MikaStars39/nano-eval huggingface dataset:
Gribble Corporation is developing a 64-bit computer architecture, which means that up to $64$-bit addresses can be used when accessing memory. Assuming virtual paged memory management is used, the corresponding address mapping mechanism now needs to be designed for this machine.
Assuming each page table entry is 8B in length, and a four-level page table structure must be adopted, with each page table in each level required to fit precisely within a single physical page, what is the maximum page size that the system can support under these conditions?

Options:
A. $$
5 1 2 \mathrm{K B}
$$
B. $$
8 1 9 2 \mathrm{K B}
$$
C. $$
2 0 \mathrm{K B}
$$
D. $$
1 2 \mathrm{K B}
$$
E. $$
3 2 \mathrm{K B}
$$
F. $$
6 4 \mathrm{K B}
$$
G. $$
1 0 2 4 \mathrm{K B}
$$
H. $$
4 8 \mathrm{K B}
$$
I. $$
2 5 6 \mathrm{B}
$$
J. $$
1 6 \mathrm{K B}
$$
Answer: E
In the index node of the file, there are 10 direct index pointers, 1 first-level index pointer, and 1 second-level index pointer. The disk block size is 1KB, and each index pointer occupies 4B. If the index node of a file is already in memory, to load the disk blocks corresponding to the file offsets (addressed in bytes) at 1234 and 307400 into memory, the number of disk blocks that need to be accessed are ( ) and ( ).

Options:
A. $$
0, ~ 2
$$
B. $$
3, ~ 4
$$
C. $$
1, ~ 5
$$
D. $$
1, ~ 3
$$
E. $$
1, ~ 1
$$
F. $$
1, ~ 4
$$
G. $$
2, ~ 1
$$
H. $$
2, ~ 2
$$
I. $$
0, ~ 3
$$
J. $$
2, ~ 3
$$
Answer: D
In a certain file system, the external storage is a hard disk. The size of a physical block is 512B. There is a file A containing 598 records, with each record taking up 255B, and each physical block holds 2 records. The directory where file A is located is shown in the following figure. The file directory adopts a multi-level tree directory structure, consisting of a root directory node, intermediate nodes serving as directory files, and leaf nodes as information files. Each directory entry occupies 127B, with each physical block holding $4$ directory entries, and the first block of the root directory is resident in memory.
```
root
├── bin
│ ├── like
│ ├── marg
│ ├── mik
│ └── za
├── dev
├── etc
├── boot
├── usr
│ ├── you
│ └── he
├── tmp
├── file1
├── dir1 (A B C )
└── dir2 (D E)
```
If the physical structure of the file is stored using a linked storage method, with each link pointer taking up 2B, how many hard disk accesses are necessary at a minimum to read file A into memory?

Options:
A. $$
3 5 1
$$
B. $$
3 1 0
$$
C. $$
3 0 6
$$
D. $$
3 0 4
$$
E. $$
2 9 8
$$
F. $$
2 8 7
$$
G. $$
2 7 9
$$
H. $$
3 1 1
$$
I. $$
3 2 3
$$
J. $$
4 0 4
$$
Answer: D
According to the temperature requirements of vegetables, Chinese cabbage belongs to () vegetables, while eggplant belongs to () vegetables.

Options:
A. Hardy, semi-hardy
B. Frost-hardy, cool-loving
C. Cool-loving, semi-hardy
D. Semi-tender, hardy
E. Cold-hardy, heat-loving
F. Heat-loving, frost-hardy
G. Warm-loving, hardy
H. Semi-hardy, warm-loving
I. Tender, semi-tender
J. Cool-season, tender
Answer: H
The formation of a cabbage head marks the end of the ____ phase.

Options:
A. Growth
B. Maturation
C. Flowering
D. Germination
E. Ripening
F. Harvesting
G. Seedling
H. Vegetative
I. Dormancy
J. Sprouting
Answer: G
Onion and green onion seeds ______ are very similar. Their main difference is that the outer skin of onion seeds is _____ and irregularly arranged, while the outer skin of green onion seeds is _____ and neatly arranged.

Options:
A. Shape, color, size, more wrinkles, fewer wrinkles
B. Size, scent, texture, packed tightly, unevenly arranged
C. Scent, layout, structure, densely packed, loosely packed
D. Hue, texture, width, smooth lines, rough patches
E. Texture, size, hue, thin layers, condensed neatly
F. Thickness, texture, color, scattered loosely, structured closely
G. Color, pattern, length, irregular surface, aligned surface
H. Pattern, shape, hue, inconsistently placed, uniformly layered
I. Texture, hue, width, fewer wrinkles, more wrinkles
J. Shape, scent, color, less regular, more regular
Answer: A
The national anthem of Austria was composed by ________.

Options:
A. Mahler
B. Brahms
C. Bruckner
D. Wagner
E. Mozart
F. Beethoven
G. Haydn
H. Offenbach
I. Strauss
J. Schubert
Answer: G
What is the maximum possible parallax of Mercury during a solar transit, as seen from either end of a 3000-km baseline on Earth?

Options:
A. $$
0. 0 3 1^{\circ}
$$
B. $$
0. 0 0 9^{\circ}
$$
C. $$
0. 0 2 0^{\circ}
$$
D. $$
0. 0 4 5^{\circ}
$$
E. $$
0. 0 0 2 2^{\circ}
$$
F. $$
0. 0 1 2^{\circ}
$$
G. $$
0. 0 1 8^{\circ}
$$
H. $$
0. 0 2 9^{\circ}
$$
I. $$
0. 0 3 5^{\circ}
$$
J. $$
0. 0 0 5^{\circ}
$$
Answer: E
How many 100-km-diameter rocky ($3000 kg/m^{3}$) planetesimals would have been needed to form Earth?

Options:
A. 490,000 planetesimals
B. 475,000 planetesimals
C. 450,000 planetesimals
D. 500,000 planetesimals
E. 480,000 planetesimals
F. 525,000 planetesimals
G. 470,000 planetesimals
H. 465,000 planetesimals
I. 460,000 planetesimals
J. 505,000 planetesimals
Answer: B
A supernova of luminosity 1 billion times the luminosity of the Sun is used as a standard candle to measure the distance to a faraway galaxy. From Earth, the supernova appears as bright as the Sun would appear from a distance of 10 kpc. What is the distance to the galaxy?

Options:
A. $$
2 8 0 \mathrm{~ M p c}
$$
B. $$
2 5 0 \mathrm{~ M p c}
$$
C. $$
3 2 0 \mathrm{~ M p c}
$$
D. $$
4 5 0 \mathrm{~ M p c}
$$
E. $$
4 2 0 \mathrm{~ M p c}
$$
F. $$
3 8 0 \mathrm{~ M p c}
$$
G. $$
5 0 0 \mathrm{~ M p c}
$$
H. $$
3 6 0 \mathrm{~ M p c}
$$
I. $$
4 0 0 \mathrm{~ M p c}
$$
J. $$
2 1 0 \mathrm{~ M p c}
$$
Answer: C
The Andromeda Galaxy is approaching our Galaxy with a radial welocity of 120 km/s. Given the galaxies' present separation of 800 kpc, and neglecting both the transverse component of the velocity and the effect of gravity in accelerating the motion, estimate when the two galaxies will collide?

Options:
A. 10.1 billion years
B. 8.3 billion years
C. 1.7 billion years
D. 6.5 billion years
E. 9.6 billion years
F. 5.4 billion years
G. 2.9 billion years
H. 7.8 billion years
I. 3.1 billion years
J. 4.2 billion years
Answer: D
If the entire universe were filled with Milky Way-like gal- p_ axies, with an average density of 0.1 galaxy per cubic mega- 30 parsec, calculate the total number of galaxies observable bythe survey in Problem 1 is____, if it covered the entire sky.

Options:
A. 350,000,000
B. 380,000,000
C. 420,000,000
D. 480,000,000
E. 470,000,000
F. 450,000,000
G. 400,000,000
H. 320,000,000
I. 500,000,000
J. 390,000,000
Answer: C
Given that the threshold temperature for the production
of electron-positron pairs is about $6 \times1 0^{9}$ K and that
a proton is 1800 times more massive than an electron,
calculate the threshold temperature for proton-antiproton
pair production is().

