MikaStars39/nano-eval huggingface dataset:
Prove the stream function equation for ideal incompressible fluid under body forces with potential energy in a plane steady flow: $\nabla^{z}\psi = -f(\psi).$ Also, consider the flow as $x\to\infty$ where $u=cy^2, v=0, \frac{\partial v}{\partial x}=0,$ with $c$ as a constant. the stream function equation for this flow is ().

Options:
A. $\nabla^{2}\psi=(22c^{3}\psi)^{\frac{1}{3}}$
B. $\nabla^{2}\psi=(30c^{2}\psi)^{\frac{1}{5}}$
C. $\nabla^{2}\psi=(10c^{2}\psi)^{\frac{1}{6}}$
D. $\nabla^{2}\psi=(20c^{4}\psi)^{\frac{1}{9}}$
E. $\nabla^{2}\psi=(14c^{2}\psi)^{\frac{1}{7}}$
F. $\nabla^{2}\psi=(24c^{2}\psi)^{\frac{1}{3}}$
G. $\nabla^{2}\psi=(16c^{5}\psi)^{\frac{1}{4}}$
H. $\nabla^{2}\psi=(12c^{3}\psi)^{\frac{1}{2}}$
I. $\nabla^{2}\psi=(18c^{4}\psi)^{\frac{1}{3}}$
J. $\nabla^{2}\psi=(28c^{6}\psi)^{\frac{1}{8}}$
Answer: F
There is a first-order cross-section combustion chamber where the known fuel-to-air ratio of the injected mixture is 1:29. The parameters of the mixed gas are $T_{1}=333.3 \, \text{K} \,, \, \, \, \, p_{1}=5516 \, \text{kN/m}^{2} \,, \, \, \, V_{1}=73.15 \, \text{m/s}$, with the fuel's calorific value at 41867 kJ/kg, and combustion is complete. Assuming that the thermodynamic properties of the mixed gas before and after combustion are identical to those of air and neglecting wall friction, the $p_{01}-p_{02}$ is ().

Options:
A. $$
8. 5 3 \mathrm{k N / m^{2}}
$$
B. $$
10. 2 9 \mathrm{k N / m^{2}}
$$
C. $$
11. 4 6 \mathrm{k N / m^{2}}
$$
D. $$
9. 6 1 \mathrm{k N / m^{2}}
$$
E. $$
6. 9 7 \mathrm{k N / m^{2}}
$$
F. $$
5. 8 8 \mathrm{k N / m^{2}}
$$
G. $$
4. 2 3 \mathrm{k N / m^{2}}
$$
H. $$
12. 0 5 \mathrm{k N / m^{2}}
$$
I. $$
7. 4 2 \mathrm{k N / m^{2}}
$$
J. $$
3. 7 5 \mathrm{k N / m^{2}}
$$
Answer: A
A factory has installed a cast iron water supply pipe that is 500 meters long with a diameter of 200 millimeters and a flow rate of 100 liters per second. The water source is a well 5 meters deep; the inlet is equipped with a water filter screen (ζ=5.2), and the pipeline contains 10 elbows at 90° (ζ=0.48), and two gates (ζ=0.08). The kinematic viscosity, v, is 1.3 mm²/s, and the roughness coefficient, Kc, is 1.3 mm.The height of the pump head is ().

Options:
A. 48.6m
B. 49.7m
C. 47.9m
D. 53.3m
E. 57.1m
F. 54.2m
G. 50.4m
H. 55.8m
I. 52.1m
J. 60.5m
Answer: D
Transonic airfoil flow is to be initially analyzed using a water tunnel test through air-water analogy. To simulate an airflow with $M_{\infty}$=1.1,a water depth of 50mm is chosen for the water tunnel test. The water depth at the stagnation point $( \zeta+h )$ is ().

Options:
A. 0.1057m
B. 0.0803m
C. 0.0704m
D. 0.1451m
E. 0.1158m
F. 0.0906m
G. 0.0605m
H. 0.1259m
I. 0.1552m
J. 0.1350m
Answer: B
The absolute pressure of the compressed air in the air storage tank of the power workshop is $p = 10.8 \times 10^{3} \, \mathrm{Pa}$, and the temperature $t = 30°C$. The storage tank consists of a cylindrical section and two hemispherical ends. Given $l = 3\,\mathrm{m}$, $d = 2\,\mathrm{m}$, and the specific gas constant for the compressed air, $R = 287\,\mathrm{J/(kg \cdot K)}$, The mass of the compressed air in the tank is ().

Options:
A. 170kg
B. 155kg
C. 180kg
D. 175kg
E. 190kg
F. 150kg
G. 200kg
H. 185kg
I. 165kg
J. 160kg
Answer: A
The diameter of a horizontal pipeline suddenly expands from $d_{1} = 24 \mathrm{~cm}$ to $d_{2} = 48 \mathrm{~cm}$. A pressure measurement pipe is installed before and after the sudden expansion, and it is observed that the water column in the pressure measurement pipe after the local resistance is higher than that before the local resistance by $h = 1 \mathrm{~cm}$. The flow rate $q_{V}$ in the pipe is ().

Options:
A. $q_{V}$=$0.0283{\mathrm{m}}^{3} / {\mathrm{s}}$
B. $q_{V}$=$0.0189{\mathrm{m}}^{3} / {\mathrm{s}}$
C. $q_{V}$=$0.0398{\mathrm{m}}^{3} / {\mathrm{s}}$
D. $q_{V}$=$0.0225{\mathrm{m}}^{3} / {\mathrm{s}}$
E. $q_{V}$=$0.0250{\mathrm{m}}^{3} / {\mathrm{s}}$
F. $q_{V}$=$0.0452{\mathrm{m}}^{3} / {\mathrm{s}}$
G. $q_{V}$=$0.0361{\mathrm{m}}^{3} / {\mathrm{s}}$
H. $q_{V}$=$0.0556{\mathrm{m}}^{3} / {\mathrm{s}}$
I. $q_{V}$=$0.0327{\mathrm{m}}^{3} / {\mathrm{s}}$
J. $q_{V}$=$0.0507{\mathrm{m}}^{3} / {\mathrm{s}}$
Answer: I
A circular plane with a radius of $R$ is uniformly positively charged, and its surface charge density is $\sigma$. In the middle of this circular plane, a small circular hole with a radius $a$, which is extremely small compared to $R$, has been drilled. Assuming $l \gg a$, what is the electric field intensity at point $G$, located on the axis through the center of the hole and a distance $l$ from the center $O$?

Options:
A. $$
E=\frac{\delta} {3 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l^{4} \,+R^{2} \, )^{\frac{1} {2}}} ]
$$
B. $$
E=\frac{\sigma} {2 \varepsilon_{0}} [ \, 1 \,-\frac{a} {( \, l^{3} \,+R^{2} \, )^{\frac{1} {2}}} ]
$$
C. $$
E=\frac{\sigma} {2 \varepsilon_{0}} [ \, 1 \,-\frac{a} {( \, l^{2} \,+R \, )^{\frac{1} {2}}} ]
$$
D. $$
E=\frac{\delta} {2 \varepsilon_{0}} [ \, 1 \,-\frac{l} {( \, a^{2} \,+R^{2} \, )^{\frac{1} {2}}} ]
$$
E. $$
E=\frac{\delta} {2 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l^{2} \,+R^{2} \, )^{\frac{1} {2}}} ]
$$
F. $$
E=\frac{\sigma} {2 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l \,+R^{2} \, )^{\frac{3} {2}}} ]
$$
G. $$
E=\frac{\sigma} {3 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l^{2} \,+R^{2} \, )^{\frac{1} {3}}} ]
$$
H. $$
E=\frac{\delta} {3 \varepsilon_{0}} [ \, 1 \,-\frac{a} {( \, l^{2} \,+R^{2} \, )^{\frac{1} {2}}} ]
$$
I. $$
E=\frac{\sigma} {4 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l^{2} \,+R^{3} \, )^{\frac{1} {2}}} ]
$$
J. $$
E=\frac{\delta} {4 \varepsilon_{0}} [ \, 1 \,-\frac{1} {( \, l^{2} \,+a^{2} \, )^{\frac{1} {2}}} ]
$$
Answer: E
A battery pack consisting of $N$ batteries is used to power an external circuit with a resistance of $R$. Each battery has an electromotive force of $E_{0}$ and an internal resistance of $r_{0}$. The battery pack is composed of several identical groups of batteries connected in series, and each group is composed of several batteries connected in parallel. To achieve the maximum current intensity in the circuit, how many battery packs $n$ and how many batteries per group $m$ should there be?

