MikaStars39/nano-eval huggingface dataset:
The measured data baud rate of an Ethernet is 40 MBaud, so its data rate is ().

Options:
A. $$
2 5 \mathrm{M b} / \mathrm{s}
$$
B. $$
8 0 \mathrm{M b} / \mathrm{s}
$$
C. $$
3 0 \mathrm{M b} / \mathrm{s}
$$
D. $$
2 0 \mathrm{M b} / \mathrm{s}
$$
E. $$
1 2 \mathrm{M b} / \mathrm{s}
$$
F. $$
1 5 \mathrm{M b} / \mathrm{s}
$$
G. $$
6 0 \mathrm{M b} / \mathrm{s}
$$
H. $$
1 0 \mathrm{M b} / \mathrm{s}
$$
I. $$
4 0 \mathrm{M b} / \mathrm{s}
$$
J. $$
5 0 \mathrm{M b} / \mathrm{s}
$$
Answer: D
A simple telephone system is composed of two end offices connected to a long-distance office, with a 1 MHz full-duplex trunk line between the end offices and the long-distance office. During an 8-hour workday, a telephone is used on average 4 times, with each use lasting an average of 6 minutes. In all calls, 10% are long-distance (i.e., through the end offices). Assuming that each call line has a bandwidth of 4 kHz, please analyze what is the maximum number of telephones an end office can support?

Options:
A. 30,000 units
B. 50,000 units
C. 60,000 units
D. 20,000 units
E. 10,000 units
F. 80,000 units
G. 100,000 units
H. 90,000 units
I. 40,000 units
J. 70,000 units
Answer: B
If the backward $N$ frame ARQ protocol is used for flow control, and the frame numbering field is 7 bits, then the maximum length of the sending window is ().

Options:
A. 127
B. 255
C. 128
D. 32
E. 512
F. 256
G. 63
H. 64
I. 120
J. 130
Answer: A
For the sliding window protocol, if the packet sequence number uses 3-bit numbering and the send window size is 5, then the maximum receive window size is ().

Options:
A. 11
B. 6
C. 12
D. 5
E. 8
F. 9
G. 4
H. 3
I. 7
J. 10
Answer: H
For CSMA/CD Ethernet with a length of 1 km and a data relocation rate of 10 Mb/s, where the signal in the cable travels at 200,000 km/s, determine the minimum frame length required for proper network operation as ().

Options:
A. 150 bits
B. 300 bits
C. 220 bits
D. 180 bits
E. 50 bits
F. 200 bits
G. 100 bits
H. 80 bits
I. 250 bits
J. 120 bits
Answer: G
A certain local area network uses the CSMA/CD protocol for media access control, with a data transmission rate of 10Mb/s. The distance between Host A and Host B is 2 km, and the signal propagation speed is 200,000 km/s. If there is a collision when Host A and Host B send data, from the moment data transmission begins, how long does it take at minimum for both hosts to detect the collision? How long does it take at maximum (assuming that no other hosts are transmitting data while Host A and Host B are transmitting)?

Options:
A. 0.3ms, 0.6ms
B. 0.2ms, 0.4ms
C. 0.1ms, 0.2ms
D. 0.08ms, 0.16ms
E. 0.05ms, 0.1ms
F. 0.15ms, 0.3ms
G. 0.25ms, 0.5ms
H. 0.12ms, 0.24ms
I. 0.2ms, 0.3ms
J. 0.18ms, 0.36ms
Answer: C
For a 100Mb/s Ethernet switch, when the output port is not queued and an Ethernet frame (excluding the preamble) is forwarded using cut-through switching, the introduced forwarding delay is at least ( ).

Options:
A. $$
1. 8 3 \mu s
$$
B. $$
0. 6 9 \mu s
$$
C. $$
0. 5 5 \mu s
$$
D. $$
0. 0 7 \mu s
$$
E. $$
0. 4 8 \mu s
$$
F. $$
3. 2 9 \mu s
$$
G. $$
0. 1 5 \mu s
$$
H. $$
2. 0 0 \mu s
$$
I. $$
0. 9 4 \mu s
$$
J. $$
1. 2 4 \mu s
$$
Answer: E
There are $4$ routes 172.18.129.0/24, 172.18.130.0/24, 172.18.132.0/24, and 172.18.133.0/24. If these routes are aggregated, the address that can cover these 4 routes is ().

Options:
A. $$
1 7 2. 1 8. 1 3 3. 0 / 2 2
$$
B. $$
1 7 2. 1 8. 1 3 1. 0 / 2 5
$$
C. $$
1 7 2. 1 8. 1 2 4. 0 / 2 6
$$
D. $$
1 7 2. 1 8. 1 3 0. 0 / 2 4
$$
E. $$
1 7 2. 1 8. 1 2 7. 0 / 3 2
$$
F. $$
1 7 2. 1 8. 1 2 8. 0 / 2 1
$$
G. $$
1 7 2. 1 8. 1 3 2. 0 / 2 5
$$
H. $$
1 7 2. 1 8. 1 2 9. 0 / 2 3
$$
I. $$
1 7 2. 1 8. 1 7 8. 0 / 2 1
$$
J. $$
1 7 2. 1 8. 1 3 5. 0 / 2 2
$$
Answer: F
The tea leaves contain more than () kinds of chemical components.

Options:
A. 800
B. 650
C. 300
D. 700
E. 450
F. 550
G. 400
H. 600
I. 500
J. 350
Answer: H
A system of particles consists of a particle located at point $(a, -a, a)$ with a mass of $4m$, a particle located at point $(-a, a, a)$ with a mass of $3m$, and a particle located at point $(a, a, a)$ with a mass of $2m$. What is the moment of inertia about the axis passing through the origin and in the direction of (1, 1, 0)?

Options:
A. $$
8 m a^{2}
$$
B. $$
3 1 m a^{2}
$$
C. $$
6 m a^{2}
$$
D. $$
9 m a^{2}
$$
E. $$
5 m a^{2}
$$
F. $$
2 3 m a^{2}
$$
G. $$
2 7 m a^{2}
$$
H. $$
4 1 m a^{2}
$$
I. $$
7 m a^{2}
$$
J. $$
3 2 m a^{2}
$$
Answer: F
A smooth rigid parabolic $R^{2} = 2 p z$ rotates around a vertical axis $z$ with a constant angular velocity $a$, with a small ring of mass $m$ sliding on it. Find the Lagrangian function and the equation of motion for the small ring. If the small ring stabilizes at some point on the parabola, what is $\omega$?

Options:
A. $$
\omega=\sqrt{\frac{g} {a}}
$$
B. $$
\omega=\sqrt{\frac{2a} {g}}
$$
C. $$
\omega=\sqrt{\frac{p} {g}}
$$
D. $$
\omega=\sqrt{\frac{p+a} {g}}
$$
E. $$
\omega=\sqrt{\frac{a^2} {2p}}
$$
F. $$
\omega=\sqrt{\frac{2g} {p}}
$$
G. $$
\omega=\sqrt{\frac{g} {p}}
$$
H. $$
\omega=\sqrt{\frac{g} {2p}}
$$
I. $$
\omega=\sqrt{\frac{g+a} {p}}
$$
J. $$
\omega=\sqrt{\frac{a} {p}}
$$
Answer: G
When four hydrogen nuclei (protons) combine to form a helium nucleus ($\alpha$ particle), a mass of $5.0 \times 10^{-29} \, \mathrm{kg}$ is converted into energy. What is the energy released when all the hydrogen nuclei in one kilogram of water combine to form helium nuclei (the mass of a hydrogen nucleus is 1.0081 atomic mass units, 1 atomic mass unit $= 1.66 \times 10^{-27} \, \mathrm{kg}$)?

Options:
A. $$
8. 6 {\times} 1 0^{1 4} ( \mathrm{J} )
$$
B. $$
2. 9 {\times} 1 0^{1 4} ( \mathrm{J} )
$$
C. $$
4. 5 {\times} 1 0^{1 4} ( \mathrm{J} )
$$
D. $$
8. 0 {\times} 1 0^{1 3} ( \mathrm{J} )
$$
E. $$
1. 2 {\times} 1 0^{1 5} ( \mathrm{J} )
$$
F. $$
7. 5 {\times} 1 0^{1 3} ( \mathrm{J} )
$$
G. $$
3. 7 {\times} 1 0^{1 2} ( \mathrm{J} )
$$
H. $$
5. 4 {\times} 1 0^{1 2} ( \mathrm{J} )
$$
I. $$
9. 3 {\times} 1 0^{1 3} ( \mathrm{J} )
$$
J. $$
6. 8 {\times} 1 0^{1 3} ( \mathrm{J} )
$$
Answer: F
At the bottom of a large container, there is a small hole. The ratio of the container's cross-sectional area to the hole area is 100. The container holds water with a height of $h=0.80 \mathrm{m}$. How much time is required for the water inside the container to completely flow out?

