MikaStars39/nano-eval huggingface dataset:
Find the value of $\sqrt{\sqrt[3]{28} - \sqrt[3]{27}}$.

Options:
A. $$
{\frac{1} {3}} ( \sqrt[3] {9 8}-\sqrt[3] 2 8-1 )
$$
B. $$
{\frac{1} {7}} ( \sqrt[3] {8 2}-\sqrt[3] 2 3-1 )
$$
C. $$
{\frac{1} {2}} ( \sqrt[3] {9 4}-\sqrt[3] 2 7-1 )
$$
D. $$
{\frac{1} {4}} ( \sqrt[3] {7 7}-\sqrt[3] 2 6-1 )
$$
E. $$
{\frac{2} {3}} ( \sqrt[3] {9 5}-\sqrt[3] 2 5-1 )
$$
F. $$
{\frac{1} {5}} ( \sqrt[3] {8 7}-\sqrt[3] 3 1-1 )
$$
G. $$
{\frac{2} {5}} ( \sqrt[3] {9 1}-\sqrt[3] 2 1-1 )
$$
H. $$
{\frac{1} {6}} ( \sqrt[3] {8 6}-\sqrt[3] 2 9-1 )
$$
I. $$
{\frac{3} {4}} ( \sqrt[3] {9 7}-\sqrt[3] 2 4-1 )
$$
J. $$
{\frac{1} {8}} ( \sqrt[3] {7 2}-\sqrt[3] 2 8-1 )
$$
Answer: A
Let $\alpha_{1}, \cdots, \alpha_{k}$ be positive numbers with the following property: in the sequence of numbers $[ \, \alpha_{1} \, ], \cdots, [ \, \alpha_{k} \, ]$, $\left[ \, 2 \alpha_{1} \, \right], \cdots, \left[ \, 2 \alpha_{k} \, \right], \left[ \, 3 \alpha_{1} \, \right], \cdots, \left[ \, 3 \alpha_{k} \, \right], \cdots$, each natural number appears exactly once. What is the value of $k$?

Options:
A. $$
k \geq2
$$
B. $$
k \geq4
$$
C. $$
k \leq5
$$
D. $$
k = 3
$$
E. $$
k \leq4
$$
F. $$
k \geq3
$$
G. $$
k \leq3
$$
H. $$
k \leq2
$$
I. $$
k \leq1
$$
J. $$
k \geq5
$$
Answer: H
Express the polynomial $( x+1 ) ( x+2 ) ( x+3 ) ( x+4 )$ as a perfect square form.

Options:
A. $$
x^{2}-5x+10
$$
B. $$
x^{3}+5x+1
$$
C. $$
x^{2}+5x+4
$$
D. $$
x^{4}+5x+2
$$
E. $$
x^{3}+5x+8
$$
F. $$
x^{2}+5 x \pm5
$$
G. $$
x^{3}+6x-7
$$
H. $$
x^{2}-4x+3
$$
I. $$
x^{2}-5x+9
$$
J. $$
x^{2}+3x-5
$$
Answer: F
Allow the expression $x \left( x-a \right) \left( x-b \right) \left( x-c \right) +1$ to be factored into the product of two polynomials whose coefficients are all integers. Find the integers $a, b, c$ that are non-zero and distinct from each other.

Options:
A. a=2, b=2, c=3 or
a=3, b=-2, c=-1 or
a=-2, b=1, c=0 or
a=2, b=4, c=6
B. a=0, b=0, c=3 or
a=1, b=1, c=4 or
a=2, b=3, c=5 or
a=-2, b=-3, c=-1
C. a=2, b=0, c=1 or
a=1, b=-3, c=-2 or
a=3, b=4, c=6 or
a=2, b=3, c=5
D. a=3, b=2, c=1 or
a=-1, b=-3, c=-4 or
a=2, b=5, c=7 or
a=0, b=1, c=2
E. a=0, b=2, c=3 or
a=4, b=3, c=2 or
a=1, b=3, c=2 or
a=-1, b=-2, c=0
F. a=1, b=3, c=5 or
a=-2, b=2, c=2 or
a=2, b=6, c=7 or
a=3, b=4, c=5
G. a=1, b=4, c=5 or
a=-3, b=-1, c=-2 or
a=3, b=5, c=7 or
a=0, b=1, c=3
H. a=1, b=1, c=2 or
a=-1, b=2, c=-1 or
a=2, b=3, c=4 or
a=-3, b=-2, c=-1
I. a=1, b=2, c=3 or
a=-1, b=-2, c=-3 or
a=1, b=-2, c=-1 or
a=2, b=-1, c=1
J. a=1, b=2, c=4 or
a=-2, b=-1, c=-3 or
a=4, b=5, c=6 or
a=-1, b=1, c=0
Answer: I
The specific weight of water at 20°C is $\gamma = 9.789 \, \mathrm{kN/m}^{3}$, and its dynamic viscosity is $\mu = 1.005 \times 10^{-3} \, \mathrm{N} \cdot \mathrm{s}/\mathbf{m}^{2}$. Find its kinematic viscosity $\nu$. The specific weight of air at 20°C is $\gamma = 11.82 \, \mathrm{N/m}^{3}$, and its kinematic viscosity is $\nu = 0.150 \, \mathrm{cm}^{2}/\mathrm{s}$. Find its dynamic viscosity.

Options:
A. $$
2. \, 8 0 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
B. $$
2. \, 7 0 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
C. $$
1. \, 8 0 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
D. $$
1. \, 8 0 1 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
E. $$
1. \, 9 9 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
F. $$
1. \, 7 1 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
G. $$
1. \, 9 0 5 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
H. $$
1. \, 8 0 4 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
I. $$
1. \, 3 0 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
J. $$
1. \, 6 5 9 \times1 0^{-5} \ \ ( \mathrm{N \cdot s / m^{2}} )
$$
Answer: C
A cylindrical inner nozzle is fitted on the water tank, with a cross-sectional area of $A$ for the nozzle. The contracted cross-sectional area of the water jet flowing through the nozzle is $A_{\mathrm{r}}$, so the contraction coefficient is $\varepsilon={\frac{A_{\mathrm{c}}}{A}}$. Assuming the volume of the tank is very large and the water level remains constant, the pressure distribution along the tank wall can be considered according to the distribution of hydrostatic pressure. Additionally, water head loss is ignored. What is the reaction force of the water jet on the water tank?

Options:
A. 0.7
B. 0.5
C. 1.6
D. 1.8
E. 0.0
F. 1.2
G. 1.1
H. 2.3
I. 1.5
J. 0.9
Answer: B
A pipeline with a diameter of $d = 300 \, \mathrm{mm}$ and a length of $L=5 \, \mathrm{km}$ is transporting heavy oil with a specific weight $\gamma = 9.66 \, \mathrm{kN/m}^{3}$. The weight flow rate is $G = 2373 \, \mathrm{kN/h}$ (where h is hours). Calculate the change in power loss of the pipeline when the oil temperature changes from $t_{1}=10^\circ\mathrm{C}$ (kinematic viscosity $v_{1}=25 \, \mathrm{cm}^{2}/\mathrm{s}$) to $t_{2}=40^\circ\mathrm{C}$ (kinematic viscosity $u_{2}=1.5 \, \mathrm{cm}^{2}/\mathrm{s}$).

Options:
A. $$
N_{1} \! \approx\! 2 8 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 7. \, 5 0 \, ( \mathrm{k W} )
$$
B. $$
N_{1} \! \approx\! 2 6 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 8. \, 8 0 \, ( \mathrm{k W} )
$$
C. $$
N_{1} \! \approx\! 3 2 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 5. \, 5 0 \, ( \mathrm{k W} )
$$
D. $$
N_{1} \! \approx\! 3 1 5 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 6. \, 7 0 \, ( \mathrm{k W} )
$$
E. $$
N_{1} \! \approx\! 2 7 5 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 6. \, 4 5 \, ( \mathrm{k W} )
$$
F. $$
N_{1} \! \approx\! 3 0 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 2 0. \, 0 0 \, ( \mathrm{k W} )
$$
G. $$
N_{1} \! \approx\! 2 9 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 9. \, 0 5 \, ( \mathrm{k W} )
$$
H. $$
N_{1} \! \approx\! 3 0 1 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 8. \, 5 5 \, ( \mathrm{k W} )
$$
I. $$
N_{1} \! \approx\! 2 8 9 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 1 7. \, 3 5 \, ( \mathrm{k W} )
$$
J. $$
N_{1} \! \approx\! 2 5 0 ( \mathrm{k W} )
$$
$$
N_{2} \! \approx\! 2 0. \, 4 0 \, ( \mathrm{k W} )
$$
Answer: I
There is a pressurized tunnel with a circular cross-section, with a length of $l = 200 \mathrm{m}$, carrying a flow rate of $Q = 700 \mathrm{m}^{3} / \mathrm{s}$. If the inner wall of the tunnel is not lined, its average diameter is $d_{1} = 7.8 \mathrm{m}$, with a roughness coefficient of $n = 0.033$. If lined with concrete, the diameter is $d_{2} = 7.0 \mathrm{m}$, and the roughness coefficient is $n = 0.014$. How much is the head loss reduced for the lined option compared to the unlined option along the length of the tunnel?

