MikaStars39/nano-eval huggingface dataset:
What is the power series expansion of $f ( \, z \, )={\frac{z^{2}-8 z+5} {( \, z+2 \, ) \, ( \, z-3 \, )^{2}}}$ at the point $z=0$ in terms of $z$?

Options:
A. $$
= \frac{4} {1 9}-\frac{2 8 z} {9 7}+\frac{1 2 z^{2}} {2 2 8}-\frac{3 1 1 z^{3}} {3 6 8 8}+\cdots, \qquad\mid z \mid< 6.
$$
B. $$
= \frac{2} {2 6}-\frac{5 1 z} {1 0 4}+\frac{9 z^{2}} {2 0 6}-\frac{4 9 1 z^{3}} {3 4 8 8}+\cdots, \qquad\mid z \mid< 3.
$$
C. $$
= \frac{7} {1 6}-\frac{2 5 z} {9 6}+\frac{1 5 z^{2}} {2 0 8}-\frac{2 8 3 z^{3}} {3 5 9 9}+\cdots, \qquad\mid z \mid< 1.
$$
D. $$
= \frac{4} {2 4}-\frac{4 6 z} {9 5}+\frac{1 8 z^{2}} {2 7 6}-\frac{3 7 1 z^{3}} {3 6 8 8}+\cdots, \qquad\mid z \mid< 3.
$$
E. $$
= \frac{8} {2 0}-\frac{1 9 z} {9 8}+\frac{1 7 z^{2}} {1 9 6}-\frac{3 6 1 z^{3}} {3 7 9 9}+\cdots, \qquad\mid z \mid< 1.
$$
F. $$
= \frac{6} {1 2}-\frac{3 4 z} {1 2 0}+\frac{1 3 z^{2}} {2 3 6}-\frac{3 2 1 z^{3}} {2 9 8 8}+\cdots, \qquad\mid z \mid< 4.
$$
G. $$
= \frac{3} {2 3}-\frac{5 3 z} {1 1 8}+\frac{8 z^{2}} {2 4 6}-\frac{3 9 1 z^{3}} {3 7 9 8}+\cdots, \qquad\mid z \mid< 2.
$$
H. $$
= \frac{1} {2 1}-\frac{2 9 z} {8 8}+\frac{1 0 z^{2}} {2 5 6}-\frac{3 3 1 z^{3}} {3 5 8 8}+\cdots, \qquad\mid z \mid< 5.
$$
I. $$
= \frac{9} {1 7}-\frac{4 1 z} {1 2 8}+\frac{1 4 z^{2}} {2 1 8}-\frac{3 5 1 z^{3}} {3 9 8 8}+\cdots, \qquad\mid z \mid< 4.
$$
J. $$
= \frac{5} {1 8}-\frac{4 3 z} {1 0 8}+\frac{1 1 z^{2}} {2 1 6}-\frac{3 7 1 z^{3}} {3 \ 8 8 8}+\cdots, \qquad\mid z \mid< 2.
$$
Answer: J
What is the characteristic of the function $f(z) = \sec \left(\frac{1}{z-1}\right)$ at the point $z = 1$?

Options:
A. The fixed point of these poles is $z=1$.
B. The limit point of these poles is $z=1$.
C. The accumulation point of these poles is $z=1$.
D. The saddle point of these poles is $z=1$.
E. The asymptotic behavior of these poles is $z=1$.
F. The oscillation point of these poles is $z=1$.
G. The critical point of these poles is $z=1$.
H. The stationary point of these poles is $z=1$.
I. The node of these poles is $z=1$.
J. The removable singularity of these poles is $z=1$.
Answer: B
The residue of the function $f(z) = \sin \frac{z}{z+1}$ at its finite singular points is ().

Options:
A. $$
- \operatorname{c o s} \left(-1\right)
$$
B. $$
- \operatorname{c o s} \ 1
$$
C. $$
- \frac{1}{2} \operatorname{c o s} \ 1
$$
D. $$
\frac{1}{2} \operatorname{c o s} \ 1
$$
E. $$
\operatorname{c o s} \ 0
$$
F. $$
- \operatorname{c o s} \ 0
$$
G. $$
\operatorname{s i n} \ 1
$$
H. $$
\operatorname{c o s} \ (-1)
$$
I. $$
2 \operatorname{c o s} \ 1
$$
J. $$
\frac{1}{3} \operatorname{c o s} \ 0
$$
Answer: B
What is the residue of the function $f ( \, z \, )={\frac{\mathrm{e}^{\frac{1} {z}}} {1-z}}$ at the singularity $z=0$?

Options:
A. $1-e$
B. $e-1$
C. $-1$
D. $e^2-1$
E. $e+1$
F. $0$
G. $2-e$
H. $-e$
I. $e$
J. $1$
Answer: B
The value of $\oint_{| z |=1} \frac{\operatorname{tan} \pi z} {z^{3}} \mathrm{d} z$ is ().

Options:
A. 3
B. 0
C. 1
D. 2
E. -1
F. e
G. ∞
H. -π
I. π
J. -∞
Answer: B
Let the number of collisions be $t$. What is the average number of scattering collisions experienced by thermal neutrons (0.025 electron volts) before being absorbed when moving through light water, heavy water, and cadmium?

Options:
A. $$156,~13600,~2.86\times10^{-3}$$
B. $$145,~14000,~2.90\times10^{-3}$$
C. $$132,~12900,~2.50\times10^{-3}$$
D. $$148,~13400,~2.70\times10^{-3}$$
E. $$140,~13300,~2.65\times10^{-3}$$
F. $$158,~13700,~2.85\times10^{-3}$$
G. $$160,~14200,~3.00\times10^{-3}$$
H. $$153,~13800,~2.80\times10^{-3}$$
I. $$165,~13650,~2.95\times10^{-3}$$
J. $$175,~15200,~3.15\times10^{-3}$$
Answer: A
Given that the enrichment level of uranium-235 in the nuclear fuel is 3.2% by weight, what is the ratio of the nucleon numbers of uranium-235 to uranium-238?

Options:
A. $$0.0340$$
B. $$0.0287$$
C. $$0.0312$$
D. $$0.0335$$
E. $$0.0299$$
F. $$0.0321$$
G. $$0.0350$$
H. $$0.0305$$
I. $$0.0378$$
J. $$0.0365$$
Answer: D
In a certain fission reactor, the fast neutron multiplication factor is 1.05, the probability of escaping resonance capture is 0.9, the probability of non-leakage during moderation is 0.952, the probability of non-leakage during diffusion is 0.94, the number of effective fission neutrons is 1.335, and the thermal neutron utilization factor is 0.882. What are the infinite medium multiplication factor and the effective multiplication factor ?

Options:
A. $$1.1127,~0.9957$$
B. $$1.1589,~1.0383$$
C. $$1.0925,~1.0659$$
D. $$1.0678,~1.1024$$
E. $$1.1352,~0.9828$$
F. $$1.1754,~1.0441$$
G. $$1.0062,~1.0856$$
H. $$1.0031,~0.9902$$
I. $$1.1203,~1.0176$$
J. $$1.0476,~0.9734$$
Answer: A
Consider a cubic reactor with side length $a = 9 \mathrm{~ m}$. The neutron flux density distribution is given by
$$
\phi(x, y, z) = 3 \times 10^{13} \cos \left( \frac{\pi x}{a} \right) \cos \left( \frac{\pi y}{a} \right) \cos \left( \frac{\pi z}{a} \right) \quad (\mathrm{cm}^{-2} \cdot \mathrm{s}^{-1})
$$
It is known that $D = 0.84 \times 10^{-2} \mathrm{~m}, \; \; L = 0.175 \; \mathrm{m}$. What is the number of neutrons absorbed per second (assuming the extrapolation distance is small enough to be neglected)?

