MikaStars39/nano-eval huggingface dataset:
What is the Jordan measure of the graph of the function $f(x) = \sin \frac{1}{x}$ on the interval (0,1] in $\mathbb{R}^{2}$?

Options:
A. $$
\sigma( \Gamma)=0
$$
B. $$
\sigma( \Gamma)=2
$$
C. $$
\sigma( \Gamma)=\frac{1}{2}
$$
D. $$
\sigma( \Gamma)=1
$$
E. $$
\sigma( \Gamma)=\frac{1}{3}
$$
F. $$
\sigma( \Gamma)=e
$$
G. $$
\sigma( \Gamma)=\frac{3}{4}
$$
H. $$
\sigma( \Gamma)=-1
$$
I. $$
\sigma( \Gamma)=\infty
$$
J. $$
\sigma( \Gamma)=\pi
$$
Answer: A
The brightest galaxy in the cluster A1060 has an apparent visual magnitude of $V=1 0. 9 9$ Estimate the distance to the cluster. Use the uncertainty in the average absolute magnitude of the brightest galaxy to determine how far off your answer could be ?

Options:
A. $$
8 3. 7 \mathrm{~ M p c}
$$
B. $$
7 0. 5 \mathrm{~ M p c}
$$
C. $$
7 2. 8 \mathrm{~ M p c}
$$
D. $$
8 5. 0 \mathrm{~ M p c}
$$
E. $$
7 8. 1 \mathrm{~ M p c}
$$
F. $$
8 0. 2 \mathrm{~ M p c}
$$
G. $$
6 5. 4 \mathrm{~ M p c}
$$
H. $$
7 6. 9 \mathrm{~ M p c}
$$
I. $$
6 7. 8 \mathrm{~ M p c}
$$
J. $$
6 9. 3 \mathrm{~ M p c}
$$
Answer: H
Calculate the time of decoupling , $t_{\mathrm{d e c}}$ as() , for a universe of matter and radiation using the WMAP values for $z_{\mathrm{d e c}}=1 0 8 9$ and other quantities.

Options:
A. $$
t \,=\, 1. 1 9 4 \times1 0^{1 2} \mathrm{~ s} \,=\, 3 7 8 \mathrm{~ y r}
$$
B. $$
t \,=\, 1. 1 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 8 0 \mathrm{~ k y r}
$$
C. $$
t \,=\, 1. 2 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 9 8 \mathrm{~ k y r}
$$
D. $$
t \,=\, 1. 1 9 5 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 8 8 \mathrm{~ k y r}
$$
E. $$
t \,=\, 8. 1 9 4 \times1 0^{1 2} \mathrm{~ s} \,=\, 3 7 8 \mathrm{~ y r}
$$
F. $$
t \,=\, 1. 1 9 4 \times1 0^{1 4} \mathrm{~ s} \,=\, 3 7 8 \mathrm{~ M y r}
$$
G. $$
t \,=\, 2. 1 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 4 7 8 \mathrm{~ k y r}
$$
H. $$
t \,=\, 9. 1 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 7 9 \mathrm{~ k y r}
$$
I. $$
t \,=\, 1. 0 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 6 8 \mathrm{~ k y r}
$$
J. $$
t \,=\, 1. 1 9 4 \times1 0^{1 3} \mathrm{~ s} \,=\, 3 7 8 \mathrm{~ k y r}
$$
Answer: J
Use the acceleration equation to show that the acceleration of the universe changed sign (from negative to positive) when the scale factor was
$$
R_{\mathrm{a c c e l}}=\left( \frac{\Omega_{m, 0}} {2 \Omega_{\Lambda, 0}} \right)^{1 / 3}.
$$
Evaluate the value of $R_{\mathrm{a c c e l}}$ and $z_{\mathrm{a c c e l}}$ at this time with WMAP values as ().

Options:
A. $R=0. 6 5$, and $z=1 / R-1=0. 5 4$
B. $R=0. 6 0$, and $z=1 / R-1=0. 6 7$
C. $R=0. 5 9$, and $z=1 / R-1=0. 6 9$
D. $R=0. 5 3$, and $z=1 / R-1=0. 7 9$
E. $R=0. 5 0$, and $z=1 / R-1=0. 9 0$
F. $R=0. 4 9$, and $z=1 / R-1=0. 8 4$
G. $R=0. 4 8$, and $z=1 / R-1=0. 8 5$
H. $R=0. 5 7$ , and $z=1 / R-1=0. 7 6$
I. $R=0. 6 2$, and $z=1 / R-1=0. 6 1$
J. $R=0. 4 4$, and $z=1 / R-1=0. 8 2$
Answer: H
The globular cluster IAU C0923--545 has an integrated apparent visual magnitude of $V=$ $+ 1 3. 0$ and an integrated absolute visual magnitude of $M_{V}=-4. 1 5. 1$ t is located 9.0 kpc from Earth and is 11.9 kpc from the Galactic center, just 0.5 kpc south of the Galactic midplane. What is the amount of interstellar extinction per kiloparsec?

Options:
A. $$
0. 2 6 \mathrm{~ m a g ~ k p c}^{-1}
$$
B. $$
0. 2 3 \mathrm{~ m a g ~ k p c}^{-1}
$$
C. $$
0. 3 0 \mathrm{~ m a g ~ k p c}^{-1}
$$
D. $$
0. 3 2 \mathrm{~ m a g ~ k p c}^{-1}
$$
E. $$
0. 1 8 \mathrm{~ m a g ~ k p c}^{-1}
$$
F. $$
0. 2 2 \mathrm{~ m a g ~ k p c}^{-1}
$$
G. $$
0. 2 7 \mathrm{~ m a g ~ k p c}^{-1}
$$
H. $$
0. 3 1 \mathrm{~ m a g ~ k p c}^{-1}
$$
I. $$
0. 3 5 \mathrm{~ m a g ~ k p c}^{-1}
$$
J. $$
0. 2 9 \mathrm{~ m a g ~ k p c}^{-1}
$$
Answer: A
Using Newtonian gravity, estimate the amount of energy required to move 107 $\mathrm{M_{\odot}}$ from a position just above the event horizon of the supermassive black hole at the center of the Galaxy to 3 kpc, the present location of the expanding arm.

Options:
A. $$
1.2 \times 10^{54} \mathrm{~J}
$$
B. $$
1.1 \times 10^{54} \mathrm{~J}
$$
C. $$
7.5 \times 10^{53} \mathrm{~J}
$$
D. $$
9 \times1 0^{5 3} \mathrm{~ J}
$$
E. $$
8.5 \times 10^{53} \mathrm{~J}
$$
F. $$
6 \times 10^{53} \mathrm{~J}
$$
G. $$
8 \times 10^{53} \mathrm{~J}
$$
H. $$
1 \times 10^{54} \mathrm{~J}
$$
I. $$
7 \times 10^{53} \mathrm{~J}
$$
J. $$
9.5 \times 10^{53} \mathrm{~J}
$$
Answer: D
It is estimated that M31 has approximately 350 globular clusters. If its absolute visual magnitude is $- 2 1. 7$ , estimate the specific frequency for its clusters.

Options:
A. $$
0. 7 3
$$
B. $$
0. 6 8
$$
C. $$
1. 1 2
$$
D. $$
1. 0 5
$$
E. $$
0. 8 9
$$
F. $$
0. 6 5
$$
G. $$
0. 9 5
$$
H. $$
1. 0 0
$$
I. $$
1. 2 0
$$
J. $$
0. 7 9
$$
Answer: A
Assuming that the least massive galaxies could have formed out of a gas with an initial temperature near the ionization temperature of hydrogen, estimate the lower mass cufoff.

Options:
A. $$
3. 2 \times10^{10} \, \mathrm{M}_{\odot}
$$
B. $$
2. 5 \times10^{8} \, \mathrm{M}_{\odot}
$$
C. $$
6. 5 \times10^{6} \, \mathrm{M}_{\odot}
$$
D. $$
5. 3 \times10^{9} \, \mathrm{M}_{\odot}
$$
E. $$
7. 1 \times10^{11} \, \mathrm{M}_{\odot}
$$
F. $$
8. 7 \times10^{8} \, \mathrm{M}_{\odot}
$$
G. $$
4. 8 \times10^{7} \, \mathrm{M}_{\odot}
$$
H. $$
10. 2 \times10^{12} \, \mathrm{M}_{\odot}
$$
I. $$
9. 4 \times10^{7} \, \mathrm{M}_{\odot}
$$
J. $$
1. 1 \times1 0^{9} \, \mathrm{M}_{\odot}
$$
Answer: J
The smallest integer that can be represented by an 8-bit binary complement consisting of three "1"s and five "0"s is ().