Options:
A. $$
1 2. 5 0 ( \mathrm{m} )
$$
B. $$
1 1. 9 0 ( \mathrm{m} )
$$
C. $$
1 3. 0 5 ( \mathrm{m} )
$$
D. $$
1 5. 4 0 ( \mathrm{m} )
$$
E. $$
1 4. 0 0 ( \mathrm{m} )
$$
F. $$
1 3. 8 0 ( \mathrm{m} )
$$
G. $$
1 1. 2 0 ( \mathrm{m} )
$$
H. $$
1 2. 9 5 ( \mathrm{m} )
$$
I. $$
1 0. 7 5 ( \mathrm{m} )
$$
J. $$
1 4. 5 0 ( \mathrm{m} )
$$
Answer: C
A trapezoidal cross-section earth channel has a roughness coefficient $n=0.025$, a bottom width $b=8.2 \, \mathrm{m}$, a side slope coefficient $m=1.5$, and a bottom slope $i=0.0004$, with a flow rate $Q=35 \, \mathrm{m}^3/\mathrm{s}$. A regulating sluice gate is built on the channel. When the water depth in front of the gate is $H=4 \, \mathrm{m}$, please calculate the water depth 5000 m upstream of the sluice gate.

Options:
A. 3.050m
B. 2.850m
C. 2.620m
D. 3.015m
E. 2.769m
F. 3.100m
G. 3.250m
H. 2.800m
I. 2.945m
J. 2.500m
Answer: E
The normal high water level of the reservoir is 72.0m, and the extreme flood level is 75.6m. The spillway is an uncontrolled wide-crested weir with a steep chute and an energy dissipation basin downstream. The weir has a width of 60m, with a weir crest height of $P=2 \mathrm{m}$ and a crest elevation of 70.6m, featuring a rounded crest inlet. If the weir is the same width as the upstream canal, what is the flow rate for the two aforementioned water levels?

Options:
A. $$
Q_{1}=1 6 9. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 2 3 4. 8 \, \mathrm{m}^{3} / \mathrm{s}
$$
B. $$
Q_{1}=1 5 5. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 2 6 7. 2 \, \mathrm{m}^{3} / \mathrm{s}
$$
C. $$
Q_{1}=1 5 8. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 3 4 5. 9 \, \mathrm{m}^{3} / \mathrm{s}
$$
D. $$
Q_{1}=1 7 0. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 2 5 8. 3 \, \mathrm{m}^{3} / \mathrm{s}
$$
E. $$
Q_{1}=1 6 0. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 3 5 0. 4 \, \mathrm{m}^{3} / \mathrm{s}
$$
F. $$
Q_{1}=1 7 2. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 2 2 9. 5 \, \mathrm{m}^{3} / \mathrm{s}
$$
G. $$
Q_{1}=1 6 5. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 2 4 7. 6 \, \mathrm{m}^{3} / \mathrm{s}
$$
H. $$
Q_{1}=1 7 5. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 3 5 7. 6 \, \mathrm{m}^{3} / \mathrm{s}
$$
I. $$
Q_{1}=1 6 3. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 3 6 0. 8 \, \mathrm{m}^{3} / \mathrm{s}
$$
J. $$
Q_{1}=1 6 8. \ 0 \mathrm{m^{3} / s}
$$
$$
Q_{2} \!=\! 1 3 2 1. 7 \, \mathrm{m}^{3} / \mathrm{s}
$$
Answer: G
The diameter of a penstock in a hydropower station is $D = 2.5 \mathrm{m}$, with a wall thickness of $e = 25 \mathrm{mm}$. If the steel pipe conveys water from the reservoir to the hydropower station with a pipeline length $L = 2000 \mathrm{m}$, and the valve at the end of the pipeline closes in $T_{s} = 3 \mathrm{s}$, will direct or indirect water hammer occur? Water head loss is not considered in the calculation.

Options:
A. Dynamic Fluid Compression
B. Direct Water Hammer
C. Steady Flow Condition Decay
D. Sudden Flow Surge Effect
E. Mechanical Impact Phenomenon
F. Rapid Pressure Fluctuation
G. Acoustic Pulsation Wave
H. Indirect Water Hammer
I. Hydraulic Resonance Initiation
J. Pipe Structural Vibration
Answer: H
The flow velocity of high-speed water is very high, and the pressure is very low, making the water prone to cavitating into bubbles, causing cavitation erosion on hydraulic structures.If small bubbles are combined together, the harm caused by the bubbles can be reduced. Now, 50 bubbles with a radius of $R_{1}=0.1 \text{ mm}$ are combined into a larger bubble. Given that the pressure of the water surrounding the bubbles is $p_{0}=6000 \text{ N/m}^{2}$ and the water temperature is 20°C,try to find the radius $R$ of the resulting bubble. (Hint: Assume the pressure of the small bubbles is $p_{1}$,the density is $\rho_{1}$, the pressure of the large bubble is $p$,and the density is $\rho$,then $p / p_{1} = \rho / \rho_{1}$.)

Options:
A. 0.4052 (mm)
B. 0.3658 (mm)
C. 0.4587 (mm)
D. 0.3883(mm)
E. 0.3774 (mm)
F. 0.4726 (mm)
G. 0.3995 (mm)
H. 0.3499 (mm)
I. 0.4210 (mm)
J. 0.3421 (mm)
Answer: D
A vertically oriented rectangular gate with a height of $h_{2}$ is submerged with its top at a depth of $h_{1}$ below the water surface. Determine the total hydrostatic pressure acting on the gate.

Options:
A. $$
h_{1}+\frac{4 h_{1}+h_{2}} {3 ( 3 h_{1}+h_{2} )} h_{2}
$$
B. $$
h_{1}+\frac{2 h_{1}+3 h_{2}} {2 ( h_{1}+3 h_{2} )} h_{2}
$$
C. $$
h_{1}+\frac{h_{1}+3 h_{2}} {4 ( h_{2}+h_{1} )} h_{2}
$$
D. $$
h_{1}+\frac{3 h_{2}+h_{1}} {4 ( h_{1}+h_{2} )} h_{2}
$$
E. $$
h_{1}+\frac{3 h_{2}+h_{1}} {3 ( 2 h_{2}+h_{1} )} h_{2}
$$
F. $$
h_{1}+\frac{3 h_{1}+2 h_{2}} {3 ( 2 h_{1}+h_{2} )} h_{2}
$$
G. $$
h_{1}+\frac{2 h_{2}+3 h_{1}} {3 ( h_{2}+2 h_{1} )} h_{2}
$$
H. $$
h_{1}+\frac{2 h_{1}+h_{2}} {4 ( h_{1}+2 h_{2} )} h_{2}
$$
I. $$
h_{1}+\frac{h_{2}+4 h_{1}} {2 ( h_{1}+h_{2} )} h_{2}
$$
J. $$
h_{1}+\frac{h_{1}+2 h_{2}} {3 ( 2 h_{1}+h_{2} )} h_{2}
$$
Answer: F
A closed container filled with water has its bottom circular hole sealed by a metal sphere. The sphere weighs 19.6 N and has a diameter $D = 0.1 \text{ m}$, while the diameter of the circular hole is $d = 0.08 \text{ m}$. The water depth is $h_{1} = 0.5 \text{ m}$, and the external container water surface is 0.1 m lower, with $h_{2} = 0.4 \text{ m}$. The water surface is at atmospheric pressure, and the pressure on the water surface inside the container is $p_{0}$. Determine the water pressure on the sphere when $p_{0}$ is equal to atmospheric pressure.

