MikaStars39/nano-eval huggingface dataset:
The velocity along the centreline of a nozzle of length $L$ is given by
$$
V=2 t \left( 1-{\frac{x} {2 L}} \right)^{2}
$$
where $V=$ velocity in m/s, t = time in seconds from commencement of flow, $x=$ distance from inlet to the nozzle. Find the convective acceleration, local acceleration and the total acceleration when $t=3 \ \mathrm{s},$ $x=0. 5$ m and $L=0. 8$ m.

Options:
A. $$
1. 0 3 9 \ \mathrm{m / s}^{2}
$$
$$
- 1 4. 9 6 5 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 4. 3 1 \ \mathrm{m / s}^{2}
$$
B. $$
0. 8 3 2 \ \mathrm{m / s}^{2}
$$
$$
- 1 3. 9 4 0 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 2. 7 0 \ \mathrm{m / s}^{2}
$$
C. $$
1. 0 2 5 \ \mathrm{m / s}^{2}
$$
$$
- 1 5. 0 1 2 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 4. 0 8 \ \mathrm{m / s}^{2}
$$
D. $$
0. 8 5 4 \ \mathrm{m / s}^{2}
$$
$$
- 1 3. 8 6 1 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 2. 8 3 \ \mathrm{m / s}^{2}
$$
E. $$
0. 7 7 3 \ \mathrm{m / s}^{2}
$$
$$
- 1 3. 4 5 5 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 1. 8 4 \ \mathrm{m / s}^{2}
$$
F. $$
1. 0 4 7 \ \mathrm{m / s}^{2}
$$
$$
- 1 4. 8 2 9 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 4. 2 0 \ \mathrm{m / s}^{2}
$$
G. $$
0. 7 9 0 \ \mathrm{m / s}^{2}
$$
$$
- 1 3. 6 7 8 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 1. 9 5 \ \mathrm{m / s}^{2}
$$
H. $$
0. 8 1 6 \ \mathrm{m / s}^{2}
$$
$$
- 1 4. 0 5 3 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 2. 6 9 \ \mathrm{m / s}^{2}
$$
I. $$
1. 1 0 8 \ \mathrm{m / s}^{2}
$$
$$
- 1 6. 2 4 0 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 5. 1 3 \ \mathrm{m / s}^{2}
$$
J. $$
0. 9 4 5 \ \mathrm{m / s}^{2}
$$
$$
- 1 4. 6 2 3 {\mathrm{~ m / s}}^{2}
$$
$$
- 1 3. 6 8 \ \mathrm{m / s}^{2}
$$
Answer: J
Obtain a relevant stream function to each of the following sets of velocity components of steady, incompressible flow:
$$
\begin{array} {c} {{{\mathrm{( i )} \ \ u=-\ c x / y}}} \\ {{{\mathrm{~ v=c ~ l n ~ x y}}}} \\ \end{array}
$$
$$
\begin{array} {r l} {{{\mathrm{( i i )}}}} & {{{} {{u}=x+y}}} \\ {{{}}} & {{{} {{v}=x-y}}} \end{array}
$$
$$
\begin{array} {l} {{{\mathrm{( i i i )}} \; \; u=2 \; {\mathrm{c} x}}} \\ {{{\nu=-\; 2 {\mathrm{c}} y}}} \\ \end{array}
$$

Options:
A. $$
\psi=-\mathbf{c} x \ \mathbf{l n} (x+y)+y x^{2}+\mathbf{c}_{2}
$$
$$
\psi=x^{2}-y^{2}-2 x y+c
$$
$$
\psi=\mathbf{c} x y^{2}+\mathbf{c}_{6}
$$
B. $$
\psi=-\ \mathbf{c} y \ \mathbf{l n} \ x+\mathbf{c} x y+\mathbf{c}_{3}
$$
$$
\psi=x y-y^{2}-c
$$
$$
\psi=2 c y x^{2}+\mathbf{c}_{1}
$$
C. $$
\psi=-\ \mathbf{c} x \ \mathbf{l n} \ x y+\mathbf{c} x+\mathbf{c}_{2}
$$
$$
\psi={\frac{1} {2}} \ ( y^{2}-x^{2} )+x y+c
$$
$$
\psi=2 \mathbf{c} x y+\mathbf{c}_{1}
$$
D. $$
\psi=-y \ \mathbf{l n} \ x y+c x+x^{2}
$$
$$
\psi={\frac{1} {2}} \ ( x+y )^{2}+x c+c
$$
$$
\psi=2 \mathbf{c} y^{2} x+\mathbf{c}_{4}
$$
E. $$
\psi=\mathbf{c} x y \ \mathbf{l n} \ y+x+\mathbf{c}
$$
$$
\psi=-{\frac{1} {2}} \ ( y^{2}-x^{2} )-x y+c
$$
$$
\psi=-\mathbf{c} x^{2} y+c
$$
F. $$
\psi=-c y \ \mathbf{l n} \ x+c x^{2}+\mathbf{c}
$$
$$
\psi=x y+y^{2}-x^{2}-c
$$
$$
\psi=-2 y x+\mathbf{c} x y
$$
G. $$
\psi=\mathbf{c} x+\mathbf{c} y \ \mathbf{l n} \ x y+\mathbf{c}_{1}
$$
$$
\psi=-{\frac{1} {2}} \ ( y^{2}+x^{2} )+x y+c
$$
$$
\psi=2 \mathbf{c} x^{2}-\mathbf{c}_{2}
$$
H. $$
\psi=c y+\mathbf{c} x \ \mathbf{l n} \ x y+\mathbf{c}_{5}
$$
$$
\psi=x y+{\frac{1} {2}} \ ( y^{2}-x^{2} )+c
$$
$$
\psi=2 x y-y+c
$$
I. $$
\psi=-c x^{2} \ \mathbf{l n} \ x+y+\mathbf{c}_{5}
$$
$$
\psi=-x^{2}-y^{2}+xy+c
$$
$$
\psi=2 \mathbf{c} x+yx+\mathbf{c}_{7}
$$
J. $$
\psi=y \ \mathbf{l n} \ x y-c x+\mathbf{c}_{3}
$$
$$
\psi=-\ ( x^{2}+y^{2} )+2 x y+c
$$
$$
\psi=2 c x y-\mathbf{c}_{4}
$$
Answer: C
For the following functions in two-dimensional irrotational flow, determine the conjugate functions:
$$
(\mathrm{i}) \ \ \psi = m \ \mathrm{l n} \ r
$$
$$
(\mathrm{ii}) \ \ \phi = \frac{m \operatorname{cos} \theta}{r}
$$

Options:
A. $$
- \mathrm{c o t} \ m \ \theta+\mathrm{c}
$$
$$
- \frac{m^2} {r} \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
B. $$
\mathrm{a r c} \ \mathrm{t a n} \ (m \ \theta) \ + \ \mathrm{c}
$$
$$
m \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
C. $$
- m^2 \ \theta+\mathrm{c}
$$
$$
\frac{m} {r^3} \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
D. $$
m \ \mathrm{r}+\mathrm{c}
$$
$$
\frac{m} {r} \ \mathrm{c o s} \ \theta+\mathrm{c}
$$
E. $$
-\mathrm{ln} \ (m \ \theta)+\mathrm{c}
$$
$$
- \frac{m \ \mathrm{c o s} \ \theta} {r}+\mathrm{c}
$$
F. $$
- m \ \theta+\mathrm{c}
$$
$$
- \frac{m} {r} \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
G. $$
m \ \theta^2+\mathrm{c}
$$
$$
\frac{m \ \mathrm{s i n} \ \theta} {r}+\mathrm{c}
$$
H. $$
\sqrt{m \ \theta}+\mathrm{c}
$$
$$
m^2 \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
I. $$
- m \ \theta^2+\mathrm{c}
$$
$$
- \frac{m \ \mathrm{s i n} \ \theta} {r^2}+\mathrm{c}
$$
J. $$
m \ \mathrm{l n} \ \theta+\mathrm{c}
$$
$$
- \frac{m} {r^2} \ \mathrm{s i n} \ \theta+\mathrm{c}
$$
Answer: F
Manning's formula for velocity of flow $V$ in uniform flow in a open channel is written as
$$
V=\frac{1} {n} \; R^{2 / 3} \; S_{0}^{1 / 2}
$$
where $R$ = hydraulic radius, $S_{0}$ = bed slope and
n = Manning's coefficient. Determine the dimen- sions of $n$ .