Options:
A. 74
B. 114
C. 63
D. 12
E. 48
F. 34
G. -96
H. 50
I. 57
J. -125
Answer: J
Let $x=0.1000$, $y=-0.1011$, use the method of alternating between addition and subtraction with two's complement to find $x / y =$ ().

Options:
A. $+0.1001$ and the remainder is $0.1100 \times 2^{-4}$.
B. $+0.0110$ and the remainder is $0.1111 \times 2^{-4}$.
C. $-0.1101$ and the remainder is $0.0110 \times 2^{-4}$.
D. $+0.1101$ and the remainder is $0.0111 \times 2^{-4}$.
E. $+0.0101$ and the remainder is $0.1011 \times 2^{-4}$.
F. $+0.1111$ and the remainder is $0.0001 \times 2^{-4}$.
G. $+0.1100$ and the remainder is $0.0011 \times 2^{-4}$.
H. $+0.1110$ and the remainder is $0.1010 \times 2^{-4}$.
I. $+0.1010$ and the remainder is $0.0101 \times 2^{-4}$.
J. $-0.0111$ and the remainder is $0.1001 \times 2^{-4}$.
Answer: D
Given two real numbers, $\mathbf{x} = -6.8$ and $\mathbf{y} = -8.25$, defined in the C language as float variables, they are stored in registers $A$ and $B$, respectively. Additionally, there are two other registers, $C$ and $D$. $A, B, C, D$ are all 32-bit registers. What is the content of register $\mathrm{D}$ where the result of $\mathbf{x}$ subtracted by $\mathbf{y}$ is stored? (The content should be represented as a hexadecimal binary sequence.)

Options:
A. $$
\mathbf{D} 0 \mathrm{F} 1 9 7 0 0 \mathrm{H}
$$
B. $$
\mathbf{B} 7 7 3 5 \mathrm{C} 0 0 \mathrm{H}
$$
C. $$
\mathbf{C} 2 6 \mathrm{F} 0 0 0 0 \mathrm{H}
$$
D. $$
\mathbf{B} 0 4 5 \mathrm{C} 0 8 0 \mathrm{H}
$$
E. $$
\mathbf{A} 3 9 1 2 3 0 0 \mathrm{H}
$$
F. $$
\mathbf{E} 3 \mathrm{F} 6 4 0 0 0 \mathrm{H}
$$
G. $$
\mathbf{D} 4 8 2 1 \mathrm{A} 0 0 \mathrm{H}
$$
H. $$
\mathbf{C} 1 0 0 \mathrm{F} 0 0 0 \mathrm{H}
$$
I. $$
\mathbf{F} 2 9 \mathrm{D} 1 0 6 0 \mathrm{H}
$$
J. $$
\mathbf{F} 6 2 \mathrm{E} 1 0 0 0 \mathrm{H}
$$
Answer: C
Assume a memory capacity of 32 words, a word length of $64$ bits, and the number of modules $m=4$. The organization is done in both sequential and interleaved ways. The memory cycle $T=200 \mathrm{ns}$, the data bus width is $64$ bits, and the bus transfer cycle $r=50 \mathrm{ns}$. In the situation of continuously reading out 4 words, what is the bandwidth of the sequential memory and the interleaved memory, respectively?

Options:
A. $$
3 0 \times1 0^{7} \mathrm{b / s}
$$
$$
7 5 \times1 0^{7} \mathrm{b / s}
$$
B. $$
3 3 \times1 0^{7} \mathrm{b / s}
$$
$$
7 1 \times1 0^{7} \mathrm{b / s}
$$
C. $$
2 6 \times1 0^{7} \mathrm{b / s}
$$
$$
7 2 \times1 0^{7} \mathrm{b / s}
$$
D. $$
2 9 \times1 0^{7} \mathrm{b / s}
$$
$$
6 8 \times1 0^{7} \mathrm{b / s}
$$
E. $$
3 5 \times1 0^{7} \mathrm{b / s}
$$
$$
7 8 \times1 0^{7} \mathrm{b / s}
$$
F. $$
2 8 \times1 0^{7} \mathrm{b / s}
$$
$$
6 9 \times1 0^{7} \mathrm{b / s}
$$
G. $$
3 0 \times1 0^{7} \mathrm{b / s}
$$
$$
7 0 \times1 0^{7} \mathrm{b / s}
$$
H. $$
3 4 \times1 0^{7} \mathrm{b / s}
$$
$$
8 0 \times1 0^{7} \mathrm{b / s}
$$
I. $$
3 1 \times1 0^{7} \mathrm{b / s}
$$
$$
7 4 \times1 0^{7} \mathrm{b / s}
$$
J. $$
3 2 \times1 0^{7} \mathrm{b / s}
$$
$$
7 3 \times1 0^{7} \mathrm{b / s}
$$
Answer: J
A certain computer has a word length of 32 bits, and the memory cycle time is 200 ns. Four-way interleaving is used, employing the lower 2 bits of the address as the memory bank address, and the data is stored sequentially by address. What is the shortest time the main system can read out one data word?

Options:
A. 200ns
B. 40ns
C. 60ns
D. 70ns
E. 80ns
F. 150ns
G. 120ns
H. 50ns
I. 100ns
J. 110ns
Answer: H
A certain 32-bit computer has a CPU clock speed of 800 MHz, with a CPI of 4 when there is a cache hit. The cache block size is 32B. The main memory uses an 8-way interleaved storage method, with each bank having a word length of 32 bits and a memory cycle time of 40 ns. The memory bus width is 32 bits, with a bus clock frequency of 200 MHz, and it supports burst transfer bus transactions. Each burst transfer bus transaction includes: sending the initial address and command, memory preparation of data, and transmitting data. Each burst transfer transmits 32B, and transmitting an address or 32-bit data requires one bus clock cycle. If during the execution of program BP, a total of 100 instructions are executed, with an average of 1.2 memory accesses per instruction, and a cache miss rate of 5%, ignoring replacement and other overheads, what is the CPU execution time for BP?

Options:
A. 1000ns
B. 1040ns
C. 1020ns
D. 1080ns
E. 1010ns
F. 950ns
G. 990ns
H. 1150ns
I. 980ns
J. 1070ns
Answer: E
A certain computer has a word length of 16 bits. In the FLAGS register, $\mathrm{ZF}$, SF, and OF stand for the zero flag, sign flag, and overflow flag, respectively. It uses a double byte word length instruction word. Assume that the first byte of the bgt (branch if greater than zero) instruction specifies the opcode and addressing mode, while the second byte is the offset address Imm8, represented in two's complement. The instruction function is:
If ($\mathrm{ZF} + (\mathrm{SF} \oplus \mathrm{OF}) = 0$), then PC = PC + 2 + Imm8 × 2; otherwise, PC = PC + 2. What is the addressing unit of this computer?

Options:
A. Kilobyte
B. Quadword
C. Bit
D. Terabyte
E. Octet
F. Sector
G. Word
H. Byte
I. Megabyte
J. Nibble
Answer: H
The clock frequency of a certain bus is 66MHz. In a $64$-bit bus, the data transmission cycle is 7 clock cycles to transmit a 6-word data block. What is the data transfer rate of the bus?

Options:
A. $$
6 8 / 2 \mathrm{M H z}
$$
B. $$
7 3 / 2 \mathrm{M H z}
$$
C. $$
5 2 / 4 \mathrm{M H z}
$$
D. $$
5 7 / 3 \mathrm{M H z}
$$
E. $$
7 4 / 8 \mathrm{M H z}
$$
F. $$
4 9 / 6 \mathrm{M H z}
$$
G. $$
6 1 / 5 \mathrm{M H z}
$$
H. $$
5 5 / 9 \mathrm{M H z}
$$
I. $$
6 9 / 1 \mathrm{M H z}
$$
J. $$
6 6 / 7 \mathrm{M H z}
$$
Answer: J
In a star topology with $n$ nodes, there are () physical links.

Options:
A. $$
n^2
$$
B. $$
n+1
$$
C. $$
4n
$$
D. $$
n-1
$$
E. $$
n+2
$$
F. $$
n/2
$$
G. $$
2n
$$
H. $$
\sqrt{n}
$$
I. $$
3n
$$
J. $$
n^3
$$
Answer: D
Binary signals transmitted over a 4kHz channel with a signal-to-noise ratio of 127:1 can achieve a maximum data transmission rate of ().

