MikaStars39/nano-eval huggingface dataset:
A body weighs 20 N and 10 N when weighed under submerged conditions in liquids of relative densities 0.8 and 1.2 respectively. Determine its volume and weight in air.

Options:
A. $$
2. 8 8 4 0 \mathrm{~ L}
$$
$$
4 5 \ \mathrm{N}
$$
B. $$
2. 6 0 3 9 \mathrm{~ L}
$$
$$
3 2 \ \mathrm{N}
$$
C. $$
2. 7 0 5 3 \mathrm{~ L}
$$
$$
3 7 \ \mathrm{N}
$$
D. $$
2. 4 7 3 8 \mathrm{~ L}
$$
$$
3 9 \ \mathrm{N}
$$
E. $$
2. 5 0 4 6 \mathrm{~ L}
$$
$$
4 3 \ \mathrm{N}
$$
F. $$
2. 5 5 3 6 \mathrm{~ L}
$$
$$
4 0 \ \mathrm{N}
$$
G. $$
2. 3 8 4 5 \mathrm{~ L}
$$
$$
4 1 \ \mathrm{N}
$$
H. $$
2. 4 4 5 3 \mathrm{~ L}
$$
$$
4 2 \ \mathrm{N}
$$
I. $$
2. 3 5 5 8 \mathrm{~ L}
$$
$$
3 8 \ \mathrm{N}
$$
J. $$
2. 6 0 5 0 \mathrm{~ L}
$$
$$
3 5 \ \mathrm{N}
$$
Answer: F
Calculate the unknown velocity component in the following, so that the equation of continuity is satisfied. $(a)u=A ( x^2+y^2); (b) u=Ae^x; (c)u=-A \operatorname{l n} \, \frac{x} {L}$

Options:
A. $$
2x \mathbf{A} y+f ( x )
$$
$$
- \mathbf{A} y^{2} e^{x}+f ( x )
$$
$$
\mathbf{A} (x/y)+f ( x )
$$
B. $$
- 3 \mathbf{A} x y+f ( x )
$$
$$
- 2 \mathbf{A} e^{x} y+f ( x )
$$
$$
\mathbf{A} / (xy)+f ( x )
$$
C. $$
- 2 \mathbf{A} x y+f ( x )
$$
$$
- \mathbf{A} e^{x} y+f ( x )
$$
$$
\mathbf{A} y / x+f ( x )
$$
D. $$
2 \mathbf{A} x^{2} y+f ( x )
$$
$$
- \mathbf{A} xy e^{x}+f ( x )
$$
$$
(xy) / \mathbf{A} +f ( x )
$$
E. $$
- \mathbf{A} x^{2} y+f ( x )
$$
$$
\mathbf{A} e^{x} +y+f ( x )
$$
$$
\mathbf{A} xy +f ( x )
$$
F. $$
- \mathbf{A} y x^{2}+f ( x )
$$
$$
e^{x} + y\mathbf{A}+f ( x )
$$
$$
\mathbf{A} (x y)+f ( x )
$$
G. $$
- \mathbf{A} x^{3} y+f ( x )
$$
$$
- 3 \mathbf{A} e^{x} y+f ( x )
$$
$$
\mathbf{A} (y/x)+f ( x )
$$
H. $$
0 \mathbf{A} x y (x)+f
$$
$$
\mathbf{A} ye^{x}+f ( x )
$$
$$
y/\mathbf{A} x +f ( x )
$$
I. $$
2 \mathbf{A} x y+f ( x )
$$
$$
\mathbf{A} e^{x} y+f ( x )
$$
$$
\mathbf{A} x/y+f ( x )
$$
J. $$
0 \mathbf{A} xy+f ( x )
$$
$$
- \mathbf{A} e^{x} x+f ( x )
$$
$$
\mathbf{A} x + y+f ( x )
$$
Answer: C
Values of $\phi$ for various flows are given. Determine the corresponding values of $\psi$
$$
( \mathrm{i} ) \ \ \phi=3 x y
$$
$$
( \mathrm{i i} ) \; \; \phi=4 ( x^{2}-y^{2} )
$$
$$
( \mathrm{i i i} ) \ \ \phi=x+y+3
$$

Options:
A. $$
-3 / 2 ( 2x^{2}-y ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
4 x - y + c
$$
$$
3x+y+c\mathrm{ ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
B. $$
2 / 3 ( x^{2}-y^{2} ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
-6xy+c
$$
$$
y + x + c + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
C. $$
3 / 2 ( x^{2}+y^{2} ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
-8 x y + c
$$
$$
y-x+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
D. $$
-4 ( y^{2}-x ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
3x y+c
$$
$$
x + y^2 + c + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
E. $$
3 / 2 ( x^{2}-4y^{2} ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
12x y + c
$$
$$
x-3y+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
F. $$
3 / 2 ( x+y^{2} ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
-6 x y+c
$$
$$
2y-4x+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
G. $$
-3 / 2 ( x^{2}-y^{2} )+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
8 x y+c
$$
$$-x+y+c {~w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}$$
H. $$
-3 / 2 ( x^{2}-2y^{2} ) {~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
8 x^2 - y + c
$$
$$
x+y+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
I. $$
-3 / 2 ( x^{2}-3y ) + \mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
-2x y + c
$$
$$
x+4y+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
J. $$
3 / 2 ( y^{2}-x^{2} ) + \mathrm{ ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
$$
2 x y + c
$$
$$
-x-2y+c+\mathrm{c ~ w h e r e ~ c ~ i s ~ a ~ c o n s t a n t}
$$
Answer: G
Given that a laminar boundary layer at zero pressure gradient is described by the velocity profile
$$
{\frac{u} {U_{0}}} \,=\, \operatorname{s i n} \left( {\frac{\pi} {2}} \cdot{\frac{y} {\delta}} \right)
$$
determine the momentum correction coefficient and energy correction coefficient.

Options:
A. $$
1. 2 4 4 6
$$
$$
1. 7 2 1 3
$$
B. $$
1. 2 1 8 6
$$
$$
1. 6 3 5 4
$$
C. $$
1. 2 3 1 4
$$
$$
1. 6 6 5 5
$$
D. $$
1. 3 2 4 5
$$
$$
1. 6 1 7 8
$$
E. $$
1. 2 3 3 7
$$
$$
1. 6 4 4 9
$$
F. $$
1. 2 5 1 9
$$
$$
1. 5 4 3 3
$$
G. $$
1. 1 3 6 8
$$
$$
1. 5 3 2 4
$$
H. $$
1. 1 1 3 3
$$
$$
1. 4 4 6 7
$$
I. $$
1. 1 2 5 0
$$
$$
1. 7 4 6 2
$$
J. $$
1. 3 4 2 1
$$
$$
1. 5 5 3 8
$$
Answer: E
The drag force $F_{\mathrm{D}}$ on a sphere in laminar flow is known to depend on its diameter $D$ , velocity of flow $V$ density of fluid $\rho$ and coefficient of viscosity $\mu.$ Obtain an expression for $F_{\mathrm{D}}$ using Raleigh's method.

