \left( h_{1}+h_{2} \right)}
$$
E. $$
\frac{\rho g ( h_{1}(h_{2})^{3}} {2 \left( h_{1}+h_{2} \right)}
$$
F. $$
\frac{\rho g ( h_{1}+h_{2} )^{3}} {2 \left( h_{1}-h_{2} \right)}
$$
G. $$
\frac{\rho g ( h_{1}-h_{2} )^{3}} {h_{1}+h_{2}}
$$
H. $$
\frac{\rho g ( h_{1}+h_{2} )^{2}} {2 \left( h_{1}-h_{2} \right)}
$$
I. $$
\frac{\rho g ( h_{2}-h_{1} )^{3}} {2 \left( h_{1}+h_{2} \right)}
$$
J. $$
\frac{\rho g ( h_{1}-h_{2} )^{3}} {3 \left( h_{1}+h_{2} \right)}
$$
Answer: C
In a fluid with density $\rho = 850 \, \mathrm{kg/m^3}$ and viscosity coefficient $\mu = 0.0128 \, \mathrm{Pa} \cdot \mathrm{s}$, there is a rectangular plate. The width of this rectangle is $b = 0.6 \, \mathrm{m}$ and the length is $L = 50 \, \mathrm{m}$. If the rectangular plate moves at a constant speed of $U = 10 \, \mathrm{m/s}$ along its length within its own plane, and the known rotational Reynolds number is $Re_{or} = 5 \times 10^5$, what is the resistance force $F$ on the plate ?

Options:
A. $$
7. 0 8 8 \, ( \mathrm{k N} )
$$
B. $$
7. 5 9 2 \, ( \mathrm{k N} )
$$
C. $$
5. 0 6 8 \, ( \mathrm{k N} )
$$
D. $$
6. 9 2 7 \, ( \mathrm{k N} )
$$
E. $$
4. 7 5 0 \, ( \mathrm{k N} )
$$
F. $$
6. 8 4 3 \, ( \mathrm{k N} )
$$
G. $$
5. 3 8 9 \, ( \mathrm{k N} )
$$
H. $$
4. 3 1 9 \, ( \mathrm{k N} )
$$
I. $$
5. 7 7 6 \, ( \mathrm{k N} )
$$
J. $$
6. 3 0 4 \, ( \mathrm{k N} )
$$
Answer: I
When a vertical plate is inserted into a stationary fluid, the surface tension causes the fluid near the plate to rise along it. If the density of the fluid is $10^{\circ} \ \mathrm{kg/m}^{\circ}$, find: If the height of the fluid rise on the plate is $0.35 \ \mathrm{cm}$ and the surface tension coefficient of the fluid is $8.11 \times 10^{-5} \ \mathrm{N/cm}$, what should the contact angle be?

Options:
A. $$
4 5^{\circ}
$$
B. $$
3 0^{\circ}
$$
C. $$
6 0^{\circ}
$$
D. $$
5 0^{\circ}
$$
E. $$
2 0^{\circ}
$$
F. $$
1 0^{\circ}
$$
G. $$
1 5^{\circ}
$$
H. $$
8 0^{\circ}
$$
I. $$
9 0^{\circ}
$$
J. $$
7 5^{\circ}
$$
Answer: G
The given velocity field is
$$
v = 2 x \mathbf{i} - a y \mathbf{j} + (3 t - b z) \mathbf{k},
$$
where $a$ and $b$ are constants. What are the equations of the streamlines passing through the spatial point $(x, y, z) = (1, 1, 1)$ at times $t = 0$ and $t = 1$?

Options:
A. $$
\left\{\begin{matrix} {{| y |=| x |^{b / 2}}} \\ {{| y |=| z |^{-b / a}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{a / 3},}}} \\ {{{\left| y \right|=\left| \frac{8-b z} {b-4} \right|^{-a / b}}}} \\ \end{array} \right.
$$
B. $$
\left\{\begin{matrix} {{| y |=| x |^{b / a}}} \\ {{| y |=| z |^{-a / 2}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{-a / 3},}}} \\ {{{\left| y \right|=\left| \frac{8-a z} {4-b} \right|^{b / 3}}}} \\ \end{array} \right.
$$
C. $$
\left\{\begin{matrix} {{| y |=| x |^{-b / a}}} \\ {{| y |=| z |^{a / 2}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{a / b},}}} \\ {{{\left| y \right|=\left| \frac{8-3z} {b-3} \right|^{b / 2}}}} \\ \end{array} \right.
$$
D. $$
\left\{\begin{matrix} {{| y |=| x |^{-b / 3}}} \\ {{| y |=| z |^{b / a}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{a / 2},}}} \\ {{{\left| y \right|=\left| \frac{8-b z} {2-b} \right|^{b / 3}}}} \\ \end{array} \right.
$$
E. $$
\left\{\begin{matrix} {{| y |=| x |^{-a / 3}}} \\ {{| y |=| z |^{b / 2}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{a / b},}}} \\ {{{\left| y \right|=\left| \frac{8-2z} {b-2} \right|^{b / a}}}} \\ \end{array} \right.
$$
F. $$
\left\{\begin{matrix} {{| y |=| x |^{b / 3}}} \\ {{| y |=| z |^{-2 / a}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{-a / b},}}} \\ {{{\left| y \right|=\left| \frac{8-a z} {3-2} \right|^{a / 3}}}} \\ \end{array} \right.
$$
G. $$
\left\{\begin{matrix} {{| y |=| x |^{-b / a}}} \\ {{| y |=| z |^{a / 2}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{-a / 3},}}} \\ {{{\left| y \right|=\left| \frac{8-3z} {b-3} \right|^{a / b}}}} \\ \end{array} \right.
$$
H. $$
\left\{\begin{matrix} {{| y |=| x |^{a / b}}} \\ {{| y |=| z |^{-a / 2}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{b / a},}}} \\ {{{\left| y \right|=\left| \frac{8-2z} {3-b} \right|^{a / 2}}}} \\ \end{array} \right.
$$
I. $$
\left\{\begin{matrix} {{| y |=| x |^{-a / b}}} \\ {{| y |=| z |^{3 / a}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{b / 3},}}} \\ {{{\left| y \right|=\left| \frac{8-b z} {2-a} \right|^{3 / b}}}} \\ \end{array} \right.
$$
J. $$
\left\{\begin{matrix} {{| y |=| x |^{-a / 2}}} \\ {{| y |=| z |^{a / b}}} \\ \end{matrix} \right.
$$
$$
\left\{\begin{array} {l} {{{\left| y \right|=\left| x \right|^{-a / 2},}}} \\ {{{\left| y \right|=\left| \frac{3-b z} {3-b} \right|^{a / b}}}} \\ \end{array} \right.
$$
Answer: J
Given $\psi=r^{4}$, where $x$ represents the stream function in an axisymmetric steady flow under the influence of a gravitational field in cylindrical coordinates $( r, \theta, x )$, what is the difference in total mechanical energy per unit mass at the points $( r, x )=( 1, 1 )$ and $( r, x )=( 2, 2 )$ ?

