MikaStars39/nano-eval huggingface dataset:
Somewhere in space, there are two magnetic fields with intensities $H_{1}$ and $H_{2}$. Both $H_{1}$ and $H_{2}$ are in the horizontal direction, with $H_{1}$ pointing north and having a magnitude of $H_{1} = 1.73 \, \mathrm{Oe}$; $H_{2}$ is pointing east with a magnitude of $H_{2} = 1.00 \, \mathrm{Oe}$. Now, how should a current-carrying straight conductor be placed at this location to ensure that the resultant force exerted by the two magnetic fields on it is zero?

Options:
A. Set the wire parallel to the horizon, west by north $30^{\circ} 55^{\prime}$.
B. Tilt the wire downward, west by south $30^{\circ} 10^{\prime}$.
C. Position the wire vertically, north by west $45^{\circ} 11^{\prime}$.
D. Adjust the wire vertically, east by south $60^{\circ} 03^{\prime}$.
E. Place the wire in a horizontal position, north by east $30^{\circ} 02^{\prime}$.
F. Place the wire horizontally, south by west $60^{\circ} 15^{\prime}$.
G. Stand the wire vertically, north by west $60^{\circ} 09^{\prime}$.
H. Orient the wire in a diagonal position, east by north $45^{\circ} 20^{\prime}$.
I. Align the wire in a vertical plane, south by east $30^{\circ} 02^{\prime}$.
J. Lay the wire horizontally, north by east $45^{\circ} 30^{\prime}$.
Answer: E
A cylindrical capacitor consists of a straight conductor with a radius of $a$ and a coaxial conductive cylinder surrounding it, with the cylinder's inner radius being $b$. The space between the wire and the cylinder is air, and both the wire and the cylinder have a length of $L$. Neglecting edge effects, what is the capacitance of this capacitor?

Options:
A. $$
\frac{2 \pi\varepsilon_{0} b} {\operatorname{l n} ( a / b )}
$$
B. $$
\frac{4 \pi\varepsilon_{0} L a} {\operatorname{l n} ( b / a )} $$
C. $$
\frac{2 \pi\varepsilon_{0} a} {\operatorname{l n} ( b / a )} $$
D. $$
\frac{\pi\varepsilon_{0} L} {\operatorname{l n} ( b / a )} $$
E. $$
\frac{2 \pi b \varepsilon_{0} L} {\operatorname{l n} ( a / b )}
$$
F. $$
\frac{2 \pi\varepsilon_{0} L} {\operatorname{l n} ( b / a )}
$$
G. $$
\frac{\varepsilon_{0} L} {\operatorname{l n} ( b / a )}
$$
H. $$
\frac{2 \varepsilon_{0} L} {\operatorname{l n} ( b / a )}
$$
I. $$
\frac{2 \pi\varepsilon_{0} L a} {b - a} $$
J. $$
\frac{2 \pi\varepsilon_{0} L} {b - a} $$
Answer: F
Using a cooling crystallizer to produce crystallized potassium carbonate with two water molecules of crystallization $(K_{2}CO_{3} \cdot 2 \cdot H_{2}O)$. The feed solution is a 59.8% potassium carbonate aqueous solution (mass fraction of solute, the same below) with a processing capacity of 1200 kg/h, and the mother liquor content is 53%. Calculate the amount of crystallized product is ().

Options:
A. 350.60kg/h
B. 500.20kg/h
C. 278.95kg/h
D. 320.75kg/h
E. 309.85kg/h
F. 295.40kg/h
G. 400.25kg/h
H. 450.15kg/h
I. 415.90kg/h
J. 280.50kg/h
Answer: E
In a kettle-type shell and tube heat exchanger, a certain liquid is heated and vaporized using saturated steam at 280 kPa (the steam only releases its latent heat of condensation). The specific heat of the liquid is $4.0 \, \text{kJ/kg} \cdot \degree \text{C}$, with an inlet temperature of 50°C, a boiling point of 88°C, and a latent heat of vaporization of 2200 kJ/kg. The liquid flow rate is 1000 kg/h. Ignoring heat loss, calculate the consumption of the heating steam is ().

Options:
A. 1070.2kg/h
B. 1081.4kg/h
C. 1112.5kg/h
D. 950.6kg/h
E. 1170.0kg/h
F. 990.0kg/h
G. 980.7kg/h
H. 1030.9kg/h
I. 1135.8kg/h
J. 1245.3kg/h
Answer: B
The diameter of the piston of a certain single-acting reciprocating pump is 160 mm, and the stroke is $200 \, \mathrm{mm}$. This pump is intended to transport a liquid with a specific gravity of 930 kg/ $\mathrm{m}^{3}$ from an open storage tank to a certain device, with a required flow rate of $25.8 \, \mathrm{m}^{3}/\mathrm{h}$ . The liquid inlet of the device is 19.5 m above the liquid surface of the storage tank; the pressure above the liquid surface inside the device is 314 kPa (gauge), and the external atmospheric pressure is 98.1 kPa. The head loss in the pipeline is 10.3 m. Calculate the shaft power of this pump's piston while assuming a 15% liquid leakage and a total efficiency of 72% is ().

Options:
A. 8.67kW
B. 5.14kW
C. 8.13kW
D. 6.05kW
E. 5.83kW
F. 7.92kW
G. 6.49kW
H. 5.36kW
I. 4.91kW
J. 7.25kW
Answer: E
A certain liquid flows as laminar flow within a circular straight pipe. If the length of the pipe and the properties of the liquid remain unchanged, but the diameter of the pipe is reduced to half of its original size, how many times greater is the energy loss due to flow resistance compared to the original?

Options:
A. 10 times
B. 1 time
C. 8 times
D. 64 times
E. 12 times
F. 4 times
G. 32 times
H. 16 times
I. 20 times
J. 2 times
Answer: H
The water depth in a rectangular channel is 0.5 m, with a flow velocity of $6 \mathrm{m/s}$. What is the critical depth relative to this flow rate?

Options:
A. 1.107m
B. 0.824m
C. 0.937m
D. 0.890m
E. 1.015m
F. 1.056m
G. 0.653m
H. 0.972m
I. 0.759m
J. 1.202m
Answer: H
Given that the coefficient of variation of the material strength of a certain component is $C_{r}=0.08$, and the coefficient of variation of stress is $C_{s}=0.10$, if the required reliability of this component is 95%, try to estimate the random safety factor of this component.

Options:
A. $$
1 \leq n \leq2. 5 4 4
$$
B. $$
1 \leq n \leq3. 0 0 8
$$
C. $$
1 \leq n \leq3. 1 1 0
$$
D. $$
1 \leq n \leq4. 2 2 1
$$
E. $$
1 \leq n \leq2. 7 3 2
$$
F. $$
1 \leq n \leq3. 6 5 5
$$
G. $$
1 \leq n \leq3. 4 7 7
$$
H. $$
1 \leq n \leq2. 8 9 9
$$
I. $$
1 \leq n \leq4. 9 0 0
$$
J. $$
1 \leq n \leq4. 0 5 6
$$
Answer: F
Calibrate the concentration of the $\mathrm{N a_{2} S_{2} O_{3}}$ solution using the standard $\mathbf{K}_{2} \mathbf{C r}_{2} \mathbf{O}_{2}$. Conduct six parallel measurements and obtain concentrations of 0.1033, 0.1060, 0.1035, 0.1031, 0.1022, and 0.1037 $mol/L$. What is the confidence interval at a 95% confidence level for the above six measured values?

Options:
A. $$
0. 1 0 3 7 \pm0. 0 0 1 5
$$
B. $$
0. 1 0 3 2 \pm0. 0 0 1 0
$$
C. $$
0. 1 0 3 4 \pm0. 0 0 1 2
$$
D. $$
0. 1 0 4 0 \pm0. 0 0 1 9
$$
E. $$
0. 1 0 3 0 \pm0. 0 0 1 8
$$
F. $$
0. 1 0 3 6 \pm0. 0 0 1 4
$$
G. $$
0. 1 0 3 5 \pm0. 0 0 1 1
$$
H. $$
0. 1 0 3 8 \pm0. 0 0 1 3
$$
I. $$
0. 1 0 3 9 \pm0. 0 0 1 7
$$
J. $$
0. 1 0 3 3 \pm0. 0 0 1 6
$$
Answer: F
Determination of the inorganic phosphorus content (mmol/L) in a person's serum sample using spectrophotometry. The results of 8 parallel measurements are: 1.25, 1.34, 1.28, 1.17, 1.33, 1.24, 1.31, 1.32$. If the standard value of inorganic phosphorus content in the serum of a normal adult is 1.20 mmol/L, is the phosphorus content in this sample considered abnormal ($P=95\%$)?