Options:
A. 24000b/s
B. 1000b/s
C. 8000b/s
D. 32000b/s
E. 12000b/s
F. 4000b/s
G. 16000b/s
H. 2000b/s
I. 6000b/s
J. 500b/s
Answer: C
The data to be sent is 1101 0110 11. Using CRC verification with the generating polynomial 10011, the final data to be sent should be ().

Options:
A. 1101 0110 1001 01
B. 1101 0110 1100 11
C. 1101 0110 1110 00
D. 1101 0110 1011 10
E. 1101 0110 1111 10
F. 1101 0110 1110 01
G. 1101 0110 1000 10
H. 1101 0110 1010 11
I. 1101 0110 1101 01
J. 1101 0110 1101 00
Answer: E
For a sliding window $a$ with a window size of $n$, there can be a maximum of () frames sent but not acknowledged.

Options:
A. $$
n \times 2
$$
B. $$
n \div 2
$$
C. $$
\frac{n}{2}
$$
D. $$
\sqrt{n}
$$
E. $$
n-1
$$
F. $$
4n + 1
$$
G. $$
n+1
$$
H. $$
2n
$$
I. $$
3n - 1
$$
J. $$
n^2
$$
Answer: E
On a certain satellite channel, the sender transmits frames with a length of 512B from one direction, and the data transmission rate of the sender is 64kb/s. The receiver at the other end returns a very short acknowledgment frame. Given that the one-way propagation delay of the satellite channel from end to end is 270 ms, what is the throughput of the channel for sending window sizes of 1, 7, 17, and 117, respectively?

Options:
A. $$
6. 0 \mathrm{k b / s}
$$
$$
38.6 \mathrm{k b / s}
$$
$$
50\mathrm{k b / s}
$$
$$
56\mathrm{k b / s}
$$
B. $$
3. 3 \mathrm{k b / s}
$$
$$
45.7 \mathrm{k b / s}
$$
$$
53\mathrm{k b / s}
$$
$$
58\mathrm{k b / s}
$$
C. $$
6. 8 \mathrm{k b / s}
$$
$$
47.5 \mathrm{k b / s}
$$
$$
64\mathrm{k b / s}
$$
$$
64\mathrm{k b / s}
$$
D. $$
5. 1 \mathrm{k b / s}
$$
$$
37.5 \mathrm{k b / s}
$$
$$
48\mathrm{k b / s}
$$
$$
54\mathrm{k b / s}
$$
E. $$
7. 2 \mathrm{k b / s}
$$
$$
51.0 \mathrm{k b / s}
$$
$$
62\mathrm{k b / s}
$$
$$
63\mathrm{k b / s}
$$
F. $$
4. 4 \mathrm{k b / s}
$$
$$
40.8 \mathrm{k b / s}
$$
$$
52\mathrm{k b / s}
$$
$$
55\mathrm{k b / s}
$$
G. $$
9. 1 \mathrm{k b / s}
$$
$$
44.7 \mathrm{k b / s}
$$
$$
57\mathrm{k b / s}
$$
$$
61\mathrm{k b / s}
$$
H. $$
8. 5 \mathrm{k b / s}
$$
$$
42.1 \mathrm{k b / s}
$$
$$
54\mathrm{k b / s}
$$
$$
60\mathrm{k b / s}
$$
I. $$
6. 5 \mathrm{k b / s}
$$
$$
43.3 \mathrm{k b / s}
$$
$$
51\mathrm{k b / s}
$$
$$
53\mathrm{k b / s}
$$
J. $$
5. 9 \mathrm{k b / s}
$$
$$
39.0 \mathrm{k b / s}
$$
$$
49\mathrm{k b / s}
$$
$$
59\mathrm{k b / s}
$$
Answer: C
A group of $N$ stations share a 56 kb/s pure ALOHA channel, with each station outputting a 1000-bit frame on average every 100 seconds, even if the previous frame has not been completed. What is the maximum possible value for $N$?

Options:
A. $$
\approx8 7 5 1
$$
B. $$
\approx9 9 8
$$
C. $$
\approx8 0 5
$$
D. $$
\approx1 2 1 8
$$
E. $$
\approx9 1 0 2
$$
F. $$
\approx1 0 3 0
$$
G. $$
\approx1 0 5 9
$$
H. $$
\approx1 2 2 0
$$
I. $$
\approx1 1 4 5
$$
J. $$
\approx9 3 4
$$
Answer: F
According to the IEEE 802.3 standard, if coaxial cable is used as the transmission medium, the maximum length of the transmission medium without any repeaters cannot exceed ().

Options:
A. 600m
B. 250m
C. 550m
D. 500m
E. 450m
F. 700m
G. 1000m
H. 800m
I. 200m
J. 400m
Answer: D
Which of the following addresses belongs to the subnet 86.32.0.0/12?

Options:
A. $$
8 6. 3 3. 2 2 4. 1 2 3
$$
B. $$
8 6. 3 8. 0 0 0. 7 7 7
$$
C. $$
8 6. 4 4. 1 1 0. 2 4 8
$$
D. $$
8 7. 3 3. 2 2 4. 1 2 3
$$
E. $$
8 6. 2 2. 1 1 2. 0 0 0
$$
F. $$
8 6. 5 5. 2 2 4. 3 6 3
$$
G. $$
8 5. 3 3. 2 2 4. 1 2 3
$$
H. $$
8 6. 7 7. 7 7 7. 8 8 8
$$
I. $$
8 6. 3 3. 5 5 5. 5 5 5
$$
J. $$
8 6. 3 3. 3 3 3. 6 6 6
$$
Answer: A
The subnet mask for a certain subnet is 255.255.255.224, with $\mathrm{IP}$ addresses allocated to 4 hosts in total, one of which has a communication failure due to improper $IP$ address assignment. The $IP$ address of this host is ( ).

Options:
A. $$
192.168.1.1
$$
B. $$
192.168.1.100
$$
C. $$
10.0.1.5
$$
D. $$
2 0 2. 3. 1. 6 5
$$
E. $$
10.1.2.3
$$
F. $$
172.16.0.10
$$
G. $$
172.16.254.1
$$
H. $$
10.0.0.255
$$
I. $$
192.168.0.5
$$
J. $$
172.16.255.254
$$
Answer: D
The raw materials used for high-quality Pu-erh tea are generally selected ().

Options:
A. Sprouts
B. Leaf veins
C. Flower petals
D. Tree bark
E. Root nodules
F. Green shoots
G. Outer branches
H. Seed pods
I. Mature leaves
J. Stem buds
Answer: A
The Lagrangian function of a system is $L=\dot{q}^{2}+q$. Solve for $q(t)$ of this system using Hamilton's canonical equations is().

Options:
A. $$
q={\frac{1} {4}} t^{2}+A+B t
$$
B. $$
q={\frac{1} {8}} t^{2}+B t+C
$$
C. $$
q={\frac{1} {3}} t^{2}+B t+A
$$
D. $$
q={\frac{1} {6}} t^{2}+C t+B
$$
E. $$
q={\frac{1} {2}} t^{2}+A+B t
$$
F. $$
q={\frac{1} {4}} t^{2}+A t+B
$$
G. $$
q={\frac{1} {4}} t+B t+C
$$
H. $$
q={\frac{1} {2}} t^{2}+C t+A
$$
I. $$
q={\frac{1} {4}} At^{2}+B t+C
$$
J. $$
q={\frac{1} {5}} t^{2}+A t+C
$$
Answer: F
A disk with a radius $r$ and mass $m$ can roll without slipping on a thin rod, while the rod rotates with a constant angular velocity $o$ around a fixed point $O$. Both the rolling and rotation occur in the same plane. What is the expression for the total kinetic energy of the disk in the form $T(q, \dot{q})$?