Options:
A. 128
B. 100
C. 200
D. 32
E. 0
F. 400
G. 64
H. 72
I. 256
J. 512
Answer: A
Consider the spatially steady irrotational motion of an incompressible fluid. If there are point sources with intensities $Q_{1}, Q_{2}, \cdots, Q_{n}$ at the points $\boldsymbol{r}_{1}, \boldsymbol{r}_{2}, \cdots, \boldsymbol{r}_{n}$, give the force acting on the point source located at $\boldsymbol{r}_1$.

Options:
A. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {4 \pi\mid\boldsymbol{r}_{b}+\boldsymbol{r}_{1} \mid^{3}}
$$
B. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {2 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
C. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}+\boldsymbol{r}_{1} )} {2 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{2}}
$$
D. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}+\boldsymbol{r}_{1} )} {4 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
E. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{1}-\boldsymbol{r}_{b} )} {4 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
F. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {4 \pi\mid\boldsymbol{r}_{1}-\boldsymbol{r}_{b} \mid^{2}}
$$
G. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( 2\boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {4 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
H. $$
Q_{1} \rho\sum_{k=2}^{n} \frac{Q_{k} ( \boldsymbol{r}_{k}-\boldsymbol{r}_{1} )} {4 \pi\mid\boldsymbol{r}_{k}-\boldsymbol{r}_{1} \mid^{3}}
$$
I. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {6 \pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
J. $$
Q_{1} \rho\sum_{b=1}^{n} \frac{Q_{b} ( \boldsymbol{r}_{b}-\boldsymbol{r}_{1} )} {\pi\mid\boldsymbol{r}_{b}-\boldsymbol{r}_{1} \mid^{3}}
$$
Answer: H
Consider the long-wave motion in a shallow water channel with a depth of $A$. At one end of the channel $(x=0)$ is a fixed vertical plane wall, and at the other end is a piston. The length of the piston is $l$, and its motion (displacement) is given by $\xi\!=\!a \cos{\omega t}$. The water in the channel undergoes forced oscillations. What is the displacement of the water surface at the piston relative to the equilibrium position ?

Options:
A. $$
\omega\, \sqrt{\frac{g} {h}} \, \xi\tan\, \sqrt{\frac{\omega t} {g h}}
$$
B. $$
\omega\, \sqrt{\frac{h} {g}} \, \xi\cot\, \sqrt{\frac{\omega x} {h g}}
$$
C. $$
\omega\, \sqrt{\frac{g} {h}} \, \xi\cot\, \sqrt{\frac{\omega t} {h g}}
$$
D. $$
\sqrt{\omega}\, \sqrt{\frac{h} {g}} \, \xi \, \sin \sqrt{\frac{\omega t} {g h}}
$$
E. $$
\omega\, \sqrt{\frac{h} {g}} \, \xi\cot\, \frac{\omega t} {\sqrt{g h}}
$$
F. $$
\omega\, \sqrt{\frac{h} {g}} \, \xi\tan\, \sqrt{\frac{\omega t} {h g}}
$$
G. $$
\sqrt{\omega h}\, \sqrt{\frac{h} {g}} \, \xi \cos \sqrt{\frac{\omega t} {g h}}
$$
H. $$
\omega\, \sqrt{\frac{g} {h}} \, \xi\, \cot \sqrt{\frac{\omega t} {g h^2}}
$$
I. $$
\omega\, \sqrt{\frac{h} {g}} \, \xi\cot\, \sqrt{\frac{\omega a} {h g}}
$$
J. $$
\omega\, \sqrt{\frac{h} {g}} \, a \, \tan\, \sqrt{\frac{\omega t} {g h}}
$$
Answer: E
At 25°C, what will be the conductivity of water after removing all impurities? At 25°C, the ionic product of water $K_{\mathrm{W}} = c_{\mathrm{H}+} \, c_{\mathrm{OH}-} = 1.008 \times 10^{-14}$. The limiting molar conductivities of the following electrolytes are given as: $\lambda_{0} (\mathrm{KOH}) = 274.4 \, \mathrm{S \, cm^{2} / mol}$, $\lambda_{0} (\mathrm{HCl}) = 426.04 \, \mathrm{S \, cm^{2} / mol}$, and $M(\mathrm{KCl}) = 149.82 \, \mathrm{S \, cm^{2} / mol}$ is().