Options:
A. $$
44.3s
$$
B. $$
39.5s
$$
C. $$
35.8s
$$
D. $$
38.9s
$$
E. $$
40.4s
$$
F. $$
37.2s
$$
G. $$
43.1s
$$
H. $$
42.0s
$$
I. $$
36.5s
$$
J. $$
41.7s
$$
Answer: E
Let $F_{1}, F_{2}, F_{3}$ be three mutually disjoint closed sets in $R^{n}$, such that $0 \leq f(x) \leq 1$; $f(x) = 0$ for $x \in F_{1}$; $f(x) = 1/2$ for $x \in F_{2}$; $f(x) = 1$ for $x \in F_{3}$. What is $f \in C(R^{n})$?

Options:
A. $$
f ( x )=\frac{1} {3} \left[ \frac{d ( x, F_{3} )} {d ( x, F_{3} )+d ( x, F_{2} \cup F_{1} )}+\frac{d ( x, F_{1} \cup F_{2} )} {d ( x, F_{1} \cup F_{2} )+d ( x, F_{3} )} \right]
$$
B. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{1} )} {d ( x, F_{1} )+d ( x, F_{2} \cup F_{3} )}+\frac{d ( x, F_{1} \cup F_{2} )} {d ( x, F_{1} \cup F_{2} )+d ( x, F_{3} )} \right]
$$
C. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{2} )} {d ( x, F_{2} )+d ( x, F_{1} \cup F_{3} )}+\frac{d ( x, F_{3} \cup F_{1} )} {d ( x, F_{3} \cup F_{1} )+d ( x, F_{2} )} \right]
$$
D. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{1} )} {d ( x, F_{1} )+d ( x, F_{3} \cup F_{2} )}+\frac{d ( x, F_{2} \cup F_{3} )} {d ( x, F_{2} \cup F_{3} )+d ( x, F_{1} )} \right]
$$
E. $$
f ( x )=\frac{1} {3} \left[ \frac{d ( x, F_{2} )} {d ( x, F_{2} )+d ( x, F_{1} \cup F_{3} )}+\frac{d ( x, F_{1} \cup F_{3} )} {d ( x, F_{1} \cup F_{3} )+d ( x, F_{2} )} \right]
$$
F. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{2} )} {d ( x, F_{2} )+d ( x, F_{3} \cup F_{1} )}+\frac{d ( x, F_{3} \cup F_{2} )} {d ( x, F_{3} \cup F_{2} )+d ( x, F_{1} )} \right]
$$
G. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{1} )} {d ( x, F_{1} )+d ( x, F_{3} \cup F_{2} )}+\frac{d ( x, F_{2} \cup F_{1} )} {d ( x, F_{2} \cup F_{1} )+d ( x, F_{3} )} \right]
$$
H. $$
f ( x )=\frac{1} {3} \left[ \frac{d ( x, F_{2} )} {d ( x, F_{2} )+d ( x, F_{3} \cup F_{1} )}+\frac{d ( x, F_{3} \cup F_{1} )} {d ( x, F_{3} \cup F_{1} )+d ( x, F_{2} )} \right]
$$
I. $$
f ( x )=\frac{1} {3} \left[ \frac{d ( x, F_{3} )} {d ( x, F_{3} )+d ( x, F_{1} \cup F_{2} )}+\frac{d ( x, F_{1} \cup F_{3} )} {d ( x, F_{1} \cup F_{3} )+d ( x, F_{2} )} \right]
$$
J. $$
f ( x )=\frac{1} {2} \left[ \frac{d ( x, F_{3} )} {d ( x, F_{3} )+d ( x, F_{1} \cup F_{2} )}+\frac{d ( x, F_{2} \cup F_{1} )} {d ( x, F_{2} \cup F_{1} )+d ( x, F_{3} )} \right]
$$
Answer: B
Attempt to construct a function $f(x)$ on (0,1) such that on any $(\alpha, \beta) \subset (0, 1)$, both the set of continuity points and the set of discontinuity points of $f(x)$ are uncountable. What is $f(x)$?

Options:
A. Let $D \subset [0, 1]$ be the rational points. Let $f(x) = \chi_{D}(x)$.
B. Let $C \subset [0, 1]$ be the Cantor set. Let $f(x) = \chi_{C}(x)$.
C. Let $M \subset [0, 1]$ be the Luzin set. Let $f(x) = \chi_{M}(x)$.
D. Let $K \subset [0, 1]$ be the Sierpinski set. Let $f(x) = \chi_{K}(x)$.
E. Let $G \subset [0, 1]$ be the Smith-Volterra set. Let $f(x) = \chi_{G}(x)$.
F. Let $R \subset [0, 1]$ be the irrational set. Let $f(x) = \chi_{R}(x)$.
G. Let $N \subset [0, 1]$ be the perfect set. Let $f(x) = \chi_{N}(x)$.
H. Let $F \subset [0, 1]$ be the fat Cantor set. Let $f(x) = \chi_{F}(x)$.
I. Let $E \subset [0, 1]$ be the Vitali set. Let $f(x) = \chi_{E}(x)$.
J. Let $H \subset [0, 1]$ be the Bernstein set. Let $f(x) = \chi_{H}(x)$.
Answer: B
Let there be a binary function defined on $E=[0,1]\times[0,1]$:
$$
f(x,y)=\begin{cases}1,&x\cdot y\text{ is an irrational number,}\\2,&x\cdot y\text{ is a rational number.}\end{cases}
$$
Find the value of the integral $\int_E f(x,y) \, dx \, dy$.

Options:
A. π
B. e
C. 0.5
D. 3/2
E. -1
F. 1
G. 2
H. 0
I. 3
J. 1/2
Answer: F
What is the value of $I \!=\! \int_{0}^{+\infty} \! \mathrm{e}^{-x} \! \mathrm{cos} \ \sqrt{x} \, \mathrm{d} x$?

Options:
A. $$
I= \mathrm{e}^{-1}+\mathrm{sin}^{2} \, {\frac{1} {2}}-\mathrm{cos} \, {\frac{3} {5}}+\mathrm{tan} \, {\frac{4} {8}}-\cdots
$$
B. $$
I= \mathrm{arctan} \, {\frac{1} {3}}-{\frac{2} {5}}+{\frac{3 !} {7 !}}-{\frac{4 !} {9 !}}+\cdots
$$
C. $$
I= {\frac{3 !} {5 !}}-{\frac{4 !} {6 !}}+{\frac{5 !} {7 !}}-{\frac{6 !} {8 !}}+\cdots
$$
D. $$
I= \mathrm{cos} \, {\frac{\pi}{6}}-\mathrm{sin} \, {\frac{\pi}{4}}+{\frac{5 !} {10 !}}-\cdots
$$
E. $$
I= {\frac{\pi}{4}}-{\frac{\pi}{6}}+{\frac{1}\cdot{2}}-{\frac{3 \cdot 4}{5}}+{\frac{6}{11}}-\cdots
$$
F. $$
I= \mathrm{e}^{0}-\mathrm{ln} \, 2+\mathrm{log} \, 3-\mathrm{sin} \, 4+\cdots
$$
G. $$
I= \mathrm{ln} \, {\frac{2} {1}}+{\frac{1 !} {2 !}}-{\frac{3 !} {4 !}}+{\frac{4 !} {5 !}}-\cdots
$$
H. $$
I= {\frac{1} {2}}+{\frac{3} {4}}-{\frac{5} {6}}+{\frac{7} {8}}-{\frac{9} {10}}+\cdots
$$
I. $$
I= 1-{\frac{1} {2}}+{\frac{2 !} {4 !}}-{\frac{3 !} {6 !}}+{\frac{4 !} {8 !}}-\cdots
$$
J. $$
I= \mathrm{tan} \, {\frac{1} {3}}+{\frac{2} {4}}-{\frac{3 !} {5 !}}+{\frac{4 !} {6 !}}-\cdots
$$
Answer: I
Given that $f(x)$ is a non-negative integrable function on $\mathbf{R}^{1}$, and that
$$
I = \iint_{n^{2}} f(4x) f(x+y) \mathrm{d}x \mathrm{d}y = 1,
$$
try to find the value of $J = \int_{\mathbb{R}^{1}} f(x) \mathrm{d}x$.

Options:
A. J=5
B. J=3
C. J=2
D. J=8
E. J=7
F. J=4
G. J=9
H. J=11
I. J=10
J. J=6
Answer: C
What is the value of $\bigvee_{-1}^{1} ( x-x^{3} )$?

