MikaStars39/nano-eval huggingface dataset:
A space traveler with a speed of $v$ and his friend on Earth synchronized their clocks at the moment of departure, setting $t^{\prime}=0$ (traveler's clock) and $t=0$ (Earth's clock), respectively. The friend on Earth observes both clocks simultaneously, directly checking $t$ and using a telescope to observe $t^{\prime}$. The question is: when the reading on $t^{\prime}$ is 1 hour, what is the reading on $t$?

Options:
A. $$
\frac{v} {\sqrt{c-v}} ( \mathrm{h} )
$$
B. $$
\sqrt{\frac{v+c} {v-c}} ( \mathrm{h} )
$$
C. $$
\sqrt{\frac{c} {v+c}} ( \mathrm{h} )
$$
D. $$
\sqrt{\frac{c} {c-v}} ( \mathrm{h} )
$$
E. $$
\sqrt{\frac{v+c} {c}} ( \mathrm{h} )
$$
F. $$
\frac{c-v} {\sqrt{c+v}} ( \mathrm{h} )
$$
G. $$
\sqrt{\frac{c+v} {c-v}} ( \mathrm{h} )
$$
H. $$
\sqrt{\frac{c-v} {c+v}} ( \mathrm{h} )
$$
I. $$
\frac{c+v} {c} ( \mathrm{h} )
$$
J. $$
\frac{c+v} {\sqrt{c-v}} ( \mathrm{h} )
$$
Answer: G
A stationary hydrogen atom emits a spectral line $\mathrm{H}_{\delta}$ with a wavelength $\lambda=0.4101 \times 10^{-6} \mathrm{m}$. When the hydrogen atom moves towards the observer at $v=5 \times 10^{5} \mathrm{m/s}$, how much change does the observer see in the wavelength?

Options:
A. $$
- 6. 8 \times1 0^{-1 0} \mathrm{m}
$$
B. $$
- 2. 6 \times 1 0^{-1 0} \mathrm{m}
$$
C. $$
- 3. 7 \times 1 0^{-1 0} \mathrm{m}
$$
D. $$
- 9. 1 \times 1 0^{-1 0} \mathrm{m}
$$
E. $$
- 7. 4 \times 1 0^{-1 0} \mathrm{m}
$$
F. $$
- 8. 5 \times 1 0^{-1 0} \mathrm{m}
$$
G. $$
- 4. 9 \times 1 0^{-1 0} \mathrm{m}
$$
H. $$
- 6. 1 \times 1 0^{-1 0} \mathrm{m}
$$
I. $$
- 2. 0 \times 1 0^{-1 0} \mathrm{m}
$$
J. $$
- 5. 3 \times 1 0^{-1 0} \mathrm{m}
$$
Answer: A
Give the equation of motion for a relativistic charged particle with a rest mass of $m_{0}$ and charge $q$, starting from rest, in a constant uniform electric field with field strength $E$.

Options:
A. $$
{\frac{m_{0}^{2} c} {q E}} \left[ {\sqrt{1+\left( {\frac{q E^{2} t} {m_{0} c}} \right)^{2}}}-1 \right]
$$
B. $$
{\frac{m_{0}^{2} c^{3}} {q E}} \left[ {\sqrt{1+\left( {\frac{q E t} {m_{0} c^{2}}} \right)^{2}}}-1 \right]
$$
C. $$
{\frac{m_{0} c^{2}} {q E}} \left[ {\sqrt{1+\left( {\frac{q E^{2} t} {m_{0} c}} \right)^{2}}}-1 \right]
$$
D. $$
{\frac{m_{0} c^{2}} {q^{2} E}} \left[ {\sqrt{1+\left( {\frac{q E t} {m_{0} c}} \right)}}-1 \right]
$$
E. $$
{\frac{m_{0} c^{2}} {q E^{2}}} \left[ {\sqrt{1+\left( {\frac{q E t} {m_{0}}} \right)^{2}}}-1 \right]
$$
F. $$
{\frac{m_{0} c^{2}} {q E}} \left[ {\sqrt{1+\left( {\frac{q^{2} E t} {m_{0} c}} \right)^{2}}}-1 \right]
$$
G. $$
x = {\frac{m_{0}^{2} c^{2}} {q E}} \left[ {\sqrt{1+\left( {\frac{q E t} {m_{0} c}} \right)^{2}}}-1 \right]
$$
H. $$
{\frac{m_{0} c^{3}} {q^{2} E}} \left[ {\sqrt{1+\left( {\frac{q E t^{2}} {m_{0} c}} \right)^{2}}}-1 \right]
$$
I. $$
{\frac{m_{0} c} {q E}} \left[ {\sqrt{1+\left( {\frac{q E t} {m_{0}}} \right)^{2}}}-1 \right]
$$
J. $$
{\frac{m_{0}^{2} c^{2}} {q E}} \left[ {\sqrt{1+\left( {\frac{q E t^{2}} {m_{0} c^{2}}} \right)}}-1 \right]
$$
Answer: G
A hypothetical flashlight can convert a significant portion of its rest mass into light, emitting a relatively collimated beam. If the flashlight starts at rest with a rest mass of $m_{0}$, and is turned on, allowing it to move freely along a straight line, what is the rest mass $m$ when it has a velocity $v$ relative to the original rest frame ? (without making any assumptions)

Options:
A. $$
m_{0} {\sqrt{\frac{c-v} {c+v}}}
$$
B. $$
m_{0} {\frac{c+v} {c-v}}
$$
C. $$
m_{0} {\frac{\sqrt{c-v}} {\sqrt{c+v}}}
$$
D. $$
m_{0} {\left(\frac{v+c} {v-c}\right)}
$$
E. $$
m_{0} {\frac{c-v^2} {c+v^2}}
$$
F. $$
m_{0} {\sqrt{\frac{v-c} {v+c}}}
$$
G. $$
m_{0} {\frac{v-c} {v+c}}
$$
H. $$
m_{0} {\frac{c-2v} {c+2v}}
$$
I. $$
m_{0} {\sqrt{\frac{v+c} {v-c}}}
$$
J. $$
m_{0} {\left(\frac{c-v} {c+v}\right)^{2}}
$$
Answer: A
A stationary $494 \mathrm{MeV}$ $K$ meson decays into a $\mu$ meson with a rest energy of $10^{6} \mathrm{MeV}$ and a neutrino with zero rest energy. Give the kinetic energy of the $\mu$ meson and neutrino resulting from the decay of the stationary $K$ meson.

Options:
A. $$
3 2 0 ( \mathrm{M e V} )
$$
$$
9 7 \mathrm{M e V}
$$
B. $$
5 0 1 ( \mathrm{M e V} )
$$
$$
2 0 \mathrm{M e V}
$$
C. $$
2 5 0 ( \mathrm{M e V} )
$$
$$
1 9 4 \mathrm{M e V}
$$
D. $$
5 8 9 ( \mathrm{M e V} )
$$
$$
3 6 \mathrm{M e V}
$$
E. $$
6 7 2 ( \mathrm{M e V} )
$$
$$
3 8 \mathrm{M e V}
$$
F. $$
1 5 2 ( \mathrm{M e V} )
$$
$$
2 3 6 \mathrm{M e V}
$$
G. $$
4 4 0 ( \mathrm{M e V} )
$$
$$
5 4 \mathrm{M e V}
$$
H. $$
3 6 5 ( \mathrm{M e V} )
$$
$$
8 1 \mathrm{M e V}
$$
I. $$
6 1 0 ( \mathrm{M e V} )
$$
$$
7 5 \mathrm{M e V}
$$
J. $$
4 7 8 ( \mathrm{M e V} )
$$
$$
1 1 2 \mathrm{M e V}
$$
Answer: F
A particle with a rest mass of $m$ collides with a stationary particle with a rest mass of $m_{1}$, resulting in a reaction that produces many particles with a total rest mass of $M$. If $m+m_{1} < M,$ then the reaction cannot occur when the kinetic energy of the incoming particle is not sufficiently large. Give the threshold kinetic energy $T_{0}$, which is the kinetic energy that the incoming particle must have for the reaction to just occur.