Options:
A. $$
\mathbf{=2. 3 \times1 0^{-5} \Omega^{-1} c m^{-1}}
$$
B. $$
\mathbf{=8. 6 \times1 0^{-6} \Omega^{-1} c m^{-1}}
$$
C. $$
\mathbf{=9. 1 \times1 0^{-7} \Omega^{-1} c m^{-1}}
$$
D. $$
\mathbf{=5. 2 \times1 0^{-8} \Omega^{-1} c m^{-1}}
$$
E. $$
\mathbf{=3. 8 \times1 0^{-7} \Omega^{-1} c m^{-1}}
$$
F. $$
\mathbf{=1. 1 \times1 0^{-9} \Omega^{-1} c m^{-1}}
$$
G. $$
\mathbf{=4. 2 \times1 0^{-6} \Omega^{-1} c m^{-1}}
$$
H. $$
\mathbf{=5. 5 \times1 0^{-8} \Omega^{-1} c m^{-1}}
$$
I. $$
\mathbf{=7. 4 \times1 0^{-8} \Omega^{-1} c m^{-1}}
$$
J. $$
\mathbf{=6. 9 \times1 0^{-9} \Omega^{-1} c m^{-1}}
$$
Answer: H
In a mixed solution containing 0.001 mol/kg Zn$SO_{\mathrm{4}}$ and 0.01 mol/kg Cu$SO_{\mathrm{4}}$, two platinum electrodes are placed, and electrolysis is conducted at 25°C with an infinitesimal current while stirring the solution thoroughly. Given that the pH value of the solution is $5_{\circ}$, roughly estimate: What is the remaining concentration of the metal ions that precipitated first when the latter metal begins to deposit?

Options:
A. $$
= 2. 1 {\times} 1 0^{-4 0} \, m o l / L
$$
B. $$
= 9. 7 {\times} 1 0^{-8 0} \, m o l / L
$$
C. $$
= 2. 9 {\times} 1 0^{-1 0} \, m o l / L
$$
D. $$
= 8. 8 {\times} 1 0^{-2 0} \, m o l / L
$$
E. $$
= 3. 4 {\times} 1 0^{-6 0} \, m o l / L
$$
F. $$
= 1. 2 {\times} 1 0^{-3 0} \, m o l / L
$$
G. $$
= 4. 5 {\times} 1 0^{-9 0} \, m o l / L
$$
H. $$
= 5. 6 {\times} 1 0^{-5 0} \, m o l / L
$$
I. $$
= 7. 9 {\times} 1 0^{-4 0} \, m o l / L
$$
J. $$
= 6. 3 {\times} 1 0^{-7 0} \, m o l / L
$$
Answer: I
If the double layer capacitance of the electrode $\mathrm{Zn} | \, \mathrm{ZnSO}_{4} \, ( \, a=1 )$ is independent of the electrode potential and has a value of 36 $\mu\mathrm{F / cm}^{2}$, with the given electrode $\varphi_{\text{pzc}}=-\, 0.763 \, \mathrm{~V}$ and $\varphi_{0}=-\, 0.63$ $\mathbf{V}$, the residual surface charge density on the electrode when the potential changes to $\varphi=0.32 \mathrm{~V}$ by a certain current is().

Options:
A. $$
= 4 1. 7 \, \mu C / c m^{2}
$$
B. $$
= 4 0. 3 \, \mu C / c m^{2}
$$
C. $$
= 3 8. 5 \, \mu C / c m^{2}
$$
D. $$
= 3 3. 1 \, \mu C / c m^{2}
$$
E. $$
= 3 2. 4 \, \mu C / c m^{2}
$$
F. $$
= 3 4. 2 \, \mu C / c m^{2}
$$
G. $$
= 3 9. 2 \, \mu C / c m^{2}
$$
H. $$
= 2 9. 1 \, \mu C / c m^{2}
$$
I. $$
= 4 5. 6 \, \mu C / c m^{2}
$$
J. $$
= 3 7. 8 \, \mu C / c m^{2}
$$
Answer: F
When electrolytic reduction of zinc ions occurs in a 0.1 mol/L Zn$Cl_{2}$ solution, the cathodic process is concentration polarization. Given that the diffusion coefficient of zinc ions is $1 \times 10^{-5} \mathrm{cm^{2}/s}$ and the effective thickness of the diffusion layer is $1.2 \times 10^{-2} \mathrm{cm}$, find: what should the corresponding cathodic current density be when the cathode overpotential is measured at 0.029 $\mathrm{V}$ at 20°C?

Options:
A. $$
= 8 \times 1 0^{-1}
$$
B. $$
= 4 \times 1 0^{-1}
$$
C. $$
= 7 \times 1 0^{-1}
$$
D. $$
= 1 \times1 0^{-1}
$$
E. $$
= 6 \times 1 0^{-1}
$$
F. $$
= 9 \times 1 0^{-1}
$$
G. $$
= 1 \times 1 0^{-2}
$$
H. $$
= 5 \times 1 0^{-1}
$$
I. $$
= 2 \times 1 0^{-1}
$$
J. $$
= 3 \times 1 0^{-1}
$$
Answer: D
The exchange current density of the electrode reaction $\mathrm{O} + 2 \mathrm{e} \to \mathrm{R}$ was measured to be $2 \times 10^{-12} \mathrm{A/cm}^{2}$ at 25°C, with a transfer coefficient $\alpha = 0.46$. What is the rate of the electrode reaction when cathodically polarized at $-1.44 \,\mathbf{V}$? It is known that the electrode process is controlled by the electron transfer step, and the electrode potential is $-0.68 \,\mathrm{V}$ when no current is applied is().