Options:
A. $$
5. 3 2 {\times} 1 0^{1 9} \, \mathrm{s}^{-1}
$$
B. $$
9. 8 1 {\times} 1 0^{1 8} \, \mathrm{s}^{-1}
$$
C. $$
6. 3 7 {\times} 1 0^{2 1} \, \mathrm{s}^{-1}
$$
D. $$
3. 1 7 {\times} 1 0^{1 9} \, \mathrm{s}^{-1}
$$
E. $$
1. 6 8 {\times} 1 0^{1 9} \, \mathrm{s}^{-1}
$$
F. $$
7. 4 7 {\times} 1 0^{1 7} \, \mathrm{s}^{-1}
$$
G. $$
8. 0 3 {\times} 1 0^{2 2} \, \mathrm{s}^{-1}
$$
H. $$
4. 5 6 {\times} 1 0^{2 0} \, \mathrm{s}^{-1}
$$
I. $$
2. 9 4 {\times} 1 0^{2 0} \, \mathrm{s}^{-1}
$$
J. $$
1. 2 4 {\times} 1 0^{2 0} \, \mathrm{s}^{-1}
$$
Answer: J
Within a uniform sphere of radius $\mathbf{R}$, where $\mathrm{S}$ neutrons are uniformly produced per unit volume per second, what is the distribution of neutron flux density inside the sphere?

Options:
A. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R-d ) \cosh ( r / L )} {r \cosh ( \frac{R-d} {L} )} ]
$$
B. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1-\frac{( R+2d ) \sinh ( r / L )} {r \sinh ( \frac{R+2d} {L} )} ]
$$
C. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R+d ) \cosh ( r / 2L )} {r \cosh ( \frac{R+d} {2L} )} ]
$$
D. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R-2d ) \sinh ( r / L )} {r \sinh ( \frac{R-2d} {L} )} ]
$$
E. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1-\frac{( R+d ) \cosh ( r / L )} {r \cosh ( \frac{R+d} {L} )} ]
$$
F. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1-\frac{( R ) \cosh ( r / 2L )} {r \cosh ( \frac{R} {2L} )} ]
$$
G. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R ) \tanh ( r / L )} {r \tanh ( \frac{R} {L} )} ]
$$
H. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R-d/2 ) \cos ( r / L )} {r \cos ( \frac{R-d/2} {L} )} ]
$$
I. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1+\frac{( R+d ) \sin ( r / L )} {r \sin ( \frac{R+d} {L} )} ]
$$
J. $$
\phi(\mathrm{r})= \frac{S} {\Sigma_{a}} [ 1-\frac{( R+d/2 ) \exp ( r / L )} {r \exp ( \frac{R+d/2} {L} )} ]
$$
Answer: E
The single-group constants are as follows:
$\mathrm{^{235}U}: \sigma_{\mathrm{f}} = 1.5~\mathrm{b},~\sigma_{\mathrm{u}} = 1.78~\mathrm{b},~\sum_{\mathrm{tr}} = 35.4~\mathrm{m}^{-1},~\nu = 2.51$;
$\mathrm{^{238}U}:~\sigma_{\mathrm{f}} = 0,~\sigma_{\mathrm{a}} = 0.18~\mathrm{b},~\sum_{\mathrm{tr}} = 35.4~\mathrm{m}^{-1}$.
A spherical fast neutron reactor made of pure $\mathrm{^{235}U}$ metal ($\rho = 18.7 \times 10^{3}~\mathrm{kg/m}^{3}$) is surrounded by an infinitely thick layer of pure $\mathrm{^{238}U}$ ($\rho = 19.0 \times 10^{3}~\mathrm{kg/m}^{3}$). What is the critical mass calculated by single-group theory?

Options:
A. $$25.0\mathrm{kg}$$
B. $$20.5\mathrm{kg}$$
C. $$24.6\mathrm{kg}$$
D. $$21.3\mathrm{kg}$$
E. $$23.8\mathrm{kg}$$
F. $$22.1\mathrm{kg}$$
G. $$26.4\mathrm{kg}$$
H. $$18.2\mathrm{kg}$$
I. $$17.7\mathrm{kg}$$
J. $$19.9\mathrm{kg}$$
Answer: D
The ratio of xenon equilibrium concentrations in two reactors with the same volume and power density, a superheated reactor $( \phi_{\mathrm{a}}=1 \times 10^{15} \mathrm{cm}^{-2} \cdot \mathrm{s}^{-1}; \sigma_{\mathrm{X e}}^{\mathrm{a}}=10 \mathrm{~b} )$ and a thermal neutron reactor $( \phi=5 \times 10^{13} \, \mathrm{cm}^{-2} \cdot \mathrm{s}^{-1}, \sigma_{\mathrm{x e}}^{a}=3 \times 10^{6} \, \mathrm{b})$?

Options:
A. $$365.9$$
B. $$50$$
C. $$102.5$$
D. $$45.8$$
E. $$16.2$$
F. $$12.3$$
G. $$78.6$$
H. $$9.7$$
I. $$243$$
J. $$0.85$$
Answer: I
Given parameters: Reactor power = $100\%$ FP, total power coefficient = $-0.020\% \Delta k / k / \%$FP, initial boron concentration = $500 \, \text{ppm}$, boron worth = $-0.010\% \Delta k / k / \text{ppm}$, control rod worth = $-0.010\% \Delta k / k / \text{cm}$. What is the required final boron concentration to reduce the plant power to 30% FP through boronization/dilution and control insertion of 50 cm (assuming other core state parameters remain unchanged)?

Options:
A. $$630~\mathrm{p p m}$$
B. $$570~\mathrm{p p m}$$
C. $$590~\mathrm{p p m}$$
D. $$620~\mathrm{p p m}$$
E. $$550~\mathrm{p p m}$$
F. $$600~\mathrm{p p m}$$
G. $$580~\mathrm{p p m}$$
H. $$610~\mathrm{p p m}$$
I. $$560~\mathrm{p p m}$$
J. $$640~\mathrm{p p m}$$
Answer: C
To ensure the effective number of fission neutrons for uranium $\eta=1.7$, what is the estimated mass enrichment of $^{235}\mathrm{U}$ in uranium when using a cross-section at $E_{\bf n}=0.0253 \, \bf eV$?

Options:
A. $$1.812\%$$
B. $$2.398\%$$
C. $$1.953\%$$
D. $$0.982\%$$
E. $$3.145\%$$
F. $$1.707\%$$
G. $$2.275\%$$
H. $$1.624\%$$
I. $$1.482\%$$
J. $$0.876\%$$
Answer: F
To obtain 1 kWh of energy, how much mass of $^{235}\text{U}$ is needed to undergo fission?

Options:
A. $$
m=4. 2 8 0 \times1 0^{-8} \, \mathrm{k g}
$$
B. $$
m=4. 3 9 0 \times1 0^{-8} \, \mathrm{k g}
$$
C. $$
m=3. 8 6 4 \times1 0^{-8} \, \mathrm{k g}
$$
D. $$
m=4. 8 7 5 \times1 0^{-8} \, \mathrm{k g}
$$
E. $$
m=4. 6 2 9 \times1 0^{-8} \, \mathrm{k g}
$$
F. $$
m=3. 9 5 8 \times1 0^{-8} \, \mathrm{k g}
$$
G. $$
m=5. 0 1 3 \times1 0^{-8} \, \mathrm{k g}
$$
H. $$
m=4. 9 3 2 \times1 0^{-8} \, \mathrm{k g}
$$
I. $$
m=4. 1 2 7 \times1 0^{-8} \, \mathrm{k g}
$$
J. $$
m=4. 4 5 5 \times1 0^{-8} \, \mathrm{k g}
$$
Answer: B
Using two Carnot refrigerators, obtain a cooling capacity of 0.1W at a temperature level of 0.1K. The first Carnot refrigerator operates between $0.1 K$ and 2K temperatures, while the second Carnot refrigerator operates between 2K and the high temperature level of 300K. How much work is required to transfer 0.1W of energy from the low temperature level to the high temperature level using these two refrigerators?

Options:
A. $$
2 8 8. 0 W
$$
B. $$
3 2 0. 1 W
$$
C. $$
3 0 1. 4 W
$$
D. $$
2 9 6. 3 W
$$
E. $$
3 1 2. 0 W
$$
F. $$
2 9 9. 9 W
$$
G. $$
2 9 7. 7 W
$$
H. $$
2 7 8. 9 W
$$
I. $$
3 1 0. 5 W
$$
J. $$
3 0 3. 6 W
$$
Answer: F
In the vapor compression system using refrigerant R22, there is a heat exchanger, where the saturated vapor from the evaporator is heated from $-10^{\circ} C$ to $5^{\circ} C$ using the $30^{\circ} C$ condensed liquid. Assuming that the compression process is isentropic, calculate the refrigeration coefficient of a system without a heat exchanger, where the condensation temperature is $30^{\circ} C$ and the evaporation temperature is $-10^{\circ} C$.