Options:
A. $$
F_{D}=\mu D^{2} V^{2} \, \mathrm{f n} \left( {\frac{\rho} {V D \mu}} \right)
$$
B. $$
F_{D}=\rho^{2} D V^{2} \, \mathrm{f n} \left( {\frac{\mu} {\rho V D}} \right)
$$
C. $$
F_{D}=\rho^{2} D^{2} V \, \mathrm{f n} \left( {\frac{\mu} {V \rho D^{2}}} \right)
$$
D. $$
F_{D}=\mu D^{2} V \, \mathrm{f n} \left( {\frac{\rho D} {V \mu}} \right)
$$
E. $$
F_{D}=\rho D^{3} V \, \mathrm{f n} \left( {\frac{\mu} {D^{2} V \rho}} \right)
$$
F. $$
F_{D}=\rho D^{2} V^{2} \, \mathrm{f n} \left( {\frac{\mu} {V D \rho}} \right)
$$
G. $$
F_{D}=\rho D V^{2} \, \mathrm{f n} \left( {\frac{\mu} {V^{2} \rho D}} \right)
$$
H. $$
F_{D}=\mu^{2} D V \, \mathrm{f n} \left( {\frac{\rho V^{2}} {D^{2} \mu}} \right)
$$
I. $$
F_{D}=\rho D^{2} V^{3} \, \mathrm{f n} \left( {\frac{\mu} {\rho D^{2} V}} \right)
$$
J. $$
F_{D}=\mu D V \, \mathrm{f n} \left( {\frac{\rho V} {D \mu}} \right)
$$
Answer: F
Our genetic information or genome is stored in the parts of the DNA molecule called base pairs. Our genome contains about 3 billion $( 3 \times1 0^{9} )$ base pairs, and there are two copies in each cell. Along the DNA molecule, there is one base pair every one-third of a nanometer. How long would the DNA helix from one cell be if it were stretched out in a line? If the entire DNA molecule were wrapped up into $a$ sphere, what would be the diameter of that sphere?

Options:
A. $$
4 \mathrm{m}
$$
$$
5.6 \mathrm{\mu m}
$$
B. $$
3 \mathrm{m}
$$
$$
3.5 \mathrm{\mu m}
$$
C. $$
2.5 \mathrm{m}
$$
$$
3.0 \mathrm{\mu m}
$$
D. $$
0.5 \mathrm{m}
$$
$$
1.8 \mathrm{\mu m}
$$
E. $$
5 \mathrm{m}
$$
$$
7.2 \mathrm{\mu m}
$$
F. $$
6.8 \mathrm{m}
$$
$$
8.4 \mathrm{\mu m}
$$
G. $$
1.5 \mathrm{m}
$$
$$
2.0 \mathrm{\mu m}
$$
H. $$
3.2 \mathrm{m}
$$
$$
4.1 \mathrm{\mu m}
$$
I. $$
6.2 \mathrm{m}
$$
$$
10.7 \mathrm{\mu m}
$$
J. $$
2 \mathrm{m}
$$
$$
2.8 \mathrm{\mu m}
$$
Answer: J
A sphere of radius $R$ moving with speed $v$ in laminar flow through a viscous fluid erperiences a drag force $F_{v i s c}=6 \pi\eta R v$ . At higher speeds inertial effects (the acceleration of the fluid displaced by the sphere) become important, and the drag becomes proportional to $v^{2}$ and to $\rho$ , the density of the fluid. The force also depends on the radius of the sphere to some unknown power: $F_{d r a g}=$ $K \rho v^{2} R^{a}$ , where $K$ is a dimensionless constant.
Make a dimensional analysis to find the power $a$. How much is it?

Options:
A. 3
B. 6
C. 5
D. 2
E. -2
F. 1
G. 7
H. 0
I. 4
J. -1
Answer: D
Suppose that the amount of a resource at time $t$ is $y ( t )$ . At $t=0$ the amount is $y_{0}$. The rate at which it is consumed is $r=-d y / d t$ . Let $r=r_{0} e^{b t}$
is, the rate of use increases exponentially with time. For example, the world use of crude oil has been increasing about 7% per vear since 1890.
Prove that the quantity remaining at time $t$ is $y(t)$=______.

Options:
A. $y_{0}+( b / r_{0} ) ( e^{b t}-1 )$
B. $y_{0}-( r_{0} b ) ( e^{b t}+1 )$
C. $y_{0}+( r_{0} b ) ( 1-e^{-b t} )$
D. $y_{0}-( r_{0} / b ) ( 1-e^{b t} )$
E. $y_{0}+( r_{0} / b ) ( 1-e^{b t} )$
F. $y_{0}+( r_{0} / b ) ( e^{-b t}-1 )$
G. $y_{0}-( r_{0} / b ) ( e^{-b t}-1 )$
H. $y_{0}+( r_{0} / b ) ( e^{b t}+1 )$
I. $y_{0}-( b r_{0} ) ( e^{b t}-1 )$
J. $y_{0}-( r_{0} / b ) ( e^{b t}-1 )$
Answer: J
There are $N$ people in a class $( N \,=\, 2 5 )$ . What is the probability that no one in the class has $a$ birthday on a particular day? Ignore seasonal variations in birth rate and ignore leap years.

Options:
A. 93%
B. 85%
C. 97%
D. 87%
E. 94%
F. 92%
G. 89%
H. 96%
I. 91%
J. 95%
Answer: A
The goal of this problem is to estimate how large a cell living in an oxygenated medium can be before it is limited by oxygen transport. Assume the extracellular space is well-stirred with uniform oxygen concentration $C_{0}$. The cell is a sphere of radius $R$. Inside the cell oxygen is consumed at a rate $Q$ molecule $m^{-3} ~ s^{-1}$ . The diffusion constant for oxygen in the cell is $D$. Assume that if the cell is to survive the oxygen concentration at the center of the cell cannot become negative. Use this constraint to estimate the marimum size of the cell is______.