Options:
A. $$
\frac{1} {2} m \dot{q}^{2}+\frac{3} {2} m \omega q r-\frac{1} {4} m \omega^{2} q^{2}-\frac{3} {4} m r^{2} \omega^{2}
$$
B. $$
\frac{1} {2} m \dot{q}^{2}-\frac{3} {2} m \omega\dot{q} r+\frac{3} {4} m \omega^{2} q^{2}-\frac{1} {4} m r^{2} \omega^{2}
$$
C. $$
\frac{1} {2} m \dot{q}^{2}-\frac{1} {2} m \omega\dot{q} r+\frac{1} {4} m \omega^{2} q^{2}+\frac{5} {4} m r^{2} \omega^{2}
$$
D. $$
\frac{3} {4} m q^{2} \omega^{2}+\frac{1} {2} m \dot{q}^{2}-\frac{1} {2} m \omega r q-\frac{1} {4} m r^{2} \omega^{2}
$$
E. $$
\frac{1} {2} m \dot{q}^{2}+\frac{1} {4} m \omega^{2} q^{2}+\frac{1} {2} m \omega\dot{q} r-\frac{3} {4} m r^{2} \omega^{2}
$$
F. $$
\frac{5} {4} m \dot{q}^{2}-\frac{1} {2} m \omega^{2} q r+\frac{3} {4} m r^{2} \omega^{2}-\frac{1} {4} m \omega^{2} q^{2}
$$
G. $$
\frac{5} {4} m \omega^{2} q^{2}+\frac{3} {4} m \dot{q}^{2}-\frac{1} {2} m \omega\dot{q} r-\frac{1} {4} m r^{2} \omega^{2}
$$
H. $$
\frac{3} {4} m \dot{q}^{2}-\frac{3} {2} m \omega\dot{q} r+\frac{1} {2} m \omega^{2} q^{2}+\frac{3} {4} m r^{2} \omega^{2}
$$
I. $$
\frac{3} {4} m q^{2} \omega^{2}+\frac{3} {2} m \omega\dot{q} r-\frac{1} {2} m \dot{q}^{2}+\frac{1} {4} m r^{2} \omega^{2}
$$
J. $$
\frac{3} {4} m \dot{q}^{2}+\frac{1} {2} m \omega q r-\frac{1} {2} m r^{2} \omega^{2}+\frac{1} {4} m \omega^{2} q^{2}
$$
Answer: H
Given that the rest mass of an electron is $9.11 \times 10^{-31} \mathrm{kg}$, $1.0 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{J}$, what is its speed when the electron's kinetic energy is 1,000,000 eV?

Options:
A. $$
2. 9 2 \times1 0^{8} \mathrm{m / s}
$$
B. $$
2. 9 7 \times1 0^{8} \mathrm{m / s}
$$
C. $$
3. 1 5 \times1 0^{8} \mathrm{m / s}
$$
D. $$
2. 6 9 \times1 0^{8} \mathrm{m / s}
$$
E. $$
3. 0 2 \times1 0^{8} \mathrm{m / s}
$$
F. $$
2. 7 8 \times1 0^{8} \mathrm{m / s}
$$
G. $$
3. 0 5 \times1 0^{8} \mathrm{m / s}
$$
H. $$
2. 8 2 \times1 0^{8} \mathrm{m / s}
$$
I. $$
2. 7 3 \times1 0^{8} \mathrm{m / s}
$$
J. $$
2. 8 9 \times1 0^{8} \mathrm{m / s}
$$
Answer: H
A mass is suspended from a spring, and the natural oscillation period of the mass-spring system is $T_{0} = 0.5 \mathrm{s}$. A vertical sinusoidal force is applied to the mass, with a maximum value of $F = 100 \mathrm{dyn}$. Additionally, a small frictional force is present, and it's assumed that the amplitude of the system at resonance is $A = 5.0 \mathrm{cm}$. The frictional force is proportional to the velocity, i.e., $f = \alpha \dot{x}$. What are the frictional resistance coefficient $\alpha$ and the value of the maximum frictional force $f_{\operatorname* {max}}$?

Options:
A. $$\alpha=1.6\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.001\mathrm{N}$$
B. $$\alpha=1.4\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.009\mathrm{N}$$
C. $$\alpha=2.0\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.008\mathrm{N}$$
D. $$\alpha=0.9\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.006\mathrm{N}$$
E. $$\alpha=3.0\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.003\mathrm{N}$$
F. $$\alpha=2.5\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.005\mathrm{N}$$
G. $$\alpha=2.2\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.002\mathrm{N}$$
H. $$\alpha=1.2\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.004\mathrm{N}$$
I. $$\alpha=1.0\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.010\mathrm{N}$$
J. $$\alpha=1.8\times10^{-3}\:\mathrm{N}\cdot\mathrm{s/m}$$
$$f_\mathrm{m}=0.007\mathrm{N}$$
Answer: A
Find the sequence of sets
$E_{1} = [0, 1/2]$, $E_{2} = [0, 1/2^{2}] \cup [2/2^{2}, 3/2^{2}]$,
$E_{3} = [0, 1/2^{3}] \cup [2/2^{3}, 3/2^{3}] \cup [4/2^{3}, 5/2^{3}] \cup [6/2^{3}, 7/2^{3}]$,
$...$ $...$ $...$ $...$ $...$ $...$ $...$ $...$ $...$
$E_{n} = [0, 1/2^{n}] \bigcup [2/2^{n}, 3/2^{n}] \cup \ldots \cup [(2^{n} - 2)/2^{n},$
$(2^{n} - 1)/2^{n}].$
$...$ $...$ $...$ $...$ $...$ $...$ $...$ $...$ $...$
What are the upper and lower limit sets?

Options:
A. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,0.5), \varliminf_{n \rightarrow \infty} E_{n}=[0,1).\mathrm{{\normalsize } }
$$
B. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0.5,1), \varliminf_{n \rightarrow \infty} E_{n}=\{0\}.\mathrm{{\normalsize } }
$$
C. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,0.9), \varliminf_{n \rightarrow \infty} E_{n}=\{0.5\}.\mathrm{{\normalsize } }
$$
D. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,1/4), \varliminf_{n \rightarrow \infty} E_{n}=[1/4,1).\mathrm{{\normalsize } }
$$
E. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,1], \varliminf_{n \rightarrow \infty} E_{n}=\emptyset.\mathrm{{\normalsize } }
$$
F. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,0.8), \varliminf_{n \rightarrow \infty} E_{n}=\{1\}.\mathrm{{\normalsize } }
$$
G. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,1), \varliminf_{n \rightarrow \infty} E_{n}=\left\{m / 2^{k}\right\}(k, m \geqslant 0) .\mathrm{{\normalsize } }
$$
H. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,0.75), \varliminf_{n \rightarrow \infty} E_{n}=\{1/n\}.\mathrm{{\normalsize } }
$$
I. $$
\varlimsup_{n \rightarrow \infty} E_{n}=[0,2), \varliminf_{n \rightarrow \infty} E_{n}=[0,1].\mathrm{{\normalsize } }
$$
Answer: G
Let $A=( n+1 / 2 \,, \ n=0 \,, \pm1 \,, \cdots) \,, B=( m \, \sqrt{2 \,} \,, \ m=0 \,, \pm1 \,,$ ...}, find $d ( A, B )$.

Options:
A. $$
d ( A, B )=2 \sqrt{2 \,}
$$
B. $$
d ( A, B )=m
$$
C. $$
d ( A, B )=\sqrt{3 \,}
$$
D. $$
d ( A, B )=m^2
$$
E. $$
d ( A, B )= \sqrt{2 \,}
$$
F. $$
d ( A, B )=0
$$
G. $$
d ( A, B )=\pm \, \sqrt{5 \,}
$$
H. $$
d ( A, B )=n
$$
I. $$
d ( A, B )=n+1
$$
J. $$
d ( A, B )=1
$$
Answer: F
Find the non-decreasing rearrangement of $f(x, y) = x^{2} + y^{2}$ on the set $x^{2} + y^{2} \leq 1$.

Options:
A. $$
f^{·} \left( \tau\right) \!=\! \cos\left(\tau\right) \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
B. $$
f^{·} \left( \tau\right) \!=\! 1/\left(\pi\!-\!\tau\right) \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
C. $$
f^{·} \left( \tau\right) \!=\! 2\pi \tau^3 \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
D. $$
f^{·} \left( \tau\right) \!=\! \sqrt{1 \!-\! \tau/ \pi} \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
E. $$
f^{·} \left( \tau\right) \!=\! \sin\left(\pi \tau\right) \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
F. $$
f^{·} \left( \tau\right) \!=\! \tau + \pi/2 \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
G. $$
f^{·} \left( \tau\right) \!=\! \pi - \sqrt{\tau} \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
H. $$
f^{·} \left( \tau\right) \!=\! e^{-\tau} \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
I. $$
f^{·} \left( \tau\right) \!=\! \tau^2 \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
J. $$
f^{·} \left( \tau\right) \!=\! \tau/\pi^2 \left( 0 \! \leq\! \tau\! \leq\! \pi\right)
$$
Answer: D
Given the function
$f(x)=\begin{cases}\sin x,&\text{if } \cos x \text{ is a rational number,}\\\cos^2x,&\text{if } \cos x \text{ is an irrational number,}\end{cases}$
find the value of the integral
$\int_0^{\pi/2}f(x)$d$x$.

