MikaStars39/nano-eval huggingface dataset:
Using WMAP values for a universe of matter and radiation, estimate the time interval $\Delta t$ between when recombination began (say, when 99% of the hydrogen atoms were ionized) and when recombination ended (say, when 1% of the hydrogen atoms were ionized). What is the difference $\Delta z$ between the values of the redshift at these two times?

Options:
A. $$
2 0 0 \ \mathrm{k y r}
$$
B. $$
1 7 2 \ \mathrm{k y r}
$$
C. $$
1 5 0 \ \mathrm{k y r}
$$
D. $$
1 5 1 \ \mathrm{k y r}
$$
E. $$
1 4 5 \ \mathrm{k y r}
$$
F. $$
1 6 0 \ \mathrm{k y r}
$$
G. $$
2 3 8 \ \mathrm{k y r}
$$
H. $$
1 4 9 \ \mathrm{k y r}
$$
I. $$
1 3 8 \ \mathrm{k y r}
$$
J. $$
1 5 3 \ \mathrm{k y r}
$$
Answer: D
Consider material that is ejected from a quasar directly toward Earth.
If the redshift of the quasar is $z_{Q}$ and the redshift of the ejecta is $z_{\mathrm{e j}}$ , show that the speed of the ejecta relative to the quasar is given by as().

Options:
A. $$
\frac{v} {c}=\frac{( z_{Q} )-( z_{\mathrm{e j}} )} {( z_{Q} )+( z_{\mathrm{e j}} )}.
$$
B. $$
\frac{v} {c}=\frac{( 1+z_{Q} )^{2}+( 1+z_{\mathrm{e j}} )^{2}} {( 1+z_{Q} )^{2}-( 1+z_{\mathrm{e j}} )^{2}}.
$$
C. $$
\frac{v} {c}=\frac{( 1+z_{\mathrm{e j}} )^{2}-( z_{Q} )} {( 1+z_{\mathrm{e j}} )^{2}+( z_{Q} )}.
$$
D. $$
\frac{v} {c}=\frac{( 1+z_{Q} )^{3}-( 1+z_{\mathrm{e j}} )^{3}} {( 1+z_{Q} )^{3}+( 1+z_{\mathrm{e j}} )^{3}}.
$$
E. $$
\frac{v} {c}=\frac{( 1+z_{Q} )-( 1+z_{\mathrm{e j}} )} {( 1+z_{Q} )+( 1+z_{\mathrm{e j}} )}.
$$
F. $$
\frac{v} {c}=\frac{( 1+z_{\mathrm{e j}} )-( 1+z_{Q} )^{2}} {( 1+z_{Q} )^{2}+( 1+z_{\mathrm{e j}} )}.
$$
G. $$
\frac{v} {c}=\frac{( 1+z_{\mathrm{e j}} )^{2}-( 1+z_{Q} )^{2}} {( 1+z_{\mathrm{e j}} )^{2}+( 1+z_{Q} )^{2}}.
$$
H. $$
\frac{v} {c}=\frac{( 1+z_{Q} )^{2}-( z_{\mathrm{e j}} )^{3}} {( 1+z_{Q} )^{2}+( z_{\mathrm{e j}} )^{3}}.
$$
I. $$
\frac{v} {c}=\frac{( z_{Q} )^{2}-( z_{\mathrm{e j}} )^{2}} {( z_{Q} )^{2}+( z_{\mathrm{e j}} )^{2}}.
$$
J. $$
\frac{v} {c}=\frac{( 1+z_{Q} )^{2}-( 1+z_{\mathrm{e j}} )^{2}} {( 1+z_{Q} )^{2}+( 1+z_{\mathrm{e j}} )^{2}}.
$$
Answer: J
Assuming a constant density of stars out to at least $V=1 1$ , how much interstellar extinction is present to that limit (express your answer in magnitudes)?

Options:
A. $$
a_{V}=0.5
$$
B. $$
a_{V}=1
$$
C. $$
a_{V}=0.1
$$
D. $$
a_{V}=0.2
$$
E. $$
a_{V}=0.05
$$
F. $$
a_{V}=3
$$
G. $$
a_{V}=5
$$
H. $$
a_{V}=0
$$
I. $$
a_{V}=2
$$
J. $$
a_{V}=4
$$
Answer: H
Compute the lowest possible density of Sgr $\mathrm{A}^{\star}$ based on the data obtained from the orbit of S2. Assume a spherically symmetric mass distribution.

Options:
A. $$
4 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
B. $$
8.9 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
C. $$
5 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
D. $$
2 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
E. $$
9.3 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
F. $$
3 \times1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
G. $$
1.5 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
H. $$
1 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
I. $$
7.2 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
J. $$
6 \times 1 0^{-4} \mathrm{~ k g ~ m}^{-3}
$$
Answer: F
NGC 3311 is a cD galaxy with an estimated 17,000 globular clusters and an absolute visual magnitude of $- 2 2. 4$ . Estimate the specific frequency of clusters in this galaxy.

Options:
A. 17.3
B. 21.4
C. 12.4
D. 13.9
E. 18.6
F. 20.7
G. 16.5
H. 15.8
I. 22.1
J. 19.2
Answer: E
M87 is a giant elliptical galaxy near the center of the Virgo cluster of galaxies. It is estimated that its mass is approximately $3 \, \times\, 1 0^{1 3} \mathrm{~ M}_{\odot}$ within a radius of 300 kpc. How long would it take for a star near the outer edge of the galaxy to orbit the center once?

Options:
A. $$
4. 5 \mathrm{~ G y r}
$$
B. $$
6. 3 \mathrm{~ G y r}
$$
C. $$
7. 2 \mathrm{~ G y r}
$$
D. $$
4. 2 \mathrm{~ G y r}
$$
E. $$
7. 8 \mathrm{~ G y r}
$$
F. $$
5. 6 \mathrm{~ G y r}
$$
G. $$
6. 9 \mathrm{~ G y r}
$$
H. $$
3. 9 \mathrm{~ G y r}
$$
I. $$
4. 8 \mathrm{~ G y r}
$$
J. $$
8. 1 \mathrm{~ G y r}
$$
Answer: F
If the register content is 11111111, if it equals +127, then it is ().

Options:
A. Address resolution
B. Cache replacement
C. Memory mapping
D. Logic synthesis
E. Instruction cycle
F. System integration
G. Power sequencing
H. Data encryption
I. Code migration
J. Hardware abstraction
Answer: I
The binary complements $[ X ]_{complement}=0.1011$ and $[ Y ]_{complement}=1.1110$ are given. What are the values of $[ X+Y ]_{complement}$ and $[ X-Y ]_{complement}$?

Options:
A. 1.0010, 0.1111
B. 0.1000, 0.1100
C. 0.0110,0.1010
D. 0.0001, 0.0111
E. 1.0100, 1.1101
F. 0.1010, 0.1110
G. 0.1111,1.0010
H. 0.1100, 1.0101
I. 1.0000, 0.0001
J. 0.1001,0.1101
Answer: J
Given that $$f(n) = \sum_{i=0}^{n} 2^{i} = 2^{n+1} - 1 = \overbrace{1 1 \cdots 1 1}^{n+1 \text{ digits}}$$, compute the C language function $f1$ of $f(n)$ as follows:
```c
int f1(unsigned n){
int sum = 1, power = 1;
for (unsigned i = 0, i < n - 1; i++) {
power *= 2;
sum += power;
}
return sum;
}
```
If all `int` in `f1` are changed to `float`, another function `f2` for computing $f(n)$ can be obtained. Assume both `unsigned` and `int` data types occupy 32 bits, and `float` uses the IEEE 754 single-precision standard. The machine number of `f2(127)` is 7F80 0000H. What value does this correspond to? What is the largest $\mathbf{n}$ such that the result of $\mathrm{f2}(n)$ does not overflow? What is the largest $\mathbf{n}$ for which the result of $f2(n)$ is exact (without rounding)?