Options:
A. $$
0. 2 1 0 \ \mathrm{( N )}
$$
B. $$
0. 2 5 3 \ \mathrm{( N )}
$$
C. $$
0. 2 0 5 \ \mathrm{( N )}
$$
D. $$
0. 1 7 5 \ \mathrm{( N )}
$$
E. $$
0. 1 9 1 \ \mathrm{( N )}
$$
F. $$
0. 2 3 0 \ \mathrm{( N )}
$$
G. $$
0. 2 0 0 \ \mathrm{( N )}
$$
H. $$
0. 2 4 7 \ \mathrm{( N )}
$$
I. $$
0. 1 8 3 \ \mathrm{( N )}
$$
J. $$
0. 1 5 9 \ \mathrm{( N )}
$$
Answer: C
A square iron gate with a side length of 30 cm is hinged at point $o$ on the top. Its weight is $G = 117.7$ N. The centerline of a water jet with a diameter $d = 0.02 \mathrm{m}$ passes through the center $C$ of the gate, with a jet velocity of $v_{0} = 15 \mathrm{m/s}$. What force $F$ needs to be applied at the bottom to keep the gate in a vertical position?

Options:
A. $$
34.167(N)
$$
B. $$
35.343(N)
$$
C. $$
28.500(N)
$$
D. $$
39.300(N)
$$
E. $$
42.789(N)
$$
F. $$
37.921(N)
$$
G. $$
40.452(N)
$$
H. $$
31.223(N)
$$
I. $$
32.678(N)
$$
J. $$
33.875(N)
$$
Answer: B
Discuss the conditions for the similarity of turbulent resistance. In general, resistance can be expressed as
$$
F = \tau \chi l
$$
where $\tau$ is the additional shear stress due to turbulence, which can be written as $\tau = \gamma R J$, where $\gamma$ is the specific weight of water; $R$ is the hydraulic radius, defined as $R = A / \chi$, with $A$ being the cross-sectional area of water flow, and $\chi$ the wetted perimeter, i.e., the length of the boundary wetted by the flow. The flow velocity, expressed using the Chezy formula, is
$$
v = C \ \sqrt{R J}
$$
where $C$ is the Chezy coefficient. According to the relationship between the drag coefficient and the Chezy coefficient, we have
$$
C = \sqrt{8 g / \lambda}
$$
where $\lambda$ is the drag coefficient. Substituting the above formulas into the resistance formula yields
$$
F = \gamma R J \; \frac{A}{R} l = \gamma R \; \frac{v^{2}}{C^{2} R} \frac{A}{R} l = \gamma A l \; \frac{v^{2}}{(8 g / \lambda) R} = \gamma A \lambda\; \frac{l}{4 R} \frac{v^{2}}{2 g} = \rho A \lambda\; \frac{l}{4 R} \frac{v^{2}}{2}
$$
The relationship between the Chezy coefficient $C$ and the roughness $n$ can be expressed via the Manning formula as $C = R^{1 / 6} / n$. Determine the scale of roughness $\lambda_{n}$.

Options:
A. $$
\lambda_{n}=\lambda_{l}^{4 / 6}
$$
B. $$
\lambda_{n}=\lambda_{l}^{8 / 6}
$$
C. $$
\lambda_{n}=\lambda_{l}^{1 / 3}
$$
D. $$
\lambda_{n}=\lambda_{l}^{6 / 6}
$$
E. $$
\lambda_{n}=\lambda_{l}^{3 / 6}
$$
F. $$
\lambda_{n}=\lambda_{l}^{9 / 6}
$$
G. $$
\lambda_{n}=\lambda_{l}^{1 / 6}
$$
H. $$
\lambda_{n}=\lambda_{l}^{7 / 6}
$$
I. $$
\lambda_{n}=\lambda_{l}^{2 / 6}
$$
J. $$
\lambda_{n}=\lambda_{l}^{5 / 6}
$$
Answer: G
The diameter of the circular pipe $d = 2.5 \mathrm{cm}$, equivalent roughness $\Delta = 0.4 \mathrm{mm}$, average flow velocity in the pipe $v = 2.5 \mathrm{m/s}$, kinematic viscosity of water $\nu = 0.01 \mathrm{cm}^{2}/\mathrm{s}$. Find the wall shear stress $\tau_{0}$.

Options:
A. $$
37.15 (N/m^{2})
$$
B. $$
28.75 (N/m^{2})
$$
C. $$
48.22 (N/m^{2})
$$
D. $$
29.42 (N/m^{2})
$$
E. $$
33.56 (N/m^{2})
$$
F. $$
35.47 (N/m^{2})
$$
G. $$
45.32 (N/m^{2})
$$
H. $$
30.69 (N/m^{2})
$$
I. $$
39.81 (N/m^{2})
$$
J. $$
41.03 (N/m^{2})
$$
Answer: F
The known velocity distribution formula for smooth turbulent flow in a circular pipe is $u / u_{*}=2. 5 \operatorname{l n} ( u_{*} \ y / \nu)+5. 5$. Determine the formula for the average velocity of the smooth circular pipe.

Options:
A. $$
v \!=\! u_{*} \left[ 4. 0 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+2. 3 5 \right]
$$
B. $$
v \!=\! u_{*} \left[ 2. 9 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+1. 5 0 \right]
$$
C. $$
v \!=\! u_{*} \left[ 3. 0 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+2. 0 0 \right]
$$
D. $$
v \!=\! u_{*} \left[ 3. 5 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+1. 2 5 \right]
$$
E. $$
v \!=\! u_{*} \left[ 2. 8 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+2. 5 0 \right]
$$
F. $$
v \!=\! u_{*} \left[ 3. 1 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+1. 9 5 \right]
$$
G. $$
v \!=\! u_{*} \left[ 2. 2 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+3. 0 0 \right]
$$
H. $$
v \!=\! u_{*} \left[ 3. 7 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+2. 1 0 \right]
$$
I. $$
v \!=\! u_{*} \left[ 1. 5 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+2. 8 5 \right]
$$
J. $$
v \!=\! u_{*} \left[ 2. 5 \! \operatorname{l n} \left( u_{*} r_{0} / \nu\right)+1. 7 5 \right]
$$
Answer: J
Given that the starting point of the straight-line segment on the $e-\operatorname{log} p$ curve of a standard consolidation test for silt in a certain area is between 50 and 100 kPa. In this area, a silt sample has a measured compression coefficient $a_{1-2}$ of 1.66 $MPa^{-1}$ in the $100 \sim 200 \mathrm{kPa}$ pressure range. Which of the following numerical values is closest to the compression index $C_{e}$?

Options:
A. 1.45
B. 0.65
C. 0.55
D. 1.25
E. 0.75
F. 0.95
G. 1.15
H. 1.35
I. 0.85
J. 1.05
Answer: C
The earth dam is experiencing serious seepage in the dam foundation, and it is proposed to use jet grouting pile technology along the dam axis at the dam crest to construct a vertical impermeable core wall. The wall extends into the impermeable layer underlying the dam foundation. It is known that the foundation of the dam base is a sand layer, with a thickness of 10m, and a length along the dam axis of 100m. The permeability coefficient of the jet grouting pile wall is $1 \times 10^{-7} \mathrm{cm/s}$, and the wall width is 2m. When the upstream water level is 40m and the downstream water level is 10m, which of the following values is closest to the seepage volume of the reinforced foundation of this earth-rock dam (excluding the seepage volume of the dam body)?

Options:
A. $2.45 m^{3}/d$
B. $1.00 m^{3}/d$
C. $3.12 m^{3}/d$
D. $1.75 m^{3}/d$
E. $1.50 m^{3}/d$
F. $2.00 m^{3}/d$
G. $0.98 m^{3}/d$
H. $0.75 m^{3}/d$
I. $1.29 m^{3}/d$
J. $2.90 m^{3}/d$
Answer: I
The highway uses a high embankment on the soft soil foundation in the bridgehead section, with an average roadbed width of 30 meters. The uniformly distributed load transferred to the bottom surface of the roadbed from the roadbed's own weight and the pavement load is 120 kPa. The foundation soil is uniform, with an average $E_{\mathrm{a}} = 6 \mathrm{MPa}$. The thickness of the compressible layer considered in the settlement calculation is 24 meters, and the settlement calculation correction coefficient is 1.2, What is the final settlement amount of the bridgehead roadbed closest to?