Options:
A. The determination of unusual phosphorus in the given blood sample is $4 5 \%$.
B. The likelihood of inorganic phosphorus irregularity in this person's blood sample is $9 5 \%$.
C. The probability of a phosphorus imbalance in this serum analysis is $6 5 \%$.
D. The inference about phosphorus deviation in this blood investigation is $3 5 \%$.
E. The conclusion regarding phosphorus excess in this serum examination is $8 0 \%$.
F. The perception of phosphate irregularity in this particular blood sample is $7 5 \%$.
G. The assessment of phosphorus anomaly in this individual's blood test is $5 9 \%$.
H. The discernment of a phosphorus issue within this serum analysis is $6 0 \%$.
I. The estimation of phosphorus discrepancy in this test sample is $7 0 \%$.
J. The evaluation of phosphorous disturbance in the test results is $8 5 \%$.
Answer: B
Weigh out 0.5490g of acetylsalicylic acid sample, add 50.00ml of 0.1660mol/L NaOH, and boil:
$HOOCC_6H_4OCOCH_3 + 2NaOH = {CH}_3COONa + C_6H_4OHCOONa + H_2O$
The excess alkali was titrated with 27.14ml HCl. It is known that 1.00ml HCl is equivalent to $0.03814g \mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 10 \mathrm{H}_{2} \mathrm{O}$, with a titer $T = 0.03814mg/ml$. Calculate the percentage content of the acetylsalicylic acid is().
($M_{acetylsalicylic acid} = 180.16, \, K_{a_1} = 1.0 \times 10^{-3}, \, K_{a_2} = 2.4 \times 10^{-12}, \, M_{Borax} = 381.4$)

Options:
A. 46.13%
B. 46.78%
C. 45.47%
D. 48.02%
E. 48.89%
F. 47.65%
G. 47.12%
H. 49.53%
I. 48.45%
J. 45.99%
Answer: G
The dissociation constants of EDTA are known to be $10^{-0.9}, 10^{-1.6}, 10^{-2.0}, 10^{-2.67}, 10^{-6.16}$, and $10^{-10.26}$. Therefore, in a solution with $\text{pH 2.67-6.16}$, the predominant form of EDTA is ( ).

Options:
A. $$
\mathrm{H_{5} Y^{+}}
$$
B. $$
\mathrm{Y^{4-}}
$$
C. $$
\mathrm{H_{3} Y^{+}}
$$
D. $$
\mathrm{H_{9} Y^{5+}}
$$
E. $$
\mathrm{H_{7} Y^{3+}}
$$
F. $$
\mathrm{H_{8} Y^{4+}}
$$
G. $$
\mathrm{H_{6} Y^{2+}}
$$
H. $$
\mathrm{H_{4} Y}
$$
I. $$
\mathrm{H_{2} Y^{2-}}
$$
J. $$
\mathrm{HY^{3-}}
$$
Answer: I
A sample mixture of pure $\mathrm{LiCl}$ and $\mathrm{BaBr_{2}}$ weighing 0.7000 g is treated with 45.15 ml of 0.2017 mol/L $\mathrm{AgNO_3}$ standard solution. The excess $\mathrm{AgNO_3}$ is titrated back using 25.00 ml of 0.1000 mol/L $\mathrm{NH_{4}SCN}$ with ferric ammonium alum as the indicator. What is the content of $\mathrm{BaBr_{2}}$ in the sample?

Options:
A. $$
7 9. 6 5 \%
$$
B. $$
7 4. 8 9 \%
$$
C. $$
8 6. 1 0 \%
$$
D. $$
8 3. 9 4 \%
$$
E. $$
9 2. 5 6 \%
$$
F. $$
7 1. 4 2 \%
$$
G. $$
8 8. 7 3 \%
$$
H. $$
6 6. 9 8 \%
$$
I. $$
9 0. 2 7 \%
$$
J. $$
8 5. 3 1 \%
$$
Answer: D
Known absorption wavelengths for electronic transitions in acetone are 188 nm and 279 nm. What is the energy of the electronic transitions?(expressed in joules and electron volts).

Options:
A. $$
E_{1}=5.80(eV)
$$
$$
E_{2}=4.30(eV)
$$
B. $$
E_{1}=6.75(eV)
$$
$$
E_{2}=4.12(eV)
$$
C. $$
E_{1}=7.10(eV)
$$
$$
E_{2}=5.20(eV)
$$
D. $$
E_{1}=6.00(eV)
$$
$$
E_{2}=3.50(eV)
$$
E. $$
E_{1}=6.55(eV)
$$
$$
E_{2}=4.44(eV)
$$
F. $$
E_{1}=5.12(eV)
$$
$$
E_{2}=3.67(eV)
$$
G. $$
E_{1}=4.99(eV)
$$
$$
E_{2}=3.20(eV)
$$
H. $$
E_{1}=5.55(eV)
$$
$$
E_{2}=4.55(eV)
$$
I. $$
E_{1}=7.30(eV)
$$
$$
E_{2}=6.10(eV)
$$
J. $$
E_{1}=6.03(eV)
$$
$$
E_{2}=5.00(eV)
$$
Answer: E
Using an instrument with $H_0=2.3487T$ to measure $^{19}$F and $^{31}$P, knowing that their gyromagnetic ratios are $2.518\times10^{8}$$T^{- 1}\cdot s^{- 1}$ and $1.0841\times 10^8$ $T^{- 1}\cdot s^{- 1}$ respectively.What are their resonance frequencies?

Options:
A. $$
6 8. 7 3 5 \mathrm{( M H z )}
$$
$$
2 3. 9 8 7 ( \mathrm{M H z} )
$$
B. $$
7 4. 9 8 6 \mathrm{( M H z )}
$$
$$
1 9. 3 2 5 ( \mathrm{M H z} )
$$
C. $$
9 4. 1 2 8 \mathrm{( M H z )}
$$
$$
4 0. 5 2 4 ( \mathrm{M H z} )
$$
D. $$
9 1. 5 8 2 \mathrm{( M H z )}
$$
$$
4 7. 3 0 1 ( \mathrm{M H z} )
$$
E. $$
9 3. 1 0 4 \mathrm{( M H z )}
$$
$$
4 8. 5 6 9 ( \mathrm{M H z} )
$$
F. $$
9 9. 2 4 1 \mathrm{( M H z )}
$$
$$
6 0. 1 5 8 ( \mathrm{M H z} )
$$
G. $$
8 6. 3 4 9 \mathrm{( M H z )}
$$
$$
3 7. 8 9 0 ( \mathrm{M H z} )
$$
H. $$
9 8. 7 5 3 \mathrm{( M H z )}
$$
$$
5 2. 6 3 4 ( \mathrm{M H z} )
$$
I. $$
8 2. 6 7 4 \mathrm{( M H z )}
$$
$$
3 5. 1 4 9 ( \mathrm{M H z} )
$$
J. $$
7 1. 5 2 4 \mathrm{( M H z )}
$$
$$
1 4. 9 7 6 ( \mathrm{M H z} )
$$
Answer: C
There is a gas-liquid chromatography column that is $2 \mathbf{m}$ long. When the linear velocity of the mobile phase is 4.0, 6.0, and 8.0 cm/s respectively, the corresponding theoretical plate numbers are 781, 985, and 1105. What is the number of theoretical plates of the chromatographic columnthe at the optimal linear velocity of the mobile phase?

Options:
A. $$
1. 4 5 \times1 0^{2}
$$
B. $$
1. 1 9 \times1 0^{3}
$$
C. $$
2. 2 2 \times1 0^{3}
$$
D. $$
2. 0 1 \times1 0^{3}
$$
E. $$
1. 7 8 \times1 0^{2}
$$
F. $$
1. 3 4 \times1 0^{3}
$$
G. $$
1. 6 7 \times1 0^{3}
$$
H. $$
9 . 5 0 \times1 0^{2}
$$
I. $$
1. 4 7 \times1 0^{3}
$$
J. $$
1. 0 5 \times1 0^{3}
$$
Answer: B
Determine the content of berberine and coptisine in the alkaloid sample by weighing 0.2000g each of the internal standard, berberine, and coptisine reference substances to prepare a mixed solution. The measured peak areas are 3.60, 3.43, and 4.04 cm', respectively. After weighing 0.2400g of the internal standard and 0.8560g of the sample and preparing the solution by the same method, the peak areas measured under the same chromatographic conditions are 4.16, 3.71, and 4.54 cm°. What is the content of berberine and coptisine in the sample?