Options:
A. $$
8. 0 \times1 0^{1 2} \mathrm{K}
$$
B. $$
4. 8 \times1 0^{1 3} \mathrm{K}
$$
C. $$
1. 1 \times1 0^{1 3} \mathrm{K}
$$
D. $$
1. 0 \times1 0^{1 4} \mathrm{K}
$$
E. $$
5. 0 \times1 0^{1 1} \mathrm{K}
$$
F. $$
2. 5 \times1 0^{1 2} \mathrm{K}
$$
G. $$
3. 5 \times1 0^{1 3} \mathrm{K}
$$
H. $$
9. 1 \times1 0^{1 2} \mathrm{K}
$$
I. $$
6. 2 \times1 0^{1 1} \mathrm{K}
$$
J. $$
7. 0 \times1 0^{1 4} \mathrm{K}
$$
Answer: C
The radius of an atom of krypton (Kr) is about 1.9 $^ {}\AA$ . If the atom is assumed to be a sphere, what is the volume in cm³ of a single Kr atom?

Options:
A. $2. 7 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
B. $2. 4 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
C. $2. 0 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
D. $3. 5 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
E. $3. 0 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
F. $1. 8 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
G. $2. 9 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
H. $3. 1 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
I. $2. 2 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
J. $3. 3 \times1 0^{-2 3} \, \mathrm{c m}^{3}$
Answer: G
The natural abundance of He is 0.000137%. (a) How many protons, neutrons, and electrons are in an atom of 3He?

Options:
A. 3 protons, 2 neutrons, and 3 electrons
B. 2 protons, 3 neutrons, and 2 electrons
C. 3 protons, 1 neutron, and 3 electrons
D. 2 protons, 1 neutron, and 2 electrons
E. 1 proton, 1 neutron, and 1 electron
F. 2 protons, 0 neutrons, and 2 electrons
G. 2 protons, 1 neutron, and 1 electron
H. 2 protons, 2 neutrons, and 2 electrons
I. 1 proton, 2 neutrons, and 1 electron
J. 2 protons, 1 neutron, and 3 electrons
Answer: D
The first atoms of seaborgium (Sg) were identified in 1974. The longest-lived isotope of Sg has a mass number of 266.Atoms of Sg are very unstable,and it is therefore difficult to study this element's prop- erties. Based on the position of Sg in the periodic table, what element should it most closely resemble in its chemical properties?

Options:
A. Mo
B. Re
C. Nb
D. W
E. Cr
F. Os
G. Co
H. Ru
I. Ti
J. Ta
Answer: D
When hydrogensulfide gas is bubbled into a solution of sodium hydroxide, the reaction forms sodium sulfide and water. How many grams of sodium sulfide are formed if 1.50 $g$ of hydrogen sulfide is bubbled into a solution containing 2.00 $g$ of sodium hydroxide, assuming that the sodium sulfide is made in 92.0% yield?

Options:
A. $$
1. 9 5 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
B. $$
2. 1 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
C. $$
2. 8 5 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
D. $$
1. 7 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
E. $$
0. 9 5 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
F. $$
1. 4 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
G. $$
1. 2 5 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
H. $$
1. 3 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
I. $$
1. 8 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
J. $$
2. 2 0 \, \mathrm{g \, N a_{2} S \, a c t u a l}
$$
Answer: I
An element $\mathsf{X}$ forms an iodide (XI$_{3}$) and a chloride (XCl$_{3}$). The iodide is quantitatively converted to the chloride when it is heated in a stream of chlorine:
$$
2X \mathrm{I}_{3}+3 \ \mathrm{C l}_{2} \longrightarrow2X \mathrm{C l}_{3}+3 \ \mathrm{I}_{2}
$$
If 0.5000 $g$ of XI5 is treated, 0.2360 $\mathbf{g}$ of XCl$_{3}$ is obtained.Calculate the atomic weight of the element X.

Options:
A. $$
1 2 9. 8 \, \mathrm{g}
$$
B. $$
1 3 8. 9 \, \mathrm{g}
$$
C. $$
1 3 4. 2 \, \mathrm{g}
$$
D. $$
1 1 9. 6 \, \mathrm{g}
$$
E. $$
1 4 5. 4 \, \mathrm{g}
$$
F. $$
1 6 3. 5 \, \mathrm{g}
$$
G. $$
1 4 2. 3 \, \mathrm{g}
$$
H. $$
1 4 9. 1 \, \mathrm{g}
$$
I. $$
1 2 5. 7 \, \mathrm{g}
$$
J. $$
1 5 0. 8 \, \mathrm{g}
$$
Answer: B
Hydrogen cyanide, HCN, is a poisonous gas. The lethal dose is approximately 300 mg HCN per kilogram of air when inhaled.Calculate the amount of HCN that gives the lethal dose in a small laboratory room measuring $1 2 \, \times\, 1 5 \, \times\, 8. 0$ ft is(). The density of air at 26°C is 0.00118 g/cm$^{3}$.

Options:
A. $$
\mathrm{2 0 g ~ H C N}
$$
B. $$
\mathrm{1 3 g ~ H C N}
$$
C. $$
\mathrm{1 4 g ~ H C N}
$$
D. $$
\mathrm{1 5 g ~ H C N}
$$
E. $$
\mathrm{1 0 g ~ H C N}
$$
F. $$
\mathrm{1 2 g ~ H C N}
$$
G. $$
\mathrm{1 1 g ~ H C N}
$$
H. $$
\mathrm{9 g ~ H C N}
$$
I. $$
\mathrm{1 6 g ~ H C N}
$$
J. $$
\mathrm{1 8 g ~ H C N}
$$
Answer: C
Which was the earliest ballet produced in the New China after its establishment?

Options:
A. "Red Detachment of Women"
B. "Peony Pavilion Revival"
C. "Great Leap Forward Ballet"
D. "Cultural Blossoms Scene"
E. "Revolutionary Spring Ballet"
F. "Song of the Arirang"
G. "White-Haired Girl Dance"
H. "Shanghai Dawn Ensemble"
I. "Swan Lake Performance"
J. "Phoenix Saga Ballet"
Answer: A
Which ethnic group's "jumping string" dance has inspired the "rocking step" seen in Yunnan flower lantern performances?

Options:
A. Lisu Ethnic Group
B. Hani Ethnic Group
C. Yi Ethnic Group
D. Naxi Ethnic Group
E. Tibetan Ethnic Group
F. Dai Ethnic Group
G. Zhuang Ethnic Group
H. Jingpo Ethnic Group
I. Miao Ethnic Group
J. Bai Ethnic Group
Answer: F
The melody "Two Violations of Xiqin Opera" originates from ________.

Options:
A. "The Rain in the Valley"
B. "The Petals in the Well"
C. "The Whisper of the Pines"
D. "The Flower on the Path"
E. "The Lotus in the Bowl"
F. "The Moon over the Bridge"
G. "The Sparrow in the Forest"
H. "The Shadow of the Lantern"
I. "The Blossom by the Window"
J. "The Breeze through the Garden"
Answer: E
The Sichuan opera actor who won the Plum Blossom Award for Chinese Theatre is ______.

Options:
A. Liu Xiaoli
B. Zhou Mingjie
C. Li Deyun
D. Zhang Huijin
E. Wang Yuyan
F. Xu Minzhi
G. Shen Tiemei
H. Fang Weihao
I. Chen Linghong
J. Ye Guanghua
Answer: G
What is the composition of the equilibrium system of $NaNO_3$ and KCL dissolved in water? How many phases can coexist in equilibrium at most?