Options:
A. $${\frac{1} {2 M}} ( m_{1}+m )^{2} c^{2}$$
B. $${\frac{m} {2}} ( M+m_{1} )^{2} c^{2}$$
C. $${\frac{M+m_{1}} {2}} ( M-m ) c^{2}$$
D. $${\frac{1} {2 m_{1}}} ( M+m_{1}+m ) ( M-m_{1}-m ) c^{2}
$$
E. $${\frac{1} {2 m}} ( M+m_{1} ) ( M-m_{1} ) c^{2}$$
F. $${\frac{2m} {m_{1}}} ( M-m_{1} ) c^{2}$$
G. $${\frac{M} {m_{1}}} ( m_{1}+m )^{2} c^{2}$$
H. $${\frac{m} {m_{1}^{2}}} ( M + m_{1} ) ( M-m ) c^{2}$$
I. $${m_{1}} ( M-m_{1}-m )^{2} c^{2}$$
J. $${\frac{1} {m_{1}}} ( M+m_{1})^{2} c^{2}$$
Answer: D
Given that the rest lifetime of a $\mu$ particle is $1 \times 10^{-6} \mathrm{s}$ and its rest mass is approximately $100 \mathrm{MeV} / c^{2}$, if it is produced in the upper atmosphere (above $10^{4} \mathrm{m}$), how much energy does the $\mu$ particle need to reach the ground?

Options:
A. 9.2 (GeV)
B. 2.7 (GeV)
C. 6.0 (GeV)
D. 3.3 GeV
E. 2.2 (MeV)
F. 8.7 (MeV)
G. 4.5 (MeV)
H. 1.8 (GeV)
I. 5.0 (MeV)
J. 3.1 (GeV)
Answer: D
A moving proton collides with a stationary proton, resulting in the reaction $\mathrm{p+p \to p+p+\pi^{0}}$. What is the threshold energy for the reaction ?

Options:
A. $$3 0 0 ( \mathrm{M e V} )
$$
B. $$2 8 0 ( \mathrm{M e V} )
$$
C. $$3 5 0 ( \mathrm{M e V} )
$$
D. $$2 7 0 ( \mathrm{M e V} )
$$
E. $$3 2 0 ( \mathrm{M e V} )
$$
F. $$2 4 0 ( \mathrm{M e V} )
$$
G. $$2 6 0 ( \mathrm{M e V} )
$$
H. $$2 9 0 ( \mathrm{M e V} )
$$
I. $$3 1 0 ( \mathrm{M e V} )
$$
J. $$2 5 0 ( \mathrm{M e V} )
$$
Answer: B
From the analysis spectrometer, an energy of $E=2 \mathrm{GeV}$ K meson beam is obtained. How far does it travel before the beam intensity decays to $10\%$ of its original value? Assume the rest lifetime of the K meson is $\tau_{0}=1.2 \times 10^{-8} \mathrm{s}$.

Options:
A. 48m
B. 20m
C. 33m
D. 42m
E. 25m
F. 55m
G. 60m
H. 50m
I. 28m
J. 40m
Answer: C
The decay of the charmed meson $\mathrm{D}^{0}$ to $$ \mathrm{K}^{-} + \pi^{+}$$ involves the masses of $\mathrm{D}$, $\mathrm{K}$, and $\pi$, which are 1.8, 0.50, and $0.15 \, \mathrm{GeV}/c^{2}$, respectively. What is the momentum of the $\mathrm{K}$ meson in the rest frame of $\mathrm{D}^{0}$ ?

Options:
A. $$
1. 3 2 \mathrm{G e V / c}
$$
B. $$
0. 7 0 \mathrm{G e V / c}
$$
C. $$
0. 6 8 \mathrm{G e V / c}
$$
D. $$
0. 6 5 \mathrm{G e V / c}
$$
E. $$
0. 8 2 \mathrm{G e V / c}
$$
F. $$
1. 0 5 \mathrm{G e V / c}
$$
G. $$
0. 5 0 \mathrm{G e V / c}
$$
H. $$
0. 9 0 \mathrm{G e V / c}
$$
I. $$
1. 1 0 \mathrm{G e V / c}
$$
J. $$
0. 7 9 \mathrm{G e V / c}
$$
Answer: E
A classic hydrogen atom, at time $t=0$, has an electron in the first Bohr orbit. Give the expression for the time required for the orbit radius to decay to zero due to radiation (assuming the energy consumed weekly is very small compared to the remaining energy).

Options:
A. $$
\frac{2 \pi \varepsilon_{0}^{2} c^{3} m^{3} a_{0}^{4}} {e^{6}}
$$
B. $$
\frac{3 \pi^{2} \varepsilon_{0}^{3} c^{2} m^{2} a_{0}^{3}} {e^{4}}
$$
C. $$
\frac{4 \pi \varepsilon_{0}^{2} c^{4} m^{2} a_{0}^{3}} {e^{5}}
$$
D. $$
\frac{5 \pi^{2} \varepsilon_{0}^{2} c^{3} m^{2} a_{0}^{3}} {e^{6}}
$$
E. $$
\frac{6 \pi^{3} \varepsilon_{0}^{2} c^{2} m^{2} a_{0}} {e^{4}}
$$
F. $$
\frac{4 \pi^{2} \varepsilon_{0}^{2} c^{3} m^{2} a_{0}^{3}} {e^{4}}
$$
G. $$
\frac{8 \pi^{2} \varepsilon_{0}^{3} c^{2} m a_{0}^{2}} {e^{5}}
$$
H. $$
\frac{4 \pi^{2} \varepsilon c^{3} m^{2} a_{0}^{4}} {e^{4}}
$$
I. $$
\frac{4 \pi^{2} \varepsilon_{0} c^{3} m a_{0}^{3}} {e^{3}}
$$
J. $$
\frac{4 \pi^{3} \varepsilon_{0}^{2} c^{3} m a_{0}^{2}} {e^{4}}
$$
Answer: F
A stationary observer at the Schwarzschild radius $r_{1}$ releases free-falling hail inward with an initial velocity $V$. What is the expression for the velocity measured by a local stationary observer when the hail reaches the Schwarzschild radius $r_{2}$ ?

Options:
A. $$
\sqrt{1+\frac{1-2 M / r_{2}} {1-2 M / r_{1}} \left( 1+V^{2} \right)}
$$
B. $$
\sqrt{1+\frac{1-2 M / r_{1}} {1-2 M / r_{2}} \left( 1-V^{2} \right)}
$$
C. $$
V_2 = \sqrt{1-\frac{1-2 M / r_{2}} {1-2 M / r_{1}} \left( 1-V^{2} \right)}
$$
D. $$
\sqrt{1-\frac{1+2 M / r_{1}} {1-2 M / r_{2}} \left( 1+V^{2} \right)}
$$
E. $$
\sqrt{1+\frac{2 M / r_{2}} {2 M / r_{1}} \left( 1-V^{2} \right)}
$$
F. $$
\sqrt{1+\frac{1-2 M / r_{2}} {1+2 M / r_{1}} \left( 1-V^{2} \right)}
$$
G. $$
\sqrt{1-\frac{2 M / r_{1}} {2 M / r_{2}} \left( 1-V^{2} \right)}
$$
H. $$
\sqrt{1+\frac{2 M / r_{2}} {1-2 M / r_{1}} ( 1-V^{2} )}
$$
I. $$
\sqrt{1+\frac{1-2 M / r_{1}} {1+2 M / r_{2}} ( 1-V^{2} )}
$$
J. $$
\sqrt{1-\frac{1-2 M / r_{1}} {1-2 M / r_{2}} \left( 1+V^{2} \right)}
$$
Answer: C
Let the sum of all the digits of $3^{10000}$ be $\mathrm{A}$, the sum of all the digits of $\mathrm{A}$ be $\mathbf{B}$, and the sum of all the digits of $\mathbf{B}$ be $\mathrm{C}$. Find the value of $\mathrm{C}$.

Options:
A. 5
B. 11
C. 2
D. 6
E. 10
F. 4
G. 9
H. 7
I. 3
J. 8
Answer: G
Find the prime factorization of 30!.