Options:
A. $$
[ \mathrm{L}^{0} \mathrm{T}^{1 / 3} ]
$$
B. $$
[ \mathrm{L}^{-2 / 3} \mathrm{T} ]
$$
C. $$
[ \mathrm{L}^{1 / 2} \mathrm{T}^{-1} ]
$$
D. $$
[ \mathrm{L}^{1 / 3} \mathrm{T} ]
$$
E. $$
[ \mathrm{L}^{-1} \mathrm{T}^{-1 / 2} ]
$$
F. $$
[ \mathrm{L}^{-2 / 3} \mathrm{T}^{2} ]
$$
G. $$
[ \mathrm{L}^{2 / 3} \mathrm{T}^{-1} ]
$$
H. $$
[ \mathrm{L}^{-1 / 3} \mathrm{T}^{2} ]
$$
I. $$
[ \mathrm{L}^{-1 / 3} \mathrm{T} ]
$$
J. $$
[ \mathrm{L}^{1 / 3} \mathrm{T}^{-1} ]
$$
Answer: I
The inspiratory pressure difference $p_{in}$ that the lung can generate is about 86 torr. What would be the absolute maximum depth at which a person could breathe through a snorkel device? (A safe depth is only about half this maximum, since lung ventilation becomes very minimal at the maximum depth. Assume the lungs are 30 cm below the mouth.)

Options:
A. $$
1 2 7 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
B. $$
1 0 5 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
C. $$
1 1 7 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
D. $$
1 2 3 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
E. $$
1 2 0 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
F. $$
1 0 8 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
G. $$
1 3 2 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
H. $$
9 8 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
I. $$
1 0 3 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
J. $$
8 7 \; \mathrm{c m \; f o r \; t h e \; l u n g s}
$$
Answer: C
A child with acute lymphocytic leukemia (ALL) has approximately $1 0^{1 2}$ leukemic cells when the disease is clinically apparent.
If a cell is about $8$ $\mu m$ in diameter, estimate the total mass of leukemic cells is ().

Options:
A. 0.32kg
B. 0.39kg
C. 0.12kg
D. 0.15kg
E. 0.67kg
F. 0.09kg
G. 0.58kg
H. 0.45kg
I. 0.27kg
J. 0.21kg
Answer: I
The following data were taken from Berg et $a l$ (1982). At $t=0$ , a 70-kg subject was given an intravenous injection of 200 mg of phenobarbital. The initial concentration in the blood was $6$ mg $l^{-1}$ . The concentration decayed exponentially with a half-life of 110 h. The erperiment was repeated, but this time the subject was fed 200 $g$ of activated charcoal every $6 \; h$ . The concentration of phenobarbital again fell exponentially, but with a half-life of 45 $h$ .What was the clearance due to charcoalf?

Options:
A. $$
0. 3 1 2 \, \mathrm{h r}^{-1}
$$
B. $$
0. 3 2 0 \, \mathrm{h r}^{-1}
$$
C. $$
0. 3 0 1 \, \mathrm{h r}^{-1}
$$
D. $$
0. 2 7 5 \, \mathrm{h r}^{-1}
$$
E. $$
0. 2 9 5 \, \mathrm{h r}^{-1}
$$
F. $$
0. 3 0 5 \, \mathrm{h r}^{-1}
$$
G. $$
0. 2 6 8 \, \mathrm{h r}^{-1}
$$
H. $$
0. 3 4 0 \, \mathrm{h r}^{-1}
$$
I. $$
0. 2 8 0 \, \mathrm{h r}^{-1}
$$
J. $$
0. 3 5 0 \, \mathrm{h r}^{-1}
$$
Answer: C
Oxygen and carbon monoride compete for binding to hemoglobin. If enough CO binds to hemoglobin,
the ability of the blood to deliver oxygen is impaired, and carbon monoride poisoning ensues. Consider the hemoglobin molecule to be a two-state system: the heme group is bound either to $O_{2}$ or to CO. Calculate the probability of binding to CO is(). Let the $G$ factor of Eq. 3.25 be equal to the ratio of the concentrations of CO and $O_{2}$ . Assume CO is 100 times less abundant than $O_{2}$ . CO is more tightly bound than $O_{2}$ to the heme group by about $0. 1 5 \, \mathrm{e V}$ . Let $T=3 0 0 \, \mathrm{K}$ .

Options:
A. 5.29
B. 1.34
C. 0.42
D. 6.47
E. 2.18
F. 4.89
G. 9.13
H. 8.66
I. 3.72
J. 7.01
Answer: I
A spherical cell has radius $R$. The flur density through the surface is given by $j_{s}=-D$ grad $C$. Suppose that the substance in question has concentration $C ( t )$ inside the cell and zero outside. The material outside is removed fast enough so that the concentration remains zero. Using spherical coordinates, find a differential equation for $C ( t )$ inside the cell is ______. The thickness of the cell wall is $\Delta r \ll R$ .

Options:
A. $$
- \frac{2 D} {R \Delta r} C
$$
B. $$
- \frac{3 D} {R \Delta r} C^2
$$
C. $$
- \frac{3 D} {R^2 \Delta r} C
$$
D. $$
- \frac{D} {R \Delta r} C
$$
E. $$
- \frac{3 D} {R \Delta r} C \log
$$
F. $$
- \frac{3 D} {2 R \Delta r} C
$$
G. $$
- \frac{3 D^2} {R \Delta r} C
$$
H. $$
- \frac{4 D} {R \Delta r} C
$$
I. $$
- \frac{3 D} {R \Delta r^2} C
$$
J. $$
- \frac{3 D} {R \Delta r} C
$$
Answer: J
Relative to the stars, what is the angle of the Moon's movement in one hour?

Options:
A. about 35 minutes of arc
B. about 31 minutes of arc
C. about 50 minutes of arc
D. about 45 minutes of arc
E. about 15 minutes of arc
F. about 42 minutes of arc
G. about 20 minutes of arc
H. about 25 minutes of arc
I. about 60 minutes of arc
J. about 48 minutes of arc
Answer: B
What is the energy (in electron volts--see More Precisely $4-1 )$ of a 450-nm blue photon? A 200-nm ultraviolet photon?

Options:
A. $$
2. 4 \mathrm{~ e V},
$$
$$
5. 7 \ \mathrm{e V}
$$
B. $$
2. 6 \mathrm{~ e V},
$$
$$
6. 9 \ \mathrm{e V}
$$
C. $$
2. 9 \mathrm{~ e V},
$$
$$
5. 6 \ \mathrm{e V}
$$
D. $$
3. 3 \mathrm{~ e V},
$$
$$
6. 3 \ \mathrm{e V}
$$
E. $$
2. 5 \mathrm{~ e V},
$$
$$
5. 9 \ \mathrm{e V}
$$
F. $$
3. 2 \mathrm{~ e V},
$$
$$
6. 0 \ \mathrm{e V}
$$
G. $$
3. 1 \mathrm{~ e V},
$$
$$
5. 8 \ \mathrm{e V}
$$
H. $$
3. 0 \mathrm{~ e V},
$$
$$
6. 5 \ \mathrm{e V}
$$
I. $$
2. 8 \mathrm{~ e V},
$$
$$
6. 2 \ \mathrm{e V}
$$
J. $$
2. 7 \mathrm{~ e V},
$$
$$
6. 8 \ \mathrm{e V}
$$
Answer: I
The Moon lies about 380,000 km away. To what distances do the angular resolutions of SST (3"), HST (0.05"), and a radio interferometer (0.001") correspond at that distance?

Options:
A. 5.5km,92m,1.8m.
B. 5.8km,88m,2.2m.
C. 6.0km,86m,2.4m.
D. 5.6km,90m,2.0m.
E. 5.2km,91m,1.6m.
F. 5.1km,95m,1.5m.
G. 5.3km,93m,1.7m.
H. 5.4km,94m,1.9m.
I. 5.7km,89m,2.1m.
J. 5.9km,87m,2.3m.
Answer: A
Calculate the solar energy flux (energy received per unit area per unit time), as seen from a distance of 10 pc from the Sun.

Options:
A. $$
5. 1 \times1 0^{-1 1} \mathrm{~ W / m}^{2}
$$
B. $$
3. 8 \times1 0^{-1 1} \mathrm{~ W / m}^{2}
$$
C. $$
3. 3 \times1 0^{-1 2} \mathrm{~ W / m}^{2}
$$
D. $$
2. 3 \times1 0^{-1 0} \mathrm{~ W / m}^{2}
$$
E. $$
4. 9 \times1 0^{-1 1} \mathrm{~ W / m}^{2}
$$
F. $$
3. 3 \times1 0^{-1 0} \mathrm{~ W / m}^{2}
$$
G. $$
4. 2 \times1 0^{-1 0} \mathrm{~ W / m}^{2}
$$
H. $$
3. 1 \times1 0^{-9} \mathrm{~ W / m}^{2}
$$
I. $$
2. 7 \times1 0^{-1 0} \mathrm{~ W / m}^{2}
$$
J. $$
6. 8 \times1 0^{-1 0} \mathrm{~ W / m}^{2}
$$
Answer: F
The largest interstellar clouds are().

