MikaStars39/nano-eval huggingface dataset:
To express $\operatorname{cos} 3\theta$ in terms of $\operatorname{sin} \theta$ and $\operatorname{cos} \theta$ is ().

Options:
A. $$
cos3 \theta\!=\! \operatorname{s i n}^{2} \, \theta\!-\! 3 \operatorname{c o s} \, \theta\! \operatorname{s i n} \, \theta\!=\! 4 \operatorname{s i n}^{2} \, \theta\!-\! 3 \operatorname{c o s} \, \theta\
$$
B. $$
cos3 \theta\!=\! \operatorname{s i n}^{2} \, \theta\!+\! 3 \operatorname{c o s}^{2} \, \theta\! \operatorname{s i n} \, \theta\!=\! 4 \operatorname{s i n}^{2} \, \theta\!+\! 3 \operatorname{c o s}^{2} \, \theta\
$$
C. $$
cos3 \theta\!=\! \operatorname{c o s}^{3} \, \theta\!-\! 3 \operatorname{c o s} \, \theta\! \operatorname{s i n}^{2} \, \theta\!=\! 4 \operatorname{c o s}^{3} \, \theta\!-\! 3 \operatorname{c o s} \, \theta\
$$
D. $$
cos3 \theta\!=\! \operatorname{s i n}^{3} \, \theta\!+\! 3 \operatorname{c o s} \, \theta\! \operatorname{s i n} \, \theta\!=\! 4 \operatorname{s i n}^{3} \, \theta\!+\! 3 \operatorname{c o s} \, \theta\
$$
E. $$
cos3 \theta\!=\! \operatorname{c o s}^{2} \, \theta\!+\! 3 \operatorname{c o s} \, \theta\! \operatorname{s i n}^{2} \, \theta\!=\! 4 \operatorname{c o s}^{2} \, \theta\!+\! 3 \operatorname{c o s} \, \theta\
$$
F. $$
cos3 \theta\!=\! \operatorname{c o s}^{3} \, \theta\!+\! 3 \operatorname{s i n} \, \theta\! \operatorname{c o s}^{2} \, \theta\!=\! 4 \operatorname{c o s}^{3} \, \theta\!+\! 3 \operatorname{s i n} \, \theta\
$$
G. $$
cos3 \theta\!=\! \operatorname{c o s}^{3} \, \theta\!+\! 3 \operatorname{s i n}^{2} \, \theta\! \operatorname{c o s} \, \theta\!=\! 4 \operatorname{c o s}^{3} \, \theta\!+\! 3 \operatorname{s i n}^{2} \, \theta\
$$
H. $$
cos3 \theta\!=\! \operatorname{s i n}^{3} \, \theta\!-\! 3 \operatorname{s i n} \, \theta\! \operatorname{c o s}^{2} \, \theta\!=\! 4 \operatorname{s i n}^{3} \, \theta\!-\! 3 \operatorname{s i n} \, \theta\
$$
I. $$
cos3 \theta\!=\! \operatorname{s i n}^{3} \, \theta \!+\! 3 \operatorname{c o s} \, \theta\! \operatorname{s i n}^{2} \, \theta\!=\! 4 \operatorname{s i n}^{3} \, \theta\!+\! 3 \operatorname{c o s} \, \theta\
$$
J. $$
cos3 \theta\!=\! \operatorname{c o s}^{2} \, \theta\!-\! 3 \operatorname{s i n} \, \theta\! \operatorname{c o s} \, \theta\!=\! 4 \operatorname{c o s}^{2} \, \theta\!-\! 3 \operatorname{s i n} \, \theta\
$$
Answer: C
The function value of $\tan(3-i)$ is ().

Options:
A. $\frac{\cos 6 - \cos 2i}{2 (\cosh^2 1 + \sin^2 3)}$
B. $\frac{\sin 6 - \sin 2i}{2 (\cosh^2 1 - \sin^2 3)}$
C. $\frac{\cos 6 + \sin 2i}{2 (\sinh^2 1 + \cos^2 3)}$
D. $\frac{\sin 9 - \sin 3i}{2 (\sinh^2 2 - \cos^2 1)}$
E. $\frac{\sin 6 + \cos 2i}{2 (\cosh^2 2 - \sin^2 4)}$
F. $\frac{\sin 3 - \cos 3i}{2 (\sinh^2 3 - \tan^2 1)}$
G. $\frac{\tan 6 - \tan 2i}{2 (\sinh^2 1 + \sin^2 3)}$
H. $\frac{\tan 3 + \sin 3i}{2 (\cosh^2 2 + \tan^2 2)}$
I. $\frac{\tan 6 + \cos 2i}{2 (\sinh^2 1 - \tan^2 3)}$
J. $\frac{\cos 9 - \tan 3i}{2 (\cosh^2 1 + \sin^2 9)}$
Answer: B
What is the value of the integral $\oint_{c} \frac{2 z^{2}-z+1} {z-1} \mathrm{d} z$,Where $C \colon\mid z \mid=2$?

Options:
A. $$
-4 \pi\mathrm{i}
$$
B. $$
4 \pi\mathrm{i}
$$
C. $$
\pi\mathrm{i}
$$
D. $$
-8 \pi\mathrm{i}
$$
E. $$
-2 \pi\mathrm{i}
$$
F. $$
2 \pi\mathrm{i}
$$
G. $$
12 \pi\mathrm{i}
$$
H. $$
6 \pi\mathrm{i}
$$
I. $$
0
$$
J. $$
8 \pi\mathrm{i}
$$
Answer: B
Let $C$ denote the circle $( \, x^{2}+y^{2}=3 \, )$, $f(z) = \oint_{C} \frac{3 \zeta^{2} \!+\! 7 \zeta\!+\! 1}{\zeta\!-\! z} \mathrm{d} \zeta\,$. $f^{\prime} ( \, 1+\mathrm{i} \, ) \,$is ().

Options:
A. $$
4 \pi\left(-5+1 4 \mathrm{i} \right)
$$
B. $$
8 \pi\left(-1+1 8 \mathrm{i} \right)
$$
C. $$
2 \pi\left(-6+1 3 \mathrm{i} \right)
$$
D. $$
7 \pi\left(-2+1 9 \mathrm{i} \right)
$$
E. $$
\pi\left(-8+1 6 \mathrm{i} \right)
$$
F. $$
3 \pi\left(-7+1 5 \mathrm{i} \right)
$$
G. $$
6 \pi\left(-3+1 0 \mathrm{i} \right)
$$
H. $$
9 \pi\left(0+1 7 \mathrm{i} \right)
$$
I. $$
5 \pi\left(-4+1 1 \mathrm{i} \right)
$$
J. $$
2 \pi\left(-9+1 2 \mathrm{i} \right)
$$
Answer: C
What is the value of the integral $\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}$? Where $a \neq 0$, $a \neq \rho$, $\rho > 0$.

Options:
A. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{\pi \rho} {\mid a \mid^{2}-\rho},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{\pi \rho} {\rho^{2}-\mid a \mid},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
B. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{\pi \rho^{2}} {\rho^{2}-\mid a \mid^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{\pi \rho^{2}} {\mid a \mid^{2}-\rho^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
C. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{\pi \rho} {\rho^{2}-\mid a \mid^{2}},}}} & {{{if \mid a \mid> \rho,}}} \\ {{{\frac{\pi \rho} {\mid a \mid^{2}-\rho^{2}},}}} & {{{ if \mid a \mid< \rho.}}} \\ \end{array} \right.
$$
D. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{2 \pi\rho} {\mid a \mid^{2}+\rho^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{2 \pi\rho} {\rho^{2}+\mid a \mid^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
E. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{2 \pi\rho} {\rho^{2}-\mid a \mid^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{2 \pi\rho} {\mid a \mid^{2}-\rho^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
F. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{2 \pi} {\rho^{2}+\mid a \mid^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{2 \pi} {\mid a \mid^{2}+\rho^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
G. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{2 \pi\rho} {\rho^{2}+\mid a \mid^{2}},}}} & {{{if \mid a \mid> \rho,}}} \\ {{{\frac{2 \pi\rho} {\mid a \mid^{2}+\rho^{2}},}}} & {{{ if \mid a \mid< \rho.}}} \\ \end{array} \right.
$$
H. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{\pi \rho} {\rho^{2}+\mid a \mid^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{\pi \rho} {\mid a \mid^{2}+\rho^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
I. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{2 \pi} {\rho^{2}-\mid a \mid^{2}},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{2 \pi} {\mid a \mid^{2}-\rho^{2}},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
J. $$
\int_{\mid z \mid=\rho} \frac{\mid\mathrm{d} z \mid} {\mid z-a \mid^{2}}=\left\{\begin{array} {c c} {{{\frac{\pi \rho} {\rho-\mid a \mid},}}} & {{{if \mid a \mid< \rho,}}} \\ {{{\frac{\pi \rho} {\mid a \mid-\rho},}}} & {{{ if \mid a \mid> \rho.}}} \\ \end{array} \right.
$$
Answer: E
What is the power series expansion of the function $f(z) = \left(3 + \frac{z^{2}}{2}\right) \sin z$ up to the term containing $z^{5}$?

