MikaStars39/nano-eval huggingface dataset:
Consider the following one-dimensional wave function:
$$
\psi(x) = A \left( \frac{x}{x_{0}} \right)^{n} \mathrm{e}^{-x / x_{0}}
$$
where $A$, $n$, and $x_{0}$ are constants.
Using the Schrödinger equation, find the potential $V(x)$ and the energy $E$, for which this wave function is an eigenfunction (assuming that as $x \to \infty$, $V(x) \to 0$).

Options:
A. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{n ( n-1 )} {x_{0}^{2}}}-{\frac{2 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{n^{2}} {x_{0}^{2}}
$$
B. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{( n-1 ) n} {x^{2}}}+{\frac{n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {m} \frac{1} {x_{0}^{2}}
$$
C. $$
V(x) = - {\frac{\hbar^{2}} {m}} \left[ {\frac{n ( n+1 )} {x^{2}}}-{\frac{2 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{n} {x_{0}^{2}}
$$
D. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{n ( n-1 )} {x^{2}}}-{\frac{2 n} {x x_{0}}} \right],
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{1} {x_{0}^{2}}
$$
E. $$
V(x) = - {\frac{\hbar^{2}} {m}} \left[ {\frac{n ( n-1 )} {x^{2}}}-{\frac{3 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{1} {x_{0}^{2}}
$$
F. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{n n} {x^{2}}}-{\frac{2 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{1} {x_{0}}
$$
G. $$
V(x) = - {\frac{\hbar^{2}} {m}} \left[ {\frac{n ( n-1 )} {x^{2}}}-{\frac{4 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {m} \frac{1} {x_{0}^{2}}
$$
H. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{n ( n+2 )} {x^{2}}}-{\frac{n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {m} \frac{1} {x_{0}^{2}}
$$
I. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{( n+1 ) n} {x^{2}}}-{\frac{3 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {m} \frac{1} {x_{0}^{2}}
$$
J. $$
V(x) = - {\frac{\hbar^{2}} {2 m}} \left[ {\frac{n^{2}} {x^{2}}}+{\frac{2 n} {x x_{0}}} \right]
$$
$$
E = - \frac{\hbar^{2}} {2 m} \frac{n} {x_{0}^{2}}
$$
Answer: D
The force constant is $k$, and the ground state wave function of a harmonic oscillator with mass $m$ is
$$
\psi_{0} ( x )=\left( \frac{\alpha} {\pi} \right)^{\frac{1} {4}} \mathrm{e}^{-\frac{1} {2} \alpha x^{2}}, \quad\alpha=\frac{m \omega_{0}} {\hbar}, \quad\omega_{0}^{2}=\frac{k} {m}
$$
Determine the probability of finding the particle outside the classical region.

Options:
A. $$
P\approx22\%
$$
B. $$
P\approx12\%
$$
C. $$
P\approx5\%
$$
D. $$
P\approx30\%
$$
E. $$
P\approx20\%
$$
F. $$
P\approx8\%
$$
G. $$
P\approx10\%
$$
H. $$
P\approx16\%
$$
I. $$
P\approx18\%
$$
J. $$
P\approx25\%
$$
Answer: H
In the non-relativistic case, the magnetic moment of a free electron is $\mu$, and it is in a uniform and constant external magnetic field.
Question: What is the form of the Hamiltonian of the system (assuming the magnetic field is $B_{z}$)?

Options:
A. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {c} B_{z} x \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{y}
$$
B. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {c} B_{z} x \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{x} \sigma_{z}
$$
C. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {c} B_{z} z \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{x}
$$
D. $$
H=\frac{1} {2 m} \left( p_{x}+\frac{e} {c} B_{y} z \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{z}
$$
E. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {c} B_{z} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{z}
$$
F. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {2c} B_{z} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{x}
$$
G. $$
H=\frac{1} {2 m} \left( p_{x}+\frac{e} {c} B_{z} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{z} \sigma_{z}
$$
H. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{2e} {c} B_{z} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {m c} B_{z} \sigma_{z}
$$
I. $$
H=\frac{1} {2 m} \left( p_{x}-\frac{e} {c} B_{y} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {2 m c} B_{x} \sigma_{z}
$$
J. $$
H=\frac{1} {4 m} \left( p_{x}-\frac{e} {c} B_{z} y \right)^{2}+\frac{1} {2 m} p_{y}^{2}+\frac{1} {2 m} p_{z}^{2}+\frac{e \hbar} {4 m c} B_{z} \sigma_{z}
$$
Answer: E
Expand the two-body operator $\mathrm{e}^{\mathrm{i} \alpha\sigma_{1} \cdot\sigma_{2}}$ for two spin-$\frac{1}{2}$ particles.

Options:
A. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{c o s} 2 \alpha+\frac{\mathrm{i}} {2} \operatorname{s i n} 2 \alpha\left( 1+\sigma_{1} \cdot\sigma_{2} \right) \right]
$$
B. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} 2 \alpha+\frac{\mathrm{i}} {2} \operatorname{sin} \alpha\left( \sigma_{1} \cdot\sigma_{2} \right) \right]
$$
C. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{sin} \alpha+\frac{\mathrm{i}} {2} \operatorname{cos} \alpha\left( \sigma_{1} +\sigma_{2} \right) \right]
$$
D. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} \alpha+\frac{\mathrm{i}} {2} \operatorname{sin} \alpha\left( \sigma_{1} \cdot\sigma_{2} -1 \right) \right]
$$
E. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} 2 \alpha-\frac{\mathrm{i}} {2} \operatorname{sin} 2 \alpha\left( 1+\sigma_{1} -\sigma_{2} \right) \right]
$$
F. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} \alpha+\frac{\mathrm{i}} {2} \operatorname{sin} \alpha\left( 1+\sigma_{1} +\sigma_{2} \right) \right]
$$
G. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} \alpha+\frac{\mathrm{i}} {3} \operatorname{sin} \alpha\left( 1+\sigma_{1} \cdot\sigma_{2} \right) \right]
$$
H. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} 3 \alpha+\frac{\mathrm{i}} {2} \operatorname{sin} 3 \alpha\left( 1+\sigma_{1} \cdot\sigma_{2} \right) \right]
$$
I. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ 1+\frac{\mathrm{i}} {2} \operatorname{sin} 2 \alpha\left( \sigma_{1} \cdot\sigma_{2} \right) \right]
$$
J. $$
\mathrm{e}^{-\mathrm{i} \alpha} \left[ \operatorname{cos} 2 \alpha+\frac{1} {2} \operatorname{sin} 2 \alpha\left( 1+\sigma_{1} \times \sigma_{2} \right) \right]
$$
Answer: A
The constant pressure expansion coefficient and isothermal compression coefficient of a certain gas are given by
$$
a=\frac{n R}{p V} \qquad \kappa=\frac{1}{p}+\frac{a}{V}
$$
where $, n, R$, and $a$ are all constants. Try to determine the equation of state for this gas as().

Options:
A. $$
p V+\frac{d}{3} p^{2}=n R T
$$
B. $$
p V^{2}+\frac{a}{3} p^{2}=n R T
$$
C. $$
p V^{2}+\frac{a}{5} p^{2}=n R T
$$
D. $$
p V+\frac{b}{2} p^{2}=n R T
$$
E. $$
p V-\frac{a}{3} V^{2}=n R T
$$
F. $$
p V+\frac{c}{4} p^{2}=n R T
$$
G. $$
p V+\frac{a} {2} p^{2}=n R T
$$
H. $$
p V-\frac{a}{2} V^{2}=n R T
$$
I. $$
p V-\frac{a}{4} V^{2}=n R T
$$
J. $$
p V^{2}+\frac{a}{4} p^{2}=n R T
$$
Answer: G
Ice, water, and water vapor, each 1g, are in thermal equilibrium in a closed container, with a pressure of 4.58 mmHg and a temperature of 0.01°C. When this system is heated by 60 cal, the total volume remains constant, accurate to within 2%. What is the weight of the ice, water, and water vapor once equilibrium is reached ?