Options:
A. $$
\mathrm{n e u t r a l ~ g a s ~ c l u m p s}
$$
B. $$
\mathrm{h e l i o s p h e r i c ~ p l a s m a}
$$
C. $$
\mathrm{p r o t o ~ s t a r ~ s h e l l s}
$$
D. $$
\mathrm{m o l e c u l a r ~ c l o u d s}
$$
E. $$
\mathrm{g a l a c t i c ~ f i l a m e n t s}
$$
F. $$
\mathrm{c o s m i c ~ d u s t ~ c l o u d s}
$$
G. $$
\mathrm{i o n i z e d ~ n u b e s}
$$
H. $$
\mathrm{a ~ t h e r m a l ~ w a v e s}
$$
I. $$
\mathrm{d i f f u s e ~ n e b u l a e}
$$
J. $$
\mathrm{s t e l l a r ~ w i n d s}
$$
Answer: D
A star of apparent magnitude 10 lies 500 pc from Earth. If interstellar absorption results in an average extinction of $2$ mag/kpc, calculate the star's absolute magnitude and luminosity.

Options:
A. $$
\operatorname{l o g}_{1 0} L=1. 7 2 \mathrm{~ a n d ~} L=5 3. 3 L
$$
B. $$
\operatorname{l o g}_{1 0} L=1. 8 0 \mathrm{~ a n d ~} L=6 3. 0 L
$$
C. $$
\operatorname{l o g}_{1 0} L=1. 7 8 \mathrm{~ a n d ~} L=5 9. 7 L
$$
D. $$
\operatorname{l o g}_{1 0} L=1. 6 7 \mathrm{~ a n d ~} L=4 9. 9 L
$$
E. $$
\operatorname{l o g}_{1 0} L=1. 7 3 \mathrm{~ a n d ~} L=5 3. 7 L
$$
F. $$
\operatorname{l o g}_{1 0} L=1. 6 5 \mathrm{~ a n d ~} L=4 8. 3 L
$$
G. $$
\operatorname{l o g}_{1 0} L=1. 6 0 \mathrm{~ a n d ~} L=4 2. 5 L
$$
H. $$
\operatorname{l o g}_{1 0} L=1. 9 0 \mathrm{~ a n d ~} L=7 1. 3 L
$$
I. $$
\operatorname{l o g}_{1 0} L=1. 9 5 \mathrm{~ a n d ~} L=7 4. 2 L
$$
J. $$
\operatorname{l o g}_{1 0} L=1. 7 5 \mathrm{~ a n d ~} L=5 6. 2 L
$$
Answer: E
A protostar on the Hayashi track evolves from a temperature of 3500 K and a luminosity 5000 times that of the Sun to a temperature of 5000 $\mathrm{K}$ and a luminosity of 3 solar units. What is the protostar's radius (b) at the end of the evolution?

Options:
A. 1.5 times the radius of the Sun
B. 1.8 times the radius of the Sun
C. 5.0 times the radius of the Sun
D. 2.3 times the radius of the Sun
E. 2.0 times the radius of the Sun
F. 4.2 times the radius of the Sun
G. 3.5 times the radius of the Sun
H. 2.7 times the radius of the Sun
I. 4.5 times the radius of the Sun
J. 3.0 times the radius of the Sun
Answer: D
If the evolutionary track in Overlay 3, showing a Sun- like star, were instead illustrating a significantly more mas- sive star, its starting point (stage 7) would be().

Options:
A. $$
\mathrm{d o w n ~ a n d ~ t o ~ t h e ~ r i g h t}
$$
B. $$
\mathrm{t o ~ t h e ~ u p p e r ~ c e n t e r ~ p o i n t}
$$
C. $$
\mathrm{a l o n g ~ t h e ~ c u r v e ~ u p w a r d}
$$
D. $$
\mathrm{s h i f t ~ d i r e c t l y ~ t o ~ t h e ~ r i g h t}
$$
E. $$
\mathrm{m o v e ~ f a r t h e r ~ u p ~ t o ~ c o r n e r}
$$
F. $$
\mathrm{b a c k ~ t o ~ t h e ~ l o w e r ~ l e f t}
$$
G. $$
\mathrm{u p ~ a n d ~ t o ~ t h e ~ l e f t}
$$
H. $$
\mathrm{o v e r ~ t h e ~ l e f t ~ h o r i z o n t a l}
$$
I. $$
\mathrm{o v e r ~ a n d ~ t o ~ t h e ~ r i g h t}
$$
J. $$
\mathrm{u p ~ i n ~ a ~ d i a g o n a l ~ d i r e c t i o n}
$$
Answer: G
The Sun will reside on the main sequence for $1 0^{1 0}$ years. If the luminosity of a main-sequence star is proportional to
the fourth power of the star's mass, what is the mass of a star that is just now leaving the main sequence in a cluster that formed 400 million years ago?

Options:
A. the star is 4.1 times more massive than the Sun
B. the star is 5.2 times more massive than the Sun
C. the star is 2.9 times more massive than the Sun
D. the star is 1.7 times more massive than the Sun
E. the star is 4.3 times more massive than the Sun
F. the star is 7.4 times more massive than the Sun
G. the star is 0.9 times more massive than the Sun
H. the star is 6.8 times more massive than the Sun
I. the star is 2.3 times more massive than the Sun
J. the star is 3.5 times more massive than the Sun
Answer: C
Nuclear fusion in the Sun will().

Options:
A. create stable compounds like water and methane
B. generate only hydrogen and helium
C. generate primarily radioactive isotopes only
D. produce heavier elements like silicon and sulfur
E. produce helium exclusively without other elements
F. form carbon and stop further reactions
G. develop elements beyond iron in large amounts
H. create elements up to and including oxygen
I. release vast amounts of solid gold
J. create elements up to and including uranium
Answer: H
A neutron star is about the same size as().

Options:
A. $$
\mathrm{a ~ massive ~ mountain}
$$
B. $$
\mathrm{a ~ U. S. ~ c i t y}
$$
C. $$
\mathrm{a ~ regional ~ highway}
$$
D. $$
\mathrm{a ~ typical ~ island}
$$
E. $$
\mathrm{a ~ small ~ planet}
$$
F. $$
\mathrm{a ~ soccer ~ stadium}
$$
G. $$
\mathrm{a ~ large ~ skyscraper}
$$
H. $$
\mathrm{a ~ large ~ continent}
$$
I. $$
\mathrm{a ~ small ~ country}
$$
J. $$
\mathrm{a ~ large ~ spaceship}
$$
Answer: B
Use the radius-luminosity-temperature relation to calculate the luminosity of a 10-km-radius neutron star for temperatures of $1 0^{5} \, \mathrm{K}, 1 0^{7} \, \mathrm{K},$ and $1 0^{9} \mathrm{K}$.

Options:
A. 99nm and 0.99nm and 0.0099nm
B. 39nm and 0.39nm and 0.0039nm
C. 79nm and 0.79nm and 0.0079nm
D. 69nm and 0.69nm and 0.0069nm
E. 59nm and 0.59nm and 0.0059nm
F. 109nm and 1.09nm and 0.0109nm
G. 49nm and 0.49nm and 0.0049nm
H. 19nm and 0.19nm and 0.0019nm
I. 29nm and 0.29nm and 0.0029nm
J. 89nm and 0.89nm and 0.0089nm
Answer: I
Let $U = [0, 10]$ be the universe of discourse. For $\lambda \in [0, 1]$, if the $\lambda$-cut sets of fuzzy set $A$, denoted as $\mathrm{F}$, are given by
$$
A_{\lambda} =\begin{cases}
[0, 10] & \lambda = 0 \\
[3, 10] & 0 < \lambda \leqslant \frac{3}{5} \\
[5\lambda, 10] & \frac{3}{5} < \lambda < 1 \\
[5, 10] & \lambda = 1
\end{cases}.
$$
what are $\mathrm{Supp} \, A$ and $\mathrm{Ker} \, A$?