Options:
A. $$
\begin{array} {l} {{n={\sqrt{\frac{N r_{0}} {R}}}}} \\ {{m={\sqrt{\frac{N R} {r_{0}}}}}} \\ \end{array}
$$
B. $$
\begin{array} {l} {{n={\sqrt{\frac{r_{0} N} {R}}}}} \\ {{m={\sqrt{\frac{R N} {r_{0}}}}}} \\ \end{array}
$$
C. $$
\begin{array} {l} {{n={\sqrt{\frac{N} {r_{0} R}}}}} \\ {{m={\sqrt{\frac{N} {R r_{0}}}}}} \\ \end{array}
$$
D. $$
\begin{array} {l} {{n={\frac{N}{R^{2}}}}} \\ {{m={\frac{N}{r_{0}^{2}}}}} \\ \end{array}
$$
E. $$
\begin{array} {l} {{n={\frac{\sqrt{N R}} {r_{0}}}}} \\ {{m={\frac{\sqrt{N r_{0}}} {R}}}} \\ \end{array}
$$
F. $$
\begin{array} {l} {{n={\sqrt{\frac{N R} {r_{0}}}}}} \\ {{m={\sqrt{\frac{N r_{0}} {R}}}}} \\ \end{array}
$$
G. $$
\begin{array} {l} {{n={\frac{R N} {r_{0}^2}}}} \\ {{m={\frac{r_{0} R} {N^2}}}} \\ \end{array}
$$
H. $$
\begin{array} {l} {{n={\frac{R} {N r_{0}}}}} \\ {{m={\frac{r_{0}} {N R}}}} \\ \end{array}
$$
I. $$
\begin{array} {l} {{n={\frac{N R} {r_{0}}}}} \\ {{m={\frac{N r_{0}} {R}}}} \\ \end{array}
$$
J. $$
\begin{array} {l} {{n={\sqrt{\frac{R} {N r_{0}}}}}} \\ {{m={\sqrt{\frac{r_{0}} {N R}}}}} \\ \end{array}
$$
Answer: F
Heat loss due to thermal conduction is negligible. The specific heat capacity of lead can be considered constant at $c_{1} = 130 \; \mathrm{J} / (\mathrm{kg} \cdot \mathrm{K})$, while the specific heat capacity of copper is $c_{2} = 388 \; \mathrm{J} / (\mathrm{kg} \cdot \mathrm{K})$. The resistivity of lead is $\rho_{1} = 22 \times 10^{-6} \, (\Omega \cdot \mathrm{cm})$, and the resistivity of copper is $\rho_{2} = 1.7 \times 10^{-6} \, (\Omega \cdot \mathrm{cm})$. The melting point of lead is $T_{1} = 327~\mathrm{^\circ C}$, and before the short circuit, the temperature of the conductor is $T_{0} = 20~\mathrm{^\circ C}$. The density of lead is $d_{1} = 11.34 \times 10^{3} \, \mathrm{kg/m}^{3}$, and the density of copper is $d_{2} = 8.9 \times 10^{3} \, \mathrm{kg/m}^{3}$. A copper wire with a cross-sectional area of $S_{1} = 2 \, \mathrm{mm}^{2}$ is used to create the circuit, connected with a lead fuse with a cross-sectional area of $S_{2} = 0.2 \, \mathrm{mm}^{2}$. The current intensity during the short circuit reaches $I = 30 \, \mathrm{A}$. How much time will it take for the fuse to begin melting after the short circuit? How many degrees will the temperature of the copper wire increase by then?

Options:
A. $$0.01 \mathrm{~s},\ 0.7^{\circ}\mathrm{C}$$
B. $$0.18 \mathrm{~s},\ 5.0^{\circ}\mathrm{C}$$
C. $$0.03 \mathrm{~s},\ 2.3^{\circ}\mathrm{C}$$
D. $$0.21 \mathrm{~s},\ 4.1^{\circ}\mathrm{C}$$
E. $$
0. 0 9 {\mathrm{~ s}},\ 0.1^{\circ}\mathrm{C}
$$
F. $$0.06 \mathrm{~s},\ 0.4^{\circ}\mathrm{C}$$
G. $$0.12 \mathrm{~s},\ 3.8^{\circ}\mathrm{C}$$
H. $$0.15 \mathrm{~s},\ 1.7^{\circ}\mathrm{C}$$
I. $$0.30 \mathrm{~s},\ 1.5^{\circ}\mathrm{C}$$
J. $$0.27 \mathrm{~s},\ 3.2^{\circ}\mathrm{C}$$
Answer: E
In a small hydroelectric power station, water flows into the turbine at a speed of 3 m/s and exits at a speed of 1 m/s. The outflow water level is 1.6 m lower than the inflow water level, and the water flow rate is 1 m³/s. If 75% of the water's energy is provided to the generator and 20% raises the water's temperature, with the generator output voltage at 240 V and the transmission line resistance at $16 \frac{2}{3} \; \Omega$, assuming power consumption is 5%, and the required voltage for users is 220 V, what are the ratios of the primary to secondary coil turns of the step-up and step-down transformers used?

Options:
A. $\frac{2}{25},\ \frac{95}{10}$
B. $\frac{4}{25},\ \frac{95}{12}$
C. $\frac{3}{23},\ \frac{98}{11}$
D. $\frac{3}{30},\ \frac{96}{11}$
E. $\frac{3}{29},\ \frac{97}{11}$
F. $\frac{3}{25},\ \frac{95}{11}$
G. $\frac{3}{27},\ \frac{92}{11}$
H. $\frac{3}{24},\ \frac{93}{11}$
I. $\frac{3}{26},\ \frac{91}{11}$
J. $\frac{3}{28},\ \frac{94}{11}$
Answer: F
When sunlight is perpendicular to the Earth's surface, each square meter of the ground receives solar power at 1.35 kW. It is known that the current mass of the sun is $2.0 \times 10^{30} \mathrm{kg}$. If this mass is converted into solar energy according to Einstein's mass-energy equivalence formula $E=mc^{2}$ and is radiated outward at the current power, how many years can the sun sustain by consuming one percent of its current mass?

Options:
A. $$
1. 8 \times1 0^{1 1}
$$
B. $$
2. 2 \times1 0^{1 1}
$$
C. $$
2. 5 \times1 0^{1 1}
$$
D. $$
4. 7 \times1 0^{1 1}
$$
E. $$
0. 6 \times1 0^{1 1}
$$
F. $$
1. 5 \times1 0^{1 1}
$$
G. $$
3. 3 \times1 0^{1 1}
$$
H. $$
1. 0 \times1 0^{1 1}
$$
I. $$
1. 2 \times1 0^{1 1}
$$
J. $$
3. 9 \times1 0^{1 1}
$$
Answer: F
The input tuning circuit of a medium-wave radio consists of a variable capacitor $C$ in series with a loop coil $L$. The range of $C$ is from 30 pF to 300 pF. $L$ is made by winding an insulated fine wire around a ferrite ring. The ferrite is known to have a relative permeability $\mu_{r} = 200$, a magnetic path length of 3.0 cm, and a cross-sectional area of 0.10 cm². The wire's cross-sectional diameter is 0.50 mm, with a resistivity of $1.7 \times 10^{-8} \; \Omega \cdot m$. If you want to tune into radio stations in the 500 kHz to 1500 kHz band, how many turns should $L$ have?

Options:
A. $60$
B. $70$
C. $65$
D. $50$
E. $83$
F. $55$
G. $100$
H. $75$
I. $90$
J. $85$
Answer: C
At a certain frequency, the impedance ratio of the capacitor $C$ to the resistor $R$ is $Z_{C} : Z_{R} = 3 : 4$. Now, after connecting them in series, apply an AC voltage $U = 100 \mathrm{V}$ at this frequency. What is the phase difference $\varphi$ between the current $I$ and the voltage $U$?

Options:
A. $$
- 2 3^{\circ}
$$
B. $$
- 5 0^{\circ}
$$
C. $$
- 4 5^{\circ}
$$
D. $$
- 1 9^{\circ}
$$
E. $$
- 2 5^{\circ}
$$
F. $$
- 5 5^{\circ}
$$
G. $$
- 2 8^{\circ}
$$
H. $$
- 6 0^{\circ}
$$
I. $$
- 3 7^{\circ}
$$
J. $$
- 4 0^{\circ}
$$
Answer: I
According to Bohr's theory, when a hydrogen atom is in its ground state, its electron moves in a circular orbit of radius $a = 5.29 \times 10^{-11} \mathrm{m}$ around the hydrogen nucleus (proton). It is known that the mass of the electron is $m = 9.11 \times 10^{-31} \, \mathrm{kg}$ and the charge is $e = -1.60 \times 10^{-19} \mathrm{C}$. What is the energy density of the magnetic field produced at the center of the orbit by this motion of the electron?