Options:
A. $$w_{berberine}(\%)=25.35\%\\w_{coptisine}(\%)=28.10\%$$
B. $$w_{berberine}(\%)=25.50\%\\w_{coptisine}(\%)=28.60\%$$
C. $$w_{berberine}(\%)=25.00\%\\w_{coptisine}(\%)=27.50\%$$
D. $$w_{berberine}(\%)=27.00\%\\w_{coptisine}(\%)=26.40\%$$
E. $$w_{berberine}(\%)=26.75\%\\w_{coptisine}(\%)=26.05\%$$
F. $$w_{berberine}(\%)=27.40\%\\w_{coptisine}(\%)=25.89\%$$
G. $$w_{berberine}(\%)=24.95\%\\w_{coptisine}(\%)=26.14\%$$
H. $$w_{berberine}(\%)=24.80\%\\w_{coptisine}(\%)=27.95\%$$
I. $$w_{berberine}(\%)=26.25\%\\w_{coptisine}(\%)=27.26\%$$
J. $$w_{berberine}(\%)=28.10\%\\w_{coptisine}(\%)=29.35\%$$
Answer: I
There is a rectangular orifice on a clapboard. The height of the orifice is $a=1. 0$ m, and the width is $b=3 \mathrm{m}$. A cylinder with a diameter of $d=2 \mathrm{m}$ was used to block the orifice. Water is filled in both sides of the clapboard. $h=2 \mathrm{m}$ and $z=$ 0. 6m. Calculate the total hydrostatic pressure exerted on the cylinder is ().

Options:
A. $$
9 3. 1 8 \mathrm{~ k N}
$$
B. $$
9 4. 0 4 \mathrm{~ k N}
$$
C. $$
9 4. 7 3 \mathrm{~ k N}
$$
D. $$
1 0 0 . 7 4 \mathrm{~ k N}
$$
E. $$
8 9. 4 5 \mathrm{~ k N}
$$
F. $$
1 0 1 . 0 3 \mathrm{~ k N}
$$
G. $$
9 2. 8 9 \mathrm{~ k N}
$$
H. $$
9 6. 5 6 \mathrm{~ k N}
$$
I. $$
8 8. 5 7 \mathrm{~ k N}
$$
J. $$
9 5. 2 0 \mathrm{~ k N}
$$
Answer: B
A trapezoidal 2 m wide at the bottom and 1 m deep has side slopes 1:1. Determine centre of pressure on the vertical gate closing the channel when it is full of water is ().

Options:
A. 0.62m
B. 1.01m
C. 0.70m
D. 1.10m
E. 0.88m
F. 0.80m
G. 0.95m
H. 1.25m
I. 0.75m
J. 1.15m
Answer: A
Using a V-notch weir ($q_{\mathrm{V}} = 0.316 \sqrt{2g} H^{2.5}$) to measure a flow rate of $q_{\mathrm{V}} = 0.015 \mathrm{m}^{3} / \mathrm{s}$, if there is a 1 mm error in reading the head over the weir, determine the relative error in the calculated flow rate $\frac{\mathrm{d} q_{\mathrm{V}}}{q_{\mathrm{V}}}$ is().

Options:
A. 2.5%
B. 3.0%
C. 5.0%
D. 6.0%
E. 7.0%
F. 1.0%
G. 1.5%
H. 2.0%
I. 0.5%
J. 4.5%
Answer: G
In a triangular cross-section long straight channel, the water depth is $h=1.5 \mathrm{m}$; the side slope coefficient is $m=1.5$, the channel bed slope is $i=0.002$, and the channel wall roughness coefficient is $n=0.015$. Find the flow rate in the channel is ().

Options:
A. $7.75m^{3}/s$
B. $8.90m^{3}/s$
C. $7.15m^{3}/s$
D. $9.10m^{3}/s$
E. $8.50m^{3}/s$
F. $7.35m^{3}/s$
G. $5.80m^{3}/s$
H. $6.95m^{3}/s$
I. $6.20m^{3}/s$
J. $9.55m^{3}/s$
Answer: F
Use Rayleigh's method to analyze the relationship formula for the free-fall distance $S$, which is known to be related to gravitational acceleration $g$ and time $\ell$?

Options:
A. $$
s=k g^{1/2} t^{3}$$
B. $$
s=k g t^{3/2}$$
C. $$
s=k t g^{3/2}$$
D. $$
s=k t^{2} g$$
E. $$
s=k g^{-1} t^{2}$$
F. $$
s=k g t^{2}
$$
G. $$
s=k g^{3} t$$
H. $$
s=k g^{2} t$$
I. $$
s=k g^{-1/2} t^{3}$$
J. $$
s=k g^{1/3} t^{4}$$
Answer: F
An oil reservoir transports petroleum with a kinematic viscosity of $\nu_{\mathrm{p}} = 0.75 \mathrm{cm^2/s}$ through a pipeline with a diameter of $d_{\mathrm{p}} = 250 \mathrm{mm}$ at a flow rate of $q_{\mathrm{V p}} = 140 \mathrm{L/s}$. To determine the minimum oil depth $h_{\mathrm{min p}}$ necessary to prevent air entrainment into the pipeline due to vortex formation, experiments are conducted using a model with a scale ratio of $\lambda_{1} = 5$. Determine the kinematic viscosity $\nu$ and the flow rate $q_{\mathrm{V m}}$ of the liquid in the model is ().

Options:
A. 4.0L/s
B. 1.2L/s
C. 5.0L/s
D. 1.5L/s
E. 0.8L/s
F. 3.5L/s
G. 3.0L/s
H. 2.7L/s
I. 2.5L/s
J. 1.8L/s
Answer: I
A fluid flows steadily through a horizontal, constant-diameter circular pipe of length $\ell$. Given that the friction factor along the path is $f$, the wall shear stress is $\tau$, the cross-sectional average velocity is $\nu$, and the fluid density is $\rho$, find the value of $\tau$ is ().

Options:
A. $\tau={\frac{\lambda}{8}} \omega^{2} \rho$
B. $\tau={\frac{f}{8}} \rho \omega^{2}$
C. $\tau={\frac{\lambda}{6}} \rho v^{2}$
D. $\tau={\frac{f}{12}} \rho \omega^{2}$
E. $\tau={\frac{\lambda}{4}} \omega v \rho$
F. $\tau={\frac{\lambda} {8}} \rho v^{2}$
G. $\tau={\frac{\lambda}{10}} \rho v^{2}$
H. $\tau={\frac{f}{8}} \omega \rho v$
I. $\tau={\frac{f}{6}} \rho v^{3}$
J. $\tau={\frac{f}{4}} \rho v^{3}$
Answer: F
Using a seamless steel pipe with a diameter $d = 200 \mathrm{mm}$ and a length $l = 3000 \mathrm{m}$ (roughness $\Delta = 2 \mathrm{mm}$) to transport oil with a density $\rho = 900 \mathrm{kg} / \mathrm{m}^{3}$. Given the flow rate $q_{\mathrm{v}} = 27.8 \mathrm{L} / \mathrm{s}$, and the kinematic viscosity of the oil: winter $\nu = 1.092 \mathrm{cm}^{2} / \mathrm{s}$; summer $\nu = 0.355 \mathrm{cm}^{2} / \mathrm{s}$. Determine the along-the-way head loss $h_{f}$ of the flow is ().

Options:
A. 19.34m
B. 24.35m
C. 25.72m
D. 22.49m
E. 21.57m
F. 26.48m
G. 20.88m
H. 18.67m
I. 23.09m
J. 17.96m
Answer: D
There is a standard complete well with a radius of $r_{0}=0.36 \mathrm{m}$, an aquifer thickness of $H=7 \mathrm{m}$, a soil permeability coefficient $k=0.0016 \mathrm{m/s}$, and a water depth in the well of $h=4 \mathrm{m}$. Try to calculate the discharge of the well is ().