Options:
A. $K=1,\Phi_{\mathrm{m a x}}=2$
B. $K=2,\Phi_{\mathrm{m a x}}=4$
C. $K=3,\Phi_{\mathrm{m a x}}=4$
D. $K=1,\Phi_{\mathrm{m a x}}=3$
E. $K=2,\Phi_{\mathrm{m a x}}=5$
F. $K=3,\Phi_{\mathrm{m a x}}=5$
G. $K=4,\Phi_{\mathrm{m a x}}=7$
H. $K=4,\Phi_{\mathrm{m a x}}=6$
I. $K=5,\Phi_{\mathrm{m a x}}=6$
J. $K=3,\Phi_{\mathrm{m a x}}=6$
Answer: F
At 298K, when an amount of substance $n$ of sodium chloride $(NaCl)$ is dissolved in 1000g of water, the constant-pressure thermal effect is related to $n (NaCl)$ as follows:
$$
\Delta H \,=\, {\bigg[} 3858.14 {\bigg(} {\frac{n} {\mathrm{mol}}} {\bigg)} \,+\, 1989.18 {\bigg(} {\frac{n} {\mathrm{mol}}} {\bigg)}^{3 / 2}-3009.60 {\bigg(} {\frac{n} {\mathrm{mol}}} {\bigg)}^{2} \,+\, 1017.83 {\bigg(} {\frac{n} {\mathrm{mol}}} {\bigg)}^{5 / 2}]J
$$
Calculate the differential heat of dissolution for a solution concentration of $1mol \cdot dm^{-3}$;

Options:
A. $$
3 6 0 0. 0 7 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
B. $$
3 2 5 8. 7 4 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
C. $$
3 5 0 0. 1 2 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
D. $$
3 3 2 1. 5 2 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
E. $$
3 4 9 8. 9 3 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
F. $$
3 3 6 9. 0 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
G. $$
3 2 9 9. 0 8 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
H. $$
3 3 7 0. 9 9 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
I. $$
3 4 5 6. 0 5 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
J. $$
3 3 4 8. 7 1 \; \mathrm{J} \cdot\mathrm{m o l}^{-1}
$$
Answer: F
At 298K and $p^{\Theta}$, 0.782 mol of $N_{2}(g)$, 0.209 mol of $O_{2}(g)$, and 0.009 mol of $Ar(g)$ are mixed under isothermal and isobaric conditions. What are the values of $\Delta_{\operatorname*{mix}}V$, $\Delta_{\operatorname*{mix}}H$, $\Delta_{\operatorname*{mix}}S$, and $\Delta_{\operatorname*{mix}}G$ for the gas mixture?

Options:
A. $$
0, \ 0, \ 3. 7 8 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 2 4 9 \mathrm{k J}
$$
B. $$
0, \ 0, \ 4. 8 5 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 4 2 8 \mathrm{k J}
$$
C. $$
0, \ 0, \ 4. 6 7 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 3 9 2 \mathrm{k J}
$$
D. $$
0, \ 0, \ 4. 3 0 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 3 6 9 \mathrm{k J}
$$
E. $$
0, \ 0, \ 5. 8 4 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 5 7 9 \mathrm{k J}
$$
F. $$
0, \ 0, \ 5. 6 1 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 5 3 7 \mathrm{k J}
$$
G. $$
0, \ 0, \ 4. 5 0 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 4 0 0 \mathrm{k J}
$$
H. $$
0, \ 0, \ 6. 7 2 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 6 2 4 \mathrm{k J}
$$
I. $$
0, \ 0, \ 5. 2 3 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 5 0 8 \mathrm{k J}
$$
J. $$
0, \ 0, \ 3. 9 1 \mathrm{J} \cdot\mathrm{K}^{-1}, \ -1. 2 8 5 \mathrm{k J}
$$
Answer: C
Given $\theta_{\mathrm{r}}=12.1 \mathrm{~K}$ and $\theta_{\mathrm{v}}=3700 \mathrm{~K}$, what is the ratio of the number of HBr ideal gas molecules in the state $v=2, J=5$ to those in the state $v=1, J=2$ at 1000 K?

Options:
A. 0.0406
B. 0.0523
C. 0.0338
D. 0.0561
E. 0.0274
F. 0.0489
G. 0.0382
H. 0.0451
I. 0.0637
J. 0.0425
Answer: A
The Apollo 12 probe collected a rock sample from the Moon, which contains potassium at a concentration of $m(K) = 0.0166 kg$ per kilogram. The fraction of ${^{40}K}$ in the sample is $1.18 \times 10^{-4}$, and the volume of ${^{40} Ar}$ gas collected from the rock sample is $81.9 \times 10^{-7} m^3$ (Equivalent to standard condition STP). What is the age of the rock?

Options:
A. $$
7. 0 1 \times1 0^{9} \mathrm{~ a}
$$
B. $$
1. 0 4 \times1 0^{9} \mathrm{~ a}
$$
C. $$
6. 7 5 \times1 0^{9} \mathrm{~ a}
$$
D. $$
5. 4 2 \times1 0^{9} \mathrm{~ a}
$$
E. $$
8. 1 6 \times1 0^{9} \mathrm{~ a}
$$
F. $$
7. 7 4 \times1 0^{9} \mathrm{~ a}
$$
G. $$
9. 2 8 \times1 0^{9} \mathrm{~ a}
$$
H. $$
6. 5 9 \times1 0^{9} \mathrm{~ a}
$$
I. $$
8. 9 3 \times1 0^{9} \mathrm{~ a}
$$
J. $$
7. 9 3 \times1 0^{9} \mathrm{~ a}
$$
Answer: J
At 298K, it is known that $\phi^{\Theta} ( Fe^{2+} \mid Fe) = -0.440 V, \phi^{\Theta} ( Fe^{3+}, Fe^{2+} \mid Pt) = 0.771 V.$ What is the value of $\phi^{\Theta} ( Fe^{3+} \mid Fe )$?

Options:
A. -0.150V
B. 0.224V
C. 0.0841V
D. -0.0363V
E. -0.637V
F. -0.442V
G. 0.302V
H. -0.498V
I. 0.187V
J. -0.120V
Answer: D
Two capillary tubes with radii of 0.20 mm and 0.10 mm, respectively, are inserted into $H_{2}O_{2}(l)$. The height difference of the liquid rise in the capillary tubes is $\Delta h = 5.50 cm$ at $298 K$, and the density of $H_{2}O_{2}$ is $\rho(H_{2}O_{2}) = 1.41 g \cdot cm^{-3}$. What is the value of $\gamma$?

Options:
A. $$
6. 8 4 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
B. $$
6. 2 7 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
C. $$
5. 6 1 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
D. $$
8. 3 2 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
E. $$
7. 6 1 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
F. $$
4. 9 8 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
G. $$
7. 0 4 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
H. $$
7. 4 3 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
I. $$
9. 1 5 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
J. $$
7. 9 0 \times1 0^{-2} \, \mathrm{N} \cdot{\mathrm{m}}^{-1}
$$
Answer: E
Consider a flat plate at zero angle of attack in a hypersonic flow at Mach 10 at standard sea level conditions. At a point 0.5 m downstream from the leading edge, the local shear stress at the wall is 282 N/m$^{2}$. The gas temperature at the wall is equal to standard sea level temperature. At this point, calculate the velocity gradient at the wall normal to the wall.

Options:
A. $$
1. 5 2 5 \times1 0^{7} \ s e c^{-1}
$$
B. $$
1. 5 9 0 \times1 0^{7} \ s e c^{-1}
$$
C. $$
1. 6 2 0 \times1 0^{7} \ s e c^{-1}
$$
D. $$
1. 6 0 0 \times1 0^{7} \ s e c^{-1}
$$
E. $$
1. 5 7 6 \times1 0^{7} \ s e c^{-1}
$$
F. $$
1. 5 3 0 \times1 0^{7} \ s e c^{-1}
$$
G. $$
1. 4 9 0 \times1 0^{7} \ s e c^{-1}
$$
H. $$
1. 5 5 0 \times1 0^{7} \ s e c^{-1}
$$
I. $$
1. 5 7 0 \times1 0^{7} \ s e c^{-1}
$$
J. $$
1. 5 8 0 \times1 0^{7} \ s e c^{-1}
$$
Answer: E
A Pitot tube on an airplane flying at standard sea level reads $1. 0 7 \times$ $1 0^{5} \, \mathrm{N / m}^{2}$ . What is the velocity of the airplane?

Options:
A. $$
8 4. 3 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
B. $$
9 8. 8 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
C. $$
7 7. 1 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
D. $$
1 0 1. 2 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
E. $$
1 2 0. 7 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
F. $$
9 5. 5 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
G. $$
9 3. 6 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
H. $$
1 0 5. 1 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
I. $$
6 9. 9 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
J. $$
8 7. 4 \ {\frac{\mathrm{m}} {\mathrm{s e c}}}
$$
Answer: B
Consider a finite wing with an aspect ratio of 6. Assume an elliptical lift distribution. The lift slope for the airfoil section is 0.1/degree. Calculate and compare the lift slopes for a straight wing.