Options:
A. -2z^2
B. z^4
C. 2z^2
D. 4z
E. 5z^5
F. -7
G. -3z^3
H. 6z
I. z^3
J. 3z^4
Answer: F
The Laurent series expansion of $1 < \mid z \mid<+\infty$
in the annular region $0 < \mid z \mid < 1$ is ().

Options:
A. $$
= \frac{1} {z}+\frac{2} {z^{2}}+\frac{2} {z^{3}}+\cdots.
$$
B. $$
= \frac{1} {z^{4}}+\frac{2} {z^{5}}+\frac{2} {z^{6}}+\cdots.
$$
C. $$
= \frac{2} {z^{3}}+\frac{2} {z^{4}}+\frac{2} {z^{5}}+\cdots.
$$
D. $$
= \frac{1} {z^{2}}+\frac{1} {z^{3}}+\frac{1} {z^{4}}+\cdots.
$$
E. $$
= \frac{1} {z^{2}}+\frac{2} {z^{3}}+\frac{2} {z^{4}}+\cdots.
$$
F. $$
= \frac{1} {z^{3}}+\frac{2} {z^{4}}+\frac{2} {z^{5}}+\cdots.
$$
G. $$
= \frac{1} {z^{2}}+\frac{3} {z^{3}}+\frac{1} {z^{4}}+\cdots.
$$
H. $$
= \frac{1} {z^{2}}+\frac{2} {z^{4}}+\frac{2} {z^{6}}+\cdots.
$$
I. $$
= \frac{2} {z^{2}}+\frac{2} {z^{3}}+\frac{2} {z^{4}}+\cdots.
$$
J. $$
= \frac{1} {z^{2}}+\frac{3} {z^{3}}+\frac{3} {z^{4}}+\cdots.
$$
Answer: E
If $m, n$ are positive integers and $a \neq b$, what is the residues of the function $f(z) = \frac{1}{(z-a)^{m}(z-b)^{n}}$ at $a$ ?

Options:
A. $$
\frac{( \,-1 \, )^{\, m+1} \, n \, ( \, n+1 \, ) \cdots( \, n+m-2 \, )} {( \, m+1 \, ) \,! \quad( \, a-b \, )^{\, n+m-3}}
$$
B. $$
\frac{( \, 1 \, )^{\, m-1} \, n \, ( \, n-1 \, ) \cdots( \, n-m+1 \, )} {( \, m-1 \, ) \,! \quad( \, a+b \, )^{\, n+m-1}}
$$
C. $$
\frac{( \,-1 \, )^{\, m-1} \, n \, ( \, n+1 \, ) \cdots( \, n+m-2 \, )} {( \, m-1 \, ) \,! \quad( \, a-b \, )^{\, n+m-1}}
$$
D. $$
\frac{( \,-1 \, )^{\, m-1} \, ( \, m \, ) \, n \cdots( \, n+m-3 \, )} {( \, m \, ) \,! \quad( \, b-a \, )^{\, n+m-2}}
$$
E. $$
\frac{( \,-1 \, )^{\, n-1} \, m \, ( \, n \, ) \cdots( \, n+m-2 \, )} {( \, n-1 \, ) \,! \quad( \, a-b \, )^{\, n+m-1}}
$$
F. $$
\frac{( \, -1 \, )^{\, m-2} \, n \, ( \, n-1 \, ) \cdots( \, n-m+2 \, )} {( \, m-2 \, ) \,! \quad( \, a+b \, )^{\, n+m-2}}
$$
G. $$
\frac{( \,-1 \, )^{\, m} \, n \, ( \, n+2 \, ) \cdots( \, n+m-1 \, )} {( \, m \, ) \,! \quad( \, a-b \, )^{\, n+m-2}}
$$
H. $$
\frac{( \,-1 \, )^{\, n-1} \, m \, ( \, m+1 \, ) \cdots( \, m+n-2 \, )} {( \, n-1 \, ) \,! \quad( \, b-a \, )^{\, m+n-1}}
$$
I. $$
\frac{( \,-1 \, )^{\, n} \, m \, ( \, m+2 \, ) \cdots( \, m+n-1 \, )} {( \, n \, ) \,! \quad( \, a-b \, )^{\, m+n-2}}
$$
J. $$
\frac{( \,-1 \, )^{\, m-1} \, n \, ( \, n+1 \, ) \cdots( \, n+m-1 \, )} {( \, m-1 \, ) \,! \quad( \, a-b \, )^{\, n+m-2}}
$$
Answer: C
What kind of singularity is the point $\infty$ for the function $f(z) = \frac{z-5}{2z+3}$? Find the residue of the function at the point $\infty$.

Options:
A. Removable pole, $\frac{8 5}{4}$
B. Essential singularity, $\frac{7 1}{2}$
C. Jump discontinuity, $\frac{0 8}{9}$
D. Essential discontinuity, $\frac{4 7}{3}$
E. Removable singularity, ${\frac{1 3} {4}}$
F. Branch point, $\frac{9 6}{7}$
G. Limit point, $\frac{1 2}{6}$
H. Pole of order two, $\frac{2 5}{3}$
I. Isolated singularity, $\frac{3 8}{2}$
J. Accumulation point, $\frac{5 4}{1}$
Answer: E
Using residues, what is the integral of $\int_{| z |=n} \tan \pi z \, \mathrm{d} z$ (where $n$ is a positive integer)?

Options:
A. $$
- 2 n \mathrm{i}
$$
B. $$
n \mathrm{i}
$$
C. $$
- 8 n \mathrm{i}
$$
D. $$
0
$$
E. $$
- 4 n \mathrm{i}
$$
F. $$
3 n \mathrm{i}
$$
G. $$
4 n \mathrm{i}
$$
H. $$
2 n \mathrm{i}
$$
I. $$
5 n \mathrm{i}
$$
J. $$
- n \mathrm{i}
$$
Answer: E
What is the integral $\oint_{| z |=1} \frac{2 \mathrm{i}} {z^{2}+2 a z+1} \mathrm{d} z \,, a \! > \! 1$?

Options:
A. $$-\frac{2 \pi} {a^{2}-1}$$
B. $$-\frac{2 \pi i} {\sqrt{a^{2}-1}}$$
C. $$-\frac{2 \pi} {\sqrt{2a^{2}-1}}$$
D. $$-\frac{2 \pi} {\sqrt{a^{2}-1}}$$
E. $$\frac{2 \pi} {\sqrt{a^{2}-1}}$$
F. $$\frac{\pi} {\sqrt{a^{2}-1}}$$
G. $$-\frac{2 \pi} {\sqrt{a^{2}+1}}$$
H. $$-\frac{2 \pi} {\sqrt{a^{2}-2}}$$
I. $$-\frac{\pi} {2\sqrt{a^{2}-1}}$$
J. $$-\frac{4 \pi} {\sqrt{a^{2}-1}}$$
Answer: D
There is a small nuclear power plant with an electric power output of 150,000 kilowatts. Assuming the efficiency of the power plant is 27%, how much uranium-235 is consumed when the reactor operates at its rated power for one hour?

Options:
A. $$
4 2. 3 g
$$
B. $$
5 4. 7 g
$$
C. $$
1 1. 9 g
$$
D. $$
6 8. 1 g
$$
E. $$
2 8. 5 g
$$
F. $$
1 7. 2 g
$$
G. $$
7 3. 6 g
$$
H. $$
3 6. 8 g
$$
I. $$
2 0. 4 g
$$
J. $$
9 5. 4 g
$$
Answer: E
A nuclear power plant uses 20% enriched U-235 as fuel, with a thermal power of 900 MW and an annual load factor (actual annual power output/rated annual power output) of 0.85. The U-235 capture-to-fission ratio is 0.169. What is the mass of nuclear fuel consumed in one year?

Options:
A. $$
1 7 5 0~ k g
$$
B. $$
1 8 1 0~ k g
$$
C. $$
1 6 5 0~ k g
$$
D. $$
1 5 9 0~ k g
$$
E. $$
1 9 0 0~ k g
$$
F. $$
1 8 4 0~ k g
$$
G. $$
1 8 9 0~ k g
$$
H. $$
1 5 7 0~ k g
$$
I. $$
1 6 9 0~ k g
$$
J. $$
1 7 2 0~ k g
$$
Answer: J
What is the average macroscopic absorption cross-section for thermal neutrons in $H_2O$ at a temperature of 535.5K and a density of $0.802 \times 10^{3} \, \mathrm{kg} / \mathrm{m}^{3}$?