Options:
A. $$
0,125,22
$$
B. $$
NaN,125,23
$$
C. $$
0,126,23
$$
D. $$
+\infty,124,21
$$
E. $$
+\infty,127,22
$$
F. $$
+\infty,126,22
$$
G. $$
0,128,25
$$
H. $$
-0,127,24
$$
I. $$
+\infty,126,23
$$
J. $$
NaN,127,24
$$
Answer: I
A certain DRAM chip uses address multiplexing technology, with a capacity of 1024×8 bits. Apart from the power supply and ground pins, the minimum number of pins for this chip is (). (There are two read/write control lines)

Options:
A. 21
B. 18
C. 14
D. 16
E. 15
F. 19
G. 20
H. 23
I. 17
J. 22
Answer: I
There are 32 pieces of 256K×1-bit SRAM chips. Using a 32-bit word addressing scheme, how many address lines does this memory require?

Options:
A. 23
B. 25
C. 22
D. 19
E. 18
F. 21
G. 20
H. 17
I. 26
J. 24
Answer: E
A certain computer uses byte addressing, with a fixed instruction word length and only two types of instruction formats. There are 29 three-address instructions and 107 two-address instructions, with each address field being 6 bits. The instruction word length should be at least ().

Options:
A. 65 bits
B. 35 bits
C. 40 bits
D. 54 bits
E. 48 bits
F. 60 bits
G. 18 bits
H. 30 bits
I. 24 bits
J. 72 bits
Answer: I
Assume the instruction pipeline divides an instruction into three parts: fetching, decoding, and executing, with their respective times being $t_{\mathrm{fetch}}=$ 2 ns, $t_{\mathrm{decode}}=2$ ns, and $t_{\mathrm{execute}}=1$ ns. Then, the time required to complete the execution of 100 instructions is ( ).

Options:
A. 205ns
B. 203ns
C. 200ns
D. 212ns
E. 208ns
F. 204ns
G. 206ns
H. 202ns
I. 215ns
J. 210ns
Answer: B
The CPU clock speed of a certain computer is 500 MHz, with a CPI of 5 (i.e., each instruction requires an average of 5 clock cycles to execute). Suppose a peripheral device has a data transfer rate of 5 MB/s and uses an interrupt method to transfer data with the host, using 32 bits as the transmission unit. The corresponding interrupt service routine includes 18 instructions, and other overheads of interrupt servicing are equivalent to the execution time of 2 instructions.
Under the interrupt method, what percentage of the CPU time is used for this peripheral device's $I/O$ operations in relation to the entire CPU time?

Options:
A. $$
0. 5 \%
$$
B. $$
8. 0 \%
$$
C. $$
9. 0 \%
$$
D. $$
4. 0 \%
$$
E. $$
5. 0 \%
$$
F. $$
1. 0 \%
$$
G. $$
3. 5 \%
$$
H. $$
2. 5 \%
$$
I. $$
7. 0 \%
$$
J. $$
6. 5 \%
$$
Answer: H
Assume there is a communication protocol where each packet introduces an overhead of 100 bytes for headers and framing. Now, using this protocol to send $1 0^{6}$ bytes of data, however, one byte is corrupted during the transmission, and hence the packet containing that byte is discarded. Try to calculate the total number of "overhead + lost" bytes for effective data sizes of 1000 bytes and 20000 bytes per packet, respectively. What is the optimal packet data size?

Options:
A. 15000B
B. 5000B
C. 18000B
D. 9500B
E. 10000B
F. 7000B
G. 2500B
H. 12000B
I. 3000B
J. 8500B
Answer: E
A channel samples once every 1/8 second, and the transmission signal has a total of 16 different states. The maximum data transmission rate is ().

Options:
A. 48b/s
B. 64b/s
C. 96b/s
D. 12b/s
E. 4b/s
F. 8b/s
G. 128b/s
H. 16b/s
I. 32b/s
J. 24b/s
Answer: I
The data link layer employs the Go-Back-$N$ protocol for flow control and error control. The sender has already transmitted frames numbered $0-6$. Upon timer expiration, only acknowledgments for frames 1, 3, and 5 have been received. The number of frames the sender needs to retransmit is ( ).

Options:
A. 10
B. 6
C. 9
D. 3
E. 2
F. 5
G. 1
H. 4
I. 8
J. 7
Answer: G
Between two hosts, the data link layer uses the Go-Back-N (GBN) protocol for data transmission. The data transmission rate is 16 kb/s, with a one-way propagation delay of 270 ms. The data frame length ranges from 128 to 512 bytes, and the receiver always acknowledges with frames of equal length to the data frames. To maximize channel utilization, the minimum number of bits required for the frame sequence number is ().

Options:
A. 5
B. 10
C. 11
D. 12
E. 3
F. 6
G. 4
H. 7
I. 9
J. 8
Answer: G
Three stations $\mathrm{A, ~ B, ~ C}$ are connected to a broadcast channel, and the medium access control uses a channel division method. The channel is divided using code division multiplexing technology. A and B need to send data to C. Assume A's code sequence is $+1, -1, -1, +1, +1, +1, +1, -1.$ The available chip sequence for station $\mathbf{B}$ is ().

Options:
A. $$
+ 1,-1,+1,-1,+1,-1,-1,+1
$$
B. $$
+ 1,+1,-1,+1,+1,-1,-1,+1
$$
C. $$
+ 1,+1,-1,-1,-1,+1,+1,-1
$$
D. $$
- 1,+1,-1,+1,-1,+1,+1,+1
$$
E. $$
- 1,+1,-1,-1,+1,+1,+1,-1
$$
F. $$
- 1,-1,+1,+1,-1,+1,-1,+1
$$
G. $$
+ 1,+1,+1,-1,+1,-1,+1,-1
$$
H. $$
+ 1,-1,-1,+1,-1,-1,+1,+1
$$
I. $$
- 1,+1,+1,-1,+1,+1,-1,-1
$$
J. $$
- 1,-1,+1,-1,+1,+1,-1,+1
$$
Answer: D
Consider establishing a CSMA/CD network with a cable length of 1 km, without using repeaters, and an operating rate of 1 Gb/s. The signal propagation rate in the cable is 200,000 km/s. What is the minimum frame length?

Options:
A. $$
5 0 0 0 0 \mathrm{b i t }
$$
B. $$
6 0 0 0 0 \mathrm{b i t }
$$
C. $$
9 0 0 0 0 \mathrm{b i t }
$$
D. $$
4 0 0 0 0 \mathrm{b i t }
$$
E. $$
7 0 0 0 0 \mathrm{b i t }
$$
F. $$
1 0 0 0 0 \mathrm{b i t }
$$
G. $$
2 0 0 0 0 \mathrm{b i t }
$$
H. $$
8 0 0 0 0 \mathrm{b i t }
$$
I. $$
3 0 0 0 0 \mathrm{b i t }
$$
J. $$
2 5 0 0 0 \mathrm{b i t }
$$
Answer: F
For a 10Mb/s shared Ethernet connected by a switch, if there are 10 users, the bandwidth available to each user is ().

Options:
A. 5Mb/s
B. 100Mb/s
C. 10Mb/s
D. 1Mb/s
E. 0.5Mb/s
F. 12Mb/s
G. 25Mb/s
H. 20Mb/s
I. 2Mb/s
J. 100Kb/s
Answer: C
Which of the following addresses is a unicast address?

Options:
A. $$
1 9 2. 1 6 8. 0 . 0 / 2 4
$$
B. $$
1 2 7. 0 . 0 . 1 / 8
$$
C. $$
2 5 5. 2 5 5. 2 5 5. 2 5 5 / 3 2
$$
D. $$
F F 0 0 : : 0 0 0 1 / 1 2 8
$$
E. $$
2 4 0 . 0 . 0 . 1 / 3 2
$$
F. $$
1 6 9. 2 5 4. 0 . 0 / 1 6
$$
G. $$
F E 8 0 : : / 6 4
$$
H. $$
2 2 4. 0 . 0 . 0 / 3 6
$$
I. $$
1 7 2. 3 1. 1 2 8. 2 5 5 / 1 8
$$
J. $$
2 2 7. 1 . 0 . 1 / 2 5
$$
Answer: I
If the 101.200.16.0/20 network is divided into 5 subnets, the smallest possible subnet has an allocatable IP address count of ().

Options:
A. 14
B. 2046
C. 30
D. 126
E. 510
F. 254
G. 6
H. 62
I. 511
J. 1022
Answer: F
A Lagrangian gyroscope with a mass of $m$ and a distance $l$ from the center of mass to the fixed point. The conditions for stable rotation of the gyroscope axis about the vertical direction is ().