Options:
A. $$
3. 2 7 \mathrm{~ p e r ~ r a d i a n}
$$
B. $$
2. 8 8 \mathrm{~ p e r ~ r a d i a n}
$$
C. $$
2. 7 9 \mathrm{~ p e r ~ r a d i a n}
$$
D. $$
3. 5 4 \mathrm{~ p e r ~ r a d i a n}
$$
E. $$
3. 6 2 \mathrm{~ p e r ~ r a d i a n}
$$
F. $$
3. 1 5 \mathrm{~ p e r ~ r a d i a n}
$$
G. $$
3. 4 0 \mathrm{~ p e r ~ r a d i a n}
$$
H. $$
2. 9 3 \mathrm{~ p e r ~ r a d i a n}
$$
I. $$
2. 8 5 \mathrm{~ p e r ~ r a d i a n}
$$
J. $$
3. 0 8 \mathrm{~ p e r ~ r a d i a n}
$$
Answer: A
Consider the isentropic flow over an airfoil. The freestream conditions are $T_{\infty}=2 4 5 \mathrm{~ K}$ and $p_{\infty}=4. 3 5 \times1 0^{4} \; \mathrm{N / m}^{2}$ . At a point on the airfoil, the pressure is $3. 6 \times1 0^{4} \, \mathrm{N / m}^{2}$ . Calculate the density at this point.

Options:
A. $$
0. 5 8 1 2 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
B. $$
0. 6 0 2 5 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
C. $$
0. 5 9 0 1 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
D. $$
0. 5 3 7 8 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
E. $$
0. 4 8 2 3 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
F. $$
0. 5 4 0 4 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
G. $$
0. 5 6 7 6 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
H. $$
0. 4 7 8 9 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
I. $$
0. 5 2 4 0 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
J. $$
0. 4 9 3 7 \ {\frac{\mathrm{k g}} {\mathrm{m}^{3}}}
$$
Answer: F
Consider air at a temperature of 230 K. Calculate the speed of sound.

Options:
A. $$
2 7 5 \mathrm{~ m / s e c}
$$
B. $$
3 1 5 \mathrm{~ m / s e c}
$$
C. $$
2 8 5 \mathrm{~ m / s e c}
$$
D. $$
2 8 0 \mathrm{~ m / s e c}
$$
E. $$
3 1 0 \mathrm{~ m / s e c}
$$
F. $$
2 9 0 \mathrm{~ m / s e c}
$$
G. $$
3 6 0 \mathrm{~ m / s e c}
$$
H. $$
3 2 0 \mathrm{~ m / s e c}
$$
I. $$
3 2 5 \mathrm{~ m / s e c}
$$
J. $$
3 0 4 \mathrm{~ m / s e c}
$$
Answer: J
A slender missile is flying at Mach 1.5 at low altitude. Assume the wave generated by the nose of the missile is a Mach wave. This wave intersects the ground 559 ft behind the nose. At what altitude is the missile flying?

Options:
A. $$
{h=5 1 0 ~ f t}
$$
B. $$
{h=5 4 0 ~ f t}
$$
C. $$
{h=5 0 0 ~ f t}
$$
D. $$
{h=6 0 0 ~ f t}
$$
E. $$
{h=5 5 0 ~ f t}
$$
F. $$
{h=4 9 0 ~ f t}
$$
G. $$
{h=6 2 0 ~ f t}
$$
H. $$
{h=4 6 0 ~ f t}
$$
I. $$
{h=4 8 0 ~ f t}
$$
J. $$
{h=4 5 0 ~ f t}
$$
Answer: C
Consider a subsonic compressible flow in cartesian coordinates where the velocity potential is given by
$$
\phi( x, y )=V_{\infty} x+\frac{7 0 \operatorname{s i n} ( 2 \pi x )} {\sqrt{1-M_{\infty}^{2}}} e^{-2 \pi y \sqrt{1-M_{\infty}^{2}}}
$$
If the freestream properties are given by $V_{\infty}=7 0 0$ ft/s, $p_{\infty}=1$ atm, and $T_{\infty}=5 1 9^{\circ} \mathrm{R}$ , calculate the following properties at the location $( x, y )=$ (0.2 ft, 0.2 ft): $M, \, p$ , and $T$ is ().

Options:
A. $$
0. 7 0 4 5
$$
$$
0. 9 4 0 \mathrm{~ a t m}
$$
$$
5 0 7. 0^{\circ} \mathrm{R}
$$
B. $$
0. 7 0 6 7
$$
$$
0. 9 3 3 \mathrm{~ a t m}
$$
$$
5 0 8. 6^{\circ} \mathrm{R}
$$
C. $$
0. 7 0 6 0
$$
$$
0. 9 2 5 \mathrm{~ a t m}
$$
$$
5 0 9. 1^{\circ} \mathrm{R}
$$
D. $$
0. 7 0 9 1
$$
$$
0. 8 9 0 \mathrm{~ a t m}
$$
$$
5 0 5. 8^{\circ} \mathrm{R}
$$
E. $$
0. 7 0 7 2
$$
$$
0. 8 8 5 \mathrm{~ a t m}
$$
$$
5 0 6. 5^{\circ} \mathrm{R}
$$
F. $$
0. 7 0 6 5
$$
$$
0. 9 3 6 \mathrm{~ a t m}
$$
$$
5 0 7. 7^{\circ} \mathrm{R}
$$
G. $$
0. 7 0 5 6
$$
$$
0. 9 4 7 \mathrm{~ a t m}
$$
$$
5 0 8. 0^{\circ} \mathrm{R}
$$
H. $$
0. 7 0 8 0
$$
$$
0. 9 7 0 \mathrm{~ a t m}
$$
$$
5 1 1. 0^{\circ} \mathrm{R}
$$
I. $$
0. 7 0 5 0
$$
$$
0. 9 5 0 \mathrm{~ a t m}
$$
$$
5 1 0. 2^{\circ} \mathrm{R}
$$
J. $$
0. 7 0 8 3
$$
$$
0. 9 6 7 \mathrm{~ a t m}
$$
$$
5 1 3. 4^{\circ} \mathrm{R}
$$
Answer: B
Consider a hypersonic vehicle with a spherical nose flying at Mach 20 at a standard altitude of 150,000 ft, where the ambient temperature and pressure are 500*R and 3.06 1b/ft?, respectively. At the point on the surface of the nose located $2 0^{\circ}$ away from the stagnation point, estimate the pressure.

Options:
A. $$
{1 9 0. 5 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
B. $$
{1 8 5. 6 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
C. $$
{1 8 4. 7 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
D. $$
{1 9 3. 2 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
E. $$
{1 8 3. 4 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
F. $$
{1 8 7. 2 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
G. $$
{1 8 8. 8 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
H. $$
{1 9 1. 1 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
I. $$
{1 8 6. 9 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
J. $$
{1 8 9. 3 {\mathrm{lb}} / {\mathrm{f t}}^{2}}
$$
Answer: F
A thin plate is placed between two flat surfaces $h$ cm apart such that the viscosity of liquids on the top and bottom of the plate are $\mu_{1}$ and $\mu_{2}$ respectively. Determine the position of the thin plate such that the viscous resistance to uniform motion of the thin plate is minimum. [Assume $h$ to be very small].

Options:
A. $$
{\frac{y} {( h-y )}}={(h + y) / \mu_{2}}
$$
B. $$
{\frac{y} {( h-y )}}={\mu_{1} + \mu_{2}}
$$
C. $$
{\frac{y} {( h-y )}}={\sqrt{\mu_{1} / \mu_{2}}}
$$
D. $$
{\frac{y} {( h-y )}}={\mu_{2} / \mu_{1}}
$$
E. $$
{\frac{y} {( h-y )}}={(h - y) / \mu_{1}}
$$
F. $$
{\frac{y} {( h-y )}}={1 / (\mu_{1} + \mu_{2})}
$$
G. $$
{\frac{y} {( h-y )}}={(\mu_{1} \cdot \mu_{2}) / h}
$$
H. $$
{\frac{y} {( h-y )}}={h / (\mu_{1} + \mu_{2})}
$$
I. $$
{\frac{y} {( h-y )}}={h / \sqrt{\mu_{1} \cdot \mu_{2}}}
$$
J. $$
{\frac{y} {( h-y )}}={\sqrt{\mu_{1} \cdot \mu_{2}}}
$$
Answer: C
Using perfect gas law, estimate the density of (i) air and (ii) $\mathrm{C O_{2}}$ , at 60°C and 120 kPa (abs) pressure.