Options:
A. 275mm
B. 260mm
C. 235mm
D. 295mm
E. 230mm
F. 265mm
G. 248mm
H. 220mm
I. 280mm
J. 240mm
Answer: G
In a certain office building, the length is $46.0 \mathrm{m}$, the width is $12.8 \mathrm{m}$, and the total construction area is $2860 \mathrm{m}^{2}$. The foundation soil is mixed fill, and the characteristic value of the foundation bearing capacity is $f_{\mathrm{a}}=86 \mathrm{kPa}$. It is planned to use lime-soil compacted piles, with a designed pile diameter of $d=400 \mathrm{mm}$, and the maximum dry density of the filler within the pile is $\rho_{\mathrm{dmax}}=1.67 \mathrm{t/m}^{3}$. Before site treatment, the average dry density is $\bar{\rho}_{\mathrm{d}}=1.33 \mathrm{t/m}^{3}$, and after compaction, the average dry density requirement between the pile and soil should reach $\bar{\rho}_{\mathrm{d1}}=1.54 \mathrm{t/m}^{3}$. A lime-soil compacted pile design is to be conducted. Pile holes are arranged in an equilateral triangle pattern, and the most suitable center-to-center distance between pile holes is ( )m. With the piles arranged in a square pattern, assuming a pile spacing of $s=900 \mathrm{mm}$, the most suitable number of pile holes is ( ).

Options:
A. 1.50m,1000
B. 1.25m,1100
C. 1.10m,1200
D. 1.35m,1075
E. 0.90m,1150
F. 0.80m,1120
G. 1.03m,1035
H. 1.20m,1080
I. 2.00m,950
J. 1.40m,1050
Answer: G
Male, 21 years old, suddenly experienced left chest pain while lifting a barbell 2 hours ago, accompanied by difficulty breathing, sweating, and palpitations. The pulse is weak at 120 beats per minute. Breathing is shallow and rapid, 24-30 breaths per minute, blood pressure is 90/60. He quickly went to the campus hospital, lying on his right side with an elevated pillow, and exhibited cyanosis of the lips and distended neck veins. The main diagnosis of the patient's condition is ().

Options:
A. Spontaneous Pneumonia
B. Acute Bronchitis
C. Chronic Obstructive Disease
D. Aortic Dissection
E. Cardiac Tamponade
F. Pulmonary Embolism
G. Tension Hemothorax
H. Congestive Heart Failure
I. Pneumothorax
J. Myocardial Infarction
Answer: I
Female, 26 years old, experiencing fatigue and skin bruising for 2 weeks, diagnosed with acute myeloid leukemia and admitted for treatment. Blood routine tests show white blood cells at $15 \times 10^9$/L, platelets at 15 or $10^9$/L, and hemoglobin at 83 g/L. Bone marrow examination reveals 65% blasts, 2% promyelocytes, 18% other cells, and 12% monocytes. If a chromosomal examination is conducted, the correct finding is _____.

Options:
A. $$
\mathfrak{t} ( 1 1 ; 2 0 ) \left( \mathsf{q 1 3} ; \mathsf{q 2 5} \right)
$$
B. $$
\mathfrak{t} ( 9 ; 2 2 ) \left( \mathsf{q 2 3} ; \mathsf{q 1 1} \right)
$$
C. $$
\mathfrak{t} ( 6 ; 1 8 ) \left( \mathsf{q 2 1} ; \mathsf{q 1 5} \right)
$$
D. $$
\mathfrak{t} ( 5 ; 5 ) \left( \mathsf{q 2 4} ; \mathsf{q 2 0} \right)
$$
E. $$
\mathfrak{t} ( 1 4 ; 1 5 ) \left( \mathsf{q 1 2} ; \mathsf{q 2 3} \right)
$$
F. $$
\mathfrak{t} ( 1 5 ; 1 7 ) \left( \mathsf{q 2 2} ; \mathsf{q 2 1} \right)
$$
G. $$
\mathfrak{t} ( 3 ; 3 ) \left( \mathsf{q 4 2} ; \mathsf{q 1 1} \right)
$$
H. $$
\mathfrak{t} ( 7 ; 1 3 ) \left( \mathsf{q 1 1} ; \mathsf{q 3 3} \right)
$$
I. $$
\mathfrak{t} ( 8 ; 2 1 ) \left( \mathsf{q 3 2} ; \mathsf{q 2 2} \right)
$$
J. $$
\mathfrak{t} ( 1 0 ; 2 3 ) \left( \mathsf{q 2 5} ; \mathsf{q 1 0} \right)
$$
Answer: F
At a certain point in the pipe, the water flow speed is $2 \mathrm{m/s}$, and the pressure is $10^{4} \mathrm{Pa}$ greater than atmospheric pressure. At another point in the pipe, where the height has decreased by 1m, the cross-sectional area of the pipe is $\frac{1}{2}$ of the initial area. Find how much the pressure at this point exceeds atmospheric pressure?

Options:
A. $$
3. 2 \times1 0^{3} \mathrm{P a}
$$
B. $$
1. 4 \times1 0^{4} \mathrm{P a}
$$
C. $$
5. 6 \times1 0^{4} \mathrm{P a}
$$
D. $$
7. 5 \times1 0^{2} \mathrm{P a}
$$
E. $$
3. 1 \times1 0^{4} \mathrm{P a}
$$
F. $$
6. 8 \times1 0^{2} \mathrm{P a}
$$
G. $$
2. 9 \times1 0^{5} \mathrm{P a}
$$
H. $$
9. 0 \times1 0^{6} \mathrm{P a}
$$
I. $$
8. 4 \times1 0^{5} \mathrm{P a}
$$
J. $$
4. 7 \times1 0^{3} \mathrm{P a}
$$
Answer: B
The water with a surface tension coefficient of $7.27 \times 10^{-2} \, \mathrm{N/m}$ $(\rho_{1}=999 \, \mathrm{kg/m^{3}})$ rises 2.5 cm in a capillary tube. Acetone $(\rho_{2}=792 \, \mathrm{kg/m^{3}})$ rises 1.4 cm in the same capillary tube. Assuming both completely wet the capillary tube, what is the surface tension coefficient of acetone?

Options:
A. $$ 5 . 8 7 \times 1 0^{-2} \mathrm{N / m} $$
B. $$ 2 . 9 5 \times 1 0^{-2} \mathrm{N / m} $$
C. $$ 6 . 5 4 \times 1 0^{-2} \mathrm{N / m} $$
D. $$ 6 . 8 1 \times 1 0^{-2} \mathrm{N / m} $$
E. $$ 7 . 1 0 \times 1 0^{-2} \mathrm{N / m} $$
F. $$ 4 . 1 9 \times 1 0^{-2} \mathrm{N / m} $$
G. $$ 4 . 7 6 \times 1 0^{-2} \mathrm{N / m} $$
H. $$ 4 . 0 2 \times 1 0^{-2} \mathrm{N / m} $$
I. $$ 5 . 3 2 \times 1 0^{-2} \mathrm{N / m} $$
J. $$
3. 2 3 \times1 0^{-2} \mathrm{N / m}
$$
Answer: J
A thin wire is bent into a semicircle with a radius $a$, and the charge $Q$ is uniformly distributed on it. What is the electric potential at the center of curvature?

Options:
A. $$
k \, {\frac{Q} {a}}
$$
B. $$
k \, {\frac{Q} {a^2}}
$$
C. $$
k \, {\frac{Q} {a + \pi}}
$$
D. $$
k \, {\frac{Q} {\pi a}}
$$
E. $$
k \, {\frac{Q} {a^3}}
$$
F. $$
k \, {\frac{Q^2} {a^2}}
$$
G. $$
k \, {\frac{3Q} {a}}
$$
H. $$
k \, {\frac{Q} {4a}}
$$
I. $$
k \, {\frac{Q} {2a}}
$$
J. $$
k \, {\frac{2Q} {a}}
$$
Answer: A
Place a glass rod with a refractive index of 1.50, a diameter of $8.0 \times 10^{-2} \mathrm{m}$, and a convex hemispherical end face into a liquid. An object is located $6.0 \times 10^{-1} \mathrm{m}$ from the end face along the rod's axis and forms an image 1.00 m inside the rod. What is the refractive index of the liquid?

Options:
A. 1.40
B. 1.30
C. 1.60
D. 1.35
E. 1.25
F. 1.65
G. 1.55
H. 1.20
I. 1.45
J. 1.70
Answer: D
A bullet with a mass of $10 \, \mathrm{g}$ is traveling at a speed of $1000 \, \mathrm{m/s}$. If the uncertainty in measuring the bullet's position is $0.10 \, \mathrm{cm}$, what is the uncertainty in its speed?