Options:
A. 4.424
B. 3.929
C. 5.312
D. 3.504
E. 6.205
F. 2.781
G. 4.018
H. 4.841
I. 5.876
J. 3.318
Answer: A
The coherence length of a helium-neon laser is 1 km, and the radius of the emitted light spot is $\mathbf{r = 0.3 \, \text{mm}}$. Find the linewidth of the light source as well as the coherence area and coherence volume at 1 km is___.

Options:
A. $$
1. 4 2 m^{2}
$$
$$
1. 4 2 \times1 0^{3} m^{3}
$$
B. $$
8. 2 3 9 m^{2}
$$
$$
8. 2 3 9 \times1 0^{3} m^{3}
$$
C. $$
9. 3 4 0 m^{2}
$$
$$
9. 3 4 0 \times1 0^{3} m^{3}
$$
D. $$
5. 8 6 m^{2}
$$
$$
5. 8 6 \times1 0^{3} m^{3}
$$
E. $$
7. 1 0 8 m^{2}
$$
$$
7. 1 0 8 \times1 0^{3} m^{3}
$$
F. $$
6. 9 7 m^{2}
$$
$$
6. 9 7 \times1 0^{3} m^{3}
$$
G. $$
3. 6 4 m^{2}
$$
$$
3. 6 4 \times1 0^{3} m^{3}
$$
H. $$
4. 7 5 m^{2}
$$
$$
4. 7 5 \times1 0^{3} m^{3}
$$
I. $$
2. 5 3 m^{2}
$$
$$
2. 5 3 \times1 0^{3} m^{3}
$$
J. $$
1. 0 1 1 m^{2}
$$
$$
1. 0 1 1 \times1 0^{3} m^{3}
$$
Answer: A
The ruby laser is a three-level system, with $\mathbf{S}_{3 2}=0.5 \times 10^{7} \, \text{s}^{-1}, \quad \mathbf{A}_{3 1}=3 \times 10^{5} \, \text{s}^{-1}, \quad \mathbf{A}_{2 1}=0.3 \times 10^{3} \, \text{s}^{-1}$. The transition probabilities for all other transitions are negligible. What is the pumping rate $\mathbf{W}_{1 3}$ at which the ruby crystal becomes transparent to light with a wavelength of 0.6943 μm?

Options:
A. $$
5 6 0 s^{-1}
$$
B. $$
2 7 5 s^{-1}
$$
C. $$
3 1 8 s^{-1}
$$
D. $$
8 2 1 s^{-1}
$$
E. $$
9 0 7 s^{-1}
$$
F. $$
4 2 3 s^{-1}
$$
G. $$
0 5 8 s^{-1}
$$
H. $$
1 3 4 s^{-1}
$$
I. $$
6 4 9 s^{-1}
$$
J. $$
7 5 2 s^{-1}
$$
Answer: C
Ruby laser cavity length $\mathbf{L}=11.25 \mathrm{cm}$, ruby rod length $l=10 \mathrm{cm}$, refractive index $\mathbf{n}=1.75$, fluorescence linewidth $\Delta\mathbf{v}_{\mathrm{F}}=2 \times 10^{5} \mathrm{MHz}$, find the number of longitudinal modes that meet the threshold condition when the excitation parameter $\alpha=1.16$. Answer is____.

Options:
A. 140
B. 80
C. 75
D. 112
E. 101
F. 130
G. 90
H. 85
I. 120
J. 95
Answer: E
$\mathrm{CO_{2}}$ Laser cavity length $l=1 \mathrm{m}$, discharge tube diameter $\mathbf{d=} 10 \mathrm{mm}$, single-pass diffraction loss rate $\delta_{\mathrm{d}}=0.5\%$, scattering loss rate of both mirrors is 1.5%, transmittance of the two mirrors is 2% and 10% respectively, other losses are negligible. When the laser operates at room temperature (300K), find: the excitation parameters; calculate the intracavity light intensity under stable operation at optimal discharge conditions; if the effective output area is considered as 80% of the discharge tube's cross-sectional area, what is the maximum output power of this laser? Relevant formulas: $\mathbf{G}_{\mathrm{m}}=1.4 \times 10^{-2} / \mathbf{d}$ (1/mm), $I_{0}=72/\mathbf{d}^{2}$ (w/mm$^{2}$), $\mathbf{p} \cdot \mathbf{d}=2.67 \times 10^{4} \mathrm{Pa} \cdot \mathrm{m}$, $\Delta\mathbf{v}_{\mathrm{L}}=0.049 \mathbf{p} (\mathrm{MHz})$.
$$
\Delta{\bf v}_{\mathrm{D}}=7.16 \times 10^{-7} {\bf v}_{0} \left( \frac{T}{M} \right)^{\frac{1}{2}}
$$
Answer is___.

Options:
A. $$
1 7. 3
$$
$$
1 1. 3 0 w / m m^{2}
$$
$$
4 5 w
$$
B. $$
1 5. 9
$$
$$
1 1. 7 5 w / m m^{2}
$$
$$
5 3 w
$$
C. $$
1 5. 5
$$
$$
1 0. 9 0 w / m m^{2}
$$
$$
5 2 w
$$
D. $$
1 6. 5
$$
$$
1 2. 0 0 w / m m^{2}
$$
$$
4 8 w
$$
E. $$
1 6. 8
$$
$$
1 2. 2 5 w / m m^{2}
$$
$$
4 4 w
$$
F. $$
1 6. 0
$$
$$
1 0. 5 0 w / m m^{2}
$$
$$
5 0 w
$$
G. $$
1 5. 8
$$
$$
9 9. 8 5 w / m m^{2}
$$
$$
5 1 w
$$
H. $$
1 6. 9
$$
$$
1 1. 4 5 w / m m^{2}
$$
$$
4 9 w
$$
I. $$
1 6. 3
$$
$$
1 1. 8 0 w / m m^{2}
$$
$$
4 6 w
$$
J. $$
1 7. 0
$$
$$
1 2. 3 5 w / m m^{2}
$$
$$
4 7 w
$$
Answer: H
In a symmetrical concave cavity with length $\mathbf{L}$ and mirror curvature radius $R=2.5L$, given the wavelength of light as $\lambda$, determine the radius of the fundamental mode spot on the mirror surface.
Answer is___.

Options:
A. $${\frac{2} {3}} {\sqrt{\frac{7 \lambda L} {\pi}}}
$$
B. $${\frac{2} {4}} {\sqrt{\frac{10 \lambda L} {\pi}}}
$$
C. $${\frac{1} {2}} {\sqrt{\frac{5 \lambda L} {\pi}}}
$$
D. $${\frac{1} {4}} {\sqrt{\frac{6 \lambda L} {\pi}}}
$$
E. $${\frac{1} {3}} {\sqrt{\frac{3 \lambda L} {\pi}}}
$$
F. $${\frac{1} {6}} {\sqrt{\frac{11 \lambda L} {\pi}}}
$$
G. $${\frac{3} {5}} {\sqrt{\frac{4 \lambda L} {\pi}}}
$$
H. $${\frac{3} {4}} {\sqrt{\frac{2 \lambda L} {\pi}}}
$$
I. $${\frac{1} {5}} {\sqrt{\frac{8 \lambda L} {\pi}}}
$$
J. $${\frac{2} {5}} {\sqrt{\frac{9 \lambda L} {\pi}}}
$$
Answer: C
When an ultrasonic field is established in a molten quartz acousto-optic medium (n=1.54) with an ultrasound frequency of $f_{s}=40 \text{ MHz}$, calculate the angle of incidence $\theta$ at which incident light with a wavelength of $\lambda=1.06 \mu \text{m}$ satisfies the Bragg condition.
Answer is____.