Options:
A. $$
R={\sqrt{\frac{7 D C_{0}} {Q}}}
$$
B. $$
R={\sqrt{\frac{4 D C_{0}} {Q}}}
$$
C. $$
R={\sqrt{\frac{6 D C_{0}} {Q}}}
$$
D. $$
R={\sqrt{\frac{2 D C_{0}} {Q}}}
$$
E. $$
R={\sqrt{\frac{9 D C_{0}} {Q}}}
$$
F. $$
R={\sqrt{\frac{3 D C_{0}} {Q}}}
$$
G. $$
R={\sqrt{\frac{8 D C_{0}} {Q}}}
$$
H. $$
R={\sqrt{\frac{10 D C_{0}} {Q}}}
$$
I. $$
R={\sqrt{\frac{5 D C_{0}} {Q}}}
$$
J. $$
R={\sqrt{\frac{12 D C_{0}} {Q}}}
$$
Answer: C
The protein concentration in serum is made up of two main components: albumin (molecular weight 75,000) 4.5 g per 100 ml and globulin (molecular weight 170,000) 2.0 g per 100 ml. Calculate the osmotic pressure due to each constituent is_____. (These results are inaccurate because of electrical effects. )

Options:
A. 12.0torr ; 2.4torr
B. 14.8torr ; 3.0torr
C. 11.0torr ; 2.5torr
D. 12.5torr ; 2.1torr
E. 15.5torr ; 3.1torr
F. 13.7torr ; 2.6torr
G. 11.7torr ; 2.3torr
H. 11.4torr ; 2.2torr
I. 10.5torr ; 1.9torr
J. 13.2torr ; 2.7torr
Answer: G
The Moon lies 384,000 km from Earth, and the Sun lies 150,000,000 km away. If both have the same angular size as seen from Earth, how many times larger than the Moon is the Sun?

Options:
A. $$
4 1 0 \mathrm{t i m e s}
$$
B. $$
3 8 0 \mathrm{t i m e s}
$$
C. $$
4 0 0 \mathrm{t i m e s}
$$
D. $$
3 9 1 \mathrm{t i m e s}
$$
E. $$
4 3 0 \mathrm{t i m e s}
$$
F. $$
4 2 0 \mathrm{t i m e s}
$$
G. $$
4 5 0 \mathrm{t i m e s}
$$
H. $$
3 6 5 \mathrm{t i m e s}
$$
I. $$
3 5 0 \mathrm{t i m e s}
$$
J. $$
3 7 5 \mathrm{t i m e s}
$$
Answer: D
A certain telescope has a $1 0^{\prime} \times 1 0^{\prime}$ field of view that is recorded using a CCD chip having 2048 X 2048 pixels. What would be the diameter, in pixels, of a typical seeing disk (1" radius)?

Options:
A. Approximately 11
B. Approximately 10
C. Approximately 3
D. Approximately 5
E. Approximately 9
F. Approximately 7
G. Approximately 8
H. Approximately 6
I. Approximately 12
J. Approximately 15
Answer: F
Compared with a star of absolute magnitude $- 2$ at a distance of 100 pc, a star of absolute magnitude $5$ at a distance of 10 pc will appear is().

Options:
A. $$
\operatorname{c o m p a r a b l y \, b r i g h t}
$$
B. $$
\operatorname{s i m i l a r \, t o t a l \, f l u x}
$$
C. $$
\operatorname{e q u a l \, b r i g h t n e s s}
$$
D. $$
\operatorname{l e s s \, b r i g h t}
$$
E. $$
\operatorname{m o r e \, i l l u m i n a t e d}
$$
F. $$
\operatorname{l e s s \, r a d i a n t}
$$
G. $$
\operatorname{u n c h a n g e d}
$$
H. $$
\operatorname{i n f i n i t e l y \, f a i n t}
$$
I. $$
\operatorname{f a i n t e r}
$$
J. $$
\operatorname{b r i g h t e r}
$$
Answer: I
A dark interstellar globule is about the same size as().

Options:
A. $$
\mathrm{a \, p l a n e t s i m a l}
$$
B. $$
\mathrm{a \, n e b u l a \, c l u m p}
$$
C. $$
\mathrm{a \, g i a n t \, s t a r}
$$
D. $$
\mathrm{t h e \, O o r t \, c l o u d}
$$
E. $$
\mathrm{t h e \, K u i p e r \, b e l t}
$$
F. $$
\mathrm{t h e \, m i l k y \, w a y}
$$
G. $$
\mathrm{a \, w h i t e \, d w a r f}
$$
H. $$
\mathrm{t h e \, s o l a r \, c o r o n a}
$$
I. $$
\mathrm{a \, s u p e r n o v a \, r e m n a n t}
$$
J. $$
\mathrm{a \, t y p i c a l \, m o o n}
$$
Answer: D
Interstellar extinction is sometimes measured in mag- nitudes per kiloparsec (1 kpc = 1000 pc). Light from a star 1500 pc away is observed to be diminished in intensity by a factor of 20 over and above the effect of the inverse-square law. What is the average interstellar extinction, in mag/kpc, along the line of sight?

Options:
A. $2. 7$ magnitudes $f$ kiloparsec
B. $0. 8$ magnitudes $f$ kiloparsec
C. $4. 0$ magnitudes $f$ kiloparsec
D. $2. 9$ magnitudes $f$ kiloparsec
E. $1. 5$ magnitudes $f$ kiloparsec
F. $2. 2$ magnitudes $f$ kiloparsec
G. $3. 5$ magnitudes $f$ kiloparsec
H. $1. 2$ magnitudes $f$ kiloparsec
I. $3. 0$ magnitudes $f$ kiloparsec
J. $2. 0$ magnitudes $f$ kiloparsec
Answer: F
A protostar that will eventually turn into a star like the Sun is significantly().

Options:
A. $$
\mathrm{m o r e \ c o m p a c t}
$$
B. $$
\mathrm{m o r e \ l u m i n o u s}
$$
C. $$
\mathrm{m o r e \ t u r b u l e n t}
$$
D. $$
\mathrm{m o r e \ d i s f u s e}
$$
E. $$
\mathrm{l e s s \ l u m i n o u s}
$$
F. $$
\mathrm{c o o l e r \ a n d \ d i m m e r}
$$
G. $$
\mathrm{m o r e \ u n s t a b l e}
$$
H. $$
\mathrm{l e s s \ m a s s i v e}
$$
I. $$
\mathrm{m o r e \ m a g n e t i c}
$$
J. $$
\mathrm{l e s s \ d e n s e}
$$
Answer: B
A brown dwarf star with a radius of 0.1 solar radii and a surface temperature of 600K (0.1 times that of the Sun). What is the maximum distance at which the brown dwarf could be observed by a telescope of limiting apparent magnitude (a) 18, (b) 30?

Options:
A. $$
0. 8 \, \mathrm{k p c}
$$
B. $$
1. 2 0 \, \mathrm{k p c}
$$
C. $$
2. 5 0 \, \mathrm{k p c}
$$
D. $$
3. 0 0 \, \mathrm{k p c}
$$
E. $$
1. 8 \, \mathrm{k p c}
$$
F. $$
0. 5 \, \mathrm{k p c}
$$
G. $$
1. 1 5 \, \mathrm{k p c}
$$
H. $$
1. 2 5 \, \mathrm{k p c}
$$
I. $$
2. 0 \, \mathrm{k p c}
$$
J. $$
1. 1 0 \, \mathrm{k p c}
$$
Answer: G
What would be the luminosity of the Sun if its surface temperature were 3000 K and its were (a) 1 AU?