Options:
A. $$
x \approx0. 4 5 \mathrm{g}, \ \ y \approx1. 5 0 \mathrm{g}, \ \ z \approx1. 0 5 \mathrm{g}
$$
B. $$
x \approx0. 1 9 \mathrm{g}, \ \ y \approx1. 6 0 \mathrm{g}, \ \ z \approx1. 2 1 \mathrm{g}
$$
C. $$
x \approx0. 2 5 \mathrm{g}, \ \ y \approx1. 7 5 \mathrm{g}, \ \ z \approx1. 0 0 \mathrm{g}
$$
D. $$
x \approx0. 3 0 \mathrm{g}, \ \ y \approx1. 5 0 \mathrm{g}, \ \ z \approx1. 2 0 \mathrm{g}
$$
E. $$
x \approx0. 2 7 \mathrm{g}, \ \ y \approx1. 6 9 \mathrm{g}, \ \ z \approx1. 0 4 \mathrm{g}
$$
F. $$
x \approx0. 3 5 \mathrm{g}, \ \ y \approx1. 4 5 \mathrm{g}, \ \ z \approx1. 2 0 \mathrm{g}
$$
G. $$
x \approx0. 2 8 \mathrm{g}, \ \ y \approx1. 6 7 \mathrm{g}, \ \ z \approx1. 0 5 \mathrm{g}
$$
H. $$
x \approx0. 2 2 \mathrm{g}, \ \ y \approx1. 8 0 \mathrm{g}, \ \ z \approx0. 9 8 \mathrm{g}
$$
I. $$
x \approx0. 1 0 \mathrm{g}, \ \ y \approx1. 7 0 \mathrm{g}, \ \ z \approx1. 2 0 \mathrm{g}
$$
J. $$
x \approx0. 1 5 \mathrm{g}, \ \ y \approx1. 8 0 \mathrm{g}, \ \ z \approx1. 0 5 \mathrm{g}
$$
Answer: C
White dwarfs are composed of a strongly degenerate electron gas with a temperature far lower than the Fermi temperature. As long as the electrons are non-relativistic, this system can resist gravitational collapse and remain stable. When the Fermi momentum is $\frac{m_{0} c}{10}$, what is the pressure of the electron gas?

Options:
A. $$
5. 1 \times1 0^{1 8} ( \mathrm{N / m^{2}} )
$$
B. $$
2. 9 \times1 0^{1 5} ( \mathrm{N / m^{2}} )
$$
C. $$
1. 5 \times1 0^{1 3} ( \mathrm{N / m^{2}} )
$$
D. $$
7. 4 \times1 0^{1 7} ( \mathrm{N / m^{2}} )
$$
E. $$
3. 3 \times1 0^{1 7} ( \mathrm{N / m^{2}} )
$$
F. $$
9. 5 \times1 0^{1 6} ( \mathrm{N / m^{2}} )
$$
G. $$
8. 6 \times1 0^{1 4} ( \mathrm{N / m^{2}} )
$$
H. $$
6. 2 \times1 0^{1 3} ( \mathrm{N / m^{2}} )
$$
I. $$
4. 7 \times1 0^{1 4} ( \mathrm{N / m^{2}} )
$$
J. $$
1. 8 \times1 0^{1 5} ( \mathrm{N / m^{2}} )
$$
Answer: F
If there is an artificial atmosphere with a composition of 20% oxygen and 80% nitrogen by volume, at a state of 14.7 psia and $60\mathrm{^\circ F}$, what is the density of the mixed gas?

Options:
A. $$
0. 0 7 6 0 \mathrm{l b f} / \mathrm{f t}^{3}
$$
B. $$
0. 0 7 8 4 \mathrm{l b f} / \mathrm{f t}^{3}
$$
C. $$
0. 0 6 5 3 \mathrm{l b f} / \mathrm{f t}^{3}
$$
D. $$
0. 0 7 9 2 \mathrm{l b f} / \mathrm{f t}^{3}
$$
E. $$
0. 0 7 2 9 \mathrm{l b f} / \mathrm{f t}^{3}
$$
F. $$
0. 0 8 1 5 \mathrm{l b f} / \mathrm{f t}^{3}
$$
G. $$
0. 0 6 8 0 \mathrm{l b f} / \mathrm{f t}^{3}
$$
H. $$
0. 0 8 0 1 \mathrm{l b f} / \mathrm{f t}^{3}
$$
I. $$
0. 0 7 0 5 \mathrm{l b f} / \mathrm{f t}^{3}
$$
J. $$
0. 0 6 9 2 \mathrm{l b f} / \mathrm{f t}^{3}
$$
Answer: A
A smooth flat plate measuring 6 inches (150 mm) wide and 18 inches (500 mm) long is placed in a crude oil flow with a stable velocity of 2 feet per second (fps) and a temperature of 60°F, where the specific gravity $s = 0.925$. What is the shear stress at the trailing edge of the plate?

Options:
A. $$
0. 0 2 9 9 9 \mathrm{b f} / \mathrm{f t}^{2}
$$
B. $$
0. 0 6 7 8 5 \mathrm{b f} / \mathrm{f t}^{2}
$$
C. $$
0. 0 1 2 7 5 \mathrm{b f} / \mathrm{f t}^{2}
$$
D. $$
0. 0 7 6 4 1 \mathrm{b f} / \mathrm{f t}^{2}
$$
E. $$
0. 1 5 4 3 9 \mathrm{b f} / \mathrm{f t}^{2}
$$
F. $$
0. 0 4 4 5 \mathrm{l b f} / \mathrm{f t}^{2}
$$
G. $$
0. 0 3 5 6 8 \mathrm{b f} / \mathrm{f t}^{2}
$$
H. $$
0. 1 1 0 2 3 \mathrm{b f} / \mathrm{f t}^{2}
$$
I. $$
0. 0 9 8 7 4 \mathrm{b f} / \mathrm{f t}^{2}
$$
J. $$
0. 0 5 3 2 6 \mathrm{b f} / \mathrm{f t}^{2}
$$
Answer: F
In the experiment to determine the Hall coefficient of a semiconductor, the current induced by the external electric field $\mathscr{E}_x$ is called the primary current. Under the influence of the Lorentz force and the Hall electric field, a transverse electron current $J_{\mathrm{ey}}$ and a transverse hole current $J_{\mathrm{py}}$ are generated. At equilibrium, what is the ratio of the transverse electron current and hole current to the primary current?

Options:
A. $$
B \frac{\sigma_{\mathrm{h}} \mu_{\mathrm{e}}}{ \sigma_{\mathrm{e}} + \sigma_{\mathrm{h}}} \frac{\mu_{\mathrm{h}}}{ \mu_{\mathrm{e}}}
$$
B. $$
B \frac{\sigma_{\mathrm{h}} + \mu_{\mathrm{h}} }{ \sigma_{\mathrm{e}} } \frac{\mu_{\mathrm{e}} }{ \sigma_{\mathrm{e}} }
$$
C. $$
B \frac{\sigma_{\mathrm{e}} \sigma_{\mathrm{h}}}{ \sigma_{\mathrm{h}} } \frac{\mu_{\mathrm{e}} }{ \sigma_{\mathrm{e}} + \mu_{\mathrm{h}} }
$$
D. $$
B ( \mu_{\mathrm{h}}+\mu_{\mathrm{e}} ) \; \frac{\sigma_{\mathrm{h}} \sigma_{\mathrm{e}}} {( \sigma_{\mathrm{h}}+\sigma_{\mathrm{e}} )^{2}}
$$
E. $$
B \frac{\sigma_{\mathrm{h}} \mu_{\mathrm{e}}}{ \mu_{\mathrm{h}} } \frac{\sigma_{\mathrm{e}} }{ \sigma_{\mathrm{h}} + \sigma_{\mathrm{e}} }
$$
F. $$
B \frac{\sigma_{\mathrm{e}} \sigma_{\mathrm{h}}}{ \mu_{\mathrm{e}}} \frac{\mu_{\mathrm{h}}}{ \sigma_{\mathrm{h}} + \sigma_{\mathrm{e}}}
$$
G. $$
B \frac{\sigma_{\mathrm{h}} + \mu_{\mathrm{e}}}{ \sigma_{\mathrm{e}}} \frac{\mu_{\mathrm{h}} \sigma_{\mathrm{e}}}{ \sigma_{\mathrm{h}} + \mu_{\mathrm{h}}}
$$
H. $$
B (\sigma_{\mathrm{h}}+\sigma_{\mathrm{e}}) \; \frac{\mu_{\mathrm{h}} \mu_{\mathrm{e}}}{(\mu_{\mathrm{e}} + \sigma_{\mathrm{h}})^{2}}
$$
I. $$
B \frac{\mu_{\mathrm{h}}}{ \sigma_{\mathrm{e}} } \frac{\sigma_{\mathrm{h}} \sigma_{\mathrm{e}}}{ \mu_{\mathrm{e}} + \sigma_{\mathrm{h}} }
$$
J. $$
B (\mu_{\mathrm{h}}+\sigma_{\mathrm{e}}) \; \frac{\sigma_{\mathrm{e}} \mu_{\mathrm{h}}}{(\sigma_{\mathrm{h}}+\mu_{\mathrm{e}})^{2}}
$$
Answer: D
There are $\mathbf{n}$-type CdS square chips with a side length of 1 mm and a thickness of 0.1 mm, with a wavelength absorption limit of 5100 Å. Now, the square surface is irradiated with violet light of intensity $1 \mathrm{mW/cm}^{2}$ ($\lambda = 4096 \dot{\mathrm{A}}$), and the quantum yield is $\beta = 1$. Assume all photogenerated holes are trapped, the photogenerated electron lifetime is $\tau_{n} = 10^{-5}$ s, and the electron mobility is $\mu_{n} = 100 \mathrm{cm}^{2}/(\mathrm{V} \cdot \mathrm{s})$; also assume the light energy is completely absorbed by the chip. How many electron-hole pairs are generated per second in the sample?