Options:
A. $$
p_{\psi}^{2} \geq 6 I_{2} m g l \Rightarrow\omega_{z}^{2} \geq \frac{6 I_{2} m g l} {I_{3}^{2}}
$$
B. $$
p_{\psi}^{2} = I_{3} m g l \Rightarrow\omega_{z}^{2} = \frac{I_{3} m g l} {I_{1} I_{2}}
$$
C. $$
p_{\psi}^{2} \leq 5 I_{3} m g l \Rightarrow\omega_{z}^{2} \leq \frac{5 I_{3} m g l} {I_{2}^{2}}
$$
D. $$
p_{\psi}^{2} < 2 I_{3} m g l \Rightarrow\omega_{z}^{2} < \frac{2 I_{3} m g l} {I_{1}^{2}}
$$
E. $$
p_{\psi}^{2} = 2 I_{3} m g l \Rightarrow\omega_{z}^{2} = \frac{2 I_{3} m g l} {I_{1}^{2}}
$$
F. $$
p_{\psi}^{2} < 4 I_{1} m g l \Rightarrow\omega_{z}^{2} < \frac{4 I_{1} m g l} {I_{3}^{2}}
$$
G. $$
p_{\psi}^{2} > 3 I_{2} m g l \Rightarrow\omega_{z}^{2} > \frac{3 I_{2} m g l} {I_{3}^{2}}
$$
H. $$
p_{\psi}^{2} > 4 I_{1} m g l \Rightarrow\omega_{z}^{2} > \frac{4 I_{1} m g l} {I_{3}^{2}}
$$
I. $$
p_{\psi}^{2} < I_{1} m g l \Rightarrow\omega_{z}^{2} < \frac{I_{1} m g l} {4 I_{3}^{2}}
$$
J. $$
p_{\psi}^{2} = 3 I_{2} m g l \Rightarrow\omega_{z}^{2} = \frac{3 I_{2} m g l} {I_{1} I_{3}}
$$
Answer: H
The inductor $L$ is connected in series with the resistor $R_{1}$, and then in parallel with the capacitor $C$. They are powered by a simple harmonic source with an internal resistance of $R_{2}$ and a frequency of $\omega$. What is the differential equation for the circuit?

Options:
A. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}+R_{1} \dot{e}_{1}-\frac{e_{2}-e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{2}+\frac{e_{2}-e_{1}} {C}=A \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
B. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}-R_{1} \dot{e}_{2}+\frac{e_{2}-e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{2}+\frac{e_{2}+e_{1}} {C}=A \operatorname{c o s} \omega t}} \\ \end{array} \right.
$$
C. $$
\left\{\begin{array} {c} {{L \ddot{e}_{2}-R_{1} \dot{e}_{1}+\frac{e_{2}-e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{1}+\frac{e_{1}-e_{2}} {C}=B \operatorname{c o s} \omega t}} \\ \end{array} \right.
$$
D. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}+R_{1} \dot{e}_{2}-\frac{e_{1}+e_{2}} {C}=0}} \\ {{R_{2} \dot{e}_{2}-\frac{e_{2}-e_{1}} {C}=B \operatorname{c o s} \omega t}} \\ \end{array} \right.
$$
E. $$
\left\{\begin{array} {c} {{L \ddot{e}_{2}+R_{1} \dot{e}_{2}-\frac{e_{2}-e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{2}+\frac{e_{2}+e_{1}} {C}=B \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
F. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}+R_{2} \dot{e}_{1}-\frac{e_{2}+e_{1}} {C}=A}} \\ {{R_{1} \dot{e}_{2}+\frac{e_{2}-e_{1}} {C}=A \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
G. $$
\left\{\begin{array} {c} {{L \ddot{e}_{2}+R_{1} \dot{e}_{1}-\frac{e_{2}+e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{1}+\frac{e_{2}-e_{1}} {C}=B \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
H. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}-R_{1} \dot{e}_{1}+\frac{e_{2}-e_{1}} {C}=0}} \\ {{R_{2} \dot{e}_{1}+\frac{e_{2}+e_{1}} {C}=A \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
I. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}+R_{1} \dot{e}_{1}+\frac{e_{1}-e_{2}} {C}=0}} \\ {{R_{1} \dot{e}_{2}+\frac{e_{2}-e_{1}} {C}=A \operatorname{c o s} \omega t}} \\ \end{array} \right.
$$
J. $$
\left\{\begin{array} {c} {{L \ddot{e}_{1}+R_{2} \dot{e}_{1}+\frac{e_{2}-e_{1}} {C}=B}} \\ {{R_{2} \dot{e}_{2}-\frac{e_{2}+e_{1}} {C}=A \operatorname{s i n} \omega t}} \\ \end{array} \right.
$$
Answer: A
Alpha Centauri is 4.31 light-years away from Earth. One of the twin brothers, $A$, travels to the star in a spaceship moving at 0.8c, and during his journey there and back, he sends a radio signal every 0.01 years (in the spaceship's rest frame). The other brother, $B$, who stays on Earth, also sends a radio signal every 0.01 years (in Earth's rest frame). How many signals do $A$ and $B$ each receive from the other?

Options:
A. $A$ received 1,075 signals, and $B$ received 645 signals.
B. $A$ received 1,150 signals, and $B$ received 680 signals.
C. $A$ received 1,025 signals, and $B$ received 700 signals.
D. $A$ received 1,200 signals, and $B$ received 900 signals.
E. $A$ received 1,100 signals, and $B$ received 600 signals.
F. $A$ received 950 signals, and $B$ received 785 signals.
G. $A$ received 1,300 signals, and $B$ received 540 signals.
H. $A$ received 1,000 signals, and $B$ received 650 signals.
I. $A$ received 1,050 signals, and $B$ received 670 signals.
J. $A$ received 1,020 signals, and $B$ received 645 signals.
Answer: A
An electron (rest mass $9.11 \times 10^{-31} \mathrm{kg}$) is moving at a speed of $0.99c$. What is the ratio of the kinetic energy calculated using Newtonian mechanics to that calculated using relativistic mechanics?

Options:
A. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=1.10
$$
B. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=0.08
$$
C. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=0.65
$$
D. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=2.02
$$
E. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=1.22
$$
F. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=0.33
$$
G. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=2.25
$$
H. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=0.50
$$
I. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=1.55
$$
J. $$
\frac{E_k(\text{Newtonian mechanics})}{E_k(\text{relativistic mechanics})}=0.75
$$
Answer: B
Two simple harmonic oscillations with the same direction and frequency have a resultant amplitude of 10 cm, with the phase of the resultant vibration lagging behind the phase of the first vibration by 30°. If the amplitude of the first vibration is $A_{1}=8.0 \text{ cm}$, what are the amplitude $A_{2}$ of the second vibration and the phase difference between the first and the second oscillations?

Options:
A. $A_2\approx2.0\mathrm{cm}$
The phase difference is $180°$.
B. $A_2\approx1.0\mathrm{cm}$
The phase difference is $210°$.
C. $A_2\approx7.0\mathrm{cm}$
The phase difference is $60°$.
D. $A_2\approx8.0\mathrm{cm}$
The phase difference is $90°$.
E. $A_2\approx0.0\mathrm{cm}$
The phase difference is $360°$.
F. $A_2\approx6.0\mathrm{cm}$
The phase difference is $30°$.
G. $A_2\approx3.0\mathrm{cm}$
The phase difference is $150°$.
H. $A_2\approx4.0\mathrm{cm}$
The phase difference is $120°$.
I. $A_2\approx9.0\mathrm{cm}$
The phase difference is $15°$.
J. $A_2\approx5.0\mathrm{cm}$
The phase difference is $45°$.
Answer: F
Find
$$
E_{n} = \left\{m / n \, ; \; m \in \mathbf{Z} \right\} ( n = 1, 2, \cdots).
$$
What are the upper and lower limit sets of the sequence of sets?