Options:
A. $$
j=0. 1 4 2 \mathrm{~ A / c m}^{2}
$$
B. $$
j=0. 1 5 5 \mathrm{~ A / c m}^{2}
$$
C. $$
j=0. 1 2 0 \mathrm{~ A / c m}^{2}
$$
D. $$
j=0. 1 3 0 \mathrm{~ A / c m}^{2}
$$
E. $$
j=0. 1 1 5 \mathrm{~ A / c m}^{2}
$$
F. $$
j=0. 1 1 8 \mathrm{~ A / c m}^{2}
$$
G. $$
j=0. 1 4 9 \mathrm{~ A / c m}^{2}
$$
H. $$
j=0. 1 3 5 \mathrm{~ A / c m}^{2}
$$
I. $$
j=0. 1 2 9 \mathrm{~ A / c m}^{2}
$$
J. $$
j=0. 1 6 3 \mathrm{~ A / c m}^{2}
$$
Answer: H
When a copper rod is immersed in a Cu$SO_{\mathrm{2}}$ solution at 18°C, the equilibrium potential of the system is measured at 0.31V, with an exchange current density of $1.3 \times 10^{-9} \mathrm{A/cm}^{2}$ and a transfer coefficient $\overrightarrow{\alpha}=1$. The polarization current density when the electrode potential is polarized to -0.23 $\mathbf{V}$ is().(assuming electrochemical polarization occurs)

Options:
A. $$
= 2. 9 7 A / c m^{2}
$$
B. $$
= 0. 9 8 A / c m^{2}
$$
C. $$
= 5. 3 3 A / c m^{2}
$$
D. $$
= 1. 4 4 A / c m^{2}
$$
E. $$
= 4. 0 1 A / c m^{2}
$$
F. $$
= 1. 8 6 A / c m^{2}
$$
G. $$
= 3. 5 4 A / c m^{2}
$$
H. $$
= 2. 5 7 A / c m^{2}
$$
I. $$
= 2. 1 2 A / c m^{2}
$$
J. $$
= 3. 9 5 A / c m^{2}
$$
Answer: A
m = 3, n = 2. Consider the following mapping $f = (f_{1}, f_{2}) : R^{5} \to R^{2}$
$$
\begin{aligned}
f_{1}(x_{1}, x_{2}, y_{1}, y_{2}, y_{3}) &= 2 e^{x_{1}} + x_{2} y_{1} - 4 y_{2} + 3 \\
f_{2}(x_{1}, x_{2}, y_{1}, y_{2}, y_{3}) &= x_{2} \cos x_{1} - 6 x_{1} + 2 y_{1} - y_{3}
\end{aligned}
$$
If $a = (0, 1)$, $b = (3, 2, 7)$ then $f(a, b) = 0$
Verify that $A_{x}$ is invertible. Therefore, by the implicit function theorem, a $C^{1}$ mapping defined in a neighborhood of (3, 2, 7) can be obtained. What is the $C^{1}$ mapping?

Options:
A. $$
\left( \begin{array} {c c c} {{\frac{3} {5}}} & {{\frac{4} {8}}} & {{-\frac{1} {7}}} \\ {{\frac{5} {9}}} & {{\frac{-2} {3}}} & {{\frac{6} {1 5}}} \\ \end{array} \right)
$$
B. $$
\left( \begin{array} {c c c} {{\frac{2} {5}}} & {{\frac{-1} {6}}} & {{\frac{3} {5}}} \\ {{\frac{4} {8}}} & {{\frac{-3} {7}}} & {{\frac{1} {4}}} \\ \end{array} \right)
$$
C. $$
\left( \begin{array} {c c c} {{\frac{2} {9}}} & {{\frac{5} {7}}} & {{-\frac{1} {4}}} \\ {{\frac{3} {5}}} & {{\frac{-2} {6}}} & {{\frac{1} {8}}} \\ \end{array} \right)
$$
D. $$
\left( \begin{array} {c c c} {{\frac{3} {4}}} & {{\frac{1} {3}}} & {{-\frac{2} {5}}} \\ {{\frac{-1} {3}}} & {{\frac{4} {5}}} & {{\frac{1} {2}}} \\ \end{array} \right)
$$
E. $$
\left( \begin{array} {c c c} {{\frac{1} {4}}} & {{\frac{1} {5}}} & {{-\frac{3} {2 0}}} \\ {{\frac{-1} {2}}} & {{\frac{6} {5}}} & {{\frac{1} {1 0}}} \\ \end{array} \right)
$$
F. $$
\left( \begin{array} {c c c} {{\frac{1} {2}}} & {{\frac{2} {5}}} & {{-\frac{1} {1 0}}} \\ {{\frac{1} {4}}} & {{\frac{3} {4}}} & {{\frac{-1} {5}}} \\ \end{array} \right)
$$
G. $$
\left( \begin{array} {c c c} {{\frac{1} {3}}} & {{\frac{2} {7}}} & {{-\frac{4} {5}}} \\ {{\frac{0} {2}}} & {{\frac{5} {1 2}}} & {{\frac{3} {8}}} \\ \end{array} \right)
$$
H. $$
\left( \begin{array} {c c c} {{\frac{-2} {3}}} & {{\frac{3} {5}}} & {{-\frac{1} {6}}} \\ {{\frac{4} {6}}} & {{\frac{-5} {2}}} & {{\frac{3} {4}}} \\ \end{array} \right)
$$
I. $$
\left( \begin{array} {c c c} {{\frac{-1} {5}}} & {{\frac{2} {6}}} & {{\frac{1} {3}}} \\ {{\frac{2} {7}}} & {{\frac{-3} {1 0}}} & {{\frac{5} {9}}} \\ \end{array} \right)
$$
J. $$
\left( \begin{array} {c c c} {{\frac{4} {1 0}}} & {{\frac{1} {4}}} & {{-\frac{3} {6}}} \\ {{\frac{5} {3}}} & {{-\frac{2} {7}}} & {{\frac{1} {5}}} \\ \end{array} \right)
$$
Answer: E
If the solid angle over which the data were collected is 0.75 square degrees, or 2.3 x 10-4 sr, estimate the number density $n_{M} ( M, S )$ of A stars out to $V=11$?