Options:
A. $$
5. 8 9 \times1 0^{8}~\mathrm{J / m^{3}}
$$
B. $$
9. 0 2 \times1 0^{5}~\mathrm{J / m^{3}}
$$
C. $$
4. 5 7 \times1 0^{4}~\mathrm{J / m^{3}}
$$
D. $$
7. 3 4 \times1 0^{6}~\mathrm{J / m^{3}}
$$
E. $$
3. 6 8 \times1 0^{3}~\mathrm{J / m^{3}}
$$
F. $$
2. 5 1 \times1 0^{1}~\mathrm{J / m^{3}}
$$
G. $$
8. 4 9 \times1 0^{2}~\mathrm{J / m^{3}}
$$
H. $$
7. 3 0 \times1 0^{0}~\mathrm{J / m^{3}}
$$
I. $$
1. 1 5 \times1 0^{9}~\mathrm{J / m^{3}}
$$
J. $$
6. 2 1 \times1 0^{7}~\mathrm{J / m^{3}}
$$
Answer: J
Given that $B = | B | = 5.0 \, \mathrm{Gs}$, the self-inductance of the loop is $L = \mu_{0} R \left( \ln \frac{8R}{a} - 1.75 \right)$. A superconducting wire with a cross-sectional radius of $a = 1.0 \, \mathrm{mm}$ is used to form the loop, and the loop's radius is $R = 25.0 \, \mathrm{mm}$. When the temperature is higher than the critical temperature $T_{c}$, the loop is placed in a uniform magnetic field with a magnetic flux density of $B$, such that the loop's axis is parallel to $B$. The current inside the loop is zero. Then, the temperature is lowered below $T_{c}$, and the magnetic field is removed. What is the current in the loop?

Options:
A. $$
2. 2~\mathrm{A}
$$
B. $$
15. 0~\mathrm{A}
$$
C. $$
10. 2~\mathrm{A}
$$
D. $$
9. 4~\mathrm{A}
$$
E. $$
8. 8~\mathrm{A}
$$
F. $$
6. 8~\mathrm{A}
$$
G. $$
3. 3~\mathrm{A}
$$
H. $$
0. 5~\mathrm{A}
$$
I. $$
12. 5~\mathrm{A}
$$
J. $$
5. 5~\mathrm{A}
$$
Answer: E
In an electronic induction accelerator, the acceleration time of the electron is $4.2\mathrm{ms}$. The maximum magnetic flux within the electron's orbit is $1.8\mathrm{Wb}$. If the orbit radius is $\mathrm{84cm}$, how much distance has it covered during the acceleration process?

Options:
A. $$
4. 0 \times1 0^{3}~\mathrm{k m}
$$
B. $$
7. 8 \times1 0^{3}~\mathrm{k m}
$$
C. $$
0. 5 \times1 0^{3}~\mathrm{k m}
$$
D. $$
9. 0 \times1 0^{3}~\mathrm{k m}
$$
E. $$
6. 8 \times1 0^{3}~\mathrm{k m}
$$
F. $$
3. 4 \times1 0^{3}~\mathrm{k m}
$$
G. $$
1. 2 \times1 0^{3}~\mathrm{k m}
$$
H. $$
8. 2 \times1 0^{3}~\mathrm{k m}
$$
I. $$
5. 6 \times1 0^{3}~\mathrm{k m}
$$
J. $$
2. 6 \times1 0^{3}~\mathrm{k m}
$$
Answer: G
A large ferromagnetic medium with a relative permeability of $\mu_{\mathrm{r}} = 200$ has a magnetic field with a magnetic flux density $B = 2.0 \, \text{T}$ inside. There are two small cavities within it: one is a thin disk shape with both surfaces perpendicular to $B$, and the other is a slender needle shape parallel to $B$. What are the magnetic field strengths $H$ at the centers of these two cavities, respectively?

Options:
A. $$
1. 6 \times1 0^{6}~\mathrm{A / m},~8. 0 \times1 0^{3}~\mathrm{A / m}
$$
B. $$
1. 4 \times1 0^{6}~\mathrm{A / m},~9. 1 \times1 0^{3}~\mathrm{A / m}
$$
C. $$
2. 4 \times1 0^{6}~\mathrm{A / m},~7. 6 \times1 0^{3}~\mathrm{A / m}
$$
D. $$
2. 0 \times1 0^{6}~\mathrm{A / m},~9. 0 \times1 0^{3}~\mathrm{A / m}
$$
E. $$
1. 2 \times1 0^{6}~\mathrm{A / m},~9. 5 \times1 0^{3}~\mathrm{A / m}
$$
F. $$
1. 7 \times1 0^{5}~\mathrm{A / m},~8. 1 \times1 0^{3}~\mathrm{A / m}
$$
G. $$
1. 5 \times1 0^{6}~\mathrm{A / m},~8. 8 \times1 0^{3}~\mathrm{A / m}
$$
H. $$
1. 6 \times1 0^{5}~\mathrm{A / m},~8. 5 \times1 0^{3}~\mathrm{A / m}
$$
I. $$
1. 9 \times1 0^{5}~\mathrm{A / m},~8. 3 \times1 0^{3}~\mathrm{A / m}
$$
J. $$
1. 8 \times1 0^{5}~\mathrm{A / m},~9. 2 \times1 0^{3}~\mathrm{A / m}
$$
Answer: A
A spherical coil is formed by closely winding a thin wire with insulated surface along a fixed diameter on the surface of a sphere with radius $R$, with a number of turns per unit length $n$ along the diameter, and $n$ is the same everywhere. When a current $I$ flows through the wire, what is the magnetic field intensity $B$ at a point located at a distance $r$ from the center of the sphere (where $r > R$) on the diameter?

Options:
A. $$
B=\frac{3} {2} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{3} e_{I} \,, \quad r > R
$$
B. $$
B=\frac{1} {3} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{2} e_{I} \,, \quad r > R
$$
C. $$
B=\frac{4} {3} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{2} e_{I} \,, \quad r > R
$$
D. $$
B=\frac{2} {3} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{3} e_{I} \,, \quad r > R
$$
E. $$
B=\frac{2} {7} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{4} e_{I} \,, \quad r > R
$$
F. $$
B=\frac{2} {5} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{4} e_{I} \,, \quad r > R
$$
G. $$
B=\frac{2} {3} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{6} e_{I} \,, \quad r > R
$$
H. $$
B=\frac{5} {6} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{5} e_{I} \,, \quad r > R
$$
I. $$
B=\frac{2} {3} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{2} e_{I} \,, \quad r > R
$$
J. $$
B=\frac{3} {4} \mu_{0} \, n I \Big( \frac{R} {r} \Big)^{3} e_{I} \,, \quad r > R
$$
Answer: D
An electronic diode consists of a cylindrical cathode $K$ with a radius of $a = 5.0 \times 10^{-4} \, \mathrm{m}$, and a coaxial cylindrical anode $A$ surrounding $K$, with a radius of $b = 4.5 \times 10^{-3} \, \mathrm{m}$. The potential of $A$ is 300V higher than $K$. When electrons leave $K$, their speed is negligible and can be disregarded. The mass of an electron is known to be $9.1 \times 10^{-31} \, \mathrm{kg}$, and its charge is $-1.6 \times 10^{-19} \, \mathrm{C}$. What is the speed of the electron when it reaches $A$?

Options:
A. $$
8. 0 \times1 0^{7}~\mathrm{m / s}
$$
B. $$
4. 0 \times1 0^{7}~\mathrm{m / s}
$$
C. $$
2. 5 \times1 0^{7}~\mathrm{m / s}
$$
D. $$
1. 2 \times1 0^{7}~\mathrm{m / s}
$$
E. $$
3. 0 \times1 0^{7}~\mathrm{m / s}
$$
F. $$
6. 0 \times1 0^{7}~\mathrm{m / s}
$$
G. $$
1. 0 \times1 0^{7}~\mathrm{m / s}
$$
H. $$
9. 0 \times1 0^{7}~\mathrm{m / s}
$$
I. $$
2. 0 \times1 0^{7}~\mathrm{m / s}
$$
J. $$
1. 5 \times1 0^{7}~\mathrm{m / s}
$$
Answer: G
Saturated steam at 100°C condenses outside the tubes of a shell-and-tube heat exchanger. The overall heat transfer coefficient is $2039 \, \text{W/m}^2 \cdot \text{°C}$, and the heat transfer area is 12.75 $m^2$. Cooling water at 15°C flows through the tubes at a rate of $2.25 \times 10^5 \, \text{kg/h}$. Assuming the mean temperature difference can be calculated using the arithmetic mean, determine the amount of steam condensed is ().