Options:
A. $$
1. 9 0 7 \ \mathrm{k g} / \mathrm{m}^{3}
$$
B. $$
1. 9 8 2 \ \mathrm{k g} / \mathrm{m}^{3}
$$
C. $$
2. 0 1 0 \ \mathrm{k g} / \mathrm{m}^{3}
$$
D. $$
1. 9 2 5 \ \mathrm{k g} / \mathrm{m}^{3}
$$
E. $$
1. 8 3 5 \ \mathrm{k g} / \mathrm{m}^{3}
$$
F. $$
1. 7 3 5 \ \mathrm{k g} / \mathrm{m}^{3}
$$
G. $$
1. 9 4 2 \ \mathrm{k g} / \mathrm{m}^{3}
$$
H. $$
1. 7 0 1 \ \mathrm{k g} / \mathrm{m}^{3}
$$
I. $$
1. 8 2 0 \ \mathrm{k g} / \mathrm{m}^{3}
$$
J. $$
1. 5 4 5 \ \mathrm{k g} / \mathrm{m}^{3}
$$
Answer: A
Find the pressure represented by a column of\
$$
2 \mathrm{~ c m ~ o f ~ m e r c u r y}.
$$

Options:
A. $$
2 8 0 0. 3 \ \mathrm{N / m}^{2}
$$
B. $$
2 7 1 0. 8 \ \mathrm{N / m}^{2}
$$
C. $$
2 5 9 9. 9 \ \mathrm{N / m}^{2}
$$
D. $$
2 6 5 0. 7 \ \mathrm{N / m}^{2}
$$
E. $$
2 7 0 0. 0 \ \mathrm{N / m}^{2}
$$
F. $$
2 6 8 0. 1 \ \mathrm{N / m}^{2}
$$
G. $$
2 6 6 2. 9 \ \mathrm{N / m}^{2}
$$
H. $$
2 6 1 0. 5 \ \mathrm{N / m}^{2}
$$
I. $$
2 7 5 0. 2 \ \mathrm{N / m}^{2}
$$
J. $$
2 6 4 5. 0 \ \mathrm{N / m}^{2}
$$
Answer: G
A hydrometer has a stem of diameter $D$ and a weight $W$ Find the distance on the stem $h$ between markings corresponding to relative densities of 1.0 and $S$ .

Options:
A. $$
\frac{W}{\gamma\left(\pi D^{3}\right)}\left(\frac{1}{S}+1\right)
$$
B. $$
\frac{W}{\gamma\left(\pi D^{2}\right)}\left(\frac{S+1}{S}\right)
$$
C. $$
\frac{W}{\gamma\left(\pi D^{2} / 2\right)}\left(1-\frac{1}{S}\right)
$$
D. $$
\frac{W} {\gamma\left( \pi D^{2} / 4 \right)} \left( \frac{1} {S}-1 \right)
$$
E. $$
\frac{W}{\pi \gamma \cdot D}\left(\frac{1}{S}-1\right)
$$
F. $$
\frac{\gamma \cdot W}{\left(\pi D^{2} / 4\right)}\left(\frac{1}{S}-1\right)
$$
G. $$
\frac{\pi \cdot W}{\gamma\left(D^{2} / 4\right)}\left(1-\frac{1}{S}\right)
$$
H. $$
\frac{W}{\gamma\left(\pi D^{2} / 8\right)}\left(\frac{S}{1}-1\right)
$$
I. $$
\frac{W}{\gamma \cdot D^{2}}\left(\frac{1}{S}+S\right)
$$
J. $$
\frac{\gamma \cdot W}{D^{2}}\left(1-\frac{1}{S}\right)
$$
Answer: D
The velocity vector in an incompressible flow is given by
$$
\mathbf{v}=( 6 x t+y z^{2} ) \mathbf{i}+( 3 t+x y^{2} ) \mathbf{j}+( x y-2 x y z-6 t z ) \mathbf{k}
$$
Determine the acceleration vector at point $\mathrm{A} \, ( 1, 1, 1 )$ at $t=1. 0.$

Options:
A. $$
3 6 \mathbf{i}+1 8 \mathbf{j}+3 0 \mathbf{k}
$$
B. $$
4 0 \mathbf{i}+1 8 \mathbf{j}+3 6 \mathbf{k}
$$
C. $$
3 8 \mathbf{i}+2 0 \mathbf{j}+4 2 \mathbf{k}
$$
D. $$
3 4 \mathbf{i}+1 5 \mathbf{j}+4 0 \mathbf{k}
$$
E. $$
3 0 \mathbf{i}+2 0 \mathbf{j}+4 0 \mathbf{k}
$$
F. $$
2 8 \mathbf{i}+1 2 \mathbf{j}+3 7 \mathbf{k}
$$
G. $$
3 8 \mathbf{i}+1 8 \mathbf{j}+3 9 \mathbf{k}
$$
H. $$
2 9 \mathbf{i}+1 9 \mathbf{j}+4 3 \mathbf{k}
$$
I. $$
4 0 \mathbf{i}+1 6 \mathbf{j}+3 8 \mathbf{k}
$$
J. $$
3 2 \mathbf{i}+1 4 \mathbf{j}+3 5 \mathbf{k}
$$
Answer: G
The stream function $\psi$ for uniform flow past a circular cylinder of radius $a$ in polar co-ordinates is maw $\psi=U_{\infty} \left( 1-{\frac{a^{2}} {r^{2}}} \right) r \, \operatorname{s i n} \, \theta\, \mathrm{w h e r e} \, \, U_{\infty}$ iste free stream velocity parallel to the horizontal axis Determine the velocity on the cylinder at $\theta=9 0^{\circ}$ .

Options:
A. $$
- \frac{U_{∞}}{\sqrt{2}}
$$
B. $$
U_{∞}
$$
C. $$
0
$$
D. $$
U_{∞} \ \cos{\theta}
$$
E. $$
- 2 \ U_{∞}
$$
F. $$
2 \ U_{∞}
$$
G. $$
- U_{∞} \ \sin{\theta}
$$
H. $$
- U_{∞}
$$
I. $$
\frac{U_{∞}}{2}
$$
J. $$
- \frac{U_{∞}}{2}
$$
Answer: E
Kerosene $( \rho=8 0 0 \, \mathrm{k g} / \mathrm{m}^{3} )$ . The outflow is to the atmosphere.
$$
Z_{0}=2 0. 0 \mathrm{~ m, ~} D_{1}=1 0 \mathrm{~ c m}
$$
$$
\mathrm{a n d} \qquad\qquad h_{1}=4 \ \mathrm{c m}
$$
$$
\begin{array} {l} {{{D_{2}=7 \mathrm{~ c m}}}} \\ {{{D_{3}=D_{4}=1 0 \mathrm{~ c m}}}} \\ {{{D_{5}=5 \mathrm{~ c m}}}} \\ \end{array}
$$
The pipe is horizontal. The fluid in the manometer is mercury. Neglecting friction, calculate (i) dis- charge $Q$ (ii) velocity $V_{5}$ (ii) $Z_{3}$ and (iv) $h_{4}$