Options:
A. $$
6 7 0 0 L
$$
B. $$
8 8 0 0 L
$$
C. $$
3 5 0 5 L
$$
D. $$
2 3 0 0 L
$$
E. $$
4 1 5 0 L
$$
F. $$
5 0 0 0 L
$$
G. $$
1 9 0 0 L
$$
H. $$
7 4 0 0 L
$$
I. $$
3 3 7 0 L
$$
J. $$
4 0 0 0 L
$$
Answer: I
A white dwarf can dramatically increase in brightness only if().

Options:
A. $$
\mathrm{i t ~ h e a t s ~ w i t h o u t ~ o u t w a r d ~ r a d i a t i o n}
$$
B. $$
\mathrm{i t ~ h a s ~ b i n a r y ~ c o m p a n i o n}
$$
C. $$
\mathrm{i t ~ u n d e r g o e s ~ r a p i d ~ m a s s ~ l o s s}
$$
D. $$
\mathrm{i t ~ c o l l a p s e s ~ i n t o ~ a ~ n e u t r o n ~ s t a r}
$$
E. $$
\mathrm{i t ~ r e a c h e s ~ t h e ~ E d d i n g t o n ~ l i m i t}
$$
F. $$
\mathrm{i t ~ s t e a d i l y ~ g a i n s ~ h e l i u m ~ f r o m ~ u p l i f t}
$$
G. $$
\mathrm{i t ~ h a s ~ a ~ s t a b l e ~ n u c l e a r ~ r e a c t i o n}
$$
H. $$
\mathrm{i t ~ e x p e r i e n c e s ~ n o v a ~ b u r s t}
$$
I. $$
\mathrm{i t ~ a c c u m u l a t e s ~ f u e l ~ o v e r ~ t i m e}
$$
J. $$
\mathrm{i t ~ f o r m s ~ a ~ s p i r a l ~ n e b u l a ~ c l o u d}
$$
Answer: B
Calculate the orbital speed of matter in an accretion disk just above the surface of a 0.6-solar-mass, 15,000-km-diameter white dwarf.

Options:
A. $$
2 3 0 0 \, \mathrm{k m / s}
$$
B. $$
2 3 8 0 \, \mathrm{k m / s}
$$
C. $$
2 4 5 0 \, \mathrm{k m / s}
$$
D. $$
1 8 0 0 \, \mathrm{k m / s}
$$
E. $$
2 5 0 0 \, \mathrm{k m / s}
$$
F. $$
2 3 5 0 \, \mathrm{k m / s}
$$
G. $$
2 2 0 0 \, \mathrm{k m / s}
$$
H. $$
2 6 0 0 \, \mathrm{k m / s}
$$
I. $$
1 9 5 0 \, \mathrm{k m / s}
$$
J. $$
2 1 0 0 \, \mathrm{k m / s}
$$
Answer: A
Black holes result from stars having initial masses().

Options:
A. more than $5 0$ times the mass of the Sun
B. more than $2 5$ times the mass of the Sun
C. less than $2 5$ times the mass of the Sun
D. exactly $2 5$ times the mass of the Sun
E. within $1 5$ times the mass of the Sun
F. less than $5 0$ times the mass of the Sun
G. matching $2 0$ to $3 0$ times the mass of the Sun
H. ranging between $1 0$ and $2 5$ times the mass of the Sun
I. under $2 0$ times the mass of the Sun
J. surpassing $1 0 0$ times the mass of the Sun
Answer: B
Calculate the tidal acceleration on a 2-m-tall human falling feet-first into a 1-solar-mass black hole.

Options:
A. $$
\mathrm{t h r e e ~ b i l l i o n ~ g^{\prime} s}
$$
B. $$
\mathrm{f i v e ~ t h o u s a n d ~ g^{\prime} s}
$$
C. $$
\mathrm{t e n ~ b i l l i o n ~ g^{\prime} s}
$$
D. $$
\mathrm{s e v e n ~ h u n d r e d ~ g^{\prime} s}
$$
E. $$
\mathrm{f o u r ~ m i l l i o n ~ g^{\prime} s}
$$
F. $$
\mathrm{t w o ~ b i l l i o n ~ g^{\prime} s}
$$
G. $$
\mathrm{o n e ~ m i l l i o n ~ n / kg^{\prime} s}
$$
H. $$
\mathrm{o n e ~ t r i l l i o n ~ g^{\prime} s}
$$
I. $$
\mathrm{s i x ~ m i l l i o n ~ g^{\prime} s}
$$
J. $$
\mathrm{f i v e ~ b i l l i o n ~ m / s^{\prime} s}
$$
Answer: F
Let $U = \{1, 2, 3, 4, 5, 6\}$, and $H$ be a set-valued mapping such that
$$
H(\lambda) =
\begin{cases}
\{1, 2, 3, 4, 5, 6\} & 0 \leq \lambda < 0.2, \\
\{1, 2, 4, 5, 6\} & 0.2 \leq \lambda < 0.5, \\
\{2, 4, 5, 6\} & 0.5 \leq \lambda < 0.6, \\
\{2, 5, 6\} & 0.6 \leq \lambda < 0.8, \\
\{5, 6\} & 0.8 \leq \lambda \leq 1.
\end{cases}.
$$
Find the corresponding set $\mathbf{F}$ from $H$-what is set $A$?

Options:
A. $$A=(0.3,0.9,0.2,0.6,1.1)$$
B. $$A=(0.2,0.9,0.4,0.7,1.0)$$
C. $$A=(0.5,0.7,0.3,0.4,1.0)$$
D. $$A=(0.5,0.8,0.2,0.6,1.1)$$
E. $$A=(0.4,0.8,0.2,0.7,1.1)$$
F. $$A=(0.2,0.6,0.4,0.7,1.1)$$
G. $$A=(0.0,0.7,0.3,0.6,1.0)$$
H. $$A=(0.6,0.8,0.1,0.7,1.0)$$
I. $$A=(0.1,0.8,0.3,0.5,1.2)$$
J. $$A=(0.3,0.7,0.2,0.5,1.0)$$
Answer: D
Let
$\xi$: the boundary point between light rain and moderate rain,
$\eta$: the boundary point between moderate rain and heavy rain, with the respective distributions:
$$
\xi \! : \! N(1, 4), \qquad \eta \! : \! N(2, 4).
$$
Try using the trisection method to determine the membership functions of the three fuzzy concepts: light rain, moderate rain, and heavy rain.