Options:
A. $$
10 \, \sqrt{2} \, / 9
$$
B. $$
5 \, \sqrt{3} \, / 9
$$
C. $$
7 \, \sqrt{3} \, / 9
$$
D. $$
8 \, \sqrt{2} \, / 9
$$
E. $$
11 \, \sqrt{2} \, / 9
$$
F. $$
9 \, \sqrt{3} \, / 9
$$
G. $$
6 \, \sqrt{3} \, / 9
$$
H. $$
15 \, \sqrt{2} \, / 9
$$
I. $$
8 \, \sqrt{3} \, / 9
$$
J. $$
12 \, \sqrt{3} \, / 9
$$
Answer: I
Let $f(x)$ be a monotonically increasing function defined on $[a, b]$. Let $E = \{x \in [a, b] : f'(x)$ exists$\}$. Find the value of $\int_{a}^{b} f'(x) \, \mathrm{d}x$.

Options:
A. $m \left( f' ( E ) \right)$
B. $m \left( f ( b ) \right)$
C. $m \left( E \right)$
D. $m^{*} \left( f ( [a, b] ) \right)$
E. $m \left( f ( E ) \right)$
F. $m^{*} \left( f ( a ) \right)$
G. $m^{*} \left( E \right)$
H. $m \left( f' ( [a, b] ) \right)$
I. $m^{*} \left( f' ( E ) \right)$
J. $m^{*} \left( f ( E ) \right)$
Answer: J
Let $f(x)$ be absolutely continuous on any interval $\left[ a, b \right] \subset \mathbf{R}^{1}$. Then, for each $y \in \mathbf{R}^{1}$, find the value of $\frac{\mathrm{d}}{\mathrm{d} y} \!\int_{a}^{b} \! f(x+y) \mathrm{d} x$.

Options:
A. $$
\frac{\mathrm{d}}{\mathrm{d} y} \int_{a}^{b} x f ( y ) \mathrm{d} x
$$
B. $$
\int_{a}^{b} {f ( x+y ) \cdot y} \mathrm{d} x
$$
C. $$
\int_{a}^{b} {\frac{\mathrm{d}} {\mathrm{d} x}} f ( x+y ) \mathrm{d} x
$$
D. $$
x \cdot \int_{a}^{b} {y f(x)} \mathrm{d} x
$$
E. $$
\int_{a}^{b} y \cdot {\frac{\mathrm{d}} {\mathrm{d} x}} f ( x ) \mathrm{d} x
$$
F. $$
\int_{a}^{b} {\frac{\mathrm{d}} {\mathrm{d} y}} f ( x+y ) \mathrm{d} x
$$
G. $$
f ( a+y ) \cdot \int_{a}^{b} 1 \mathrm{d} x
$$
H. $$
y \cdot \int_{a}^{b} {f(x)} \mathrm{d} x
$$
I. $$
\int_{a}^{b} {f ( x+y )} \mathrm{d} x
$$
J. $$
\int_{a}^{b} {f' ( x+y )} \mathrm{d} x
$$
Answer: F
A 4-pole shunt generator rated at 82 kW, 230 V, and 930 r/min has an armature circuit resistance of $\mathrm{R}_{a}=0.0259 \, \Omega$ and a field winding resistance of $\mathrm{R_{f}}=22.8 \, \Omega$ at 75°C. Under rated load, the excitation circuit includes an adjusting resistor $\mathrm{R}_{\mathrm{f}}=3.5 \Omega$, with a brush voltage drop of $2 \Delta\mathbf{U}_{\mathrm{b}}=2 \mathbf{V}$. The iron and mechanical losses are $p_{\mathrm{Fe}}+p_{\Omega}=2.3 \mathrm{kW}$, and the stray losses are $p_{s}=0.05 \mathrm{P}_{\mathrm{N}}$. What are the input power, electromagnetic power, electromagnetic torque, and efficiency of the generator under rated load?

Options:
A. 96.0kw,86.2kw,911.5N·m,86.2%
B. 92.9kw,88.5kw,902.7N·m,86.3%
C. 93.5kw,90.1kw,904.1N·m,87.8%
D. 93.8kw,89.0kw,900.2N·m,87.1%
E. 94.6kw,88.2kw,905.7N·m,86.7%
F. 95.7kw,87.3kw,906.9N·m,85.4%
G. 92.1kw,86.8kw,899.6N·m,84.5%
H. 94.1kw,85.9kw,907.3N·m,88.0%
I. 95.3kw,87.5kw,910.4N·m,85.9%
J. 94.3kw,89.7kw,908.0N·m,85.6%
Answer: E
A three-phase transformer, $\mathrm{S_{N}=750 \, kV \cdot A \, , \, \ U_{N1} / U_{N2}=10000 \, V / 400 \, V \, , \, \ Y, \, \ yn0 \,}$ connection, yielded $\mathrm{I}_{0}=60 \, \mathrm{A} \; , \; \; \mathrm{p}_{0}=3800 \, \mathrm{w}$ during a no-load test on the low-voltage side, and $\mathbf{U}_{k}=440 \, \mathbf{V}$, $\mathrm{P_{k}=10900 \, w \, \, \, (I_{k1}=I_{N1})}$ during a short-circuit test on the high-voltage side, at a room temperature of $20^{\circ} \mathrm{C}$. Determine the magnetizing resistance and short-circuit impedance referred to the high-voltage side is ().

Options:
A. 3000Ω,6.50Ω
B. 2100Ω,6.25Ω
C. 2400Ω,4.85Ω
D. 1500Ω,6.00Ω
E. 2394Ω,5.85Ω
F. 2350Ω,6.10Ω
G. 2450Ω,5.95Ω
H. 2600Ω,5.75Ω
I. 2750Ω,5.05Ω
J. 2200Ω,5.50Ω
Answer: E
In a mixing tank, there is originally $2000 \mathrm{kg}$ of brine with a mass fraction of 10%. Now, brine with a mass fraction of 0.2% is added to the tank at a mass flow rate of 100 kg/min, while at the same time, the mixed solution is discharged from the tank at a mass flow rate of 60 kg/min. Assuming good stirring and the solution in the tank is well mixed, find the time required for the mass fraction of the solution in the tank to decrease to 1%.

Options:
A. $$
9 1. 5 \operatorname* {m i n}
$$
B. $$
9 3. 6 \operatorname* {m i n}
$$
C. $$
7 9. 8 \operatorname* {m i n}
$$
D. $$
8 8. 9 \operatorname* {m i n}
$$
E. $$
1 0 0. 4 \operatorname* {m i n}
$$
F. $$
9 5. 2 \operatorname* {m i n}
$$
G. $$
8\, 6. 2 \operatorname* {min}
$$
H. $$
7 5. 3 \operatorname* {m i n}
$$
I. $$
8 4. 1 \operatorname* {m i n}
$$
J. $$
8\, 2. 7 \operatorname* {min}
$$
Answer: G
Use the pump to transport water from the storage tank to the top of the absorption tower. It is known that the temperature of the water in the tank is 20°C, the height from the water surface in the tank to the top of the tower is 30m, and the delivery pipeline is adiabatic with an inner diameter of $7.5 \mathrm{cm}$. The pump's water delivery flow rate is $0.8 \mathrm{m}^3/\mathrm{min}$, and the shaft power is 10kW. Try to calculate the temperature increase value ∆$t$ of the water delivered to the bottom of the tower, assuming $\alpha = 1$.

Options:
A. $$
= 0. 2 0 0^{\circ} \, \mathrm{C}
$$
B. $$
= 0. 1 0 8^{\circ} \, \mathrm{C}
$$
C. $$
= 0. 0 9 2^{\circ} \, \mathrm{C}
$$
D. $$
= 0. 0 7 6^{\circ} \, \mathrm{C}
$$
E. $$
= 0. 0 5 0^{\circ} \, \mathrm{C}
$$
F. $$
= 0. 1 3 9^{\circ} \, \mathrm{C}
$$
G. $$
= 0. 1 2 5^{\circ} \, \mathrm{C}
$$
H. $$
= 0. 1 5 4^{\circ} \, \mathrm{C}
$$
I. $$
= 0. 0 8 4^{\circ} \, \mathrm{C}
$$
J. $$
= 0. 1 1 2^{\circ} \, \mathrm{C}
$$
Answer: B
A certain fluid descends in a film-like manner along a vertical flat wall 1 meter wide, with a mass flow rate of $0.15 \mathrm{kg/s}$. The kinematic viscosity of the fluid is known to be $1 \times 10^{-4} \mathrm{m}^{2}/\mathrm{s}$, and the density is $1000 \mathrm{kg/m}^{3}$. Determine the thickness of the liquid film formed once the flow stabilizes.