Options:
A. $$
2. 9 3 \times1 0^{-2}
$$
B. $$
4. 0 1 \times1 0^{-4}
$$
C. $$
7. 8 5 \times1 0^{-3}
$$
D. $$
3. 4 5 \times1 0^{-3}
$$
E. $$
5. 9 2 \times1 0^{-4}
$$
F. $$
1. 2 6 \times1 0^{-2}
$$
G. $$
4. 7 8 \times1 0^{-3}
$$
H. $$
8. 4 6 \times1 0^{-3}
$$
I. $$
6. 5 3 \times1 0^{-2}
$$
J. $$
2. 3 1 \times1 0^{-3}
$$
Answer: J
The monochromaticity of a photon with a wavelength of $$\lambda=5000 \, \text{Å}$$ is $\Delta\lambda/\lambda=10^{-7}$. Find the position uncertainty of this photon, $$\Delta\mathbf{x}$$.
Answer is___.

Options:
A. 3m
B. 8m
C. 10m
D. 1m
E. 7m
F. 5m
G. 2m
H. 4m
I. 6m
J. 9m
Answer: F
The transition probability matrix of a Markov chain is
$$
P=\left[ \begin{matrix} {{p}} & {{q}} & {{0}} \\ {{q}} & {{0}} & {{p}} \\ {{0}} & {{q}} & {{p}} \\ \end{matrix} \right]
$$
Find its stationary probability distribution is ().

Options:
A. $$
\pi_{0}={\frac{1} {1+{\frac{q} {p}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {1+\frac{p} {q}+\left( \frac{p} {q} \right)^{3}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {2+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
B. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {1+\frac{p} {q}+\left( \frac{p} {q} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
C. $$
\pi_{0}={\frac{1} {3+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {1+\frac{p} {q}+\left( \frac{q} {p} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
D. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {2+\frac{p} {q}+\left( \frac{p} {q} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{q} {p}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
E. $$
\pi_{0}={\frac{1} {2+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {1+\frac{p} {q}+\left( \frac{q} {p} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{3}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
F. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {2+\frac{p} {q}+\left( \frac{p} {q} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
G. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{3}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {1+\frac{p} {q}+\left( \frac{p} {q} \right)^{3}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{3}}}
$$
H. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{q} {p}} {2+\frac{p} {q}+\left( \frac{p} {q} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{3}} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{3}}}
$$
I. $$
\pi_{0}={\frac{1} {1+{\frac{p} {q}}+\left( {\frac{p} {q}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{p} {q}} {2+\frac{p} {q}+\left( \frac{q} {p} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{p} {q}} \right)^{2}} {1+{\frac{q} {p}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
J. $$
\pi_{0}={\frac{1} {1+{\frac{q} {p}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
$$
\pi_{1}=\frac{\frac{q} {p}} {1+\frac{q} {p}+\left( \frac{q} {p} \right)^{2}}
$$
$$
\pi_{2}={\frac{\left( {\frac{q} {p}} \right)^{2}} {2+{\frac{q} {p}}+\left( {\frac{q} {p}} \right)^{2}}}
$$
Answer: B
Place 3 balls randomly into 3 boxes, let $X$ be the number of boxes containing balls. Find $E(X)$ and $D(X)$.

Options:
A. $$
E( X ) \!=\! \frac{1 9} {9}
$$
$$
D ( X ) \!=\! \frac{2 6} {8 1}
$$
B. $$
E( X ) \!=\! \frac{1 7} {9}
$$
$$
D ( X ) \!=\! \frac{2 0} {8 1}
$$
C. $$
E( X ) \!=\! \frac{2 1} {9}
$$
$$
D ( X ) \!=\! \frac{4 2} {8 1}
$$
D. $$
E( X ) \!=\! \frac{2 9} {9}
$$
$$
D ( X ) \!=\! \frac{1 2} {8 1}
$$
E. $$
E( X ) \!=\! \frac{1 5} {9}
$$
$$
D ( X ) \!=\! \frac{3 1} {8 1}
$$
F. $$
E( X ) \!=\! \frac{2 8} {9}
$$
$$
D ( X ) \!=\! \frac{3 0} {8 1}
$$
G. $$
E( X ) \!=\! \frac{2 4} {9}
$$
$$
D ( X ) \!=\! \frac{2 4} {8 1}
$$
H. $$
E( X ) \!=\! \frac{2 6} {9}
$$
$$
D ( X ) \!=\! \frac{4 0} {8 1}
$$
I. $$
E( X ) \!=\! \frac{2 3} {9}
$$
$$
D ( X ) \!=\! \frac{2 8} {8 1}
$$
J. $$
E( X ) \!=\! \frac{2 5} {9}
$$
$$
D ( X ) \!=\! \frac{3 6} {8 1}
$$
Answer: A
Let the random variable $X \sim U(0, \pi)$, $Y = \sin X$. Use the characteristic function to find $f_{Y}(y)$ is ().

Options:
A. $$
f_{Y} ( y )=\frac{1} {\pi} \cdot\frac{2} {\sqrt{1-y^{2}}}, \quad0 < y < 1
$$
B. $$
f_{Y} ( y )=\frac{2} {\pi} \cdot\frac{1} {1-y^{2}}, \quad-1 < y < 1
$$
C. $$
f_{Y} ( y )=\frac{3} {\pi} \cdot\frac{1} {1-y^{2}}, \quad-1 < y < 1
$$
D. $$
f_{Y} ( y )=\frac{3} {\pi} \cdot\frac{1} {\sqrt{1-y^{2}}}, \quad0 < y < 1
$$
E. $$
f_{Y} ( y )=\frac{2} {\pi} \cdot\frac{2} {\sqrt{1-y^{2}}}, \quad-1 < y < 1
$$
F. $$
f_{Y} ( y )=\frac{1} {\pi} \cdot\frac{1} {\sqrt{1-y^{3}}}, \quad-1 < y < 1
$$
G. $$
f_{Y} ( y )=\frac{2} {e} \cdot\frac{1} {\sqrt{1-y^{2}}}, \quad0 < y < 1
$$
H. $$
f_{Y} ( y )=\frac{1} {\pi} \cdot\frac{1} {\sqrt{1+y^{2}}}, \quad0 < y < 1
$$
I. $$
f_{Y} ( y )=\frac{2} {\pi} \cdot\frac{1} {\sqrt{1-y^{4}}}, \quad-1 < y < 1
$$
J. $$
f_{Y} ( y )=\frac{2} {\pi} \cdot\frac{1} {\sqrt{1-y^{2}}}, \quad0 < y < 1
$$
Answer: J
Consider the random process $\left\{X(t) = A \cos(\beta t + \Theta)\right., -\infty < t < +\infty\}$ where $\beta$ is a constant, the random variable $A \sim U(0, 1)$, $\Theta \sim U(0, 2\pi)$ and they are independent of each other. Find the mean function $m(t)$, the correlation function $R(s, t)$, and the variance function $D(t)$ of the random process $\left\{X(t), -\infty \leq t \leq +\infty\right\}$ is ().

Options:
A. $$
m ( t ) \!=\cos(\Theta)
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {9} \operatorname{c o s} ( \beta ( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {9}
$$
B. $$
m ( t ) \!=A
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {8} \operatorname{c o s} ( \beta ( s+t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {4}
$$
C. $$
m ( t ) \!=0
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {6} \operatorname{c o s} ( \beta( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {6}
$$
D. $$
m ( t ) \!=\frac{1} {3}
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {3} \operatorname{c o s} ( \beta ( s-t+1 ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! 1
$$
E. $$
m ( t ) \!=\sin(\Theta)
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{2} {6} \operatorname{c o s} ( \beta ( 2s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {10}
$$
F. $$
m ( t ) \!=\! \frac{1} {6}
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {8} \operatorname{c o s} ( \beta ( s+t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {8}
$$
G. $$
m ( t ) \!=\cos(\beta t)
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{1} {6} \sin( \beta ( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {3}
$$
H. $$
m ( t ) \!=\frac{1} {2}
$$
$$
\begin{array} {c} {{{R ( s, t )=A \operatorname{c o s} ( \beta ( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {2}
$$
I. $$
m ( t ) \!=\frac{1} {4}
$$
$$
\begin{array} {c} {{{R ( s, t )=\beta \operatorname{c o s} ( \beta ( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {5}
$$
J. $$
m ( t ) \!=\sin(\beta t)
$$
$$
\begin{array} {c} {{{R ( s, t )=\frac{2} {6} \operatorname{c o s} ( \beta ( s-t ) )}}} \\ \end{array}
$$
$$
D ( t ) \!=\! \frac{1} {7}
$$
Answer: C
When an object moves uniformly in a straight line at speed $U$ within an initially stationary incompressible homogeneous fluid, and if the effect of body forces can be neglected, let $\tau$ be the entire space outside the object and $\phi$ be the dissipation function. The resistance experienced by the object is ( ) .