Options:
A. $$1$$
B. $$\frac{3\pi}{8}$$
C. $$\frac{\pi}{2}$$
D. $$\frac{\pi}{6}$$
E. $$\frac{5\pi}{12}$$
F. $$
\frac{\pi} {4}
$$
G. $$2$$
H. $$\frac{\pi}{3}$$
I. $$0$$
J. $$\frac{1}{2}$$
Answer: F
Find the value of $I = \int_{0}^{+\infty} \frac{\sin(ax)}{e^{z} - 1} \mathrm{d}x \quad (a > 0)$.

Options:
A. $$
I=\sum_{n=1}^{\infty} {\frac{1} {n^{2}+a^{2}}} \quad( a > 0 )
$$
B. $$
I=\sum_{n=1}^{\infty} {\frac{a} {n^{2}+a^{2}}} \quad( a > 0 )
$$
C. $$
I=\int_{0}^{+\infty} {\frac{\sin(ax)} {e^{x} - a}} \quad( a > 0 )
$$
D. $$
I=\sum_{n=1}^{\infty} {\frac{a} {n^{3}+a^{2}}} \quad( a > 0 )
$$
E. $$
I=\int_{0}^{+\infty} {\frac{\sin(ax)} {x^{2}+z^{2}}} \quad( a > 0 )
$$
F. $$
I=\sum_{n=1}^{\infty} {\frac{a^{2}} {n^{2}+1}} \quad( a > 0 )
$$
G. $$
I=\int_{0}^{+\infty} {\frac{\cos(ax)} {e^{z} - 1}} \quad( a > 0 )
$$
H. $$
I=2\sum_{n=1}^{\infty} {\frac{a} {n^{2}+a}} \quad( a > 0 )
$$
I. $$
I=\sum_{n=1}^{\infty} {\frac{a} {(n+a)^{2}}} \quad( a > 0 )
$$
J. $$
I=\sum_{n=1}^{\infty} {\frac{a} {n^{2}+a}} \quad( a > 0 )
$$
Answer: B
Given $E \subset [0, 1] \times [0, 1]$, let
$$
A = \left\{\left( x_{1} / 2, x_{2} / 2 \right) ; \left( x_{1}, x_{2} \right) \in E \right\},
$$
$$
B = \left\{( t x_{1}, t x_{2}, t ) \in [0, 1]^{3}, \ ( x_{1}, x_{2} ) \in E, t \in [0, 1] \right\},
$$
where $[0, 1]^{3} \triangleq [0, 1] \times [0, 1] \times [0, 1]$. Find the values of $m(A)$ and $m(B)$.

Options:
A. $m(A)=\frac{m(E)}2$,
$m(B)=\frac{4m(E)}3$
B. $m(A)=0$,
$m(B)=\frac{3m(E)}4$
C. $m(A)=\frac{m(E)}3$,
$m(B)=\frac{m(E)}4$
D. $m(A)=\frac 14m(E)$,
$m(B)=2m(E)$
E. $m(A)=m(E)$,
$m(B)=m(E)$
F. $m(A)=2m(E)$,
$m(B)=\frac{m(E)}3$
G. $m(A)=\frac{m(E)}3$,
$m(B)=m(E)$
H. $m(A)=\frac{3m(E)}4$,
$m(B)=\frac{m(E)}2$
I. $m(A)=\frac 14m(E)$,
$m(B)=\frac{m(E)}3$
J. $m(A)=m(E)$,
$m(B)=\frac{2m(E)}3$
Answer: I
Given $f \! \in\! L^{p}(\mathbb{R}^{1}) \, (p \! > \! 1)$, and $1/p \!+\! 1/p^{\prime} \!=\! 1$, let $F(x) = \int_{0}^{t} f(t) \, dt$, $x \in \mathbb{R}^{1}$. Find the value of $| F(x+h) - F(x) |$.

Options:
A. $$
O ( \left| h \right|^{2 / p^{\prime}} ) \,, \quad h \to0
$$
B. $$
o ( \left| h \right|^{1 / p^{\prime}} ) \,, \quad h \to0
$$
C. $$
o ( \left| h \right|^{1 / p} ) \,, \quad h \to0
$$
D. $$
o ( \left| h \right|^{1} ) \,, \quad h \to0
$$
E. $$
O ( \left| h \right|^{2 - 1/p} ) \,, \quad h \to0
$$
F. $$
o ( \left| h \right|^{p - 1} ) \,, \quad h \to0
$$
G. $$
O ( \left| h \right|^{p} ) \,, \quad h \to0
$$
H. $$
O ( \left| h \right|^{1 - p} ) \,, \quad h \to0
$$
I. $$
O ( \left| h \right|^{p^{\prime}} ) \,, \quad h \to0
$$
J. $$
o ( \left| h \right|^{2 / p} ) \,, \quad h \to0
$$
Answer: B
Let $E \subset \mathbf{R}$ be a measurable set with $m(E) < +\infty$, and let $g(x)$ be bounded and measurable on $E$. Define $a = \operatorname*{inf} \left( g(x) : x \in E \right)$ and $b = \operatorname*{sup} \left( g(x) : x \in E \right)$, where $h \in \mathrm{AC} \left( [a, b] \right)$. If we define $f(x) = h[g(x)]$ for $x \in E$, then $f \in L(E)$. Determine the value of $\int_{E} f(x) \, \mathrm{d}x$.

Options:
A. $$
h ( b ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ t, b \right] ) ) \mathrm{d} t
$$
B. $$
h ( b ) \cdot m ( E )+\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
C. $$
h ( b ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, b \right] ) ) \mathrm{d} t
$$
D. $$
h ( b ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
E. $$
h ( b ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a+1, t+1 \right] ) ) \mathrm{d} t
$$
F. $$
h ( b ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
G. $$
h ( b ) \cdot m ( E )-\! \! \int_{b}^{a} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
H. $$
h ( a ) \cdot m ( E )+\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
I. $$
h ( b-1 ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ a, t \right] ) ) \mathrm{d} t
$$
J. $$
h ( a ) \cdot m ( E )-\! \! \int_{a}^{b} \! \! h^{\prime} ( t ) \cdot m ( g^{-1} ( \left[ t, b \right] ) ) \mathrm{d} t
$$
Answer: D
Assume $f(x, y)$ is a binary function defined on $[a, b] \times [c, d]$, and there exists $y_{0} \in (c, d)$, such that $f(x, y_{0})$ is integrable over $[a, b]$; moreover, for each $x \in [a, b]$, $f(x, y)$ is an absolutely continuous function with respect to $y$ over $[c, d]$, and $f_{y}^{\prime}(x, y)$ is integrable on $[a, b] \times [c, d]$. Then the function $F(y) = \int_{a}^{b} f(x, y) \mathrm{d}x$ is an absolutely continuous function defined on $[c, d]$, and for almost every $y \in [c, d]$, what is the value of $F^{\prime}(y)$?

Options:
A. $$
\int_{a}^{b} \frac{\partial}{\partial x} f ( x, y ) \mathrm{d} y
$$
B. $$
\int_{a}^{b} f_{y}^{\prime} ( x, a ) \mathrm{d} x
$$
C. $$
\int_{c}^{d} f_{x}^{\prime} ( x, y ) \mathrm{d} y
$$
D. $$
\int_{c}^{d} \frac{\partial}{\partial y} f ( x, y ) \mathrm{d} x
$$
E. $$
\int_{a}^{b} f(x, y) \mathrm{d} y
$$
F. $$
\int_{c}^{d} f(x, y) \mathrm{d} x
$$
G. $$
\int_{a}^{b} \frac{\partial}{\partial y} f ( x, y ) \mathrm{d} x
$$
H. $$
\int_{a}^{b} f_{y}^{\prime} ( x, y ) \mathrm{d} x
$$
I. $$
\int_{a}^{b} f_{y}^{\prime} ( a, y ) \mathrm{d} x
$$
J. $$
\int_{a}^{b} f_{x}^{\prime} ( x, y ) \mathrm{d} x
$$
Answer: H
A planer with a gantry, where gear 1 is directly connected to the motor, provides data for each gear and the worktable. Given: cutting force $\mathrm{F}_{\mathrm{z}}=9810 \, \mathrm{N}$, cutting speed $\nu_{\mathrm{z}}=43 \, \mathrm{m/min}$, transmission efficiency $\eta_{c}=0.8$, the pitch of gear 6 is $20 \, \mathrm{mm}$, the flywheel moment of inertia of the motor armature is $230 \, \mathrm{N} \cdot \mathrm{m}$, and the coefficient of friction between the worktable and the bed is 0.1. What is the output power of the motor during cutting?