Options:
A. $$
2. 9 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
B. $$
2. 7 \times1 0^{-{2}} \ \mathrm{p c}^{-3}
$$
C. $$
1. 5 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
D. $$
3. 5 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
E. $$
3. 0 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
F. $$
3. 2 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
G. $$
4. 0 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
H. $$
2. 4 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
I. $$
1. 8 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
J. $$
2. 6 \times1 0^{-{3}} \ \mathrm{p c}^{-3}
$$
Answer: A
When less than () of lead is added to the tea infusion, it results in a light flavor with a slightly sour taste.

Options:
A. 0.05mgH
B. 0.02mgL
C. 0.10mgC
D. 0.07mgE
E. 0.01mgM
F. 0.06mgF
G. 0.09mgG
H. 0.03mgK
I. 0.04mgI
J. 0.08mgN
Answer: I
A gyroscope undergoes regular precession around the center, with $I_{1}=I_{2}=2I_{3}$. Given that its spin angular velocity is $\omega_{1}$ and the angle between the spin axis and the precession axis is 60°, the precession angular velocity $\omega_{2}$ is ().

Options:
A. $$
\omega_{2}=\sqrt{3} \omega_{1} \,
$$
B. $$
\omega_{2}=\frac{1} {2} \omega_{1} \,
$$
C. $$
\omega_{2}=\omega_{1} \,
$$
D. $$
\omega_{2}=3 \omega_{1} \,
$$
E. $$
\omega_{2}=2 \omega_{1} \,
$$
F. $$
\omega_{2}=\frac{1} {\sqrt{3}} \omega_{1} \,
$$
G. $$
\omega_{2}=\frac{2} {\sqrt{5}} \omega_{1} \,
$$
H. $$
\omega_{2}=\frac{1} {\sqrt{2}} \omega_{1} \,
$$
I. $$
\omega_{2}=\frac{1} {4} \omega_{1} \,
$$
J. $$
\omega_{2}=\frac{1} {3} \omega_{1} \,
$$
Answer: B
Using the Hamilton-Jacobi equation to find the motion of a system with the Hamiltonian function $H=p^{2}-q$ is ().

Options:
A. $$
\xi=t-\frac{\partial W} {\partial E}=t \mp\sqrt{E+q}
$$
$$
p=\frac{\partial W} {\partial q}=\mp\sqrt{E+q}=\xi+t, \quad q=( \xi+t )^{2}+E
$$
B. $$
\xi=-t-\frac{\partial W} {\partial E}=-t \mp\sqrt{E-q}
$$
$$
p=\frac{\partial W} {\partial q}=\mp\sqrt{E-q}=\xi-t, \quad q=( \xi+t )^{2}+E
$$
C. $$
\xi=-t+\frac{\partial W} {\partial E}=-t \pm\sqrt{q-E}
$$
$$
p=\frac{\partial W} {\partial q}=\pm\sqrt{q-E}=\xi+t, \quad q=( \xi+t )^{2}+E
$$
D. $$
\xi=t+\frac{\partial W} {\partial E}=t \pm\sqrt{E-q}
$$
$$
p=\frac{\partial W} {\partial q}=\pm\sqrt{E-q}=\xi-t, \quad q=( \xi-t )^{2}+E
$$
E. $$
\xi=-t-\frac{\partial W} {\partial E}=-t \pm\sqrt{E-q}
$$
$$
p=\frac{\partial W} {\partial q}=\pm\sqrt{E-q}=\xi+t, \quad q=( \xi-t )^{2}+E
$$
F. $$
\xi=t+\frac{\partial W} {\partial E}=t \mp\sqrt{q+E}
$$
$$
p=\frac{\partial W} {\partial q}=\mp\sqrt{q+E}=\xi-t, \quad q=( \xi-t )^{2}+E
$$
G. $$
\xi=t+\frac{\partial W} {\partial E}=t \mp\sqrt{E+q}
$$
$$
p=\frac{\partial W} {\partial q}=\mp\sqrt{E+q}=\xi-t, \quad q=( \xi-t )^{2}-E
$$
H. $$
\xi=-t+\frac{\partial W} {\partial E}=-t \mp\sqrt{q-E}
$$
$$
p=\frac{\partial W} {\partial q}=\mp\sqrt{q-E}=\xi+t, \quad q=( \xi+t )^{2}+E
$$
I. $$
\xi=-t+\frac{\partial W} {\partial E}=-t \pm\sqrt{E+q}
$$
$$
p=\frac{\partial W} {\partial q}=\pm\sqrt{E+q}=\xi+t, \quad q=( \xi+t )^{2}-E
$$
J. $$
\xi=-t+\frac{\partial W} {\partial E}=-t \mp\sqrt{E-q}
$$
$$
p=\frac{\partial W} {\partial q}=\pm\sqrt{E+q}=\xi+t, \quad q=( \xi+t )^{2}+E
$$
Answer: I
A stationary particle with a mass number of 42 decays into two fragments, one of which has a rest mass of 20 and moves with a velocity of $0.6 c$. What is the rest mass $m_{0}$ of the other fragment? (1 atomic mass unit $= 1.66 \times 10^{-27} \, \mathrm{kg}$).