Options:
A. $$
0. 0 2 4 m^{3} / \mathrm{s}
$$
B. $$
0. 0 3 6 m^{3} / \mathrm{s}
$$
C. $$
0. 0 6 4 m^{3} / \mathrm{s}
$$
D. $$
0. 0 2 8 m^{3} / \mathrm{s}
$$
E. $$
0. 0 2 0 m^{3} / \mathrm{s}
$$
F. $$
0. 0 4 2 m^{3} / \mathrm{s}
$$
G. $$
0. 0 1 8 m^{3} / \mathrm{s}
$$
H. $$
0. 0 5 0 m^{3} / \mathrm{s}
$$
I. $$
0. 0 3 0 m^{3} / \mathrm{s}
$$
J. $$
0. 0 1 2 m^{3} / \mathrm{s}
$$
Answer: A
Given an antenna input power of $10W$, a directivity coefficient $D=3$, and an efficiency $\eta_{\bf A}=0.5$, find the magnitude of the electric field at $r=10km$ and the antenna gain $G$, expressed in decibels.

Options:
A. $$
1. 6 8 \ \mathrm{d B}
$$
B. $$
1. 6 2 \ \mathrm{d B}
$$
C. $$
1. 9 0 \ \mathrm{d B}
$$
D. $$
1. 7 4 \ \mathrm{d B}
$$
E. $$
1. 5 4 \ \mathrm{d B}
$$
F. $$
1. 8 4 \ \mathrm{d B}
$$
G. $$
1. 7 6 \ \mathrm{d B}
$$
H. $$
1. 7 0 \ \mathrm{d B}
$$
I. $$
1. 7 8 \ \mathrm{d B}
$$
J. $$
1. 8 0 \ \mathrm{d B}
$$
Answer: G
Given that the transmitter operates at a frequency of $f = 10 \mathrm{GHz}$ with an output power of $P_{0} = 100 \mathrm{W}$, using a BJ-100 rectangular waveguide (filled with air as the dielectric, with dimensions $a \times b = 22.86 \times 10.16 \mathrm{mm}^{2}$) in the fundamental mode, the transmitter power is directed to a parabolic antenna with a diameter of $d = 2 \mathrm{m}$ (aperture efficiency of $\nu = 0.5$). The ratio of the antinode electric field to the node electric field in the waveguide is 2, and the propagation loss in the waveguide is $\mathrm{1 \, dB}$. At a distance of 10 km from the transmitting antenna, a horn antenna with an aperture size of $20 \times 10 \mathrm{cm^{2}}$ (aperture efficiency of $\nu = 0.4$) is precisely aligned to receive the signal. Determine the maximum field strength of the traveling wave in the rectangular waveguide at the receiving end .

Options:
A. $$
1. 9 8 \mathrm{V} / \mathrm{m}
$$
B. $$
1. 0 6 \mathrm{V} / \mathrm{m}
$$
C. $$
1. 2 3 \mathrm{V} / \mathrm{m}
$$
D. $$
1. 4 5 \mathrm{V} / \mathrm{m}
$$
E. $$
1. 0 0 \mathrm{V} / \mathrm{m}
$$
F. $$
1. 8 2 \mathrm{V} / \mathrm{m}
$$
G. $$
1. 3 4 \mathrm{V} / \mathrm{m}
$$
H. $$
1. 7 1 \mathrm{V} / \mathrm{m}
$$
I. $$
1. 5 7 \mathrm{V} / \mathrm{m}
$$
J. $$
1. 0 9 \mathrm{V} / \mathrm{m}
$$
Answer: B
An elliptically polarized wave traveling in the $z$ direction in the air has components in the $x$ and $y$ directions.
$$
E_{x}=3 \sin ( \omega t-k x ) \,, \mathrm{( V m^{-1} )}
$$
$$
E_{y}=6 \sin ( \omega\, t-k x+7 5^{\circ} ) \,, \mathrm{( V m^{-1} )}
$$
Find: the average power transmitted by the wave per unit area,

Options:
A. $$
7 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
B. $$
7 0 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
C. $$
9 0 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
D. $$
5 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
E. $$
8 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
F. $$
2 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
G. $$
6 0 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
H. $$
5 0 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
I. $$
4 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
J. $$
3 5 \mathrm{m} \mathrm{W} \mathrm{m}^{-2}
$$
Answer: G
Given that the maximum value of the first sidelobe of a certain antenna is 0.01 of the maximum value of the main lobe, find the decibel value of the first sidelobe level ($FSLL$). Also, given that $D=100$ for this antenna, find the value of $D(\theta)$ in the direction of the first sidelobe.

Options:
A. $$
- 1 5 \mathrm{d B},0.07
$$
B. $$
- 4 0 \mathrm{d B},0.01
$$
C. $$
- 1 0 \mathrm{d B},0.04
$$
D. $$
- 2 0 \mathrm{d B},0.06
$$
E. $$
- 4 0 \mathrm{d B},0.02
$$
F. $$
- 3 0 \mathrm{d B},0.02
$$
G. $$
- 3 5 \mathrm{d B},0.03
$$
H. $$
- 5 0 \mathrm{d B},0.03
$$
I. $$
- 2 5 \mathrm{d B},0.05
$$
J. $$
- 4 5 \mathrm{d B},0.05
$$
Answer: B
Within the range of ±45° on the equatorial plane, what percentage of the total radiation power is the radiation power of the electric fundamental oscillator?

Options:
A. 89.47%
B. 72.10%
C. 81.65%
D. 85.30%
E. 78.93%
F. 92.74%
G. 67.51%
H. 84.22%
I. 95.17%
J. 88.39%
Answer: J
The standard $3 \, \text{cm}$ rectangular waveguide $a \times b = 22.86 \times 10.16 \, \text{mm}^2$ is to be connected to a rectangular waveguide with an equivalent impedance of $100 \, \Omega$ through a quarter-wavelength transformer (their dimension $a$ is the same). Find the dimension $b^{\prime}$ of the transformer section when $\lambda = 3.2 \, \text{cm}$.

Options:
A. 4.8(mm)
B. 5.6(mm)
C. 8.2(mm)
D. 6.6(mm)
E. 7.6(mm)
F. 13.8(mm)
G. 12.6(mm)
H. 9.0(mm)
I. 11.4(mm)
J. 10.2(mm)
Answer: D
Using a $\mathrm{BJ-100} (a \times b = 22.86 \times 10.16 \mathrm{mm}^{2})$ rectangular waveguide as a feed line to transmit $\mathrm{TE}_{10}$ waves. If the breakdown field strength of the air is $3 \times 10^{6} \mathrm{V/m}$ and the breakdown field strength when filled with a gas of $\varepsilon_{r}=2.25$ is $6 \times 10^{6} \mathrm{V/m}$, what is the power capacity of the waveguide under both conditions?

Options:
A. $$
\approx3 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx2. 9 \times1 0^{6} ( \mathbf{W} )
$$
B. $$
\approx5 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx3. 4 \times1 0^{6} ( \mathbf{W} )
$$
C. $$
\approx2 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx1. 7 \times1 0^{6} ( \mathbf{W} )
$$
D. $$
\approx6 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx6. 1 \times1 0^{6} ( \mathbf{W} )
$$
E. $$
\approx4 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx5. 2 \times1 0^{6} ( \mathbf{W} )
$$
F. $$
\approx7 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx6. 8 \times1 0^{6} ( \mathbf{W} )
$$
G. $$
\approx10 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx9. 1 \times1 0^{6} ( \mathbf{W} )
$$
H. $$
\approx8 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx7. 5 \times1 0^{6} ( \mathbf{W} )
$$
I. $$
\approx9 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx8. 6 \times1 0^{6} ( \mathbf{W} )
$$
J. $$
\approx7. 5 \times1 0^{\mathfrak{s}} \left( \mathbf{W} \right)
$$
$$
\approx7. 2 \times1 0^{6} ( \mathbf{W} )
$$
Answer: F
The air-filled rectangular waveguide has cross-sectional dimensions of $a \times b = 23 \, \text{mm} \times 10 \, \text{mm}$; if a probe excites the $\text{TE}_{10}$ mode within it, the frequency of the signal source is $f = 10.87 \, \text{GHz}$. If the electric field of the excited $\text{TE}_{10}$ wave is $E_{y} = 50 \sin \left( \frac{\pi}{a} x \right) e^{-\mathrm{i} \beta z} \, (\text{V/m})$, find the maximum value of the longitudinal surface current density on the waveguide wall.