Options:
A. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-2}{2}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-1}{2}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-2x}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
B. $$
A_{1}\left(x\right)=\phi\left(\frac{x-1}{2}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-2}{3}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\mathrm{e}^{-t^{2}}\mathrm{d}t
$$
C. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-1}{2}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-2}{2}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
D. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-1}{3}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-2}{4}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
E. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-2}{4}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-3}{4}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{2\pi}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
F. $$
A_{1}\left(x\right)=\phi\left(\frac{x-1}{2}\right), \\A_{2}\left(x\right)=1-\phi\left(\frac{x-3}{2}\right)
\\A_{3}\left(x\right)=A_{1}\left(x\right)A_{2}\left(x\right)
\\\phi(x)=\frac{1}{2\pi}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t}{2}}\mathrm{d}t
$$
G. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-1}{4}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-2}{4}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
H. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-1}{2}\right), \\A_{2}\left(x\right)=1-\phi\left(\frac{x-2}{2}\right)
\\A_{3}\left(x\right)=A_{1}\left(x\right)A_{2}\left(x\right)
\\\phi(x)=\frac{1}{2\pi}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t
$$
I. $$
A_{1}\left(x\right)=1-\phi\left(\frac{x-2}{2}\right), \\A_{2}\left(x\right)=\phi\left(\frac{x-3}{2}\right)
\\A_{3}\left(x\right)=1-A_{1}\left(x\right)-A_{2}\left(x\right)
\\\phi(x)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t}{2}}\mathrm{d}t
$$
J. $$
A_{1}\left(x\right)=\phi\left(\frac{x-1}{2}\right), \\A_{2}\left(x\right)=1-\phi\left(\frac{x-2}{2}\right)
\\A_{3}\left(x\right)=A_{1}\left(x\right)A_{2}\left(x\right)
\\\phi(x)=\frac{1}{2\pi}\int_{-\infty}^{x}\mathrm{e}^{-\frac{t^{2}}{4}}\mathrm{d}t
$$
Answer: C
Let $U=\left\{ \, u_{1} \,, u_{2} \,, u_{3} \, \right\}$, $V=\left\{ \, v_{1} \,, v_{2} \,, v_{3} \,, v_{4} \, \right\}$, $R \in\mathcal{F} \left( \, U \times V \right)$ and
$$
R=\left\{ \begin{matrix} {{{0. 1}}} & {{{0. 3}}} & {{{0. 5}}} & {{{0. 9}}} \\ {{{1}}} & {{{0. 7}}} & {{{0. 2}}} & {{{0. 3}}} \\ {{{0}}} & {{{0. 2}}} & {{{0. 1}}} & {{{1}}} \\ \end{matrix} \right\}
$$
What is $f ( \, u_{i} \, )$ for $i = 1 \,, 2 \,, 3?$

Options:
A. $$f(u_1)=(1,0.3,0.5,0.9),f(u_2)=(0.7,1,0.2,0.3),f(u_3)=(1,0.3,0.1,1)$$
B. $$f(u_1)=(0.1,0.3,0.5,0.9),f(u_2)=(1,0.7,0.2,0.3),f(u_3)=(1,0.2,0.1,1)$$
C. $$f(u_1)=(0.1,0.2,0.5,0.9),f(u_2)=(1,0.9,0.2,0.3),f(u_3)=(1,0.2,0.1,0.3)$$
D. $$f(u_1)=(0.1,0.3,0.5,0.9),f(u_2)=(1,0.7,0.2,1),f(u_3)=(0,0.2,0.1,1)$$
E. $$f(u_1)=(0.1,0.3,0.2,0.9),f(u_2)=(1,0.7,0.1,0.3),f(u_3)=(0,0.2,0.1,1)$$
F. $$f(u_1)=(0.1,0.3,0.5,0.2),f(u_2)=(1,0.7,0.9,0.3),f(u_3)=(1,0.2,0.1,1)$$
G. $$f(u_1)=(0.1,0.3,0.5,1),f(u_2)=(0.7,1,0.2,0.3),f(u_3)=(0,0.2,0.1,1)$$
H. $$f(u_1)=(0.1,0.5,0.3,0.9),f(u_2)=(0.7,1,0.2,0.3),f(u_3)=(0,0.2,0.1,1)$$
I. $$f(u_1)=(0.3,0.1,0.5,0.9),f(u_2)=(1,0.7,0.3,0.2),f(u_3)=(1,0.2,0.1,0.9)$$
J. $$f(u_1)=(0.1,0.3,0.9,0.5),f(u_2)=(1,0.3,0.7,0.2),f(u_3)=(1,0.2,1,0.1)$$
Answer: B
Let $\\U=\{ \, u_{1} \,, u_{2} \,, u_{3} \, \} \,, \\V=\{ \, v_{1} \,, v_{2} \,, v_{3} \,, v_{4} \, \}$
$\\f \colon U \longrightarrow{\cal F} ( V )$
$f ( u_{1} ) \,=\, ( 0.5, 0.2, 0, 1 ) \,, \quad f ( u_{2} ) \,=\, ( 1, 0.3, 0.5, 0 ) \,, \quad f ( u_{3} ) \,=\, ( 0.7, 0.1, 0.2, 0.9 )$
$\\A \;=\; \{\, u_{1} \,, u_{2} \, \} \;, \quad B \,=\, \frac{0.9} {u_{1}}+\frac{0.3} {u_{2}}+\frac{1} {u_{3}}$
What are $T_{f} ( A ), T_{f} ( B )$?

Options:
A. $T_{f}(A)=(\begin{array}{c}{1,0.3,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.6,0.3,0.3,0.5}\end{array})$
B. $T_{f}(A)=(\begin{array}{c}{1,0.3,0.2,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.5,0.3,0.3,0.9}\end{array})$
C. $T_{f}(A)=(\begin{array}{c}{1,0.2,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.8,0.3,0.3,0.9}\end{array})$
D. $T_{f}(A)=(\begin{array}{c}{0.7,0.3,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.3,0.1,0.7}\end{array})$
E. $T_{f}(A)=(\begin{array}{c}{1,0.3,0.5,0}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.2,0.3,0.9}\end{array})$
F. $T_{f}(A)=(\begin{array}{c}{1,1.0,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.1,0.6,0.9}\end{array})$
G. $T_{f}(A)=(\begin{array}{c}{1,0.1,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.2,0.3,0.9}\end{array})$
H. $T_{f}(A)=(\begin{array}{c}{1,0.3,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.3,0.3,0.9}\end{array})$
I. $T_{f}(A)=(\begin{array}{c}{0.5,0.2,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.9,0.3,0.1,0.7}\end{array})$
J. $T_{f}(A)=(\begin{array}{c}{0.5,0.3,0.5,1}\end{array}),T_{f}(B)=(\begin{array}{c}{0.7,0.3,0.2,1}\end{array})$
Answer: H
Let the variables $x, y, z$ satisfy the following conditions:
$$
\left\{\begin{matrix} x \geqslant a_{1} \\ \text{and } y \leqslant 1-a_{1} \\ \end{matrix} \right. \ \ \mathrm{if} \ \ \left\{\begin{matrix} x \geqslant a_{1} \\ \text{and } y \geqslant a_{1} \\ \text{and } z \leqslant 1-a_{1} \\ \end{matrix} \right. \ . \ \ \mathrm{if} \ \ \left\{\begin{matrix} x \leqslant 1-a_{1} \\ \text{and } y \leqslant 1-a_{1} \\ \text{and } z \geqslant a_{1} \\ \end{matrix} \right.
$$
What is the expression for the function $f(x, y, z)$ that satisfies $f(x, y, z) \geqslant a_{1}$?