Options:
A. $$
3 0!=2^{2 2} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{2} \cdot1 8 \cdot1 9 \cdot2 3 \cdot2 9
$$
B. $$
3 0!=2^{2 3} \cdot3^{1 3} \cdot5^{5} \cdot7^{5} \cdot1 1^{2} \cdot1 3^{2} \cdot1 7 \cdot1 9 \cdot2 4 \cdot2 9
$$
C. $$
3 0!=2^{2 3} \cdot3^{1 4} \cdot5^{4} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{2} \cdot1 6 \cdot1 9 \cdot2 3 \cdot2 9
$$
D. $$
3 0!=2^{2 4} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{3} \cdot1 3^{1} \cdot1 7 \cdot1 9 \cdot2 3 \cdot2 9
$$
E. $$
3 0!=2^{2 3} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{2} \cdot1 7 \cdot1 9 \cdot2 3 \cdot2 9
$$
F. $$
3 0!=2^{2 3} \cdot3^{1 5} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{2} \cdot1 7 \cdot1 8 \cdot2 3 \cdot2 9
$$
G. $$
3 0!=2^{2 3} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{2} \cdot1 7 \cdot1 9 \cdot2 2 \cdot2 8
$$
H. $$
3 0!=2^{2 3} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 2^{2} \cdot1 7 \cdot1 9 \cdot2 3 \cdot2 9
$$
I. $$
3 0!=2^{2 4} \cdot3^{1 4} \cdot5^{5} \cdot7^{4} \cdot1 1^{2} \cdot1 3^{1} \cdot1 7 \cdot1 9 \cdot2 3 \cdot2 9
$$
J. $$
3 0!=2^{2 3} \cdot3^{1 4} \cdot5^{5} \cdot7^{3} \cdot1 1^{3} \cdot1 3^{2} \cdot1 7 \cdot1 9 \cdot2 3 \cdot2 9
$$
Answer: E
Let \( n \) be any positive integer, and alpha be a real number, then
$$
[ \alpha]+\left[ \alpha+\frac{1} {n} \right]+\cdots+\left[ \alpha+\frac{n-1} {n} \right]=?
$$

Options:
A. $$
\left[ n + \alpha \right]
$$
B. $$
( \alpha n )
$$
C. $$
[ n \alpha]
$$
D. $$
\left( n \alpha \right)
$$
E. $$
n \left( \alpha \right)
$$
F. $$
[ \alpha + n]
$$
G. $$
(n \alpha)
$$
H. $$
[ \alpha n]
$$
I. $$
n \left[ \alpha \right]
$$
Answer: C
The distance between points $A$ and $B$ is 5 km. Person A walks from $A$ towards $B$, while Person B rides a bicycle from $B$ towards $A$ at the same time. The speed of the bicycle is twice the walking speed. After meeting Person A, Person B turns around and return to $B$. Once arriving back at $B$, Person B again rides the bicycle towards $A$, repeating this back-and-forth motion. What is the total distance traveled by Person B when Person A arrives at $B$?

Options:
A. 20km
B. 15km
C. 14km
D. 25km
E. 22km
F. 18km
G. 12km
H. 10km
I. 8km
J. 5km
Answer: H
How much is $\begin{aligned}\sqrt{\sqrt[3]{4}-1}\:+\:\sqrt{\sqrt[3]{16}\:-\sqrt[3]{4}}\:\end{aligned}$?

Options:
A. 7
B. 3
C. 2
D. 6
E. 1
F. 9
G. 5
H. 4
I. 0
J. 8
Answer: B
What is the value of $\sqrt[4] {\frac{3+2^{4} \sqrt{5}} {3-2^{4} \sqrt{5}}}$?

Options:
A. $$
\frac{\sqrt[4]{3}+1} {\sqrt[4]{3}-1}
$$
B. $$
\frac{\sqrt[4]{2}+1} {\sqrt[4]{2}-1}
$$
C. $$
\frac{2+\sqrt{5}} {2-\sqrt{5}}
$$
D. $$
\frac{\sqrt[4] {5}+1} {\sqrt[4] {5}-1}
$$
E. $$
\frac{3+\sqrt{5}} {3-\sqrt{5}}
$$
F. $$
\frac{\sqrt{5}+1} {\sqrt{5}-1}
$$
G. $$
\frac{\sqrt[4] {5}-1} {\sqrt[4] {5}+1}
$$
H. $$
\frac{4+\sqrt{5}} {4-\sqrt{5}}
$$
I. $$
\frac{\sqrt[3]{5}+1} {\sqrt[3]{5}-1}
$$
J. $$
\frac{\sqrt[4] {5}-\sqrt{5}} {\sqrt[4] {5}+\sqrt{5}}
$$
Answer: D
In the Farey sequence $F_{n}$, how many fractions are adjacent to $\frac{1}{2}$?

Options:
A. When $n$ is even, $\frac{3} {2} \pm\frac{1} {n}$; when $n$ is odd, ${\frac{3} {2}} \pm{\frac{1} {n-2}}$
B. When $n$ is even, $\frac{1} {3} \pm\frac{1} {2 n}$; when $n$ is odd, ${\frac{1} {3}} \pm{\frac{1} {2 n-2}}$
C. When $n$ is odd, $\frac{1} {6} \pm\frac{1} {n}$; when $n$ is even, ${\frac{1} {6}} \pm{\frac{1} {n-2}}$
D. When $n$ is odd, $\frac{1} {4} \pm\frac{1} {2 n}$; when $n$ is even, ${\frac{1} {4}} \pm{\frac{1} {2 n-2}}$
E. When $n$ is even, $\frac{2} {7} \pm\frac{1} {4 n}$; when $n$ is odd, ${\frac{2} {7}} \pm{\frac{1} {4 n-2}}$
F. When $n$ is even, $\frac{5} {2} \pm\frac{1} {2 n}$; when $n$ is odd, ${\frac{5} {2}} \pm{\frac{1} {2 n-2}}$
G. When $n$ is odd, $\frac{2} {5} \pm\frac{1} {3 n}$; when $n$ is even, ${\frac{2} {5}} \pm{\frac{1} {3 n-2}}$
H. When $n$ is odd, $\frac{1} {2} \pm\frac{1} {2 n}$; when $n$ is even, ${\frac{1} {2}} \pm{\frac{1} {2 n-2}}$
I. When $n$ is even, $\frac{1} {2} \pm\frac{3} {4 n}$; when $n$ is odd, ${\frac{1} {2}} \pm{\frac{3} {4 n-2}}$
J. When $n$ is odd, $\frac{1} {2} \pm\frac{2} {3 n}$; when $n$ is even, ${\frac{1} {2}} \pm{\frac{2} {3 n-2}}$
Answer: H
For which positive integers $n$ can there exist positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the system of equations
$x_{1}+x_{2}+\cdots+x_{n}=3 \,, {\frac{1} {x_{1}}}+{\frac{1} {x_{2}}}+\cdots+{\frac{1} {x_{n}}}=3$?

Options:
A. 6,7
B. 3,4
C. 4,5
D. 5,6
E. 2,3
F. 1,4
G. 2,5
H. 7,9
I. 1,5
J. 7,8
Answer: E
The expression $\left( \, x-a \, \right) \left( \, x-10 \, \right) +1$ can be factored into the product $( x+b ) ( x+c )$ with integers $b$ and $c$. Determine the integer $a$.

Options:
A. a=10 or a=14
B. a=6 or a=16
C. a=2 or a=20
D. a=4 or a=18
E. a=1 or a=19
F. a=7 or a=13
G. a=3 or a=17
H. a=8 or a=12
I. a=5 or a=15
J. a=9 or a=11
Answer: H
How many solutions does the equation $\sin x = \frac{x}{100}$ have?

Options:
A. 35
B. 12
C. 40
D. 15
E. 7
F. 31
G. 10
H. 23
I. 48
J. 50
Answer: F
At the end of a horizontal pipeline, an upward-bending nozzle is installed, with a curvature angle of $\alpha = 45^{\circ}$. The inlet diameter of the nozzle $\boldsymbol{d}_{A} = 0.20 \mathrm{m}$ and the outlet diameter $d_{B} = 0.10 \mathrm{m}$. The average flow velocity at the outlet cross-section $v_{B} = 10 \mathrm{m/s}$, and the height difference between the centers of cross-sections $A$ and $B$ is $\Delta z = 0.2 \mathrm{m}$. The head loss through the nozzle is $h_{w}=0.5 \ \frac{v_{B}^{2}}{2g}$, and the weight of the water in the nozzle is $G=196 \mathrm{N}$. Find the force $F$ exerted by the water flow on the nozzle.

Options:
A. $$
1. \, 9 \, \mathrm{k N}
$$
B. $$
3. \, 2 \, \mathrm{k N}
$$
C. $$
3. \, 0 \, \mathrm{k N}
$$
D. $$
2. \, 7 \, \mathrm{k N}
$$
E. $$
2. \, 1 \, \mathrm{k N}
$$
F. $$
1. \, 8 \, \mathrm{k N}
$$
G. $$
1. \, 5 \, \mathrm{k N}
$$
H. $$
2. \, 0 \, \mathrm{k N}
$$
I. $$
2. \, 4 \, \mathrm{k N}
$$
J. $$
2. \, 6 \, \mathrm{k N}
$$
Answer: E
There is a cylindrical tube used for experiments with a diameter of 15mm. A test is being conducted to measure the head loss along the pipe's length, which is 4. $0 \mathrm{m}$. What is the flow velocity in the measuring section when the water flow is at the critical transition point from turbulent to laminar? Assume the water temperature is 40°C.