Options:
A. $$
1 0 6. 1 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
B. $$
7 9. 2 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
C. $$
9 8. 4 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
D. $$
1 0 0. 0 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
E. $$
1 1 5. 3 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
F. $$
8 5. 6 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
G. $$
1 2 7. 5 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
H. $$
1 1 0. 9 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
I. $$
1 3 3. 7 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
J. $$
9 2. 1 \times1 0^{-3} \quad( \mathrm{A / m} )
$$
Answer: A
A two-element antenna array consisting of two half-wave dipoles, where one has a feed current of $I_{m1}$, and the other element is passive (short-circuited). The two antennas are spaced $\lambda/4$ apart. Find the directivity factor of this antenna.

Options:
A. 3.4
B. 2.9
C. 3.7
D. 4.8
E. 3.1
F. 4.5
G. 5.8
H. 3.6
I. 3.5
J. 5.0
Answer: C
The wind speed at a distance of 8,000 meters from the typhoon center is 13.33 m/s, and the barometer reading is 98,200 Pa. What are the wind speed and wind pressure at a distance of 800 meters from the typhoon center? Assume the flow field is point vortex-induced flow (air density is $1.29 \mathrm{kg}/\mathrm{m}^{3}$).

Options:
A. $$
1 5 6. 4 {\mathrm{m / s}},\ 79000 \mathrm{P a}
$$
B. $$
1 3 9. 8 {\mathrm{m / s}},\ 89000 \mathrm{P a}
$$
C. $$
1 8 0. 1 {\mathrm{m / s}},\ 81000 \mathrm{P a}
$$
D. $$
1 3 3. 3 {\mathrm{m / s}},\ 86853 \mathrm{P a}
$$
E. $$
1 4 2. 5 {\mathrm{m / s}},\ 85000 \mathrm{P a}
$$
F. $$
1 6 8. 9 {\mathrm{m / s}},\ 80000 \mathrm{P a}
$$
G. $$
1 1 9. 2 {\mathrm{m / s}},\ 92000 \mathrm{P a}
$$
H. $$
1 5 0. 7 {\mathrm{m / s}},\ 78000 \mathrm{P a}
$$
I. $$
1 2 3. 7 {\mathrm{m / s}},\ 91000 \mathrm{P a}
$$
J. $$
1 0 7. 3 {\mathrm{m / s}},\ 94000 \mathrm{P a}
$$
Answer: D
A 45 kN weight is dropped from an airplane, with the requirement that the landing speed does not exceed $10 \mathrm{m} / \mathrm{s}$. The weight is attached to a parachute with a drag coefficient of $C_{\mathrm{D}}=2$. Ignoring the weight of the parachute, and assuming the air density is $\rho=1.2 \mathrm{kg} / \mathrm{m}^{3}$, what should be the diameter of the parachute?

Options:
A. $$
d \geqslant1 9 . 5 0 \mathrm{m}
$$
B. $$
d \geqslant1 4 . 5 9 \mathrm{m}
$$
C. $$
d \geqslant1 6 . 7 5 \mathrm{m}
$$
D. $$
d \geqslant3 2 . 9 4 \mathrm{m}
$$
E. $$
d \geqslant2 0 . 7 2 \mathrm{m}
$$
F. $$
d \geqslant2 3 . 4 1 \mathrm{m}
$$
G. $$
d \geqslant1 5 . 8 8 \mathrm{m}
$$
H. $$
d \geqslant2 1. 8 5 \mathrm{m}
$$
I. $$
d \geqslant2 6 . 3 8 \mathrm{m}
$$
J. $$
d \geqslant1 8 . 6 3 \mathrm{m}
$$
Answer: H
The volumetric flow rate of oil through the pipeline is $q_{V} = 1000 \, \mathrm{m}^{3}/\mathrm{h}$, with a kinematic viscosity $\nu = 1 \times 10^{-5} \, \mathrm{m}^{2}/\mathrm{s}$. The length of the pipeline is $l = 200 \, \mathrm{m}$, and the absolute roughness is $\Delta = 0.046 \, \mathrm{mm}$. The maximum allowable head loss is $h_{1} = 20 \, \mathrm{m}$. What should be the diameter $d$ of the new low carbon steel pipeline?

Options:
A. $$
d_{g}=3 8 0 \mathrm{m m}
$$
B. $$
d_{g}=5 0 0 \mathrm{m m}
$$
C. $$
d_{g}=4 0 0 \mathrm{m m}
$$
D. $$
d_{g}=1 5 0 \mathrm{m m}
$$
E. $$
d_{g}=3 0 0 \mathrm{m m}
$$
F. $$
d_{g}=2 0 0 \mathrm{m m}
$$
G. $$
d_{g}=3 5 0 \mathrm{m m}
$$
H. $$
d_{g}=4 5 0 \mathrm{m m}
$$
I. $$
d_{g}=2 5 0 \mathrm{m m}
$$
J. $$
d_{g}=4 2 0 \mathrm{m m}
$$
Answer: E
The pressure of the air in the high-pressure tank is $2.5 \times 10^{5} \, \text{Pa}$, with a density of $2.64 \, \text{kg/m}^{3}$, and a temperature of 330K. A converging nozzle is attached to the wall of the container, with an outlet area $A_{2} = 20 \, \text{cm}^{2}$. The back pressure at the outlet is $\phi_{\mathrm{b}} = 10^{5} \, \text{Pa}$. What is the mass flow rate $q_{m}$ of the nozzle?

Options:
A. $$
1. 1 1 2 2 \mathrm{k g} / \mathrm{s}
$$
B. $$
3. 1 6 4 8 \mathrm{k g} / \mathrm{s}
$$
C. $$
0. 2 3 8 5 \mathrm{k g} / \mathrm{s}
$$
D. $$
2. 7 5 9 0 \mathrm{k g} / \mathrm{s}
$$
E. $$
1. 9 8 7 4 \mathrm{k g} / \mathrm{s}
$$
F. $$
4. 0 1 5 6 \mathrm{k g} / \mathrm{s}
$$
G. $$
2. 3 0 1 2 \mathrm{k g} / \mathrm{s}
$$
H. $$
2. 6 3 2 1 \mathrm{k g} / \mathrm{s}
$$
I. $$
0. 9 4 5 7 \mathrm{k g} / \mathrm{s}
$$
J. $$
1. 4 8 6 3 \mathrm{k g} / \mathrm{s}
$$
Answer: A
Let $X(t) = A \cos \beta t + B \sin \beta t \,(-\infty < t < +\infty)$, where $A \sim N(0, \sigma^{2})$ and $B \sim N(0, \sigma^{2})$ are independent of each other, and $\beta$ is a constant. Find the one-dimensional probability density $f(t, x)$ of the stochastic process $\{X(t), -\infty < t < +\infty\}$.

Options:
A. $$
f ( t, x )=\frac{1}{\sqrt{2 \pi} \sigma} \mathrm{e}^{-\frac{t^{2}}{2 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
B. $$
f ( t, x )=\frac{1}{\sqrt{5 \pi} \sigma} \mathrm{e}^{-\frac{t^{2}}{5 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
C. $$
f ( t, x )=\frac{1}{\sqrt{3 \pi} \sigma} \mathrm{e}^{-\frac{x^{2}}{3 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
D. $$
f ( t, x )=\frac{1}{\sqrt{2 \pi} \sqrt{\sigma}} \mathrm{e}^{-\frac{t^{2}}{4 \sigma}}, \quad x \in \mathbb{R}
$$
E. $$
f ( t, x )=\frac{2}{\sqrt{\pi} \sigma} \mathrm{e}^{-\frac{t^{2}}{\sigma^{2}}}, \quad x \in \mathbb{R}
$$
F. $$
f ( t, x )=\frac{1}{3\pi \sigma} \mathrm{e}^{-\frac{x^{2}}{8 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
G. $$
f ( t, x )=\frac{1}{\pi \sigma} \mathrm{e}^{-\frac{t^{2}}{\pi \sigma^{2}}}, \quad x \in \mathbb{R}
$$
H. $$
f ( t, x )=\frac{1}{2\pi \sigma} \mathrm{e}^{-\frac{x^{2}}{4 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
I. $$
f ( t, x )=\frac{3}{\sqrt{2 \pi} \sigma} \mathrm{e}^{-\frac{x^{2}}{6 \sigma^{2}}}, \quad x \in \mathbb{R}
$$
J. $$
f ( t, x )=\frac{1}{\sqrt{2 \pi} \sigma} \mathrm{e}^{-\frac{x^{2}}{2 \sigma^{2}}}, \quad t \in \mathbb{R}
$$
Answer: A
Given $\{N(t), t \geq 0\}$ is a Poisson process with parameter $\lambda$, find the two-dimensional probability distribution is ().