Options:
A. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{N}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{Z} .
$$
B. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{R}, \varliminf_{n \rightarrow \infty} E_{n}=\emptyset .
$$
C. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{Q}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{Z} .
$$
D. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{N}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{C} .
$$
E. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{C}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{R} .
$$
F. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{C}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{Z} .
$$
G. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{Z}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{Q} .
$$
H. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{R}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{N} .
$$
I. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\mathbf{Z}, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{R} .
$$
J. $$
\varlimsup_{n \rightarrow \infty} E_{n}=\emptyset, \varliminf_{n \rightarrow \infty} E_{n}=\mathbf{Q} .
$$
Answer: C
Expanding numbers in the interval (0,1) into decimal form, consider the set of all points where the digit at a specified decimal place is 4, denoted as $E$. Find the measure $m(E)$ of the set of points $E$.

Options:
A. $$
m ( E )=1 / 2 0
$$
B. $$
m ( E )=4 / 1 0
$$
C. $$
m ( E )=5 / 1 0
$$
D. $$
m ( E )=1 / 5 0
$$
E. $$
m ( E )=3 / 1 0
$$
F. $$
m ( E )=1 / 1 5
$$
G. $$
m ( E )=1 / 1 1
$$
H. $$
m ( E )=0 / 1 0
$$
I. $$
m ( E )=2 / 1 0
$$
J. $$
m ( E )=1 / 1 0
$$
Answer: J
Find the value of the integral
$$ I=\! \int_{0}^{1} \frac{x^{m-1}} {1+x^{n}} \mathrm{d} x \
$$.

Options:
A. $$
I= \frac{1} {n} \,-\, \frac{1} {n \,+\, m} \,+\, \frac{2} {n \,+\, 2 m}-\frac{2} {n \,+\, 3 m} \,+\, \cdots
$$
B. $$
I= \frac{1} {m} \,-\, \frac{2} {m \,+\, n} \,+\, \frac{3} {m \,+\, 2 n}-\frac{4} {m \,+\, 3 n} \,+\, \cdots
$$
C. $$
I= \frac{1} {n} \,-\, \frac{1} {n \,+\, m} \,+\, \frac{1} {n \,+\, 2 m}-\frac{1} {n \,+\, 3 m} \,+\, \cdots
$$
D. $$
I= \frac{1} {m} \,-\, \frac{1} {m \,+\, 3n} \,+\, \frac{1} {m \,+\, 5n}-\frac{1} {m \,+\, 7n} \,+\, \cdots
$$
E. $$
I= \frac{1} {m} \,-\, \frac{1} {m \,+\, n} \,+\, \frac{1} {m \,+\, 2 n}-\frac{1} {m \,+\, 3 n} \,+\, \cdots
$$
F. $$
I= \frac{1} {m \,+\, n} \,-\, \frac{1} {m} \,+\, \frac{1} {n}-\frac{1} {m \,+\, 2n} \,+\, \cdots
$$
G. $$
I= \frac{1} {m \,+\, n} \,-\, \frac{1} {m \,+\, 2n} \,+\, \frac{1} {m \,+\, 3n}-\frac{1} {m \,+\, 4n} \,+\, \cdots
$$
H. $$
I= \frac{1} {m^{2}} \,-\, \frac{1} {m^{2} \,+\, n} \,+\, \frac{1} {m^{2} \,+\, 2 n}-\frac{1} {m^{2} \,+\, 3 n} \,+\, \cdots
$$
I. $$
I= \frac{1} {2m} \,-\, \frac{1} {2m \,+\, n} \,+\, \frac{1} {2m \,+\, 2 n}-\frac{1} {2m \,+\, 3 n} \,+\, \cdots
$$
J. $$
I= \frac{1} {n} \,-\, \frac{1} {n \,+\, 2m} \,+\, \frac{1} {n \,+\, 3m}-\frac{1} {n \,+\, 4m} \,+\, \cdots
$$
Answer: E
Let $E_{k} \subset [0, 1] \, (k=1, 2, \ldots, N)$ be measurable sets, where the union of any three of them is $[0, 1]$. Define $S_{1}=\sum_{k=1}^{N} m(E_{k})$, and
$$
S_{2}=\sum_{j=1}^{N} \sum_{k=1}^{N} m(E_{j} \cap E_{k}), \quad S_{3}=\sum_{1 \leq i < j < k \leq N} m(E_{i} \cap E_{j} \cap E_{k}),
$$
what is the value of $I$?

Options:
A. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=S_{1}+\, 3 S_{2}+\, 3 S_{3}
$$
B. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=4 S_{1}+\, 4 S_{2}+\, 2 S_{3}
$$
C. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=2 S_{1}+\, 6 S_{2}+\, 2 S_{3}
$$
D. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=S_{1}+\, 6 S_{2}+\, 6 S_{3}
$$
E. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=S_{1}+\, 4 S_{2}+\, 4 S_{3}
$$
F. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=2 S_{1}+\, 4 S_{2}+\, 2 S_{3}
$$
G. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=3 S_{1}+\, 5 S_{2}+\, 3 S_{3}
$$
H. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=3 S_{1}+\, 3 S_{2}+\, 6 S_{3}
$$
I. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=2 S_{1}+\, 2 S_{2}+\, 9 S_{3}
$$
J. $$
\int_{0}^{1} \bigl( \sum_{k=1}^{N} \chi_{E_{k}} ( x ) \bigr)^{3} \mathrm{d} x=S_{1}+\, 2 S_{2}+\, 8 S_{3}
$$
Answer: D
Find the value of $I=\int_{0}^{1}f(x)$d$x$ is (), where
$$f(x)=\begin{cases}x\mathrm{ln}x/(x-1),&0<x<1,\\0,&x=0,\\1,&x=1.\end{cases}$$

Options:
A. $$
I \!=\! \frac{3}{4} \!-\! \sum_{n=2}^{\infty} \frac{1} {n ( n \!+\! 1 )}
$$
B. $$
I \!=\! \ln 2 \!-\! \sum_{n=1}^{\infty} \frac{1} {n^{2} ( n \!+\! 2 )}
$$
C. $$
I \!=\! \frac{1}{2} \!+\! \sum_{n=3}^{\infty} \frac{1} {n^{3} ( n \!+\! 1 )}
$$
D. $$
I \!=\! 1 \!-\! \sum_{n=2}^{\infty} \frac{1} {n^{2} ( n \!-\! 1 )}
$$
E. $$
I \!=\! e \!-\! \sum_{n=2}^{\infty} \frac{1} {n^{3} ( n \!+\! 1 )}
$$
F. $$
I \!=\! \frac{1}{4} \!+\! \sum_{n=2}^{\infty} \frac{1} {n^{2} ( n \!-\! 2 )}
$$
G. $$
I \!=\! \ln 3 \!-\! \sum_{n=3}^{\infty} \frac{1} {n^{2} ( n \!-\! 3 )}
$$
H. $$
I \!=\! \frac{1}{3} \!-\! \sum_{n=2}^{\infty} \frac{1} {n^{2} ( n \!+\! 1 )}
$$
I. $$
I \!=\! \pi \!-\! \sum_{n=2}^{\infty} \frac{1} {n^{3} ( n \!-\! 1 )}
$$
J. $$
I \!=\! 0 \!+\! \sum_{n=1}^{\infty} \frac{1} {n^{2} ( n \!-\! 2 )}
$$
Answer: D
Find the Dini derivative of
$$ f(x)=\begin{cases}ax\sin^2\frac{1}{x}+bx\cos^2\frac{1}{x},&x>0,\\0,&x=0,\quad(a<b,a^{\prime}<b^{\prime}).\\\\a^{\prime}x\sin^2\frac{1}{x}+b^{\prime}x\cos^2\frac{1}{x},&x<0\end{cases}$$.