Options:
A. 3500kg/h
B. 3700kg/h
C. 3900kg/h
D. 3332kg/h
E. 3100kg/h
F. 3400kg/h
G. 3200kg/h
H. 3600kg/h
I. 4000kg/h
J. 3800kg/h
Answer: D
The submergence level of a certain drum vacuum filter is $1/3$, with a rotation speed of 2 r/min. It operates under a pressure differential of 50 kPa, and the experimental measured filtration constant is $K=4.24 \times 10^{-5} \, \text{m}^2/\text{s}$. The filter cake is incompressible, and the medium resistance is negligible. If the vacuum degree decreases to 25 kPa while other conditions remain unchanged, what will be its production capacity?

Options:
A. $3.05 {m}^{3} /{h}$
B. $4.12 {m}^{3} /{h}$
C. $3.78 {m}^{3} /{h}$
D. $3.54 {m}^{3} /{h}$
E. $5.23 {m}^{3} /{h}$
F. $4.78 {m}^{3} /{h}$
G. $3.22 {m}^{3} /{h}$
H. $2.48 {m}^{3} /{h}$
I. $2.87 {m}^{3} /{h}$
J. $4.50 {m}^{3} /{h}$
Answer: D
In an experiment to determine the performance of a centrifugal pump using water, when the flow rate is 26 m³/h, the reading on the pump outlet pressure gauge is 152 kPa, and the reading on the vacuum gauge at the pump inlet is 24.8 kPa. The vertical distance between the pressure tapping points of the vacuum gauge and the pressure gauge is $0.4 \mathrm{m}$. The diameters of the pump's inlet and outlet pipes are equal. If the shaft power of the pump under operating conditions is 2.45 kW, try to calculate the mechanical efficiency of the pump at this time is ().

Options:
A. 61.04%
B. 59.72%
C. 53.28%
D. 54.19%
E. 48.15%
F. 49.35%
G. 60.56%
H. 57.89%
I. 50.28%
J. 45.67%
Answer: C
Methane flows through a horizontal reducing pipe with an inner diameter gradually decreasing from 200mm to 100mm. A U-tube differential manometer, with water as the indicating fluid, is connected at the wide and narrow ends of the pipe to measure the flow rate of the methane gas. Assuming no energy loss during the reduction process and that the density of methane is 0.645 $kg/m^{3}$, what is the flow rate of the methane gas in the pipeline when the water level difference on the two sides of the U-tube is 50mm?

Options:
A. $1322m^{3}/h$
B. $1125m^{3}/h$
C. $1138m^{3}/h$
D. $1065m^{3}/h$
E. $980m^{3}/h$
F. $1188m^{3}/h$
G. $1300m^{3}/h$
H. $1450m^{3}/h$
I. $1010m^{3}/h$
J. $1200m^{3}/h$
Answer: C
There is a circular rod made of A6 steel, with an average diameter $d$ mean $\mu_{d}=30 \text{mm}$, and a standard deviation $\sigma_{d}=3 \text{mm}$. The average yield limit $r$ of the circular rod is $\mu_{r}=290 \text{N/mm}^{2}$, with a standard deviation $\sigma_{r}=25 \text{N/mm}^{2}$. When the rod is subjected to an axial tensile force $P = 10^{5} \, \text{N}$ (considered to be constant), determine the reliability of the rod.

Options:
A. 0.9982
B. 0.9967
C. 0.9999
D. 0.9978
E. 0.9986
F. 0.9948
G. 0.9954
H. 0.9971
I. 0.9959
J. 0.9949
Answer: C
After the quenching and tempering treatment of 45# steel, its $H B=2 4 1 \! \sim\! 2 8 6$, with an allowable torsional shear stress of $\left[ \tau\right]=$ $4 4 \! \sim\! 5 4 \mathrm{M P a}$. If the pump shaft is made of this material, the pump power is $N \!=\! \ ( 2 2 \pm2 ) \! \ \ \mathbf{k} \mathbf{W}$, and the rotation speed is 2900 $r/min$. What is the minimum required diameter of the pump shaft? (Ignoring the impact of stress concentration factors)

Options:
A. $$
( 2 9. 5 \! \sim\! 1 6. 2 7 ) \, \mathrm{m m}
$$
B. $$
( 2 5. 1 \! \sim\! 1 9. 4 3 ) \, \mathrm{m m}
$$
C. $$
( 1 5. 9 \! \sim\! 2 1. 5 3 ) \, \mathrm{m m}
$$
D. $$
( 3 2. 0 \! \sim\! 1 4. 5 2 ) \, \mathrm{m m}
$$
E. $$
( 2 3. 7 \! \sim\! 2 0. 1 4 ) \, \mathrm{m m}
$$
F. $$
( 2 8. 2 \! \sim\! 1 7. 9 8 ) \, \mathrm{m m}
$$
G. $$
( 2 7. 8 \! \sim\! 2 5. 3 9 ) \, \mathrm{m m}
$$
H. $$
( 2 2. 6 \! \sim\! 1 5. 1 1 ) \, \mathrm{m m}
$$
I. $$
( 2 0. 8 \! \sim\! 1 8. 8 6 ) \, \mathrm{m m}
$$
J. $$
( 3 5. 4 \! \sim\! 2 7. 6 9 ) \, \mathrm{m m}
$$
Answer: I
Given that a certain method has a standard deviation $\sigma=0.15$ for measuring iron, and the iron content in a sample measured by this method is 12.87%. Suppose this result is obtained from 1 measurement, 5 measurements, or 10 measurements. Calculate the confidence interval for the mean at 95% confidence level in each case is(). knowing that for a 95% confidence level $u=1.96$.

Options:
A. When $n=1$, $\mu = 12.87 \pm 0.28$
When $n=5$, $\mu = 12.87 \pm 0.14$
When $n=10$, $\mu = 12.87 \pm 0.09$
B. When $n=1$, $\mu = 12.87 \pm 0.30$
When $n=5$, $\mu = 12.87 \pm 0.15$
When $n=10$, $\mu = 12.87 \pm 0.07$
C. When $n=1$, $\mu = 12.87 \pm 0.31$
When $n=5$, $\mu = 12.87 \pm 0.12$
When $n=10$, $\mu = 12.87 \pm 0.09$
D. When $n=1$, $\mu = 12.87 \pm 0.28$
When $n=5$, $\mu = 12.87 \pm 0.14$
When $n=10$, $\mu = 12.87 \pm 0.08$
E. When $n=1$, $\mu = 12.87 \pm 0.29$
When $n=5$, $\mu = 12.87 \pm 0.13$
When $n=10$, $\mu = 12.87 \pm 0.09$
F. When $n=1$, $\mu = 12.87 \pm 0.31$
When $n=5$, $\mu = 12.87 \pm 0.13$
When $n=10$, $\mu = 12.87 \pm 0.08$
G. When $n=1$, $\mu = 12.87 \pm 0.27$
When $n=5$, $\mu = 12.87 \pm 0.15$
When $n=10$, $\mu = 12.87 \pm 0.11$
H. When $n=1$, $\mu = 12.87 \pm 0.30$
When $n=5$, $\mu = 12.87 \pm 0.14$
When $n=10$, $\mu = 12.87 \pm 0.10$
I. When $n=1$, $\mu = 12.87 \pm 0.32$
When $n=5$, $\mu = 12.87 \pm 0.13$
When $n=10$, $\mu = 12.87 \pm 0.08$
J. When $n=1$, $\mu = 12.87 \pm 0.29$
When $n=5$, $\mu = 12.87 \pm 0.12$
When $n=10$, $\mu = 12.87 \pm 0.10$
Answer: E
Given that 1 ml of a certain standard HCl solution contains 0.004374 g of HCl, calculate the titer of this standard HCl solution with respect to NaOH.

Options:
A. $$
0. 0 0 3 9 5 0 ( \mathrm{g / m l} )
$$
B. $$
0. 0 0 4 8 0 0 ( \mathrm{g / m l} )
$$
C. $$
0. 0 0 5 0 0 0 ( \mathrm{g / m l} )
$$
D. $$
0. 0 0 4 4 0 0 ( \mathrm{g / m l} )
$$
E. $$
0. 0 0 4 1 5 0 ( \mathrm{g / m l} )
$$
F. $$
0. 0 0 4 3 0 0 ( \mathrm{g / m l} )
$$
G. $$
0. 0 0 4 6 0 0 ( \mathrm{g / m l} )
$$
H. $$
0. 0 0 3 8 0 0 ( \mathrm{g / m l} )
$$
I. $$
0. 0 0 4 0 0 0 ( \mathrm{g / m l} )
$$
J. $$
0. 0 0 5 2 0 0 ( \mathrm{g / m l} )
$$
Answer: B
Using a 0.1000 mol/L NaOH solution to titrate a 0.1000 mol/L formic acid solution, what is the pH at the stoichiometric point?