Options:
A. $$\frac{1}{U} \iiint_{\tau} \phi d\tau$$
B. $$U \iiint_{r} \phi d\tau$$
C. $$\frac{1}{2} \iiint_{r} \phi d\tau$$
D. $$\frac{3}{U} \iiint_{r} \phi d\tau$$
E. $$\frac{1}{U^2} \iiint_{r} \phi d\tau$$
F. $$\frac{2}{U} \iiint_{r} \phi d\tau$$
G. $$\frac{1}{U} \iiint_{r} \tau d\phi$$
H. $$\frac{1}{2U} \iiint_{r} \phi d\tau$$
I. $$\iiint_{r} \phi d\tau$$
J. $$U^2 \iiint_{r} \phi d\tau$$
Answer: A
In an initially stationary, ideally unbounded, incompressible homogeneous fluid, there is a balloon with a radius of $a$. At the initial moment, the internal pressure of the balloon is $p_{0}$, and the velocity at the surface of the balloon is zero. Neglecting the effects of body forces and surface tension, with the pressure at infinity being zero, the law governing the change in the balloon's radius over time under isothermal conditions is ( ) .

Options:
A. $$
t=\frac{1} {\sqrt{\frac{3 P_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{1 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
B. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
C. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} a^{2}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
D. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{2 / 3} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
E. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {b}}
$$
F. $$
t=\frac{1} {\sqrt{\frac{2 p_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\sqrt{\operatorname{l n} \frac{R} {a}}}
$$
G. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} a^{3}} {\sigma}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
H. $$
t=\frac{1} {\sqrt{\frac{2 P_{1} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
I. $$
t=\frac{1} {\sqrt{\frac{2 P_{0} b^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
J. $$
t=\frac{1} {\sqrt{\frac{4 P_{0} a^{3}} {\rho}}} \int_{a}^{R} \frac{R^{3 / 2} \mathrm{d} R} {\operatorname{l n} \frac{R} {a}}
$$
Answer: F
A homogeneous sphere with a radius of $a$ and a density of $\rho_1$ is placed in a stratified liquid. It is known that the density of the liquid at the surface is $\rho_0$. The distribution of the liquid's density is given by
$$
\rho = \rho_{\circ} \left( 1 - \beta z \right),
$$
where the $z$-axis is vertically upward, and the origin of the coordinate is at the liquid surface. $\beta$ is a known positive constant, and $\rho_{1} > \rho_{0} \left( 1+\beta a \right)$. At what position of the sphere's center can the equilibrium be achieved between the sphere and the surrounding liquid?

Options:
A. $$
\frac{\rho_{0}-\rho}{\beta+1},
$$
B. $$
\frac{\rho_{1}\beta}{\rho_{0}},
$$
C. $$
\frac{\rho_{0}a}{\rho} - \beta,
$$
D. $$
\frac{\rho\rho_{0}}{\beta a},
$$
E. $$
\frac{\rho+\rho_{0}}{\beta a},
$$
F. $$
\frac{\rho_{1}a}{\beta\rho_{0}},
$$
G. $$
\frac{a-\rho_{0}}{\beta\rho},
$$
H. $$
\frac{\rho_{1}}{\rho_{0}} - a\beta,
$$
I. $$
z = \frac{\rho-\rho_{0}} {\beta},
$$
J. $$
\rho_{0}\beta - \frac{\rho}{a},
$$
Answer: I
Given a horizontal circular pipe with a radius of $a$, the volumetric flow rate of the fluid is $Q$. Assuming the fluid has a viscosity coefficient of $u$ and the pipe length is $L$, with laminar flow within the pipe, give the velocity distribution of the fluid in the pipe.

Options:
A. $$
\frac{Q} {\pi a^{4}} \left( a^{2}-r \right)
$$
B. $$
\frac{2 Q} {\pi a^{2}} \left( a-r \right)
$$
C. $$
\frac{2 u} {\pi Q a^{4}} \left( a^{2}-r^{2} \right)
$$
D. $$
\frac{Q^{2}} {\pi a^{3}} \left( a^{2}+r^{2} \right)
$$
E. $$
\frac{Q} {2 \pi a^{3}} \left( a^{2}-r^{2} \right)
$$
F. $$
\frac{2 Q} {\pi a^{4}} \left( r^{2}-a^{2} \right)
$$
G. $$
\frac{2 \pi Q} {a^{4}} \left( a^{2}-u \right)
$$
H. $$
\frac{Q} {\pi a^{2}} \left( a^{4}-r^{4} \right)
$$
I. $$
\frac{2 Q u} {\pi a^{4}} \left( r^{2} \right)
$$
J. $$
u = \frac{2 Q} {\pi a^{4}} \left( a^{2}-r^{2} \right)
$$
Answer: J
Set the complex potential as
$$
w \left( z \right)=m \operatorname{ln} \left( z-\frac{1} {z} \right),
$$
give the volume of fluid passing through the line connecting the points $z=\mathrm{i}$ and $z=1 / 2$ per unit time.

Options:
A. $$
\ \ 3 m \frac{\pi} {2} $$
B. $$
\ \ \pi m $$
C. $$
\ \ \frac{\pi} {2} m
$$
D. $$
\ \ m \frac{\pi} {4} $$
E. $$
\ \ \frac{\pi} {3} m $$
F. $$
\ \ \pi \frac{m} {4} $$
G. $$
\ \ \frac{m^2} {2} \pi $$
H. $$
\ \ 2 m \pi $$
I. $$
\ \ \frac{\pi} {5} m $$
J. $$
\ \ \frac{2\pi} {3} m $$
Answer: C
In a rectangular water trough with a width of $a$ and a length of $b$, where the water depth is $h$, determine the possible standing wave frequencies of the fluid.

Options:
A. $$
\pi\sqrt{\frac{m^{2}} {a^{2}}+\frac{n^{2}} {b^{2}}}
$$
B. $$
\pi\sqrt{\frac{h^{2}} {a^{2}}+\frac{b^{2}} {n^{2}}}
$$
C. $$
\pi\sqrt{\frac{m^{2}} {b^{2}}+\frac{n^{2}} {a^{2}}}
$$
D. $$
\pi\sqrt{\frac{m^{2}} {a^{2}}+\frac{h^{2}} {b^{2}}}
$$
E. $$
\pi\sqrt{\frac{m^{2}} {a^{2}}-\frac{n^{2}} {b^{2}}}
$$
F. $$
\pi\sqrt{\frac{n^{2}} {a^{2}}+\frac{m^{2}} {b^{2}}}
$$
G. $$
\pi\sqrt{\frac{m^{2}} {h^{2}}+\frac{n^{2}} {a^{2}}}
$$
H. $$
\pi\sqrt{\frac{n^{2}} {b^{2}}+\frac{m^{2}} {h^{2}}}
$$
I. $$
\pi\sqrt{\frac{b^{2}} {m^{2}}+\frac{a^{2}} {n^{2}}}
$$
J. $$
\pi\sqrt{\frac{m^{2}} {b^{2}}-\frac{n^{2}} {a^{2}}}
$$
Answer: A
If the $\mathrm{C-C}$ bond length $l = 1.54 \times 10^{-10}$ m, then the root mean square end-to-end distance of a free jointed chain of polyethylene with a degree of polymerization of $1,000$ is: $\langle r^{2} \rangle^{1 / 2} = ( \qquad )$