Options:
A. $$
R e={\frac{u_{\mathrm{b}} d_{\mathrm{c}}} {\nu}}={\frac{2 \delta w} {\nu\ \rho\delta}}={\frac{2 w} {\nu\ \rho}}={\frac{2 \times0. 2 0} {1 \times1 0^{-4} \times1 0 0 0}}=4
$$
B. $$
R e={\frac{u_{\mathrm{b}} d_{\mathrm{c}}} {\nu}}={\frac{6 \delta w} {\nu\ \rho\delta}}={\frac{6 w} {\nu\ \rho}}={\frac{6 \times0. 0 5} {1 \times1 0^{-4} \times1 0 0 0}}=3
$$
C. $$
R e={\frac{u_{\mathrm{b}} d_{\mathrm{c}}} {\nu}}={\frac{3 \delta w} {\nu\ \rho\delta}}={\frac{3 w} {\nu\ \rho}}={\frac{3 \times0. 1 2} {1 \times1 0^{-4} \times1 0 0 0}}=3
$$
D. $$
R e={\frac{d_{\mathrm{c}} u_{\mathrm{b}}} {\nu}}={\frac{4 \delta w} {\nu\ \rho\delta}}={\frac{4 w} {\nu\ \rho}}={\frac{4 \times0. 1 5} {1 \times1 0^{-4} \times1 0 0 0}}=6
$$
E. $$
R e={\frac{d_{\mathrm{c}} u_{\mathrm{b}}} {\nu}}={\frac{5 \delta w} {\nu\ \rho\delta}}={\frac{5 w} {\nu\ \rho}}={\frac{5 \times0. 0 5} {1 \times1 0^{-4} \times1 0 0 0}}=2.5
$$
F. $$
R e={\frac{d_{\mathrm{c}} u_{\mathrm{b}}} {\nu}}={\frac{1 \delta w} {\nu\ \rho\delta}}={\frac{1 w} {\nu\ \rho}}={\frac{1 \times0. 1 8} {1 \times1 0^{-4} \times1 0 0 0}}=1.8
$$
G. $$
R e={\frac{u_{\mathrm{b}} d_{\mathrm{c}}} {\nu}}={\frac{8 \delta w} {\nu\ \rho\delta}}={\frac{8 w} {\nu\ \rho}}={\frac{8 \times0. 0 1} {1 \times1 0^{-4} \times1 0 0 0}}=0.8
$$
H. $$
R e={\frac{u_{\mathrm{b}} d_{\mathrm{c}}} {\nu}}={\frac{9 \delta w} {\nu\ \rho\delta}}={\frac{9 w} {\nu\ \rho}}={\frac{9 \times0. 2 0} {1 \times1 0^{-4} \times1 0 0 0}}=18
$$
I. $$
R e={\frac{d_{\mathrm{c}} u_{\mathrm{b}}} {\nu}}={\frac{3 \delta w} {\nu\ \rho\delta}}={\frac{3 w} {\nu\ \rho}}={\frac{3 \times0. 1 1} {1 \times1 0^{-4} \times1 0 0 0}}=3.3
$$
J. $$
R e={\frac{d_{\mathrm{c}} u_{\mathrm{b}}} {\nu}}={\frac{7 \delta w} {\nu\ \rho\delta}}={\frac{7 w} {\nu\ \rho}}={\frac{7 \times0. 0 1} {1 \times1 0^{-4} \times1 0 0 0}}=0.7
$$
Answer: D
There is a spherical solid particle with a diameter of $0.1 \, \text{mm}$ settling in still air under standard atmospheric pressure and at 30°C. The settling velocity is known to be $0.01 \, \mathrm{m/s}$. Find the relative velocity between the air and the sphere at a point $r = 0.3 \, \text{mm}$ and $\theta = \pi/4$ from the center of the particle.

Options:
A. $$
= 1. 2 3 \times1 0^{-2} {\mathrm{m / s}}
$$
B. $$
= 4. 4 4 \times1 0^{-3} {\mathrm{m / s}}
$$
C. $$
= 5. 5 5 \times1 0^{-3} {\mathrm{m / s}}
$$
D. $$
= 6. 3 3 \times1 0^{-3} {\mathrm{m / s}}
$$
E. $$
= 7. 9 8 \times1 0^{-3} {\mathrm{m / s}}
$$
F. $$
= 9. 0 5 \times1 0^{-3} {\mathrm{m / s}}
$$
G. $$
= 8. 6 3 \times1 0^{-2} {\mathrm{m / s}}
$$
H. $$
= 8. 1 5 \times1 0^{-3} {\mathrm{m / s}}
$$
I. $$
= 7. 2 0 \times1 0^{-3} {\mathrm{m / s}}
$$
J. $$
= 6. 4 7 \times1 0^{-3} {\mathrm{m / s}}
$$
Answer: H
Water at 20°C flows over a flat plate at a velocity of $2 \, \mathrm{m/s}$. Determine the flow velocity at a point $0.2 \, \mathrm{m}$ from the leading edge of the plate and at a vertical distance of $1 \times 10^{-3} \, \mathrm{m}$ from the plate surface. It is known that the kinematic viscosity of water is $\nu = 1.006 \times 10^{-6} \, \mathrm{m}^{2}/\mathrm{s}$, and $Re_{x_{c}} = 5 \times 10^{5}$.

Options:
A. $$
= 1. 6 9 9 {\mathrm{~ m / s}}
$$
B. $$
= 1. 7 9 8 {\mathrm{~ m / s}}
$$
C. $$
= 1. 7 5 5 {\mathrm{~ m / s}}
$$
D. $$
= 1. 8 0 3 {\mathrm{~ m / s}}
$$
E. $$
= 1. 7 4 2 {\mathrm{~ m / s}}
$$
F. $$
= 1. 7 2 6 {\mathrm{~ m / s}}
$$
G. $$
= 1. 7 1 1 {\mathrm{~ m / s}}
$$
H. $$
= 1. 8 2 0 {\mathrm{~ m / s}}
$$
I. $$
= 1. 8 5 5 {\mathrm{~ m / s}}
$$
J. $$
= 1. 6 8 4 {\mathrm{~ m / s}}
$$
Answer: F
Incompressible fluid flows into a rectangular channel with a width of $b$ and a height of $2h$ at a velocity of $u_{0}$ (where $b \gg a$), forming a velocity boundary layer from the inlet. It is known that the thickness of the boundary layer can be approximately estimated by the expression $\delta = 5.48 \sqrt{Vx / u_{0}}$, where $x$ is the distance along the flow direction. Based on the above conditions, derive the expression for calculating the flow entrance length $L_{\mathrm{e}}$.

Options:
A. $$
R e={\frac{2u_{0} h} {\nu}}
$$
B. $$
R e={\frac{u_{0}^{2} h} {\nu}}
$$
C. $$
R e={\frac{u_{0} \overline{b}} {\nu}}
$$
D. $$
R e={\frac{u_{0} h} {b \nu}}
$$
E. $$
R e={\frac{u_{0} h} {u}}
$$
F. $$
R e={\frac{u_{0} a} {\nu}}
$$
G. $$
R e={\frac{u_{0} h} {\eta}}
$$
H. $$
R e={\frac{v_{0} h} {\nu}}
$$
I. $$
R e={\frac{b u_{0}} {\nu}}
$$
J. $$
R e={\frac{u_{0} h} {\nu}}
$$
Answer: J
20°C water is flowing turbulently inside a smooth pipe with an inner diameter of 75mm. It is known that the shear stress at the wall is $3.68 \mathrm{N/m}^{2}$. Determine the pressure drop per meter of pipe length.

Options:
A. $$
= 1 0 9. 5 \, \mathrm{P a} / \mathrm{m}
$$
B. $$
= 1 8 0. 2 \, \mathrm{P a} / \mathrm{m}
$$
C. $$
= 1 2 3. 6 \, \mathrm{P a} / \mathrm{m}
$$
D. $$
= 1 9 6. 3 \, \mathrm{P a} / \mathrm{m}
$$
E. $$
= 2 8 9. 1 \, \mathrm{P a} / \mathrm{m}
$$
F. $$
= 1 6 6. 9 \, \mathrm{P a} / \mathrm{m}
$$
G. $$
= 2 0 4. 8 \, \mathrm{P a} / \mathrm{m}
$$
H. $$
= 2 7 6. 0 \, \mathrm{P a} / \mathrm{m}
$$
I. $$
= 2 3 4. 7 \, \mathrm{P a} / \mathrm{m}
$$
J. $$
= 1 5 7. 4 \, \mathrm{P a} / \mathrm{m}
$$
Answer: D
Assuming that the velocity distribution within the flat plate turbulent boundary layer can be described using a two-layer model, that is, the velocity is a linear distribution in the laminar sublayer; while in the turbulent core, the velocity follows the $1/7$th power law distribution, try to derive the expression for the thickness of the laminar sublayer.