Options:
A. $$
6. 1 2 0 \times1 0^{-2 7} \mathrm{m / s}
$$
B. $$
6. 8 4 7 \times1 0^{-2 6} \mathrm{m / s}
$$
C. $$
6. 3 1 3 \times1 0^{-2 8} \mathrm{m / s}
$$
D. $$
6. 5 3 2 \times1 0^{-2 8} \mathrm{m / s}
$$
E. $$
6. 9 0 4 \times1 0^{-3 0} \mathrm{m / s}
$$
F. $$
6. 6 8 9 \times1 0^{-2 5} \mathrm{m / s}
$$
G. $$
6. 7 8 2 \times1 0^{-2 7} \mathrm{m / s}
$$
H. $$
6. 4 2 6 \times1 0^{-3 0} \mathrm{m / s}
$$
I. $$
6. 1 4 5 \times1 0^{-2 6} \mathrm{m / s}
$$
J. $$
6. 6 2 6 \times1 0^{-2 9} \mathrm{m / s}
$$
Answer: J
An $\alpha$ particle moves in a circular orbit with a radius of $R=0.83 \mathrm{cm}$ in a uniform magnetic field with a magnetic induction intensity of $B=0.025 \mathrm{T}$. Find the de Broglie wavelength of a small ball with mass $m=0.1 \mathrm{kg}$ moving at the same speed as the $\alpha$ particle.

Options:
A. $7. 8 1 4 \times1 0^{-3 3} \mathrm{m}$
B. $1. 2 3 4 \times1 0^{-3 7} \mathrm{m}$
C. $0. 9 7 2 \times1 0^{-3 0} \mathrm{m}$
D. $9. 0 5 1 \times1 0^{-3 2} \mathrm{m}$
E. $6. 6 2 6 \times1 0^{-3 4} \mathrm{m}$
F. $4. 3 7 9 \times1 0^{-3 6} \mathrm{m}$
G. $8. 6 7 9 \times1 0^{-3 8} \mathrm{m}$
H. $3. 5 8 6 \times1 0^{-3 1} \mathrm{m}$
I. $2. 4 6 8 \times1 0^{-3 9} \mathrm{m}$
J. $5. 9 4 2 \times1 0^{-3 5} \mathrm{m}$
Answer: E
The half-life of $^{3 2}_{1 5} P$ is 14.3 days. Find the activity of $1.0 \, \mu g_{1 5}^{3 2} P$ in $( \operatorname{Ci} )$.

Options:
A. $$
0. 2 0 {\mathrm{C i}}
$$
B. $$
0. 1 5 {\mathrm{C i}}
$$
C. $$
0. 4 1 {\mathrm{C i}}
$$
D. $$
0. 3 5 {\mathrm{C i}}
$$
E. $$
0. 2 9 {\mathrm{C i}}
$$
F. $$
0. 3 0 {\mathrm{C i}}
$$
G. $$
0. 2 2 {\mathrm{C i}}
$$
H. $$
0. 1 8 {\mathrm{C i}}
$$
I. $$
0. 4 3 {\mathrm{C i}}
$$
J. $$
0. 3 8 {\mathrm{C i}}
$$
Answer: E
The physical half-life of a radionuclide used for organ scans is 8 days, and the biological half-life is 3 days. Assuming the tested radioactive activity is $0.1 \text{Ci}$, calculate the radioactive activity remaining in the body after 24 hours.

Options:
A. $$
1. 4 5 \times1 0^{-2} {\mathrm{C i}}
$$
B. $$
2. 8 7 \times1 0^{-2} {\mathrm{C i}}
$$
C. $$
7. 2 9 \times1 0^{-2} {\mathrm{C i}}
$$
D. $$
9. 1 2 \times1 0^{-2} {\mathrm{C i}}
$$
E. $$
8. 5 7 \times1 0^{-2} {\mathrm{C i}}
$$
F. $$
4. 7 5 \times1 0^{-2} {\mathrm{C i}}
$$
G. $$
5. 2 1 \times1 0^{-2} {\mathrm{C i}}
$$
H. $$
6. 3 8 \times1 0^{-2} {\mathrm{C i}}
$$
I. $$
6. 8 3 \times1 0^{-2} {\mathrm{C i}}
$$
J. $$
3. 9 6 \times1 0^{-2} {\mathrm{C i}}
$$
Answer: C
Given that, 3DG100, $\beta=45$, $U_{\mathrm{CC}}=20 \mathrm{V}$, $R_{\mathrm{C}}=6.8 k \Omega$, $R_{\mathrm{B}}=500 k \Omega$, what is the quiescent operating point of this amplifier circuit?

Options:
A. 8.4V
B. 5.2V
C. 9.0V
D. 4.7V
E. 12.6V
F. 6.9V
G. 6.3V
H. 7.8V
I. 11.3V
J. 10.1V
Answer: H
Estimate the arterial blood pressure in the head and feet when a person is in a handstand position.

Options:
A. $$
1 7. 9 6 ( \mathrm{~ k P a ~} ) ;
1.05 ( \mathrm{~ k P a ~} )
$$
B. $$
1 8. 5 4 ( \mathrm{~ k P a ~} ) ;
0.94 ( \mathrm{~ k P a ~} )
$$
C. $$
1 7. 4 8 ( \mathrm{~ k P a ~} ) ;
0.88 ( \mathrm{~ k P a ~} )
$$
D. $$
1 5. 3 7 ( \mathrm{~ k P a ~} ) ;
0.99 ( \mathrm{~ k P a ~} )
$$
E. $$
1 8. 8 8 ( \mathrm{~ k P a ~} ) ;
1.15 ( \mathrm{~ k P a ~} )
$$
F. $$
1 9. 2 3 ( \mathrm{~ k P a ~} ) ;
1.02 ( \mathrm{~ k P a ~} )
$$
G. $$
1 6. 0 0 ( \mathrm{~ k P a ~} ) ;
0.91 ( \mathrm{~ k P a ~} )
$$
H. $$
2 2. 0 9 ( \mathrm{~ k P a ~} ) ;
1.25 ( \mathrm{~ k P a ~} )
$$
I. $$
2 0. 1 5 ( \mathrm{~ k P a ~} ) ;
1.10 ( \mathrm{~ k P a ~} )
$$
J. $$
2 3. 4 4 ( \mathrm{~ k P a ~} ) ;
0.97 ( \mathrm{~ k P a ~} )
$$
Answer: B
Given that $\mathbf{2}{g}$ of hydrogen gas is contained in a $20L$ container, if the pressure of the hydrogen gas inside the container is $3.996 \times 10^{3} \mathrm{Pa}$, what is the average translational kinetic energy of a hydrogen molecule?

Options:
A. $$
\approx7. 4 6 \times1 0^{-2 2} \mathrm{~ ( J )}
$$
B. $$
\approx4. 3 2 \times1 0^{-2 2} \mathrm{~ ( J )}
$$
C. $$
\approx3. 5 7 \times1 0^{-2 1} \mathrm{~ ( J )}
$$
D. $$
\approx9. 2 1 \times1 0^{-2 1} \mathrm{~ ( J )}
$$
E. $$
\approx6. 0 5 \times1 0^{-2 3} \mathrm{~ ( J )}
$$
F. $$
\approx2. 8 8 \times1 0^{-2 3} \mathrm{~ ( J )}
$$
G. $$
\approx1. 9 9 \times1 0^{-2 2} \mathrm{~ ( J )}
$$
H. $$
\approx5. 1 5 \times1 0^{-2 1} \mathrm{~ ( J )}
$$
I. $$
\approx8. 8 3 \times1 0^{-2 2} \mathrm{~ ( J )}
$$
J. $$
\approx1. 2 9 \times1 0^{-2 3} \mathrm{~ ( J )}
$$
Answer: G
The mass of the spring oscillator is $0.025 \, \mathrm{kg}$, and the spring constant is $k = 0.4 \, \mathrm{N} \cdot \mathrm{m}^{-1}$. At time $t = 0$, the oscillator is at position $x_{0} = 0.1 \, \mathrm{m}$ and is moving in the positive direction of the $x$-axis with an initial velocity of $v_{0} = 0.4 \, \mathrm{m} \cdot \mathrm{s}^{-1}$. Find the displacement, velocity, and acceleration of the oscillator respectively at $t = \frac{\pi}{8}$.