Options:
A. $$
D_{+} f ( 0 ) \!=\! a^{\prime} \,, \; D_{-} f ( 0 ) \!=\! b^{\prime} \,, \; D^{-} f ( 0 ) \!=\! a.
$$
B. $$
D_{+} f ( 0 ) \!=\! b^{\prime} \,, \; D_{-} f ( 0 ) \!=\! a^{\prime} \,, \; D^{-} f ( 0 ) \!=\! b.
$$
C. $$
D_{+} f ( 0 ) \!=\! a \,, \; D_{-} f ( 0 ) \!=\! b \,, \; D^{-} f ( 0 ) \!=\! a.
$$
D. $$
D_{+} f ( 0 ) \!=\! b \,, \; D_{-} f ( 0 ) \!=\! b \,, \; D^{-} f ( 0 ) \!=\! b.
$$
E. $$
D_{+} f ( 0 ) \!=\! b \,, \; D_{-} f ( 0 ) \!=\! b^{\prime} \,, \; D^{-} f ( 0 ) \!=\! a.
$$
F. $$
D_{+} f ( 0 ) \!=\! a^{\prime} \,, \; D_{-} f ( 0 ) \!=\! a \,, \; D^{-} f ( 0 ) \!=\! b.
$$
G. $$
D_{+} f ( 0 ) \!=\! b^{\prime} \,, \; D_{-} f ( 0 ) \!=\! a \,, \; D^{-} f ( 0 ) \!=\! a^{\prime}.
$$
H. $$
D_{+} f ( 0 ) \!=\! b \,, \; D_{-} f ( 0 ) \!=\! a^{\prime} \,, \; D^{-} f ( 0 ) \!=\! b^{\prime}.
$$
I. $$
D_{+} f ( 0 ) \!=\! a \,, \; D_{-} f ( 0 ) \!=\! a^{\prime} \,, \; D^{-} f ( 0 ) \!=\! b^{\prime}.
$$
J. $$
D_{+} f ( 0 ) \!=\! b^{\prime} \,, \; D_{-} f ( 0 ) \!=\! b \,, \; D^{-} f ( 0 ) \!=\! a^{\prime}.
$$
Answer: I
Let $f(x), w(x)$ be non-negative measurable functions on $E$. If
$$
\int_{E} \! w(x) \, \mathrm{d} x = 1 \,, \quad \int_{E} \! f(x) w(x) \, \mathrm{d} x < +\infty\,,
$$
find the value of $\operatorname*{lim}_{p \to 0+} \left( \int_{E} \! f^{p}(x) w(x) \, \mathrm{d} x \right)^{1/p}$.

Options:
A. $$
\sqrt{\int_{E}^{w ( x ) +1\cdot\operatorname{l n}} f ( x ) \mathrm{d} x}
$$
B. $$
\frac{1}{2}\int_{E}^{w ( x ) \cdot\operatorname{l n}} f ( x ) \mathrm{d} x
$$
C. $$
\max \left(\int_{E}^{w ( x ) \cdot\operatorname{l n}} f ( x ) \mathrm{d} x , 1\right)
$$
D. $$
\int_{E}^{w ( x )-1 \cdot\operatorname{l n}} f ( x ) \mathrm{d} x
$$
E. $$
1 + e^{\int_{E}^{w ( x )} \cdot\operatorname{l n} f ( x ) \mathrm{d} x}
$$
F. $$
e^{\int_{E}^{w ( x ) \cdot\operatorname{l n}} f ( x ) \mathrm{d} x}
$$
G. $$
e^{-\int_{E}^{w ( x ) \cdot\operatorname{l n}} f ( x ) \mathrm{d} x}
$$
H. $$
\left(\int_{E}^{w ( x ) \cdot\operatorname{sin}} f ( x ) \mathrm{d} x \right)^2
$$
I. $$
\int_{E}^{\operatorname{cos}(w ( x ))} f ( x ) \mathrm{d} x + 1
$$
J. $$
\int_{E}^{w ( x ) \cdot f(x) } \operatorname{l n} x \mathrm{d} x
$$
Answer: F
Let $f {\in} L^{1} ( [0, 1] )$, and we have
$$
\int_{0}^{1} \! x^{n} f ( x ) \mathrm{d} x=\frac{1} {n+2} \quad( n=0, 1, 2, \cdots) \,,
$$
what is the value of $f(x)$?

Options:
A. $$
x^3, \mathrm{a. e.} x \in[ 0, 1 ]
$$
B. $$
x, \mathrm{a. e.} x \in[ 0, 1 ]
$$
C. $$
x^2, \mathrm{a. e.} x \in[ 0, 1 ]
$$
D. $$
x+1, \mathrm{a. e.} x \in[ 0, 1 ]
$$
E. $$
1, \mathrm{a. e.} x \in[ 0, 1 ]
$$
F. $$
0, \mathrm{a. e.} x \in[ 0, 1 ]
$$
G. $$
3x, \mathrm{a. e.} x \in[ 0, 1 ]
$$
H. $$
-x, \mathrm{a. e.} x \in[ 0, 1 ]
$$
I. $$
2x, \mathrm{a. e.} x \in[ 0, 1 ]
$$
J. $$
x^4, \mathrm{a. e.} x \in[ 0, 1 ]
$$
Answer: B
Let $x_{1} < x_{2} < \cdots< x_{n}$ be the $n$ distinct real roots of the $n$-th degree polynomial $P(x)$, with $A>0$, and consider the set
$$
E=\left\{x \in\mathbb{R}^{1}, \, P^{\prime} \left( x \right) / P(x) > \lambda\right\},
$$
then $E$ is the union of a finite number of mutually disjoint intervals. What is the total length of these intervals?

Options:
A. $\lambda / n$
B. $n \lambda^2$
C. $1 / \lambda$
D. $n \cdot \lambda$
E. $\lambda + n$
F. $n / 2\lambda$
G. $n / \lambda$
H. $n^2 / \lambda$
I. $\lambda^2 / n$
J. $n - \lambda$
Answer: G
A separately excited DC motor has the following nameplate data: $\mathrm{P}_{\mathrm{N}}=13 \mathrm{kW}, \ \ \mathrm{U}_{\mathrm{N}}=220 \mathrm{V}, \ \ \mathrm{I}_{\mathrm{N}}=68.6 \mathrm{A} \, \ \ \mathrm{n}_{\mathrm{N}}=1500 \mathrm{r}/\mathrm{min}$. This motor is used as a hoisting motor. When lowering a load at a speed of $800 \mathrm{r}/\mathrm{min}$ with the rated load, what is the value of the series resistor?

Options:
A. 6.15Ω
B. 5.90Ω
C. 5.27Ω
D. 3.55Ω
E. 4.76Ω
F. 3.62Ω
G. 4.19Ω
H. 3.88Ω
I. 5.04Ω
J. 4.53Ω
Answer: J
A three-phase induction motor operates on a power grid with $\mathrm{U}_{\mathrm{N}}=380 \, \mathrm{V}$ and $\mathrm{f}=50 \, \mathrm{Hz}$, with its stator windings connected in a delta (triangle) configuration. It is known that the electromotive force of each phase is 92% of the rated voltage. The number of turns per phase of the stator winding is $\mathbf{N}_{1}=312$ turns, and the winding factor $\mathbf{K}_{\mathrm{w1}}=0.96$. The magnetic flux per pole $\phi_{1}$ is ().

Options:
A. 0.0048wb
B. 0.0072wb
C. 0.0081wb
D. 0.0045wb
E. 0.0090wb
F. 0.0075wb
G. 0.0105wb
H. 0.0053wb
I. 0.0069wb
J. 0.0063wb
Answer: H
Air at a temperature of 293 K and a pressure of $1.20 \times 10^{5} \, \mathrm{Pa}$ flows into a horizontal circular pipe with an inner diameter of 100 mm at a mass flow rate of 0.5 kg/s. The airflow inside the pipe is turbulent. Steam heats the exterior of the pipe with a heat transfer rate of $1 \times 10^{5} \, \mathrm{J/s}$. Assuming all the heat is absorbed by the air, the pressure at the outlet of the pipe is $1.01325 \times 10^{5} \, \mathrm{Pa}$. "Try to determine the temperature of the air at the outlet of the pipe. Assume the air can be regarded as an ideal gas, with an average specific heat capacity of $1.005 \, \mathrm{kJ/(kg \cdot K)}$.

Options:
A. $$
T_{2}=5 0 1 \mathrm{K}
$$
B. $$
T_{2}=4 7 5 \mathrm{K}
$$
C. $$
T_{2}=4 6 0 \mathrm{K}
$$
D. $$
T_{2}=4 5 0 \mathrm{K}
$$
E. $$
T_{2}=5 2 3 \mathrm{K}
$$
F. $$
T_{2}=5 0 8 \mathrm{K}
$$
G. $$
T_{2}=4 4 9 \mathrm{K}
$$
H. $$
T_{2}=4 8 9 \mathrm{K}
$$
I. $$
T_{2}=5 1 5 \mathrm{K}
$$
J. $$
T_{2}=4 3 2 \mathrm{K}
$$
Answer: H
The velocity vector of a certain flow field can be expressed as $\boldsymbol{u} ( x, y )=5 x i-5 y j$. Try to write out the expression for the convective acceleration vector $\frac{\mathrm{D} u} {\mathrm{D} \theta}$ of this flow field.