Options:
A. $$
\mathrm{Supp}A=
\begin{bmatrix}
0,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
5,10
\end{bmatrix}
$$
B. $$
\mathrm{Supp}A=
\begin{bmatrix}
3,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
0,5
\end{bmatrix}
$$
C. $$
\mathrm{Supp}A=
\begin{bmatrix}
3,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
0,10
\end{bmatrix}
$$
D. $$
\mathrm{Supp}A=
\begin{bmatrix}
3,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
5,10
\end{bmatrix}
$$
E. $$
\mathrm{Supp}A=
\begin{bmatrix}
0,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
3,10
\end{bmatrix}
$$
F. $$
\mathrm{Supp}A=
\begin{bmatrix}
3,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
0,3
\end{bmatrix}
$$
G. $$
\mathrm{Supp}A=
\begin{bmatrix}
0,3
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
3,10
\end{bmatrix}
$$
H. $$
\mathrm{Supp}A=
\begin{bmatrix}
5,10
\end{bmatrix},\mathrm{Ker}A=
\begin{bmatrix}
0,5
\end{bmatrix}
$$
Answer: D
Let the domain of discourse $R = \{ 1, 2, 3, 4, 5 \}, \; A, B \in \mathcal{F}(R)$, and
$A = \left( 0.2, 0.3, 0.6, 0.1, 0.9 \right), \; B = \left( 0.1, 0.2, 0.7, 0.2, 0 \right)$.
What is the Euclidean nearness degree?

Options:
A. 0.3
B. 0.8
C. 0.6
D. 0.7
E. 0.5
F. 0.1
G. 0.9
H. 0.4
I. 0.65
J. 0.2
Answer: C
Let $R_{1} \! \in\! {\mathcal{F}} ( U \! \times\! V ), R_{2} \! \in\! {\mathcal{F}} ( V \! \times\! W )$
$$
R_{1}=\left\{ \begin{matrix} {{{0. 7}}} & {{{0. 4}}} & {{{0. 1}}} & {{{1}}} \\ {{{0. 8}}} & {{{0. 3}}} & {{{0. 6}}} & {{{0. 3}}} \\ {{{0. 4}}} & {{{0. 7}}} & {{{0. 2}}} & {{{0. 9}}} \\ \end{matrix} \right\}, \quad R_{2}=\left\{ \begin{matrix} {{{0. 6}}} & {{{0. 5}}} \\ {{{0. 2}}} & {{{0. 8}}} \\ {{{0. 9}}} & {{{0. 3}}} \\ {{{0. 8}}} & {{{1}}} \\ \end{matrix} \right\}
$$
Find $R_{1} \circ R_{2} \,,( R_{1} )_{0. 6} \circ( R_{2} )_{0. 6} \,.$

Options:
A. $$
R_1\circ R_2=\left\{\begin{matrix}0.7&0.6\\0.3&0.8\\1&0.8\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\0&1\\1&0\end{matrix}\right\}.
$$
B. $$
R_1\circ R_2=\left\{\begin{matrix}0.8&1\\0.6&0.5\\0.8&0.9\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\1&0\\1&1\end{matrix}\right\}.
$$
C. $$
R_1\circ R_2=\left\{\begin{matrix}0.8&1\\0.6&0.4\\0.9&0.8\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}0&1\\1&1\\1&1\end{matrix}\right\}.
$$
D. $$
R_1\circ R_2=\left\{\begin{matrix}0.8&0.5\\0.5&0.9\\0.6&0.7\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\1&1\\0&0\end{matrix}\right\}.
$$
E. $$
R_1\circ R_2=\left\{\begin{matrix}0.6&0.8\\0.4&0.7\\0.9&0.4\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&0\\1&1\\0&1\end{matrix}\right\}.
$$
F. $$
R_1\circ R_2=\left\{\begin{matrix}0.7&0.9\\0.5&0.6\\0.7&0.8\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\0&1\\1&0\end{matrix}\right\}.
$$
G. $$
R_1\circ R_2=\left\{\begin{matrix}0.9&1\\0.8&0.3\\0.6&0.7\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\0&1\\1&1\end{matrix}\right\}.
$$
H. $$
R_1\circ R_2=\left\{\begin{matrix}0.8&0.6\\0.4&0.5\\0.9&1\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&1\\1&0\\0&1\end{matrix}\right\}.
$$
I. $$
R_1\circ R_2=\left\{\begin{matrix}0.5&0.7\\0.8&0.6\\0.8&0.9\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}1&0\\1&1\\1&1\end{matrix}\right\}.
$$
J. $$
R_1\circ R_2=\left\{\begin{matrix}0.9&0.6\\0.8&0.4\\0.7&1\end{matrix}\right\},
$$
$$
(R_1)_{0.6}\circ(R_2)_{0.6}=\left\{\begin{matrix}0&1\\1&1\\1&1\end{matrix}\right\}.
$$
Answer: B
Let $U=\{\, u_{1} \,, u_{2} \,, u_{3} \,\}$, $V=\{\, v_{1} \,, v_{2} \,, v_{3} \,, v_{4} \,\}$, and
$$
R=\left[ \begin{matrix} {{{1}}} & {{{0}}} & {{{1}}} & {{{0}}} \\ {{{0}}} & {{{1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} & {{{1}}} \\ \end{matrix} \right], \quad R \in{\mathcal{F}} \left( \begin{matrix} {{{U}}} & {{{V}}} \\ {{{U}}} & {{{V}}} \\ \end{matrix} \right)
$$
$$
A=\{\, u_{1} \,, u_{3} \,\} \,, \quad B=\frac{0.5}{u_{1}}+\frac{0.3}{u_{2}}
$$
What are $T_{R}(A)$ and $T_{R}(B)$?

Options:
A. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,0,1,0}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1},v_{3}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.6,0.3,0.4,0}\end{pmatrix}$
B. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,1,0,0}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1},v_{2}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.4,0.3,0.5,0.2}\end{pmatrix}$
C. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,0,1,1}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1},v_{3},v_{4}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.5,0.3,0.5,0}\end{pmatrix}$
D. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,0,0,1}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1},v_{4}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.5,0.2,0.5,0}\end{pmatrix}$
E. $\left.T_{R}\left(A\right)=\begin{pmatrix}{0,0,1,0}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{3}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.3,0.2,0.5,0.1}\end{pmatrix}$
F. $\left.T_{R}\left(A\right)=\begin{pmatrix}{0,1,1,1}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{2},v_{3},v_{4}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.4,0.3,0.4,0}\end{pmatrix}$
G. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,1,1,0}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1},v_{2},v_{3}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.5,0.3,0.6,0}\end{pmatrix}$
H. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,1,0,1}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{2},v_{3},v_{4}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.5,0.3,0.0,0}\end{pmatrix}$
I. $\left.T_{R}\left(A\right)=\begin{pmatrix}{1,0,0,0}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{1}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.5,0.3,0.0,0}\end{pmatrix}$
J. $\left.T_{R}\left(A\right)=\begin{pmatrix}{0,0,1,1}\end{pmatrix}\text{or }T_{R}\begin{pmatrix}{A}\end{pmatrix}=\left\{\begin{array}{c}{v_{3},v_{4}}\end{array}\right.\right\}, T_{R}\begin{pmatrix}{B}\end{pmatrix}=\begin{pmatrix}{0.4,0.3,0.5,0}\end{pmatrix}$
Answer: C
Let $U=\left\{ a, b, c, d, e, f \right\}$, $V=\{ x, y, z \}$,
$$
f(u)=v=\left\{\begin{array} {ll}
x & u \in \{a, d, e, f\}, \\
y & u \in \{b, c\}, \\
\end{array} \right.
$$
$$
A = \frac{0.1}{a} + \frac{0.4}{b} + \frac{1}{c} + \frac{0.6}{d} + \frac{0.3}{e}, \qquad A^{\prime} = \frac{1}{a} + \frac{0.8}{b} + \frac{0.5}{c} + \frac{0.2}{f}.
$$
What are $f(A)$, $f^{-1}(f(A))$, $f(A^{c})$, and $f(A \cup A^{\prime})$, respectively?