Options:
A. $$f(\:x\:,y\:,z\:)=\bar{x}\bar{y}+\bar{x}y\bar{z}+x\:\bar{y}z$$
B. $$f(\:x\:,y\:,z\:)=x\bar{y}+\bar{x}y\bar{z}+\bar{x}\:\bar{y}z$$
C. $$f(\:x\:,y\:,z\:)=xy+xy\bar{z}+\bar{x}y\:\bar{z}$$
D. $$f(\:x\:,y\:,z\:)=x\bar{y}+xy\bar{z}+\bar{x}\:yz$$
E. $$f(\:x\:,y\:,z\:)=x\bar{z}+xy\bar{z}+\bar{x}\:y\bar{z}$$
F. $$f(\:x\:,y\:,z\:)=x\bar{y}+\bar{x}\bar{y}+x\:\bar{y}z$$
G. $$f(\:x\:,y\:,z\:)=x\bar{y}+xy\bar{z}+x\:\bar{y}z$$
H. $$f(\:x\:,y\:,z\:)=x\bar{y}+xy\bar{z}+\bar{x}\:\bar{y}z$$
I. $$f(\:x\:,y\:,z\:)=x\bar{y}+xy\bar{z}+\bar{x}\:y\bar{z}$$
Answer: H
For a certain product, let $u_{1}$ represent durability, $u_{2}$ represent color and style, $u_{3}$ represent price, and $u_{4}$ represent the impression of the product. The satisfaction level of the evaluator for the product is expressed as:
$$
h = \frac{0.6}{u_{1}} + \frac{0.7}{u_{2}} + \frac{0.9}{u_{3}} + \frac{1}{u_{4}}
$$
The weight of importance for the product is given as:
$$
g(u_{1}) = 0.2, \quad g(u_{2}) = 0.1, \quad g(u_{3}) = 0.3, \quad g(u_{4}) = 0.4
$$
What is the evaluation of this product using the $\mathrm{F}$ integration method?

Options:
A. 0.8
B. 0.5
C. 0.65
D. 0.7
E. 0.4
F. 1.0
G. 0.3
H. 0.6
I. 0.2
J. 0.9
Answer: H
The fluorocarbon compound C$_{2}$Cl$_{3}$F$_{3}$ has a normal boiling point of 47.6°C. The specific heats of C2Cl$_3F_3(l)$ and $\mathrm{C}_{2}\mathrm{Cl}_{3}\mathrm{F}_{3}(g)$ are 0.91 J/g-K and 0.67 J/g-K, respectively. The heat of vaporization for the compound is 27.49 kJ/mol.What is the amount of heat required to convert 50.0 g of C₂Cl₃F₃ from a liquid at 10.00 °C to a gas at 85.00 °C?

Options:
A. $$
0. 9 6 \ \mathrm{k J}
$$
B. $$
0. 8 4 \ \mathrm{k J}
$$
C. $$
1 2. 7 \ \mathrm{k J}
$$
D. $$
9. 7 2 \ \mathrm{k J}
$$
E. $$
1 3. 2 \ \mathrm{k J}
$$
F. $$
1 0. 9 \ \mathrm{k J}
$$
G. $$
1 1. 8 \ \mathrm{k J}
$$
H. $$
1 0. 3 \ \mathrm{k J}
$$
I. $$
1 4. 5 \ \mathrm{k J}
$$
J. $$
1 5. 0 \ \mathrm{k J}
$$
Answer: H
Spinel is a mineral that contains 37.9% Al, 17.1 % Mg, and 45.0% O, by mass, and has a density of 3.57 g/cm³. The unit cell is cubic, withan edge length of 809 pm. How many atoms of each type are in the unit cell?

Options:
A. 12 Al atoms, 14 Mg atoms, and 34 O atoms.
B. 19 Al atoms, 5 Mg atoms, and 27 O atoms.
C. 16 Al atoms, 8 Mg atoms, and 32 O atoms.
D. 14 Al atoms, 9 Mg atoms, and 35 O atoms.
E. 18 Al atoms, 7 Mg atoms, and 30 O atoms.
F. 17 Al atoms, 11 Mg atoms, and 29 O atoms.
G. 13 Al atoms, 12 Mg atoms, and 33 O atoms.
H. 11 Al atoms, 15 Mg atoms, and 36 O atoms.
I. 20 Al atoms, 6 Mg atoms, and 28 O atoms.
J. 15 Al atoms, 10 Mg atoms, and 31 O atoms.
Answer: C
A 1.800-g sample of phenol ($C_6H_5OH$) was burned in a bomb calorimeter whose total heat capacity is 11.66 $\mathrm{k J} / ^\circ C$ The temperature of the calorimeter plus contents in- creased from 21.36 °C to 26.37°C. What is the heat of combustion per gram of phenol?

Options:
A. $$
- 3. 0 5 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
B. $$
- 3. 2 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
C. $$
- 3. 4 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
D. $$
- 2. 8 5 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
E. $$
- 3. 5 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
F. $$
- 2. 7 5 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
G. $$
- 2. 5 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
H. $$
- 2. 9 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
I. $$
- 3. 6 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
J. $$
- 3. 1 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_{6} H_{5} O H}
$$
Answer: A
A house is designed to have passive solar energy features. Brickwork incorporated into the interior of the house acts as a heat absorber. Each brick weighs approximately 18 kg. The specific heat of the brick is 0.85 J/g-K. How many bricks must be incorporated into the interior of the house to provide the same total heat capacity as $1. 7 \times1 0^{3}$ gal of water?