Options:
A. $$
0. 2 5 4 3 ( \mathrm{m / s} )
$$
B. $$
0. 1 9 3 6 ( \mathrm{m / s} )
$$
C. $$
0. 1 4 7 2 ( \mathrm{m / s} )
$$
D. $$
0. 3 2 0 1 ( \mathrm{m / s} )
$$
E. $$
0. 2 0 9 6 ( \mathrm{m / s} )
$$
F. $$
0. 3 1 5 8 ( \mathrm{m / s} )
$$
G. $$
0. 2 7 2 5 ( \mathrm{m / s} )
$$
H. $$
0. 2 3 6 0 ( \mathrm{m / s} )
$$
I. $$
0. 2 8 1 7 ( \mathrm{m / s} )
$$
J. $$
0. 1 6 8 9 ( \mathrm{m / s} )
$$
Answer: E
There is a trapezoidal cross-section channel with the following known parameters: bottom width $b=10 \,\mathrm{m}$, uniform water depth $h=3 \,\mathrm{m}$, side slope ratio $1:1$, channel roughness coefficient (i.e., Manning's roughness coefficient) $n=0.02$, and flow rate $Q=39 \,\mathrm{m}^3/\mathrm{s}$. Determine the head loss per kilometer of the channel length.

Options:
A. $$
0.137m
$$
B. $$
0.153m
$$
C. $$
0.162m
$$
D. $$
0.172m
$$
E. $$
0.156m
$$
F. $$
0.148m
$$
G. $$
0.139m
$$
H. $$
0.134m
$$
I. $$
0.145m
$$
J. $$
0.165m
$$
Answer: F
Given a trapezoidal section channel with a flow rate of $Q = 30 \, \mathrm{m}^{3} / \mathrm{s}$, a roughness coefficient $n = 0.025$, a side slope coefficient $m = 1.5$, and a bottom slope $i = 0.0009$, try to select the width-to-depth ratio $\beta$ according to the conditions of the hydraulically optimal cross-section and the practical economic cross-section condition $A / A_{m} = 1.04$.

Options:
A. 4.315
B. 3.202
C. 2.912
D. 3.508
E. 2.105
F. 2.745
G. 3.698
H. 3.124
I. 3.406
J. 4.001
Answer: B
Given that the height of the weir $P = 50 \mathrm{m}$, with a vertical upstream face, a design head $H_{d} = 8.0 \mathrm{m}$, and a downstream slope of the weir face $m_{1}=0.65$. If the maximum operating head is 10m, the sluice has 12 openings, each with a net width $b = 15 \mathrm{m}$. The heads of the piers and end piers are sharp and rounded. The total width of the upstream diversion channel is $B_{0} = 220 \mathrm{m}$, with a pier thickness of $d = 2.5 \mathrm{m}$. Using the WES weir type, what should be the flow rate?

Options:
A. $$
1 2 6 0 6 \, ( \mathrm{m^{3} / s} )
$$
B. $$ 1 2 9 0 8 \, ( \mathrm{m^{3} / s} ) $$
C. $$ 1 2 7 8 5 \, ( \mathrm{m^{3} / s} ) $$
D. $$ 1 2 6 9 2 \, ( \mathrm{m^{3} / s} ) $$
E. $$ 1 2 5 4 3 \, ( \mathrm{m^{3} / s} ) $$
F. $$ 1 2 4 7 5 \, ( \mathrm{m^{3} / s} ) $$
G. $$ 1 2 8 3 6 \, ( \mathrm{m^{3} / s} ) $$
H. $$ 1 2 8 1 2 \, ( \mathrm{m^{3} / s} ) $$
I. $$ 1 2 6 3 0 \, ( \mathrm{m^{3} / s} ) $$
J. $$ 1 2 9 9 4 \, ( \mathrm{m^{3} / s} ) $$
Answer: A
The elevation of the bottom slab of the Jinjiji Barrage is the same as the elevation of the upstream riverbed, both being $5 \mathrm{m}$. There are 28 sluice openings, each with a width of $10 \mathrm{m}$. The thickness of the sluice piers is $d=1.6 \mathrm{m}$, with the pier heads being semicircular and the side piers forming a 1/4 circle arc with a radius of $r=1.9 \mathrm{m}$. The width of the river upstream from the sluice is $B_{0}=327 \mathrm{m}$. Determine the flow rate when the sluice gates are fully open, the normal upstream water level is $9 \mathrm{m}$, and the water flowing through the sluice is in free discharge.

Options:
A. $$
\! 5 2 3 4 {\mathrm{m}}^{3} / s
$$
B. $$
\! 4 9 6 0 {\mathrm{m}}^{3} / s
$$
C. $$
\! 3 6 0 0 {\mathrm{m}}^{3} / s
$$
D. $$
\! 5 0 0 7 {\mathrm{m}}^{3} / s
$$
E. $$
\! 4 1 8 3 {\mathrm{m}}^{3} / s
$$
F. $$
\! 4 5 0 0 {\mathrm{m}}^{3} / s
$$
G. $$
\! 4 2 9 1 {\mathrm{m}}^{3} / s
$$
H. $$
\! 4 5 2 7 {\mathrm{m}}^{3} / s
$$
I. $$
\! 5 0 1 2 {\mathrm{m}}^{3} / s
$$
J. $$
\! 4 8 3 9 {\mathrm{m}}^{3} / s
$$
Answer: H
A large thin plate moves at a speed of $u$ along the centerline of an oil layer with a thickness of $h$ and a dynamic viscosity coefficient of $\mu_{0}$. If replaced by an oil layer with a smaller dynamic viscosity coefficient $\mu_{1}$, and the plate moves at the same speed at a non-centerline position, the drag force is equal. Find the distance between the plate and the closer wall.

Options:
A. $$
y \!=\! \frac{h} {\mu_{0}} \! \left( 1 \! -\! \sqrt{1 \!-\! \frac{\mu_{1}} {h}} \right)
$$
B. $$
y \!=\! h \! \left( 1 \! \pm\! \sqrt{\frac{\mu_{1}} {\mu_{0}} \!+\! 1} \right)
$$
C. $$
y \!=\! \frac{h} {2} \! \left( 1 \! \mp\! \sqrt{1 \!+\! \frac{\mu_{1}} {\mu_{0}}} \right)
$$
D. $$
y \!=\! h \! \left( \frac{1} {2} \! -\! \sqrt{\frac{\mu_{1}} {\mu_{0}}} \right)
$$
E. $$
y \!=\! \frac{h} {2} \! \left( 1 \! \pm\! \sqrt{1 \!+\! \frac{\mu_{1}} {\mu_{0}}} \right)
$$
F. $$
y \!=\! \frac{h} {2} \! \left( 1 \! \pm\! \sqrt{\frac{\mu_{1}} {\mu_{0}}} \right)
$$
G. $$
y \!=\! \frac{h} {2} \! \left( 1 \! \pm\! \sqrt{1 \!-\! \frac{\mu_{1}} {\mu_{0}}} \right)
$$
H. $$
y \!=\! \frac{h} {2} \! \left( \sqrt{1 \!-\! \frac{\mu_{1}} {\mu_{0}}} \right)
$$
I. $$
y \!=\! \frac{h} {2} \! \left( 1 \! -\! \sqrt{\frac{\mu_{1}} {\mu_{0}} \!-\! 1} \right)
$$
J. $$
y \!=\! h \! \left( \sqrt{1 \!-\! \frac{\mu_{1}} {\mu_{0}}} \! -\! \frac{1} {2} \right)
$$
Answer: G
A pipe with a length of 4m, closed at the bottom, is inclined at a 30° angle to the vertical and filled with water. It rotates around a vertical axis with an angular velocity $\omega\!=\! 8.02 \mathrm{rad} / \mathrm{s}$. Find the pressure at the midpoint $B$.

Options:
A. $$
- 2 0. 3 4 7 \ ( \mathrm{k N / m^{2}} )
$$
B. $$
- 9. 8 3 2 \ ( \mathrm{k N / m^{2}} )
$$
C. $$
- 1 5. 1 8 6 \ ( \mathrm{k N / m^{2}} )
$$
D. $$
- 1 3. 7 4 1 \ ( \mathrm{k N / m^{2}} )
$$
E. $$
- 1 4. 4 5 0 \ ( \mathrm{k N / m^{2}} )
$$
F. $$
- 1 7. 9 0 8 \ ( \mathrm{k N / m^{2}} )
$$
G. $$
- 5. 7 8 9 \ ( \mathrm{k N / m^{2}} )
$$
H. $$
- 1 6. 2 9 3 \ ( \mathrm{k N / m^{2}} )
$$
I. $$
- 1 2. 6 0 5 \ ( \mathrm{k N / m^{2}} )
$$
J. $$
- 2. 1 3 2 \ ( \mathrm{k N / m^{2}} )
$$
Answer: C
There is a water-filled bucket with a circular hole at the bottom, having a radius of $r = 0.4 \mathrm{m}$. The hole is plugged with a frustum-shaped stopper, with a diameter $D = 1.2 \mathrm{m}$. The water depth above is $H = 5 \mathrm{m}$, while $h = 0.9 \mathrm{m}$. Calculate the total hydrostatic pressure acting on this stopper.