Options:
A. $$
= 6 4. 2 R e_{x}^{-5 / 9}
$$
B. $$
= 1 8. 9 R e_{x}^{-2 / 5}
$$
C. $$
= 5 7. 6 R e_{x}^{-3 / 7}
$$
D. $$
= 4 9. 0 R e_{x}^{-6 / 13}
$$
E. $$
= 8 3. 5 R e_{x}^{-7 / 8}
$$
F. $$
= 7 2. 4 R e_{x}^{-9 / 1 0}
$$
G. $$
= 3 5. 7 R e_{x}^{-8 / 11}
$$
H. $$
= 9 1. 3 R e_{x}^{-1 / 6}
$$
I. $$
= 2 6. 8 R e_{x}^{-4 / 3}
$$
J. $$
= 0 2. 1 R e_{x}^{-7 / 12}
$$
Answer: F
Starting from Fourier's law, derive the temperature distribution equation for one-dimensional steady-state heat conduction in a single-layer plane wall. Given
$$
x=0, \ \ t=t_{1}; \ \ x=b, \ \ t=t_{2}
$$

Options:
A. $$
= \frac{t_{1}+t_{2}}{b} x+l $$
B. $$
= \frac{t_{2}+t_{1}}{l} x+t_{2}$$
C. $$
= {\frac{t_{1}-t_{2}}{b}} x+t_{2}$$
D. $$
= {\frac{b(t_{1}-t_{2})}{l}} +t_{2}$$
E. $$
= {\frac{t_{2}-t_{1}x} {b}} +t_{1}$$
F. $$
= {\frac{l(t_{2}-t_{1})}{b}} +x $$
G. $$
= {\frac{t_{2}-t_{1}} {l}} x+t_{1}
$$
H. $$
= {\frac{l(t_{1}+t_{2})}{x}} +t_{2}$$
I. $$
= {\frac{t_{1}-t_{2}}{l}} x-t_{1}$$
J. $$
= {\frac{t_{2}-t_{1}}{x}} l+t_{1}$$
Answer: G
After heating a steel ball with a diameter of 60 mm to 760°C, it is quenched by immersing it into a water tank with a temperature of 30°C, where the water temperature remains constant at 30°C. How long will it take for the temperature at the center of the ball to drop to 93°C? Given that the convective heat transfer coefficient between the surface of the steel ball and the water is $h=710 \, \mathrm{W/(m^2 \cdot K)}$, and the average thermal conductivity and thermal diffusivity of the steel ball are $k=45 \, \mathrm{W/(m \cdot K)}$ and $\alpha=9.027 \times 10^{-6} \, \mathrm{m^2/s}$, respectively.

Options:
A. $$
\theta=1. 0 \, / \, 0. 0 1=1 0 0 \mathrm{~ s=1. 8 0 ~}
$$
B. $$
\theta=1. 5 \, / \, 0. 0 1=1 5 0 \mathrm{~ s=2. 8 1 ~}
$$
C. $$
\theta=2. 0 \, / \, 0. 0 1=2 0 0 \mathrm{~ s=4. 4 0 ~}
$$
D. $$
\theta=2. 5 \, / \, 0. 0 1=2 5 0 \mathrm{~ s=5. 6 3 ~}
$$
E. $$
\theta=3. 0 \, / \, 0. 0 1=3 0 0 \mathrm{~ s=6. 7 5 ~}
$$
F. $$
\theta=1. 8 \, / \, 0. 0 1=1 8 0 \mathrm{~ s=3. 4 0 ~}
$$
G. $$
\theta=2. 8 \, / \, 0. 0 1=2 8 0 \mathrm{~ s=6. 3 0 ~}
$$
H. $$
\theta=1. 7 \, / \, 0. 0 1=1 7 0 \mathrm{~ s=3. 1 0 ~}
$$
I. $$
\theta=2. 2 \, / \, 0. 0 1=2 2 0 \mathrm{~ s=4. 9 2 ~}
$$
J. $$
\theta=1. 9 \, / \, 0. 0 1=1 9 0 \mathrm{~ s=3. 1 7 ~}
$$
Answer: J
Water at a temperature of 333 K flows through a circular pipe with an inner diameter of 25 mm at a mass flow rate of 35 kg/h. The pipe wall temperature is maintained at a constant 363 $\mathrm{K}$. It is known that the water flow is fully developed upon entering the pipe. The water flows through a pipe length of $4$ m and is heated, with the measured outlet temperature of the water being approximately 345 $\mathrm{K}$. Try to find the average convective heat transfer coefficient $h_{m}$ of the water flow inside the pipe.

Options:
A. $$
= 1 2 9. 5 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
B. $$
= 1 2 0. 5 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
C. $$
= 1 4 0. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
D. $$
= 1 2 4. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
E. $$
= 1 2 5. 5 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
F. $$
= 1 1 9. 5 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
G. $$
= 1 4 5. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
H. $$
= 1 3 0. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
I. $$
= 1 5 0. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
J. $$
= 1 3 5. 0 ~ \mathrm{W} / ( \mathrm{m}^{2} \ \ \mathrm{K} )
$$
Answer: D
A spherical container with a radius of 10 cm is maintained at a temperature of $\mathbf{t}=27^{\circ} \, \mathbf{C}$, but there is a 1 $\operatorname{cm}^{2}$ area on the container wall that is kept at an extremely low temperature. The container is filled with water vapor, with an initial pressure of $\mathbf{p}_{0}=10 \mathrm{mm} \mathbf{Hg}$. Assuming that every water molecule condenses upon impacting the cold surface and deposits on this cold surface, how long is it will take for the vapor pressure in the container to drop to $\mathbf{p}=10^{-4} \mathrm{mm} \mathbf{Hg}$ ?.

Options:
A. Approximately 4.00 seconds
B. Approximately 2.25 seconds
C. Approximately 3.25 seconds
D. Approximately 3.75 seconds
E. Approximately 5.65 seconds
F. Approximately 2.75 seconds
G. Approximately 4.50 seconds
H. Approximately 3.00 seconds
I. Approximately 5.20 seconds
J. Approximately 6.10 seconds
Answer: C
An adiabatic cylindrical chamber is divided into two parts by a piston. Initially, the piston is in the middle of the chamber. On one side of the piston, there is 0.001 m$^{3}$ of air at a temperature of 300 K and a pressure of 0.2 MPa. On the other side, there is also 0.001 m$^{3}$ of air at a temperature of 300 K, but with a pressure of 0.1 MPa. Assuming the air can be regarded as an ideal gas, what is the entropy increase of the system after the piston reaches the equilibrium position?

Options:
A. 0.133 joules per kelvin
B. 0.048 joules per kelvin
C. 0.057 joules per kelvin
D. 0.089 joules per kelvin
E. 0.063 joules per kelvin
F. 0.117 joules per kelvin
G. 0.075 joules per kelvin
H. 0.091 joules per kelvin
I. 0.102 joules per kelvin
J. 0.035 joules per kelvin
Answer: C
At $0^{\circ} \mathrm{C}$ and 1 atm, the density of air is $0.00129 \, \mathrm{g} \cdot \mathrm{cm}^{-3}$. The specific heat at constant pressure for air, $\mathrm{C_p}$, is $0.238 \, \mathrm{cal} \cdot \mathrm{g}^{-1} \cdot \mathrm{K}^{-1}$, with $\gamma = 1.41$. Given 27 m$^{3}$ of air, if the container has a crack and the external pressure is 1 atm, the amount of heat required to slowly heat the air from 0°C to 20°C is____.

Options:
A. $$
1. 5 0 \times1 0^{5} \mathrm{c a l}
$$
B. $$
1. 4 5 \times1 0^{5} \mathrm{c a l}
$$
C. $$
2. 2 5 \times1 0^{5} \mathrm{c a l}
$$
D. $$
1. 8 5 \times1 0^{5} \mathrm{c a l}
$$
E. $$
1. 9 5 \times1 0^{5} \mathrm{c a l}
$$
F. $$
2. 1 5 \times1 0^{5} \mathrm{c a l}
$$
G. $$
2. 0 0 \times1 0^{5} \mathrm{c a l}
$$
H. $$
1. 3 0 \times1 0^{5} \mathrm{c a l}
$$
I. $$
1. 6 0 \times1 0^{5} \mathrm{c a l}
$$
J. $$
1. 7 0 \times1 0^{5} \mathrm{c a l}
$$
Answer: I
Let $w = {^3\sqrt{z}}$ be defined on the complex plane with a branch cut starting from the origin $z = 0$ along the negative real axis, and $w(-2) = -^3\sqrt{2}$ (this is the function value corresponding to the boundary landing point). Find the value of $w(i)$.