Options:
A. $$
4. 5 8 \times1 0^{1 2}
$$
B. $$
1. 1 5 \times1 0^{1 3}
$$
C. $$
2. 4 4 \times1 0^{1 3}
$$
D. $$
5. 1 2 \times1 0^{1 2}
$$
E. $$
2. 0 7 \times1 0^{1 3}
$$
F. $$
3. 7 2 \times1 0^{1 3}
$$
G. $$
4. 0 9 \times1 0^{1 3}
$$
H. $$
4. 8 6 \times1 0^{1 2}
$$
I. $$
3. 5 9 \times1 0^{1 2}
$$
J. $$
3. 0 1 \times1 0^{1 3}
$$
Answer: E
Consider a uniformly doped $n\text{-GaAs}$ crystal with an equilibrium electron concentration of $n_{0}$. Due to certain inhomogeneities or thermal disturbances in the medium, there is an increment in electron concentration denoted as $\delta n$. Assume the differential mobility $\mu = 2000 \ \mathrm{cm}^{2} / ( \mathrm{V} \cdot \mathrm{s} )$, $n_{0} = 5 \times 10^{15} \ \mathrm{cm}^{-3}$, and $\varepsilon_{\gamma} = 13$. What is the value of $\tau_{\mathrm{D}}$?

Options:
A. $$
0. 5 9 \times1 0^{-9} ( \mu s )
$$
B. $$
0. 6 5 \times1 0^{-9} ( \mu s )
$$
C. $$
1. 0 0 \times1 0^{-9} ( \mu s )
$$
D. $$
0. 7 9 \times1 0^{-9} ( \mu s )
$$
E. $$
0. 9 3 \times1 0^{-9} ( \mu s )
$$
F. $$
0. 7 0 \times1 0^{-9} ( \mu s )
$$
G. $$
0. 8 4 \times1 0^{-9} ( \mu s )
$$
H. $$
0. 7 2 \times1 0^{-9} ( \mu s )
$$
I. $$
0. 8 7 \times1 0^{-9} ( \mu s )
$$
J. $$
0. 8 1 \times1 0^{-9} ( \mu s )
$$
Answer: H
The refractive index of a glass for X-rays with a wavelength of 0.70 Å is $1. 6 0 \times1 0^{-8}$ less than 1. What is the maximum refraction angle at which the X-rays can be totally reflected from the outer surface of the glass?

Options:
A. $$
\alpha_{m}=0. 0 8 7 1^{\circ}=5. 2 2 6^{\prime}
$$
B. $$
\alpha_{m}=0. 1 5 3 0^{\circ}=9. 1 8 0^{\prime}
$$
C. $$
\alpha_{m}=0. 2 0 4 0^{\circ}=1 2. 3 0 0^{\prime}
$$
D. $$
\alpha_{m}=0. 1 0 2 5^{\circ}=6. 1 5 0^{\prime}
$$
E. $$
\alpha_{m}=0. 0 9 5 8^{\circ}=5. 7 4 8^{\prime}
$$
F. $$
\alpha_{m}=0. 1 7 4 6^{\circ}=1 0. 4 7 0^{\prime}
$$
G. $$
\alpha_{m}=0. 1 9 2 0^{\circ}=1 1. 5 2 0^{\prime}
$$
H. $$
\alpha_{m}=0. 1 3 6 4^{\circ}=8. 1 8 4^{\prime}
$$
I. $$
\alpha_{m}=0. 1 1 8 0^{\circ}=7. 0 8 0^{\prime}
$$
J. $$
\alpha_{m}=0. 0 9 0 5^{\circ}=5. 4 3 0^{\prime}
$$
Answer: D
There are two wavelengths, $\lambda_{1}$ and $\lambda_{2}$, which differ by 0.001 Å near 6000 Å. To resolve them using a Fabry-Pérot interferometer, what should the spacing $h$ be? Assume the reflectivity $R = 0.95$, give the lower limit of cavity length under resolution requirement.

Options:
A. $$
h \approx2. 8 1 \mathrm{c m}
$$
B. $$
h \approx3. 4 7 \mathrm{c m}
$$
C. $$
h \approx2. 9 4 \mathrm{c m}
$$
D. $$
h \approx2. 7 9 \mathrm{c m}
$$
E. $$
h \approx3. 0 1 \mathrm{c m}
$$
F. $$
h \approx3. 3 5 \mathrm{c m}
$$
G. $$
h \approx2. 5 8 \mathrm{c m}
$$
H. $$
h \approx3. 1 8 \mathrm{c m}
$$
I. $$
h \approx3. 2 4 \mathrm{c m}
$$
J. $$
h \approx2. 6 5 \mathrm{c m}
$$
Answer: C
Given: A steel pipe with an outer diameter of 100 mm runs through a large room at a temperature of 27°C. The outer wall temperature of the pipe is 100°C, and the surface emissivity is 0.85.
Find: The heat loss per unit length of the pipe is().

Options:
A. $$ 3 7 5. 4 8 \mathrm{W} / \mathrm{m}^{2} $$
B. $$ 6 0 2. 3 5 \mathrm{W} / \mathrm{m}^{2} $$
C. $$ 4 5 6. 1 0 \mathrm{W} / \mathrm{m}^{2} $$
D. $$ 4 1 5. 8 2 \mathrm{W} / \mathrm{m}^{2} $$
E. $$ 5 5 0. 2 5 \mathrm{W} / \mathrm{m}^{2} $$
F. $$
4 5 6. 2 5 \mathrm{W} / \mathrm{m}^{2}
$$
G. $$ 5 0 2. 9 1 \mathrm{W} / \mathrm{m}^{2} $$
H. $$ 4 9 0. 4 7 \mathrm{W} / \mathrm{m}^{2} $$
I. $$ 4 8 9. 3 0 \mathrm{W} / \mathrm{m}^{2} $$
J. $$ 4 3 6. 2 5 \mathrm{W} / \mathrm{m}^{2} $$
Answer: F
A spacecraft reenters the atmosphere with a surface shell temperature of $1000^{\circ} \mathrm{C}$ and then falls into an ocean with a temperature of $5^{\circ} \mathrm{C}$. The heat transfer coefficient between the seawater and the shell surface is $1135 W / (m^{2} \cdot K)$. What is the maximum surface temperature of the spacecraft after it enters the ocean? The shell thickness is $\delta=50 \, \text{mm}$, thermal conductivity $\lambda=56.8 \, W / (m \cdot K)$, and thermal diffusivity $\alpha=4.13 \times 10^{-6} \, m^{2} / s$. The inner side can be considered adiabatic.

Options:
A. $$
4 9 8^{0} C
$$
B. $$
4 6 7^{0} C
$$
C. $$
4 8 0^{0} C
$$
D. $$
5 2 9^{0} C
$$
E. $$
5 3 5^{0} C
$$
F. $$
4 5 0^{0} C
$$
G. $$
5 2 2^{0} C
$$
H. $$
5 6 1^{0} C
$$
I. $$
5 1 0^{0} C
$$
J. $$
4 4 0^{0} C
$$
Answer: G
A steel plate with a thickness of 20 mm, after being heated to 500°C, is cooled in air at 20°C. Assume the average surface heat transfer coefficient on both sides of the steel plate during cooling is 35 W/(m²·K), and the thermal conductivity of the steel plate is 45 W/(m²·K), with a diffusivity of 1.375 × 10⁻⁵ m²/s. Determine the time required for the steel plate to cool until its temperature differs from the air by 10°C is().