Options:
A. $$
0. 1 ( \mathrm{m} )
$$
$$
- 0. 4 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
- 1.6 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
B. $$
0. 1 ( \mathrm{m} )
$$
$$
0. 3 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
0.5 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
C. $$
0. 05 ( \mathrm{m} )
$$
$$
- 0. 2 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
- 0.8 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
D. $$
0. 2 ( \mathrm{m} )
$$
$$
0. 5 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
0.9 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
E. $$
0. 2 ( \mathrm{m} )
$$
$$
0 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
0.8 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
F. $$
0. 12 ( \mathrm{m} )
$$
$$
- 0. 3 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
- 1.2 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
G. $$
0. 15 ( \mathrm{m} )
$$
$$
- 0. 2 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
- 0.6 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
H. $$
0. 15 ( \mathrm{m} )
$$
$$
0. 1 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
1.0 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
I. $$
0. 05 ( \mathrm{m} )
$$
$$
0. 4 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
1.3 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
J. $$
0. 08 ( \mathrm{m} )
$$
$$
- 0. 3 \, ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
$$
- 1.0 \, ( \mathrm{m} \cdot\mathrm{s}^{-2} )
$$
Answer: A
Visible light with a wavelength of $400-760 \, \text{nm}$ is perpendicularly incident on a thin film with a thickness of $380 \, \text{nm}$, which is placed in air and has a refractive index of $1.33$. Which wavelengths of light reflected from the film experience constructive interference?

Options:
A. $$458.1nm,719.3nm$$
B. $$385.2nm,742.5nm$$
C. $$369.4nm,451.5nm$$
D. $$452.2nm,387.4nm$$
E. $$602.1nm,413.9nm$$
F. $$527.9nm,391.6nm$$
G. $$673.9nm,404.3nm$$
H. $$533.7nm,412.8nm$$
I. $$610.4nm,320.5nm$$
J. $$299.6nm,644.2nm$$
Answer: G
The $\mathbf{X}$ rays with a wavelength of $\lambda_{0}=0.0708 \mathrm{~nm}$ are subjected to Compton scattering on paraffin. Determine the wavelength of the scattered $\mathbf{X}$ rays in the directions of $\frac{\pi}{2}$ and ${\pi}$.

Options:
A. $$0.0744(nm),0.0752(nm)$$
B. $$0.0738(nm),0.0774(nm)$$
C. $$0.0718(nm),0.0742(nm)$$
D. $$0.0725(nm),0.0749(nm)$$
E. $$0.0730(nm),0.0765(nm)$$
F. $$0.0722(nm),0.0753(nm)$$
G. $$0.0741(nm),0.0785(nm)$$
H. $$0.0727(nm),0.0761(nm)$$
I. $$0.0732(nm),0.0756(nm)$$
J. $$0.0736(nm),0.0770(nm)$$
Answer: I
Experiments have found that if the atomic nucleus is viewed as $A$ small hard spheres packed together to form a larger sphere, the relationship between the radius of the sphere and the mass number $A$ of the nucleus is $R=r_{0} A^{1 / 3}$, where $r_{0}=1.2 \times 10^{-15} \mathrm{~m}$. Calculate the density of the atomic nucleus.

Options:
A. $$
6. 2 \times1 0^{1 8} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
B. $$
3. 0 \times1 0^{1 6} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
C. $$
1. 2 \times1 0^{1 9} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
D. $$
1. 5 \times1 0^{1 7} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
E. $$
5. 4 \times1 0^{1 5} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
F. $$
7. 9 \times1 0^{1 7} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
G. $$
1. 0 \times1 0^{1 6} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
H. $$
2. 5 \times1 0^{1 8} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
I. $$
4. 8 \times1 0^{1 5} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
J. $$
2. 3 \times1 0^{1 7} \ \mathrm{k g} \cdot\mathrm{m}^{-3}
$$
Answer: J
Using BPO as an initiator, the activation energies of the elementary reactions in styrene polymerization are $E_{\mathrm{d}} \!=\! 125.6 \, \mathrm{kJ/mol}$, $E_{\mathrm{p}} = 32.6 \, \mathrm{kJ/mol}$, $E_{t} = 10 \, \mathrm{kJ/mol}$. Try comparing the changes in the total reaction rate constant and degree of polymerization when the temperature is increased from 50°C to 60°C.

Options:
A. 2.95, 0.680
B. 2.64, 0.690
C. 2.89, 0.710
D. 3.12, 0.745
E. 2.56, 0.705
F. 2.47, 0.660
G. 3.05, 0.700
H. 3.18, 0.735
I. 2.75, 0.675
J. 2.82, 0.625
Answer: I
Methyl methacrylate ($\mathbf{M}_{1}$) and butadiene ($\mathbf{M}_{2}$) undergo free radical copolymerization at 60°C. What ratio of ingredients should be used to achieve a copolymer with a uniform composition be obtained? $r_{1}=0.25, \; \; r_{2}=0.91$.

Options:
A. 50:50 (mass ratio)
B. 23:77 (mass ratio)
C. 10:90 (mass ratio)
D. 60:40 (mass ratio)
E. 45:55 (mass ratio)
F. 29:71 (mass ratio)
G. 35:65 (mass ratio)
H. 32:68 (mass ratio)
I. 27:73 (mass ratio)
J. 18:82 (mass ratio)
Answer: J
The viscosity of a polymer sample at 0°C is 1.$0 \times 10^{3} \, \mathrm{Pa} \cdot s$. If its viscosity as a function of temperature follows the WLF equation, and assuming the viscosity at the glass transition temperature $T_{g}$ is 1.$0 \times 10^{12} \, \mathrm{Pa} \cdot s$, find its viscosity at 25°C.

Options:
A. $$
0 5 \mathrm{P a} \cdot\mathrm{s}
$$
B. $$
3 0 \mathrm{P a} \cdot\mathrm{s}
$$
C. $$
8 0 \mathrm{P a} \cdot\mathrm{s}
$$
D. $$
4 0 \mathrm{P a} \cdot\mathrm{s}
$$
E. $$
6 0 \mathrm{P a} \cdot\mathrm{s}
$$
F. $$
9 5 \mathrm{P a} \cdot\mathrm{s}
$$
G. $$
5 0 \mathrm{P a} \cdot\mathrm{s}
$$
H. $$
2 5 \mathrm{P a} \cdot\mathrm{s}
$$
I. $$
1 5 \mathrm{P a} \cdot\mathrm{s}
$$
J. $$
2 0 \mathrm{P a} \cdot\mathrm{s}
$$
Answer: H
At 25°C, dissolve $10^{-5}$ mol of poly(methyl methacrylate) (PMMA) ($M_n = 10^5$, $\rho = 1.20 \text{ g/cm}^3$) in 150 g of chloroform ($\rho = 1.49 \text{ g/cm}^3$). Calculate the mixing enthalpy $\Delta H_m$ and the mixing free energy $\Delta G_m$. Given $\chi_1 = 0.377$.

Options:
A. $$
\Delta H_{m} \!=\! 3. \, 5 7 9 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 4. \, 1 ( \mathrm{J} )
$$
B. $$
\Delta H_{m} \!=\! 6. \, 7 4 2 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 8. \, 6 ( \mathrm{J} )
$$
C. $$
\Delta H_{m} \!=\! 9. \, 3 8 3 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 5. \, 7 ( \mathrm{J} )
$$
D. $$
\Delta H_{m} \!=\! 5. \, 6 2 0 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 9. \, 9 ( \mathrm{J} )
$$
E. $$
\Delta H_{m} \!=\! 1. \, 9 8 4 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-2 0. \, 5 ( \mathrm{J} )
$$
F. $$
\Delta H_{m} \!=\! 8. \, 4 3 3 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-2 3. \, 0 ( \mathrm{J} )
$$
G. $$
\Delta H_{m} \!=\! 4. \, 6 5 8 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 3. \, 2 ( \mathrm{J} )
$$
H. $$
\Delta H_{m} \!=\! 8. \, 1 7 2 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-2 1. \, 4 ( \mathrm{J} )
$$
I. $$
\Delta H_{m} \!=\! 2. \, 4 1 1 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 6. \, 4 ( \mathrm{J} )
$$
J. $$
\Delta H_{m} \!=\! 7. \, 9 8 3 \, ( \mathrm{J} )
$$
$$
\Delta G_{m} \!=\!-1 2. \, 8 ( \mathrm{J} )
$$
Answer: C
Dissolve 2 mol of vinyl acetate and 0.01 mol of lauroyl peroxide in benzene to obtain 2 L of solution. Polymerization is completed at 60°C after 6 hours, resulting in a polymer with an average molecular weight of $1.1 \times 10^{5}$. If the initiator concentration is changed to 0.004 mol/L, what is the number-average molecular weight ${\bar{M}}_{n}$ of the resulting polymer?