Options:
A. $$
\mathrm{e}^{\frac{5} {4} \pi i}
$$
B. $$
\mathrm{e}^{\frac{1} {3} \pi i}
$$
C. $$
\mathrm{e}^{\frac{3} {4} \pi i}
$$
D. $$
\mathrm{e}^{\frac{4} {3} \pi i}
$$
E. $$
\mathrm{e}^{\frac{1} {4} \pi i}
$$
F. $$
\mathrm{e}^{\frac{7} {6} \pi i}
$$
G. $$
\mathrm{e}^{\frac{5} {6} \pi i}
$$
H. $$
\mathrm{e}^{\frac{2} {5} \pi i}
$$
I. $$
\mathrm{e}^{\frac{2} {3} \pi i}
$$
J. $$
\mathrm{e}^{\frac{7} {5} \pi i}
$$
Answer: G
Find the analytic function $f(z) = u + \mathrm{i} v$ according to the following conditions:
(1) $u = x^2 + xy - y^2$,
$$
f(\mathrm{i}) = -1 + \mathrm{i};
$$

Options:
A. $$
x y+x^{2}-y^{2}+\mathrm{i} \left( x^{2}+{\frac{x y} {2}}+{\frac{y^{2}} {2}}+{\frac{1} {2}} \right)
$$
B. $$
x^{2}+x y-y^{2}+\mathrm{i} \left( 2 x y+{\frac{y^{2}} {2}}+{\frac{x^{2}} {2}}+{\frac{1} {2}} \right)
$$
C. $$
x y+y^{2}-x^{2}+\mathrm{i} \left( 2 y x+{\frac{y^{2}} {2}}+{\frac{x^{2}} {2}}+{\frac{1} {2}} \right)
$$
D. $$
x^{2}+x y+y^{2}+\mathrm{i} \left( 2 x^{2}+{\frac{y^{2}} {2}}+{\frac{y x} {2}}+{\frac{1} {2}} \right)
$$
E. $$
x^{2}+y x-x^{2}+\mathrm{i} \left( x^{2}+{\frac{2 y x} {2}}+{\frac{y^{2}} {2}}+{\frac{1} {2}} \right)
$$
F. $$
y^{2}+y x+x^{2}+\mathrm{i} \left( y^{2}+{\frac{x^{2}} {2}}+{\frac{2 y^{2}} {2}}+{\frac{1} {2}} \right)
$$
G. $$
y^{2}+x y-x^{2}+\mathrm{i} \left( 2 x^{2}+{\frac{x y} {2}}+{\frac{x^2} {2}}+{\frac{1} {2}} \right)
$$
H. $$
y^{2}+x^{2}+x y+\mathrm{i} \left( 2 x y+{\frac{x y} {2}}+{\frac{y^{2}} {2}}+{\frac{1} {2}} \right)
$$
I. $$
xy+y^{2}+x^{2}+\mathrm{i} \left( y^{2}+{\frac{2 x y} {2}}+{\frac{x^{2}} {2}}+{\frac{1} {2}} \right)
$$
J. $$
x^{2}+y^{2}-y x+\mathrm{i} \left( 2 y x+{\frac{x y} {2}}+{\frac{y^{2}} {2}}+{\frac{1} {2}} \right)
$$
Answer: B
Express the following functions as power series of $z$: (1) $f ( z )=\frac{4 z^{2}+30 z+68} {( z+4 )^{2} ( z-2 )}$ (2) $f ( z )=\frac{z}{\mathrm{e}^{z}-1}.$

Options:
A. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n+1} \, \frac{n} {2^{n+2}}-\frac{2} {4^{n-1}} \biggr] z^{n}, \mid z \mid< 4
$$
$$
1+{\frac{z} {4}}-{\frac{z^{2}} {8}}+\cdots, \mid z \mid< 4 \pi.
$$
B. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n+1} \, \frac{n} {3^{2 n}}-\frac{3} {6^{n}} \biggr] z^{n}, \mid z \mid< 6
$$
$$
1-{\frac{z} {5}}+{\frac{z^{3}} {5}}-\cdots, \mid z \mid< 5 \pi.
$$
C. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n} \, \frac{n+1} {5^{3 n}}-\frac{1} {4^{n-2}} \biggr] z^{n}, \mid z \mid< 8
$$
$$
1+{\frac{z} {7}}-{\frac{z^{2}} {3}}+\cdots, \mid z \mid< 7 \pi.
$$
D. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n} \, \frac{n+2} {8^{n+3}}-\frac{3} {2^{n}} \biggr] z^{n}, \mid z \mid< 7
$$
$$
1-{\frac{z} {6}}+{\frac{z^{2}} {4}}-\cdots, \mid z \mid< 6 \pi.
$$
E. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n} \, \frac{n+2} {7^{2 n}}-\frac{2} {5^{n}} \biggr] z^{n}, \mid z \mid< 5
$$
$$
1-{\frac{z} {3}}+{\frac{z^{3}} {10}}-\cdots, \mid z \mid< 3 \pi.
$$
F. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n} \, \frac{n+2} {4^{3 n}}-\frac{2} {3^{n+1}} \biggr] z^{n}, \mid z \mid< 1
$$
$$
1+{\frac{z} {4}}-{\frac{z^{2}} {6}}+\cdots, \mid z \mid< 9 \pi.
$$
G. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n} \, \frac{n+1} {4^{n+2}}-\frac{1} {3^{n}} \biggr] z^{n}, \mid z \mid< 3
$$
$$
1+{\frac{z} {3}}-{\frac{z^{2}} {15}}+\cdots, \mid z \mid< \pi.
$$
H. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n+1} \, \frac{n+1} {2^{2 n+3}}-\frac{1} {2^{n-1}} \biggr] z^{n}, \mid z \mid< 2
$$
$$
1-{\frac{z} {2}}+{\frac{z^{2}} {1 2}}-\cdots, \mid z \mid< 2 \pi.
$$
I. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n+1} \, \frac{n+2} {6^{2 n-1}}-\frac{3} {5^{n+2}} \biggr] z^{n}, \mid z \mid< 10
$$
$$
1-{\frac{z} {9}}+{\frac{z^{2}} {3}}-\cdots, \mid z \mid< 10 \pi.
$$
J. $$
\sum_{n=0}^{∞} \biggl[ (-1 )^{n+1} \, \frac{n+1} {3^{2 n+1}}-\frac{3} {4^{n}} \biggr] z^{n}, \mid z \mid< 9
$$
$$
1-{\frac{z} {8}}+{\frac{z^{2}} {2}}-\cdots, \mid z \mid< 8 \pi.
$$
Answer: H
Expand the function $f(z) = \frac{1}{z^2 + 1}$ into series at $z = -1$ and $z = \infty$ respectively.

Options:
A. $$
\sum_{n=0}^{\infty} \frac{( -\mathrm{i} )^{n}} {( z+\mathrm{i} )^{n+3}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n} \frac{z^2} {z^{2n+2}}.
$$
B. $$
\sum_{n=0}^{\infty} \frac{( 2\mathrm{i} )^{n+1}} {( z+\mathrm{i} )^{n+3}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n+1} \frac{1} {z^{2 \left( n+1 \right)+1}}.
$$
C. $$
\sum_{n=0}^{\infty} \frac{( 2\mathrm{i} )^{n+1}} {( z+\mathrm{i} )^{n+4}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n} \frac{1} {z^{2n+3}}.
$$
D. $$
\sum_{n=0}^{\infty} \frac{( -2 )^{n}} {( z-\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n} \frac{1} {z^{n+2}}.
$$
E. $$
\sum_{n=0}^{\infty} \frac{( \mathrm{i} )^{n}} {( z-\mathrm{i} )^{n+3}}
$$
$$
\sum_{n=0}^{\infty} \frac{1} {z^{2n+2}}.
$$
F. $$
\sum_{n=0}^{\infty} \frac{( 2 \mathrm{i} )^{n}} {( z+\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n} \frac{1} {z^{2 \left( n+1 \right)}}.
$$
G. $$
\sum_{n=0}^{\infty} \frac{1} {( z+2\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n+1} \frac{z} {z^{2n+3}}.
$$
H. $$
\sum_{n=0}^{\infty} \frac{( -\mathrm{i} )^{n+1}} {( z+2\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n+1} \frac{1} {z^{2n}}.
$$
I. $$
\sum_{n=0}^{\infty} \frac{( -2\mathrm{i} )^{n}} {( z+\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \left(-1 \right)^{n-1} \frac{1} {z^{2 \left( n \right)}}.
$$
J. $$
\sum_{n=0}^{\infty} \frac{\mathrm{i}^{n}} {( z+2\mathrm{i} )^{n+2}}
$$
$$
\sum_{n=0}^{\infty} \frac{1} {z^{2n+4}}.
$$
Answer: F
Calculate the following products:
$$
\int_{| z |-1} \frac{\mathrm{d} z} {z \operatorname{sin} z};
$$

Options:
A. 1/z
B. cos(z)
C. 2π
D. 0
E. e
F. 1
G. -1
H. ∞
I. -π
J. π
Answer: D
Find the value of the integral
$$
\int_{0}^{2 \pi} \frac{\mathrm{d}x}{\left(2+\sqrt{3} \cos x\right)^2}
$$