Options:
A. $$
= 5 i+5 j y
$$
B. $$
= 2 5 x i+2 5 y j
$$
C. $$
= 5 x i-5 y j+ 10
$$
D. $$
= 10 x i+10 y i
$$
E. $$
= 5 y i+5 x j
$$
F. $$
= 5 x y i-5 y j
$$
G. $$
= 5 x i+5 y j
$$
H. $$
= 2 y i- 2 x j
$$
I. $$
= 2 5 x j+2 5 y i
$$
J. $$
= 5 x j-5 y i
$$
Answer: B
Viscous incompressible fluid undergoes steady-state laminar flow in an infinitely long pipe with an elliptical cross-section. Assume the major and minor axes of the ellipse are $a$ and $b$, respectively. Determine the velocity distribution of the flow in this pipe. [Hint: Assume $u_{z}$ has the following form for velocity distribution $u_{z}=A \! \left( 1-\! {\frac{x^{2}} {a^{2}}} \!-\! {\frac{y^{2}} {b^{2}}} \right)$, where $A$ is a constant to be determined].

Options:
A. $$
u_{z}=-\frac{1} {2} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
B. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a b^{3}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
C. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {a} ( 1-\frac{x^{2}} {b^{2}}-\frac{y^{2}} {a^{2}} )
$$
D. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
E. $$
u_{z}=-\frac{\mu} {2} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
F. $$
u_{z}=-\frac{1} {\mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
G. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{4} b^{2}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
H. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{2} b^{2}} {ab} ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
I. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} \frac{a^{3} b^{3}} {a^{2}+b^{2}} ( 1-\frac{x^{2}} {a}-\frac{y^{2}} {b})
$$
J. $$
u_{z}=-\frac{1} {2 \mu} \frac{\partial p_{\mathrm{d}}} {\partial z} a b ( 1-\frac{x^{2}} {a^{2}}-\frac{y^{2}} {b^{2}} )
$$
Answer: D
Water at 20°C flows at a velocity of 1 m/s over the surface of a smooth flat plate with a width of 1 m. Determine: the boundary layer thickness at two points, $x = 0.15 \mathrm{m}$ and $x = 0.3 \mathrm{m}$, from the leading edge of the plate is().
$$
R e_{x_{c}}=5 \times 10^{5}
$$

Options:
A. $$
= 3. 0 0 \times 1 0^{-3} \mathrm{m}
$$
B. $$
= 2. 4 5 \times 1 0^{-3} \mathrm{m}
$$
C. $$
= 3. 0 5 \times 1 0^{-3} \mathrm{m}
$$
D. $$
= 2. 5 0 \times 1 0^{-3} \mathrm{m}
$$
E. $$
= 2. 6 5 \times 1 0^{-3} \mathrm{m}
$$
F. $$
= 3. 2 5 \times 1 0^{-3} \mathrm{m}
$$
G. $$
= 3. 1 5 \times 1 0^{-3} \mathrm{m}
$$
H. $$
= 2. 8 0 \times 1 0^{-3} \mathrm{m}
$$
I. $$
= 2. 9 5 \times 1 0^{-3} \mathrm{m}
$$
J. $$
= 2. 7 5 \times1 0^{-3} \mathrm{m}
$$
Answer: J
Assume the velocity distribution of a flat plate laminar boundary layer is given by ${\frac{u_{x}} {u_{0}}}=1-\mathrm{e}^{-y / \delta}$. Try using the boundary layer integral momentum equation to derive the formulas for boundary layer thickness and the plate drag coefficient. In the equation, $\delta=\delta( x )$ is the boundary layer thickness, and $u_{0}$ is the free stream velocity.

Options:
A. $$
= 1. 2 6 4 2 R e_{L}^{-1 / 2}
$$
B. $$
= 1. 0 7 8 9 R e_{L}^{-1 / 2}
$$
C. $$
= 1. 8 6 4 2 R e_{L}^{-1 / 2}
$$
D. $$
= 2. 5 0 0 0 R e_{L}^{-1 / 2}
$$
E. $$
= 1. 2 4 5 6 R e_{L}^{-1 / 2}
$$
F. $$
= 1. 2 6 4 2 R e_{L}^{-1 / 3}
$$
G. $$
= 1. 5 1 3 8 R e_{L}^{-1 / 2}
$$
H. $$
= 1. 2 6 4 3 R e_{L}^{-1 / 3}
$$
I. $$
= 1. 3 3 3 3 R e_{L}^{-1 / 2}
$$
J. $$
= 0. 9 8 7 6 R e_{L}^{-1 / 2}
$$
Answer: A
A certain liquid is undergoing steady turbulent flow within a smooth circular pipe with an inner diameter of $0.05 \, \text{m}$. The density of this liquid is $1200 \, \text{kg/m}^3$, and its viscosity is $0.01 \, \text{Pa} \cdot \text{s}$. The measured pressure drop per meter of pipe length is $528 \, \text{N/m}^2$. Try to find the average flow velocity of this liquid in the pipe.

Options:
A. =6050
B. =5901
C. =8500
D. =6799
E. =7400
F. =8200
G. =7122
H. =6800
I. =7550
J. =5900
Answer: G
The viscosity of a certain fluid can be estimated using fluid resistance experiments. The method involves determining the average velocity $u_{\mathrm{b}}$ under steady-state turbulence and the pressure drop $(-\Delta p)$ when the pipe length is $L$, as measured in experiments. Derive the formula for calculating the fluid viscosity in terms of the pipe's inner diameter $d$, fluid density $\rho$, average flow velocity $u_{\mathrm{b}}$, and the pressure drop per unit pipe length $- \Delta p / L$.

Options:
A. $$
\mu=\left[ \frac{d^{1. 2} (-A p / L )} {0. 0 9 2 \rho^{0. 8} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
B. $$
\mu=\left[ \frac{d^{0. 8} (-B p / L )} {0. 0 9 2 \rho^{1. 2} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
C. $$
\mu=\left[ \frac{d^{1. 0} (-E p / L )} {0. 0 9 2 \rho^{0. 8} u_{\mathrm{b}}^{2. 0}} \right]^{5}
$$
D. $$
\mu=\left[ \frac{d^{1. 5} (-C p / L )} {0. 0 9 2 \rho^{1. 0} u_{\mathrm{b}}^{1. 5}} \right]^{5}
$$
E. $$
\mu=\left[ \frac{d^{1. 1} (-I p / L )} {0. 0 9 2 \rho^{0. 8} u_{\mathrm{b}}^{1. 9}} \right]^{5}
$$
F. $$
\mu=\left[ \frac{d^{1. 3} (-G p / L )} {0. 0 9 2 \rho^{0. 9} u_{\mathrm{b}}^{1. 7}} \right]^{5}
$$
G. $$
\mu=\left[ \frac{d^{1. 2} (-H p / L )} {0. 0 9 2 \rho^{1. 1} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
H. $$
\mu=\left[ \frac{d^{0. 9} (-F p / L )} {0. 0 9 2 \rho^{0. 8} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
I. $$
\mu=\left[ \frac{d^{1. 2} (-D p / L )} {0. 0 9 2 \rho^{0. 5} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
J. $$
\mu=\left[ \frac{d^{1. 4} (-J p / L )} {0. 0 9 2 \rho^{0. 7} u_{\mathrm{b}}^{1. 8}} \right]^{5}
$$
Answer: A
A model pipe geometrically similar to the air duct is made at a scale of 1:30, using water with 50 times the viscosity and 800 times the density of air for the model experiment. If the flow velocity in the air duct is 6 m/s, what should be the flow velocity of water in the model pipe to achieve similarity with the prototype?