Options:
A. $$
\tau=3450s
$$
B. $$
\tau=3200s
$$
C. $$
\tau=3633s
$$
D. $$
\tau=2900s
$$
E. $$
\tau=2650s
$$
F. $$
\tau=3980s
$$
G. $$
\tau=2800s
$$
H. $$
\tau=4700s
$$
I. $$
\tau=2500s
$$
J. $$
\tau=4120s
$$
Answer: C
The spherical tank, with inner and outer diameters of 0.5m and 0.6m respectively, is filled with chemical waste of certain radioactivity, having a volumetric heat generation rate of $\Phi=1 \times 10^{5} W/m^{3}$. The tank is cooled by being immersed in a water flow, with a surface heat transfer coefficient of $\mathrm{h=1000} W/(m^{2}.K)$. The fluid temperature is $t_{f}=25^{\circ} \mathrm{C}$. Task: Determine the outer surface temperature of the spherical tank.

Options:
A. $$
36.05^{\circ} \mathrm{C}
$$
B. $$
35.82^{\circ} \mathrm{C}
$$
C. $$
31.22^{\circ} \mathrm{C}
$$
D. $$
34.15^{\circ} \mathrm{C}
$$
E. $$
30.78^{\circ} \mathrm{C}
$$
F. $$
29.45^{\circ} \mathrm{C}
$$
G. $$
33.60^{\circ} \mathrm{C}
$$
H. $$
29.75^{\circ} \mathrm{C}
$$
I. $$
32.50^{\circ} \mathrm{C}
$$
J. $$
28.90^{\circ} \mathrm{C}
$$
Answer: E
A shell-and-tube heat exchanger is used to condense steam at 7355 Pa. It is required to condense 18 kg of steam per hour. The water provided as the cooling medium enters at 25°C and exits at 35°C. Assuming a heat transfer coefficient of $k = 84 \, W/(m^2 \cdot K)$, what is the required heat transfer area?

Options:
A. $$
0. 5 3 m^{2}
$$
B. $$
0. 6 9 m^{2}
$$
C. $$
0. 8 1 m^{2}
$$
D. $$
1. 6 4 m^{2}
$$
E. $$
1. 0 2 m^{2}
$$
F. $$
0. 9 8 m^{2}
$$
G. $$
1. 5 0 m^{2}
$$
H. $$
1. 3 5 m^{2}
$$
I. $$
0. 7 4 m^{2}
$$
J. $$
1. 2 6 m^{2}
$$
Answer: I
Calculate the $N$-point uniformly sampled values of the Fourier transform of the following signal over the interval $\left[ 0, 2 \pi\right]$ is ().
$n ( u ( n+N )-u ( n-N-1 ) ) ;$

Options:
A. $X(k)=- N \frac{1+\mathrm{e}^{-j\frac{2\pi}{N}k}}{1-\mathrm{e}^{-j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\cos\left(\frac{\pi}{N}k\right)}{\sin\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
B. $X(k)=N \frac{1+\mathrm{e}^{j\frac{2\pi}{N}k}}{1-\mathrm{e}^{j\frac{2\pi}{N}k}}=-\mathrm{j}N \frac{\sin\left(\frac{\pi}{N}k\right)}{\cos\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
C. $X(k)=- N \frac{1-\mathrm{e}^{-j\frac{2\pi}{N}k}}{1-\mathrm{e}^{j\frac{2\pi}{N}k}}=-\mathrm{j}N \frac{\sin\left(\frac{\pi}{N}k\right)}{\cos\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
D. $X(k)=- N \frac{1+\mathrm{e}^{j\frac{2\pi}{N}k}}{1+\mathrm{e}^{-j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\cos\left(\frac{\pi}{N}k\right)}{\sin\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
E. $X(k)=- N \frac{1+\mathrm{e}^{j\frac{2\pi}{N}k}}{1-\mathrm{e}^{j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\sin\left(\frac{\pi}{N}k\right)}{\cos\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
F. $X(k)=N \frac{1-\mathrm{e}^{-j\frac{2\pi}{N}k}}{1-\mathrm{e}^{j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\sin\left(\frac{\pi}{N}k\right)}{\cos\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
G. $X(k)=N \frac{1-\mathrm{e}^{j\frac{2\pi}{N}k}}{1+\mathrm{e}^{j\frac{2\pi}{N}k}}=-\mathrm{j}N \frac{\cos\left(\frac{\pi}{N}k\right)}{\sin\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
H. $X(k)=- N \frac{1-\mathrm{e}^{-j\frac{2\pi}{N}k}}{1+\mathrm{e}^{-j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\sin\left(\frac{\pi}{N}k\right)}{\cos\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
I. $X(k)=N \frac{1+\mathrm{e}^{-j\frac{2\pi}{N}k}}{1+\mathrm{e}^{j\frac{2\pi}{N}k}}=-\mathrm{j}N \frac{\cos\left(\frac{\pi}{N}k\right)}{\sin\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
J. $X(k)=N \frac{1-\mathrm{e}^{j\frac{2\pi}{N}k}}{1-\mathrm{e}^{-j\frac{2\pi}{N}k}}=\mathrm{j}N \frac{\cos\left(\frac{\pi}{N}k\right)}{\sin\left(\frac{\pi}{N}k\right)},\quad k=0,1,2,\cdots,N-1$
Answer: A
Given that $x(n)$ is a finite-length sequence with $N$ points, and $X(k) = \operatorname{DFT}[x(n)]$. Now, by inserting $r-1$ zero points between every two points of $x(n)$, a finite-length sequence $y(n)$ with $rN$ points is obtained:
$$
y(n) = \left\{\begin{matrix} x\left(\frac{n}{r}\right), & n = ir, & i = 0, 1, 2, \cdots, N-1 \\ 0, & \text{others}& \end{matrix}\right.
$$
Try to find the relationship between the $rN$point $\operatorname{DFT}[y(n)] = Y(k)$ and $X(k)$ is ().

Options:
A. $$
Y ( k )=X \left( ( k ) \right)_{N} R_{r N} ( k )
$$
B. $$
Y ( k )=X \left( ( k/2 ) \right)_{N} U_{, N} ( k )
$$
C. $$
Y ( k )=X \left( ( rk ) \right)_{N} P_{, N} ( k )
$$
D. $$
Y ( k )=X \left( ( k - r ) \right)_{N} Q_{, N} ( k )
$$
E. $$
Y ( k )=X \left( ( k ) \right)_{N/2} V_{, N} ( k )
$$
F. $$
Y ( k )=X \left( ( k^2 ) \right)_{N} B_{, N} ( k )
$$
G. $$
Y ( k )=X \left( ( k r ) \right)_{N} M_{, N} ( k )
$$
H. $$
Y ( k )=X \left( ( k ) \right)_{r} S_{, N} ( k )
$$
I. $$
Y ( k )=X \left( ( k ) \right)_{2N} T_{, N} ( k )
$$
J. $$
Y ( k )=X \left( ( k + r ) \right)_{N} L_{, N} ( k )
$$
Answer: A
Design a high-pass digital filter with a passband cutoff frequency of $\omega_{\mathfrak{p}} = 0.8\pi$ rad, passband gain attenuation not exceeding 3dB, stopband cutoff frequency of $\omega_s = 0.5\pi$ rad, and stopband gain attenuation not less than 18dB. Try using a Butterworth filter for implementation is ().