Options:
A. $$
\mathrm{ 6. ~ 2 6 \times~ 1 0^{-9} ~ m }
$$
B. $$
\mathrm{ 6. ~ 8 9 \times~ 1 0^{-9} ~ m}
$$
C. $$
\mathrm{ 7. ~ 3 5 \times~ 1 0^{-9} ~ m }
$$
D. $$
\mathrm{ 6. ~ 3 7 \times~ 1 0^{-9} ~ m }
$$
E. $$
\mathrm{ 6. ~ 4 4 \times~ 1 0^{-9} ~ m }
$$
F. $$
\mathrm{ 7. ~ 5 1 \times~ 1 0^{-9} ~ m }
$$
G. $$
\mathrm{ 5. ~ 1 2 \times~ 1 0^{-9} ~ m }
$$
H. $$
\mathrm{ 5. ~ 7 3 \times~ 1 0^{-9} ~ m }
$$
I. $$
\mathrm{ 5. ~ 9 0 \times~ 1 0^{-9} ~ m }
$$
J. $$
\mathrm{ 7. ~ 9 8 \times~ 1 0^{-9} ~ m }
$$
Answer: B
Given that the degree of polymerization of the polyethylene sample is $5 \times 10^{4}$, with a $\mathrm{C-C}$ bond length of 0.154 nm and a bond angle of $109.5^{\circ}$, calculate the mean square end-to-end distance of the polyethylene sample when considered as a freely rotating chain is ().

Options:
A. $$
5 600. 1 nm^{2}
$$
B. $$
4 600. 7 nm^{2}
$$
C. $$
6 200. 4 nm^{2}
$$
D. $$
4 800. 5 nm^{2}
$$
E. $$
5 000. 2 nm^{2}
$$
F. $$
4 300. 8 nm^{2}
$$
G. $$
4 500. 3 nm^{2}
$$
H. $$
3 900. 9 nm^{2}
$$
I. $$
4 743. 2 nm^{2}
$$
J. $$
3 500. 0 nm^{2}
$$
Answer: I
Given that the crystal density of poly(ethylene oxide) is 1.33 g/cm³, its melting point is 80°C, and its heat of fusion is 8.29 kJ/mol (per monomer unit), with an interaction parameter with water of 0.45, what are the melting points when the water volume fractions are 0.01, 0.02, and 0.05, respectively?

Options:
A. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=349.8 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=347.9 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=343.5 \mathrm{~K}$
B. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=351.5 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=348.1 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=343.3 \mathrm{~K}$
C. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=351.3 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=349.6 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=342.5 \mathrm{~K}$
D. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=350.8 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=348.9 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=342.7 \mathrm{~K}$
E. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=351.0 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=348.7 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=343.9 \mathrm{~K}$
F. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=350.7 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=348.5 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=342.1 \mathrm{~K}$
G. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=351.2 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=349.0 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=341.8 \mathrm{~K}$
H. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=350.6 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=347.5 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=344.7 \mathrm{~K}$
I. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=349.9 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=348.2 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=344.0 \mathrm{~K}$
J. When $\varphi_{1}=0.01$, $T_{\mathrm{m}}=350.5 \mathrm{~K}$; when $\varphi_{1}=0.02$, $T_{\mathrm{m}}=349.1 \mathrm{~K}$; when $\varphi_{1}=0.05$, $T_{\mathrm{m}}=344.3 \mathrm{~K}$
Answer: F
During the actual process of mixing polymers with solvents, the change in the volume of the system is ( ).

Options:
A. $$
\Delta V_{\mathrm{M}} = 1
$$
B. $$
\Delta V_{\mathrm{M}} \ll 0
$$
C. $$
\Delta V_{\mathrm{M}} > 0
$$
D. $$
\Delta V_{\mathrm{M}} \approx 0
$$
E. $$
\Delta V_{\mathrm{M}} < 0
$$
F. $$
\Delta V_{\mathrm{M}} \neq 0
$$
G. $$
\Delta V_{\mathrm{M}} \geq 0
$$
H. $$
\Delta V_{\mathrm{M}} \leq 0
$$
I. $$
\Delta V_{\mathrm{M}} \gg 0
$$
J. $$
\Delta V_{\mathrm{M}} = 0
$$
Answer: E
Assuming a polymer sample contains three components with relative molecular masses of 10,000, 20,000, and 30,000, the number-average relative molecular mass $M_{\mathrm{n}}$ of the sample is measured to be 20,000, and the weight-average relative molecular mass $M_{\mathrm{w}}$ is 23,000. Try to calculate the mass fraction of each component in this sample is ().

Options:
A. $W_1=0.2,W_2=0.3,W_3=0.5$
B. $W_1=0.4,W_2=0.35,W_3=0.25$
C. $W_1=0.3,W_2=0.35,W_3=0.35$
D. $W_1=0.25,W_2=0.25,W_3=0.5$
E. $W_1=0.1,W_2=0.6,W_3=0.3$
F. $W_1=0.05,W_2=0.45,W_3=0.5$
G. $W_1=0.2,W_2=0.5,W_3=0.3$
H. $W_1=0.15,W_2=0.4,W_3=0.45$
I. $W_1=0.3,W_2=0.4,W_3=0.3$
J. $W_1=0.25,W_2=0.3,W_3=0.45$
Answer: H
At 30°C, a constant stress $\sigma_{0}=10^{7} \, \mathrm{N / m}^{2}$ is applied to the Voigt mechanical model. Assuming the viscosity $\eta$ follows the WLF equation, and it is known that when $T_{\mathrm{g}}=5^{\circ} \mathrm{C}$, $\eta_{\mathrm{5 C}}=4.9 \times 10^{16} \mathrm{~N \cdot~ s / m^{2} ~}$, calculate the total deformation after $10^{4} \mathrm{~s}$?

Options:
A. $$56.2\%$$
B. $$45.8\%$$
C. $$75.0\%$$
D. $$63.4\%$$
E. $$49.5\%$$
F. $$58.9\%$$
G. $$71.9\%$$
H. $$68.3\%$$
I. $$52.7\%$$
J. $$60.1\%$$
Answer: D
In the crystalline region of polytetrafluoroethylene, the conformation of its molecular chain is ().

Options:
A. Helix Chain
B. Curved Pattern
C. Bent Helix
D. Wavy Alignment
E. Tangle Structure
F. Twisted Form
G. Slanted Coil
H. Zigzag Chain
I. Linear Spiral
J. Coiled Sheet
Answer: A
In a $\theta$ solvent at $25^{\circ} \mathrm{C}$, the osmotic pressure of a polystyrene solution with a concentration of $1 \, \times \, 10^{-3} \, \mathrm{g / mL}$ is measured to be $0.5055 \ \mathrm{g} / \mathrm{cm}^{2}$. When the incident light is unpolarized and the internal interference effects are ignored, the Rayleigh factor at $90^{\circ}$ obtained by the light scattering method is $3 \times 10^{-5} / \mathrm{cm}$. The optical constant is known to be $K = 1 \times 10^{-6} \mathrm{~cm^{2} \cdot mol / g^{2}~}$. Calculate the polydispersity index of this polystyrene sample is ().

Options:
A. 1.1
B. 1.3
C. 1.4
D. 1.7
E. 1.9
F. 1.6
G. 1.5
H. 1.8
I. 1.2
J. 1.0
Answer: I
When a polymer dissolves in a good solvent, the excess chemical potential of the solvent $\Delta\mu_{1}^{\mathrm{E}} ().$

Options:
A. Significantly higher
B. Less than zero
C. Exactly zero
D. More than one
E. Double than zero
F. Equal to one
G. Greater than zero
H. Same as infinity
I. Beyond measure
J. Close to infinity
Answer: B
The better the flexibility of the polymer chain, the length of the segments of its equivalent freely jointed chain ().

Options:
A. The thicker
B. The harder
C. The heavier
D. The longer
E. The stronger
F. The taller
G. The larger
H. The broader
I. The wider
J. The smaller
Answer: J
When the relative molecular mass of a polymer is sufficiently large, the viscous flow activation energy of the polymer increases with the increase in relative molecular mass is ().