Options:
A. $$\begin{aligned}&f(A)=(0.7,1,1),\quad f^{-1}(f(A))=(1,0.4,0.7,0.3,0,0.6)\\&f(A^{c})=(1,0.8,1),\quad f(A\cup A^{\prime})=(1,1,0.7)\end{aligned}$$
B. $$\begin{aligned}&f(A)=(1,0.6,1),\quad f^{-1}(f(A))=(0.6,0.8,0.6,0,1,0.6)\\&f(A^{c})=(0.8,1,0),\quad f(A\cup A^{\prime})=(0.6,0,1)\end{aligned}$$
C. $$\begin{aligned}&f(A)=(0.6,1,1),\quad f^{-1}(f(A))=(0.6,1,0.7,0.6,1,0.6)\\&f(A^{c})=(1,1,0),\quad f(A\cup A^{\prime})=(1,0.6,0)\end{aligned}$$
D. $$\begin{aligned}&f(A)=(0,0.8,0),\quad f^{-1}(f(A))=(0,0,1,0.6,0.6,0.6)\\&f(A^{c})=(1,1,0),\quad f(A\cup A^{\prime})=(0.6,1,0)\end{aligned}$$
E. $$\begin{aligned}&f(A)=(0.6,0.9,0.3),\quad f^{-1}(f(A))=(0.6,0.9,0,0.6,0.6,0.6)\\&f(A^{c})=(0.9,0.5,0),\quad f(A\cup A^{\prime})=(1,1,1)\end{aligned}$$
F. $$\begin{aligned}&f(A)=(0.6,1,0),\quad f^{-1}(f(A))=(0.6,1,1,0.6,0.6,0.6)\\&f(A^{c})=(1,0.6,0),\quad f(A\cup A^{\prime})=(1,1,0)\end{aligned}$$
G. $$\begin{aligned}&f(A)=(0.8,0.5,0),\quad f^{-1}(f(A))=(0.8,0.5,0.8,1,0.3,0.6)\\&f(A^{c})=(1,0.5,0.3),\quad f(A\cup A^{\prime})=(1,0.8,0.3)\end{aligned}$$
H. $$\begin{aligned}&f(A)=(1,0.6,0),\quad f^{-1}(f(A))=(1,0,0.6,0.6,0.6,1)\\&f(A^{c})=(1,0.6,0.2),\quad f(A\cup A^{\prime})=(0.6,1,1)\end{aligned}$$
I. $$\begin{aligned}&f(A)=(0.4,0.5,0),\quad f^{-1}(f(A))=(0.4,0.5,0.6,1,0.7,0)\\&f(A^{c})=(0.9,0.3,1),\quad f(A\cup A^{\prime})=(0.9,1,0.6)\end{aligned}$$
J. $$\begin{aligned}&f(A)=(0.5,0.6,0),\quad f^{-1}(f(A))=(1,0.8,0.8,0.7,0.3,0.5)\\&f(A^{c})=(1,1,0),\quad f(A\cup A^{\prime})=(0.7,0.6,1)\end{aligned}$$
Answer: F
Let $V_{T}=\left\{ \left. a \right., b \right\}, V_{N}=\left\{ \left. A \right., B, C \right\}, S$ be the starting symbol. The production rules are as follows:
$$
\begin{array} {l c} \\{{{S \xrightarrow[p1]{{\mathrm{\scriptsize~ 1 ~}}} A B}}} & {{{A \xrightarrow[p2]{{\mathrm{\scriptsize~ 0. 9 ~}}} a A B}}} & {{{A \xrightarrow[p3]{{\mathrm{\scriptsize~ 1 ~}}}} a}}
\\ \\{{{A \xrightarrow[p4]{{\mathrm{\scriptsize~ 0. 5 ~}}} a B}}} & {{{A \xrightarrow[p5]{{\mathrm{\scriptsize~ 0. 2 ~}}} B}}} & {{{A \xrightarrow[p6]{{\mathrm{\scriptsize~ 0. 5 ~}}} a C}}}
\\ \\{{{B \xrightarrow[p7]{{\mathrm{\scriptsize~ 1 ~}}} a b}}} & {{{C \xrightarrow[p8]{{\mathrm{\scriptsize~ 0. 5 ~}}} a}}} & {{{C \xrightarrow[p9]{{\mathrm{\scriptsize~ 0. 2 ~}}} a a}}} \\ \end{array}
$$
What is the membership degree of the terminal string " $a b$ " and "aaabbb"?

Options:
A. $$G(ab)=0.9,G(aaabbb)=0.8$$
B. $$G(ab)=0.8,G(aaabbb)=1$$
C. $$G(ab)=0.7,G(aaabbb)=0.9$$
D. $$G(ab)=1,G(aaabbb)=0.9$$
E. $$G(ab)=0.9,G(aaabbb)=1$$
F. $$G(ab)=0.8,G(aaabbb)=0.7$$
G. $$G(ab)=1,G(aaabbb)=0.95$$
H. $$G(ab)=0.85,G(aaabbb)=0.9$$
I. $$G(ab)=1,G(aaabbb)=0.7$$
J. $$G(ab)=1,G(aaabbb)=0.8$$
Answer: D
Let $\Omega=\{\omega_{1}, \omega_{2}, \omega_{3}\}, \ A=0.6 / \omega_{1}+1 / \omega_{2}+0.7 / \omega_{3}$
$$
\begin{array}{l}
{{{\pi_{\omega_{1}} ="approximately 0.2" =\, 0.7 / 0.1 + 1 / 0.2 + 0.6 / 0.3}}} \\
{{{\pi_{\omega_{2}} ="approximately 0.7" =\, 0.6 / 0.6 + 1 / 0.7 + 0.5 / 0.8}}} \\
{{{\pi_{\omega_{3}} ="approximately 0.1" =\, 0.5 / 0 + 1 / 0.1 + 0.4 / 0.2}}} \\
\end{array}
$$
What is $p(A)$?

Options:
A. $0.4/0.87+0.5/0.92+0.6/0.93+1/0.85+0.5/0.89+0.4/0.86+0.5/0.90$
B. $0.3/0.86+0.4/0.91+0.6/0.94+1/0.88+0.5/0.89+0.5/0.84+0.7/0.85$
C. $0.3/0.91+0.5/0.94+0.4/0.93+1/0.86+0.4/0.90+0.6/0.84+0.7/0.87$
D. $0.4/0.86+0.4/0.9+0.5/0.93+1/0.89+0.5/0.92+0.5/0.88+0.6/0.85$
E. $0.4/0.83+0.5/0.90+0.6/0.91+1/0.86+0.3/0.89+0.6/0.85+0.5/0.88$
F. $0.4/0.88+0.5/0.92+0.6/0.95+1/0.86+0.4/0.93+0.5/0.87+0.5/0.90$
G. $0.5/0.85+0.3/0.93+0.4/0.92+1/0.87+0.6/0.90+0.6/0.86+0.4/0.89$
H. $0.3/0.87+0.5/0.91+0.5/0.94+1/0.88+0.4/0.91+0.6/0.87+0.7/0.84$
I. $0.3/0.84+0.4/0.89+0.6/0.92+1/0.87+0.5/0.91+0.5/0.85+0.7/0.88$
J. $0.4/0.89+0.5/0.90+0.6/0.94+1/0.85+0.4/0.92+0.5/0.87+0.6/0.88$
Answer: D
The vapor pressure of a volatile liquid can be determined by slowly bubbling a known volume of gas through it at a known temperature and pressure. In an experiment, 5.00 L of $\mathrm{N_{2}}$ gas is passed through 7.2146 g of liquid benzene, $\mathrm{C}_{6} \mathrm{H}_{6},$ at 26.0 °C. The liquid remaining after the experiment weighs 5.1493 $g$ . Assuming that the gas becomes saturated with benzene vapor and that the total gas volume and temperature remain constant, what is the vapor pressure of the benzene in torr?

Options:
A. $$
9 2. 2 \mathrm{~ t o r r}
$$
B. $$
7 5. 3 \mathrm{~ t o r r}
$$
C. $$
6 0. 4 \mathrm{~ t o r r}
$$
D. $$
9 8. 7 \mathrm{~ t o r r}
$$
E. $$
1 0 1. 3 \mathrm{~ t o r r}
$$
F. $$
1 0 5. 8 \mathrm{~ t o r r}
$$
G. $$
6 9. 5 \mathrm{~ t o r r}
$$
H. $$
1 2 1. 6 \mathrm{~ t o r r}
$$
I. $$
8 4. 9 \mathrm{~ t o r r}
$$
J. $$
1 1 4. 7 \mathrm{~ t o r r}
$$
Answer: D
When a 9.55-g sample of solid sodium hydroxide dis- solves in 100.0 g of water in a coffee-cup calorimeter (Figure 5.17), the temperature rises from 23.6 $^\circ C$ to 47.4 $^\circ C$ . Calculate $\Delta H$ is() (in kJ/mol NaOH) for the solu- tion process
$$
\mathrm{N a O H} ( s ) \, \longrightarrow\, \mathrm{N a}^{+} ( a q ) \,+\, \mathrm{O H}^{-} ( a q )
$$
Assume that the specific heat of the solution is the same as that of pure water.

Options:
A. $$
- 5 0. 1 \, \mathrm{k J / m o l \, N a O H}
$$
B. $$
- 4 3. 6 \, \mathrm{k J / m o l \, N a O H}
$$
C. $$
- 4 7. 8 \, \mathrm{k J / m o l \, N a O H}
$$
D. $$
- 3 8. 2 \, \mathrm{k J / m o l \, N a O H}
$$
E. $$
- 4 1. 9 \, \mathrm{k J / m o l \, N a O H}
$$
F. $$
- 5 2. 4 \, \mathrm{k J / m o l \, N a O H}
$$
G. $$
- 3 9. 7 \, \mathrm{k J / m o l \, N a O H}
$$
H. $$
- 4 9. 5 \, \mathrm{k J / m o l \, N a O H}
$$
I. $$
- 4 5. 7 \, \mathrm{k J / m o l \, N a O H}
$$
J. $$
- 4 0. 5 \, \mathrm{k J / m o l \, N a O H}
$$
Answer: I
A pound of plain M&M" candies contains 96 g fat, 320 $\mathrm{g}$ carbohydrate, and 21 g protein. What is the fuel value in kJ in a 42-g (about 1.5 oz) serving? How many Calories does it provide?