Options:
A. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-2z^{-1}+z^{-2}}{10.9876+14.1234z^{-1}+10.9876z^{-2}}$$
B. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-2.5z^{-1}+z^{-2}}{16.8765+19.0098z^{-1}+16.8765z^{-2}}$$
C. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1+z^{-1}+z^{-2}}{20.4321+15.2345z^{-1}+13.8765z^{-2}}$$
D. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-3z^{-1}+z^{-2}}{12.5678+18.0034z^{-1}+12.5678z^{-2}}$$
E. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1+1.5z^{-1}+z^{-2}}{15.4321+17.2345z^{-1}+15.4321z^{-2}}$$
F. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-2z^{-1}+z^{-2}}{14.8194+16.9358z^{-1}+14.8194z^{-2}}$$
G. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-4z^{-1}+z^{-2}}{11.9876+15.8765z^{-1}+11.9876z^{-2}}$$
H. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-z^{-1}+z^{-2}}{19.4321+13.2345z^{-1}+19.4321z^{-2}}$$
I. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-5z^{-1}+z^{-2}}{13.9876+12.7943z^{-1}+13.9876z^{-2}}$$
J. $$H(z)=H_a(s)\bigg|_{s=\frac{1-s^{-1}}{1+s^{-1}}}=\frac{1-1.5z^{-1}+z^{-2}}{17.5678+14.0034z^{-1}+17.5678z^{-2}}$$
Answer: F
Design a Chebyshev low-pass filter with a passband ripple not exceeding 2dB. The cutoff frequency of the filter should be 40 rad/s, and the filter gain attenuation should be greater than 20dB at a stopband frequency of 52 rad/s is ().

Options:
A. $$H_{\mathrm{d}}(s)=\frac{7.20\times10^6}{(s+7.20)(s^2+4.80s+1350)(s^2+11.0s+500)}$$
B. $$H_{\mathrm{d}}(s)=\frac{7.50\times10^6}{(s+7.50)(s^2+4.50s+1300)(s^2+12.0s+550)}$$
C. $$H_{\mathrm{d}}(s)=\frac{8.90\times10^6}{(s+8.90)(s^2+5.80s+1500)(s^2+15.0s+650)}$$
D. $$H_{\mathrm{d}}(s)=\frac{5.00\times10^6}{(s+5.00)(s^2+3.50s+1100)(s^2+10.0s+450)}$$
E. $$H_{\mathrm{d}}(s)=\frac{8.37\times10^6}{(s+8.37)(s^2+5.39s+1520)(s^2+14.1s+627)}$$
F. $$H_{\mathrm{d}}(s)=\frac{10.1\times10^6}{(s+10.1)(s^2+7.10s+1700)(s^2+18.0s+750)}$$
G. $$H_{\mathrm{d}}(s)=\frac{9.50\times10^6}{(s+9.50)(s^2+6.50s+1650)(s^2+17.0s+720)}$$
H. $$H_{\mathrm{d}}(s)=\frac{6.00\times10^6}{(s+6.00)(s^2+4.00s+1400)(s^2+12.0s+580)}$$
I. $$H_{\mathrm{d}}(s)=\frac{9.00\times10^6}{(s+9.00)(s^2+6.00s+1600)(s^2+16.0s+700)}$$
J. $$H_{\mathrm{d}}(s)=\frac{6.80\times10^6}{(s+6.80)(s^2+4.20s+1450)(s^2+13.5s+600)}$$
Answer: E
Given a uniformly charged sphere with a radius of $a$ and a total charge of $Q$, determine the divergence and curl of the electric field intensity $E$ at a point inside the sphere is ().

Options:
A. $$
\frac{\rho} {\varepsilon_{0}a},0
$$
B. $$
\frac{\rho} {\varepsilon_{0}},0
$$
C. $$
\frac{\rho a} {\varepsilon_{0}},0
$$
D. $$
\frac{Q} {\varepsilon_{0}},0
$$
E. $$
\frac{\rho} {2\varepsilon_{0}},0
$$
F. $$
\frac{Q} {a^2\varepsilon_{0}},0
$$
G. $$
\frac{Q} {4\pi\varepsilon_{0}},0
$$
H. $$
\frac{\rho} {a},0
$$
I. $$
\frac{\rho} {3\varepsilon_{0}},0
$$
J. $$
\frac{Q} {a^3\varepsilon_{0}},0
$$
Answer: B
In a rectangular waveguide filled with an ideal medium where $\mathbf{e}_{t}=\mathbf{9}$, with waveguide dimensions of $a \times b = 23 \, \mathrm{mm} \times 10 \, \mathrm{mm}$, what should be the range of the operating wavelength to allow only the $\mathrm{TE}_{10}$ mode to propagate?

Options:
A. $$
5 0 \mathrm{~ m m} < \lambda_{0} < 1 3 5 \mathrm{~ m m}
$$
B. $$
7 8 \mathrm{~ m m} < \lambda_{0} < 1 3 3 \mathrm{~ m m}
$$
C. $$
6 0 \mathrm{~ m m} < \lambda_{0} < 1 4 0 \mathrm{~ m m}
$$
D. $$
8 5 \mathrm{~ m m} < \lambda_{0} < 1 4 2 \mathrm{~ m m}
$$
E. $$
6 9 \mathrm{~ m m} < \lambda_{0} < 1 3 8 \mathrm{~ m m}
$$
F. $$
8 0 \mathrm{~ m m} < \lambda_{0} < 1 4 5 \mathrm{~ m m}
$$
G. $$
6 5 \mathrm{~ m m} < \lambda_{0} < 1 3 5 \mathrm{~ m m}
$$
H. $$
7 0 \mathrm{~ m m} < \lambda_{0} < 1 4 0 \mathrm{~ m m}
$$
I. $$
5 5 \mathrm{~ m m} < \lambda_{0} < 1 3 0 \mathrm{~ m m}
$$
J. $$
7 5 \mathrm{~ m m} < \lambda_{0} < 1 2 5 \mathrm{~ m m}
$$
Answer: E
A circular air waveguide with a radius of $R=5$ cm is used to transmit the $TE_{01}$ mode. If the operating frequency $f={\sqrt{3}} ( f_{e} )_{{TE}_{01}}$, find the waveguide wavelength, phase shift constant, and wave impedance of the $TE_{01}$ mode is ().

Options:
A. $$
5. 7 \ \mathrm{c m}
$$
$$
1. 0 7 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 7 0. 8 5 \Omega
$$
B. $$
6. 2 \ \mathrm{c m}
$$
$$
1. 4 0 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
5 1 0. 3 0 \Omega
$$
C. $$
5. 9 \ \mathrm{c m}
$$
$$
1. 0 3 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 8 5. 7 0 \Omega
$$
D. $$
4. 9 \ \mathrm{c m}
$$
$$
1. 0 2 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 2 0. 0 0 \Omega
$$
E. $$
5. 2 \ \mathrm{c m}
$$
$$
1. 1 5 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 3 0. 2 0 \Omega
$$
F. $$
6. 0 \ \mathrm{c m}
$$
$$
1. 2 0 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
5 0 0. 0 0 \Omega
$$
G. $$
5. 4 \ \mathrm{c m}
$$
$$
1. 1 0 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 5 0. 5 5 \Omega
$$
H. $$
6. 5 \ \mathrm{c m}
$$
$$
1. 3 0 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
5 2 5. 9 0 \Omega
$$
I. $$
5. 6 \ \mathrm{c m}
$$
$$
1. 0 8 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 6 1. 7 5 \Omega
$$
J. $$
5. 8 \ \mathrm{c m}
$$
$$
1. 0 5 \ \mathrm{r a d} / \mathrm{c m}
$$
$$
4 8 0. 4 5 \Omega
$$
Answer: I
In a certain pool, there is \(1 \, m^3\) of wastewater containing a total nitrogen concentration of 20 mg/L. Surface water is used for replacement, entering the pool at a flow rate of \(10 \, m^3/\text{min}\), with a total nitrogen concentration of 2 mg/L, while the same amount of water is discharged from the pool. Assuming that the mixing in the pool is good and the biodegradation process can be ignored, how much time is needed for the total nitrogen concentration in the pool to decrease to 5 mg/L?

Options:
A. $$
0. 2 0 \operatorname* {m i n}
$$
B. $$
0. 4 0 \operatorname* {m i n}
$$
C. $$
0. 4 5 \operatorname* {m i n}
$$
D. $$
0. 1 5 \operatorname* {m i n}
$$
E. $$
0. 3 5 \operatorname* {m i n}
$$
F. $$
0. 0 8 \operatorname* {m i n}
$$
G. $$
0. 3 0 \operatorname* {m i n}
$$
H. $$
0. 2 5 \operatorname* {m i n}
$$
I. $$
0. 5 0 \operatorname* {m i n}
$$
J. $$
0. 1 8 \operatorname* {m i n}
$$
Answer: J
Water flows in a laminar pattern through a circular straight pipe. If the flow rate remains constant, explain how the energy loss due to flow resistance changes under the following condition. If the pipe diameter is doubled, what will the pressure drop become?