Options:
A. 105478
B. 131247
C. 165934
D. 94852
E. 122984
F. 187493
G. 153276
H. 176390
I. 139087
J. 112561
Answer: E
Using the equilibrium swelling method to determine the crosslink density of sulfur-cured styrene-butadiene rubber (SBR), the test data are as follows: test temperature is 25°C, dry rubber weight is 0.1273g, swollen weight is 2.116g, dry rubber density is 0.941g/cm³, the density of the solvent benzene used is 0.8685g/mL, and the system's $\chi_{1}=0.398$. Please calculate the Young's modulus $E$ of the crosslinked SBR rubber.

Options:
A. $$E=7.\:5\times10^{4}\:(\mathrm{Pa})$$
B. $$E=6.\:8\times10^{4}\:(\mathrm{Pa})$$
C. $$E=6.\:4\times10^{4}\:(\mathrm{Pa})$$
D. $$E=7.\:2\times10^{4}\:(\mathrm{Pa})$$
E. $$E=7.\:0\times10^{4}\:(\mathrm{Pa})$$
F. $$E=5.\:3\times10^{4}\:(\mathrm{Pa})$$
G. $$E=6.\:6\times10^{4}\:(\mathrm{Pa})$$
H. $$E=5.\:7\times10^{4}\:(\mathrm{Pa})$$
I. $$E=6.\:1\times10^{4}\:(\mathrm{Pa})$$
J. $$E=5.\:9\times10^{4}\:(\mathrm{Pa})$$
Answer: B
When synthesizing nylon-66 from equal amounts of adipic acid and hexamethylenediamine, the monofunctional compound acetic acid is often added to end the chains and control the relative molecular weight. To obtain a polyamide with a relative molecular weight of 11318 and a reaction degree of 99.5%, what should be the molecular ratio of acetic acid to adipic acid (or hexamethylenediamine)?

Options:
A. $$
0. 1 0 : 1
$$
B. $$
0. 0 1 : 1
$$
C. $$
0. 0 6 : 1
$$
D. $$
0. 0 5 : 1
$$
E. $$
0. 0 8 : 1
$$
F. $$
0. 1 5 : 1
$$
G. $$
0. 0 2 : 1
$$
H. $$
0. 0 4 : 1
$$
I. $$
0. 0 9 : 1
$$
J. $$
0. 0 3 : 1
$$
Answer: B
To determine the crosslink density of vulcanized natural rubber using the equilibrium swelling method, the following experimental data was obtained: the mass of the rubber sample $W_{2} = 2.034 \times 10^{-3} \, \mathrm{kg}$, after being soaked in benzene in a constant temperature water bath at 298 K for $7 \sim 10 \, \mathrm{days}$, reaching swelling equilibrium, and then weighed $W_{2} + \mathbf{W}_{1} = 10.023 \times 10^{-3} \, \mathrm{kg}$. The manual lists the density of benzene at 298 K as $\rho_{\mathrm{l}}=0.868 \times 10^{3} \, \mathrm{kg \cdot m^{-3}}$, molar volume $\widetilde{V}_{1} = 89.3 \times 10^{-6} \, \mathrm{m^{3} \cdot mol^{-1}}$, the density of natural rubber $\rho_{2} = 0.9971 \times 10^{3} \, \mathrm{kg \cdot m^{-3}}$, and the interaction parameter between natural rubber and benzene $\chi_{1} = 0.437$. Using the above data, calculate the average relative molecular mass of the network chain of the crosslinked polymer $( \overline{{{M_{\mathrm{c}}}}} )$.

Options:
A. $$
4. 8 9 \times1 0^{4}
$$
B. $$
2. 4 2 \times1 0^{4}
$$
C. $$
5. 5 5 \times1 0^{4}
$$
D. $$
3. 5 6 \times1 0^{4}
$$
E. $$
9. 7 8 \times1 0^{3}
$$
F. $$
2. 3 2 \times1 0^{3}
$$
G. $$
1. 0 9 \times1 0^{5}
$$
H. $$
6. 3 3 \times1 0^{4}
$$
I. $$
7. 2 1 \times1 0^{3}
$$
J. $$
1. 6 4 \times1 0^{5}
$$
Answer: B
Using the viscosity method to determine the relative molecular mass of a polystyrene sample, the experiment is conducted in a benzene solution at 30 °C. First, weigh 0.1375 g of the sample and prepare a 25 cm³ PS-benzene solution. Using a pipette, take 10 cm³ of this solution and inject it into the viscometer. The outflow time $t_{1}=241.6$ s is measured, then respectively add benzene $5 \, \mathrm{cm}^{3}, 5 \, \mathrm{cm}^{3}, 10 \, \mathrm{cm}^{3}, 10 \, \mathrm{cm}^{3}$ to dilute, measuring the outflow times as $t_{2}=189.7 \, \mathrm{s.}, t_{3}=166.0 \, \mathrm{s.}, t_{4}=144.4 \, \mathrm{s.}, t_{5}=134.2 \, \mathrm{s.}$ Finally, the outflow time of pure benzene is measured as $t_{0}=106.8 \, \mathrm{s}$. It is found that for the PS-benzene system at 30 °C, $K=0.99 \times 10^{-2} \, \mathrm{cm}^{3} \cdot \mathrm{g}^{-1}, a=0.74$. Calculate the viscosity-average relative molecular mass of the sample.

Options:
A. $6.32 \times 10^5$
B. $7.18 \times 10^5$
C. $6.67 \times 10^5$
D. $6.05 \times 10^5$
E. $5.11 \times 10^5$
F. $4.54 \times 10^5$
G. $5.92 \times 10^5$
H. $5.25 \times 10^5$
I. $5.49 \times 10^5$
J. $4.87 \times 10^5$
Answer: I
When developing a polymer, whether a monomer can polymerize needs to be evaluated from both thermodynamic and kinetic perspectives. The main parameter for determining the tendency for polymerization in thermodynamics is().

Options:
A. Molecular weight $M_n$
B. Gibbs free energy $\Delta G$
C. Reaction temperature $T$
D. Reaction rate constant $k$
E. Crystallization temperature $T_c$
F. Polymerization enthalpy $\Delta H$
G. Colligative properties $\Delta P$
H. Polymerization entropy $\Delta S$
I. Activation energy $E_a$
J. Pressure-volume work $PV$
Answer: F
Assume the modulating wave is a sine wave with a frequency of $500 \mathrm{Hz}$, an amplitude of 5V, and an initial phase of 0. The carrier frequency is 10kHz with an amplitude of 50V. The expression of the amplitude-modulated wave is ().

Options:
A. $$
\Phi_{\mathrm{t}}=5 0+[ 1+2 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
B. $$
\Phi_{\mathrm{t}}=5 0+[ 1+5 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
C. $$
\Phi_{\mathrm{t}}=5 0+[ 1+6 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
D. $$
\Phi_{\mathrm{t}}=4 0+[ 1+5 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
E. $$
\Phi_{\mathrm{t}}=5 0+[ 1+5 \operatorname{s i n} 9 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
F. $$
\Phi_{\mathrm{t}}=5 0+[ 1+5 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 5 0 0 0 \ \pi\ \mathrm{t}
$$
G. $$
\Phi_{\mathrm{t}}=5 0+[ 1+5 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 4 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
H. $$
\Phi_{\mathrm{t}}=5 0+[5 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
I. $$
\Phi_{\mathrm{t}}=5 0+[ 1+4 \operatorname{s i n} 1 0 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
J. $$
\Phi_{\mathrm{t}}=5 0+[ 1-5 \operatorname{s i n} 5 0 0 \ \pi\ \mathrm{t} ] 5 0 \operatorname{s i n} 2 0 0 0 0 \ \pi\ \mathrm{t}
$$
Answer: B
Light with a wavelength of $1.3 \mu m$ is incident into a single-mode step-index optical fiber, where the refractive indices of the core and cladding are $n_{1}=1.52$ and $n_{2}=1.42$, respectively. If it is placed in air (with a refractive index of 1), what are its numerical aperture, maximum incident angle, and core diameter?