Options:
A. 8.3×10 $^{4}$or 83,000 bricks
B. 4.7×10 $^{4}$or 47,000 bricks
C. 1.1×10 $^{6}$or 1,100,000 bricks
D. 9.2×10 $^{3}$or 9,200 bricks
E. 6.0×10 $^{3}$or 6,000 bricks
F. 5.0×10 $^{5}$or 500,000 bricks
G. 2.5×10 $^{5}$or 250,000 bricks
H. 7.5×10 $^{2}$or 750 bricks
I. 1.8×10 $^{4}$or 18,000 bricks
J. 3.6×10 $^{3}$or 3,600 bricks
Answer: I
It is estimated that the net arnount of carbon dioxide fixed by photosynthesis on the landmass of Earth is 5.5 $\times$ $1 0^{1 6} \, \mathrm{g / yr}$ of $\mathrm{C O_{2}}$. Assume that all this carbon is converted into glucose.Calculate the energy stored by photosynthesis on land per year in kJ.

Options:
A. $$
4. 5 \times1 0^{1 6} \, \mathrm{k J}
$$
B. $$
5. 2 \times1 0^{1 8} \, \mathrm{k J}
$$
C. $$
7. 1 \times1 0^{1 8} \, \mathrm{k J}
$$
D. $$
3. 9 \times1 0^{1 6} \, \mathrm{k J}
$$
E. $$
6. 4 \times1 0^{1 9} \, \mathrm{k J}
$$
F. $$
5. 7 \times1 0^{1 9} \, \mathrm{k J}
$$
G. $$
1. 9 \times1 0^{1 6} \, \mathrm{k J}
$$
H. $$
2. 5 \times1 0^{1 7} \, \mathrm{k J}
$$
I. $$
4. 2 \times1 0^{1 5} \, \mathrm{k J}
$$
J. $$
5. 8 \times1 0^{1 7} \, \mathrm{k J}
$$
Answer: J
Let $C \subset\mathbb{P}^{3}$ be a smooth, irreducible, nondegenerate curve of degree $4$\
What is the maximum value of the genus of $C $?

Options:
A. 3
B. 4
C. 7
D. 8
E. 2
F. 9
G. 0
H. 5
I. 1
J. 6
Answer: I
Let H denote the upper half-plane; that is, $\mathbb{H}=\{z \in\mathbb{C} \ : \ \mathrm{I m} \ z > 0 \}$ , with the metric $\frac{1} {y^{2}} d x d y$ for $z=x+i y$ .Suppose $\Gamma$ is a group of isometries acting on H such that $\mathbb{H} / \Gamma$ is a smooth surface $S$ , and you are given that a fundamental domain $D$ for the action of $\Gamma$ on H is given as follows:
$$
D=\{x+i y \in\mathbb{H} \mid-\frac{3} {2} \leq x \leq\frac{3} {2}, ( x-c )^{2}+y^{2} \geq\frac{1} {9} \mathrm{~ f o r ~} c \in\{\pm\frac{1} {3}, \pm\frac{2} {3}, \pm\frac{4} {3} \} \}.
$$
What is the value of $\chi(S)$calculated using Gauss-Bonnet? You may use that the (Gaussian) curva- ture of H is identically equal to $- 1$ .

Options:
A. -5
B. -2
C. 2
D. -3
E. 0
F. 1
G. 4
H. -4
I. -1
J. 3
Answer: B
Given a smooth function $f : \mathbb{R}^{n-1} \to\mathbb{R}$ , define $F : \mathbb{R}^{n} \to\mathbb{R}$ by
$$
F ( x_{1}, \ldots, x_{n} ) :=f ( x_{1}, \ldots, x_{n-1} )-x_{n}
$$
and consider the preimage $X_{f}=F^{-1} ( 0 ) \subset\mathbb{R}^{n}$ .\
Consider the two examples $X_{f}$ and $X_{g} \subset\mathbb{R}^{3}$ with $f ( x_{1}, x_{2} )=x_{1}^{2}+x_{2}^{2}$ and $g ( x_{1}, x_{2} )=x_{1}^{2}-x_{2}^{2}$ . What is the normal vectors at every point
$( x_{1}, x_{2}, x_{3} ) \in X_{f}$ and $( x_{1}, x_{2}, x_{3} ) \in X_{g}$ ?

Options:
A. $$
{\frac{(-x, y, -1 )} {\sqrt{1+x^{2}+y^{2}}}}
$$
B. $$
{\frac{(2 x, 2 y, -1 )} {\sqrt{1+4 x^{2}+4 y^{2}}}}
$$
C. $$
{\frac{(-x,-y, 1 )} {\sqrt{1+x^{2}+y^{2}}}}
$$
D. $$
{\frac{(x, 2 y, -1 )} {\sqrt{1+x^{2}+4 y^{2}}}}
$$
E. $$
{\frac{(2x, 2y, 1)}{\sqrt{1 + 4x^2 + 4y^2}}}
$$
F. $$
{\frac{(-x, -2y, -1)}{\sqrt{1 + x^2 + 4y^2}}}
$$
G. $$
{\frac{(-2 x, 0, 1 )} {\sqrt{1+4 x^{2}}}}
$$
H. $$
{\frac{(-2 x,-2 y, 1 )} {\sqrt{1+4 x^{2}+4 y^{2}}}}
$$
I. $$
{\frac{(2 x, -2 y, 1 )} {\sqrt{1+4 x^{2}+y^{2}}}}
$$
J. $$
{\frac{(2 x,-2 y, 1 )} {\sqrt{1+4 x^{2}+4 y^{2}}}}
$$
Answer: H
There exists a unique (isomorphic) non Abelian group $G$with an order of 21.
(a) Let $G$ be this group. How many conjugacy classes does $G$ have?
(b) What are the dimensions of the irreducible representations of $G?$

Options:
A. (a)three conjugacy sets.\
(b)dimensions 2,2,3,3, and 3.
B. (a)five conjugacy sets.\
(b)dimensions 2,2,2,3, and 3.
C. (a)four conjugacy sets.\
(b)dimensions 1,1,1,1, and 3.
D. (a)three conjugacy sets.\
(b)dimensions 1,2,3,4, and 5.
E. (a)four conjugacy sets.\
(b)dimensions 1,1,2,3, and 4.
F. (a)four conjugacy sets.\
(b)dimensions 1,1,2,2, and 4.
G. (a)six conjugacy sets.\
(b)dimensions 1,1,1,2, and 2.
H. (a)six conjugacy sets.\
(b)dimensions 1,1,2,3, and 3.
I. (a)six conjugacy sets.\
(b)dimensions 1,1,1,1, and 2.
J. (a)five conjugacy sets.\
(b)dimensions 1,1,1,3, and 3.
Answer: J
Fix two positive real numbers $a, b > 0$ . What is the value of the
integral?
$$
\int_{-\infty}^{\infty} \frac{\operatorname{c o s} ( a x )-\operatorname{c o s} ( b x )} {x^{2}} d x
$$