Options:
A. $$ 0. 8 9 6 \; \mathrm{m}^{-1} $$
B. $$ 2. 1 3 4 \; \mathrm{m}^{-1} $$
C. $$ 1. 5 6 0 \; \mathrm{m}^{-1} $$
D. $$ 1. 4 5 7 \; \mathrm{m}^{-1} $$
E. $$ 1. 6 8 9 \; \mathrm{m}^{-1} $$
F. $$ 1. 0 1 2 \; \mathrm{m}^{-1} $$
G. $$ 1. 0 5 6 \; \mathrm{m}^{-1} $$
H. $$ 0. 9 7 3 \; \mathrm{m}^{-1} $$
I. $$ 1. 2 2 8 \; \mathrm{m}^{-1} $$
J. $$
1. 1 2 3 \; \mathrm{m}^{-1}
$$
Answer: J
The neutron flux density distribution inside a bare cylindrical pile is
$$
\phi( r, z )=10^{12} \cos \left( \frac{\pi z} {H} \right) J_{0} \left( \frac{2.405 r} {R} \right) \quad(\text{cm}^{-2} \cdot \text{s}^{-1})
$$
where $H$ and $R$ are the height and radius of the reactor (assuming the extrapolated distance can be neglected). Given $H=7 \ \text{m}, \ \ R=3 \ \text{m}$, and the reactor power is 10 MW, how much $^{235}\text{U}$ is loaded in the reactor?

Options:
A. $$
1 0 3 \mathrm{k g}
$$
B. $$
1 1 0 \mathrm{k g}
$$
C. $$
1 0 2 \mathrm{k g}
$$
D. $$
9 5 \mathrm{k g}
$$
E. $$
9 9 \mathrm{k g}
$$
F. $$
1 0 5 \mathrm{k g}
$$
G. $$
1 0 6 \mathrm{k g}
$$
H. $$
1 0 8 \mathrm{k g}
$$
I. $$
9 8 \mathrm{k g}
$$
J. $$
1 1 2 \mathrm{k g}
$$
Answer: H
Given a heavy water-uranium reactor core with a $k_{\infty} = 1.28$, $L^{2} = 1.8 \times 10^{-2} \mathrm{~ m^{2}}$, and $\tau = 1.20 \times 10^{-2} \mathrm{~ m^{2}}$. According to the single-group theory, correct the critical equation for the single-group theory. What are the core material curvature and the total neutron non-leakage probability at criticality?

Options:
A. $$0.7805,~0.67$$
B. $$0.7799,~0.66$$
C. $$0.7810,~0.69$$
D. $$0.7820,~0.71$$
E. $$0.7813,~0.68$$
F. $$0.7825,~0.72$$
G. $$0.7832,~0.64$$
H. $$0.7850,~0.65$$
I. $$0.7841,~0.63$$
J. $$0.7801,~0.70$$
Answer: E
For a uniform cylindrical bare core, where the material curvature is equal to $B_{m}^{2}$, what is the $R / H$ value that minimizes the critical volume?

Options:
A. $\frac{R} {H} \!=\! 1. 3 9 8 5$
B. $\frac{R} {H} \!=\! 1. 5 6 8 3$
C. $\frac{R} {H} \!=\! 2. 0 3 6 7$
D. $\frac{R} {H} \!=\! 0. 7 9 4 8$
E. $\frac{R} {H} \!=\! 3. 2 1 7 9$
F. $\frac{R} {H} \!=\! 0. 5 4 1 2$
G. $\frac{R} {H} \!=\! 0. 6 3 5 5$
H. $\frac{R} {H} \!=\! 1. 1 0 2 9$
I. $\frac{R} {H} \!=\! 1. 8 7 3 4$
J. $\frac{R} {H} \!=\! 2. 4 5 4 1$
Answer: F
During the startup process of a nuclear reactor, without any operation by the operator, the reactor power increases from $3 {\times} 10^{-6} \%~ \mathrm{FP}$ to $5 \times 10^{-6} \%~\mathrm{FP}$ within two minutes. What is the average nuclear reactor period in seconds during the power growth process?

Options:
A. $$321.48~s$$
B. $$234.91~s$$
C. $$210.37~s$$
D. $$360.92~s$$
E. $$278.52~s$$
F. $$256.74~s$$
G. $$199.45~s$$
H. $$312.88~s$$
I. $$245.89~s$$
J. $$289.65~s$$
Answer: B
A thermal neutron nuclear reactor experienced an emergency shutdown at the 42nd week while operating at a low power level. It regained criticality 12 hours later and then increased power to 60% FP at a certain rate over 6 hours. How long is required to reach xenon equilibrium at this level?

Options:
A. 30 to 40 hours
B. 100 to 110 hours
C. 40 to 50 hours
D. 50 to 60 hours
E. 70 to 80 hours
F. 60 to 70 hours
G. 20 to 30 hours
H. 80 to 90 hours
I. 90 to 100 hours
J. 10 to 20 hours
Answer: C
The rated electric power $P_{\mathrm{e}}$ of a power plant is 1000 MW, with an efficiency $\eta_{\mathrm{e}}$ (the ratio of electric power to thermal power) of 32%, and an annual load factor $\eta$ (the ratio of actual annual generation to rated annual generation) of 0.85. If the power plant is a thermal power plant, and it is known that the calorific value of standard coal is $Q = 29.271 \, \mathrm{MJ} / \mathrm{kg}$, try to estimate how many tons of standard coal are consumed by the plant in one year.

Options:
A. $$
M=3. 0 8 6 0 \times1 0^{6} \, \mathrm{t}
$$
B. $$
M=3. 0 0 1 8 \times1 0^{6} \, \mathrm{t}
$$
C. $$
M=2. 8 6 1 8 \times1 0^{6} \, \mathrm{t}
$$
D. $$
M=2. 5 1 9 3 \times1 0^{6} \, \mathrm{t}
$$
E. $$
M=2. 7 3 5 1 \times1 0^{6} \, \mathrm{t}
$$
F. $$
M=3. 0 5 7 4 \times1 0^{6} \, \mathrm{t}
$$
G. $$
M=2. 7 4 8 2 \times1 0^{6} \, \mathrm{t}
$$
H. $$
M=3. 1 2 6 5 \times1 0^{6} \, \mathrm{t}
$$
I. $$
M=2. 9 8 4 7 \times1 0^{6} \, \mathrm{t}
$$
J. $$
M=2. 9 1 9 6 \times1 0^{6} \, \mathrm{t}
$$
Answer: C
The relationship between the emitted neutron flux $J^{+}$, the incident neutron flux $J^{-}$, and the surface neutron flux density $\phi(a)$ is _____ .

Options:
A. $$ \phi( a )=J^{+}+3J^{-} $$
B. $$ \phi( a )=3 ( J^{+}+2J^{-} ) $$
C. $$ \phi( a )=2 ( J^{-}-J^{+} ) $$
D. $$ \phi( a )=J^{+}-2J^{-} $$
E. $$ \phi( a )=J^{+}+J^{-} $$
F. $$
\phi( a )=2 ( J^{+}+J^{-} )
$$
G. $$ \phi( a )=4 ( J^{+}+J^{-} ) $$
H. $$ \phi( a )=3 ( J^{+}-J^{-} ) $$
I. $$ \phi( a )=J^{+}+J^{-}+1 $$
J. $$ \phi( a )=2 ( J^{+}-2J^{-} ) $$
Answer: F
For dry air, the initial pressure is 7.0 MPa and the temperature is 40°C. After passing through the throttle expansion valve, the pressure is 0.1 MPa. Determine the temperature of the air as it leaves the throttling valve.

Options:
A. 18°C
B. 55°C
C. -22°C
D. 40°C
E. 32°C
F. 28°C
G. -8°C
H. 45°C
I. -15°C
J. 12°C
Answer: F
There is a resonant cavity with a cavity length of $\mathbf{L}=1 \mathbf{m}$. Among the two mirrors, one is fully reflective and the other is partially reflective, with the reflection coefficient of the partially reflective mirror being $\mathrm{r=0.99}$. Calculate the number of longitudinal modes contained within the 1500 MHz range and the linewidth of each longitudinal mode (excluding other losses). Answer
is___.

Options:
A. $$
1 5
$$
$$
0. 2 4 M H z
$$
B. $$
1 1
$$
$$
0. 3 5 M H z
$$
C. $$
2 0
$$
$$
0. 2 4 M H z
$$
D. $$
1 0
$$
$$
0. 2 2 M H z
$$
E. $$
1 0
$$
$$
0. 3 0 M H z
$$
F. $$
1 1
$$
$$
0. 2 4 M H z
$$
G. $$
1 1
$$
$$
0. 1 9 M H z
$$
H. $$
1 2
$$
$$
0. 1 8 M H z
$$
I. $$
9 8
$$
$$
0. 2 5 M H z
$$
J. $$
9 9
$$
$$
0. 2 3 M H z
$$
Answer: F
The naturally broadened line shape function is
$$
g_{H} ( \nu, \nu_{0} ) \frac{\frac{1} {\tau_{c}}} {( \frac{1} {2 \tau_{c}} )^{2}+4 \pi^{2} ( \nu-\nu_{0} )^{2}}
$$
Find ① the line width ② if replaced by a rectangular line shape function (with equal height to the two functions), find the line width again. Answer is___.