Options:
A. 10kW
B. 11kW
C. 23kW
D. 13kW
E. 5kW
F. 18kW
G. 7kW
H. 20kW
I. 15kW
J. 8kW
Answer: B
A three-phase 4-pole induction motor is given with a power supply frequency of $\mathrm{f=50 \, Hz}$ and a rated speed of $\mathrm{n}_{\mathrm{N}}=1450 \, \mathrm{r/min}$. What is the slip $\mathrm{S}$?

Options:
A. 2.9%
B. 2.2%
C. 1.6%
D. 6.0%
E. 3.3%
F. 4.1%
G. 4.8%
H. 3.7%
I. 5.2%
J. 5.5%
Answer: E
There is a mixing tank initially containing 100 kg of an aqueous solution of $Na_{\mathrm{2}}$$SO_{\mathrm{4}}$ with a concentration (mass fraction) of 50%. Now, a 15% mass fraction $\mathrm{Na}_{2}\mathrm{SO}_{4}$ solution is added to the tank at a mass flow rate of 12 kg/min, while 10 kg/min of the solution is simultaneously withdrawn from the tank. Assume that the liquid in the tank is thoroughly mixed. Calculate the mole fraction of $Na_{\mathrm{2}}$$SO_{\mathrm{4}}$ in the solution in the mixing tank after 10 minutes. It can be assumed that the volume of the solution does not change during the mixing process.

Options:
A. $$
3. 8 6 \%
$$
B. $$
2. 7 4 \%
$$
C. $$
6. 9 3 \%
$$
D. $$
4. 4 2 \%
$$
E. $$
5. 4 0 \%
$$
F. $$
3. 3 5 \%
$$
G. $$
4. 9 9 \%
$$
H. $$
5. 8 7 \%
$$
I. $$
5. 0 1 \%
$$
J. $$
7. 2 5 \%
$$
Answer: D
Water at 20 °C flows inside a circular pipe with a radius of $r_{\mathrm{i}}$. The velocity gradient at the wall is measured as $\frac{\mathrm{d} u}{\mathrm{d} r} \Bigg|_{r=r_{1}} = -1000 \mathrm{m} / (\mathrm{m \ s})$. Try to find the momentum flux at the wall.

Options:
A. $$
= 2. 0 0 5 \mathrm{N / m}^{2}
$$
B. $$
= 1. 0 0 3 \mathrm{N / m}^{2}
$$
C. $$
= 0. 5 0 5 \mathrm{N / m}^{2}
$$
D. $$
= 1. 1 0 5 \mathrm{N / m}^{2}
$$
E. $$
= 1. 0 0 5 \mathrm{N / m}^{2}
$$
F. $$
= 0. 2 0 5 \mathrm{N / m}^{2}
$$
G. $$
= 1. 5 0 0 \mathrm{N / m}^{2}
$$
H. $$
= 1. 0 0 8 \mathrm{N / m}^{2}
$$
I. $$
= 2. 5 0 0 \mathrm{N / m}^{2}
$$
J. $$
= 1. 2 0 5 \mathrm{N / m}^{2}
$$
Answer: E
Derive the relationship between the wall shear stress $\tau_{\mathrm{s}}$ and the average velocity $u_{\mathrm{b}}$ for incompressible fluid under one-dimensional steady-state laminar flow in a circular pipe.

Options:
A. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=\mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
B. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=5 \mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
C. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=4 \mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
D. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=6 \mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
E. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=3 \mu u_{\mathrm{b}} / r_{\mathrm{o}}
$$
F. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=8 \mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
G. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=4 \mu u_{\mathrm{b}} / \left(2r_{\mathrm{i}}\right)
$$
H. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=2 \mu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
I. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=4 \mu u_{\mathrm{b}} / D
$$
J. $$
\tau_{\mathrm{s}}=-\mu\frac{\mathrm{d} u} {\mathrm{d} r} \Big|_{r=r_{\mathrm{i}}}=4 \nu u_{\mathrm{b}} / r_{\mathrm{i}}
$$
Answer: C
What should the radius of a spherical water droplet falling at a constant speed in air be less than for it to be considered under Stokes' solution? Additionally, calculate the maximum possible value of its descent velocity. Given that the kinematic viscosity of air is $\nu=1.33 \times 10^{-5} \, \mathrm{m}^2/\mathrm{s}$, and the density ratio of water to air is 770.

Options:
A. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 2 1 4 5 \mathrm{m / s}}}} \\ {{{r=7. 5 5 \times1 0^{-6} \mathrm{m}}}} \\ \end{array}
$$
B. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 2 2 3 4 \mathrm{m / s}}}} \\ {{{r=4. 2 5 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
C. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 1 7 7 3 \mathrm{m / s}}}} \\ {{{r=3. 7 5 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
D. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 0 8 3 5 \mathrm{m / s}}}} \\ {{{r=6. 3 5 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
E. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 1 0 1 7 \mathrm{m / s}}}} \\ {{{r=8. 1 2 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
F. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 3 1 0 9 \mathrm{m / s}}}} \\ {{{r=2. 2 0 \times1 0^{-6} \mathrm{m}}}} \\ \end{array}
$$
G. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 1 9 6 2 \mathrm{m / s}}}} \\ {{{r=4. 9 8 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
H. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 2 9 2 3 \mathrm{m / s}}}} \\ {{{r=3. 5 0 \times1 0^{-6} \mathrm{m}}}} \\ \end{array}
$$
I. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 0 5 4 8 \mathrm{m / s}}}} \\ {{{r=1. 2 5 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
J. $$
\begin{array} {c} {{{u_{\mathrm{t, m a x}}=0. 3 8 4 2 \mathrm{m / s}}}} \\ {{{r=5. 1 0 \times1 0^{-5} \mathrm{m}}}} \\ \end{array}
$$
Answer: C
At atmospheric pressure, air at 40°C flows over a smooth flat plate at a velocity of 12 m/s. The plate is 0.15 m long and 1 m wide. Calculate the total drag force exerted on both the upper and lower surfaces of the plate. Assume $Re_{x_{c}} = 5 \times 10^{5}$.

Options:
A. $$
= 0. 1 3 4 7 \, \mathrm{N}
$$
B. $$
= 0. 1 0 2 5 \, \mathrm{N}
$$
C. $$
= 0. 2 5 0 3 \, \mathrm{N}
$$
D. $$
= 0. 0 6 7 9 \, \mathrm{N}
$$
E. $$
= 0. 0 9 6 5 \, \mathrm{N}
$$
F. $$
= 0. 0 8 9 1 \, \mathrm{N}
$$
G. $$
= 0. 3 0 1 8 \, \mathrm{N}
$$
H. $$
= 0. 0 4 3 6 \, \mathrm{N}
$$
I. $$
= 0. 0 5 8 2 \, \mathrm{N}
$$
J. $$
= 0. 0 7 1 4 \, \mathrm{N}
$$
Answer: E
Air at atmospheric pressure and 303K flows through a circular pipe with an inner diameter of 100mm at a volumetric flow rate of $0.1 \, \mathrm{m}^3/\mathrm{s}$. For fully developed flow, estimate the thickness of the laminar sublayer, buffer layer, and the turbulent core.

Options:
A. $$
= 0. 0 3 6 9 \mathrm{m}
$$
B. $$
= 0. 0 4 8 1 \mathrm{m}
$$
C. $$
= 0. 0 2 8 4 \mathrm{m}
$$
D. $$
= 0. 0 5 7 8 \mathrm{m}
$$
E. $$
= 0. 0 6 4 1 \mathrm{m}
$$
F. $$
= 0. 1 2 3 4 \mathrm{m}
$$
G. $$
= 0. 0 6 8 9 \mathrm{m}
$$
H. $$
= 0. 0 5 1 5 \mathrm{m}
$$
I. $$
= 0. 0 4 9 2 \mathrm{m}
$$
J. $$
= 0. 0 7 3 0 \mathrm{m}
$$
Answer: I
The water at a temperature of $20^{\circ} \mathrm{C}$ flows through a rough pipe with an internal diameter of 50mm at an average velocity of 1m/s. The circular pipe's roughness is given by ${\frac{e}{d}}=0.0005$. Find the shear stress value at $y=20\mathrm{mm}$.