Options:
A. Fixating
B. Oscillating
C. Altering
D. Decreasing
E. Rising
F. Stabilizing
G. Dropping
H. Unchanging
I. Fluctuating
J. Lowering
Answer: H
When a certain polymer undergoes tensile stress, it exhibits brittle fracture, indicating ().

Options:
A. Thermal conductivity is compromised significantly.
B. Crystal lattice exhibits no dislocation movement.
C. Material has no permanent deformation.
D. Surface energy exceeds stored energy substantially.
E. Molecular weight is inadequate for use.
F. Viscous flow dominates under tensile load.
G. Tensile strength is lower than shear strength.
H. Polymer is semi-crystalline rather than amorphous.
I. Elastic limit exceeds yield limit greatly.
J. Cross-link density is excessively high.
Answer: G
In the rubber networks in practice, there are free chain segments, which causes their elastic modulus to be () than that of an ideal network.

Options:
A. Incremented
B. Elevated
C. Ample
D. Diminished
E. Higher
F. Enlarged
G. Lower
H. Enhanced
I. Strengthened
J. Reduced
Answer: G
At 18°C, the conductivity of a saturated aqueous solution of Ca$F_{\mathrm{2}}$ was measured to be $38.9 \times 10^{-6}$ S/cm, while the conductivity of water at 18°C is $1.5 \times 10^{-6}$ S/cm. It is also known that the limiting molar conductivities of various electrolytes in aqueous solution are: $\lambda_{\mathrm{m, \, 0}} \left( \mathrm{CaCl_{2}} \right) = 233.4 \, \mathrm{S \, cm^{2}/eq}$, $\lambda_{\mathrm{m, \, 0}} \left( \mathrm{NaCl} \right) = 108.9 \, \mathrm{S \, cm^{2}/eq}$, $\lambda_{\mathrm{m, \, 0}} \left( \mathrm{NaF} \right) = 90.2 \, \mathrm{S \, cm^{2}/eq}$. Assuming the hydrolysis of $\mathrm{F}^{-}$ ions can be neglected, the solubility product $\mathrm{K_{s}}$ of calcium fluoride at 18°C is().

Options:
A. $$
= 3. 3 \times1 0^{-1 1} ( \mathrm{m o l / d m}^{3} )^{3}
$$
B. $$
= 4. 8 \times1 0^{-1 0} ( \mathrm{m o l / d m}^{3} )^{3}
$$
C. $$
= 1. 9 \times1 0^{-1 2} ( \mathrm{m o l / d m}^{3} )^{3}
$$
D. $$
= 8. 2 \times1 0^{-1 2} ( \mathrm{m o l / d m}^{3} )^{3}
$$
E. $$
= 6. 5 \times1 0^{-1 0} ( \mathrm{m o l / d m}^{3} )^{3}
$$
F. $$
= 3. 6 \times1 0^{-1 0} ( \mathrm{m o l / d m}^{3} )^{3}
$$
G. $$
= 9. 1 \times1 0^{-1 0} ( \mathrm{m o l / d m}^{3} )^{3}
$$
H. $$
= 7. 4 \times1 0^{-1 1} ( \mathrm{m o l / d m}^{3} )^{3}
$$
I. $$
= 5. 2 \times1 0^{-1 2} ( \mathrm{m o l / d m}^{3} )^{3}
$$
J. $$
= 2. 7 \times1 0^{-1 1} ( \mathrm{m o l / d m}^{3} )^{3}
$$
Answer: J
In a certain dilute solution at 18°C, the molar conductivities of ions such as $\mathrm{H^{+}, K^{+}, Cl^{-}}$ are 278 Scm$^{2}$/mol, 48 Scm$^{2}$/mol, and 49 Scm$^{2}$/mol, respectively. At 18°C, in an electric field with a field strength of 10 V/cm, what is the average velocity at which each ion moves?

Options:
A. $$
V_{H^{+}}=0. 0 2 8 0 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 1 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 0 c m \cdot s^{-1}
$$
B. $$
V_{H^{+}}=0. 0 2 9 0 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 4 9 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 2 c m \cdot s^{-1}
$$
C. $$
V_{H^{+}}=0. 0 2 8 6 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 0 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 4 c m \cdot s^{-1}
$$
D. $$
V_{H^{+}}=0. 0 2 8 2 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 2 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 3 c m \cdot s^{-1}
$$
E. $$
V_{H^{+}}=0. 0 2 8 8 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 0 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 1 c m \cdot s^{-1}
$$
F. $$
V_{H^{+}}=0. 0 2 8 5 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 0 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 4 8 c m \cdot s^{-1}
$$
G. $$
V_{H^{+}}=0. 0 2 8 7 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 3 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 1 c m \cdot s^{-1}
$$
H. $$
V_{H^{+}}=0. 0 2 8 9 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 4 8 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 4 9 c m \cdot s^{-1}
$$
I. $$
V_{H^{+}}=0. 0 2 9 1 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 5 4 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 5 c m \cdot s^{-1}
$$
J. $$
V_{H^{+}}=0. 0 3 0 0 c m \cdot s^{-1} \,, \ \ V_{K^{+}}=0. 0 0 4 8 c m \cdot s^{-1} \,, \ \ V_{C I^{-}}=0. 0 0 5 0 c m \cdot s^{-1}
$$
Answer: E
At 25°C, the diffusion-controlled rate of a certain electrode reaction is $3 \times 10^{-2} \text{ mol/m}^2 \text{s}$. If the diffusion rate is increased 1000 times through vigorous stirring, the diffusion step can become a non-controlling step. How much should the diffusion activation energy be changed for this purpose?

Options:
A. The diffusion activation energy should be stabilized at 29000 J/mol.
B. The diffusion activation energy should be adjusted to 6000 J/mol.
C. The diffusion activation energy should remain unchanged at 5000 J/mol.
D. The diffusion activation energy should be increased to 17106 J/mol.
E. The diffusion activation energy should be increased to 34000 J/mol.
F. The diffusion activation energy should decrease to 11000 J/mol.
G. The diffusion activation energy should be reduced to 17106 J/mol.
H. The diffusion activation energy should be increased to 12000 J/mol.
I. The diffusion activation energy should increase to 8000 J/mol.
J. The diffusion activation energy should be constant at 25000 J/mol.
Answer: G
When you sit in front of a color television set (with a picture tube voltage of 25kV), you are exposed to $\mathbf{X}$-ray radiation. As a result of the continuous spectrum, what is the shortest wavelength (maximum energy) of the $\mathbf{X}$-ray ?

Options:
A. 0. 52( $^ {}\AA$)
B. 0. 45( $^ {}\AA$)
C. 0. 49( $^ {}\AA$)
D. 0. 44( $^ {}\AA$)
E. 0. 48( $^ {}\AA$)
F. 0. 51( $^ {}\AA$)
G. 0. 47( $^ {}\AA$)
H. 0. 53( $^ {}\AA$)
I. 0. 46( $^ {}\AA$)
J. 0. 50( $^ {}\AA$)
Answer: E
For a one-dimensional crystal, the electron energy band can be expressed as
$$
E ( k )=\frac{\hbar^{2}} {m a^{2}} \Big( \frac{7} {8}-\cos k a+\frac{1} {8} \cos 2 k a \Big)
$$
where $a$ is the lattice constant. What is the bandwidth ?