Options:
A. $$
1/ 7 6
$$
B. $$
1/ 1 5
$$
C. $$
1/ 2 5
$$
D. $$
1/ 3 4
$$
E. $$
1/ 4 3
$$
F. $$
1/ 1 2
$$
G. $$
1/ 3 5
$$
H. $$
1/ 1 6
$$
I. $$
1 / 1 4
$$
J. $$
1/ 1 7
$$
Answer: H
Determining the thermal conductivity of a material using the flat plate method involves heating one side of the plate with an electric heater while removing heat from the other side through a water-cooled layer. The surface temperatures on both sides of the plate are measured using thermocouples, and the heat passing through the plate is equal to the power consumed by the electric heater. Suppose the heated area of a certain material is $0.02 \, \mathrm{m}^{2}$ with a thickness of $0.01 \, \mathrm{m}$. When the electric current and voltage of the heater are 2.8A and 140V respectively, the temperatures on both sides of the plate are 300°C and 100°C. When the current and voltage are 2.28A and 114V respectively, the temperatures on both sides of the plate are 200°C and 50°C. If the relationship between the material's thermal conductivity and temperature is linear, i.e., $\lambda=\lambda_{0} (1 + aT)$ (where $\mathrm{T}$ is in °C), try to determine the expression for the relationship between thermal conductivity and temperature is ().

Options:
A. $$
0. 6 8 9 \ \mathrm{( 1+2. 1 0 \times1 0^{-3} T )}
$$
B. $$
0. 6 7 7 \ \mathrm{( 1+2. 2 4 \times1 0^{-3} T )}
$$
C. $$
0. 5 9 3 \ \mathrm{( 1+1. 8 7 \times1 0^{-3} T )}
$$
D. $$
0. 7 3 3 \ \mathrm{( 1+2. 4 5 \times1 0^{-3} T )}
$$
E. $$
0. 7 1 4 \ \mathrm{( 1+2. 3 0 \times1 0^{-3} T )}
$$
F. $$
0. 6 8 2 \ \mathrm{( 1+2. 1 9 \times1 0^{-3} T )}
$$
G. $$
0. 6 0 2 \ \mathrm{( 1+2. 0 9 \times1 0^{-3} T )}
$$
H. $$
0. 5 9 8 \ \mathrm{( 1+2. 5 5 \times1 0^{-3} T )}
$$
I. $$
0. 7 0 1 \ \mathrm{( 1+2. 5 0 \times1 0^{-3} T )}
$$
J. $$
0. 6 5 9 \ \mathrm{( 1+2. 3 5 \times1 0^{-3} T )}
$$
Answer: B
The surface temperature of a water heater is 80°C, with a surface area of \(2 \, \mathrm{m}^2\) at \(20^{\circ} C\). Considering it as a black body, calculate the energy loss due to radiation is ().

Options:
A. $$
1 0 8 0. 6 8 W
$$
B. $$
9 2 9. 5 0 W
$$
C. $$
1 0 1 5. 3 0 W
$$
D. $$
9 7 2. 3 5 W
$$
E. $$
9 2 5. 0 4 W
$$
F. $$
8 4 0. 7 8 W
$$
G. $$
8 6 4. 9 0 W
$$
H. $$
8 5 2. 6 4 W
$$
I. $$
9 1 8. 2 7 W
$$
J. $$
9 9 1. 4 9 W
$$
Answer: E
Assume the concentration of the liquid phase is uniform, the surface area of bromine particles per unit volume of solution is a, the initial bromine content in the water is 0, and the saturation concentration at the surface of the bromine particles is $c_{\mathrm A S}$.
Bromine particles quickly dissolve in water under stirring. After 3 minutes, the concentration of the solution is measured at 50% saturation. Calculate the mass transfer coefficient of the system.

Options:
A. $$
6. 5 7 \times1 0^{-3} \mathrm{s}^{-1}
$$
B. $$
7. 6 8 \times1 0^{-3} \mathrm{s}^{-1}
$$
C. $$
1. 3 6 \times1 0^{-3} \mathrm{s}^{-1}
$$
D. $$
4. 4 5 \times1 0^{-3} \mathrm{s}^{-1}
$$
E. $$
3. 8 5 \times1 0^{-3} \mathrm{s}^{-1}
$$
F. $$
8. 0 9 \times1 0^{-3} \mathrm{s}^{-1}
$$
G. $$
0. 8 5 \times1 0^{-3} \mathrm{s}^{-1}
$$
H. $$
2. 7 9 \times1 0^{-3} \mathrm{s}^{-1}
$$
I. $$
5. 1 2 \times1 0^{-3} \mathrm{s}^{-1}
$$
J. $$
9. 2 3 \times1 0^{-3} \mathrm{s}^{-1}
$$
Answer: E
Using a batch reactor, a certain bacterium is cultured from $0.2 \, \mathrm{g/L}$ to $14 \, \mathrm{g/L}$ over 24 hours. Assuming the cultivation process follows a logarithmic growth phase, with $Y_{x/s} = 0.5$, $m_{x} = 0$, and $S_{0} = 10 \,\mathrm{g/L}$, calculate the substrate concentration after 10 hours of cultivation is ().

Options:
A. $$
11. 1 \ g / \mathrm{L}
$$
B. $$
8. 1 \ g / \mathrm{L}
$$
C. $$
4. 3 \ g / \mathrm{L}
$$
D. $$
5. 6 \ g / \mathrm{L}
$$
E. $$
6. 8 \ g / \mathrm{L}
$$
F. $$
3. 9 \ g / \mathrm{L}
$$
G. $$
9. 5 \ g / \mathrm{L}
$$
H. $$
7. 4 \ g / \mathrm{L}
$$
I. $$
10. 2 \ g / \mathrm{L}
$$
J. $$
12. 5 \ g / \mathrm{L}
$$
Answer: B
To transport water from a pool to a 10m open elevated tank using a centrifugal pump. The transfer flow rate is $q_V= 20 \mathrm{m}^3/h$, with a total pipeline length of $L=35$ meters (including all equivalent lengths of local resistance), and a pipe diameter of 40mm. The friction coefficient is $\lambda=0.025$. If the characteristic equation of the selected centrifugal pump within the operating range is $H_{\mathrm{e}}=60-7.9\times10^{5}q_{V}^{2}$, where units are $H_{\mathrm{e}},m;q_{\mathrm{V}}, \mathrm{m}^3/s$. What is the actual pipeline flow rate after the pump operates normally ($m^{3}/h$)?

Options:
A. $$
2 3. 0 1 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
B. $$
1 8. 9 6 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
C. $$
2 0. 6 5 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
D. $$
2 2. 9 3 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
E. $$
2 1. 0 9 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
F. $$
2 0. 4 7 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
G. $$
2 1. 3 2 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
H. $$
2 0. 8 1 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
I. $$
1 9. 7 8 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
J. $$
2 2. 5 4 {\mathrm{m}}^{3} / {\mathrm{h}}
$$
Answer: H
A filter press initially filters at a constant speed for 10 minutes, yielding 5 liters of filtrate. After that, with this maximum pressure maintained, it performs constant pressure filtration. How many liters of filtrate can be obtained during the 60 minutes of constant pressure filtration? Assume the resistance of the filtering medium $V_{\mathrm{e}}$ = 0.5 L.

Options:
A. 14.0L
B. 10.5L
C. 11.5L
D. 13.5L
E. 14.5L
F. 16.0L
G. 15.5L
H. 15.0L
I. 12.0L
J. 12.5L
Answer: D
A spherical particle with a diameter of $0.12\mathrm{mm}$ and a density of $2300\mathrm{kg}/\mathrm{m}^{3}$ freely settles in water at $20^{\circ}\mathrm{C}$. Please calculate the distance required for the particle to reach 99% of its settling velocity starting from rest.