Options:
A. $$
8. 6 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 3 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
B. $$
7. 8 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
1. 9 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
C. $$
8. 3 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 0 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
D. $$
8. 0 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 0 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
E. $$
9. 0 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 3 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
F. $$
8. 5 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 2 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
G. $$
8. 7 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 1 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
H. $$
7. 5 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
1. 5 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
I. $$
9. 1 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
2. 2 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
J. $$
6. 9 \times1 0^{2} \mathrm{k J / s e r v i n g}
$$
$$
1. 8 \times1 0^{2} \, \mathrm{C a l / s e r v i n g}
$$
Answer: G
The hydrocarbons acetylene ($C_{\mathrm{2}}$$H_{\mathrm{2}}$) and benzene ($C_{\mathrm{6}}$$H_{\mathrm{6}}$) have the same empirical formula. Benzene is an "aromatic" hydrocarbon, one that is unusually stable because of its structure.Determine the fuel value in $\mathrm{k J / g}$ for acetylene and benzene.

Options:
A. $$
- 3 6 0 0. 5 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
3 9 \mathrm{k J / g}
$$
B. $$
- 3 1 0 0. 0 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
5 0 \mathrm{k J / g}
$$
C. $$
- 3 2 8 0. 2 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 6 \mathrm{k J / g}
$$
D. $$
- 3 4 5 0. 0 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 5 \mathrm{k J / g}
$$
E. $$
- 3 2 6 7. 5 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 2 \mathrm{k J / g}
$$
F. $$
- 3 4 0 0. 7 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 1 \mathrm{k J / g}
$$
G. $$
- 3 5 7 8. 4 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 7 \mathrm{k J / g}
$$
H. $$
- 2 9 9 9. 9 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 0 \mathrm{k J / g}
$$
I. $$
- 3 5 0 0. 0 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 3 \mathrm{k J / g}
$$
J. $$
- 3 3 0 1. 3 \mathrm{~ k J / m o l ~ C_{6} H_{6} ~}
$$
$$
4 4 \mathrm{k J / g}
$$
Answer: E
Compute
$$
\int_{0}^{2 \pi} \frac{1} {( 3+\operatorname{c o s} \theta)^{2}} \, d \theta
$$
using contour integration.

Options:
A. $$
2 \pi\frac{3 \cdot2} {( 3 \sqrt{0} )^{3}}=\frac{0 \sqrt{0} \pi} {8 6}
$$
B. $$
6 \pi\frac{1 \cdot3} {( 6 \sqrt{5} )^{3}}=\frac{5 \sqrt{5} \pi} {3 8}
$$
C. $$
7 \pi\frac{3 \cdot2} {( 5 \sqrt{3} )^{3}}=\frac{4 \sqrt{3} \pi} {2 7}
$$
D. $$
9 \pi\frac{1 \cdot2} {( 8 \sqrt{8} )^{3}}=\frac{8 \sqrt{8} \pi} {6 2}
$$
E. $$
4 \pi\frac{2 \cdot3} {( 7 \sqrt{7} )^{3}}=\frac{7 \sqrt{7} \pi} {5 3}
$$
F. $$
8 \pi\frac{2 \cdot3} {( 4 \sqrt{2} )^{3}}=\frac{3 \sqrt{2} \pi} {1 6}
$$
G. $$
3 \pi\frac{2 \cdot3} {( 9 \sqrt{9} )^{3}}=\frac{9 \sqrt{9} \pi} {7 5}
$$
H. $$
1 \pi\frac{2 \cdot3} {( 1 \sqrt{1} )^{3}}=\frac{1 \sqrt{1} \pi} {9 8}
$$
I. $$
5 \pi\frac{4 \cdot1} {( 2 \sqrt{6} )^{3}}=\frac{6 \sqrt{6} \pi} {4 9}
$$
J. $$
0 \pi\frac{3 \cdot2} {( 0 \sqrt{2} )^{3}}=\frac{2 \sqrt{2} \pi} {0 4}
$$
Answer: F
Let $U \subset\mathbb{C}$ be an open set containing the closed unit disc $\overline{{{{\Delta}}}}=\{z \in$ $\mathbb{C} : | z | \leq1 \}$ , and suppose that $f$ is a function on $U$ holomorphic except for a simple pole at $z_{0}$ with $| z_{0} |=1$ . Show that if
$$
\sum_{n=0}^{\infty} a_{n} z^{n}
$$
denotes the power series expansion of $f$ in the open unit disk, then what is its limit value?

Options:
A. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+2}}=z_{0}
$$
B. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=-z_{0}
$$
C. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=z_{0}
$$
D. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=\frac{1}{z_{0}}
$$
E. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=\infty
$$
F. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=1
$$
G. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=2z_{0}
$$
H. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=z_{1}
$$
I. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n}} {a_{n+1}}=0
$$
J. $$
\operatorname* {l i m}_{n \to\infty} \frac{a_{n+1}} {a_{n}}=z_{0}
$$
Answer: C
Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. What is the number of $3 \times3$ nilpotent matrices on $\mathbb{F}_{q}$?

Options:
A. $q^{11}$
B. $q^{12}$
C. $q^{8}$
D. $q^{5}$
E. $q^{10}$
F. $q^{3}$
G. $q^{6}$
H. $q^{9}$
I. $q^{7}$
J. $q^{4}$
Answer: G
Given that $\int_{0}^{\infty} e^{-x^{2}} \, d x \,=\, {\frac{1} {2}} \sqrt{\pi}$. Using complex calculus to calculate inappropriate integrals $\int_{0}^{\infty} \operatorname{s i n} ( x^{2} ) \, d x$ and $\int_{0}^{\infty} \operatorname{c o s} ( x^{2} ) \, d x$ both converge to ().

Options:
A. $$
\frac{\pi}{4} e.
$$
B. $$
\pi \sqrt{2/3}.
$$
C. $$
\frac{4\pi}{3 \sqrt{2}}.
$$
D. $$
\sqrt{\pi/ 8}.
$$
E. $$
\frac{\pi}{3 \sqrt{2}}.
$$
F. $$
\sqrt{\pi} e^{2}.
$$
G. $$
e^{\pi/2}.
$$
H. $$
\sqrt{2\pi} e.
$$
I. $$
\frac{e \pi^{2}}{8}.
$$
J. $$
\frac{\sqrt{\pi}}{6}.
$$
Answer: D
Consider the graph $S$ of the function $F ( x, y )=\operatorname{c o s h} ( x ) \operatorname{c o s} ( y )$ in $\mathbb{R}^{3}$ and
let
$$
\Phi: \mathbb{R}^{2} \to S \subset R^{3}
$$
be its parametrization: $\Phi( x, y )=( x, y, \operatorname{c o s h} ( x ) \operatorname{c o s} ( y ) ).$ \
What is the Gaussian curvature $of a general surface embedded in
$\mathbb{R}^{3}$ ?

Options:
A. -4
B. 3
C. -2
D. 2
E. 4
F. 0
G. 5
H. -1
I. -3
J. 1
Answer: H
(DG) The symplectic group $S p ( 2 n, \mathbb{R} )$ is defined as the subgroup of $G l ( 2 n, \mathbb{R} )$ that preserves the matrix
$$
\Omega=\left( \begin{matrix} {{0}} & {{I_{n}}} \\ {{-I_{n}}} & {{0}} \\ \end{matrix} \right)
$$
where $I_{n}$ is the $n \times n$ identify matrix. That is, it is composed of elements of $G l ( 2 n, \mathbb{R} )$ that satisfy the relation
$$
M^{T} \Omega M=\Omega.
$$
What is the dimension of the symplectic group?

Options:
A. $$
n^{2}+2n
$$
B. $$
2n^{2}+n
$$
C. $$
2n(n-1)
$$
D. $$
4n+2
$$
E. $$
3n^{2}-n
$$
F. $$
n^{2}+n
$$
G. $$
n(n+2)
$$
H. $$
n(n+1)
$$
I. $$
2n^{2}-n
$$
J. $$
n(2n+1)
$$
Answer: J
What is the statement of Bézout's theorem for $\mathbb{P}^{n}$ ?