Options:
A. $\frac{3h^2}{m a}$
B. $\frac{7\hbar}{4a}$
C. $\frac{5h^{-2}}{m a^2}$
D. $\frac{h^2}{2m a}$
E. $\frac{8\hbar^{2}}{m a^2}$
F. $\frac{\hbar}{m a}$
G. $\frac{h}{8m a^{2}}$
H. $\frac{2\hbar^{2}}{ma}$
I. $\frac{\hbar^{2}}{4 m a}$
J. $\frac{5h}{4 m a}$
Answer: H
A 0.1 kg Ge single crystal, doped with $3.2 \times 10^{-9} \text{ kg}$ of Sb. Assuming all impurities are ionized, calculate the resistivity of this material as (). (given $\mu_{\mathrm{n}} = 0.39 \, \mathrm{m}^2/\mathrm{V} \cdot \mathrm{s}$, the density of Ge single crystal is 5.32 g/cm³, and the atomic mass of Sb is 121.8)

Options:
A. $$
8. 8 ( \Omega\mathrm{c m} )
$$
B. $$
6. 5 ( \Omega\mathrm{c m} )
$$
C. $$
3. 1 ( \Omega\mathrm{c m} )
$$
D. $$
7. 2 ( \Omega\mathrm{c m} )
$$
E. $$
2. 7 ( \Omega\mathrm{c m} )
$$
F. $$
9. 0 ( \Omega\mathrm{c m} )
$$
G. $$
1. 9 ( \Omega\mathrm{c m} )
$$
H. $$
2. 3 ( \Omega\mathrm{c m} )
$$
I. $$
4. 6 ( \Omega\mathrm{c m} )
$$
J. $$
5. 4 ( \Omega\mathrm{c m} )
$$
Answer: G
One important method for accurately measuring the lifetimes of multiply ionized atomic states is as follows: Use a laser beam tuned to the resonance wavelength of the ion under study to excite a given ion beam, and measure the radiation intensity of the ion beam as a function of its travel distance. When a specific ion beam with a resonance wavelength of 4885 Å is excited by an argon-ion laser with a wavelength of 4880 Å, the intensity decreases to half of its original value at a point 10 mm from the excitation point. If the velocity of the ion beam is given as $v / c = 5 \times 10^{-3}$, what is the lifetime of the ion's excited state when resonance is achieved ?

Options:
A. $$
\approx4. 7 \times1 0^{-7} ( \mathrm{s} )
$$
B. $$
\approx5. 9 \times1 0^{-6} ( \mathrm{s} )
$$
C. $$
\approx1. 1 \times1 0^{-8} ( \mathrm{s} )
$$
D. $$
\approx1. 2 \times1 0^{-6} ( \mathrm{s} )
$$
E. $$
\approx7. 4 \times1 0^{-7} ( \mathrm{s} )
$$
F. $$
\approx9. 6 \times1 0^{-9} ( \mathrm{s} )
$$
G. $$
\approx2. 8 \times1 0^{-8} ( \mathrm{s} )
$$
H. $$
\approx6. 5 \times1 0^{-9} ( \mathrm{s} )
$$
I. $$
\approx3. 5 \times1 0^{-7} ( \mathrm{s} )
$$
J. $$
\approx8. 3 \times1 0^{-8} ( \mathrm{s} )
$$
Answer: F
For an absolutely integrable function $g$ on $\mathbb{R}$, its Fourier transforms $\hat{g}$ and $\tilde{g}$ are defined as
$$
\hat{g} ( \xi)=\frac{1} {\sqrt{2 \pi}} \int_{\mathbb{R}} g ( x ) e^{-i x \xi} d x, \quad \tilde{g} ( \xi)=\frac{1} {\sqrt{2 \pi}} \int_{\mathbb{R}} g ( x ) e^{i x \xi} d x.
$$
Let $g ( x )=e^{-\lambda| x |}, \, \lambda> 0.$ What is the Fourier transform $H_{\lambda} ( \xi)$?

Options:
A. $$
{\sqrt{\frac{2} {\pi}}} {\frac{2\lambda} {\lambda^{2}+\xi^{2}}}
$$
B. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\lambda^{2}} {\lambda+\xi^{2}}}
$$
C. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\lambda} {\lambda^{2}-2\xi^{2}}}
$$
D. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\xi^{2}} {\lambda^{2}+\xi^{2}}}
$$
E. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\xi} {\lambda^{2}+\xi^{2}}}
$$
F. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\lambda} {\lambda^{2}+\xi^{2}}}
$$
G. $$
{\sqrt{\frac{\pi} {2}}} {\frac{\lambda} {\lambda^{2}-\xi^{2}}}
$$
H. $$
{\sqrt{\frac{\pi} {2}}} {\frac{\xi} {\lambda+\xi^{2}}}
$$
I. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\lambda - \xi} {\lambda^{2}+\xi^{2}}}
$$
J. $$
{\sqrt{\frac{2} {\pi}}} {\frac{\lambda} {\xi^{2}-\lambda^{2}}}
$$
Answer: F
For the galaxies in the Virgo cluster, the dispersion in the radial velocity is $\sigma_{r}=6 6 6$ km $\mathrm{s}^{-1}$ . What is the mass of the Virgo galaxy cluster estimated from the virial theorem?

Options:
A. $$
2. 8 \times1 0^{1 3} \mathrm{~ M}_{\odot}
$$
B. $$
9. 9 \times1 0^{1 3} \mathrm{~ M}_{\odot}
$$
C. $$
5. 5 \times1 0^{1 3} \mathrm{~ M}_{\odot}
$$
D. $$
3. 3 \times1 0^{1 4} \mathrm{~ M}_{\odot}
$$
E. $$
7. 7 \times1 0^{1 4} \mathrm{~ M}_{\odot}
$$
F. $$
4. 3 \times1 0^{1 4} \mathrm{~ M}_{\odot}
$$
G. $$
1. 1 \times1 0^{1 5} \mathrm{~ M}_{\odot}
$$
H. $$
6. 6 \times1 0^{1 5} \mathrm{~ M}_{\odot}
$$
I. $$
8. 2 \times1 0^{1 4} \mathrm{~ M}_{\odot}
$$
J. $$
6. 0 \times1 0^{1 5} \mathrm{~ M}_{\odot}
$$
Answer: E
Two images of the quasar $\mathrm{Q 0} 1 4 2-1 0 0$ (also known as UM 673) are formed by a gravitational lens. The redshift of the quasar is $z=2. 7 2 7$ , and the redshift of the imaging galaxy is $z=0. 4 9 3$ . The images are separated by $2. 2 2^{\prime\prime}$ , and the lensing galaxy is along the line between the two images, $0. 8^{\prime\prime}$ from one of them. Estimate what is the mass of the lensing galaxy ?

Options:
A. $$
5. 1 \times1 0^{1 0} h^{-1} \ \mathrm{M}_{\odot}
$$
B. $$
2. 8 \times1 0^{1 1} h^{-1} \ \mathrm{M}_{\odot}
$$
C. $$
2. 2 \times1 0^{1 3} h^{-1} \ \mathrm{M}_{\odot}
$$
D. $$
3. 1 \times1 0^{1 2} h^{-1} \ \mathrm{M}_{\odot}
$$
E. $$
4. 0 \times1 0^{1 2} h^{-1} \ \mathrm{M}_{\odot}
$$
F. $$
6. 3 \times1 0^{1 2} h^{-1} \ \mathrm{M}_{\odot}
$$
G. $$
1. 5 \times1 0^{1 0} h^{-1} \ \mathrm{M}_{\odot}
$$
H. $$
1. 2 \times1 0^{1 1} h^{-1} \ \mathrm{M}_{\odot}
$$
I. $$
7. 5 \times1 0^{1 1} h^{-1} \ \mathrm{M}_{\odot}
$$
J. $$
8. 9 \times1 0^{1 0} h^{-1} \ \mathrm{M}_{\odot}
$$
Answer: B
Suppose that Earth were a perfectly smooth sphere. If you drew a circle of radius $D=1 0 0 \, \mathrm{m e}$ ters on Earth's surface, what discrepancy would you find between the expected and measured values of the circle's circumference?

Options:
A. $$
9 5 . 4 \mathrm{~ n m}
$$
B. $$
2 2 . 0 \mathrm{~ n m}
$$
C. $$
1 8 . 9 \mathrm{~ n m}
$$
D. $$
1 2 . 5 \mathrm{~ n m}
$$
E. $$
7 1 . 6 \mathrm{~ n m}
$$
F. $$
2 7 . 3 \mathrm{~ n m}
$$
G. $$
2 5. 7 \mathrm{~ n m}
$$
H. $$
3 0 . 2 \mathrm{~ n m}
$$
I. $$
6 8 . 1 \mathrm{~ n m}
$$
J. $$
4 5 . 8 \mathrm{~ n m}
$$
Answer: G
Assume that the original cloud had a mass of 1 $\mathbf{M_{\odot}}$ and an initial radius of 0.5 pc. If collapse is halted at approximately 100 AU, find the initial angular velocity of the cloud?

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