Options:
A. $$
2 \pi\tau_{c}
$$
$$
{\frac{1} {8 \tau_{c}}}
$$
B. $$
\frac{1} {\pi\tau_{c}}
$$
$$
{\frac{1} {2 \tau_{c}}}
$$
C. $$
\frac{1} {4 \pi\tau_{c}}
$$
$$
{\frac{1} {7 \tau_{c}}}
$$
D. $$
\frac{1} {6 \pi\tau_{c}}
$$
$$
{\frac{1} {3 \tau_{c}}}
$$
E. $$
\frac{1} {2 \pi\tau_{c}}
$$
$$
{\frac{1} {4 \tau_{c}}}
$$
F. $$
\frac{2} {\pi\tau_{c}}
$$
$$
{\frac{1} {3 \tau_{c}}}
$$
G. $$
\frac{1} {5 \pi\tau_{c}}
$$
$$
{\frac{1} {9 \tau_{c}}}
$$
H. $$
\frac{5} {2 \pi\tau_{c}}
$$
$$
{\frac{1} {10 \tau_{c}}}
$$
I. $$
\frac{3} {4 \pi\tau_{c}}
$$
$$
{\frac{1} {6 \tau_{c}}}
$$
J. $$
\frac{2} {3 \pi\tau_{c}}
$$
$$
{\frac{1} {5 \tau_{c}}}
$$
Answer: E
Calculate the ruby laser emitter peak emission interface $\mathbf{v}=\mathbf{v}_{0}$ $\lambda_{0}=0.6943 \mu\mathrm{m}, \Delta\nu_{\mathrm{F}}=3.3 \times 10^{11} \mathrm{Hz}, \tau_{2}=4.2 \mathrm{ms}, \mathrm{n=1.76}$ Answer is__.

Options:
A. $$
9. 3 1 {\times} 1 0^{-2 4} \, m^{2}
$$
B. $$
8. 1 3 {\times} 1 0^{-2 4} \, m^{2}
$$
C. $$
6. 4 5 {\times} 1 0^{-2 4} \, m^{2}
$$
D. $$
2. 8 4 {\times} 1 0^{-2 4} \, m^{2}
$$
E. $$
4. 9 9 {\times} 1 0^{-2 4} \, m^{2}
$$
F. $$
7. 2 0 {\times} 1 0^{-2 4} \, m^{2}
$$
G. $$
5. 0 8 {\times} 1 0^{-2 4} \, m^{2}
$$
H. $$
4. 1 7 {\times} 1 0^{-2 4} \, m^{2}
$$
I. $$
3. 6 2 {\times} 1 0^{-2 4} \, m^{2}
$$
J. $$
1. 9 5 {\times} 1 0^{-2 4} \, m^{2}
$$
Answer: D
$\mathrm{CO_{2}}$ laser cavity length $L = 1 \mathrm{m}$, discharge tube diameter $\mathbf{d} = 10 \mathrm{mm}$, and the reflectivity of the two mirrors are $0.92$ and $0.8$ respectively. The discharge tube pressure is $3000 \mathrm{Pa}$. It can be considered as homogeneously broadened and is assumed to operate under optimal discharge conditions. Find (1) the excitation parameter $\alpha$, (2) the oscillation bandwidth $\Delta\mathbf{v}_{\mathrm{T}}$, (3) the number of longitudinal modes satisfying the threshold condition, and (4) the intracavity light intensity during stable operation. (Frequency is at the medium's central frequency $\mathbf{v}_{0}$) Empirical formulas: $\Delta\mathbf{v}_{\mathrm{L}} = 0.049 \mathbf{p} (\mathbf{MHz})$,
$$
\mathbf{G}_{\mathrm{m}} = 1.4 \times 10^{\circ^{2}} / \mathbf{d} \ ( 1 / \mathbf{mm} ), \ \mathbf{I}_{\mathrm{s}} = 72 / \mathbf{d}^{2} (\mathbf{w} / \mathbf{mm}^{2}).
$$Answer is___.

Options:
A. $$
8. 0 4
$$
$$
4 6 0 M H z
$$
$$
2
$$
$$
7. 1 5 w / m m^{2}
$$
B. $$
9. 1 5
$$
$$
4 2 0 M H z
$$
$$
3
$$
$$
5. 8 7 w / m m^{2}
$$
C. $$
7. 6 2
$$
$$
4 4 0 M H z
$$
$$
3
$$
$$
6. 5 9 w / m m^{2}
$$
D. $$
9. 5 7
$$
$$
3 7 5 M H z
$$
$$
6
$$
$$
5. 2 8 w / m m^{2}
$$
E. $$
8. 9 1
$$
$$
5 2 5 M H z
$$
$$
7
$$
$$
3. 6 8 w / m m^{2}
$$
F. $$
1 0. 6 8
$$
$$
3 2 0 M H z
$$
$$
5
$$
$$
2. 8 4 w / m m^{2}
$$
G. $$
1 0. 2 3
$$
$$
5 0 0 M H z
$$
$$
4
$$
$$
3. 9 5 w / m m^{2}
$$
H. $$
8. 5 2
$$
$$
3 5 0 M H z
$$
$$
2
$$
$$
4. 7 2 w / m m^{2}
$$
I. $$
7. 8 4
$$
$$
4 8 0 M H z
$$
$$
5
$$
$$
6. 3 1 w / m m^{2}
$$
J. $$
9. 3 9
$$
$$
3 9 0 M H z
$$
$$
4
$$
$$
4. 9 0 w / m m^{2}
$$
Answer: B
The 10 cm ruby rod is placed inside a 20 cm resonant cavity. It is known that its spontaneous emission lifetime $\tau_{21}=4 \times 10^{-3} \mathrm{s}$, $AV = 2 \times 10^{5}$ MHz, and the one-way cavity loss rate is 8 = 0.01. Determine ① the threshold inversion particle number density $\Delta\mathbf{n}$, and ② when optical pumping excitation produces $\Delta\mathbf{n}=1.2 \Delta\mathbf{n}$, how many longitudinal modes can oscillate (n=1.76)?

Options:
A. $$
2 \times1 0^{-3 2} \, m^{-3}
$$
$$
1 9 5
$$
B. $$
3 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 7 5
$$
C. $$
2 \times2 0^{-2 2} \, m^{-3}
$$
$$
1 7 0
$$
D. $$
2 \times5 0^{-2 2} \, m^{-3}
$$
$$
1 6 0
$$
E. $$
2 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 6 5
$$
F. $$
2 \times1 5^{-2 2} \, m^{-3}
$$
$$
1 4 5
$$
G. $$
1 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 8 0
$$
H. $$
4 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 5 8
$$
I. $$
6 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 5 0
$$
J. $$
5 \times1 0^{-2 2} \, m^{-3}
$$
$$
1 6 8
$$
Answer: E
There is a plano-concave cavity with a concave mirror of radius of curvature $\mathbf{R}=5 \mathbf{m}$, a cavity length $\mathbf{L}=1 \mathbf{m}$, and a light wavelength of 0.5 μm. Find: ① the fundamental mode spot radius on the two mirrors ② the far-field divergence angle of the fundamental mode Gaussian beam.
Answer is___.

Options:
A. $$
0. 7 0 0 m m
$$
$$
3. 5 0 {\times} 1 0^{-4} \, r a d
$$
B. $$
0. 9 3 2 m m
$$
$$
4. 8 7 {\times} 1 0^{-4} \, r a d
$$
C. $$
0. 6 2 6 m m
$$
$$
5. 6 8 {\times} 1 0^{-4} \, r a d
$$
D. $$
0. 4 8 8 m m
$$
$$
5. 3 4 {\times} 1 0^{-4} \, r a d
$$
E. $$
0. 6 7 4 m m
$$
$$
6. 0 9 {\times} 1 0^{-4} \, r a d
$$
F. $$
0. 8 1 5 m m
$$
$$
6. 4 8 {\times} 1 0^{-4} \, r a d
$$
G. $$
0. 8 6 0 m m
$$
$$
5. 1 2 {\times} 1 0^{-4} \, r a d
$$
H. $$
0. 7 5 6 m m
$$
$$
3. 9 2 {\times} 1 0^{-4} \, r a d
$$
I. $$
0. 5 9 0 m m
$$
$$
4. 2 0 {\times} 1 0^{-4} \, r a d
$$
J. $$
0. 5 0 1 m m
$$
$$
5. 9 0 {\times} 1 0^{-4} \, r a d
$$
Answer: C
If the real space volume is $\mathsf{V}$, then the volume of each mode in $\mathsf{K}$ space is $\Delta k_{m} \Delta k_{n} \Delta k_{q} = (2 \pi)^{3} / (8 V)$. Since $k = 2 \pi v / c \geq 0$ and each oscillation mode can have 2 mutually perpendicular polarization modes, what is the mode density in $\mathsf{K}$ space? What is the total number of oscillation modes? What is the mode density in real space (the number of oscillation modes per unit volume and unit frequency interval)?