Options:
A. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{s \leq t} {j \geqslant k} P \left( N ( s )=j, N ( t )-N ( s )=j-k \right)=\frac{\lambda^{j} s^{k} ( t-s )^{j-k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda s}
$$
B. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{s < t} {j \geq k} P \left( N ( s )=j, N ( t )-N ( s )=k-j \right)=\frac{\lambda^{k} t^{j} ( s-t )^{k-j}} {j! ( k-j ) \,!} \mathrm{e}^{-\lambda t}
$$
C. $$
P ( N ( s )=k, N ( t )=j ) \, \frac{t \geq s} {k \geqslant j} P \left( N ( s )=k, N ( t )-N ( s )=j-k \right)=\frac{\lambda^{k} t^{j} ( s-t )^{k-j}} {j! ( k-j ) \,!} \mathrm{e}^{-\lambda t}
$$
D. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{s < t} {j \leqslant k} P \left( N ( s )=j, N ( t )-N ( s )=k-j \right)=\frac{\lambda^{k} t^{j} ( t-s )^{k-j}} {j! ( k-j ) \,!} \mathrm{e}^{-\lambda t}
$$
E. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{t \leq s} {j \leqslant k} P \left( N ( s )=j, N ( t )-N ( s )=k+j \right)=\frac{\lambda^{j} t^{k} ( s-t )^{j+k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda s}
$$
F. $$
P ( N ( s )=k, N ( t )=j ) \, \frac{t \leq s} {k \leqslant j} P \left( N ( s )=k, N ( t )-N ( s )=j-k \right)=\frac{\lambda^{j} s^{k} ( t-s )^{j-k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda s}
$$
G. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{s \geq t} {j \geqslant k} P \left( N ( s )=j, N ( t )-N ( s )=k-j \right)=\frac{\lambda^{j} t^{k} ( s-t )^{j-k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda s}
$$
H. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{s \leq t} {j \leqslant k} P \left( N ( s )=j, N ( t )-N ( s )=k+j \right)=\frac{\lambda^{k} t^{j} ( t-s )^{k+j}} {j! ( k-j ) \,!} \mathrm{e}^{-\lambda t}
$$
I. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{t \leq s} {j \leqslant k} P \left( N ( t )=j, N ( s )-N ( t )=k-j \right)=\frac{\lambda^{j} t^{k} ( s-t )^{j-k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda s}
$$
J. $$
P ( N ( s )=j, N ( t )=k ) \, \frac{t < s} {k < j} P \left( N ( s )=j, N ( t )-N ( s )=j-k \right)=\frac{\lambda^{j} s^{k} ( s-t )^{j-k}} {k! ( j-k ) \,!} \mathrm{e}^{-\lambda t}
$$
Answer: D
Let $\mathbf{x}$ be a four-dimensional zero-mean Gaussian random vector, with its covariance matrix $Λ_{X}$. Their expressions are as follows:
$$
\mathbf{X}={\left[ \begin{matrix} {{X_{1}}} \\ {{X_{2}}} \\ {{X_{3}}} \\ {{X_{4}}} \\ \end{matrix} \right]}, \qquad\mathbf{Λ}_{\mathbf{X}}={\left[ \begin{matrix} {{\lambda_{1 1}}} & {{\lambda_{1 2}}} & {{\lambda_{1 3}}} & {{\lambda_{1 4}}} \\ {{\lambda_{2 1}}} & {{\lambda_{3 2}}} & {{\lambda_{2 3}}} & {{\lambda_{3 4}}} \\ {{\lambda_{3 1}}} & {{\lambda_{3 2}}} & {{\lambda_{3 3}}} & {{\lambda_{3 4}}} \\ {{\lambda_{4 1}}} & {{\lambda_{4 2}}} & {{\lambda_{4 3}}} & {{\lambda_{4 4}}} \\ \end{matrix} \right]}
$$ Find $E \{X_{1}^{2} X_{2}^{2} \}$ is ().

Options:
A. $$
\lambda_{1 1} \lambda_{3 3}+\lambda_{1 2} \lambda_{4 4}
$$
B. $$
\lambda_{1 2} \lambda_{3 3} + \lambda_{2 2} \lambda_{4 4}
$$
C. $$
2 \lambda_{2 3}^{2}+\lambda_{1 1} \lambda_{1 2}
$$
D. $$
2 \lambda_{1 2}^{2}+\lambda_{1 1} \lambda_{2 2}
$$
E. $$
\lambda_{1 4}^{2}+\lambda_{2 3} \lambda_{4 4}
$$
F. $$
2 \lambda_{2 2}^{2}+\lambda_{1 1} \lambda_{2 3}
$$
G. $$
\lambda_{1 1} \lambda_{2 4}+\lambda_{3 2} \lambda_{4 4}
$$
H. $$
\lambda_{3 3}^{2}+\lambda_{1 1} \lambda_{2 2}
$$
I. $$
\lambda_{1 3}^{2}+\lambda_{2 2} \lambda_{3 3}
$$
J. $$
2 \lambda_{1 3}^{2}+\lambda_{1 4} \lambda_{2 2}
$$
Answer: D
Assuming the different arrival times of a given renewal counting process follow an exponential distribution
$$
F_{z} ( t )=\left\{\begin{cases} {{{1-\mathrm{e}^{-\lambda t}}}} & {{{t \geq0}}} \\ {{{0,}}} & {{{t < 0}}} \\ \end{cases} \right.
$$
Determine the characteristic function of the $j$-th arrival time is ().

Options:
A. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{\lambda t}\sum_{i=0}^{t-1}\frac{(t)^i}{i!},&t\geqslant0\\\\0,&t<0\end{cases}$$
B. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=0}^{t}\frac{(\lambda t)^{i-1}}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
C. $$F_{T_i}(t)=\begin{cases}\mathrm{e}^{-\lambda t}\sum_{i=0}^{t-1}\frac{(\lambda)^i}{t!},&t\geqslant0\\\\0,&t<0\end{cases}$$
D. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=0}^{t-2}\frac{(\lambda t)^i}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
E. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=0}^{t-1}\frac{(t)^i}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
F. $$F_{T_i}(t)=\begin{cases}1-\lambda t\sum_{i=0}^{t-1}\frac{\mathrm{e}^{-\lambda t}}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
G. $$F_{T_i}(t)=\begin{cases}\mathrm{e}^{-\lambda t}\sum_{i=1}^{t}\frac{(\lambda t)^i}{i!},&t\geqslant0\\\\1,&t<0\end{cases}$$
H. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=1}^{t-1}\frac{(\lambda t)^i}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
I. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=0}^{t-1}\frac{(\lambda t)^i}{t!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
J. $$F_{T_i}(t)=\begin{cases}1-\mathrm{e}^{-\lambda t}\sum_{i=0}^{t-1}\frac{(\lambda t)^i}{i!},&t\geqslant0\\\\0 ,&t<0\end{cases}$$
Answer: J
Calls arriving at the exchange (with infinitely many channels) follow a Poisson process with an arrival rate of $\lambda$, and each call duration follows an exponential distribution with a mean of $1 / \mu$. Let $X(t)$ be the number of calls in progress at time $t$. What is the probability that there are no calls in the process?

Options:
A. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\mu^2} {\lambda} ( 1-\mathrm{e}^{-\lambda t} ) \Bigr]}}} \\ \end{array}
$$
B. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\mu} {\lambda} ( 1-\mathrm{e}^{-\lambda t} ) \Bigr]}}} \\ \end{array}
$$
C. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\lambda} {\mu} ( 1-\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
D. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\mu} {\lambda} ( 1+\mathrm{e}^{-\lambda t} ) \Bigr]}}} \\ \end{array}
$$
E. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\lambda \mu ( 1-\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
F. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\lambda+\mu} {\lambda} ( 1-\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
G. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\lambda} {\mu} ( 1+\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
H. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{1} {\mu} ( 1-\mathrm{e}^{-\lambda t} ) \Bigr]}}} \\ \end{array}
$$
I. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\lambda \cdot t ( 1-\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
J. $$
\begin{array} {c} {{{P \{X ( t )=0 \}=\operatorname{e x p} \Bigl[-\frac{\lambda^2} {\mu} ( 1-\mathrm{e}^{-\mu t} ) \Bigr]}}} \\ \end{array}
$$
Answer: C
Three-phase half-wave controlled rectifier circuit with back electromotive force load: to ensure continuous current, a reactor with a sufficiently large inductance is connected in series. Given $U_{2} = 100 \, V$, $R_{d} = 1 \, \Omega$, $L_{T} = 1 \, mH$, and $E = sov$, find the output voltage $U_{d}$, the current $I_{d}$, and the commutation overlap angle $\gamma$ when $\alpha = 30^{\circ}$. The answer is ().