Options:
A. $$
18.500(kN)
$$
B. $$
28.410(kN)
$$
C. $$
27.650(kN)
$$
D. $$
23.040(kN)
$$
E. $$
19.890(kN)
$$
F. $$
30.010(kN)
$$
G. $$
24.670(kN)
$$
H. $$
21.330(kN)
$$
I. $$
22.044(kN)
$$
J. $$
25.760(kN)
$$
Answer: I
Given the velocity field $u_{x}=2 t+2 x+2 y, \; \; u_{y}=t-y+z, \; \; u_{z}=t+x-z$, find the acceleration at the point (3, 2, 1) in the flow field at $t=3$.

Options:
A. $$
3 8. 0 8 \ ( \mathrm{m / s^{2}} )
$$
B. $$
3 6. 0 7 \ ( \mathrm{m / s^{2}} )
$$
C. $$
4 1. 0 6 \ ( \mathrm{m / s^{2}} )
$$
D. $$
4 0. 0 5 \ ( \mathrm{m / s^{2}} )
$$
E. $$
3 9. 1 2 \ ( \mathrm{m / s^{2}} )
$$
F. $$
3 7. 1 0 \ ( \mathrm{m / s^{2}} )
$$
G. $$
3 5. 1 1 \ ( \mathrm{m / s^{2}} )
$$
H. $$
4 4. 0 3 \ ( \mathrm{m / s^{2}} )
$$
I. $$
4 3. 0 9 \ ( \mathrm{m / s^{2}} )
$$
J. $$
4 2. 0 4 \ ( \mathrm{m / s^{2}} )
$$
Answer: D
Use the π theorem to analyze the expression for the resistance $F$ experienced by a ship navigating on water? Assume that the resistance $F$ is related to the ship's navigation speed $v$, the characteristic length of the ship $l$, the density of the water $\rho$, and the gravitational acceleration $g$ considering the effect of waves.

Options:
A. $$
F {=} C_{D} v A \left( \rho g / 2 \right)
$$
B. $$
F {=} \rho A \sin(v \pi) (g / 2)
$$
C. $$
F {=} A g \left( \rho v^{3} \right)^{2}
$$
D. $$
F {=} C_{L} l (g/ 2 ) \rho v^{2}
$$
E. $$
F {=} C_{D} \pi \left( \rho^{2} / 2 \right) l^{3}
$$
F. $$
F {=} C_{D} A \left( \rho/ 2 \right) v^{2}
$$
G. $$
F {=} \rho A v^{2} (g l / 2 )
$$
H. $$
F {=} \pi A \left( l v^{2} / 2 \right) \rho
$$
I. $$
F {=} C_{D} A \left( g v / 2 \right) \rho
$$
J. $$
F {=} C_{D} \rho \left( g / 2 \right) v A
$$
Answer: F
A pipe with a diameter $d \!=\! 0.3 \, \mathrm{m}$ transports water with a density $\rho\!=\! 999.1 \, \mathrm{kg} / \mathrm{m}^{3}$ and a kinematic viscosity $\nu=1.14 \times 10^{-6} \, \mathrm{m^{2} / s}$. When the flow rate is $Q=0.212 \, \mathrm{m}^{3} / \mathrm{s}$, it is observed that the head loss over a flow length $L=100 \, \mathrm{m}$ is $h_{f}=2.3 \, \mathrm{m}$. Additionally, the velocity gradient of the velocity distribution curve at a distance $r=0.06 \, \mathrm{m}$ from the pipe axis is $\mathrm{d} u / \mathrm{d} y=3.61 \, \mathrm{m} / (\mathrm{s \cdot m})$. Find the viscous shear stress $\tau_{1}$ at a distance $r=0.06 \, \mathrm{m}$ from the pipe axis.

Options:
A. $$
6.767(N/m^{2})
$$
B. $$
6.754(N/m^{2})
$$
C. $$
6.740(N/m^{2})
$$
D. $$
6.750(N/m^{2})
$$
E. $$
6.735(N/m^{2})
$$
F. $$
6.700(N/m^{2})
$$
G. $$
6.738(N/m^{2})
$$
H. $$
6.800(N/m^{2})
$$
I. $$
6.720(N/m^{2})
$$
J. $$
6.789(N/m^{2})
$$
Answer: B
There is a new cast iron pipeline with a length of $L = 60 \mathrm{m}$ and an originally designed diameter of $d = 0.3 \mathrm{m}$. The originally designed water flow rate is $Q = 0.3 \, \mathrm{m}^{3} / \mathrm{s}$. However, the measured diameter is only $0.29 \mathrm{m}$, and the roughness coefficient of the pipe is $n = 0.01$. Determine the actual flow rate as a percentage of the originally designed flow rate when the hydraulic head remains unchanged.

Options:
A. 89.9%
B. 94.2%
C. 88.4%
D. 91.3%
E. 87.5%
F. 92.7%
G. 93.1%
H. 86.8%
I. 90.5%
J. 95.7%
Answer: D
The pressure gauge reading of an inlet pipe during a certain water injection test is $p_{3}=0.90 \mathrm{MPa}$, with the gauge center 0.5m above the aperture. The injected flow rate is $Q=80 \mathrm{L/min}$, and the length of the test section is $L=5 \mathrm{m}$. The total pressure loss in the drill pipe and joint is 0.04MPa. The drill hole is inclined with an angle $\boldsymbol{\alpha}=60°$. The groundwater level is above the test section, and the actual length along the drill hole from the aperture to the groundwater level is $H=24.8 \mathrm{m}$. Which of the following options is closest to the permeability of the test section's geologic formation (Lugeon value $\mathbf{Lu}$) ?

Options:
A. 10.5
B. 11.6
C. 17.2
D. 12.9
E. 18.0
F. 19.4
G. 14.3
H. 15.7
I. 16.1
J. 13.8
Answer: I
Take a standard rock specimen with a diameter of 50mm and a length of 70mm, and conduct a radial point load strength test. The measured ultimate load at failure is 4000N, and at the moment of failure, there was no penetration at the loading point. Analyze and determine which of the following categories the hardness of this rock belongs to ().

Options:
A. Harder rock
B. Dense rock
C. Weak rock
D. Softer rock
E. Rough rock
F. Porous rock
G. Brittle rock
H. Heavy rock
I. Moderate rock
J. Smooth rock
Answer: A
A homogeneous earth dam is 1.2 km long, 20 m high, with a crest width of 8 m and a base width of 75 m. The required compaction degree is not less than 0.95. The soil material in the natural material field has a moisture content of 21, a specific gravity of 2.70, a unit weight of 18 kN/m³, a maximum dry density of 16.8 kN/m³, and an optimum moisture content of 20. The construction of this earth dam requires natural soil material is ()$\mathrm{m}^3$.

Options:
A. $$
1 0. 9 \times1 0^{5} \mathrm{m}^{3}
$$
B. $$
6. 9 \times1 0^{4} \mathrm{m}^{3}
$$
C. $$
9. 5 \times1 0^{4} \mathrm{m}^{3}
$$
D. $$
1 1. 1 \times1 0^{5} \mathrm{m}^{3}
$$
E. $$
1 3. 4 \times1 0^{5} \mathrm{m}^{3}
$$
F. $$
1 2. 3 \times1 0^{5} \mathrm{m}^{3}
$$
G. $$
1 0. 7 \times1 0^{5} \mathrm{m}^{3}
$$
H. $$
8. 5 \times1 0^{4} \mathrm{m}^{3}
$$
I. $$
7. 7 \times1 0^{4} \mathrm{m}^{3}
$$
J. $$
9. 8 \times1 0^{4} \mathrm{m}^{3}
$$
Answer: G
The length of a certain cement mixing pile composite foundation pile is $12 \text{ m}$, with an area replacement ratio of $m = 0.21$. The additional pressure at the top of the composite soil layer is $p_{z} = 114 \text{kPa}$, and the additional pressure at the bottom is $p_{\text{xl}} = 40 \text{kPa}$. The compressive modulus of the soil between the piles is $E_{s} = 2.25 \text{MPa}$, and the compressive modulus of the mixing pile is $E_{\text{p}} = 168 \text{MPa}$. The compression of the soil layer beneath the pile tip is 12.2 cm. According to the "Technical Code for Ground Treatment of Buildings" (JGJ 79-2002), the total settlement of this composite foundation is () cm.