Options:
A. 5.67
B. 8.23
C. 9.88
D. 12.46
E. 10.11
F. 11.07
G. 7.22
H. 6.35
I. 3.47
J. 4.55
Answer: B
Calculate the potential and potential jump range at the stoichiometric point during the titration of $\mathrm{Sn}^{2+}$ solution with $\mathrm{Fe}^{3+}$ solution in a $1 \text{ mol/L HCl}$ solution. Which redox indicator can be selected to indicate the endpoint? ($\varphi_{\mathrm{Fe^{3+} / Fe^{2+}}}^{\Theta^{\prime}} = 0.70 \mathrm{V}, \varphi_{\mathrm{Sn^{4+} / Sn^{2+}}}^{\Theta^{\prime}} = 0.14 \mathrm{V}$)

Options:
A. Methyl Orange as an Indicator
B. Phenol Red as an Indicator
C. Cresol Red as an Indicator
D. Thymol Blue as an Indicator
E. Methylene Blue as an Indicator
F. Alizarin Yellow as an Indicator
G. Bromothymol Blue as an Indicator
H. Phenolphthalein as an Indicator
I. Congo Red as an Indicator
J. Litmus Paper as an Indicator
Answer: E
0.5000g of phosphate ore sample underwent dissolution, oxidation, and other chemical treatments, during which $\mathrm{P O_{4}^{3-}}$ was precipitated as $\mathrm{MgNH_{4}PO_{4}\cdot 6H_{2}O}$ and then calcined at high temperature to form $\mathbf{M} \mathbf{g}_{2} \mathbf{P}_{2} \mathbf{O}_{7}$, with a mass of 0.2018g. What is the percentage content of $\mathbf{P}_{2} \mathbf{O}_{5}$ in the ore sample?

Options:
A. $$ 21.62 \% $$
B. $$ 26.53 \% $$
C. $$ 23.87 \% $$
D. $$
25.74 \%
$$
E. $$ 27.16 \% $$
F. $$ 30.12 \% $$
G. $$ 19.47 \% $$
H. $$ 24.08 \% $$
I. $$ 28.89 \% $$
J. $$ 22.35 \% $$
Answer: D
Take caffeic acid, dry it at 105°C to a constant weight, and accurately weigh 10.00 mg. Dissolve in a small amount of ethanol, transfer to a 200 ml volumetric flask, and add water to the mark. Take 5.00 ml of this solution and place it in a 50 ml volumetric flask, add 4 ml of 6 mol/L $\mathrm{HCl}$, and add water to the mark. Place this solution in a 1 cm quartz absorption cell, and measure the absorbance at 323 nm, which is 0.463. Given that the molar absorptivity of caffeic acid $E_{\mathrm{l cm}}^{1\%}=927.9$. What is the percentage content of caffeic acid?

Options:
A. $$
92.11 \%
$$
B. $$
81.95 \%
$$
C. $$
77.65 \%
$$
D. $$
99.80 \%
$$
E. $$
66.32 \%
$$
F. $$
70.48 \%
$$
G. $$
88.67 \%
$$
H. $$
79.50 \%
$$
I. $$
85.74 \%
$$
J. $$
95.23 \%
$$
Answer: D
The results of separating a sample on a 3 m long chromatography column are as follows: the dead time is 1 min, the retention time for component 1 is 14 min, the retention time for component 2 is 17 min, and the peak width is 1 min. What is the theoretical plate number $n$ and the height equivalent to a theoretical plate $H$ for the column using component 2?

Options:
A. $$\begin{aligned}&n_{2}\:=\:7.0\:\times\:10^{3}\\&H_{2}\:=\:0.\:39\:(\:mm\:)\end{aligned}$$
B. $$\begin{aligned}&n_{2}\:=\:2.7\:\times\:10^{3}\\&H_{2}\:=\:0.\:92\:(\:mm\:)\end{aligned}$$
C. $$\begin{aligned}&n_{2}\:=\:3.9\:\times\:10^{3}\\&H_{2}\:=\:0.\:80\:(\:mm\:)\end{aligned}$$
D. $$\begin{aligned}&n_{2}\:=\:4.1\:\times\:10^{3}\\&H_{2}\:=\:0.\:72\:(\:mm\:)\end{aligned}$$
E. $$\begin{aligned}&n_{2}\:=\:8.4\:\times\:10^{3}\\&H_{2}\:=\:0.\:30\:(\:mm\:)\end{aligned}$$
F. $$\begin{aligned}&n_{2}\:=\:3.3\:\times\:10^{3}\\&H_{2}\:=\:0.\:75\:(\:mm\:)\end{aligned}$$
G. $$\begin{aligned}&n_{2}\:=\:6.1\:\times\:10^{3}\\&H_{2}\:=\:0.\:45\:(\:mm\:)\end{aligned}$$
H. $$\begin{aligned}&n_{2}\:=\:5.2\:\times\:10^{3}\\&H_{2}\:=\:0.\:54\:(\:mm\:)\end{aligned}$$
I. $$\begin{aligned}&n_{2}\:=\:4.6\:\times\:10^{3}\\&H_{2}\:=\:0.\:65\:(\:mm\:)\end{aligned}$$
J. $$\begin{aligned}&n_{2}\:=\:5.7\:\times\:10^{3}\\&H_{2}\:=\:0.\:51\:(\:mm\:)\end{aligned}$$
Answer: I
In the rate theory equation of a certain chromatographic column, the three parameters A, B, and C are 0.013 $cm$, 0.40 $cm^2/s$, and 0.0053 $s$ respectively. What is the optimal carrier gas linear velocity?

Options:
A. $$
9. 4 ( \mathrm{c m / s} )
$$
B. $$
10. 2 ( \mathrm{c m / s} )
$$
C. $$
12. 0 ( \mathrm{c m / s} )
$$
D. $$
5. 7 ( \mathrm{c m / s} )
$$
E. $$
6. 5 ( \mathrm{c m / s} )
$$
F. $$
7. 0 ( \mathrm{c m / s} )
$$
G. $$
11. 1 ( \mathrm{c m / s} )
$$
H. $$
9. 9 ( \mathrm{c m / s} )
$$
I. $$
7. 8 ( \mathrm{c m / s} )
$$
J. $$
8. 8 ( \mathrm{c m / s} )
$$
Answer: J
Given that the relative retention value between components A and B is 1.5. When substance B is developed on a certain thin-layer plate, the spot is 8.3 cm from the origin, and the solvent front is 16 cm from the origin. If A is developed on the same plate simultaneously, what would be the migration distance of component A?

Options:
A. $$
1 4 . 0 ( \mathrm{{\, c m )}}
$$
B. $$
1 0 . 8 ( \mathrm{{\, c m )}}
$$
C. $$
1 5 . 7 ( \mathrm{{\, c m )}}
$$
D. $$
7 . 5 ( \mathrm{{\, c m )}}
$$
E. $$
1 6 . 5 ( \mathrm{{\, c m )}}
$$
F. $$
1 3 . 6 ( \mathrm{{\, c m )}}
$$
G. $$
8 . 9 ( \mathrm{{\, c m )}}
$$
H. $$
9 . 2 ( \mathrm{{\, c m )}}
$$
I. $$
1 1 . 3 ( \mathrm{{\, c m )}}
$$
J. $$
1 2. 4 ( \mathrm{{\, c m )}}
$$
Answer: J
A new water pipeline, pipe diameter $d=0 . 4m$,tube length $l=1 0 0 \mathrm{m}$ , roughness factor $n=0 . 011$,linear loss $h_{f}=0 . 4m$,water flow is turbulent rough region. How much is the flow?

Options:
A. $0.150m^{3}/s$
B. $0.140m^{3}/s$
C. $0.180m^{3}/s$
D. $0.165m^{3}/s$
E. $0.170m^{3}/s$
F. $0.145m^{3}/s$
G. $0.135m^{3}/s$
H. $0.175m^{3}/s$
I. $0.160m^{3}/s$
J. $0.156m^{3}/s$
Answer: J
A shaft 100 mm diameter (D) runs in a bearing 200 mm long (L). The two surfaces are separated by an oil film 2.5 mm thick (c). Take the oil viscosity (μ) as 0.25 kg/ms. If the shaft rotates at a speed of (N) revolutions per minute.Calculate the torque necessary to rotate the shaft at 600 rpm is ().