Options:
A. $$7.2V,39.50A,13.14°$$
B. $$5.8V,43.50A,9.25°$$
C. $$5.4V,47.75A,14.92°$$
D. $$6.5V,40.00A,14.50°$$
E. $$8.0V,42.15A,10.31°$$
F. $$5.9V,45.80A,11.50°$$
G. $$6.1V,41.90A,12.85°$$
H. $$6.7V,44.63A,11.28°$$
I. $$7.0V,46.00A,15.00°$$
J. $$7.3V,48.25A,8.63°$$
Answer: H
A construction company undertook a tunnel project with a length of 700 meters, using an anchor net shotcrete support design. Geological data indicated that a fault would be exposed at 200 meters into the tunnel. To ensure the pullout resistance of the anchor bolts, technical measures called for the use of two resin cartridges (one rapid and one medium-speed cartridge) to secure the anchor bolts. To ensure the safe and reliable passage through the fault during construction, after obtaining approval from the design unit, technicians changed the support form in the fault section to a combined support of anchor net shotcrete and steel frames, with a spacing of 0.8 meters between the steel frames.
When the tunnel construction reached the fault section, quality control personnel discovered during a check of the tunnel support quality that the rock surfaces were uneven, and there were many rock chips falling around the openings of the anchor bolt holes. Although the anchor plates could still tightly adhere to the rock surface, many anchor bolts were excessively protruding, measuring $200 \sim 250 \mathrm{mm}$. Which of the following is the reason for the excessive protrusion of the tunnel anchor bolts?

Options:
A. The depth of the anchor bolt hole has not been inspected.
B. The resin cartridges were expired, reducing their efficacy.
C. Engineers installed the steel frames earlier than required.
D. Structural engineers neglected changes in weather impacts.
E. Insufficient training for workers on tunnel support systems.
F. The type of concrete mix for shotcrete was unsuitable here.
G. Incorrect diameter of resin cartridges was selected.
H. Anchor bolts were positioned at a wrong angle improperly.
I. The anchor bolt length calculations were incorrect always.
J. Geological surveys were omitted during initial planning.
Answer: A
At $912^\circ C$, the volume of the unit cell of bcc iron is $0.02464 \mathrm{~nm}^{3}$, while at the same temperature, the volume of the unit cell of fcc iron is $0.0486 \mathrm{~nm}^{3}$. What is the percentage change in density when iron transitions from bcc to fcc? It is known that the relative atomic mass of $\mathrm{Fe}$ is $55.85$.

Options:
A. $$
\approx0. 9 8 7 \%
$$
B. $$
\approx2. 5 0 1 \%
$$
C. $$
\approx1. 7 2 8 \%
$$
D. $$
\approx1. 3 9 9 \%
$$
E. $$
\approx0. 6 5 4 \%
$$
F. $$
\approx3. 4 5 6 \%
$$
G. $$
\approx1. 1 5 0 \%
$$
H. $$
\approx2. 1 3 2 \%
$$
I. $$
\approx3. 0 4 9 \%
$$
J. $$
\approx4. 9 8 3 \%
$$
Answer: D
It is known that when $\mathrm{Cu}$ is heated from 300 °C to 1000 °C, the equilibrium concentration of vacancies in the crystal increases by a factor of $1.36 \times 10^{5}$ at 300 °C. Try to calculate the vacancy formation energy $E_{V}$ in the Cu crystal.

Options:
A. $$
9. 1 \times1 0^{-1 9} \ \mathrm{J}
$$
B. $$
5. 8 \times1 0^{-1 9} \ \mathrm{J}
$$
C. $$
1. 7 \times1 0^{-1 9} \ \mathrm{J}
$$
D. $$
6. 4 \times1 0^{-1 9} \ \mathrm{J}
$$
E. $$
7. 9 \times1 0^{-1 9} \ \mathrm{J}
$$
F. $$
4. 6 \times1 0^{-1 9} \ \mathrm{J}
$$
G. $$
3. 5 \times1 0^{-1 9} \ \mathrm{J}
$$
H. $$
6. 2 \times1 0^{-1 9} \ \mathrm{J}
$$
I. $$
8. 3 \times1 0^{-1 9} \ \mathrm{J}
$$
J. $$
2. 4 \times1 0^{-1 9} \ \mathrm{J}
$$
Answer: C
The tensile axis direction of the iron single crystal is [011]. If $\tau_{c} = 33.8 \mathrm{~MPa}$, what stress is required for the material to yield?

Options:
A. $$
\approx6 4. 9 \mathrm{M P a}
$$
B. $$
\approx7 2. 6 \mathrm{M P a}
$$
C. $$
\approx9 1. 5 \mathrm{M P a}
$$
D. $$
\approx8 0. 1 \mathrm{M P a}
$$
E. $$
\approx8 2. 8 \mathrm{M P a}
$$
F. $$
\approx9 5. 7 \mathrm{M P a}
$$
G. $$
\approx9 8. 4 \mathrm{M P a}
$$
H. $$
\approx7 5. 3 \mathrm{M P a}
$$
I. $$
\approx6 1. 2 \mathrm{M P a}
$$
J. $$
\approx8 4. 7 \mathrm{M P a}
$$
Answer: E
Given a thick-walled pipe with an inner diameter of 30 mm, separated by an iron membrane with a thickness of 0.1 mm. Nitrogen gas is introduced into the pipe from one end to maintain a nitrogen concentration of $1200 \ \mathrm{mol}/\mathrm{m}^3$ on one side of the membrane, while the nitrogen concentration on the other side is $100 \ \mathrm{mol}/\mathrm{m}^3$. If the nitrogen flow rate through the pipe is measured to be $2.8 \times 10^{-8} \ \mathrm{mol/s}$ at 700°C, calculate the diffusion coefficient of nitrogen in iron at this condition.

Options:
A. $$
6. 3 \times1 0^{-1 3} \mathrm{~ m^{2} / s}
$$
B. $$
8. 4 \times1 0^{-1 1} \mathrm{~ m^{2} / s}
$$
C. $$
3. 8 \times1 0^{-1 2} \mathrm{~ m^{2} / s}
$$
D. $$
5. 5 \times1 0^{-1 2} \mathrm{~ m^{2} / s}
$$
E. $$
5. 1 \times1 0^{-1 1} \mathrm{~ m^{2} / s}
$$
F. $$
1. 9 \times1 0^{-1 2} \mathrm{~ m^{2} / s}
$$
G. $$
4. 9 \times1 0^{-1 4} \mathrm{~ m^{2} / s}
$$
H. $$
3. 6 \times1 0^{-1 2} \mathrm{~ m^{2} / s}
$$
I. $$
2. 4 \times1 0^{-1 3} \mathrm{~ m^{2} / s}
$$
J. $$
7. 2 \times1 0^{-1 0} \mathrm{~ m^{2} / s}
$$
Answer: H
Using the Cu $K_{a} (\lambda=0.154056 \ \mathrm{nm})$ X-ray, the diffraction pattern of the quenched 2Cr13 martensitic stainless steel sample was measured. The $2 \theta$ values corresponding to the four diffraction peaks were measured as $44.479^{\circ}, 64.740^{\circ}, 82.000^{\circ}, 98.519^{\circ}$, respectively. Calculate its lattice constant $a$.

Options:
A. $a\approx0.3411(nm)$
B. $a\approx0.2555(nm)$
C. $a\approx0.3722(nm)$
D. $a\approx0.2999(nm)$
E. $a\approx0.4231(nm)$
F. $a\approx0.2877(nm)$
G. $a\approx0.1955(nm)$
H. $a\approx0.3802(nm)$
I. $a\approx0.3018(nm)$
J. $a\approx0.3125(nm)$
Answer: F
The angle between the tensile axis and the slip direction of the zinc single crystal before stretching is $45^{\circ}$. After stretching, the angle between the force axis and the slip direction is $30^{\circ}$. Try to calculate the elongation of the specimen after stretching.