Options:
A. 18.25cm
B. 15.72cm
C. 14.69cm
D. 11.56cm
E. 16.48cm
F. 13.53cm
G. 12.34cm
H. 20.39cm
I. 19.07cm
J. 17.85cm
Answer: C
Mr. Wang, male, 45 years old, with no previous history of heart disease. Over the past week, he has frequently experienced sudden palpitations. Physical examination: heart rate: $150bpm$, regular rhythm. After carotid sinus pressure, the heart rate suddenly slows to $5bpm$, but after exercise, it increases again to $150bpm$. Based on the above medical history, what is the most likely diagnosis for this patient?

Options:
A. Premature Ventricular Contraction
B. Atrial Flutter
C. Mitral Valve Prolapse
D. Ventricular Fibrillation
E. Atrial Fibrillation
F. Long QT Syndrome
G. Paroxysmal Supraventricular Tachycardia
H. Hypertrophic Cardiomyopathy
I. Sinus Bradycardia
J. Sick Sinus Syndrome
Answer: G
Female, 28 years old. Suffers from rheumatic heart disease. Mitral stenosis with insufficiency. Palpitations, shortness of breath, edema of the lower limbs. Takes digoxin orally every day at $0.25mg$, and has intermittently used hydrochlorothiazide for 2 months. The electrocardiogram shows ventricular premature contractions and bigeminy. Among the following options, the first choice of treatment is ().

Options:
A. Switch to amiodarone, accompany with aspirin and nitrates.
B. Discontinue digoxin, administer potassium salts and phenytoin sodium.
C. Increase digoxin dose, prescribe beta-blockers and diuretics.
D. Maintain digoxin, add spironolactone and reduce sodium intake.
E. Initiate diltiazem, supplement with calcium and vitamin D.
F. Start metoprolol, complement with garlic extract and melatonin.
G. Prescribe verapamil, enhance with fish oil and multivitamins.
H. Adjust digoxin dosage, incorporate adenosine and probiotics.
I. Prescribe lisinopril, fortify with echinacea and vitamin B complex.
J. Continue digoxin, recommend magnesium sulfate and digitalis.
Answer: B
Two barrels, labeled as 1 and 2, each with a large opening at the top and filled with different liquids, both have a small hole opened on the side at the same depth $h$ below the liquid surface in each. However, the area of the small hole in barrel 1 is half of that in barrel 2. The question is: how much height of liquid needs to be added or removed from the second barrel to make the flow rates of the two barrels equal?

Options:
A. $$
{\frac{2} {3}} h
$$
B. $$
{\frac{9} {8}} h
$$
C. $$
{\frac{1} {4}} h
$$
D. $$
{\frac{5} {4}} h
$$
E. $$
{\frac{1} {2}} h
$$
F. $$
{\frac{3} {4}} h
$$
G. $$
{\frac{4} {5}} h
$$
H. $$
{\frac{5} {6}} h
$$
I. $$
{\frac{7} {8}} h
$$
J. $$
{\frac{1} {3}} h
$$
Answer: F
In a liquid, there is an air bubble with a diameter of 1 mm. The viscosity of the liquid is 0.15 Pa·s, and the density is $0.9 \times 10^{3} \mathrm{kg/m}^{3}$. If this air bubble rises in water, what is its terminal velocity? (Take the density of water as $10^{3} \mathrm{kg/m}^{3}$ and the viscosity as $1 \times 10^{-3} \mathrm{Pa·s}$.)

Options:
A. $$
\approx1. 7 {\mathrm{m / s}}
$$
B. $$
\approx1. 2 {\mathrm{m / s}}
$$
C. $$
\approx1. 9 {\mathrm{m / s}}
$$
D. $$
\approx0. 2 {\mathrm{m / s}}
$$
E. $$
\approx2. 0 {\mathrm{m / s}}
$$
F. $$
\approx0. 1 {\mathrm{m / s}}
$$
G. $$
\approx1. 5 {\mathrm{m / s}}
$$
H. $$
\approx0. 3 {\mathrm{m / s}}
$$
I. $$
\approx0. 8 {\mathrm{m / s}}
$$
J. $$
\approx0. 6 {\mathrm{m / s}}
$$
Answer: J
In a rectangular coordinate system, there is a charge of $25 \times 10^{-9} \mathrm{C}$ at the origin, and another charge of $-25 \times 10^{-9} \mathrm{C}$ at $x=6 \mathrm{m}, y=0$. What the electric field intensity at $x=3 \mathrm{m}, y=4 \mathrm{m}$ is().

Options:
A. 15.6 N/C, direction is negative y-direction
B. 10.8 N/C, direction is positive x-direction
C. 5.4 N/C, direction is positive y-direction
D. 10.8 N/C, direction is negative y-direction
E. 8.1 N/C, direction is positive y-direction
F. 7.5 N/C, direction is positive x-direction
G. 10.8 N/C, direction is negative x-direction
H. 12.5 N/C, direction is positive x-direction
I. 10.8 N/C, direction is positive z-direction
J. 15.6 N/C, direction is negative z-direction
Answer: B
Connect a capacitor with capacitance $C=318 \mu\mathrm{F}$ to an AC power source with voltage $u(t)=311 \sin(314t)$ $\mathbf{V}$, and the expression for the instantaneous current is ().

Options:
A. $$
2 0 0 \operatorname{s i n} ( 3 1 4 t+{0} ) \ {\mathrm{~ A}}
$$
B. $$
3 1 1 \operatorname{cos}( 1 5 7 t+{\pi} ) \ {\mathrm{~ A}}
$$
C. $$
2 5 0 \operatorname{s i n} ( 3 1 4 t+{\frac{3\pi} {2}} ) \ {\mathrm{~ A}}
$$
D. $$
2 5 7 \operatorname{s i n} ( 3 1 4 t+{\frac{\pi} {2}} ) \ {\mathrm{~ A}}
$$
E. $$
3 1 1 \operatorname{s i n} ( 1 5 7 t+{\frac{\pi} {3}} ) \ {\mathrm{~ A}}
$$
F. $$
3 1 1 \operatorname{cosine}( 3 1 4 t+{\frac{\pi} {2}} ) \ {\mathrm{~ A}}
$$
G. $$
3 1 1 \operatorname{cosine}( 3 1 4 t ) \ {\mathrm{~ A}}
$$
H. $$
3 1 1 \operatorname{s i n} ( 3 1 4 t+{\frac{\pi} {4}} ) \ {\mathrm{~ A}}
$$
I. $$
3 1 1 \operatorname{s i n} ( 3 1 4 t+{\frac{\pi} {2}} ) \ {\mathrm{~ A}}
$$
J. $$
3 1 1 \operatorname{s i n} ( 3 1 4 t+{\pi} ) \ {\mathrm{~ A}}
$$
Answer: I
What is the de Broglie wavelength of an electron with a kinetic energy of $500 \mathrm{eV}$?

Options:
A. $$
4. 4 \times1 0^{-1 2} {\mathrm{m}}
$$
B. $$
5. 5 \times1 0^{-1 1} {\mathrm{m}}
$$
C. $$
6. 1 \times1 0^{-1 0} {\mathrm{m}}
$$
D. $$
8. 2 \times1 0^{-1 2} {\mathrm{m}}
$$
E. $$
2. 3 \times1 0^{-1 0} {\mathrm{m}}
$$
F. $$
9. 8 \times1 0^{-1 2} {\mathrm{m}}
$$
G. $$
7. 4 \times1 0^{-1 2} {\mathrm{m}}
$$
H. $$
3. 6 \times1 0^{-1 1} {\mathrm{m}}
$$
I. $$
5. 9 \times1 0^{-1 1} {\mathrm{m}}
$$
J. $$
1. 1 \times1 0^{-1 0} {\mathrm{m}}
$$
Answer: B
In a television picture tube, the accelerating voltage is $9 \mathrm{kV}$, and the diameter of the electron gun's muzzle is 0.1 mm. What is the uncertainty in the transverse velocity of the electrons as they are emitted from the electron gun?

Options:
A. 6.88 m/s
B. 7.16 m/s
C. 7.38 m/s
D. 6.84 m/s
E. 6.92 m/s
F. 6.71 m/s
G. 6.44 m/s
H. 5.55 m/s
I. 7.27m/s
J. 7.30 m/s
Answer: I
In a hydrogen atom, when the electron is in the state $n=4$, what are the possible values of the corresponding angular quantum number? What is the corresponding angular momentum?