Options:
A. $$
\frac{8 V}{\left(11 \pi\right)^{3}}
$$
$$
\frac{5 \pi v^{3}}{4 c^{3}} V
$$
$$
\frac{3 \pi v^{2}}{2 c^{3}}
$$
B. $$
\frac{12 V}{\left(7 \pi\right)^{3}}
$$
$$
\frac{6 \pi v^{3}}{2 c^{3}} V
$$
$$
\frac{4 \pi v^{2}}{3 c^{3}}
$$
C. $$
\frac{20 V}{\left(6 \pi\right)^{3}}
$$
$$
\frac{3 \pi v^{3}}{2 c^{3}} V
$$
$$
\frac{10 \pi v^{2}}{c^{3}}
$$
D. $$
\frac{14 V}{\left(8 \pi\right)^{3}}
$$
$$
\frac{2 \pi v^{3}}{5 c^{3}} V
$$
$$
\frac{8 \pi v^{2}}{c^{3}}
$$
E. $$
\frac{8 V} {\left( 2 \pi\right)^{3}}
$$
$$
\frac{8 \pi v^{3}} {3 c^{3}} V
$$
$$
\frac{8 \pi v^{2}} {c^{3}}
$$
F. $$
\frac{6 V}{\left(10 \pi\right)^{3}}
$$
$$
\frac{9 \pi v^{3}}{3 c^{3}} V
$$
$$
\frac{6 \pi v^{2}}{5 c^{3}}
$$
G. $$
\frac{16 V}{\left(4 \pi\right)^{3}}
$$
$$
\frac{4 \pi v^{3}}{c^{3}} V
$$
$$
\frac{12 \pi v^{2}}{c^{2}}
$$
H. $$
\frac{4 V}{\left(3 \pi\right)^{3}}
$$
$$
\frac{10 \pi v^{3}}{2 c^{3}} V
$$
$$
\frac{6 \pi v^{2}}{c^{2}}
$$
I. $$
\frac{18 V}{\left(9 \pi\right)^{3}}
$$
$$
\frac{8 \pi v^{3}}{6 c^{3}} V
$$
$$
\frac{2 \pi v^{2}}{c^{3}}
$$
J. $$
\frac{10 V}{\left(5 \pi\right)^{3}}
$$
$$
\frac{5 \pi v^{3}}{3 c^{3}} V
$$
$$
\frac{14 \pi v^{2}}{2 c^{3}}
$$
Answer: E
A stationary neon atom with a spectral line transition from $3 \mathbf{S}_{2} \to2 \mathbf{P}_{4}$ has a central wavelength of $0.6328 \mu\mathrm{m}$. What will the central wavelength become when it moves towards an observer at a speed of 0.1c?

Options:
A. $$
0. 6 7 5 4 \mu m
$$
B. $$
0. 5 6 9 5 \mu m
$$
C. $$
0. 7 5 0 6 \mu m
$$
D. $$
0. 5 5 0 2 \mu m
$$
E. $$
0. 4 8 2 3 \mu m
$$
F. $$
0. 7 0 4 2 \mu m
$$
G. $$
0. 6 0 1 3 \mu m
$$
H. $$
0. 6 3 9 7 \mu m
$$
I. $$
0. 5 8 8 1 \mu m
$$
J. $$
0. 6 1 2 8 \mu m
$$
Answer: B
Given $X \sim N(0, 1), Y \sim N(0, 1)$ and they are independent. Let $U = X^{2} + Y^{2}$, $V = \frac{X}{Y}$. Find the joint probability density $f_{UV}(u, v)$ of the two-dimensional random variable $(U, V)$ is ().

Options:
A. $$
f_{UV}(u, v)= \frac{1} {2 \pi} \cdot\frac{1} {1+v^{2}} \mathrm{e}^{u} \,, \quad u < 0 \,, v \in\, {R}
$$
B. $$
f_{UV}(u, v)= \frac{1} {4 \pi} \cdot\frac{1} {1+u^{2}} \mathrm{e}^{-\frac{u} {v}} \,, \quad u > 0 \,, v \in\, {R}
$$
C. $$
f_{UV}(u, v)= \frac{1} {2 \pi} \cdot u \cdot\frac{1} {1+v^{2}} \mathrm{e}^{-\frac{1} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
D. $$
f_{UV}(u, v)= \frac{1} {4 \pi} \cdot\frac{v} {1+u^{2}} \mathrm{e}^{-\frac{v} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
E. $$
f_{UV}(u, v)= \frac{1} {2 \pi} \cdot\frac{1} {1+v^{2}} \mathrm{e}^{-\frac{u} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
F. $$
f_{UV}(u, v)= \frac{1} {\pi} \cdot\frac{1} {1+v^{3}} \mathrm{e}^{-\frac{1} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
G. $$
f_{UV}(u, v)= \frac{1} {2 \pi} \cdot\frac{u} {1+v^{2}} \mathrm{e}^{-\frac{v} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
H. $$
f_{UV}(u, v)= \frac{1} {\pi} \cdot\frac{1} {u+v^{2}} \mathrm{e}^{-\frac{u} {2}} \,, \quad u > 0 \,, v \in\, {R}
$$
I. $$
f_{UV}(u, v)= \frac{u} {2 \pi} \cdot\frac{1} {1+v} \mathrm{e}^{-\frac{u^{2}} {2}} \,, \quad u \geq 0 \,, v \in\, {R}
$$
J. $$
f_{UV}(u, v)= \frac{1} {2 \pi} \cdot\frac{1} {1+v^{2}} \mathrm{e}^{-u} \,, \quad u > 0 \,, v \in\, {R}
$$
Answer: E
Let the number of people taking the elevator from the ground floor follow a Poisson distribution with a mean of 10. Suppose this building has a total of $N+1$ floors. Each passenger has an equal probability of choosing any floor to exit the elevator, independent of the floor choices of other passengers. Calculate the expected number of stops the elevator makes before all passengers have exited is ().

Options:
A. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{5 0} {N}} )
$$
B. $$
E ( X )=N \cdot(\mathrm{e}^{-\frac{1 0} {N}})
$$
C. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{2 0} {N}} )
$$
D. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{1 0} {2}} )
$$
E. $$
E ( X )=N^{2} \cdot( 1-\mathrm{e}^{-\frac{1 0} {N}} )
$$
F. $$
E ( X )=2N \cdot( 1-\mathrm{e}^{-\frac{1 0} {N}} )
$$
G. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{1 0} {N}} )
$$
H. $$
E ( X )=\frac{N}{2} \cdot( 1-\mathrm{e}^{-\frac{1 0} {N}} )
$$
I. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{1 0} {2N}} )
$$
J. $$
E ( X )=N \cdot( 1-\mathrm{e}^{-\frac{1 0} {N^{2}}} )
$$
Answer: G
The motion of the fluid is expressed in Lagrange variables $(a, b, c)$ as
$$
x = a \mathrm{e}^{-2t}, \qquad y = b (1+t)^{2}, \qquad z = c \mathrm{e}^{2t} (1+t)^{-2}.
$$
The smoke line passing through the point (1, 1, 1) in space at time t is ( ) .

Options:
A. $$
\left\{
\begin{array}{l}
x = e^{-2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{2} (1 + t)^{-2} \\
z = e^{2 t_{0}} (1 + t_{0})^{2t} e^{2t} (1 + t)^{2}
\end{array}
\right.
$$
B. $$
\left\{
\begin{array}{l}
x = e^{2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{2} (1 + t)^{-2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{-t} e^{2t} (1 + t)^{2}
\end{array}
\right.
$$
C. $$
\left\{
\begin{array}{l}
x = e^{-2 t_{0}} e^{2t} \\
y = (1 + t_{0})^{-2} (1 + t)^{2} \\
z = e^{2 t_{0}} (1 + t_{0})^{-t} e^{-2t} (1 + t)^{-2}
\end{array}
\right.
$$
D. $$
\left\{
\begin{array}{l}
x = e^{-2 t_{0}} e^{2t} \\
y = (1 + t_{0})^{-2} (1 + t)^{-2} \\
z = e^{2 t_{0}} (1 + t_{0})^{-t} e^{-2t} (1 + t)^{2}
\end{array}
\right.
$$
E. $$
\left\{
\begin{array}{l}
x = e^{2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{2} (1 + t)^{2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{t} e^{-2t} (1 + t)^{2}
\end{array}
\right.
$$
F. $$
\left\{
\begin{array}{l}
x = e^{2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{-2} (1 + t)^{2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{t} e^{2t} (1 + t)^{-2}
\end{array}
\right.
$$
G. $$
\left\{
\begin{array}{l}
x = e^{2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{-2} (1 + t)^{-2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{t} e^{-2t} (1 + t)^{2}
\end{array}
\right.
$$
H. $$
\left\{
\begin{array}{l}
x = e^{-2 t_{0}} e^{2t} \\
y = (1 + t_{0})^{-2} (1 + t)^{-2} \\
z = e^{2 t_{0}} (1 + t_{0})^{t} e^{2t} (1 + t)^{-2}
\end{array}
\right.
$$
I. $$
\left\{
\begin{array}{l}
x = e^{-2 t_{0}} e^{-2t} \\
y = (1 + t_{0})^{2} (1 + t)^{2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{t} e^{-2t} (1 + t)^{-2}
\end{array}
\right.
$$
J. $$
\left\{
\begin{array}{l}
x = e^{2 t_{0}} e^{2t} \\
y = (1 + t_{0})^{2} (1 + t)^{2} \\
z = e^{-2 t_{0}} (1 + t_{0})^{-t} e^{-2t} (1 + t)^{-2}
\end{array}
\right.
$$
Answer: F
A cylindrical piston with a diameter of $D$ and a length of $L$, weighing $\omega$, is placed in an upright cylindrical oil cylinder with a diameter of $D+2\delta$. If the central axis of the piston always coincides with the central axis of the oil cylinder, and $2\delta/D \ll 1$, $2\delta/L \ll 1$, what is the sinking speed of the piston under the influence of gravity ?
( Assume the viscosity coefficient of the oil is $\mu$ and the piston approximately moves at a constant speed while sinking. )