Options:
A. 1.234Nm
B. 1.210Nm
C. 1.045Nm
D. 0.712Nm
E. 1.001Nm
F. 1.095Nm
G. 0.987Nm
H. 0.980Nm
I. 0.965Nm
J. 0.876Nm
Answer: G
A large long straight water and soil channel with a trapezoidal cross-section is given, where the bottom width of the channel $b=10.0 \mathrm{m}$, side slope coefficient $m=1.5$, bottom slope $i=0.0003$, and channel wall roughness coefficient $n=0.025$. Try to calculate the flow rate $q_{\mathrm{V}}$ when the water depth $h=2.65 \mathrm{m}$?

Options:
A. $$
3 6. 4 9 m^{3} / \mathrm{s}
$$
B. $$
3 8. 9 7 m^{3} / \mathrm{s}
$$
C. $$
4 0. 6 1 m^{3} / \mathrm{s}
$$
D. $$
4 1. 0 0 m^{3} / \mathrm{s}
$$
E. $$
3 9. 2 8 m^{3} / \mathrm{s}
$$
F. $$
3 8. 2 1 m^{3} / \mathrm{s}
$$
G. $$
3 9. 5 2 m^{3} / \mathrm{s}
$$
H. $$
3 7. 6 5 m^{3} / \mathrm{s}
$$
I. $$
3 6. 8 3 m^{3} / \mathrm{s}
$$
J. $$
4 2. 4 3 m^{3} / \mathrm{s}
$$
Answer: E
Rectangular cross-section channel, roughness coefficient $n = 0.025$, with a flow rate $q_{\mathrm{V}} = 5.0 \mathrm{m}^{3} / \mathrm{s}$, bottom width $b = 4 \mathrm{m}$, and water depth $h = 2 \mathrm{m}$. What should the bottom slope of this channel be?

Options:
A. $$
q_{\mathrm{V}}=2. 0 1 2 \times1 0^{-4}
$$
B. $$
q_{\mathrm{V}}=1. 9 8 3 \times1 0^{-4}
$$
C. $$
q_{\mathrm{V}}=2. 4 4 1 \times1 0^{-4}
$$
D. $$
q_{\mathrm{V}}=2. 9 8 5 \times1 0^{-4}
$$
E. $$
q_{\mathrm{V}}=6. 1 5 7 \times1 0^{-4}
$$
F. $$
q_{\mathrm{V}}=3. 4 4 6 \times1 0^{-4}
$$
G. $$
q_{\mathrm{V}}=5. 3 6 0 \times1 0^{-4}
$$
H. $$
q_{\mathrm{V}}=4. 5 1 9 \times1 0^{-4}
$$
I. $$
q_{\mathrm{V}}=3. 7 2 5 \times1 0^{-4}
$$
J. $$
q_{\mathrm{V}}=4. 2 7 3 \times1 0^{-4}
$$
Answer: C
A model ship made according to the length scale $\lambda_{l}=30$ is subjected to a towing experiment in a towing tank to determine the resistance of the full-scale ship. If the speed of the full-scale ship is $v_{\mathrm{p}}=27 \mathrm{km/h}$, what should be the speed $v_{\mathrm{m}}$ of the model ship?

Options:
A. 1.410m/s
B. 1.370m/s
C. 1.456m/s
D. 1.320m/s
E. 1.369m/s
F. 1.348m/s
G. 1.500m/s
H. 1.290m/s
I. 1.200m/s
J. 1.380m/s
Answer: E
Conduct a model test study on the layout of an overflow dam for a specific project. It is known that the maximum discharge flow of the overflow dam is $197 \mathrm{m}^{3}/\mathrm{s},$ and the laboratory's water supply capacity is $0.04 \mathrm{m}^{3}/\mathrm{s}$. Try to determine the model's maximum length scale ratio based on the water supply capacity is ().

Options:
A. $$
\lambda_{l}=5 5. 7 8
$$
B. $$
\lambda_{l}=4 8. 9 4
$$
C. $$
\lambda_{l}=5 4. 3 6
$$
D. $$
\lambda_{l}=4 9. 8 7
$$
E. $$
\lambda_{l}=5 9. 1 0
$$
F. $$
\lambda_{l}=6 2. 0 3
$$
G. $$
\lambda_{l}=6 0. 5 2
$$
H. $$
\lambda_{l}=5 6. 2 1
$$
I. $$
\lambda_{l}=5 7. 0 9
$$
J. $$
\lambda_{l}=5 3. 1 5
$$
Answer: I
A certain section of water pipe has a diameter $d = 100 \mathrm{mm}$ and a flow velocity $v = 1.0 \mathrm{m/s}$, with a water temperature of 10°C. What is the maximum flow velocity to maintain laminar flow?

Options:
A. 2.50cm/s
B. 4.20cm/s
C. 1.50cm/s
D. 6.10cm/s
E. 2.90cm/s
F. 3.01cm/s
G. 4.80cm/s
H. 3.75cm/s
I. 5.00cm/s
J. 1.80cm/s
Answer: F
To lower the groundwater level for a railway cutting, a collection gallery with a length of $l=100 \mathrm{m}$ was constructed on the side of the cutting to drain the groundwater. Given that the aquifer head $H$ is 6m, the gallery water depth $h=0.3 \mathrm{m}$, the influence range $L=135 \mathrm{m}$, and the single-side seepage flow rate of the gallery $q_{\mathrm{V}}=0.297 \mathrm{m}^{3} / \mathrm{h}$, try to determine the seepage coefficient $k$ of the aquifer is ().

Options:
A. 0.0344cm/min
B. 0.0456cm/min
C. 0.0333cm/min
D. 0.0319cm/min
E. 0.0425cm/min
F. 0.0372cm/min
G. 0.0440cm/min
H. 0.0367cm/min
I. 0.0298cm/min
J. 0.0391cm/min
Answer: F
What is the type of well where the water production rate $q_{\mathrm{V}}$ is proportional to the square of the aquifer head $H$?

Options:
A. Enhanced recovery well
B. Step drawdown well
C. Injectivity testing well
D. Artesian flow well
E. Surface casing well
F. Complete diving well
G. Gravity-fed well
H. Directional drilling well
I. Partial diving well
J. Hydraulic fracturing well
Answer: F
There is a half-wave oscillator with a length of $L$ and a current distribution of $I(z) = I_{\mathfrak{m}} \cos kz$
where $-L/2 \le z \le L/2$.
What is the directivity factor?

Options:
A. 3.75
B. 4.10
C. 1.64
D. 6.89
E. 5.62
F. 2.33
G. 1.28
H. 0.92
I. 0.47
J. 1.00
Answer: C
Calculate the half-power beamwidth and directivity of a square aperture antenna with dimensions $5 \lambda \times 5 \lambda$ (assuming uniform amplitude and phase) and a circular aperture antenna with a diameter $D = 5 \lambda$ (assuming uniform amplitude and phase).

Options:
A. 0.180,0.203
B. 0.150,0.240
C. 0.160,0.215
D. 0.155,0.230
E. 0.190,0.210
F. 0.178,0.204
G. 0.175,0.205
H. 0.185,0.202
I. 0.170,0.200
J. 0.165,0.220
Answer: F
Given a dipole of $2l = 2 \mathrm{m}$, with operating wavelengths $\lambda = 10 \mathrm{m}$ and $4 \mathrm{m}$, determine the effective length in both cases.

Options:
A. 0.90m,1.10m
B. 1.18m,1.28m
C. 0.85m,1.50m
D. 1.10m,1.35m
E. 1.00m,1.15m
F. 1.20m,1.40m
G. 1.03m,1.27m
H. 1.05m,1.30m
I. 1.15m,1.25m
J. 0.95m,1.20m
Answer: G
Calculate the half-power beamwidth $\mathrm{BW}$ of a symmetrical dipole with $l=0.25 \lambda$.