Options:
A. $$
= 2 2 . 5 0 \mathrm{~ m / s}
$$
B. $$
= 6 . 0 0 \mathrm{~ m / s}
$$
C. $$
= 3 0 . 0 0 \mathrm{~ m / s}
$$
D. $$
= 9 . 5 0 \mathrm{~ m / s}
$$
E. $$
= 1 0 . 0 0 \mathrm{~ m / s}
$$
F. $$
= 1 8 . 0 0 \mathrm{~ m / s}
$$
G. $$
= 4 . 7 5 \mathrm{~ m / s}
$$
H. $$
= 1 1. 2 5 \mathrm{~ m / s}
$$
I. $$
= 5 . 5 0 \mathrm{~ m / s}
$$
J. $$
= 1 5 . 0 0 \mathrm{~ m / s}
$$
Answer: H
A stainless steel ingot with a length of 0.2 m and a diameter of 0.05 m has a thermal diffusivity $\alpha=0.0156 \, \mathrm{m}^2 /\mathrm{h}$ and a thermal conductivity $k=20 \, \mathrm{W}/(\mathrm{m} \, \mathrm{K})$, with an initial temperature of 363 K. This steel ingot is placed in a furnace heated to a temperature of 1473 K. The combined heat transfer coefficient between the surface of the steel ingot and the surrounding environment (including convection and radiation) is $100 \, \mathrm{W}/(\mathrm{m}^2 \, \mathrm{K})$. Calculate the time required for the steel ingot to reach a temperature of 1073 K.

Options:
A. $$
= 0. 1 1 2 \, \mathrm{h}
$$
B. $$
= 0. 0 5 4 \, \mathrm{h}
$$
C. $$
= 0. 2 4 9 \, \mathrm{h}
$$
D. $$
= 0. 0 9 8 \, \mathrm{h}
$$
E. $$
= 0. 2 6 2 \, \mathrm{h}
$$
F. $$
= 0. 0 7 1 \, \mathrm{h}
$$
G. $$
= 0. 2 0 8 \, \mathrm{h}
$$
H. $$
= 0. 1 4 4 \, \mathrm{h}
$$
I. $$
= 0. 1 7 6 \, \mathrm{h}
$$
J. $$
= 0. 3 0 3 \, \mathrm{h}
$$
Answer: H
A 0.45 m thick aluminum plate with an initial uniform temperature of $500 \mathrm{K}$ is suddenly exposed to a medium at $340 \mathrm{K}$ for cooling. The convective heat transfer coefficient between the surface of the aluminum plate and the surrounding environment is $455 \, \mathrm{W} / ( \mathrm{m}^{2} \, \mathrm{K} )$. Determine the time required for the temperature at the center of the aluminum plate to drop to 470K. The average thermal diffusivity of the aluminum plate is known to be $\alpha=0.340 \, \mathrm{m}^{2} / \mathrm{h}$, and the thermal conductivity is $k=208 \, \mathrm{W} / ( \mathrm{m} \, \mathrm{K} )$.

Options:
A. $$
\theta=0. 7 \, / \, 6. 2 5 9=0. 1 1 9 \, \mathrm{h}
$$
B. $$
\theta=1. 2 \, / \, 7. 8 6 3=0. 1 5 2 \, \mathrm{h}
$$
C. $$
\theta=0. 5 \, / \, 5. 7 3 1=0. 0 8 7 \, \mathrm{h}
$$
D. $$
\theta=1. 1 \, / \, 7. 5 4 2=0. 1 4 6 \, \mathrm{h}
$$
E. $$
\theta=1. 3 \, / \, 8. 0 7 2=0. 1 6 1 \, \mathrm{h}
$$
F. $$
\theta=0. 9 \, / \, 6. 7 1 6=0. 1 3 4 \, \mathrm{h}
$$
G. $$
\theta=0. 4 \, / \, 5. 4 6 8=0. 0 7 3 \, \mathrm{h}
$$
H. $$
\theta=0. 6 \, / \, 5. 9 2 4=0. 1 0 7 \, \mathrm{h}
$$
I. $$
\theta=1. 0 \, / \, 7. 1 8 4=0. 1 3 9 \, \mathrm{h}
$$
J. $$
\theta=0. 8 \, / \, 6. 9 5 8=0. 1 2 4 \, \mathrm{h}
$$
Answer: F
A certain type of oil flows past a heated flat plate surface at a uniform velocity of 1 m/s. The uniform temperature of the oil is 293 K, and the surface of the plate is maintained at 353 K. Assuming the critical Reynolds number $R e_{x_{c}}=5 \times 10^{5}$. It is known that at the film temperature in the boundary layer, the liquid density $\rho=750 \mathrm{kg/m}^{3}$, viscosity $\mu=3 \times 10^{-3} \mathrm{N \cdot s/m}^{2}$, thermal conductivity $k=0.15 \mathrm{W/(m \cdot K)}$, and specific heat capacity $c_{p}=200 \mathrm{J/(kg \cdot K)}$. Find the local convective heat transfer coefficient $h_{x}$ at the critical point and the temperature gradient at the wall surface.

Options:
A. $$
= 8 1 4 2 1. 7 3 \mathrm{~ K / m}
$$
B. $$
= 6 5 4 3 2. 1 0 \mathrm{~ K / m}
$$
C. $$
= 7 7 7 7 7. 7 7 \mathrm{~ K / m}
$$
D. $$
= 1 1 2 2 3. 8 7 \mathrm{~ K / m}
$$
E. $$
= 9 7 6 5 4. 2 0 \mathrm{~ K / m}
$$
F. $$
= 9 9 9 9 9. 9 9 \mathrm{~ K / m}
$$
G. $$
= 1 4 7 8 9. 0 2 \mathrm{~ K / m}
$$
H. $$
= 1 0 2 3 3. 4 8 \mathrm{~ K / m}
$$
I. $$
= 1 0 8 8 5. 9 5 \mathrm{~ K / m}
$$
J. $$
= 1 3 4 5 6. 5 1 \mathrm{~ K / m}
$$
Answer: I
For a certain non-ideal gas, the interaction potential between molecules is given by
$$
\Phi\left( r \right)=\xi\mathrm{exp} \left( -\alpha r^{2} \right)
$$
If $\xi$ is very small, the partition function can be expanded as a power series in $\xi$. The equation of state for this gas (to the order of $\xi$) is______.

Options:
A. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{3} \ \frac{1} {2} \xi\ \left( \frac{\pi} {\beta} \right)^{3 / 2}
$$
B. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{2} \ \frac{1} {2} \xi\ \left( \frac{\pi} {\alpha} \right)^{3 / 2}
$$
C. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{3} \ \frac{1} {3} \xi\ \left( \frac{\pi} {\alpha} \right)^{3 / 2}
$$
D. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{3} \ \frac{1} {2} \xi\ \left( \frac{\pi} {\alpha} \right)^{2 / 3}
$$
E. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{2} \ \frac{1} {3} \xi\ \left( \frac{\pi} {\beta} \right)^{3 / 2}
$$
F. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{2} \ \frac{1} {2} \xi\ \left( \frac{\beta} {\pi} \right)^{3 / 2}
$$
G. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{3} \ \frac{1} {3} \xi\ \left( \frac{\beta} {\pi} \right)^{3 / 2}
$$
H. $$
- \frac{\mathrm{N k T}} {\mathrm{V}}+\left( \frac{\mathrm{N}} {\mathrm{V}} \right)^{2} \ \frac{1} {2} \xi\ \left( \frac{\pi} {\beta} \right)^{2 / 3}
$$
Answer: B
The two energy levels of molecule X are $\varepsilon_{1}=6.1 \times 10^{-21}J$ and $\varepsilon_{2}=8.4 \times 10^{-21}J$, with corresponding degeneracies $\omega_{1}=3$ and $\omega_{2}=5$. In the ensemble of molecule $\mathbf{X}$, what is the ratio of the number of molecules $\mathbf{a}_{1} / \mathbf{a}_{2}$ at 3000K?

Options:
A. $$
0.815
$$
B. $$
0.491
$$
C. $$
0.862
$$
D. $$
0.603
$$
E. $$
0.419
$$
F. $$
0.634
$$
G. $$
0.729
$$
H. $$
0.587
$$
I. $$
0.536
$$
J. $$
0.774
$$
Answer: F
The known internal energy and state equation of a certain system are respectively given by
$$
\mathrm{U}=\mathrm{b V T}^{4} \qquad \mathrm{p V}=\frac{1}{3} \mathrm{U}
$$
where $b$ is a constant. Find the expression for entropy, given that the entropy is zero when $\mathbf{T}=\mathbf{0}$. The answer is ______.