Options:
A. Fourteen mass numbers
B. Thirty-two mass numbers
C. Eight mass numbers
D. Forty-eight mass numbers
E. Ninety-two mass numbers
F. Twenty-six mass numbers
G. Sixty-four mass numbers
H. Twelve mass numbers
I. Fifty-four mass numbers
J. Sixteen mass numbers
Answer: C
The amplitude of a plane simple harmonic wave is 1.0 cm, the frequency is 100 Hz, and the wave speed is 400 m/s. When the particle at the wave source moves in the positive direction from the equilibrium position, it is taken as the starting time. What is the expression for the vibration of the medium particle located 800 cm from the wave source?

Options:
A. $$
\operatorname{s i n} ( 2 0 0 \pi x ) \mathrm{c m}
$$
B. $$
\operatorname{s i n} ( 2 \pi t ) \mathrm{c m}
$$
C. $$
\operatorname{c o s} ( 2 0 0 \pi t ) \mathrm{c m}
$$
D. $$
\operatorname{s i n} ( 2 0 \pi t ) \mathrm{c m}
$$
E. $$
\operatorname{s i n} ( 2 0 0 \pi t ) \mathrm{c m}
$$
F. $$
\operatorname{c o s} ( 4 0 0 \pi t ) \mathrm{c m}
$$
G. $$
\operatorname{s i n} ( 1 0 0 \pi t ) \mathrm{c m}
$$
H. $$
\operatorname{c o s} ( 2 0 0 \pi x ) \mathrm{c m}
$$
I. $$
\operatorname{s i n} ( 4 0 0 \pi t ) \mathrm{c m}
$$
J. $$
-\operatorname{s i n} ( 2 0 0 \pi t ) \mathrm{c m}
$$
Answer: E
The moment of inertia of a flywheel is $I$, and the angular velocity at the beginning of braking is $\omega_{0}$. How much time will it take for the angular velocity to decrease to one-third of its initial value?

Options:
A. $$\frac{2I\omega_0}{k}$$
B. $$\frac{I}{2k\omega_0^2}$$
C. $$\frac{3I\omega_0}{2k}$$
D. $$\frac{4I}{k\omega_0}$$
E. $$\frac{I}{2k\omega_0}$$
F. $$\frac{3I}{k\omega_0}$$
G. $$\frac{I}{k\omega_0^2}$$
H. $$\frac{2I}{k\omega_0}$$
I. $$\frac{I}{k\omega_0}$$
J. $$\frac{2I}{k\omega_0^2}$$
Answer: H
A separately excited DC motor has the following nameplate data: $\mathrm{P}_{\mathrm{N}}=2.2 \, \mathrm{kW}$, $\mathrm{U}_{\mathrm{N}}=220 \, \mathrm{V}$, $\mathrm{I}_{\mathrm{N}}=12.6 \, \mathrm{A}$, $\mathrm{n}_{\mathrm{N}}=1500 \, \mathrm{r/min}$. When $\mathrm{n}_{\mathrm{N}}=1550 \, \mathrm{r/min}$, what is the value of the armature current $\mathbf{I_{a}}$?

Options:
A. 9.50A
B. 8.00A
C. 10.25A
D. 10.50A
E. 8.75A
F. 9.75A
G. 8.50A
H. 11.00A
I. 9.25A
J. 10.00A
Answer: I
Two transformers are operating in parallel, both connected in a $\mathrm{Y, ~ d 11}$ configuration, with a nominal voltage $\mathbf{U}_{\mathrm{N 1}} / \mathbf{U}_{\mathrm{N 2}} = 35 \mathrm{kV} / 10.5 \mathrm{kV}$. The first transformer is rated at $1250 \, \mathrm{kV} \cdot \mathrm{A}$, with an impedance of $\mathrm{Z}_{\mathrm{k (1)}}^{\bullet} = 6.5\%$, and the second transformer is rated at $2000 \, \mathrm{kV} \cdot \mathrm{A}$, with an impedance of $\mathrm{Z}_{\mathrm{k (II)}}^{\bullet} = 6\%$. The maximum output of the parallel group when neither transformer is overloaded, and calculate the utilization rate of the parallel group is ().