Options:
A. $$
1 5. 1 2 0 \; \mathrm{L / s}
$$
$$
7. 4 8 9 2 \ \mathrm{m / s}
$$
$$
1 6. 4 4 5 3 \mathrm{~ m}
$$
$$
0. 2 3 1 \mathrm{~ m}
$$
B. $$
1 5. 6 2 5 \; \mathrm{L / s}
$$
$$
7. 9 5 7 7 \ \mathrm{m / s}
$$
$$
1 5. 6 8 1 4 \mathrm{~ m}
$$
$$
0. 1 7 0 \mathrm{~ m}
$$
C. $$
1 5. 3 4 7 \; \mathrm{L / s}
$$
$$
7. 5 6 9 2 \ \mathrm{m / s}
$$
$$
1 7. 1 4 7 8 \mathrm{~ m}
$$
$$
0. 2 3 5 \mathrm{~ m}
$$
D. $$
1 5. 6 2 5 \; \mathrm{L / s}
$$
$$
7. 9 5 0 0 \ \mathrm{m / s}
$$
$$
1 6. 8 5 0 0 \mathrm{~ m}
$$
$$
0. 2 0 0 \mathrm{~ m}
$$
E. $$
1 5. 6 2 5 \; \mathrm{L / s}
$$
$$
8. 3 2 0 1 \ \mathrm{m / s}
$$
$$
1 5. 9 3 6 7 \mathrm{~ m}
$$
$$
0. 2 1 4 \mathrm{~ m}
$$
F. $$
1 5. 4 7 3 \; \mathrm{L / s}
$$
$$
8. 1 2 0 9 \ \mathrm{m / s}
$$
$$
1 6. 2 1 8 7 \mathrm{~ m}
$$
$$
0. 1 6 7 \mathrm{~ m}
$$
G. $$
1 5. 8 4 0 \; \mathrm{L / s}
$$
$$
7. 7 9 6 3 \ \mathrm{m / s}
$$
$$
1 6. 3 9 2 1 \mathrm{~ m}
$$
$$
0. 2 4 3 \mathrm{~ m}
$$
H. $$
1 5. 6 2 5 \; \mathrm{L / s}
$$
$$
7. 9 5 7 7 \ \mathrm{m / s}
$$
$$
1 6. 7 7 2 4 \mathrm{~ m}
$$
$$
0. 1 8 9 \mathrm{~ m}
$$
I. $$
1 5. 9 3 8 \; \mathrm{L / s}
$$
$$
8. 0 1 4 7 \ \mathrm{m / s}
$$
$$
1 6. 7 0 0 0 \mathrm{~ m}
$$
$$
0. 1 7 9 \mathrm{~ m}
$$
J. $$
1 5. 6 2 5 \; \mathrm{L / s}
$$
$$
7. 9 5 7 7 \ \mathrm{m / s}
$$
$$
1 7. 2 0 0 0 \mathrm{~ m}
$$
$$
0. 2 0 8 \mathrm{~ m}
$$
Answer: H
Lacey's formula for wetted perimeter $P$ in an alluvial channel is given in FPS units as
$$
P=2. 6 7 {\sqrt{Q}}
$$
where $Q$ = discharge in $\mathrm{f t^{3} / s}$ . Find the dimensions of the numerical constant and convert this formula into SI units.

Options:
A. $$
4. 4 6
$$
$$
P=4. 4 6 {\sqrt{Q}}
$$
B. $$
4. 3 8
$$
$$
P=4. 3 8 {\sqrt{Q}}
$$
C. $$
5. 2 4
$$
$$
P=5. 2 4 {\sqrt{Q}}
$$
D. $$
5. 1 7
$$
$$
P=5. 1 7 {\sqrt{Q}}
$$
E. $$
4. 6 8
$$
$$
P=4. 6 8 {\sqrt{Q}}
$$
F. $$
4. 7 5
$$
$$
P=4. 7 5 {\sqrt{Q}}
$$
G. $$
4. 9 0
$$
$$
P=4. 9 0 {\sqrt{Q}}
$$
H. $$
4. 5 9
$$
$$
P=4. 5 9 {\sqrt{Q}}
$$
I. $$
4. 8 3
$$
$$
P=4. 8 3 {\sqrt{Q}}
$$
J. $$
5. 0 2
$$
$$
P=5. 0 2 {\sqrt{Q}}
$$
Answer: I
For Froude model law, find the ratios of velocity, discharge, force, work and power in terms of the length scale.

Options:
A. $$
L_{\mathrm{r}}^{1/2}
$$
$$
( L_{\mathrm{r}}^{3} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{6} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{7 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{8} )
$$
B. $$
L_{\mathrm{r}}^{-1/2}
$$
$$
( L_{\mathrm{r}}^{9 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{3 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{10} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{1.5} )
$$
C. $$
\sqrt{L_{\mathrm{r}}}
$$
$$
( L_{\mathrm{r}}^{5 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{3} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{4} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{7 / 2} )
$$
D. $$
L_{\mathrm{r}}^{3/2}
$$
$$
( L_{\mathrm{r}} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{7} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{5 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{3 / 2} )
$$
E. $$
L_{\mathrm{r}}
$$
$$
( L_{\mathrm{r}}^{4} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{5} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{6} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}} )
$$
F. $$
\sqrt{L_{\mathrm{r}}^{3}}
$$
$$
( L_{\mathrm{r}}^{6} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{5} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{4} )
$$
G. $$
L_{\mathrm{r}}^{2}
$$
$$
( L_{\mathrm{r}}^{3.5} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{1 / 3} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{5} )
$$
H. $$
L_{\mathrm{r}}^{0}
$$
$$
( L_{\mathrm{r}}^{7 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{1} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{9} )
$$
I. $$
L_{\mathrm{r}}^{0.5}
$$
$$
( L_{\mathrm{r}}^{8 / 3} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{4} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{8 / 3} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{6} )
$$
J. $$
L_{\mathrm{r}}^{-1}
$$
$$
( L_{\mathrm{r}}^{0} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{9 / 2} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{8} )
$$
$$
( \rho_{\mathrm{r}} L_{\mathrm{r}}^{-2} )
$$
Answer: C
Suppose a fish has an average density of 1030 kg $m^{-3}$ , compared to the density of the surrounding water,1000 kg $m^{-3}$ . One way the fish can keep from slowly sinking is by using an air bladder (the density of air is 1.2 kg $m^{-3} )$ . What fraction of the fish's total volume must be air in order for the fish to be neutrally buoyant ?(The buoyant force is equal and opposite to the weight). Assume that the volume $V$ of the fish's tissue is fized, so in order to increase the volume $U$ of the air bladder, the total volume of the fish $V+U$ must increase.

Options:
A. $$1.5\%$$
B. $$2.0\%$$
C. $$4.1\%$$
D. $$4.8\%$$
E. $$2.5\%$$
F. $$5.2\%$$
G. $$1.8\%$$
H. $$2.9\%$$
I. $$3.5\%$$
J. $$3.0\%$$
Answer: H
An exponentially growing culture of bacteria increases from $1 0^{6}$ to $5 \times1 0^{8}$ cells in 6 $h$ . What is the time between successive cell divisions if there is no cell mortality?

Options:
A. 1.25hr
B. 2.50hr
C. 0.67hr
D. 1.75hr
E. 0.95hr
F. 1.00hr
G. 0.50hr
H. 0.60hr
I. 0.45hr
J. 0.85hr
Answer: C
Some systems are so small that only a few molecules of a particular type are present, and statistical arguments begin to break down. What's the estimated number of hydrogen ions inside an E. coli bacterium with a pH value of 7? (When pH = 7 the concentration of hydrogen ions is $1 0^{-7} \, \mathrm{m o l} \ \mathrm{l}^{-1} \,. )$

Options:
A. 15
B. 50
C. 25
D. 30
E. 20
F. 40
G. 10
H. 45
I. 35
J. 55
Answer: D
For diagnostic studies of the lung, it would be convenient to have radioactive particles that tag the air and that are small enough to penetrate all the way to the alveoli. It is possible to make the isotope 99m Tc into a "pseudogas" by burning a flammable aerosol containing it. The resulting particles have a radius of about 60 nm
[ W. M. Burch, I. J. Tetley, and J. L . Gras (1984). Clin. Phys. Physiol Meas 5: 79-85]. Estimate the mean free path for these particles is(). If it is small compared to the molecular diameter, then Stokes' law applies, and you can use Eq. 4.23 to obtain the diffusion constant. (The viscosity of air at body temperature is about $1.8 \times 10^{-5} \, \text{Pa} \, \text{s}$.)

Options:
A. $$
1. 6 5 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
B. $$
3. 2 8 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
C. $$
1. 9 7 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
D. $$
4. 2 3 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
E. $$
3. 5 6 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
F. $$
2. 7 1 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
G. $$
1. 8 9 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
H. $$
2. 5 0 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
I. $$
2. 1 1 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
J. $$
3. 0 0 \times1 0^{-1 0} \mathrm{{m}}^{2} {\mathrm{s}}^{-1}
$$
Answer: I
We can use the microscopic model of a random walk to derive important information about diffusion without ever using the binomial probability distribution. Let $x_{i} ( n )$ be the position of the ith particle after $n$ steps of a random walk. Then
$$
x_{i} ( n )=x_{i} ( n ) \pm\lambda,
$$
where half the time you take the + sign and half the time the -sign. Then ${\overline{{x}}} ( n )$ , the value of x averaged over $N$
particles, is
$$
\overline{{{{x}}}} ( n )=\frac{1} {N} \sum_{i=1}^{N} x_{i} ( n ).
$$
When $\overline{{{{x}}}} ( n )$ = ________.That on average the particles go nowhere.