Options:
A. $82^{\circ}$
B. $77^{\circ}$
C. $83^{\circ}$
D. $90^{\circ}$
E. $75^{\circ}$
F. $70^{\circ}$
G. $76^{\circ}$
H. $78^{\circ}$
I. $80^{\circ}$
J. $85^{\circ}$
Answer: H
Find the radiation power and radiation resistance of the electric fundamental oscillator and the magnetic fundamental oscillator (small current loop)

Options:
A. $$
\frac{1} {6} I^{2} R_{r},
$$
$$
7 5 \left( \frac{\pi d} {\lambda_{0}} \right)^{3}
$$
B. $$
\frac{2} {1} I^{2} R_{m},
$$
$$
8 5 \left( \frac{\pi e} {\lambda_{m}} \right)^{2}
$$
C. $$
\frac{3} {2} I^{2} R_{m},
$$
$$
6 0 \left( \frac{\pi c} {\lambda_{m}} \right)^{2}
$$
D. $$
\frac{1} {2} I^{2} R_{r},
$$
$$
8 0 \left( \frac{\pi l} {\lambda_{0}} \right)^{2}
$$
E. $$
\frac{1} {3} I^{2} R_{m},
$$
$$
9 0 \left( \frac{\pi a} {\lambda_{m}} \right)^{3}
$$
F. $$
\frac{2} {3} I^{2} R_{m},
$$
$$
7 0 \left( \frac{\pi b} {\lambda_{m}} \right)^{2}
$$
G. $$
\frac{1} {4} I^{2} R_{r},
$$
$$
1 0 0 \left( \frac{\pi l} {\lambda_{0}} \right)^{3}
$$
H. $$
\frac{1} {7} I^{2} R_{r},
$$
$$
6 9 \left( \frac{\pi h} {\lambda_{0}} \right)^{3}
$$
I. $$
\frac{1} {5} I^{2} R_{r},
$$
$$
8 8 \left( \frac{\pi f} {\lambda_{0}} \right)^{3}
$$
J. $$
\frac{2} {4} I^{2} R_{m},
$$
$$
9 2 \left( \frac{\pi g} {\lambda_{m}} \right)^{2}
$$
Answer: D
A rectangular resonant cavity filled with air resonates in the $\mathrm{TE}_{101}$ mode when $\lambda_{0}=10 \mathrm{cm}$ and in the $\mathrm{TE}_{102}$ mode when $\lambda_{0}=8 \mathrm{cm}$. Determine the dimensions of this rectangular cavity.

Options:
A. $$
a=\frac{2 5} {\sqrt{1 5}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {4} a=\frac{5} {\sqrt{1 5}} ( \mathrm{c m} ) \,, \quad l=\frac{2 5} {\sqrt{3}} ( \mathrm{c m} )
$$
B. $$
a=\frac{2 1} {\sqrt{1 5}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {2} a=\frac{1 0} {\sqrt{1 5}} ( \mathrm{c m} ) \,, \quad l=\frac{2 1} {\sqrt{3}} ( \mathrm{c m} )
$$
C. $$
a=\frac{1 5} {\sqrt{1 2}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {3} a=\frac{1 5} {\sqrt{1 2}} ( \mathrm{c m} ) \,, \quad l=\frac{1 5} {\sqrt{3}} ( \mathrm{c m} )
$$
D. $$
a=\frac{2 0} {\sqrt{1 3}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {2} a=\frac{1 0} {\sqrt{1 3}} ( \mathrm{c m} ) \,, \quad l=\frac{2 0} {\sqrt{3}} ( \mathrm{c m} )
$$
E. $$
a=\frac{2 3} {\sqrt{1 6}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {3} a=\frac{2 3} {\sqrt{1 6}} ( \mathrm{c m} ) \,, \quad l=\frac{2 3} {\sqrt{3}} ( \mathrm{c m} )
$$
F. $$
a=\frac{1 6} {\sqrt{1 0}} ( \mathrm{c m} ) \,, \quad b=\frac{3} {4} a=\frac{1 2} {\sqrt{1 0}} ( \mathrm{c m} ) \,, \quad l=\frac{1 8} {\sqrt{3}} ( \mathrm{c m} )
$$
G. $$
a=\frac{2 2} {\sqrt{1 2}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {2} a=\frac{2 2} {\sqrt{1 2}} ( \mathrm{c m} ) \,, \quad l=\frac{2 2} {\sqrt{3}} ( \mathrm{c m} )
$$
H. $$
a=\frac{2 4} {\sqrt{1 4}} ( \mathrm{c m} ) \,, \quad b=\frac{2} {3} a=\frac{2 4} {\sqrt{1 4}} ( \mathrm{c m} ) \,, \quad l=\frac{2 4} {\sqrt{2}} ( \mathrm{c m} )
$$
I. $$
a=\frac{1 8} {\sqrt{1 1}} ( \mathrm{c m} ) \,, \quad b=\frac{2} {3} a=\frac{1 1} {\sqrt{1 1}} ( \mathrm{c m} ) \,, \quad l=\frac{2 0} {\sqrt{2}} ( \mathrm{c m} )
$$
J. $$
a=\frac{1 9} {\sqrt{1 4}} ( \mathrm{c m} ) \,, \quad b=\frac{1} {2} a=\frac{9} {\sqrt{1 4}} ( \mathrm{c m} ) \,, \quad l=\frac{1 9} {\sqrt{2}} ( \mathrm{c m} )
$$
Answer: D
A lossless transmission line with a characteristic impedance of $50\Omega$ and a length of $2.5\lambda$ is terminated with a load $Z_{\mathrm{L}}=25+\mathrm{j}30\Omega$. Given a source $E_{g} = 20 \mathbf{V}$ and $Z_{g} = 50\Omega$, find the power transmitted from the signal source to the load. If a $\lambda/4$ impedance transformer section is used to match the load, determine the position $d$ where the transformer section is connected and the power absorbed by the load at this time.

Options:
A. $$
0. 4 8 \mathbf{W},1(W)
$$
B. $$
1. 2 0 \mathbf{W},1(W)
$$
C. $$
0. 5 9 \mathbf{W},1(W)
$$
D. $$
0. 9 4 \mathbf{W},1(W)
$$
E. $$
0. 8 1 \mathbf{W},1(W)
$$
F. $$
1. 0 3 \mathbf{W},2(W)
$$
G. $$
0. 9 9 \mathbf{W},1(W)
$$
H. $$
0. 3 3 \mathbf{W},1(W)
$$
I. $$
0. 6 8 \mathbf{W},1(W)
$$
J. $$
0. 7 7 \mathbf{W},1(W)
$$
Answer: J
In a lossless uniform transmission line with $Z_{0} = 500 \Omega$, a load of $Z_{\mathrm{L}} = (300 + \mathrm{j} 400) \Omega$, wavelength $\lambda = 3 \mathrm{m}$, transmission line length $l = 1.25 \lambda$, source $E_{\mathrm{g}} = 50 \mathrm{V}$, and source impedance $Z_{\mathrm{g}} = Z_{0} = 500 \Omega$, find the power absorbed by the load $P_{\mathrm{L}}$.

Options:
A. $$0.51(W)$$
B. $$0.49(W)$$
C. $$0.46(W)$$
D. $$0.48(W)$$
E. $$0.53(W)$$
F. $$0.45(W)$$
G. $$0.52(W)$$
H. $$0.44(W)$$
I. $$0.47(W)$$
J. $$0.50(W)$$
Answer: I
The ground station receives signals transmitted by the probe satellite at a frequency of $136 \text{MHz}$. The satellite is $500 \text{km}$ away from the ground, and the power delivered to the satellite's antenna by its transmitter is $0.5 \text{W}$. This antenna has a gain of $3 \text{dB}$ relative to an isotropic antenna. Assuming the propagation occurs in free space, where the wave impedance is $120\pi\ \Omega$, calculate the power flux density $(\text{W} / \text{m}^{2})$.

Options:
A. $$
2. 0 5 8 \times1 0^{-1 2}. \mathbf{W} / \mathbf{m}^{2}
$$
B. $$
0. 5 6 2 \times1 0^{-1 2}. \mathbf{W} / \mathbf{m}^{2}
$$
C. $$
7. 3 4 1 \times1 0^{-1 3}. \mathbf{W} / \mathbf{m}^{2}
$$
D. $$
5. 8 1 9 \times1 0^{-1 2}. \mathbf{W} / \mathbf{m}^{2}
$$
E. $$
0. 3 1 8 \times1 0^{-1 2}. \mathbf{W} / \mathbf{m}^{2}
$$
F. $$
2. 8 5 6 \times1 0^{-1 1}. \mathbf{W} / \mathbf{m}^{2}
$$
G. $$
3. 6 7 2 \times1 0^{-1 3}. \mathbf{W} / \mathbf{m}^{2}
$$
H. $$
0. 4 9 5 \times1 0^{-1 2}. \mathbf{W} / \mathbf{m}^{2}
$$
I. $$
1. 1 2 4 \times1 0^{-1 3}. \mathbf{W} / \mathbf{m}^{2}
$$
J. $$
1. 9 7 3 \times1 0^{-1 4}. \mathbf{W} / \mathbf{m}^{2}
$$
Answer: E
For a flat plate turbulent boundary layer, the expression for the velocity distribution and wall shear stress is given by
$$
\frac{u}{V_{\infty}} = \left(\frac{y}{\delta}\right)^{\frac{1}{7}} \qquad\qquad \tau_{0} = 0.0233 \left(\frac{\mu}{\rho V_{\infty} \delta}\right)^{\frac{1}{4}} \rho V_{\infty}^{2}
$$
Using the boundary layer momentum integral equation, what are the boundary layer thickness $\delta(x)$ and the drag coefficient for one side of the flat plate $C_{\mathrm{f}}$?