Options:
A. $$
\frac{-1} {4 \mu} \Big( \frac{2 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{4}} {D}
$$
B. $$
\frac{-1} {5 \mu} \Big( \frac{4 w} {\pi D^{3}}-\rho g \Big) \frac{\delta^{3}} {L}
$$
C. $$
\frac{-1} {3 \mu} \Big( \frac{4 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{4}} {D}
$$
D. $$
\frac{-1} {3 \mu} \Big( \frac{4 w} {\pi L^{2}}-\rho g \Big) \frac{\delta^{3}} {D}
$$
E. $$
\frac{-2} {3 \mu} \Big( \frac{3 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{3}} {D}
$$
F. $$
\frac{-1} {3 \mu} \Big( \frac{5 w} {\pi D^{2}}-\rho g \Big) \frac{\delta^{3}} {L}
$$
G. $$
\frac{1} {2 \mu} \Big( \frac{5 w} {\pi D^{2} L}+\rho g \Big) \frac{\delta^{3}} {D}
$$
H. $$
\frac{1} {3 \mu} \Big( \frac{4 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{3}} {D}
$$
I. $$
\frac{-1} {2 \mu} \Big( \frac{6 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{3}} {D}
$$
J. $$
\frac{-1} {3 \mu} \Big( \frac{4 w} {\pi D^{2} L}-\rho g \Big) \frac{\delta^{3}} {D}
$$
Answer: J
An adiabatic container with a valve in the middle contains a pure ideal gas, with equal gas masses of 1 on either side of the valve, equal pressure, and temperatures of $T$ and $T$ respectively. Once the valve is opened, heat conduction occurs between the gases in the left and right sections of the container, ultimately bringing the gas in the container to a state of thermodynamic equilibrium. What is the entropy change during this process ?

Options:
A. $$
c_{p} \operatorname{l n} \frac{( T_{1}+T_{2} )^{2}} {4 T_{1} T_{2}} \,
$$
B. $$
c_{P} \operatorname{l n} \frac{(2T_{1}+2T_{2})^{2}} {T_{1} T_{2}} \,.
$$
C. $$
c_{P} \operatorname{l n} \frac{4 T_{1} T_{2}} {(T_{1}-T_{2})^{2}} \,.
$$
D. $$
c_{P} \operatorname{l n} \frac{(T_{1}+2T_{2})^{2}} {4 T_{1} T_{2}} \,.
$$
E. $$
c_{P} \operatorname{l n} \frac{2 T_{1} T_{2}} {(T_{1}+T_{2})^{2}} \,.
$$
F. $$
c_{P} \operatorname{l n} \frac{(T_{1}+T_{2})} {T_{1} T_{2}} \,.
$$
G. $$
c_{P} \operatorname{l n} \frac{(T_{1} T_{2})^{2}} {(T_{1}+T_{2})^{2}} \,.
$$
H. $$
c_{P} \operatorname{l n} \frac{T_{1} T_{2}} {2 (T_{1}+T_{2})^{2}} \,.
$$
I. $$
c_{P} \operatorname{l n} \frac{T_{1} T_{2}} {(T_{1}+T_{2})^{2}} \,.
$$
J. $$
c_{P} \operatorname{l n} \frac{(T_{1}-T_{2})^{2}} {4 T_{1} T_{2}} \,.
$$
Answer: A
Given a flow field with the velocity distribution as follows:
$$
\left\{\begin{array} {l} {u=U \left[ t-\mathrm{erf} \left(-\frac{y}{2 \sqrt{v t}} \right) \right] \qquad( t > 0, \ \ y > 0 )} \\ {v=0} \\ {w=0,} \\ \end{array} \right.
$$
where $U$ and $\boldsymbol{v}$ are constants, and $\mathrm{erf}(\eta)=\frac{2}{\sqrt{\pi}} \int_{0}^{\eta} \mathrm{e}^{-\xi^{2}} \mathrm{d} \xi$ is the error function used for calculating vorticity. So what is the vorticity ?

Options:
A. $$
\frac{U} {\pi\nu t} \mathrm{e}^{-\frac{y^{2}} {2 \nu t}}
$$
B. $$
\frac{U} {\sqrt{2\pi\nu t}} \mathrm{e}^{-\frac{y^{2}} {2 \nu t}}
$$
C. $$
\frac{U} {\sqrt{\pi}} \mathrm{e}^{-\frac{y^{2}} {4 \nu t}}
$$
D. $$
\frac{U} {\pi t} \mathrm{e}^{-\frac{y} {2 \sqrt{v t}}}
$$
E. $$
\frac{2U} {\sqrt{\pi\nu t}} \mathrm{e}^{-\frac{y^{2}} {4 \nu t}}
$$
F. $$
\frac{U} {2\sqrt{\pi\nu t}} \mathrm{e}^{-\frac{y^{2}} {8 \nu t}}
$$
G. $$
\frac{U} {\sqrt{\pi v}} \mathrm{e}^{-\frac{y^{2}} {4 v t}}
$$
H. $$
U e^{-\frac{y^{2}} {4 \nu t}}
$$
I. $$
\frac{U} {\sqrt{\pi\nu t}} \mathrm{e}^{-\frac{y^{2}} {4 \nu t}}
$$
J. $$
\frac{U} {\pi\nu} \mathrm{e}^{-\frac{y^{2}} {4 \nu t}}
$$
Answer: I
A small sphere with a radius of $a$ sinks in a highly viscous incompressible fluid. As the sphere rotates around its own diameter with a small constant angular velocity $\omega$, the fluid induces motion. Assuming that body forces are negligible and the pressure remains constant during the motion, give the viscous torque acting on the sphere.

Options:
A. $$
- 10 \pi\mu a^{3} \omega
$$
B. $$
- 8 \pi\mu a^{4} \omega
$$
C. $$
- 5 \pi\mu a^{3} \omega
$$
D. $$
- 8 \pi\mu a^{3} \omega
$$
E. $$
- 8 \pi\mu a^{3} \nu
$$
F. $$
- 8 \pi\mu a^{3} \theta
$$
G. $$
- 8 \pi a^{3} \omega
$$
H. $$
4 \pi\mu a^{3} \omega
$$
I. $$
- 6 \pi\mu a^{2} \omega
$$
J. $$
- 8 \mu a^{3} \omega
$$
Answer: D
Consider the steady flow of an ideal incompressible fluid around an elliptical cylinder. The major and minor axes of the elliptical cylinder are $a$ and $b$, respectively. The free-stream velocity is $U$, and the angle between its direction and the major axis is $a$. What is the moment of force for the focal point ?

Options:
A. $$
- 2 \pi\rho U^{2} a b \operatorname{c o s} \alpha
$$
B. $$
- 2 \pi\rho U^{2} c^{2} \operatorname{s i n} 2 \alpha
$$
C. $$
\pi \rho U^{2} a b \operatorname{s i n} 2 \alpha
$$
D. $$
- 2 \pi\rho U^{2} a b \operatorname{s i n} \alpha
$$
E. $$
- \pi \rho U^{2} c^{2} \operatorname{s i n} \alpha
$$
F. $$
2 \pi\rho U^{2} c^{2} \operatorname{s i n} 2 \alpha
$$
G. $$
- 2 \pi\rho U^{2} a^{2} b^{2} \operatorname{s i n} 2 \alpha
$$
H. $$
- 2 \pi\rho U^{2} c^{2} \operatorname{c o s} 2 \alpha
$$
I. $$
- 2 \pi\rho U c^{2} \operatorname{s i n} 2 \alpha
$$
J. $$
- 4 \pi\rho U^{2} c^{2} \operatorname{s i n} 2 \alpha
$$
Answer: B
What is the order of flexibility when the main chain is composed of the following saturated single bonds?