Options:
A. $$\dim(X\cap Y)=(\dim X)+(\dim Y)$$
B. $$\deg(XY)=(\deg X)+(\deg Y)$$
C. $$\dim(X-Y)=(\dim X)+(\dim Y)$$
D. $$\deg(X\cup Y)=(\deg X)(\deg Y)$$
E. $$\dim(X\cup Y)=(\dim X)(\dim Y)$$
F. $$\deg(X\cap Y)=(\deg X)(\deg Y)$$
G. $$\dim(X+Y)=(\deg X)(\deg Y)$$
H. $$\deg(X+Y)=(\deg X)+(\deg Y)$$
I. $$\deg(X-Y)=(\deg X)(\deg Y)$$
J. $$\deg(X\cdot Y)=(\deg X)+(\deg Y)$$
Answer: F
Let $n$ be a positive integer, what is the residue of the Gamma function at the pole $z=-n?$

Options:
A. $$\Gamma(z)=\frac{\Gamma(z+1)}{n}$$
B. $$\Gamma(z)=\frac{\Gamma(-z)}{z}$$
C. $$\Gamma(z)=\frac{\Gamma(z+1)}{z}$$
D. $$\Gamma(z)=\frac{-\Gamma(z+1)}{n}$$
E. $$\Gamma(z)=\frac{z\Gamma(z+1)}{n}$$
F. $$\Gamma(z)=\frac{\Gamma(z-n)}{z}$$
G. $$\Gamma(z)=\frac{n\Gamma(z+1)}{-z}$$
H. $$\Gamma(z)=\frac{\Gamma(n+z)}{z}$$
I. $$\Gamma(z)=\frac{\Gamma(z+2)}{z}$$
J. $$\Gamma(z)=-\frac{\Gamma(z+1)}{z}$$
Answer: C
Let $\gamma: ( 0, 1 ) \to\mathbb{R}^{3}$ be a smooth arc, with $\gamma^{\prime} \neq0$ everywhere.\
What is the curvature and torsion of the arc?

Options:
A. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
b^{\prime}=\kappa v-\tau n.
$$
B. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
n^{\prime}=-\kappa v+\tau b.
$$
C. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime\prime} ( t ) |^{2},
$$
Finally the torsion $\tau$ is defined by
$$
n^{\prime}=-v+\tau b.
$$
D. The curvature of $\gamma$ is
$$
\kappa( t )=| v(t) |,
$$
Finally the torsion $\tau$ is defined by
$$
n^{\prime}=-b+\tau \kappa.
$$
E. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
n^{\prime}=\kappa v-\tau b.
$$
F. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
b^{\prime}=\kappa n+\tau v.
$$
G. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
v^{\prime}=-\kappa b+\tau n.
$$
H. The curvature of $\gamma$ is
$$
\kappa( t )=| \gamma^{\prime\prime} ( t ) |,
$$
Finally the torsion $\tau$ is defined by
$$
b^{\prime}=-\kappa v+\tau n.
$$
I. The curvature of $\gamma$ is
$$
\kappa( t )=\gamma^{\prime\prime} ( t ),
$$
Finally the torsion $\tau$ is defined by
$$
v^{\prime}=-\kappa b+\tau v.
$$
J. The curvature of $\gamma$ is
$$
\kappa( t )=\gamma^{\prime} ( t ),
$$
Finally the torsion $\tau$ is defined by
$$
n^{\prime}=\kappa n+\tau b.
$$
Answer: B
Let $X=( S^{1} \times S^{1} ) \setminus\{p \}$ be a once-punctured torus.
How many connected, 3-sheeted covering spaces $f : Y \to X$ are there?

Options:
A. 14
B. 9
C. 13
D. 5
E. 11
F. 7
G. 12
H. 10
I. 6
J. 8
Answer: F
Motivate the name "screening length" for $\kappa_{\mathrm{s c}}^{-1}$ by finding the electro- static potential of a single ion at rest at the origin of a degenerate, neutral plasma in the Thomas-Fermi approximation is().

Options:
A. $$
Z e \operatorname{log} (\kappa_{s c} r ) / r ]$$
B. $$
Z e \operatorname{sin} (\kappa_{s c} r ) / r ]$$
C. $$
Z e \operatorname{e x p} (-\kappa_{s c} r ) / r ]
$$
D. $$
Z e \operatorname{e x p} (\kappa_{s c} r^2 ) / r ]$$
E. $$
Z e \operatorname{e x p} (\kappa_{s c} r ) \cdot r ]$$
F. $$
Z r^{-1} \operatorname{exp} (\kappa_{s c} r ) ]$$
G. $$
Z e^{-1} \operatorname{e x p} (-\kappa_{s c} r ) ]$$
H. $$
Z e \operatorname{e x p} (-r / \kappa_{s c} ) / r ]$$
I. $$
Z e \operatorname{cos} (\kappa_{s c} r ) / r ]$$
J. $$
Z \operatorname{e x p} (-\kappa_{s c} r ) \cdot r ]$$
Answer: C
The value of $\Delta E$ is sensitive to the value of $\varepsilon_{0}-\varepsilon$ at the time of the quake. Estimate $\Delta E$ for a Vela corequake is(), assuming $\varepsilon_{0}-\varepsilon-2 \times1 0^{-3}$ . For a 1.93 $M_{\odot}$ TI stan with a solid core Pandharipande, Pines, and Smith give $A \,=\, 1 7. 8 \, \times\, 1 0^{5 2}$ erg, $B_{\mathrm{c o r e}}=1 4. 4$ $\times1 0^{5 2}$ erg.

Options:
A. $$
\Delta E\sim4 \times1 0^{4 5} \mathrm{~ e r g}
$$
B. $$
\Delta E\sim6 \times1 0^{4 2} \mathrm{~ e r g}
$$
C. $$
\Delta E-3 \times1 0^{4 3} \mathrm{~ e r g}
$$
D. $$
\Delta E\sim7 \times1 0^{4 1} \mathrm{~ e r g}
$$
E. $$
\Delta E\sim1 \times1 0^{4 6} \mathrm{~ e r g}
$$
F. $$
\Delta E\sim8 \times1 0^{4 8} \mathrm{~ e r g}
$$
G. $$
\Delta E-5 \times1 0^{4 6} \mathrm{~ e r g}
$$
H. $$
\Delta E-9 \times1 0^{4 7} \mathrm{~ e r g}
$$
I. $$
\Delta E\sim1 \times1 0^{4 5} \mathrm{~ e r g}
$$
J. $$
\Delta E\sim2 \times1 0^{4 4} \mathrm{~ e r g}
$$
Answer: I
The entropy of a $M_{\odot}$black holein units of $k$ , Boltzmann's constant is().

Options:
A. $$
7. 0 \times1 0^{3 3} \, k
$$
B. $$
8. 0 \times1 0^{2 2} \, k
$$
C. $$
1. 0 \times1 0^{7 7} \, k
$$
D. $$
1. 1 \times1 0^{7 7} \, k
$$
E. $$
5. 0 \times1 0^{9 9} \, k
$$
F. $$
9. 0 \times1 0^{4 4} \, k
$$
G. $$
3. 0 \times1 0^{5 5} \, k
$$
H. $$
4. 0 \times1 0^{8 8} \, k
$$
I. $$
6. 0 \times1 0^{1 2} \, k
$$
J. $$
2. 0 \times1 0^{6 6} \, k
$$
Answer: C
Consider a neutral beam of cold plasma, intially moving with velocity $\upsilon$ in a field-free region, incident normally to a plane surface, beyond which there is a uniform $B$ field parallel to the plane (Lamb, Pethick, and Pines, 1973). Neglect collisions between particles.Estimate the ion and electron contributions to the surface current density $ℊ$ and the corresponding jump in the magnetic field that the current generates is().

Options:
A. $$
\Delta B \sim7 \pi( ℊ_{e}^2-ℊ_{i}^2 ) / c \simeq7 \pi\rho v^{4} / B
$$
B. $$
\Delta B \sim1 \pi( ℊ_{i}^3+ℊ_{e}^3 ) / c \simeq1 \pi\rho v^{0} / B
$$
C. $$
\Delta B \sim8 \pi( ℊ_{i}+ℊ_{e} ) / c \simeq8 \pi\rho v^{5} / B
$$
D. $$
\Delta B \sim9 \pi( ℊ_{e}^3-ℊ_{i}^3 ) / c \simeq9 \pi\rho v^{1} / B
$$
E. $$
\Delta B \sim2 \pi( ℊ_{e}+ℊ_{i} ) / c \simeq2 \pi\rho v^{1} / B
$$
F. $$
\Delta B \sim6 \pi( ℊ_{i}-ℊ_{e} ) / c \simeq6 \pi\rho v^{2} / B
$$
G. $$
\Delta B \sim4 \pi( ℊ_{i}+ℊ_{e} ) / c \simeq4 \pi\rho v^{2} / B
$$
H. $$
\Delta B \sim3 \pi( ℊ_{e}-ℊ_{i} ) / c \simeq3 \pi\rho v^{2} / B
$$
I. $$
\Delta B \sim10 \pi( ℊ_{i}^4+ℊ_{e}^4 ) / c \simeq10 \pi\rho v^{3} / B
$$
J. $$
\Delta B \sim5 \pi( ℊ_{i}^2+ℊ_{e}^2 ) / c \simeq5 \pi\rho v^{3} / B
$$
Answer: G
Find the central temperature at the onset of instability is().