Options:
A. $$
0
$$
B. $$
\pi^2
$$
C. $$
3 \pi
$$
D. $$
12 \pi
$$
E. $$
4 \pi
$$
F. $$
6 \pi
$$
G. $$
8 \pi
$$
H. $$
\pi
$$
I. $$
2 \pi
$$
J. $$
10 \pi
$$
Answer: E
Calculate the following integral:
$$
\int_{0}^{+\infty} {\frac{\operatorname{s i n} x} {x ( x^{2}+1 )^{2}}} \mathrm{d} x
$$

Options:
A. $$
{\frac{\pi} {2}} \left( 1-{\frac{3} {2 \mathrm{e}}} \right)
$$
B. $$
{\frac{\pi} {2}} \left( 1-{\frac{3} {\mathrm{e}}} \right)
$$
C. $$
{\frac{\pi} {2}} \left( 1-{\frac{2} {2 \mathrm{e}}} \right)
$$
D. $$
{\frac{\pi} {2}} \left( 1-{\frac{3} {4 \mathrm{e}}} \right)
$$
E. $$
{\frac{\pi} {4}} \left( 1-{\frac{3} {2 \mathrm{e}}} \right)
$$
F. $$
{\frac{\pi^{2}} {2}} \left( 1-{\frac{3} {2 \mathrm{e}}} \right)
$$
G. $$
{\frac{\pi} {2}} \left( 2-{\frac{3} {2 \mathrm{e}}} \right)
$$
H. $$
{\frac{\pi} {2}} \left( 1+{\frac{3} {2 \mathrm{e}}} \right)
$$
I. $$
{\frac{\pi} {6}} \left( 1-{\frac{3} {2 \mathrm{e}}} \right)
$$
J. $$
{\frac{\pi} {3}} \left( 1-{\frac{5} {2 \mathrm{e}}} \right)
$$
Answer: A
Using Rouche's theorem, determine the number of roots of the following equation within $B=\{z \mid \lvert z \rvert < 1 \}$:
$$
z^{8}-5z^{5}-2z+1=0
$$

Options:
A. $$7$$
B. $$9$$
C. $$1$$
D. $$8$$
E. $$5$$
F. $$3$$
G. $$4$$
H. $$6$$
I. $$0$$
J. $$2$$
Answer: E
Find the fractional linear transformation that maps the disk $|z| < \rho$ conformally onto the disk $|w| < R$, such that $z = a$ ($|a| < \rho$) is mapped to $w = 0$.

Options:
A. $$
R \mathrm{e}^{i a \rho} \cdot\frac{z+a} {\rho^{2}+a \bar{z}}
$$
B. $$
R \rho\mathrm{e}^{i z} \cdot\frac{z+a} {\rho^{2}+\bar{a} z}
$$
C. $$
R \rho^2 \cdot\frac{z - a} {\rho^{2} - \bar{a} z}
$$
D. $$
R \rho\mathrm{e}^{i a} \cdot\frac{z-a} {\rho^{2}-\bar{a} z}
$$
E. $$
R \rho \mathrm{e}^{i } \cdot\frac{a-z} {\rho^{2}-\bar{z} a}
$$
F. $$
\rho R \cdot\mathrm{e}^{i a} \frac{z-\bar{a}} {\rho^{2}-a z}
$$
G. $$
R \rho\mathrm{e}^{-i a} \cdot\frac{z+a} {\rho^{2}+\bar{a} z}
$$
H. $$
R \rho \cdot\frac{z-a \mathrm{e}^{i \rho}} {\rho^{2}-a z}
$$
I. $$
R \rho\mathrm{e}^{i z} \cdot\frac{\bar{z}-a} {\rho^{2} + a \bar{z}}
$$
J. $$
R \mathrm{e}^{i \rho} \cdot\frac{z-a} {\rho^{2}-\bar{z} a}
$$
Answer: D
Assume that the spectral density of the vertical acceleration transmitted to the human body through the seat support surface is a white noise, $Ga(f) = 0.1 \, \text{m}^2 \cdot \text{s}^{-3}$. Find the root mean square weighted acceleration $a_w$ within the range of 0.5~80HZ and the weighted vibration level $L_{aw}$ is().

Options:
A. $$
= 1 1 0. 4 7
$$
B. $$
= 1 4 2. 8 9
$$
C. $$
= 1 4 7. 7 0
$$
D. $$
= 1 2 8. 9 4
$$
E. $$
= 1 5 5. 1 2
$$
F. $$
= 1 2 3. 5 0
$$
G. $$
= 1 5 9. 0 0
$$
H. $$
= 1 6 0. 3 3
$$
I. $$
= 1 3 8. 6 5
$$
J. $$
= 1 3 5. 2 5
$$
Answer: C
In addition to using saline solution to disinfect fish fries, they can also be soaked in a ____ potassium permanganate solution for 10 to 20 minutes.

Options:
A. 25-40ppm
B. 2-8ppm
C. 20-35ppm
D. 30-50ppm
E. 15-30ppm
F. 50-60ppm
G. 10-20ppm
H. 5-10ppm
I. 1-5ppm
J. 0.5-2ppm
Answer: G
The transportation of fish species is best _____ transportation.

Options:
A. Swimming
B. Stocking
C. Aging
D. Fasting
E. Feeding
F. Cooling
G. Dipping
H. Breeding
I. Heating
J. Packing
Answer: D
Please prove that the Lorentz force density (the Lorentz force acting on a unit volume of charged material) $\boldsymbol{f}=\rho\boldsymbol{E}+\boldsymbol{J} \times\boldsymbol{B}$ is the first three components of a covariant vector in special relativity. Find the fourth component of this covariant vector.

Options:
A. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{B} \cdot\boldsymbol{E}
$$
B. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{E} \times\boldsymbol{B}
$$
C. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{J} \cdot\boldsymbol{B}
$$
D. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{B} \times\boldsymbol{J}
$$
E. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{E} \cdot\boldsymbol{B}
$$
F. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{E} \cdot\boldsymbol{J}
$$
G. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{J} \cdot\boldsymbol{E}
$$
H. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{B} \cdot\boldsymbol{J}
$$
I. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{E} \cdot\boldsymbol{E}
$$
J. $$
{\frac{\mathrm{i}} {c}} \boldsymbol{J} \times\boldsymbol{E}
$$
Answer: G
What is the other name for the Franco-Flemish School?

Options:
A. The Polyphonic Burgundian School, the Flemish Classic School
B. The Second Netherlandish School, the Third Netherlandish School
C. The Gothic Polyphonic School, the Franco-Flamand Movement
D. The Renaissance Vocal School, the Flemish Renaissance Group
E. The Burgundian Tradition, the Netherlandish Motet Group
F. The Early Netherlandic School, the Franco-German Style
G. The Renaissance Court School, the Burgundian Vocal Style
H. The Franco-Dutch School, the Netherlandish Ars Nova
I. The Burgundy Court School, the Franco-Gothic Style
J. The High Flemish Style, the Franco-Classical School
Answer: B
What was the predecessor of the Baroque chalumeau?

Options:
A. Recorder
B. Tenor Cornett
C. Sackbut
D. Serpent
E. Clarinet
F. Bass Dulcian
G. Flauto Dolce
H. Shawm
I. Crumhorn
J. Oboe d'Amore
Answer: E
What are the characteristics of Haydn's string quartets?

Options:
A. The four instruments are equally important, each with its own personality. The viola is like a talkative woman, frequently interrupting; the second violin primarily serves as the cello's companion, expressing itself less often.
B. The four instruments achieve balance by emphasizing exact repetition. The viola alternates between counterpoints and silences, while the second violin redundantly doubles the cello's parts for uniformity.
C. The four instruments operate with strict uniformity, blending into a single voice. The viola never plays independently, only echoing others; the second violin moves in unison with the viola, avoiding unique contributions.
D. The four instruments display a rigid hierarchy, with the violin leading every movement. The viola provides purely functional support, and the second violin mirrors the first, showing no independence at all.
E. The four instruments are always dominated by the violin, suppressing other voices. The viola plays a minimal role, simply mirroring the cello; the second violin often avoids complex harmonies, staying silent.
F. The four instruments highlight the first violin as the central figure of expression. The viola subtly shadows the first violin's melody, while the second violin sticks to imitative harmonic lines shared with the cello.
G. The four instruments rely heavily on the cello for guidance, creating a bass-driven harmony. The viola tends to follow the melody tightly, while the second violin imitates the first violin directly, offering no contrast.
H. The four instruments are divided into pairs that alternate in dialogue. The viola cooperates strictly with the first violin, and the second violin teams with the cello, creating predictable contrasts over harmony.
I. The four instruments focus on symmetry through paired gestures and patterns. The viola consistently supports the harmony, and the second violin contrasts minimally, adding rhythmic ornamentations sparingly.
J. The four instruments prioritize melodic harmony over individuality. The viola avoids standing out, harmonizing with the cello; the second violin defers to the first violin's leadership, enhancing its melodies.
Answer: A
What are the similarities between French Grand Opera and Italian Opera Seria?