Options:
A. $$
= 0. 9 8 2 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
B. $$
= 0. 5 5 8 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
C. $$
= 0. 4 7 3 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
D. $$
= 0. 3 1 9 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
E. $$
= 0. 2 4 1 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
F. $$
= 0. 6 5 2 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
G. $$
= 0. 7 4 1 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
H. $$
= 0. 8 3 9 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
I. $$
= 0. 5 8 7 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
J. $$
= 0. 6 2 3 \, \mathrm{N} / {\mathrm{m}}^{2}
$$
Answer: B
Within the range of $R e < 2 \times 10^{5}$, the Fanning friction factor for fully developed turbulent flow in a pipe can be represented by the Blasius formula: $f=0.046 R e^{-1/5}$. Attempt to derive the expression for the laminar sublayer thickness.

Options:
A. $$
=4 2. 0 R e^{-3 / 5}
$$
B. $$
=4 5. 0 R e^{-7 / 1 0}
$$
C. $$
=2 7. 0 R e^{-1 / 5}
$$
D. $$
=3 3. 0 R e^{-9 / 1 0}
$$
E. $$
=1 8. 0 R e^{-4 / 5}
$$
F. $$
=4 0. 0 R e^{-1 / 3}
$$
G. $$
=5 0. 0 R e^{-8 / 1 0}
$$
H. $$
=5 5. 0 R e^{-2 / 5}
$$
I. $$
=2 9. 0 R e^{-6 / 1 0}
$$
J. $$
=6 0. 0 R e^{-5 / 1 0}
$$
Answer: D
There is a naturally cooled cylindrical metal conductor with an outer diameter of $100 \ \mathrm{mm}$ and a wall thickness of $20 \ \mathrm{mm}$ . There is a uniform internal heat source within the conductor, with a value of $\dot{q}=1.0 \times 10^{7} \, \mathrm{J} / (\mathrm{m}^{3} \cdot \mathrm{s})$ . It is known that only one-dimensional radial heat conduction occurs within the conductor, and after reaching a steady state, the measured external surface temperature remains constant at 100°C, with an average thermal conductivity of $50 \ \mathrm{W / (m \cdot K)}$. Please select an appropriate generalized thermal transfer to measure the external surface temperature, which remains constant at $100^{\circ} \mathrm{C}$, with an average thermal conductivity of $50 \ \mathrm{W / (m \cdot K)}$.

Options:
A. $$
=-5 \times1 0^{4} r^{2}+8 0 \operatorname{l n} r+5 5 9. 6
$$
B. $$
=-7 \times1 0^{4} r^{2}+9 2 \operatorname{l n} r+4 5 0. 6
$$
C. $$
=-5 \times1 0^{4} r^{2}+9 0 \operatorname{l n} r+4 6 0. 6
$$
D. $$
=-4 \times1 0^{4} r^{2}+9 5 \operatorname{l n} r+4 5 6. 6
$$
E. $$
=-5 \times1 0^{5} r^{2}+9 0 \operatorname{l n} r+4 8 0. 6
$$
F. $$
=-5 \times1 0^{5} r^{2}+9 5 \operatorname{l n} r+4 7 5. 6
$$
G. $$
=-1 \times1 0^{4} r^{2}+9 8 \operatorname{l n} r+4 9 0. 6
$$
H. $$
=-3 \times1 0^{4} r^{2}+9 0 \operatorname{l n} r+5 0 4. 6
$$
I. $$
=-5 \times1 0^{4} r^{2}+9 0 \operatorname{l n} r+4 9 4. 6
$$
J. $$
=-6 \times1 0^{4} r^{2}+8 5 \operatorname{l n} r+5 0 0. 6
$$
Answer: I
There is a very long steel rod, with an initial uniform temperature of 600°C throughout. All surfaces except for the two end faces perpendicular to the axial direction are insulated. The temperature of one end face of the steel rod is suddenly reduced to 300°C and maintained at this temperature. It is known that the average thermal diffusivity of the steel rod is $\alpha=1.47 \times 10^{-5} \, \mathrm{m}^{2} / \mathrm{s}$ and the thermal conductivity is $k=54 \mathrm{W /} (\mathrm{m} \, \mathrm{K})$. Calculate the amount of heat transferred through the unit cooling surface during the specified time.

Options:
A. $$
=-7. 1 \times1 0^{5} \, \mathrm{J / m}^{2}
$$
B. $$
=-4. 5 \times1 0^{9} \, \mathrm{J / m}^{2}
$$
C. $$
=-5. 0 \times1 0^{7} \, \mathrm{J / m}^{2}
$$
D. $$
=-9. 0 \times1 0^{6} \, \mathrm{J / m}^{2}
$$
E. $$
=-1. 0 \times1 0^{8} \, \mathrm{J / m}^{2}
$$
F. $$
=-3. 2 \times1 0^{8} \, \mathrm{J / m}^{2}
$$
G. $$
=-6. 2 \times1 0^{7} \, \mathrm{J / m}^{2}
$$
H. $$
=-1. 5 \times1 0^{7} \, \mathrm{J / m}^{2}
$$
I. $$
=-8. 4 \times1 0^{6} \, \mathrm{J / m}^{2}
$$
J. $$
=-2. 0 \times1 0^{6} \, \mathrm{J / m}^{2}
$$
Answer: E
Air at atmospheric pressure and 303K flows over a flat surface with a uniform velocity of 20 m/s, with a width of $1 \; \mathrm{m}$ and a length of 2 m. The surface temperature is maintained at 373K. Calculate the rate of heat exchange between the entire surface of the plate and the air. Assume $R e_{x_{c}}=5 \times 10^{5}$.

Options:
A. $$
= 4 9 8 0 \mathrm{W}
$$
B. $$
= 6 3 2 0 \mathrm{W}
$$
C. $$
= 6 0 5 0 \mathrm{W}
$$
D. $$
= 5 9 0 0 \mathrm{W}
$$
E. $$
= 5 6 0 0 \mathrm{W}
$$
F. $$
= 4 7 6 0 \mathrm{W}
$$
G. $$
= 5 2 5 0 \mathrm{W}
$$
H. $$
= 5 8 8 0 \mathrm{W}
$$
I. $$
= 4 5 0 0 \mathrm{W}
$$
J. $$
= 6 2 0 0 \mathrm{W}
$$
Answer: H
At a temperature of 18.8°C, observe the Brownian motion of a particle with a radius of $0.4 \times 10^{-6}$ m in a liquid with a viscosity coefficient of $2.78 \times 10^{-3} \, \text{Pa} \cdot \text{s}$. The measured mean square displacement of the particle over a time interval of 10 seconds is
$$
\overline{{{\mathbf{x}^{2}}}} = 3.3 \times 10^{-12} \, \text{m}^{2}
$$
Based on this data, calculate the value of the Boltzmann constant $k$.
The answer is ___.

Options:
A. $$
3.44 \times 10^{-23}\ joules\ per\ degree\ Kelvin
$$
B. $$
9.11 \times 10^{-22}\ joules\ per\ degree\ Kelvin
$$
C. $$
2.67 \times 10^{-24}\ joules\ per\ degree\ Kelvin
$$
D. $$
8.87 \times 10^{-24}\ joules\ per\ degree\ Kelvin
$$
E. $$
1.19 \times 10^{-23}\ joules\ per\ degree\ Kelvin
$$
F. $$
6.72 \times 10^{-23}\ joules\ per\ degree\ Kelvin
$$
G. $$
5.29 \times 10^{-22}\ joules\ per\ degree\ Kelvin
$$
H. $$
4.58 \times 10^{-25}\ joules\ per\ degree\ Kelvin
$$
I. $$
7.33 \times 10^{-21}\ joules\ per\ degree\ Kelvin
$$
J. $$
1.25 \times 10^{-20}\ joules\ per\ degree\ Kelvin
$$
Answer: E
What percentage of the total number of molecules have a speed that does not differ from the most probable speed by more than 1%?

Options:
A. $$
1. 5 2 \%
$$
B. $$
2. 5 7 \%
$$
C. $$
2. 0 4 \%
$$
D. $$
2. 1 9 \%
$$
E. $$
1. 3 9 \%
$$
F. $$
1. 4 1 \%
$$
G. $$
1. 6 6 \%
$$
H. $$
1. 8 3 \%
$$
I. $$
1. 9 5 \%
$$
J. $$
2. 3 2 \%
$$
Answer: G
The constant pressure expansion coefficient and the constant volume pressure coefficient of 1 mole of gas are respectively
$$
\alpha={\frac{\mathrm{R}} {\mathrm{p V}}} \text{ and } \beta={\frac{1} {\mathrm{T}}}
$$
Try to find the equation of state for this gas. The answer is ______.