Options:
A. $$\overline{{{{x}}}} ( n-1 )$$
B. $$\overline{{{{x}}}} ( n+2 )$$
C. $$\overline{{{{x}}}} ( n+1 )$$
D. $$\sum_{i=0}^{N} x_{i} ( n )$$
E. $$\overline{{{{x}}}} ( 2n )$$
F. $$\overline{{{{x}}}} ( 2n+1 )$$
G. $$\overline{{{{x}}}} ( n )$$
H. $$n \times \overline{{{{x}}}} ( 1 )$$
I. $$0.5 \times \overline{{{{x}}}} ( n )$$
J. $$2 \times \overline{{{{x}}}} ( n )$$
Answer: A
Given that the angular size of Venus is $5 5^{\prime\prime}$ when the planet is 45,000,000 km from Earth, What is the diameter of Venus? (in kilometers)

Options:
A. 100km
B. 600km
C. 12000km
D. 4000km
E. 2000km
F. 1400km
G. 900km
H. 1500km
I. 1100km
J. 8000km
Answer: C
What is the energy (in electron volts) of a 100-GHz (1 gigahert $\mathrm{z}=1 0^{9}$ Hz) microwave photon?

Options:
A. $$
4. 5 \times1 0^{-6} \mathrm{e V}.
$$
B. $$
1. 9 \times1 0^{-5} \mathrm{e V}.
$$
C. $$
5. 0 \times1 0^{-5} \mathrm{e V}.
$$
D. $$
8. 8 \times1 0^{-6} \mathrm{e V}.
$$
E. $$
4. 1 \times1 0^{-4} \mathrm{e V}.
$$
F. $$
3. 2 \times1 0^{-4} \mathrm{e V}.
$$
G. $$
2. 8 \times1 0^{-3} \mathrm{e V}.
$$
H. $$
2. 4 \times1 0^{-4} \mathrm{e V}.
$$
I. $$
7. 4 \times1 0^{-3} \mathrm{e V}.
$$
J. $$
6. 2 \times1 0^{-4} \mathrm{e V}.
$$
Answer: E
What are the angular resolutions for a radio interferometer with a baseline of 5000 km operating at a frequency of 5 GHz, and an infrared interferometer with a baseline of 50 m operating at a wavelength of 1 μm?

Options:
A. 0.015 arc seconds, 0.018 arc seconds
B. 0.008 arc seconds, 0.012 arc seconds
C. 0.020 arc seconds, 0.035 arc seconds
D. 0.002 arc seconds, 0.007 arc seconds
E. 0.001 arc seconds, 0.002 arc seconds
F. 0.003
arc seconds ,
0.005
arc seconds
G. 0.025 arc seconds, 0.030 arc seconds
H. 0.005 arc seconds, 0.010 arc seconds
I. 0.010 arc seconds, 0.025 arc seconds
J. 0.004 arc seconds, 0.009 arc seconds
Answer: F
Given that the Sun's lifetime is about 10 billion years.What is the lifespan of a red dwarf star with a mass of 0.2 solar masses and a luminosity of 0.01 solar luminosities?

Options:
A. the star will live ${\frac{2 0 0} {2}}=2 0$ times longer than the Sun, or 2 trillion years.
B. the star will live ${\frac{2 0 0} {5}}=1 0$ times longer than the Sun, or 40 billion years.
C. the star will live ${\frac{1 0 0} {1}}=1 0 0$ times longer than the Sun, or 1 trillion years.
D. the star will live ${\frac{1 0 0} {2}}=1 0$ times longer than the Sun, or 100 trillion years.
E. the star will live ${\frac{5 0} {1 0}}=2 0$ times longer than the Sun, or 50 billion years.
F. the star will live ${\frac{1 0 0} {2}}=5 0$ times longer than the Sun, or 500 billion years.
G. the star will live ${\frac{1 0 0} {1 0}}=1 0$ times longer than the Sun, or 100 billion years.
H. the star will live ${\frac{5 0} {5}}=2 0$ times longer than the Sun, or 10 billion years.
I. the star will live ${\frac{1 0 0} {5}}=2 0$ times longe than the Sun, or 200 billion years.
J. the star will live ${\frac{1 0 0} {1}}=2 0$ times longer than the Sun, or 20 trillion years.
Answer: I
The average density of interstellar gas within the Local Bubble is much lower than the value mentioned in the text-- in fact, it is roughly $1 0^{3}$ hydrogen atoms/m?. Given that the mass of a hydrogen atom is $1. 7 \times1^{-2 7}$ kg, calculate the total mass of interstellar matter contained within a bubble volume equal in size to planet Earth.

Options:
A. $$
2. 1 \mathrm{~ g r a m s}
$$
B. $$
1. 6 \mathrm{~ g r a m s}
$$
C. $$
1. 5 \mathrm{~ g r a m s}
$$
D. $$
2. 3 \mathrm{~ g r a m s}
$$
E. $$
1. 8 \mathrm{~ g r a m s}
$$
F. $$
1. 9 \mathrm{~ g r a m s}
$$
G. $$
2. 0 \mathrm{~ g r a m s}
$$
H. $$
1. 7 \mathrm{~ g r a m s}
$$
I. $$
1. 2 \mathrm{~ g r a m s}
$$
J. $$
2. 5 \mathrm{~ g r a m s}
$$
Answer: F
To carry enough energy to ionize a hydrogen atom, a photon must have a wavelength of less than 9.12 $\times1 0^{-8}$ m (91.2 nm). Using Wien's law, calculate the temperature a star must have for the peak wavelength of its blackbody curve to equal this value.

Options:
A. $$
\mathrm{T}=2 5, 7 0 0 \mathrm{~ K}
$$
B. $$
\mathrm{T}=3 5, 4 0 0 \mathrm{~ K}
$$
C. $$
\mathrm{T}=4 2, 1 0 0 \mathrm{~ K}
$$
D. $$
\mathrm{T}=3 7, 2 0 0 \mathrm{~ K}
$$
E. $$
\mathrm{T}=4 8, 0 0 0 \mathrm{~ K}
$$
F. $$
\mathrm{T}=3 1, 8 0 0 \mathrm{~ K}
$$
G. $$
\mathrm{T}=3 9, 5 0 0 \mathrm{~ K}
$$
H. $$
\mathrm{T}=2 8, 3 0 0 \mathrm{~ K}
$$
I. $$
\mathrm{T}=2 9, 7 0 0 \mathrm{~ K}
$$
J. $$
\mathrm{T}=4 5, 2 0 0 \mathrm{~ K}
$$
Answer: F
Use the radius-luminosity-temperature relation to explain how a protostar's luminosity changes as it moves from stage 4 (temperature 3000 $\mathrm{K}$ , radius $2 \times1 0^{8}$ km) to stage $6$ (temperature 4500 $\mathrm{K}$ , radius $1 0^{6}$ km). What is the change in absolute magnitude?

Options:
A. $$
8. 9
$$
B. $$
10. 8
$$
C. $$
12. 5
$$
D. $$
9. 2
$$
E. $$
9. 7
$$
F. $$
8. 5
$$
G. $$
6. 9
$$
H. $$
7. 3
$$
I. $$
11. 4
$$
J. $$
10. 2
$$
Answer: E
Use the radius-luminosity-temperature relation to calculate the radius of a red supergiant with temperature 3000 K (half the solar value) and total luminosity 10,000 times that of the Sun.How many planets of our solar system would this star engulf?

Options:
A. $$
\operatorname{n i n e}
$$
B. $$
\operatorname{o n e}
$$
C. $$
\operatorname{s i x}
$$
D. $$
\operatorname{t e n}
$$
E. $$
\operatorname{t h r e e}
$$
F. $$
\operatorname{f o u r}
$$
G. $$
\operatorname{s e v e n}
$$
H. $$
\operatorname{t w o}
$$
I. $$
\operatorname{e i g h t}
$$
J. $$
\operatorname{f i v e}
$$
Answer: F
Calculate the average density of a red-giant core of 0.25 solar mass and radius 15,000 km.

Options:
A. $$
2. 8 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
B. $$
1. 5 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
C. $$
6. 2 \times1 0^{6} \, \frac{k g} {m^{3}}
$$
D. $$
8. 4 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
E. $$
9. 7 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
F. $$
3. 9 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
G. $$
5. 6 \times1 0^{6} \, \frac{k g} {m^{3}}
$$
H. $$
3. 5 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
I. $$
4. 2 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
J. $$
7. 3 \times1 0^{7} \, \frac{k g} {m^{3}}
$$
Answer: H
Repeat the previous calculation for a supernova having a peak luminosity $1 0^{1 0}$ times that of the Sun. What would be the apparent magnitude of the explosion if it occurred at a distance of 10,000 Mpc?