Options:
A. $$
1. 0 4 \times1 0^{-5} \, \mathrm{m}
$$
B. $$
4. 5 5 \times1 0^{-5} \, \mathrm{m}
$$
C. $$
5. 7 2 \times1 0^{-5} \, \mathrm{m}
$$
D. $$
0. 5 3 \times1 0^{-5} \, \mathrm{m}
$$
E. $$
9. 8 1 \times1 0^{-5} \, \mathrm{m}
$$
F. $$
6. 7 5 \times1 0^{-5} \, \mathrm{m}
$$
G. $$
3. 9 6 \times1 0^{-5} \, \mathrm{m}
$$
H. $$
7. 1 8 \times1 0^{-5} \, \mathrm{m}
$$
I. $$
2. 3 4 \times1 0^{-5} \, \mathrm{m}
$$
J. $$
8. 2 9 \times1 0^{-5} \, \mathrm{m}
$$
Answer: F
In a certain double-pipe heat exchanger, the inner pipe is a $\phi\,30\,\mathrm{mm}\times3\,\mathrm{mm}$ copper tube, and the outer pipe is a $\phi48\mathrm{mm}\times3\mathrm{mm}$ steel tube, with a length of $2\mathrm{m}$. Saturated steam at 105°C flows between the pipes to heat the air inside the inner pipe, raising the air temperature from 20°C to 75°C. Given that the air flow rate is 50 kg/h and the ambient temperature is 20°C, what is the heat transfer coefficient on the air side inside the pipe?

Options:
A. $$
1 1 2. \ 6 \mathrm{W / ( m^{2} \cdot K )}
$$
B. $$
8 7. \ 5 \mathrm{W / ( m^{2} \cdot K )}
$$
C. $$
9 6. \ 3 \mathrm{W / ( m^{2} \cdot K )}
$$
D. $$
1 0 0. \ 5 \mathrm{W / ( m^{2} \cdot K )}
$$
E. $$
1 0 4. \ 7 \mathrm{W / ( m^{2} \cdot K )}
$$
F. $$
8 9. \ 9 \mathrm{W / ( m^{2} \cdot K )}
$$
G. $$
9 9. \ 8 \mathrm{W / ( m^{2} \cdot K )}
$$
H. $$
9 0. \ 2 \mathrm{W / ( m^{2} \cdot K )}
$$
I. $$
7 5. \ 3 \mathrm{W / ( m^{2} \cdot K )}
$$
J. $$
8 2. \ 0 \mathrm{W / ( m^{2} \cdot K )}
$$
Answer: C
A shell-and-tube heat exchanger with single shell and double tube passes uses 120°C saturated steam on the shell side to heat atmospheric pressure air. The air flows through the tubes at a velocity of 12 m/s with a tube diameter of $\phi\, 38\, \mathrm{mm} \times 2.5\, \mathrm{mm}$, and there are a total of 200 tubes. Given that the inlet air temperature is 26°C, it is required that the outlet air temperature be 86°C. Determine the necessary tube length for this heat exchanger.

Options:
A. 4.07m
B. 0.86m
C. 2.87m
D. 2.15m
E. 1.92m
F. 3.46m
G. 0.95m
H. 5.18m
I. 3.26m
J. 1.08m
Answer: J
A certain shell-and-tube heat exchanger uses steam at an absolute pressure of 0.2 MPa to heat a certain amount of cold water from 20°C to 60°C. If the pressure of the heating steam and the inlet temperature of the water remain unchanged, and the outlet temperature of the water rises to 62°C, what is the change in the heat transfer rate of this heat exchanger compared to the original operating conditions? (Expressed as a relative ratio to the original condition.) In the calculation, you may ignore the fouling resistance, the thermal resistance of the tube wall, and the steam side. The water flows turbulently inside the tubes. The saturation temperature of steam at an absolute pressure of 0.2 MPa is 120°C.

Options:
A. 0.845
B. 0.793
C. 0.761
D. 0.923
E. 0.679
F. 0.707
G. 0.789
H. 0.882
I. 0.740
J. 0.611
Answer: C
In a certain factory, a one-pass shell-and-tube heat exchanger is used to heat an ethanol-water solution from 25°C to 80°C with saturated steam at 130°C. The processing capacity of the ethanol-water solution is 48,000 kg/h, flowing inside the tubes. Saturated steam condenses between the tubes. The heat exchanger consists of 60 steel tubes with dimensions of φ 25 mm x 2.5 mm, each 4.5 meters long. The thermal conductivity of the steel is 45 W/(m·°C). The density of the ethanol-water solution is 880 kg/m³, its viscosity is 1.2 mPa·s, its specific heat capacity is 4.02 kJ/(kg·°C), and its thermal conductivity is 0.42 W/(m·°C). The heat transfer coefficient for the condensation of the steam is 10⁴ W/(m²·°C). Under operating conditions, the thermal resistance of fouling and heat loss is negligible. It is suggested that the one-pass system be changed to a two-pass system without changing the number of tubes. If the inlet and outlet temperatures of the ethanol-water solution remain unchanged, to what temperature should the saturated steam be adjusted?

Options:
A. 123.3°C
B. 136.4°C
C. 119.9°C
D. 118.2°C
E. 129.5°C
F. 127.6°C
G. 121.8°C
H. 134.1°C
I. 115.7°C
J. 128.9°C
Answer: G
A certain packed absorption tower is 2.7 meters high and uses clean water under atmospheric pressure to counter-currently absorb ammonia from a mixed gas. The molar flow rate of the mixed gas entering the tower is $0.03 \, \mathrm{kmol}/(\mathrm{m}^{2}\cdot\mathrm{s})$. The spray density of the clean water is $0.018 \, \mathrm{kmol}/(\mathrm{m}^{2}\cdot\mathrm{s})$. The inlet gas contains 2% ammonia (by volume). It is known that the overall gas phase mass transfer coefficient $K_{y}a$ is $0.1 \, \mathrm{kmol}/(\mathrm{m}^{3}\cdot\mathrm{s})$, and the Henry's constant under the operating conditions is $60\,\mathrm{kPa}$. What is the concentration of ammonia in the exhaust gas?

Options:
A. $$
y_{\text{out}}=0.002
$$
B. $$
y_{\text{out}}=0.010
$$
C. $$
y_{\text{out}}=0.0015
$$
D. $$
y_{\text{out}}=0.007
$$
E. $$
y_{\text{out}}=0.006
$$
F. $$
y_{\text{out}}=0.0035
$$
G. $$
y_{\text{out}}=0.005
$$
H. $$
y_{\text{out}}=0.009
$$
I. $$
y_{\text{out}}=0.0085
$$
J. $$
y_{\text{out}}=0.004
$$
Answer: A
To eliminate all microorganisms, including bacterial spores. How long is the autoclave steam sterilization method typically maintained at a pressure of $1.034 \times 10^3 \text{kPa}$?

Options:
A. 1 to 3 minutes
B. 40 to 45 minutes
C. 10 to 15 minutes
D. 50 to 55 minutes
E. 60 to 65 minutes
F. 15 to 20 minutes
G. 25 to 30 minutes
H. 2 to 4 minutes
I. 30 to 35 minutes
J. 5 to 10 minutes
Answer: F
Which of the following concepts about abortion is correct?

Options:
A. Therapeutic abortion-primarily occurs without clinical indications
B. Missed miscarriage-immediately presents with heavy bleeding
C. Recurrent abortion-never involves genetic counseling
D. Spontaneous abortion-always requires surgical intervention
E. Elective termination-only performed after gestational viability
F. Inevitable miscarriage-generally develops from threatened miscarriage
G. Septic abortion-routinely results from medical treatments
H. Early miscarriage-frequently resolves with no medical attention
I. Complete miscarriage-typically necessitates hormone therapy
J. Induced miscarriage-exclusively due to nutritional deficiencies
Answer: F
26-year-old, pregnant. (G,P.), the uterus stopped growing at 4 months of pregnancy. Examination revealed that the uterus was smaller than expected for the gestational age, but there was no natural miscarriage even after 6 weeks. Which of the following should be closely monitored?

Options:
A. Amniotic fluid
B. Fibrinogen
C. Progesterone
D. HCG levels
E. Thyroid function
F. White blood cells
G. Blood calcium
H. Estrogen
I. Placental growth
J. Red blood cells
Answer: B
28 years old, gravida para (G.P.). At 35 weeks of pregnancy with severe preeclampsia, pregnancy needs to be terminated. Which of the following indicates fetal immaturity?