Options:
A. $$
0 \,, \, \, 1 \,, \, \, 2 \,, \, \, 3
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {2 \pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
B. $$
0 \,, \, \, 2 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{5} h} {2 \pi}, \, \, \frac{\sqrt{1 4} h} {\pi}
$$
C. $$
0 \,, \, \, 1 \,, \, \, 2 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {2 \pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
D. $$
1 \,, \, \, 2 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {\pi}, \, \, \frac{\sqrt{1 0} h} {\pi}
$$
E. $$
0 \,, \, \, 1 \,, \, \, 2 \,, \, \, 3
$$
$$
0 \,, \, \, \frac{\sqrt{3} h} {2 \pi}, \, \, \frac{\sqrt{5} h} {2 \pi}, \, \, \frac{\sqrt{1 5 } h} {\pi}
$$
F. $$
0 \,, \, \, 1 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{5} h} {2 \pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
G. $$
1 \,, \, \, 2 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{3} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {\pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
H. $$
0 \,, \, \, 1 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{3} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {2 \pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
I. $$
0 \,, \, \, 1 \,, \, \, 2 \,, \, \, 3
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {\pi}, \, \, \frac{\sqrt{1 4} h} {\pi}
$$
J. $$
0 \,, \, \, 2 \,, \, \, 3 \,, \, \, 4
$$
$$
0 \,, \, \, \frac{\sqrt{2} h} {2 \pi}, \, \, \frac{\sqrt{6} h} {2 \pi}, \, \, \frac{\sqrt{1 2} h} {\pi}
$$
Answer: A
After administering $100 \mu\mathrm{Ci}^{131} \mathrm{I}$ orally to the patient, the urine collected within 24 hours showed an activity of 52.1 $\mu\mathrm{Ci}$. Assuming that $^{131}I$ is only excreted through urine, how much $^{131} \mathrm{I}$ remains in the patient's body in $\mu\mathrm{Ci}$?

Options:
A. $$
8 4. 3 \mu\mathrm{C i}
$$
B. $$
8 6. 5 \mu\mathrm{C i}
$$
C. $$
9 1. 7 \mu\mathrm{C i}
$$
D. $$
7 8. 4 \mu\mathrm{C i}
$$
E. $$
7 2. 2 \mu\mathrm{C i}
$$
F. $$
8 0. 9 \mu\mathrm{C i}
$$
G. $$
9 4. 6 \mu\mathrm{C i}
$$
H. $$
6 5. 9 \mu\mathrm{C i}
$$
I. $$
5 7. 8 \mu\mathrm{C i}
$$
J. $$
8 1. 0 \mu\mathrm{C i}
$$
Answer: C
Given the magnetic induction intensity of the external magnetic field $B = 1.5 \mathrm{T}$, what is the additional potential energy of the $^{6}\mathrm{Li}$ nucleus in this magnetic field?

Options:
A. $$
\pm8. 4 1 \times1 0^{-2 6} \mathrm{J}
$$
B. $$
\pm3. 4 7 \times1 0^{-2 6} \mathrm{J}
$$
C. $$
\pm4. 2 6 \times1 0^{-2 8} \mathrm{J}
$$
D. $$
\pm2. 8 1 \times1 0^{-2 9} \mathrm{J}
$$
E. $$
\pm7. 3 5 \times1 0^{-2 8} \mathrm{J}
$$
F. $$
\pm4. 5 4 \times1 0^{-2 7} \mathrm{J}
$$
G. $$
\pm6. 2 3 \times1 0^{-2 7} \mathrm{J}
$$
H. $$
\pm1. 2 3 \times1 0^{-2 9} \mathrm{J}
$$
I. $$
\pm5. 9 8 \times1 0^{-2 7} \mathrm{J}
$$
J. $$
\pm9. 2 9 \times1 0^{-2 8} \mathrm{J}
$$
Answer: G
Let the mass of the Earth be $m$, the radius be $R$, the rotational period be $T$, the mass of the Sun be $M$, the distance from the center of the Earth to the center of the Sun be $r$, and the gravitational constant between the Earth and the Sun be $G$. If the motion of the Earth around the Sun is regarded as circular motion, and the Earth is considered a sphere with uniformly distributed density, find the angular momentum of the earth's rotation and the orbital angular momentum of the earth moving around the sun.

Options:
A. $$
\frac{2} {5} m R \cdot\frac{2 \pi} {T}
$$
$$
m R \sqrt{G M r}
$$
B. $$
\frac{2} {5} m R^{2} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{G M r}
$$
C. $$
\frac{2} {5} m R^{3} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{G r}
$$
D. $$
\frac{1} {5} m R^{2} \cdot\frac{4 \pi} {T}
$$
$$
M \sqrt{G M r}
$$
E. $$
\frac{2} {5} m R^{2} \cdot\frac{2 \pi} {T}
$$
$$
\sqrt{m G M r}
$$
F. $$
\frac{3} {5} m R^{2} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{G R r}
$$
G. $$
\frac{2} {3} m R^{2} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{G M R}
$$
H. $$
\frac{2} {5} m R^{2} \cdot\frac{4 \pi} {T}
$$
$$
M \sqrt{G M r}
$$
I. $$
\frac{3} {5} m R^{3} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{G r M}
$$
J. $$
\frac{2} {7} m R^{2} \cdot\frac{2 \pi} {T}
$$
$$
m \sqrt{M G r}
$$
Answer: B
Under isothermal conditions, a large drop of water with a radius of $r_{0}$ = 0.5 cm is split into several smaller droplets, each with a radius of 0.1 cm. Calculate the work required for this process.

Options:
A. $$
2. 5 4 \times1 0^{-5} \mathrm{~ J}
$$
B. $$
7. 3 5 \times1 0^{-5} \mathrm{~ J}
$$
C. $$
7. 9 2 \times1 0^{-5} \mathrm{~ J}
$$
D. $$
6. 2 1 \times1 0^{-5} \mathrm{~ J}
$$
E. $$
5. 6 8 \times1 0^{-5} \mathrm{~ J}
$$
F. $$
4. 9 9 \times1 0^{-5} \mathrm{~ J}
$$
G. $$
9. 1 2 \times1 0^{-5} \mathrm{~ J}
$$
H. $$
6. 7 3 \times1 0^{-5} \mathrm{~ J}
$$
I. $$
8. 0 5 \times1 0^{-5} \mathrm{~ J}
$$
J. $$
3. 8 4 \times1 0^{-5} \mathrm{~ J}
$$
Answer: G
Two simple harmonic oscillations in the same direction have their motion equations as $x_{1}=0.03 \cos (10t+\frac{3\pi}{4})$ and $x_{2}=0.04 \cos (10t+\frac{\pi}{4})$, where the unit of $x$ is meters, and the unit of $t$ is seconds. Find the amplitude and initial phase of their resultant oscillation.

Options:
A. $$
0. 0 1 \left( \mathrm{m} \right)
$$
$$
1. 0 0 \, ( \mathrm{r a d} )
$$
B. $$
0. 0 4 \left( \mathrm{m} \right)
$$
$$
1. 7 5 \, ( \mathrm{r a d} )
$$
C. $$
0. 0 3 \left( \mathrm{m} \right)
$$
$$
1. 1 0 \, ( \mathrm{r a d} )
$$
D. $$
0. 0 2 \left( \mathrm{m} \right)
$$
$$
1. 1 0 \, ( \mathrm{r a d} )
$$
E. $$
0. 0 7 \left( \mathrm{m} \right)
$$
$$
1. 2 0 \, ( \mathrm{r a d} )
$$
F. $$
0. 0 5 \left( \mathrm{m} \right)
$$
$$
1. 4 3 \, ( \mathrm{r a d} )
$$
G. $$
0. 0 8 \left( \mathrm{m} \right)
$$
$$
0. 9 0 \, ( \mathrm{r a d} )
$$
H. $$
0. 0 9 \left( \mathrm{m} \right)
$$
$$
1. 3 0 \, ( \mathrm{r a d} )
$$
I. $$
0. 0 5 \left( \mathrm{m} \right)
$$
$$
2. 0 0 \, ( \mathrm{r a d} )
$$
J. $$
0. 0 6 \left( \mathrm{m} \right)
$$
$$
1. 5 0 \, ( \mathrm{r a d} )
$$
Answer: F
Using the Doppler effect to monitor the speed of a car, a fixed wave source emits ultrasound at a frequency of 100 kHz. When the car approaches the wave source, the receiver installed together with the wave source receives the ultrasound frequency reflected back from the car as 110 kHz. Given that the speed of sound in air is $300 \, m \cdot s^{-1}$, find the speed of the car.