Options:
A. $$
3 1 5 4 \, \mathrm{k V} \cdot\mathrm{A},0.97
$$
B. $$
3 1 4 0 \, \mathrm{k V} \cdot\mathrm{A},0.95
$$
C. $$
3 3 0 0 \, \mathrm{k V} \cdot\mathrm{A},0.94
$$
D. $$
3 2 5 0 \, \mathrm{k V} \cdot\mathrm{A},0.98
$$
E. $$
3 0 2 5 \, \mathrm{k V} \cdot\mathrm{A},0.96
$$
F. $$
3 4 0 0 \, \mathrm{k V} \cdot\mathrm{A},0.90
$$
G. $$
3 1 8 0 \, \mathrm{k V} \cdot\mathrm{A},0.91
$$
H. $$
3 2 6 0 \, \mathrm{k V} \cdot\mathrm{A},0.89
$$
I. $$
3 2 0 0 \, \mathrm{k V} \cdot\mathrm{A},0.92
$$
J. $$
3 1 7 5 \, \mathrm{k V} \cdot\mathrm{A},0.93
$$
Answer: A
Under atmospheric pressure, water at a temperature of 20°C flows over a smooth flat surface at a speed of 5 m/s. Determine the range for the critical distance $x_{\mathrm{c}}$ at which the laminar boundary layer transitions to a turbulent boundary layer.

Options:
A. $$
0. 0 6 \sim0. 8 5 \mathrm{m}
$$
B. $$
0. 0 5 \sim0. 7 2 \mathrm{m}
$$
C. $$
0. 0 7 \sim0. 8 0 \mathrm{m}
$$
D. $$
0. 0 3 \sim0. 5 8 \mathrm{m}
$$
E. $$
0. 0 9 \sim1. 0 0 \mathrm{m}
$$
F. $$
0. 2 0 \sim1. 5 0 \mathrm{m}
$$
G. $$
0. 1 5 \sim0. 9 0 \mathrm{m}
$$
H. $$
0. 0 2 \sim0. 5 0 \mathrm{m}
$$
I. $$
0. 0 4 \sim0. 6 0 \mathrm{m}
$$
J. $$
0. 1 0 \sim0. 7 0 \mathrm{m}
$$
Answer: I
A horizontal circular pipe with a length of 2 m and a diameter of $\phi 19 \mathrm{mm} \times 2 \mathrm{mm}$ is heated to a surface temperature of 250°C. The pipe is exposed to an atmosphere at 20°C and a pressure of 101.3 kPa. Calculate the natural convection heat transfer rate of the pipe

Options:
A. $$
= 3 1 9. 6 \, \mathrm{W}
$$
B. $$
= 4 5 7. 2 \, \mathrm{W}
$$
C. $$
= 4 1 5. 7 \, \mathrm{W}
$$
D. $$
= 2 8 3. 8 \, \mathrm{W}
$$
E. $$
= 3 7 2. 5 \, \mathrm{W}
$$
F. $$
= 4 0 2. 3 \, \mathrm{W}
$$
G. $$
= 3 8 3. 1 \, \mathrm{W}
$$
H. $$
= 2 9 8. 4 \, \mathrm{W}
$$
I. $$
= 3 9 4. 4 \, \mathrm{W}
$$
J. $$
= 3 6 2. 9 \, \mathrm{W}
$$
Answer: J
Find the value of the integral
$$
\int_{0}^{2 \pi a} ( 2 z^{2}+8 z+1 ) \mathrm{d} z
$$
where the integration path is the cycloid connecting $0$ to $2 \pi a$:
$$
x=a ( \theta-\sin \theta) \,, y=a ( 1-\cos \theta).
$$

Options:
A. $$
{\frac{1 9} {3}} \pi^{3} a^{3}-1 8 \pi^{2} a^{2}+5 \pi a
$$
B. $$
{\frac{1 6} {3}} \pi^{3} a^{3}+1 6 \pi^{2} a^{2}+2 \pi a
$$
C. $$
{\frac{1 4} {3}} \pi^{3} a^{3}+1 6 \pi^{2} a^{2}+4 \pi a
$$
D. $$
{\frac{1 1} {3}} \pi^{3} a^{3}+1 6 \pi^{2} a^{2}-4 \pi a
$$
E. $$
{\frac{1 9} {3}} \pi^{3} a^{3}-1 6 \pi^{2} a^{2}+3 \pi a
$$
F. $$
{\frac{1 7} {3}} \pi^{3} a^{3}+1 4 \pi^{2} a^{2}-2 \pi a
$$
G. $$
{\frac{1 3} {3}} \pi^{3} a^{3}+1 2 \pi^{2} a^{2}-3 \pi a
$$
H. $$
{\frac{1 3} {3}} \pi^{3} a^{3}-1 6 \pi^{2} a^{2}+2 \pi a
$$
I. $$
{\frac{1 0} {3}} \pi^{3} a^{3}+1 5 \pi^{2} a^{2}+0 \pi a
$$
J. $$
{\frac{1 8} {3}} \pi^{3} a^{3}+1 0 \pi^{2} a^{2}+3 \pi a
$$
Answer: B
A particle with charge $q$ moves in uniform circular motion around the $z$-axis in the $xy$ plane, with an angular frequency of $\omega$ and a radius of $R_{0}$. Assuming $\omega R_{0} \ll c$, calculate the frequency and energy flux density of the radiation field. Discuss the amplitude of the electromagnetic field at $\theta=0, \frac{\pi}{4}, \frac{\pi}{2}$, and $\pi$.