Options:
A. Amniotic fluid phosphatidylglycerol absence confirms insufficient lung maturity
B. Amniotic fluid glucose concentration measured below optimal neonatal levels
C. Amniotic fluid bilirubin substances measured by $\Delta_{450}$OD...value <0.02
D. Amniotic fluid creatinine concentration indicates insufficient renal development
E. Amniotic fluid lactate indicates inefficient fetal metabolic activity
F. Amniotic fluid bilirubin substances measured by $\Delta OD_{450}$...value >0.02
G. Amniotic fluid lecithin-to-sphingomyelin ratio remains under adequate threshold
H. Amniotic fluid volume lower suggests restricted fetal movement capacity
I. Amniotic fluid uric acid levels suggest inadequate fetal digestive maturity
J. Amniotic fluid acetylcholine esterase test suggests neural tube closure defect
Answer: F
A 48-year-old woman, GP, has been experiencing dysmenorrhea for 20 years, with worsening symptoms in the past year. Her menstrual cycle is still regular, with abdominal pain starting on the first day of menstruation, worsening during the period, and disappearing after the period ends. During menstruation, she experiences a sensation of heaviness in the anus, making sitting difficult, and an increased frequency of bowel movements. Gynecological examination reveals the uterus is the size consistent with an 8-week pregnancy, with a uniformly firm texture. A nodular tenderness the size of a soybean can be felt on the uterosacral ligaments during a combined examination. The clinical diagnosis is endometriosis. To alleviate the abdominal pain symptoms, the following medication is administered immediately; which is the only choice?

Options:
A. Diclofenac (Voltaren)
B. Celecoxib (Celebrex)
C. Acetaminophen (Tylenol)
D. Ketoprofen (Orudis)
E. Ibuprofen (Advil)
F. Aspirin (Acetylsalicylic Acid)
G. Indomethacin (Indocin)
H. Mefenamic acid (Ponstel)
I. Naproxen (Aleve)
J. Flufenamic acid (Flumetacin)
Answer: J
Let $X$ follow a Poisson distribution with parameter $\lambda = 3$. Then $P(|X - 3| \geq 2)$ = ().

Options:
A. $1-\frac{66}{8}e^{-3}=0.294$
B. $1-\frac{99}{8}e^{-3}=0.384$
C. $1-\frac{22}{8}e^{-3}=0.359$
D. $1-\frac{55}{8}e^{-3}=0.262$
E. $1-\frac{77}{8}e^{-3}=0.358$
F. $1-\frac{88}{8}e^{-3}=0.413$
G. $1-\frac{44}{8}e^{-3}=0.317$
H. $1-\frac{99}{5}e^{-3}=0.298$
I. $1-\frac{11}{5}e^{-3}=0.239$
J. $1-\frac{27}{8}e^{-3}=0.473$
Answer: B
Let $X_1, X_2, X_3$ be random variables, and $X_1 \sim N(0,1), X_2 \sim N(0,2^2), X_3 \sim N(5,3^2)$. Let $P_{i} = P\left\{-2 \leqslant X_{i} \leqslant 2\right\}$ for ($i = 1,2,3$). Then ().

Options:
A. $P_{3}>P_{1}>P_{2}$
B. $P_{2}=P_{1}>P_{3}$
C. $P_{1}>P_{3}>P_{2}$
D. $P_{1}>P_{2}=P_{3}$
E. $P_{2}>P_{3}>P_{1}$
F. $P_{1}>P_{2}>P_{3}$
G. $P_{3}=P_{2}>P_{1}$
H. $P_{3}>P_{2}>P_{1}$
I. $P_{1}=P_{3}>P_{2}$
J. $P_{2}>P_{1}>P_{3}$
Answer: F
Given that the population $X$ follows a normal distribution $N(\mu, \sigma^2)$, and $X_1, X_2, \cdots, X_n$ are samples, where $\overline{X}$ and $S^2$ are the sample mean and sample variance respectively. Also assume $X_{n+1} \sim N(\mu, \sigma^2)$ and is independent from $X_1, X_2, \cdots, X_n$. Determine the distribution of the statistic $T = \frac{X_{n+1} - \overline{X}}{S} \sqrt{\frac{n}{n+1}}$ is().

Options:
A. The statistic $T$ follows a $t$-distribution with $n$ degrees of freedom.
B. The statistic $T$ follows a beta distribution with parameters $n-1$.
C. The statistic $T$ follows a chi-squared distribution with $n-1$ degrees.
D. The statistic $T$ follows an F-distribution with $n-1$ degrees of freedom.
E. The statistic $T$ follows a $t$-distribution with $n-1$ degrees of freedom.
F. The statistic $T$ follows a binomial distribution with $n-1$ trials.
G. The statistic $T$ follows a $t$-distribution with $n+1$ degrees of freedom.
H. The statistic $T$ follows an exponential distribution with rate $n-1$.
I. The statistic $T$ follows a normal distribution with $n-1$ degrees of freedom.
J. The statistic $T$ follows a normal distribution with $\sigma$ squared variance.
Answer: E
There is a certain compression ignition internal combustion engine with an output power of 3700 kW. The calorific value of the fuel is $4.65 \times 10^{4} \ \mathrm{kJ/kg}$. Assuming the cycle can be simplified to an ideal cycle with constant pressure heating of air, the cycle compression ratio is 14, and the pre-expansion ratio is 2. Determine the fuel consumption is(). The specific heat of air is given as a constant value, $R_{\mathrm{g}}=287 \ \mathrm{J/(kg \cdot K)}$, and $\kappa=1.4$.

Options:
A. 0.145 kg/s
B. 0.134 kg/s
C. 0.140 kg/s
D. 0.122 kg/s
E. 0.138 kg/s
F. 0.129 kg/s
G. 0.131 kg/s
H. 0.137 kg/s
I. 0.128 kg/s
J. 0.142 kg/s
Answer: B
A container with a volume of $2.8 \, \mathrm{m}^3$ is filled with 1,350 kg of water and steam in equilibrium, with a temperature and pressure of 230 °C and 2.79505 MPa, respectively. Now, 900 kg of water at 65 °C is pumped into the container. How much heat should be added during this process to maintain the initial temperature and pressure inside the container?

Options:
A. $$ 1 9 6 5 3 2. 9 \ \mathrm{k J} $$
B. $$ 7 5 8 3 2 1. 6 \ \mathrm{k J} $$
C. $$ 8 2 7 6 9 0. 3 \ \mathrm{k J} $$
D. $$ 2 0 7 4 8 1. 7 \ \mathrm{k J} $$
E. $$ 3 5 4 1 9 7. 8 \ \mathrm{k J} $$
F. $$ 4 6 2 7 8 6. 5 \ \mathrm{k J} $$
G. $$ 5 4 9 8 4 3. 2 \ \mathrm{k J} $$
H. $$ 6 8 4 3 2 5. 0 \ \mathrm{k J} $$
I. $$ 9 3 1 2 5 4. 1 \ \mathrm{k J} $$
J. $$
6 1 5 \ 9 5 4. 4 \ \mathrm{k J}
$$
Answer: J
A mixture of hydrogen and nitrogen gases produces ammonia under chemical equilibrium at 400 °C and 10 atm, resulting in 3.85% ammonia (by volume). Find: the equilibrium constant $K_{p}$ for the reaction $3 \mathrm{H}_{2}+\mathrm{N}_{2} \rightleftharpoons2 \mathrm{N} \mathrm{H}_{3}$ at 400 °C is().

Options:
A. $$
5. 4 3 8 \times1 0^{-6} \mathrm{~ a t m}^{-2}
$$
B. $$
6. 8 2 3 \times1 0^{-4} \mathrm{~ a t m}^{-2}
$$
C. $$
2. 9 3 5 \times1 0^{-5} \mathrm{~ a t m}^{-2}
$$
D. $$
2. 1 0 7 \times1 0^{-3} \mathrm{~ a t m}^{-2}
$$
E. $$
3. 6 0 9 \times1 0^{-3} \mathrm{~ a t m}^{-2}
$$
F. $$
7. 0 1 4 \times1 0^{-6} \mathrm{~ a t m}^{-2}
$$
G. $$
4. 5 7 2 \times1 0^{-3} \mathrm{~ a t m}^{-2}
$$
H. $$
8. 7 5 9 \times1 0^{-5} \mathrm{~ a t m}^{-2}
$$
I. $$
1. 6 4 4 \times1 0^{-4} \mathrm{~ a t m}^{-2}
$$
J. $$
9. 1 8 2 \times1 0^{-5} \mathrm{~ a t m}^{-2}
$$
Answer: I