Options:
A. $$
( 2. 0 1 \times1 0^{1 3} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
B. $$
( 3. 5 2 \times1 0^{1 2} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
C. $$
( 1. 7 5 \times1 0^{1 2} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
D. $$
( 1. 8 3 \times1 0^{1 4} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
E. $$
( 3. 9 4 \times1 0^{1 2} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
F. $$
( 4. 3 1 \times1 0^{1 3} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
G. $$
( 2. 6 7 \times1 0^{1 3} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
H. $$
( 2. 4 9 \times1 0^{1 3} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
I. $$
( 1. 9 8 \times1 0^{1 3} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
J. $$
( 2. 5 0 \times1 0^{1 4} \ \mathrm{K} ) \frac{M_{\odot}} {M}
$$
Answer: H
Show that
$$
x \frac{d} {d x} ( \delta( x ) )=-\delta( x ).
$$
[Hint: Use integration by parts.]

Options:
A. $$
x \cdot \ln(x) \cdot \delta(x).
$$
B. $$
-x \cdot \delta(x).
$$
C. $$
x \cdot e^{x} \cdot \delta(x).
$$
D. $$
\int x \delta(x) dx.
$$
E. $$
-\delta( x ).
$$
F. $$
x \cdot \delta(x^2).
$$
G. $$
x \cdot \frac{d}{dx}(\delta(x)).
$$
H. $$
\frac{d}{dx}(\delta'(x)).
$$
I. $$
x \cdot \delta'(x).
$$
J. $$
\delta''(x) \cdot x.
$$
Answer: E
The potential $V_{0} ( \theta)$ is specified on the surface of a hollow sphere, of radius $R$ Find the potential inside the sphere.

Options:
A. $$
{\frac{k} {2}} \left( 2-{\frac{R} {r}} \operatorname{s i n} \theta\right)
$$
B. $$
{\frac{k} {2}} \left( 1+{\frac{r} {R}} \operatorname{s i n} \theta\right)
$$
C. $$
{\frac{k} {4}} \left( 1-{\frac{R} {r}} \operatorname{t a n} \theta\right)
$$
D. $$
{\frac{k} {2}} \left( 1-{\frac{r} {R}} \operatorname{c o s} \theta\right)
$$
E. $$
{\frac{2k} {3}} \left( 1-{\frac{r} {R}} \operatorname{c o s} \theta^{2}\right)
$$
F. $$
{\frac{k} {5}} \left( 1-{\frac{r} {R}} \operatorname{s i n} \theta\right)
$$
G. $$
{\frac{k} {2}} \left( 1-{\frac{r^{2}} {R}} \operatorname{c o s} \theta\right)
$$
H. $$
{\frac{k} {3}} \left( 1-{\frac{r} {R}} \operatorname{c o s} 2\theta\right)
$$
I. $$
{\frac{k} {2}} \left( 2-{\frac{r} {R}} \operatorname{c o s} \theta\right)
$$
J. $$
k \left( 1-{\frac{R} {r}} \sin \theta\right)
$$
Answer: D
If the electric field in some region is given (in spherical coordinates) by the
expression
$$
\mathbf{E} ( \mathbf{r} )={\frac{A \, {\hat{\mathbf{r}}}+B \, \operatorname{s i n} \theta\, \operatorname{c o s} \phi\, {\hat{\boldsymbol{\phi}}}} {r}},
$$
where $A$ and $B$ are constants, what is the charge depsity?

Options:
A. $$
\frac{\epsilon_{0}} {r^{2}} ( B-A \operatorname{s i n} \phi).
$$
B. $$
\frac{\epsilon_{0}} {r^{3}} ( A-B \operatorname{s i n} \phi).
$$
C. $$
\frac{\epsilon_{0}} {r^{2}} (A-B \operatorname{c o s} \phi).
$$
D. $$
\frac{\epsilon_{0}} {r} ( A+B \operatorname{c o s} \phi).
$$
E. $$
\frac{\epsilon_{0}} {r^{2}} ( A+B \operatorname{c o s} \phi).
$$
F. $$
\frac{\epsilon_{0}} {r^{3}} ( A+B \operatorname{s i n} \phi).
$$
G. $$
\frac{2 \epsilon_{0}}{r^{2}} ( A-B \operatorname{s i n} \phi).
$$
H. $$
\frac{\epsilon_{0}} {r^{2}} (A+B \operatorname{s i n} \phi).
$$
I. $$
\frac{\epsilon_{0}} {r} ( A-B \operatorname{s i n} \phi).
$$
J. $$
\frac{\epsilon_{0}} {r^{2}} ( A-B \operatorname{s i n} \phi).
$$
Answer: J
Suppose you had a detection device that could count every decay event from a radioactive sample of plutonium-239 $( t_{1 / 2}$ is 24,000 yr) Sa How many counts per second would you obtain from a sample containing $0.385 \mathbf{g}$ plutonium-239?

Options:
A. $$
8. 9 \times1 0^{8} {\mathrm{~ d i s}} / {\mathrm{s}}
$$
B. $$ 9. 8 \times1 0^{5} {\mathrm{~ d i s}} / {\mathrm{s}} $$
C. $$ 7. 2 \times1 0^{6} {\mathrm{~ d i s}} / {\mathrm{s}} $$
D. $$ 3. 4 \times1 0^{7} {\mathrm{~ d i s}} / {\mathrm{s}} $$
E. $$ 3. 9 \times1 0^{8} {\mathrm{~ d i s}} / {\mathrm{s}} $$
F. $$ 5. 1 \times1 0^{9} {\mathrm{~ d i s}} / {\mathrm{s}} $$
G. $$ 2. 7 \times1 0^{7} {\mathrm{~ d i s}} / {\mathrm{s}} $$
H. $$ 1. 5 \times1 0^{9} {\mathrm{~ d i s}} / {\mathrm{s}} $$
I. $$ 6. 8 \times1 0^{6} {\mathrm{~ d i s}} / {\mathrm{s}} $$
J. $$ 4. 6 \times1 0^{8} {\mathrm{~ d i s}} / {\mathrm{s}} $$
Answer: A
If the equilibrium constant for a two-electron redox re- action at 298 $\mathbf{K}$ is $1. 5 \times1 0^{-4}$ , the correspond- ing $\Delta G^{\circ}$ and $E_{\mathrm{c e l l}}^{\mathrm{o}}$ is______ .

Options:
A. $$
\Delta G^{\circ}=19.5kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.115V
$$
B. $$
\Delta G^{\circ}=30.2kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.125V
$$
C. $$
\Delta G^{\circ}=22.4kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.130V
$$
D. $$
\Delta G^{\circ}=18.7kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.098V
$$
E. $$
\Delta G^{\circ}=20.3kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.118V
$$
F. $$
\Delta G^{\circ}=23.1kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.107V
$$
G. $$
\Delta G^{\circ}=24.8kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.110V
$$
H. $$
\Delta G^{\circ}=21.8kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.113V
$$
I. $$
\Delta G^{\circ}=27.6kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.120V
$$
J. $$
\Delta G^{\circ}=25.0kJ
$$
$$
\mathrm{E}^{\mathrm{o}}=-0.105V
$$
Answer: H
One enzyme-catalysed reaction in a biochemical cycle has an equilibrium constant that is 8.4 times the equilibrium constant of a second reaction. If the standard Gibbs energy of the former reaction is -250 kJ mol-1, what is the standard reaction Gibbs energy of the second reaction?

Options:
A. $$
- 2 5 0 \mathrm{k J m o l}^{-1}
$$
B. $$
- 2 4 0 \mathrm{k J m o l}^{-1}
$$
C. $$
- 2 4 5 \mathrm{k J m o l}^{-1}
$$
D. $$
- 2 3 5 \mathrm{k J m o l}^{-1}
$$
E. $$
- 2 2 5 \mathrm{k J m o l}^{-1}
$$
F. $$
- 2 6 0 \mathrm{k J m o l}^{-1}
$$
G. $$
- 2 7 0 \mathrm{k J m o l}^{-1}
$$
H. $$
- 2 8 0 \mathrm{k J m o l}^{-1}
$$
I. $$
- 2 1 5 \mathrm{k J m o l}^{-1}
$$
J. $$
- 2 9 5 \mathrm{k J m o l}^{-1}
$$
Answer: C