Options:
A. $$
\delta=0. 1 8 4 0 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{2} {6}} x^{\frac{4} {4}},\ C_{\text{f}}=\frac{0. 0 6 5 4 2} {( R e_{1} )^{\frac{6} {5}}}
$$
B. $$
\delta=0. 6 4 9 5 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {8}} x^{\frac{2} {3}},\ C_{\text{f}}=\frac{0. 0 9 2 6 4} {( R e_{1} )^{\frac{5} {5}}}
$$
C. $$
\delta=0. 3 3 5 0 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {9}} x^{\frac{5} {4}},\ C_{\text{f}}=\frac{0. 0 3 7 1 5} {( R e_{1} )^{\frac{5} {6}}}
$$
D. $$
\delta=0. 5 9 7 4 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {6}} x^{\frac{3} {4}},\ C_{\text{f}}=\frac{0. 0 5 8 9 7} {( R e_{1} )^{\frac{2} {5}}}
$$
E. $$
\delta=0. 3 8 1 2 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {5}} x^{\frac{4} {5}},\ C_{\text{f}}=\frac{0. 0 7 4 1 3} {( R e_{1} )^{\frac{1} {5}}}
$$
F. $$
\delta=0. 2 3 1 8 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {7}} x^{\frac{5} {6}},\ C_{\text{f}}=\frac{0. 1 1 7 8 2} {( R e_{1} )^{\frac{4} {5}}}
$$
G. $$
\delta=0. 4 2 7 3 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{2} {5}} x^{\frac{3} {5}},\ C_{\text{f}}=\frac{0. 0 2 7 9 3} {( R e_{1} )^{\frac{3} {5}}}
$$
H. $$
\delta=0. 8 3 0 6 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{1} {4}} x^{\frac{7} {8}},\ C_{\text{f}}=\frac{0. 0 1 4 9 0} {( R e_{1} )^{\frac{7} {5}}}
$$
I. $$
\delta=0. 9 2 6 1 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{0} {5}} x^{\frac{1} {3}},\ C_{\text{f}}=\frac{0. 0 4 5 3 8} {( R e_{1} )^{\frac{8} {5}}}
$$
J. $$
\delta=0. 7 1 5 9 \Big( \frac{\mu} {\rho V_{\infty}} \Big)^{\frac{3} {5}} x^{\frac{1} {2}},\ C_{\text{f}}=\frac{0. 0 8 3 6 1} {( R e_{1} )^{\frac{2} {6}}}
$$
Answer: E
A standard galvanized steel pipe with a length of $l=1000 \, \mathrm{m}$ and an inner diameter of $d=2000 \, \mathrm{mm}$ is used to transport heavy oil with a kinematic viscosity of $\nu=0.355 \times 10^{-4} \, \mathrm{m^{2}/s}$. The flow rate measured is $q_{V}=0.038 \, \mathrm{m^{3}/s}$. What is the head loss along the pipe? $(\Delta=0.2 \, \mathrm{mm})$

Options:
A. $$0.050$$
B. $$0.070$$
C. $$0.020$$
D. $$0.030$$
E. $$0.075$$
F. $$0.035$$
G. $$0.060$$
H. $$0.080$$
I. $$0.065$$
J. $$0.045$$
Answer: F
An airplane is flying at a speed of 530 m/s in stationary air at a temperature of 20°C. Find the Mach number and Mach angle of the airplane. If the airplane passes overhead at an altitude of 800 m, how far past overhead must the airplane fly before the person on the ground can hear the sound?

Options:
A. $$
8 3 7 \mathrm{m}
$$
B. $$
9 4 2 \mathrm{m}
$$
C. $$
7 6 4 \mathrm{m}
$$
D. $$
1 1 5 0 \mathrm{m}
$$
E. $$
1 2 2 5 \mathrm{m}
$$
F. $$
1 0 0 0 \mathrm{m}
$$
G. $$
6 5 9 \mathrm{m}
$$
H. $$
6 0 6 \mathrm{m}
$$
I. $$
4 9 0 \mathrm{m}
$$
J. $$
5 2 8 \mathrm{m}
$$
Answer: B
A normal shock wave is moving at a speed of $600 \mathrm{m/s}$ in still air, where the air pressure is $p_{1}=103\ \mathrm{kPa}$ and the temperature is $T_{1}=293\ \mathrm{K}$. What is the Mach number after the shock wave passes through?

Options:
A. $$
0. 6 5 4
$$
B. $$
1. 1 8 7
$$
C. $$
1. 3 4 5
$$
D. $$
1. 0 9 8
$$
E. $$
1. 2 0 4
$$
F. $$
2. 1 5 0
$$
G. $$
0. 9 1 3
$$
H. $$
0. 7 5 6
$$
I. $$
0. 5 9 2
$$
J. $$
0. 8 0 2
$$
Answer: J
Let the number of occurrences of event A in $[0, t)$, denoted as $\{N(t), t \geq 0\}$, be a homogeneous Poisson process with parameter $\lambda$. Given that event A has happened $n$ times within $\lfloor0, t \rfloor$, find the conditional probability density function of the time $\tau_{k}$ for the $k$-th occurrence $(k < n)$ of event A is ().

Options:
A. $$
C_{a}^{k} t^{k} \left(\frac{s^{k-1}}{n-k}\right)^{1-\frac{s}{t}}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
B. $$
(k-1) C_{n}^{k-1} \left(\frac{s}{t}\right)^{k} \left(1-\frac{s}{t}\right)^{n-k}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
C. $$
\frac{k s^{k-1}}{t} \left( \frac{s}{t} \right)^{n-k}, \quad 0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
D. $$
\frac{k}{n} s^{k} \left(1-\frac{s}{t}\right)^{n-k}, \quad0\leqslant s\leqslant t, 1\leqslant k\leqslant n
$$
E. $$
k C_{a}^{k} ·\frac{s^{k-1}} {t^{k}} \left( 1-\frac{s} {t} \right)^{n-k}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
F. $$
k \lambda s^{k-1} t^{n-k} \left(1-\frac{s}{t}\right)^{k}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
G. $$
(n-k) C_{a}^{n} \left(\frac{s}{k t}\right)^{k-1} \left(1-\frac{s}{t}\right), \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
H. $$
k C_{n}^{k} \left(\frac{s}{t to the power of k}\right)\left(1-s\right)^{n-k}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
I. $$
k C_{k}^{a} \frac{s^{k-1}}{t^k} \left(1-\frac{s}{t}\right)^{k-n}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
J. $$
k \lambda \frac{s^{k}}{t^{n}} \left(1-\frac{s}{t}\right)^{n}, \quad0 \leqslant s \leqslant t, 1 \leqslant k \leqslant n
$$
Answer: E
Let the number of phone calls received by a certain telephone exchange in $t$ minutes $\left\{X(t), t \geq 0 \right\}$ be a Poisson process with a rate of $\lambda$. Find the probability $p_{2}$ that 5 calls are received in 3 minutes, and the 5th call arrives in the 3rd minute is ().

Options:
A. $$
\frac{1 9 0} {1 4 0} \lambda^{4} \, \mathrm{e}^{-3 \lambda}
$$
B. $$
\frac{1 2 5} {1 0 0} \lambda^{4} \, \mathrm{e}^{-5 \lambda}
$$
C. $$
\frac{2 2 1} {1 9 0} \lambda^{6} \, \mathrm{e}^{-3 \lambda}
$$
D. $$
\frac{1 7 0} {1 6 0} \lambda^{4} \, \mathrm{e}^{-2 \lambda}
$$
E. $$
\frac{1 5 0} {1 7 5} \lambda^{2} \, \mathrm{e}^{-4 \lambda}
$$
F. $$
\frac{1 0 0} {1 5 0} \lambda^{5} \, \mathrm{e}^{-2 \lambda}
$$
G. $$
\frac{2 1 5} {1 1 5} \lambda^{5} \, \mathrm{e}^{-4 \lambda}
$$
H. $$
\frac{1 8 5} {1 2 5} \lambda^{3} \, \mathrm{e}^{-3 \lambda}
$$
I. $$
\frac{2 1 1} {1 2 0} \lambda^{5} \, \mathrm{e}^{-3 \lambda}
$$
J. $$
\frac{2 0 5} {1 3 5} \lambda^{3} \, \mathrm{e}^{-5 \lambda}
$$
Answer: I