Options:
A. $$
\approx3 7. 5 \%
$$
B. $$
\approx4 3. 3 \%
$$
C. $$
\approx4 1. 4 \%
$$
D. $$
\approx4 5. 1 \%
$$
E. $$
\approx3 9. 8 \%
$$
F. $$
\approx4 8. 2 \%
$$
G. $$
\approx4 7. 6 \%
$$
H. $$
\approx4 2. 7 \%
$$
I. $$
\approx5 0. 0 \%
$$
J. $$
\approx3 5. 9 \%
$$
Answer: C
Assuming $\Delta G_{\mathrm{B}}, \gamma, E$ are all constants. Try to derive the nucleation work $\Delta G^{*}$ for a cubic crystal nucleus.

Options:
A. $$
\frac{1 9 \gamma^{5}} {6 ( \triangle G_{\mathrm{B}}+E_{3} ) \rho_{0}^{1}}
$$
B. $$
\frac{1 1 \gamma^{1}} {1 ( \triangle G_{\mathrm{B}}-E_{1} ) \rho_{0}^{0}}
$$
C. $$
\frac{1 7 \gamma^{7}} {9 ( \triangle G_{\mathrm{B}}+E ) \rho_{0}^{5}}
$$
D. $$
\frac{1 5 \gamma^{6}} {8 ( \triangle G_{\mathrm{B}}+2E ) \rho_{0}^{4}}
$$
E. $$
\frac{1 3 \gamma^{3}} {7 ( \triangle G_{\mathrm{B}}-E_{2} ) \rho_{0}^{3}}
$$
F. $$
\frac{1 4 \gamma^{8}} {2 ( \triangle G_{\mathrm{B}}+E_{2} ) \rho_{0}^{6}}
$$
G. $$
\frac{1 8 \gamma^{2}} {5 ( \triangle G_{\mathrm{B}}-2E ) \rho_{0}^{2}}
$$
H. $$
\frac{1 6 \gamma^{3}} {3 ( \triangle G_{\mathrm{B}}-E_{3} ) \rho_{0}^{2}}
$$
I. $$
\frac{1 2 \gamma^{4}} {4 ( \triangle G_{\mathrm{B}}-E ) \rho_{0}^{3}}
$$
J. $$
\frac{1 0 \gamma^{0}} {0 ( \triangle G_{\mathrm{B}}-2E_{3} ) \rho_{0}^{7}}
$$
Answer: H
Given a transistor with $\alpha_{0}=0.96$ and $f_{a}=5 \text{MHz}$. Calculate the operating frequency when $\alpha$ decreases to 0.6.

Options:
A. $$
= 8. 3 1 5 \mathrm{z M H z}
$$
B. $$
= 4. 9 3 5 \mathrm{z M H z}
$$
C. $$
= 4. 7 2 5 \mathrm{z M H z}
$$
D. $$
= 9. 0 6 0 \mathrm{z M H z}
$$
E. $$
= 5. 3 4 5 \mathrm{z M H z}
$$
F. $$
= 5. 1 0 0 \mathrm{z M H z}
$$
G. $$
= 5. 8 8 0 \mathrm{z M H z}
$$
H. $$
= 7. 8 2 5 \mathrm{z M H z}
$$
I. $$
= 6. 5 7 0 \mathrm{z M H z}
$$
J. $$
= 6. 2 4 5 \mathrm{z M H z}
$$
Answer: J
A single-tube power amplifier operates in an ambient temperature environment of $\boldsymbol{T}_{a} = 40^{\circ} \mathbf{C}$, with the transistor's maximum junction temperature $T_{j_{M}} = 100^{\circ} \mathbf{C}$. The thermal resistance from the chip to the casing is $R_{j c}=1.5^{\circ} \mathbf{C} / \mathbf{W}$, and the maximum allowable power dissipation at $25^{\circ} \mathbf{C}$ is $P_{C M} = 20 \mathbf{W}$. Determine the maximum distortion-free output power $P_{L \operatorname*{max}}$ that the load can obtain, and the size of the heatsink that should be selected.

Options:
A. $$
= 7 \mathbf{W}
$$
A 2.0mm thick copper heat sink with an area of 470cm² can be placed horizontally.
B. $$
= 15 \mathbf{W}
$$
A 3.5mm thick steel heat sink with an area of 260cm² can be placed vertically.
C. $$
= 6 \mathbf{W}
$$
A 2.5mm thick copper heat sink with an area of 420cm² can be placed horizontally.
D. $$
= 5 \mathbf{W}
$$
A 1.0mm thick copper heat sink with an area of 620cm² can be placed vertically.
E. $$
= 10 \mathbf{W}
$$
A 3.0mm copper heat sink with an area of 350cm² can be placed horizontally.
F. $$
= 8 \mathbf{W}
$$
A 2.5mm thick aluminum heat sink with an area of 420cm² can be placed vertically.
G. $$
= 12 \mathbf{W}
$$
A 1.5mm thick steel heat sink with an area of 550cm² can be placed vertically.
H. $$
= 11 \mathbf{W}
$$
A 2.0mm thick aluminum heat sink with an area of 500cm² can be placed horizontally.
I. $$
= 9 \mathbf{W}
$$
A 3.0mm thick aluminum heat sink with an area of 300cm² can be placed vertically.
J. $$
= 13 \mathbf{W}
$$
A 3.0mm thick plastic heat sink with an area of 390cm² can be placed horizontally.
Answer: F
When the load resistance $R_{\mathrm{L}}=1 \ \ \mathrm{k} \Omega$, the output voltage of the voltage amplifier circuit decreases by 20% compared to when the load is open-circuited ($R_{\mathrm{L}}=\infty)$. Find the output resistance $R$ of this amplifier circuit.

Options:
A. $$
7 5 0 \Omega
$$
B. $$
3 0 0 \Omega
$$
C. $$
5 0 0 \Omega
$$
D. $$
6 0 0 \Omega
$$
E. $$
2 0 0 \Omega
$$
F. $$
2 5 0 \Omega
$$
G. $$
3 5 0 \Omega
$$
H. $$
1 5 0 \Omega
$$
I. $$
1 0 0 \Omega
$$
J. $$
4 0 0 \Omega
$$
Answer: F
At room temperature (300 K), if a diode is applied a forward voltage of 0.7 V, the resulting forward current is 1 mA. Determine the reverse saturation current of the diode.

Options:
A. $$
\approx4 \times1 0^{-1 5} \mathrm{~ A}
$$
B. $$
\approx9 \times1 0^{-1 6} \mathrm{~ A}
$$
C. $$
\approx1 \times1 0^{-1 4} \mathrm{~ A}
$$
D. $$
\approx3 \times1 0^{-1 5} \mathrm{~ A}
$$
E. $$
\approx1 \times1 0^{-1 6} \mathrm{~ A}
$$
F. $$
\approx5 \times1 0^{-1 7} \mathrm{~ A}
$$
G. $$
\approx2 \times1 0^{-1 5} \mathrm{~ A}
$$
H. $$
\approx7 \times1 0^{-1 7} \mathrm{~ A}
$$
I. $$
\approx2 \times1 0^{-1 6} \mathrm{~ A}
$$
J. $$
\approx6 \times1 0^{-1 5} \mathrm{~ A}
$$
Answer: G
Solve the following problem: Given an NPN-type BJT with $I_{\mathrm{EQ}} = 0.1 \ \ \mathrm{mA}$ and $I_{\mathrm{ES}} = 6 \times 10^{-15} \ \mathrm{mA}$, where the base width modulation effect can be ignored. Find the value of $V_{\mathrm{BEQ}}$.

Options:
A. $$
\approx0. 8 0 \mathrm{~ V}
$$
B. $$
\approx1. 0 3 \mathrm{~ V}
$$
C. $$
\approx0. 6 8 \mathrm{~ V}
$$
D. $$
\approx0. 5 9 \mathrm{~ V}
$$
E. $$
\approx0. 9 1 \mathrm{~ V}
$$
F. $$
\approx0. 7 9 \mathrm{~ V}
$$
G. $$
\approx0. 6 4 \mathrm{~ V}
$$
H. $$
\approx0. 7 8 \mathrm{~ V}
$$
I. $$
\approx0. 7 5 \mathrm{~ V}
$$
J. $$
\approx0. 7 1 \mathrm{~ V}
$$
Answer: F