Options:
A. $$
2 7
$$
B. $$
2 6
$$
C. $$
2 8
$$
D. $$
2 4
$$
E. $$
3 0
$$
F. $$
3 2
$$
G. $$
2 2
$$
H. $$
2 3
$$
I. $$
2 5
$$
J. $$
2 9
$$
Answer: I
A neutron star's immense gravitational attraction is due primarily to its small radius and().

Options:
A. $$
\mathrm {r e l a t i v e l y \, l o w \, m a s s}
$$
B. $$
\mathrm{s t r o n g \, m a g n e t i c \, f i e l d}
$$
C. $$
\mathrm{l o w \, t o t a l \, g r a v i t y}
$$
D. $$
\mathrm{h i g h \, d e n s i t y}
$$
E. $$
\mathrm{s m a l l \, r a d i u s \, o n l y}
$$
F. $$
\mathrm{c o l l a p s i n g \, n u c l e a r \, c o r e}
$$
G. $$
\mathrm{l o w \, t e m p e r a t u r e }
$$
H. $$
\mathrm{i n t e n s e \, f u s i o n \, r a t e s}
$$
I. $$
\mathrm{l a r g e \, m a s s}
$$
J. $$
\mathrm{r a p i d \, s p i n }
$$
Answer: I
Supermassive black holes are thought to exist in the centers of many galaxies. What would be the Schwarzschild radii of black holes of 1 million and 1 billion solar masses, respectively?

Options:
A. For 1 million solar masses, the star's radius hits 2.9 million km, aligning with Earth's orbit. A billion masses form a 2.9 billion km radius.
B. At 1 million solar mass, expect a 3.4 million km radius, more than Neptune. Meanwhile, a billion masses yield a 3.6 billion km radius.
C. In a 1 million solar mass scenario, its radius becomes 3.1 million km, matching Mercury's size. A billion solar mass implies 3.1 billion km.
D. A 1 million solar mass predicts a 3.5 million km radius larger than Venus, while the billion mass suggests a 3.4 billion km radius.
E. With 1 million solar masses, the Schwarzschild radius totals 2.8 million km, close to Mars. Likewise, 1 billion solar mass totals 3.2 billion km.
F. The 1 million solar mass black hole would vary with a 3.3 million km radius, double Pluto's span. In contrast, 1 billion masses create 2.8 billion km.
G. The 1 million solar mass star measures a 3.2 million km radius, larger than the Moon. Similarly, 1 billion solar mass has a radius of 3.1 billion km.
H. Therefore, the 1 million solar mass star has a Schwarzschild radius of 3 million km, which is 4.3 times the radius of the Sun. The 1 billion solar mass black hole would have a Schwarzschild radius of 3 billion km.
I. For 1 million solar masses, expect a radius of 2.6 million km matching Saturn, whereas 1 billion masses offer a 2.5 billion km radius.
J. Schwarzschild estimates a 1 million solar mass gives a 2.7 million km radius, near Jupiter's orbit. Comparatively, 1 billion masses grow to 2.7 billion km.
Answer: H
Let $U=\{a, b, c, d, e\}$, and
$$
A_{\lambda}=\begin{cases}
{{\{ d \}}} & {{{0.7 < \lambda \leqslant 0.8}}} \\
{{\{ c, d \}}} & {{{0.5 < \lambda \leqslant 0.7}}} \\
{{\{ c, d, e \}}} & {{{0.3 < \lambda \leqslant 0.5}}} \\
{{\{ b, c, d, e \}}} & {{{0.1 < \lambda \leqslant 0.3}}} \\
{{\{ a, b, c, d, e \}}} & {{{0 \leqslant \lambda \leqslant 0.1}}} \\
\end{cases}.
$$
What is the $\mathrm{F}$ set of $A$?

Options:
A. $$A=\frac{0.1}{a}+\frac{0.4}{b}+\frac{0.5}{c}+\frac{0.8}{d}+\frac{0.6}{e}$$
B. $$A=\frac{0.2}{a}+\frac{0.3}{b}+\frac{0.7}{c}+\frac{0.8}{d}+\frac{0.5}{e}$$
C. $$A=\frac{0.1}{b}+\frac{0.3}{a}+\frac{0.5}{c}+\frac{0.7}{d}+\frac{0.8}{e}$$
D. $$A=\frac{0.3}{a}+\frac{0.1}{b}+\frac{0.4}{c}+\frac{0.8}{d}+\frac{0.5}{e}$$
E. $$A=\frac{0.1}{a}+\frac{0.3}{b}+\frac{0.6}{c}+\frac{0.7}{d}+\frac{0.4}{e}$$
F. $$A=\frac{0.1}{a}+\frac{0.3}{b}+\frac{0.7}{c}+\frac{0.8}{d}+\frac{0.5}{c}$$
G. $$A=\frac{0.1}{a}+\frac{0.2}{b}+\frac{0.6}{c}+\frac{0.8}{d}+\frac{0.5}{e}$$
H. $$A=\frac{0.1}{a}+\frac{0.3}{c}+\frac{0.6}{b}+\frac{0.8}{d}+\frac{0.5}{e}$$
I. $$A=\frac{0.1}{a}+\frac{0.2}{b}+\frac{0.5}{c}+\frac{0.8}{d}+\frac{0.7}{e}$$
J. $$A=\frac{0.2}{a}+\frac{0.3}{b}+\frac{0.4}{c}+\frac{0.8}{d}+\frac{0.6}{e}$$
Answer: F
Let the domain of discourse $R$ be the real number field, and $A$ be a triangular $\mathrm{F}$ set parameterized by $(a, \sigma)$, where
$$
A(x) =
\begin{cases}
\frac{1}{\sigma}x + \frac{\sigma-a}{\sigma} & a-\sigma \leqslant x \leqslant a, \\
-\frac{1}{\sigma}x + \frac{\sigma+a}{\sigma} & a < x \leqslant a+\sigma, \\
0 & \text{others}.
\end{cases}
$$
If $A(x) = 0$ or $1$, what are the possible values of $x$?

Options:
A. $$
x=a+\sigma,x=a-\frac{\sigma}{2},x=a
$$
B. $$
x=a-\sigma,x=a+\sigma,x=a-\frac{\sigma}{2}
$$
C. $$
x=a-\sigma,x=a+\frac{\sigma}{2},x=a
$$
D. $$
x=a-\sigma,x=a+\frac{\sigma}{3},x=a
$$
E. $$
x=a-\frac{\sigma}{3},x=a+\sigma,x=a
$$
F. $$
x=a+\sigma,x=a+\frac{\sigma}{2},x=a
$$
G. $$
x=a-\sigma,x=a+\sigma,x=a
$$
H. $$
x=a-\frac{\sigma}{2},x=a+\sigma,x=a
$$
I. $$
x=a-\sigma,x=a+\sigma,x=a+\frac{\sigma}{2}
$$
J. $$
x=a-\sigma,x=a+\sigma,x=a-\frac{\sigma}{3}
$$
Answer: G
Let $R_{1} \! \in\! {\mathcal{F}} \left( \ U \times V \right)$, $R_{2} \! \in\! {\mathcal{F}} \left( \ V \times W \right)$, and $U, V, W$ are all real number domains.
$$
R_{1} \left( \, u, v \, \right) = \mathrm{e}^{-k \left( u - v \right)^{2}} \,, \quad R_{2} \left( \, v, w \, \right) = \mathrm{e}^{-k \left( v - w \right)^{2}}
$$
What is the membership function of $R_{1} \circ R_{2}$?

Options:
A. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(u-w^{2}\right)}$$
B. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(u^{2}+w^{2}\right)}$$
C. $$
(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(\frac{u-w}{2}\right)^{2}}
$$
D. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(u^{2}-w^{2}\right)}$$
E. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(u+w\right)^{2}}$$
F. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(2u-2w\right)^{2}}$$
G. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(\frac{u+w}{4}\right)^{2}}$$
H. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-2k\left(u-w\right)^{2}}$$
I. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(\frac{u+w}{2}\right)^{2}}$$
J. $$(R_{1}\circ R_{2})(u,w)=\mathrm{e}^{-k\left(\lvert u-w\rvert\right)^{2}}$$
Answer: C