Options:
A. $$
p(V+T)=RT
$$
B. $$
pV=RT
$$
C. $$
p(V+C)=R(T-C)
$$
D. $$
pV=R(T-C)
$$
E. $$
pV=R
$$
F. $$
p(V-C)=RT
$$
G. $$
pV=RT+C
$$
H. $$
p(V+C)=R(T+C)
$$
I. $$
pV=R(T+p)
$$
J. $$
p(V-D)=R(T+C)
$$
Answer: F
Perform the following operations on the given complex numbers.
(1) Express the complex number $z = -1 + 3 \mathrm{i}$ in exponential form;
(2) Calculate the complex number $\left( \frac{1-\sqrt{3} \, \mathrm{i}}{2} \right)^{3}$.

Options:
A. $$
\sqrt{3 0} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (- 7 )+\pi]}
$$
$$
4 + \mathrm{i}
$$
B. $$
\sqrt{1 0} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-3 )+\pi]}
$$
$$
- 1
$$
C. $$
\sqrt{1 4} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (- 2 )+\pi/4]}
$$
$$
0
$$
D. $$
\sqrt{2} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-8 )+\pi/2]}
$$
$$
- 5
$$
E. $$
\sqrt{1 2} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-1 )+\pi/3]}
$$
$$
1 + \mathrm{i}
$$
F. $$
\sqrt{5} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (2 )+\pi /2]}
$$
$$
- 8
$$
G. $$
\sqrt{2 5} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-5 )+\pi/3]}
$$
$$
1
$$
H. $$
\sqrt{9} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-4 )-\pi]}
$$
$$
- 1 + \mathrm{i}
$$
I. $$
\sqrt{7} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (-6 )]}
$$
$$
- 1 -\mathrm{i}
$$
J. $$
\sqrt{6} \, \mathrm{e}^{i [ \operatorname{a r c t a n} (3 )+2\pi]}
$$
$$
2 - \mathrm{i}
$$
Answer: B
Find the values of $(1+i)^i$ and $3^i$.

Options:
A. $$
\mathrm{e}^{\mathrm{i l n} 3} \cdot\mathrm{e}^{-\left( \frac{\pi} {8}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 7} \cdot\mathrm{e}^{-4 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
B. $$
\mathrm{e}^{\mathrm{i l n} 1} \cdot\mathrm{e}^{-\left( \frac{\pi} {12}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 6} \cdot\mathrm{e}^{-5 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
C. $$
\mathrm{e}^{\mathrm{i l n} 4} \cdot\mathrm{e}^{-\left( \frac{\pi} {3}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 2} \cdot\mathrm{e}^{-3 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
D. $$
\mathrm{e}^{\mathrm{i l n} 7} \cdot\mathrm{e}^{-\left( \frac{5\pi} {4}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 2} \cdot\mathrm{e}^{-8 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
E. $$
\mathrm{e}^{\mathrm{i l n} 6} \cdot\mathrm{e}^{-\left( \frac{3\pi} {5}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 8} \cdot\mathrm{e}^{-7 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
F. $$
\mathrm{e}^{\mathrm{i l n} 8} \cdot\mathrm{e}^{-\left( \frac{\pi} {10}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 2} \cdot\mathrm{e}^{-9 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
G. $$
\mathrm{e}^{\mathrm{i l n} \pi} \cdot\mathrm{e}^{-\left( \frac{\pi} {6}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 5} \cdot\mathrm{e}^{- k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
H. $$
\mathrm{e}^{\mathrm{i l n} \sqrt{2}} \cdot\mathrm{e}^{-\left( \frac{\pi} {4}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 3} \cdot\mathrm{e}^{-2 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
I. $$
\mathrm{e}^{\mathrm{i l n} 2} \cdot\mathrm{e}^{-\left( \frac{\pi} {4}+3 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 4} \cdot\mathrm{e}^{- k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
J. $$
\mathrm{e}^{\mathrm{i l n} 5} \cdot\mathrm{e}^{-\left( \frac{\pi} {2}+2 k \pi\right)} \quad( k=0, \pm1, \pm2, \cdots)
$$
$$
\mathrm{e}^{\operatorname{i l n} 1} \cdot\mathrm{e}^{-6 k \pi} \quad( k=0, \pm1, \pm2, \cdots)
$$
Answer: H
For a given fluid flow potential $f(z) = \frac{1}{z^{2}-1}$, try to respectively determine the flow rate and circulation along the circles (1) $C_{1}: \mid z-1 \mid=\frac{1}{2}$, (2) $C_{2}: \mid z+1 \mid=\frac{1}{2}$, and (3) $C_{3} : \mid z \mid=3$.

Options:
A. $$2$$
$$2$$
$$2$$
B. $$-3$$
$$-3$$
$$-3$$
C. $$i$$
$$i$$
$$i$$
D. $$-2$$
$$-2$$
$$-2$$
E. $$-\frac{1}{2}$$
$$-\frac{1}{2}$$
$$-\frac{1}{2}$$
F. $$3$$
$$3$$
$$3$$
G. $$1$$
$$1$$
$$1$$
H. $$0$$
$$0$$
$$0$$
I. $$0.5$$
$$0.5$$
$$0.5$$
J. $$-1$$
$$-1$$
$$-1$$
Answer: H
Write the power series expansion of $\mathrm{e}^{z} \operatorname{ln}(1+z)$ up to the term containing $z^{5}$, where $\operatorname{ln}(1+z) \, \bigg|_{z=0}=0.$

Options:
A. $$
z+\frac{2 z^{2}} {5}+\frac{z^{3}} {4}+\frac{z^{5}} {18}+\cdots\quad( \mid z \mid< 1 )
$$
B. $$
z+\frac{z^{2}} {5}+\frac{3 z^{3}} {7}+\frac{3 z^{4}} {8}+\cdots\quad( \mid z \mid< 1 )
$$
C. $$
z+\frac{z^{2}} {7}+\frac{z^{4}} {10}+\frac{2 z^{3}} {5}+\cdots\quad( \mid z \mid< 1 )
$$
D. $$
z+\frac{z^{2}} {2}+\frac{z^{3}} {3}+\frac{3 z^{5}} {4 0}+\cdots\quad( \mid z \mid< 1 )
$$
E. $$
z+\frac{z^{2}} {3}+\frac{2 z^{3}} {5}+\frac{z^{4}} {6}+\cdots\quad( \mid z \mid< 1 )
$$
F. $$
z+\frac{z^{2}} {3}+\frac{z^{3}} {5}+\frac{z^{5}} {15}+\cdots\quad( \mid z \mid< 1 )
$$
G. $$
z+\frac{z^{3}} {3}+\frac{z^{4}} {8}-\frac{2 z^{5}} {5}+\cdots\quad( \mid z \mid< 1 )
$$
H. $$
z+\frac{z^{2}} {4}+\frac{z^{3}} {9}+\frac{z^{5}} {25}+\cdots\quad( \mid z \mid< 1 )
$$
I. $$
z+\frac{z^{2}} {6}+\frac{z^{3}} {8}-\frac{z^{5}} {10}+\cdots\quad( \mid z \mid< 1 )
$$
J. $$
z+\frac{z^{4}} {3}-\frac{z^{3}} {6}+\frac{z^{5}} {9}+\cdots\quad( \mid z \mid< 1 )
$$
Answer: D
Examine the singularity type of the function
$$
f(z) = \sin\left(\frac{1}{\sin\frac{1}{z}}\right).
$$

Options:
A. $$
z=\frac{1}{k \pi^2}(k=\pm 1,\pm 2, \cdots) $$
and $$z=\infty
$$
B. $$
z=\frac{1}{k \pi}(k=\pm 1,\pm 2, \cdots) $$
and$$ z=0
$$
C. $$
z=\frac{1}{k}(k=\pm 1,\pm 2, \cdots) $$
and $$ z=\infty
$$
D. $$
z=\frac{k}{\pi}(k=\pm 1,\pm 2, \cdots) $$
and $$ z=\infty
$$
E. $$
z=\frac{\pi}{k}(k=\pm 1,\pm 2, \cdots) $$
and$$ z=\infty
$$
F. $$
z=\frac{1} {k \pi} ( k=\pm1, \pm2, \cdots) $$
and$$ z=\infty
$$
G. $$
z=\frac{1}{2k \pi}(k=\pm 1,\pm 2, \cdots) $$
and$$ z=\infty
$$
H. $$
z=\frac{1}{k \pi}(k=\pm 1,\pm 2, \cdots) $$
and$$ z=-\infty
$$
I. $$
z=\frac{1}{k \pi}(k=0,\pm 1,\pm 2, \cdots) $$
and
$$ z=0
$$
J. $$
z=\frac{1}{k \pi}(k=\pm 1,\pm 2, \cdots) $$
and$$ z=1
$$
Answer: F