Options:
A. $$
1 3. 8 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
B. $$
1 5. 7 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
C. $$
1 4. 3 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
D. $$
1 5. 0 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
E. $$
1 2. 4 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
F. $$
1 2. 9 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
G. $$
1 1. 9 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
H. $$
1 6. 5 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
I. $$
1 3. 5 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
J. $$
1 4. 8 ( \mathrm{m} \cdot\mathrm{s}^{-1} )
$$
Answer: C
Given that the threshold wavelength of the photoelectric effect for cesium is $660 \, \mathrm{nm}$, find the speed of the photoelectrons emitted by illuminating the cesium layer with light of wavelength $\lambda=400 \, \mathrm{nm}$.

Options:
A. $$
1. 9 4 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
B. $$
2. 7 3 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
C. $$
9. 0 8 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
D. $$
3. 2 1 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
E. $$
4. 5 6 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
F. $$
6. 5 6 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
G. $$
5. 4 3 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
H. $$
3. 6 8 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
I. $$
8. 1 2 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
J. $$
7. 8 9 \times1 0^{5} \ \mathrm{m} \cdot\mathrm{s}^{-1}
$$
Answer: F
In a particular working substance, atoms have the following energy levels: -13.2 eV (ground state), -11.1 eV, -10.6 eV, $-9.8 \, \mathrm{eV}$. The -10.6 eV state primarily transitions to the -11.1 eV state, and the -9.8 eV state primarily transitions to the ground state. What wavelength of light should be used to pump this laser for optimal operation? What is the wavelength of the laser emitted by this laser?

Options:
A. $$477.8(nm),591.6(nm)$$
B. $$442.1(nm),562.8(nm)$$
C. $$468.8(nm),591.0(nm)$$
D. $$492.0(nm),573.7(nm)$$
E. $$455.3(nm),579.2(nm)$$
F. $$486.5(nm),584.3(nm)$$
G. $$465.7(nm),595.4(nm)$$
H. $$488.9(nm),602.7(nm)$$
I. $$479.9(nm),600.2(nm)$$
J. $$499.5(nm),613.9(nm)$$
Answer: A
The concentration of the monomer solution is 0.20 mol/L, and the concentration of the peroxide initiator is $4.0 \times 10^{-3} \, \mathrm{mol/L}$. It is heated to polymerize at 60°C. If the half-life of the initiator is 44 hours, the initiator efficiency $f = 0.8$, $k_{\mathrm{p}} = 145 \mathrm{L / (mol \cdot s)}$, $k_{\mathrm{t}} = 7.0 \times 10^{7} \, \mathrm{L / mol}$, how long is needed to achieve 50% conversion?

Options:
A. 155.4h
B. 170.8h
C. 185.0h
D. 165.9h
E. 175.6h
F. 190.1h
G. 160.7h
H. 180.2h
I. 145.3h
J. 200.5h
Answer: B
Calculate the reactivity ratios of styrene-butadiene and styrene-methyl methacrylate is().It is known that for butadiene-styrene, $Q_{1}=1 \,, \, \, \, Q_{2}=2. \, 3 9 \,, \, \, \, e_{1}=-\, 0. \, 8 \,, \, \, \, e_{2}=-\, 1. \, 0 5$; for styrene-methyl methacrylate, $Q_{1}=1 \,, \, \, \, Q_{2}=$ 0.74, $e_{1}=-\, 0. \, 8 \,, \, \, \, e_{2}=0. \, 4$.

Options:
A. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 0 1 1}}} \\ {{{r_{2}=0. 9 9}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 5 8 2}}} \\ {{{r_{2}=0. 7 3}}} \\ \end{array}
$$
B. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 4 9 2}}} \\ {{{r_{2}=1. 1 6}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 4 7 3}}} \\ {{{r_{2}=0. 3 9}}} \\ \end{array}
$$
C. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=0. 5 1 1}}} \\ {{{r_{2}=1. 8 4}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 5 1 7}}} \\ {{{r_{2}=0. 4 6}}} \\ \end{array}
$$
D. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 2 3 4}}} \\ {{{r_{2}=2. 3 4}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 4 8 3}}} \\ {{{r_{2}=0. 5 1}}} \\ \end{array}
$$
E. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 1 2 0}}} \\ {{{r_{2}=2. 5 5}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 4 4 8}}} \\ {{{r_{2}=0. 5 8}}} \\ \end{array}
$$
F. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 0 8 3}}} \\ {{{r_{2}=1. 9 4}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 6 0 9}}} \\ {{{r_{2}=0. 4 4}}} \\ \end{array}
$$
G. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=0. 7 2 4}}} \\ {{{r_{2}=3. 0 6}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 6 7 4}}} \\ {{{r_{2}=0. 5 2}}} \\ \end{array}
$$
H. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=0. 9 3 7}}} \\ {{{r_{2}=3. 0 8}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 4 9 5}}} \\ {{{r_{2}=0. 6 6}}} \\ \end{array}
$$
I. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=0. 6 8 9}}} \\ {{{r_{2}=2. 1 7}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 6 2 3}}} \\ {{{r_{2}=0. 5 0}}} \\ \end{array}
$$
J. For the styrene-butadiene system:
$$
\begin{array} {l} {{{r_{1}=1. 3 7 5}}} \\ {{{r_{2}=2. 7 6}}} \\ \end{array}
$$
For the styrene-methyl methacrylate system:
$$
\begin{array} {l} {{{r_{1}=0. 5 6 1}}} \\ {{{r_{2}=0. 7 8}}} \\ \end{array}
$$
Answer: C
The viscosity of a certain polystyrene sample at 160°C is $8.0 \times 10^{2} \, \mathrm{Pa} \cdot \mathrm{s}$. Estimate its viscosity at the glass transition temperature of 120°C.

Options:
A. $$
2. 9 8 \times1 0^{9} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
B. $$
3. 2 1 \times1 0^{8} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
C. $$
6. 0 1 \times1 0^{8} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
D. $$
3. 4 5 \times1 0^{3} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
E. $$
7. 8 6 \times1 0^{10} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
F. $$
4. 7 9 \times1 0^{5} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
G. $$
9. 9 2 \times1 0^{5} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
H. $$
2. 5 6 \times1 0^{7} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
I. $$
1. 3 9 \times1 0^{6} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
J. $$
5. 0 8 \times1 0^{4} \ \ ( \mathrm{P a} \cdot\mathrm{s} )
$$
Answer: H
The glass transition temperature of polystyrene is known to be 358 K. Apply a certain stress to a sample of polystyrene and determine how much faster the creep rate is at 393 K compared to the creep rate at 443 K.

Options:
A. $$
5. 9 \times1 0^{3}
$$
B. $$
3. 8 \times1 0^{3}
$$
C. $$
7. 2 \times1 0^{3}
$$
D. $$
6. 4 \times1 0^{3}
$$
E. $$
9. 7 \times1 0^{3}
$$
F. $$
1. 3 \times1 0^{3}
$$
G. $$
2. 6 \times1 0^{3}
$$
H. $$
7. 8 \times1 0^{3}
$$
I. $$
4. 5 \times1 0^{3}
$$
J. $$
8. 1 \times1 0^{3}
$$
Answer: D
The density of an ideal rubber is $\rho_{2}=1.000 \, \mathrm{g/cm}^{3}$, and its elongation ratio is 2.000 at 25°C with stress of $3.389 \times 10^{3} \, \mathrm{N/m^{2}}$. This rubber is then placed in a solvent with a molar volume of $V_{1}=2.000 \times 10^{-3} \, \mathrm{cm}^{3}/\mathrm{mol}$, and the equilibrium swelling ratio $Q$ is measured to be 8.000. Calculate the interaction parameter between the rubber and the solvent, $\chi_{1}$ is().(Boltzmann constant $k = 1.381 \times 10^{-23} \, \mathrm{J/K}$).

Options:
A. 0.47
B. 0.40
C. 0.49
D. 0.45
E. 0.52
F. 0.38
G. 0.55
H. 0.42
I. 0.36
J. 0.58
Answer: D
A rectangular cross-linked rubber sample measures 100 mm in length, 40 mm in width, and 5 mm in thickness. At 26.85°C, a force of 10 N can stretch it to twice its original length along its length direction. If the density of this rubber is $900 \mathrm{kg} / \mathrm{m}^{3}$, calculate the average molecular weight of the rubber network chains.

Options:
A. $6.5\times10^{4}$
B. $8.1\times10^{4}$
C. $8.5\times10^{4}$
D. $7.3\times10^{4}$
E. $7.9\times10^{4}$
F. $8.7\times10^{4}$
G. $9.0\times10^{4}$
H. $7.0\times10^{4}$
I. $7.8\times10^{4}$
J. $7.2\times10^{4}$
Answer: E