Options:
A. 0.549,33.02°,1.866μm
B. 0.553,33.42°,1.926μm
C. 0.545,33.12°,1.906μm
D. 0.536,32.12°,1.856μm
E. 0.542,32.82°,1.836μm
F. 0.548,33.88°,1.936μm
G. 0.528,31.92°,1.746μm
H. 0.540,32.72°,1.816μm
I. 0.531,32.62°,1.776μm
J. 0.534,32.22°,1.796μm
Answer: E
A beam with a divergence angle of $\theta=1 \mathrm{mrad}$ and a diameter of $d=2 \mathrm{mm}$. What is the spot size at the terminal after collimation through the antenna when transmitted over 20 km? If the beam density is uniform and the average atmospheric attenuation is $α=1.5 \mathrm{dB}/\mathrm{km}$, how much has the intensity of the light at the terminal changed?

Options:
A. 7m,$0. 1 8 \times1 0^{-7}$
B. 6m,$0. 2 0 \times1 0^{-6}$
C. 5m,$0. 3 0 \times1 0^{-8}$
D. 4m,$0. 5 0 \times1 0^{-6}$
E. 5m,$0. 4 4 \times1 0^{-9}$
F. 3m,$0. 4 0 \times1 0^{-9}$
G. 6m,$0. 2 5 \times1 0^{-7}$
H. 4m,$0. 9 0 \times1 0^{-8}$
I. 4m,$0. 2 5 \times1 0^{-7}$
J. 5m,$0. 2 5 \times1 0^{-7}$
Answer: I
The equation $( \, 1+z \, )^{5}=( \, 1-z \, )^{5}$, the value of \(z\) is ().

Options:
A. $$
\operatorname{s i n} {\frac{\gamma} {4}}
$$;
$$
\gamma=0, \frac{\pi} {2}, \pi, \frac{3 \pi} {2}, 2 \pi
$$
B. $$
\operatorname{c o s} {\frac{\eta} {6}}
$$;
$$
\eta=\pi, \frac{5 \pi} {6}, {7 \pi} {6}, \frac{11 \pi} {6}, {13 \pi}
$$
C. $$
\operatorname{c s c} {\frac{\nu} {4}}
$$;
$$
\nu=\pi, \frac{3 \pi} {4}, \frac{5 \pi} {4}, \frac{7 \pi} {4}, {9 \pi}
$$
D. $$
\operatorname{t a n} {\frac{\theta} {4}}
$$;
$$
\theta=0, \frac{\pi} {4}, {3 \pi} {4}, \pi, \frac{5 \pi} {4}
$$
E. $$
\operatorname{i t a n} {\frac{\alpha} {2}}
$$;
$$
\alpha=0, \frac{2 \pi} {5}, \frac{4 \pi} {5}, \frac{6 \pi} {5}, \frac{8 \pi} {5}
$$
F. $$
\operatorname{s e c} {\frac{\mu} {3}}
$$;
$$
\mu=0, \frac{2 \pi} {3}, {3 \pi}, \frac{5 \pi} {3}, {11 \pi}
$$
G. $$
\operatorname{s i n} {\frac{\zeta} {5}}
$$;
$$
\zeta=\pi, \frac{2 \pi} {3}, \frac{4 \pi} {3}, \frac{5 \pi} {3}, {7 \pi}
$$
H. $$
\operatorname{c o t} {\frac{\lambda} {2}}
$$;
$$
\lambda=0, \frac{\pi} {5}, \frac{3 \pi} {5}, \pi, \frac{7 \pi} {5}
$$
I. $$
\operatorname{c o t} {\frac{\delta} {3}}
$$;
$$
\delta=0, \frac{\pi} {3}, {2 \pi} {3}, \pi, {4 \pi} {3}
$$
J. $$
\operatorname{c o s} {\frac{\beta} {3}}
$$;
$$
\beta=0, \pi, {2 \pi}, \frac{3 \pi} {4}, {5 \pi}
$$
Answer: E
Given the harmonic function $u(x, y) = x^{2} - y^{2} + x y$, what are its conjugate harmonic function $v(x, y)$ and the analytic function $f(z) = u(x, y) + \mathrm{i} v(x, y)$?

Options:
A. $$
v = y^2 - x^2 + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{2} \cdot (1 + \mathrm{i}) + C
$$
B. $$
v = xy + \frac{x^2}{2} + \frac{y^2}{2} + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
z^2 \cdot (1.5 - \mathrm{i}) + \mathrm{i} C
$$
C. $$
v = y^2 + 2xy + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{2} \cdot (1 - \mathrm{i}) + \mathrm{i} C
$$
D. $$
v = 3xy - \frac{x^2}{3} + \frac{y^2}{3} + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
z^2 \cdot (2 - \mathrm{i}) + C
$$
E. $$
v = x^2 - 2xy - y^2 + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{2} \cdot (0.5 - \mathrm{i}) + \mathrm{i} C
$$
F. $$
v = 2xy - \frac{x^2}{3} + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
z^2 \cdot (2 - \mathrm{i}) + \mathrm{i} C
$$
G. $$
v = x^2 - \frac{y^2}{2} + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{2} \cdot (3 - \mathrm{i}) + \mathrm{i} C
$$
H. $$
v = 2x^2 - xy + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{3} \cdot (2 - \mathrm{i}) + \mathrm{i} C
$$
I. $$
v = 2xy - \frac{x^2}{2} + \frac{y^2}{2} + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
\frac{z^2}{2} \cdot (2 - \mathrm{i}) + \mathrm{i} C
$$
J. $$
v = x^2 + y^2 + C \quad\mathrm{( ~ C ~ is \ an\ arbitrary\ constant)}
$$
$$
z^2 \cdot (1 - \mathrm{i}) + \mathrm{i} C
$$
Answer: I
What is the value of the integral $\oint_{| z |=3} \frac{\mathrm{e}^{z}} {z \left( z^{2} - 1 \right)} \mathrm{d} z$?

Options:
A. $$
2\pi\mathrm{i} \Big( \mathrm{e}-\frac{1} {\mathrm{e}} \Big) \;
$$
B. $$
\pi\mathrm{i} \Big( \mathrm{e}^{2} +\mathrm{e}^{-2} -1 \Big) \;
$$
C. $$
\pi\mathrm{i} \Big( \mathrm{e}^{2} + \frac{2} {\mathrm{e}} - 3 \Big) \;
$$
D. $$
\pi\mathrm{i} \Big( \mathrm{e} - \frac{2} {\mathrm{e}} + 1 \Big) \;
$$
E. $$
\pi\mathrm{i} \Big( \mathrm{e} - \frac{1} {\mathrm{e}^{2}} \Big) \;
$$
F. $$
\pi\mathrm{i} \Big( \mathrm{e}-\frac{1} {\mathrm{e}}+2 \Big) \;
$$
G. $$
\pi\mathrm{i} \Big( \mathrm{e}+\frac{1} {\mathrm{e}}-2 \Big) \;
$$
H. $$
\pi \Big( \mathrm{e} + \frac{1} {\mathrm{e}} - 2 \Big) \;
$$
I. $$
\pi\mathrm{i} \Big( \frac{\mathrm{e}} {2} +\frac{2} {\mathrm{e}} \Big) \;
$$
J. $$
\pi\mathrm{i} \Big( \frac{\mathrm{e}} {2} - \frac{2} {\mathrm{e}} \Big) \;
$$
Answer: G
What is the value of the integral $I=\oint_{C} \frac{1}{z^{3} (z+1)(z-2)} \mathrm{d}z$? Where $C$ is $\mid z \mid=r \,, r \neq 1 \,, 2$.

Options:
A. $e^{\pi i}$
B. $\pi i$
C. 1/2
D. 1
E. $-2$
F. 0
G. $2\pi i$
H. $3\pi$
I. $4$
J. $-1$
Answer: F
What is the radius of convergence for the power series $\sum_{n=1}^{\infty} \frac{z^{n}} {a^{n}+\mathrm{i} b^{n}} ( \, a > 0 \,, b > 0 \, )$?

Options:
A. $$
R=a \cdot b
$$
B. $$
R=\operatorname* {g c d} \left\{\left. a \right., b \right\}
$$
C. $$
R=\frac{1}{a} + \frac{1}{b}
$$
D. $$
R=\frac{a \cdot b}{a + b}
$$
E. $$
R=\frac{a^n + b^n}{n}
$$
F. $$
R=\left|a - b \right|
$$
G. $$
R=\operatorname* {m i n} \left\{\left. a \right., b \right\}
$$
H. $$
R=\operatorname* {m a x} \left\{\left. a \right., b \right\}
$$
I. $$
R=\sqrt {a^{2} + b^{2}}
$$
J. $$
R=\frac{a + b}{2}
$$
Answer: H