Options:
A. Both focus on religious narratives; neither includes sung recitative.
B. Both emerged in the Baroque period; neither requires a full orchestra.
C. Both originate in Germany; neither features scenic innovations.
D. Both were composed for ballet; neither emphasizes solo performances.
E. Both use mythological plots; neither showcases large choruses.
F. Both involve elaborate staging; neither employs comic elements.
G. Both emphasize improvisation; neither relies on written scores.
H. Both cater to small venues; neither incorporates grand spectacles.
I. Both primarily use historical themes; neither has dialogue.
J. Both prioritize spoken drama; neither highlights vocal virtuosity.
Answer: I
Who is the founder of the Hungarian national music school?

Options:
A. Kodály; Wagner; Liszt.
B. Dvořák; Kodály; Bartók.
C. Liszt; Dohnányi; Kodály.
D. Kodály; Liszt; Wagner.
E. Kodály; Liszt; Dohnányi.
F. Eckel; Mazzoni; Bartók.
G. Kodály; Bartók; Liszt.
H. Bartók; Liszt; Dvořák.
I. Bartók; Kodály; Erkel.
J. Liszt; Kodály; Bartók.
Answer: F
What is the first American musical that represents the American spirit?

Options:
A. "Porgy and Bess"
B. "Annie Get Your Gun"
C. "Rent"
D. "Carousel"
E. "West Side Story"
F. "Oklahoma"
G. "Hair"
H. "Show Boat"
I. "The Music Man"
J. "42nd Street"
Answer: F
An observer whose dark-adapted pupil is 5 mm in diameter can observe a star of 6th magnitude towards the zenith (overhead). He now observes with a pair of binoculars having an objective size of 40mm. The sensitivity of the binoculars compared with the eye is proportional to the ratio of the areas of the objective and pupil so that, using the binoculars, the observerwill be able to observe fainter objects. What is the faintest magnitude star that the observer might hope to observe at the zenith with their use?

Options:
A. 10.0
B. 10.5
C. 8.7
D. 8.5
E. 11.0
F. 9.8
G. 12.2
H. 11.5
I. 12.0
J. 9.3
Answer: B
Assume that Venus has a circular orbit and that it orbits the Sun every 224.7 days. As Venus passed closest to the Earth a planetary radar at Jodrell Bank measured the round trip travel time of a radar pulse to be 272 s. Estimate the value of the Astronomical Unit, namely the semi-major axis of the Earth's orbit around the Sun. (Assume that the Earth also a circular orbit and that the velocity of light is $3 \times1 0^{8} \mathrm{m s}^{-1}$. )

Options:
A. $$
1. 4 6 0 \times1 0^{8} \mathrm{k m}
$$
B. $$
1. 4 7 0 \times1 0^{8} \mathrm{k m}
$$
C. $$
1. 4 8 5 \times1 0^{8} \mathrm{k m}
$$
D. $$
1. 4 9 0 \times1 0^{8} \mathrm{k m}
$$
E. $$
1. 5 2 5 \times1 0^{8} \mathrm{k m}
$$
F. $$
1. 4 6 5 \times1 0^{8} \mathrm{k m}
$$
G. $$
1. 4 8 0 \times1 0^{8} \mathrm{k m}
$$
H. $$
1. 5 0 0 \times1 0^{8} \mathrm{k m}
$$
I. $$
1. 4 7 5 \times1 0^{8} \mathrm{k m}
$$
J. $$
1. 3 8 0 \times1 0^{8} \mathrm{k m}
$$
Answer: I
A rapidly rotating asteroid lies four times further from the Sun than the Earth. Assuming the asteroid acts like a black body and is roughly spherical, estimate its surface temperature. (The solar constant at the Earth's distance is 1370 $\mathbf{Wm}^{-2}$ and Stefan's constant is $5. 7 \times1 0^{-8} \mathrm{W m}^{-2} \mathrm{K}^{-4}$.)

Options:
A. 144K
B. 155K
C. 132K
D. 161K
E. 128K
F. 120K
G. 118K
H. 150K
I. 139K
J. 147K
Answer: I
A planet has been discovered orbiting the sun-like star HD 209458. From the period and velocity measured by the Doppler method, its mass was
calculated to be 0.69 times that of Jupiter. The plane of the planet's orbit is edge-on and it has been observed to transit across the face of the star. The light from the star was seen to drop by 1.7% when the planet's disc lay fully within that of the star which has a radius of $8 \times1 0^{5} \mathrm{k m}$ . Assuming that the brightness of the star is constant across the observed face, estimate the diameter of the planet and hence it's density in comparison with that of Jupiter which has a radius of $7. 1 \times1 0^{4} \mathrm{k m}$.

Options:
A. $$
2. 5 \times1 0^{5} \mathrm{k m}, 0.18
$$
B. $$
2. 0 \times1 0^{5} \mathrm{k m}, 0.23
$$
C. $$
2. 1 \times1 0^{5} \mathrm{k m}, 0.21
$$
D. $$
2. 7 \times1 0^{5} \mathrm{k m}, 0.17
$$
E. $$
2. 4 \times1 0^{5} \mathrm{k m}, 0.22
$$
F. $$
1. 9 \times1 0^{5} \mathrm{k m}, 0.25
$$
G. $$
2. 2 \times1 0^{5} \mathrm{k m}, 0.19
$$
H. $$
2. 3 \times1 0^{5} \mathrm{k m}, 0.24
$$
I. $$
1. 8 \times1 0^{5} \mathrm{k m}, 0.27
$$
J. $$
2. 6 \times1 0^{5} \mathrm{k m}, 0.15
$$
Answer: C
The author imaged Jupiter when closest to the Earth last year using his 230 mm aperture, $f10$, Schmidt-Cassegrain telescope with a 170 pixels per millimetre CCD camera having an effective area of $3. 9 \times2. 8 \ \mathrm{m m}$. To increase the image scale, the effective focal ratio of the telescope was increased by a factor of 2.5 by the use of a Barlow lens. Assuming that Jupiter's equator lay parallel to one axis of the CCD chip, how many pixels lay across Jupiter's image?

Options:
A. $$230$$
B. $$275$$
C. $$190$$
D. $$200$$
E. $$240$$
F. $$250$$
G. $$300$$
H. $$150$$
I. $$180$$
J. $$220$$
Answer: D
A star has a mass which is 20 times that of the Sun and a surface temperature of 30 000K. Given that the Sun has a surface temperature of 6000K and assuming both stars have a similar density, estimate the star's lifetime on the main sequence given that our Sun will remain on the main sequence for 10 000 million years. (Assume both stars act as black bodies and convert a similar percentage of their mass into energy.)

Options:
A. $$
1. 9 2 1 \times1 0^{7} \mathrm{y r s}
$$
B. $$
6. 5 4 3 \times1 0^{6} \mathrm{y r s}
$$
C. $$
4. 5 6 7 \times1 0^{8} \mathrm{y r s}
$$
D. $$
2. 1 1 1 \times1 0^{8} \mathrm{y r s}
$$
E. $$
2. 7 3 2 \times1 0^{6} \mathrm{y r s}
$$
F. $$
8. 6 8 6 \times1 0^{6} \mathrm{y r s}
$$
G. $$
4. 3 4 3 \times1 0^{7} \mathrm{y r s}
$$
H. $$
5. 4 7 3 \times1 0^{6} \mathrm{y r s}
$$
I. $$
3. 2 7 6 \times1 0^{8} \mathrm{y r s}
$$
J. $$
7. 8 3 1 \times1 0^{7} \mathrm{y r s}
$$
Answer: G
Betelgeuse is a red giant at a distance of 428 light years. In the future it will become supernova similar to 'Tycho's supernova' which was observed in 1572 and lies at a distance of 9800 light years. At its peak, its brightness was similar to that of Venus (which has a peak magnitude of $\sim -4$ ). What might we expect the peak apparent magnitude of the Betelgeuse supernova explosion to be?

Options:
A. 5.0
B. -0.5
C. 2.8
D. 2.0
E. 1.6
F. 4.7
G. 6.3
H. 3.4
I. -3.2
J. -2.1
Answer: E
Consider a one-dimensional infinite potential well with a width of $a$:
$$
\mathrm{V}(x) = \left\{
\begin{aligned}
0, & \quad \text{if } |x| < \frac{a}{2}; \\
+\infty, & \quad \text{if } |x| \geq \frac{a}{2}.
\end{aligned}
\right.
$$
At $t = 0$, a point particle is released at $x = 0$. Calculate the ratio of the probabilities for the particle to be in the second excited state to the ground state.

Options:
A. 1.5
B. 0.25
C. 5
D. 4
E. 3
F. 2
G. 0
H. 0.5
I. 0.75
J. 1
Answer: J