Options:
A. $$-C-C->-C-O->-Si-O-$$
B. $$-Si-O->-C-N->-C-C-$$
C. $$-C-G-->-Si-O->-C-C-$$
D. $$-Si-O->-C-C->-C-H-$$
E. $$-Si-O->-C-O->-C-C-$$
F. $$-C-O->-Si-S->-C-C-$$
G. $$-C-H->-C-O->-Si-O-$$
H. $$-C-C->-Si-P->-C-O-$$
I. $$-C-H->-Si-O->-C-O-$$
J. $$-Si-A->-C-O->-C-C-$$
Answer: E
The conformation adopted by the molecular chain of isotropic polyvinyl alcohol is ().

Options:
A. Spiral Curve for Lattice Framework
B. Sawtooth Chain for Planar Surfaces
C. Linear Sequence in Random Coils
D. Crisscross Web for Crystalline Cluster
E. Helical Twist on Cylindrical Core
F. Knotted Loop on Reticular Grid
G. Loop Configuration in Spiral Toroid
H. Skewed Drape on Mesh Network
I. Curved Twist on Shifted Matrix
J. Zigzag Pattern with Axial Rotation
Answer: B
Assume there are two types of polymers, A and B. It is known that $\overline{M_{\Lambda}}=2.0 \times 10^{5}$ and $\overline{M_{\mathrm{B}}}=1.8 \times 10^{6}$. The mean-square end-to-end distances measured in solution are $\overline{r_{\mathrm{A}}^{2}}=6.4 \times 10^{3} \, \mathrm{nm}^{2}$ and $\overline{r_{\mathrm{B}}^{2}}=8.1 \times 10^{4} \, \mathrm{nm}^{2}$, with expansion factors $\alpha_{\mathrm{A}}=2$ and $\alpha_{\mathrm{B}}=3$. Try to determine the flexibility of the two polymers is ().

Options:
A. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 7 9 4}}} \\ {{{A_{\mathrm{B}}=0. 0 6 2 3}}} \\ \end{array}
$$
B. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 8 3 9}}} \\ {{{A_{\mathrm{B}}=0. 0 9 1 6}}} \\ \end{array}
$$
C. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 9 9 0}}} \\ {{{A_{\mathrm{B}}=0. 0 6 8 2}}} \\ \end{array}
$$
D. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 8 7 1}}} \\ {{{A_{\mathrm{B}}=0. 0 9 0 5}}} \\ \end{array}
$$
E. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 1 0 5 3}}} \\ {{{A_{\mathrm{B}}=0. 0 9 4 2}}} \\ \end{array}
$$
F. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 9 3 0}}} \\ {{{A_{\mathrm{B}}=0. 0 8 5 0}}} \\ \end{array}
$$
G. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 8 9 4}}} \\ {{{A_{\mathrm{B}}=0. 0 7 0 7}}} \\ \end{array}
$$
H. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 6 8 8}}} \\ {{{A_{\mathrm{B}}=0. 0 7 7 3}}} \\ \end{array}
$$
I. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 0 9 5 2}}} \\ {{{A_{\mathrm{B}}=0. 0 8 0 4}}} \\ \end{array}
$$
J. $$
\begin{array} {l} {{{A_{\mathrm{A}}=0. 1 1 2 3}}} \\ {{{A_{\mathrm{B}}=0. 0 5 8 9}}} \\ \end{array}
$$
Answer: G
To reduce the viscosity of a polymer melt by half, estimate how much the weight-average molecular weight of the polymer should be reduced.

Options:
A. 0.816
B. 0.940
C. 0.810
D. 0.980
E. 0.920
F. 0.750
G. 0.890
H. 1.050
I. 0.700
J. 1.020
Answer: A
The order of the molecular chain size of sodium polyacrylate in the following sodium polyacrylate solutions is ().
(a) 0.1% aqueous solution (b) 0.3% aqueous solution (c) 0.3% aqueous solution with added sodium chloride

Options:
A. $$
( \mathrm{c} ) > ( \mathrm{b} ) > ( \mathrm{a} )
$$
B. $$
( \mathrm{b} ) > ( \mathrm{c} ) > ( \mathrm{a} )
$$
C. $$
( \mathrm{a} ) > ( \mathrm{c} ) > ( \mathrm{b} )
$$
D. $$
( \mathrm{c} ) > ( \mathrm{a} ) > ( \mathrm{b} )
$$
E. $$
( \mathrm{a} ) > ( \mathrm{b} ) > ( \mathrm{c} )
$$
F. $$
( \mathrm{b} ) > ( \mathrm{a} ) > ( \mathrm{c} )
$$
Answer: E
The density of a cross-linked polymer $\rho_{25} \circ$ is 0.94 $g·cm^{3}$, and the average relative molecular mass between cross-linking points is 28,000. Try to calculate the entropy change per unit volume when this polymer is stretched to 100% deformation at 25°C is ().

Options:
A. $$
- 2 9 3. 0 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
B. $$
- 3 0 0. 4 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
C. $$
- 3 1 1. 7 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
D. $$
- 2 8 1. 3 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
E. $$
- 3 7 1. 1 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
F. $$
- 1 7 9. 2 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
G. $$
- 2 5 9. 5 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
H. $$
- 2 8 5. 9 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
I. $$
- 2 7 9. 1 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
J. $$
- 2 9 7. 1 \mathrm{J} / ( \mathrm{K \cdot m^{3}} )
$$
Answer: I
Equation describing the stress-strain relationship of rubber is ().

Options:
A. Strain Formula of Polyethylene
B. Stress Mechanics of Polymers
C. Rubber Toughness Annotation
D. Polymer Deformation Rule
E. Law of Elasticity for Rubber
F. Rubber's Modulus Transformation
G. Strain-to-Stress Ratio Expression
H. Compression Dynamics of Rubber
I. Elasticity Framework for Plastics
J. Equation of State of Rubber
Answer: J
The greater the viscoelastic activation of a polymer, the higher its melt viscosity ( ).

Options:
A. Improved acoustic dampening
B. Higher molecular flexibility
C. More sensitive to temperature
D. Reduced chain entanglement
E. Increased thermal conductivity
F. Greater chemical reactivity
G. Superior electrical insulation
H. Enhanced tensile strength
I. Lower polymer brittleness
J. Less resistant to pressure
Answer: C
Polymers typically crystallize in very dilute solutions to form ().

Options:
A. Structured polymers
B. Monocrystalline
C. Single-phase layers
D. Liquid crystals
E. Interlinked chains
F. Aggregated molecules
G. Multifaceted crystals
H. Amorphous compounds
I. Micellar formations
J. Polycrystalline
Answer: B
When cross-linked rubber is stretched ( ).

Options:
A. Plasticized
B. Thermoplastic
C. Non-reactive
D. Exothermic
E. Hydrophobic
F. Endothermic
G. Elastoplastic
H. Non-stretchable
I. Hydrophilic
J. Brittle
Answer: D
The larger the relative molecular mass of a polymer, the lower its solubility in solution is ().

Options:
A. Higher
B. Expanded
C. Magnified
D. Enlarged
E. Strengthened
F. Smaller
G. Greater
H. Intensified
I. Amplified
J. Increased
Answer: F
The condition for a polymer to undergo forced elastomeric deformation is ().

Options:
A. Strain energy equals stored potential energy.
B. Deformation speed exceeds polymer viscosity.
C. Cross-link density surpasses glass transition.
D. Shear stress surpasses structural limit.
E. Elastic modulus exceeds tensile strength.
F. Temperature remains below melting point.
G. Crystallization rate outpaces melting rate.
H. Fracture stress is greater than yield stress.
I. Molecular weight reaches equilibrium state.
J. Strain rate falls below critical threshold.
Answer: H
The elastic modulus of rubber materials will () with the rise in temperature.

Options:
A. Become zero
B. Decrease
C. Intensify cyclically
D. Collapse entirely
E. Remain constant
F. Reach a peak
G. Reverse abruptly
H. Fluctuate randomly
I. Increase
J. Shift to static
Answer: I
Given that the molar conductivities of the ions at 25°C are: $\lambda_{\mathrm{m}} ( \mathrm{H}^{+} )=349.7 \mathrm{~S~cm}^{2}/\mathrm{mol}$, $\lambda_{\mathrm{m}} ( \mathrm{K}^{+} )=73.5 \mathrm{~S~cm}^{2}/\mathrm{mol}$, $\lambda_{\mathrm{m}} ( \mathrm{Cl}^{-} )=76.3 \mathrm{~S~cm}^{2}/\mathrm{mol}$. Calculate the conductivity of an aqueous solution containing 0.001 mol/L KCl and 0.001 mol/L HCl at 25°C. The conductivity of water can be neglected is().