Options:
A. $$
E_{\phi} \left( t \right) |_{\theta=0} \neq0 \qquad E_{\theta} \left( t \right) |_{\theta=0}=0
$$
B. $$
E_{\phi} \left( t \right) |_{\theta=\pi} =0 \qquad E_{\theta} \left( t \right) |_{\theta=\pi}\neq0
$$
C. $$
E_{\phi} \left( t \right) |_{\theta=\frac{\pi} {4}} \neq0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {4}}=0
$$
D. $$
E_{\phi} \left( t \right) |_{\theta=\frac{\pi} {4}} =0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {4}}\neq0
$$
E. $$
E_{\phi} \left( t \right) |_{\theta=\pi} =0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {2}}\neq0
$$
F. $$
E_{\phi} \left( t \right) |_{\theta=\pi} \neq0 \qquad E_{\theta} \left( t \right) |_{\theta=\pi}=0
$$
G. $$
E_{\phi} \left( t \right) |_{\theta=\frac{\pi} {2}} \neq0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {2}}=0
$$
H. $$
E_{\phi} \left( t \right) |_{\theta=0} =0 \qquad E_{\theta} \left( t \right) |_{\theta=0}\neq0
$$
I. $$
E_{\phi} \left( t \right) |_{\theta=\frac{\pi} {2}} =0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {2}}\neq0
$$
J. $$
E_{\phi} \left( t \right) |_{\theta=0} =0 \qquad E_{\theta} \left( t \right) |_{\theta=\frac{\pi} {2}}\neq0
$$
Answer: G
The most ideal structure for soil fertility among common soil aggregates is ____.

Options:
A. Compact structure
B. Crystalline structure
C. Porous structure
D. Sand-based structure
E. Clay-dense structure
F. Layered structure
G. Fragmented structure
H. Aggregate structure
I. Particle structure
J. Loamy structure
Answer: H
Suppose that the Tunguska event was caused by an asteroid colliding with Earth. Assume that the density of the object was 2000 kg $\mathrm{m}^{-3}$ and that it exploded above the surface of the planet traveling at a rate equal to Earth's escape velocity. If all of the energy of the explosion was derived from the asteroid's kinetic energy, estimate the mass and radius of the impacting body (assume spherical symmetry).

Options:
A. 20m
B. 22m
C. 10m
D. 9m
E. 25m
F. 18m
G. 7m
H. 12m
I. 15m
J. 30m
Answer: H
It is known that the potential generated by a certain spherically symmetric charge distribution in spherical coordinates is expressed as $\varphi( r )={\frac{a} {r}} \mathbf{e}^{b r}$, where $a$ and $b$ are constants. The volume charge density is ().

Options:
A. $$
- {\frac{a b^{2} \varepsilon_{0}} {r}} e^{b r}
$$
B. $$
- {\frac{a b \varepsilon_{0}} {r^{2}}} e^{b r}
$$
C. $$
{\frac{a \varepsilon_{0}} {r}} e^{-b r}
$$
D. $$
- {\frac{a b \varepsilon_{0}} {r^{2}}} e^{-b r}
$$
E. $$
{\frac{a b \varepsilon_{0}} {r^{2}}} e^{b r}
$$
F. $$
{\frac{b^{2} a \varepsilon_{0}} {r}} e^{b r}
$$
G. $$
{\frac{a b \varepsilon_{0}} {r}} e^{-b r}
$$
H. $$
{\frac{a b^{2} \varepsilon_{0}} {r^{2}}} e^{-b r}
$$
I. $$
- {\frac{a b \varepsilon_{0}} {r}} e^{b r}
$$
J. $$
- {\frac{a b \varepsilon_{0}} {r^{2}}} e^{b r^{2}}
$$
Answer: I
The inner and outer radii of a spherical capacitor are $a$ and $b$ respectively, and the dielectric constant of the material between the inner and outer conductors is $c$, with a conductivity of $\gamma$. A low-frequency voltage $u=U_{\mathrm{m}} \cos \omega t$ is applied between the inner and outer conductors. What is the total current between the inner and outer conductors?

Options:
A. $$
I={\frac{4 \pi U_{m} a b} {b+a}} {\big(} \gamma\operatorname{c o s} \omega t+\varepsilon\operatorname{s i n} \omega t {\big)}
$$
B. $$
I={\frac{2 \pi U_{m} a b} {b^2-a^2}} {\big(} \varepsilon \operatorname{c o s} \omega t-\omega\operatorname{s i n} \omega t {\big)}
$$
C. $$
I={\frac{4 \pi U_{m} a^2 b^2} {b+a}} {\big(} r \operatorname{c o s} \omega t+\varepsilon\omega\operatorname{s i n} \omega t {\big)}
$$
D. $$
I={\frac{8 \pi U_{m} a b^2} {b-a}} {\big(} \gamma \operatorname{c o s} \omega t-\varepsilon\operatorname{s i n} \omega t {\big)}
$$
E. $$
I={\frac{\pi U_{m} a^2 b} {b-a}} {\big(} r\operatorname{c o s} \omega t+\varepsilon\omega\operatorname{s i n} \omega t {\big)}
$$
F. $$
I={\frac{4 \pi U_{m} a b} {b-a}} {\big(} r \operatorname{c o s} \omega t-\varepsilon\omega\operatorname{s i n} \omega t {\big)}
$$
G. $$
I={\frac{4 \pi U_{m} a b} {b-a}} {\big(} \varepsilon \gamma \operatorname{c o s} \omega t-\varepsilon\omega\operatorname{s i n} \omega t {\big)}
$$
H. $$
I={\frac{2 \pi U_{m} a b^2} {b-a}} {\big(} \varepsilon r\operatorname{c o s} \omega t-\omega\operatorname{s i n} \omega t {\big)}
$$
I. $$
I={\frac{\pi U_{m} a b} {b-a}} {\big(} r\operatorname{c o s} \omega t-\varepsilon\omega\operatorname{s i n} \omega t {\big)}
$$
J. $$
I={\frac{4 \pi U_{m} a b} {a^2-b^2}} {\big(} r\operatorname{c o s} \omega t-\varepsilon\operatorname{s i n} \gamma t {\big)}
$$
Answer: F