Options:
A. $$
{\frac{7} {3}} \mathrm{b \, T^{3} \, V}
$$
B. $$
{\frac{6} {4}} \mathrm{b \, T^{3} \, V}
$$
C. $$
{\frac{4} {3}} \mathrm{b \, T^{3} \, V}
$$
D. $$
{\frac{5} {6}} \mathrm{b \, T^{2} \, V}
$$
E. $$
{\frac{3} {2}} \mathrm{b \, T^{4} \, V}
$$
F. $$
{\frac{3} {4}} \mathrm{b \, T^{3} \, V}
$$
G. $$
{\frac{5} {3}} \mathrm{b \, T^{4} \, V}
$$
H. $$
{\frac{8} {5}} \mathrm{b \, T^{3} \, V}
$$
I. $$
{\frac{4} {7}} \mathrm{b \, T^{5} \, V}
$$
J. $$
{\frac{2} {3}} \mathrm{b \, T^{4} \, V}
$$
Answer: C
Given the square $z_{1} z_{2} z_{3} z_{4}$ with opposite vertices $z_{1} (0, -1)$ and $z_{3} (2, 5)$, find the coordinates of vertices $z_{2}$ and $z_{4}$.

Options:
A. $$
( 4, 0 )
$$
$$
( -2, 1 )
$$
B. $$
( 2, 1 )
$$
$$
( 0, -3 )
$$
C. $$
( 3, 0 )
$$
$$
( 1, -2 )
$$
D. $$
( 5, 0 )
$$
$$
( -3, 2 )
$$
E. $$
( 4, 2 )
$$
$$
( -2, 0 )
$$
F. $$
( 3, 2 )
$$
$$
( 1, -2 )
$$
G. $$
( 5, 2 )
$$
$$
( -3, 4 )
$$
H. $$
( 5, 1 )
$$
$$
( -3, 3 )
$$
I. $$
( 3, 1 )
$$
$$
( 1, -3 )
$$
J. $$
( 4, 1 )
$$
$$
(-2, 3 )
$$
Answer: J
Given $u(x, y) = x^3 + 6x^2y - 3xy^2 - 2y^3$, find the analytic function $f(z) = u + \mathrm{i} v$ that satisfies the condition $f(0) = 0$.

Options:
A. $$
( 6+4 \mathrm{i} ) z^{5}
$$
B. $$
( 4-5 \mathrm{i} ) z^{6}
$$
C. $$
( 1-2 \mathrm{i} ) z^{3}
$$
D. $$
( 2+ \mathrm{i} ) z^{3}
$$
E. $$
( 5+3 \mathrm{i} ) z^{7}
$$
F. $$
( 1+2 \mathrm{i} ) z^{2}
$$
G. $$
( 2- \mathrm{i} ) z^{8}
$$
H. $$
( 3-2 \mathrm{i} ) z^{2}
$$
I. $$
( 2-3 \mathrm{i} ) z^{4}
$$
J. $$
( 3+4 \mathrm{i} ) z^{5}
$$
Answer: C
Let the function $f(z)$ be analytic within $|z| < 1$ and continuous on the closed circle $|z| \leq 1$, such that $f(0) = 1$. Find the value of the integral
$$
\frac{1}{2 \pi \mathrm{i}} \int_{|z|=1} \left[ 2 \pm \left( z + \frac{1}{z} \right) \right] f(z) \; \frac{\mathrm{d}z}{z}.

Options:
A. $$
5 \pm f^{\prime} ( 1 ).
$$
B. $$
1 \pm 3f ( 0 ).
$$
C. $$
2 \pm 2f ( 0 ).
$$
D. $$
3 \pm f^{\prime}( z ).
$$
E. $$
2 \pm zf^{\prime} ( 0 ).
$$
F. $$
3 \pm zf ( 0 ).
$$
G. $$
2 \pm f^{\prime} ( 0 ).
$$
H. $$
0 \pm f ( 0 ).
$$
I. $$
4 \pm f ( 1 ).
$$
J. $$
1 \pm f^{\prime} ( 1 ).
$$
Answer: G
Determine the order of the following functions at the zero point $z=0$. (1) $z^{2} ( e^{z^{2}}-1 )$; (2) 6sin $z^{3} + z^{3} ( z^{6}-6 )$.

Options:
A. $$5$$
$$10$$
B. $$2$$
$$15$$
C. $$3$$
$$12$$
D. $$4$$
$$10$$
E. $$4$$
$$15$$
F. $$4$$
$$13$$
G. $$6$$
$$15$$
H. $$3$$
$$15$$
I. $$4$$
$$12$$
J. $$5$$
$$15$$
Answer: E
Calculate the residues of the following function $f(z)$ at its isolated singular points (including the point at infinity):
$$
f(z) = {\frac{1}{\operatorname{sin} {\frac{1}{z}}}}.
$$

Options:
A. $$
-\frac{1}{4}
$$
B. $$
\frac{1}{12}
$$
C. $$
-\frac{1}{12}
$$
D. $$
\frac{1}{4}
$$
E. $$
\frac{1}{8}
$$
F. $$
-\frac{1}{3}
$$
G. $$
\frac{1}{6}
$$
H. $$
-\frac{1}{2}
$$
I. $$
-\frac{1} {6}
$$
J. $$
\frac{1}{2}
$$
Answer: I
Calculate the following products:
$$
\int_{C} \frac{d z} {( z-1 )^{2} ( z^{2}+1 )}, C_{1} x^{2}+y^{2}=2 ( x+y )
$$

Options:
A. $$
\frac{\pi} {6} i
$$
B. $$
- \frac{\pi} {2} i
$$
C. $$
\pi^{2} i
$$
D. $$
- \pi i
$$
E. $$
- \frac{\pi} {3}
$$
F. $$
\frac{\pi} {4} - i
$$
G. $$
\frac{\pi} {3} i
$$
H. $$
- \frac{i} {\pi}
$$
I. $$
\frac{1} {\pi} i
$$
J. $$
\frac{i} {2 \pi}
$$
Answer: B
Calculate the integral
$$
\int_{-\infty}^{+\infty} \frac{x^{2}}{(x^{2}+a^{2})^{2}} \mathrm{d} x \quad (a > 0)
$$

Options:
A. $$\frac{πa^{2}}{4}$$
B. $$\frac{πa}{2}$$
C. $$\frac{1}{2a^{2}}$$
D. $$\frac{π}{4a}$$
E. $$\frac{1}{a^{2}}$$
F. $$\frac{π^{2}}{2a}$$
G. $$\frac{π}{2a}$$
H. $$\frac{π}{a^{3}}$$
I. $$\frac{a}{2π}$$
J. $$\frac{2a}{π}$$
Answer: G
Calculate
$$
\int_{| z |=1} \frac{1} {\mathrm{e}^{\frac{1} {z}}-1} \mathrm{d} z.
$$

Options:
A. $$
\frac{\pi i} {3}
$$
B. $$
\frac{1} {2}
$$
C. $$
-2\pi i
$$
D. $$
\frac{\pi i} {4}
$$
E. $$
\frac{\pi} {6}
$$
F. $$
\frac{\pi i} {6}
$$
G. $$
-\frac{\pi i} {2}
$$
H. $$
\pi i
$$
I. $$
\frac{\pi i} {12}
$$
J. $$
0
$$
Answer: F
If $w={\frac{a z+b} {c z+d}}$ transforms the unit circle into a straight line, what conditions should its coefficients satisfy?

Options:
A. $$
a b+c d \neq0, \mid a \mid=\mid b \mid
$$
B. $$
c d-a b \neq0, \mid c \mid=\mid a \mid
$$
C. $$
c a-b d =0, \mid a \mid=\mid d \mid
$$
D. $$
a c-b d \neq0, \mid b \mid=\mid d \mid
$$
E. $$
b c-a d \neq0, \mid c \mid=\mid b \mid
$$
F. $$
b c+a d \neq0, \mid a \mid=\mid d \mid
$$
G. $$
c d-b a \neq0, \mid b \mid=\mid d \mid
$$
H. $$
a d-b c \neq0, \mid c \mid=\mid d \mid
$$
I. $$
a d+b c =0, \mid b \mid=\mid c \mid
$$
J. $$
a b-d c \neq0, \mid a \mid=\mid c \mid
$$
Answer: H