Options:
A. $$
= 7. 2 1 0 \times1 0^{-4} \mathrm{S / c m}
$$
B. $$
= 4. 8 5 9 \times1 0^{-4} \mathrm{S / c m}
$$
C. $$
= 4. 1 1 5 \times1 0^{-4} \mathrm{S / c m}
$$
D. $$
= 5. 7 5 8 \times1 0^{-4} \mathrm{S / c m}
$$
E. $$
= 8. 0 5 6 \times1 0^{-4} \mathrm{S / c m}
$$
F. $$
= 3. 5 8 2 \times1 0^{-4} \mathrm{S / c m}
$$
G. $$
= 6. 7 8 3 \times1 0^{-4} \mathrm{S / c m}
$$
H. $$
= 9. 0 9 8 \times1 0^{-4} \mathrm{S / c m}
$$
I. $$
= 6. 3 0 2 \times1 0^{-4} \mathrm{S / c m}
$$
J. $$
= 5. 4 1 9 \times1 0^{-4} \mathrm{S / c m}
$$
Answer: D
At 7.25°C, the equivalent conductance of an acetic acid solution with a dilution of 32I/eq is 9.02 Scm$^{2}$/eq. At this temperature, the limiting equivalent conductance of HCl, NaCl, and NaAc are 426.2 Scm$^{2}$/eq, 126.5 Scm$^{2}$/eq, and 91.0 Scm$^{2}$/eq, respectively. The dissociation constant of acetic acid is().

Options:
A. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 4 8, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 2. 8 5 \! \times\! 1 0^{-5}
$$
B. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 3 1, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 2. 3 2 \! \times\! 1 0^{-5}
$$
C. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 1 9, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 1. 4 6 \! \times\! 1 0^{-5}
$$
D. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 5 0, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 4. 2 3 \! \times\! 1 0^{-5}
$$
E. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 2 5, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 1. 8 2 \! \times\! 1 0^{-5}
$$
F. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 2 3, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 1. 6 9 \! \times\! 1 0^{-5}
$$
G. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 0 8, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 0. 3 9 \! \times\! 1 0^{-5}
$$
H. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 3 5, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 2. 7 7 \! \times\! 1 0^{-5}
$$
I. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 4 2, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 3. 4 0 \! \times\! 1 0^{-5}
$$
J. $$
\alpha=\lambda/ \lambda_{0} \!=\! 0. 0 1 0, \ \ K_{\alpha} \!=\! \alpha^{2} / ( 1 \!-\! \alpha) \bar{V} \!=\! 0. 8 7 \! \times\! 1 0^{-5}
$$
Answer: F
Given the reaction $3 \mathrm{H}_{2} (101,325 \mathrm{~Pa}) + \mathrm{Sb}_{2} \mathrm{O}_{3}(\text{solid}) \Longleftrightarrow 2 \mathrm{Sb} + 3 \mathrm{H}_{2}O$ with $\Delta G^{0} = -8,364 \mathrm{~J/mol}$ at 25°C, the electromotive force of the following cell and identify the anode and cathode is ().
Pt|$H_{\mathrm{2}}$(101,325 Pa)|$H_{\mathrm{2}}$O(pH=3)|$Sb_{\mathrm{2}}$$O_{\mathrm{3}}$(solid)|Sb is().

Options:
A. $$
= 0. 0 4 4 2 V
$$
B. $$
= 0. 1 9 5 6 V
$$
C. $$
= 0. 0 5 6 3 V
$$
D. $$
= 0. 0 1 4 3 V
$$
E. $$
= 0. 0 0 9 8 V
$$
F. $$
= 0. 0 8 4 9 V
$$
G. $$
= 0. 1 1 7 5 V
$$
H. $$
= 0. 2 3 0 1 V
$$
I. $$
= 0. 3 5 7 2 V
$$
J. $$
= 0. 1 4 9 0 V
$$
Answer: D
At 25°C, if the cathodic reaction $\mathbf{Ag}^{+} + \mathrm{e} \to \mathbf{Ag}$ is controlled by the diffusion process, the concentration polarization overpotential $\eta_{\text{concentration}} = \varphi - \varphi_{\text{eq}} = -59 \, \mathrm{mV}$ is measured. Given that the initial concentration $c_{\mathrm{Ag}^{+}}^{0} = 1 \, \mathrm{mol/L}$, the concentration gradient $\left( \, \frac{dc_{\mathrm{Ag}^{+}}}{dx} \, \right)_{x=0} = 7 \times 10^{-2} \, \mathrm{mol/cm^{4}}$, and the diffusion coefficient $D_{\mathrm{Ag}^{+}} = 6 \times 10^{-5} \, \mathrm{cm^{2}/s}$. The surface concentration of Ag$^{+}$ ions $c_{\mathrm{Ag}^{\mathrm{s}}}^{+}$ is().

Options:
A. $$
= 0. 7 m o l / L
$$
B. $$
= 0. 1 m o l / L
$$
C. $$
= 0. 9 m o l / L
$$
D. $$
= 0. 4 m o l / L
$$
E. $$
= 0. 8 m o l / L
$$
F. $$
= 0. 5 m o l / L
$$
G. $$
= 0. 0 5 m o l / L
$$
H. $$
= 0. 6 m o l / L
$$
I. $$
= 0. 3 m o l / L
$$
J. $$
= 0. 2 m o l / L
$$
Answer: B
The experiment measured that the heat capacity of iron is $c_{V}^{( 1 )}=0.054 \ \text{cal/mol}\cdot K$ at $T_{1} = 20 \ \text{K}$, and $c_{V}^{( 2 )}=0.18 \ \text{cal/mol}\cdot K$ at $T_{2} = 30 \ \text{K}$. What is the Debye temperature of iron ?

Options:
A. $$
5 9 1 ( \mathrm{K} )
$$
B. $$
2 7 8 ( \mathrm{K} )
$$
C. $$
4 1 3 ( \mathrm{K} )
$$
D. $$
5 0 7 ( \mathrm{K} )
$$
E. $$
4 2 5 ( \mathrm{K} )
$$
F. $$
3 8 4 ( \mathrm{K} )
$$
G. $$
3 2 9 ( \mathrm{K} )
$$
H. $$
4 7 6 ( \mathrm{K} )
$$
I. $$
6 3 0 ( \mathrm{K} )
$$
J. $$
4 8 2 ( \mathrm{K} )
$$
Answer: C
At room temperature, ${\frac{k_{\mathrm{B}} T} {e}} \!=\! 26 \mathrm{mV}$, the mobile carrier density of a cadmium sulfide sample is $10^{16} \mathrm{cm^{-3}}$, and the mobility is $\mu\!=\! 10^{2} \mathrm{cm}^{2} / \mathrm{V} \cdot \mathrm{s}$. What is the conductivity of this sample ?

Options:
A. $$ 1 2 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
B. $$ 1 6 \, ( \Omega\! \cdot\! {\mathrm{m}} )^{-1}
$$
C. $$ 2 5 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
D. $$ 2 2 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
E. $$ 2 0 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
F. $$ 1 4 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
G. $$ 1 8 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
H. $$ 3 5 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
I. $$ 1 0 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
J. $$ 2 8 \, ( \Omega\! \cdot\! {\mathrm{c m}} )^{-1}
$$
Answer: B
At the temperature of liquid helium, conduction electrons are generated in intrinsic germanium using light excitation, with an average density of $1 \times 10^{12} \text{ cm}^{-3}$. At this temperature, the mobilities of electrons and holes are equal, $\mu = 0.5 \times 10^{4} \text{ cm}^{2} / \text{V} \cdot \text{s}$.
If a voltage of 100 V is applied to a cubic crystal with a side length of 1 cm, what current would be approximately generated at this time?

Options:
A. $$
8 0 ( \mathrm{m A} )
$$
B. $$
7 0 ( \mathrm{m A} )
$$
C. $$
1 1 0 ( \mathrm{m A} )
$$
D. $$
9 0 ( \mathrm{m A} )
$$
E. $$
3 5 0 ( \mathrm{m A} )
$$
F. $$
1 9 0 ( \mathrm{m A} )
$$
G. $$
1 3 0 ( \mathrm{m A} )
$$
H. $$
1 6 0 ( \mathrm{m A} )
$$
I. $$
2 0 0 ( \mathrm{m A} )
$$
J. $$
2 4 0 ( \mathrm{m A} )
$$
Answer: H
Find the limit of
$$
\operatorname*{lim}_{n \to +\infty} \int_{0}^{+\infty} \frac{e^{-x} \cos x}{n x^{2} + \frac{1}{n}} \, \mathrm{d} x.
$$ is ().

Options:
A. $$
{\sqrt{\pi \over 2}}
$$
B. $$
{\frac{\pi} {2}}
$$
C. $$
{\frac{\pi} {4}}
$$
D. $$
{e}
$$
E. $$
{\frac{3\pi} {2}}
$$
F. $$
{\frac{4\pi} {3}}
$$
G. $$
{\ln{2}}
$$
H. $$
{\pi}
$$
I. $$
{0}
$$
J. $$
{\frac{1} {2}}
$$
Answer: B