Options:
A. $$\frac{\pi}{2}(a-b)$$
B. $$-\pi(a-b)$$
C. $$\pi(ab)$$
D. $$2\pi(a-b)^2$$
E. $$-\pi a b$$
F. $$\pi(b-a)$$
G. $$\pi(a+b)$$
H. $$0$$
I. $$\sqrt{\pi}(a-bo)$$
J. $$\pi a^2 - \pi b^2$$
Answer: F
Let $\omega$ and $\eta$ be two complex numbers such that $\mathrm{I m} \left( \frac{\omega} {\eta} \right) > 0$ . Let $G$ be the closed parallelogram consisting of all $z\in\mathbb{C}$ such that $z=\lambda\omega+\rho\eta$ for some $0\le \lambda ,\rho \le1$ . Let $\partial G$ be the boundary of $G$ and Let $G^{0}=G-\partial G$ be the interior of $G$ . Let $P_{1}, \cdots, P_{k}, Q_{1}, \cdots, Q_{\ell}$ be points in $G^{0}$ and let $m_{1}, \cdots, m_{k}, n_{1}, \cdots, n_{\ell}$ be positive integers. Let $f$ be a function on $G$ such that
$$
\frac{f ( z ) \prod_{j=1}^{\ell} ( z-Q_{j} )^{n_{j}}} {\prod_{p=1}^{k} ( z-P_{p} )^{m_{p}}}
$$
is continuous and nowhere zero on $G$ and is holomorphic on $G^{0}$ . Let $\varphi( z )$ and $\psi( z )$ be two polynomials on $\mathbb{C}$ . Assume that $f ( z+\omega)=e^{\varphi( z )} f ( z )$ if both $z$ and $z+\omega$ are in $G$ . Assume also that $f ( z+\eta)=e^{\psi( z )} f ( z )$ if both $z$ and $z+\eta$ are in $G$. The form of $\sum_{p=1}^{k} m_{p}-\sum_{j=1}^{\ell} n_{j}$ in terms of $\omega$ and $\eta$ and the coefficients of $\varphi( z )$ and $\psi( z )$ is ().

Options:
A. $$\frac{1}{2\pi\sqrt{-1}}\left(\psi(0)+\varphi(\eta)+\psi(\omega)-\varphi(0)\right)$$
B. $$\frac{1}{2\pi}\left(\varphi(\eta)-\psi(\omega)+\varphi(0)+\psi(0)\right)$$
C. $$\frac{1}{2\pi\sqrt{-1}}\left(\psi(\eta)-\varphi(0)+\psi(\omega)-\varphi(\omega)\right)$$
D. $$\frac{1}{2\pi\sqrt{-1}}\left(\varphi(\omega)+\varphi(\eta)-\psi(\eta)-\psi(0)\right)$$
E. $$\frac{1}{2\pi\sqrt{-1}}\left(\psi(\eta)+\varphi(0)-\varphi(\omega)-\psi(\omega)\right)$$
F. $$\frac{1}{2\pi}\left(\varphi(\omega)+\psi(\eta)-\psi(\omega)+\varphi(0)\right)$$
G. $$\frac{1}{2\pi\sqrt{-1}}\left(\psi(\omega)-\varphi(\eta)-\psi(0)+\varphi(\omega)\right)$$
H. $$\frac{1}{2\pi\sqrt{-1}}\left(\varphi(0)+\psi(0)-\varphi(\eta)+\psi(\eta)\right)$$
I. $$\frac{1}{2\pi\sqrt{-1}}\left(-\varphi(\eta)+\varphi(0)+\psi(\omega)-\psi(0)\right)$$
J. $$\frac{1}{2\pi}\left(\varphi(\omega)-\varphi(\eta)+\psi(0)-\psi(\omega)\right)$$
Answer: I
Let $p$ be a prime number, $n > 1$ an integer, and $K=\mathbb{F}_{p^{n}}$ the field with $p^{n}$ elements. Considering the field automorphism of $K$ defined by
$$
F ( x )=x^{p}
$$
as an $\mathbb{F}_{p}$ -linear endomorphism of $K$ . What is its characteristic polynomial?

Options:
A. $$
T^{n}+p
$$
B. $$
T^{p}-1
$$
C. $$
T^{n}-T
$$
D. $$
T^{n}+1
$$
E. $$
T^{n}-p
$$
F. $$
T^{p^{n}}-1
$$
G. $$
T^{p}+T
$$
H. $$
T^{n}+T
$$
I. $$
T^{p}-T
$$
J. $$
T^{n}-1
$$
Answer: J
Let $b$ be any integer with $( 7, b )=1$ and consider the polynomial
$$
f_{b} ( x )=x^{3}-2 1 x+3 5 b.
$$
Let $P$ denote the set of $b \in\mathbb{Z}$ such that $( 7, b )=1$ and the Galois group of $f_{b}$ is the alternating group $A_{3}$ . Find $P$ is ().

Options:
A. $$
P=\{-1 7 n^{3}+9 n-4 \, | \, n \in\mathbb{Z} \}. $$
B. $$
P=\{4 5 n^{2}-3 n+6 \, | \, n \in\mathbb{Z} \}. $$
C. $$
P=\{n^{2}-7 5 n-1 \, | \, n \in\mathbb{Z} \}. $$
D. $$
P=\{2 2 n^{2}+1 n+7 \, | \, n \in\mathbb{Z} \}. $$
E. $$
P=\{-7 5 n^{2}+6 n+1 \, | \, n \in\mathbb{Z} \}.
$$
F. $$
P=\{-9 8 n^{2}-5 n+8 \, | \, n \in\mathbb{Z} \}. $$
G. $$
P=\{5 7 n^{3}+4 n-2 \, | \, n \in\mathbb{Z} \}. $$
H. $$
P=\{-3 n^{2}+1 2 n+5 \, | \, n \in\mathbb{Z} \}. $$
I. $$
P=\{-6 9 n^{2}-8 n+3 \, | \, n \in\mathbb{Z} \}. $$
J. $$
P=\{1 3 n^{2}-1 5 n+4 \, | \, n \in\mathbb{Z} \}. $$
Answer: E
What is the order of Galois group $G$ of the polynomial $X^{6}+3$ over the rational number field $\mathbb{Q}$?

Options:
A. 9
B. 4
C. 8
D. 7
E. 12
F. 11
G. 5
H. 14
I. 6
J. 10
Answer: I
Let $\mathbb{P}^{5}$ be the projective space of homogeneous quadratic polynomials $F ( X, Y, Z )$ over $\mathbb{C}$ , and let $\Phi\subset\mathbb{P}^{5}$ be the set of those polynomials that are products of linear factors. Similarly, let $\mathbb{P}^{9}$ be the projective space of homogeneous cubic polynomials $F ( X, Y, Z )$ , and let $\Psi\subset\mathbb{P}^{9}$ be the set of those polynomials that are products of linear factors.\
(a) What is the dimensions of $\Phi$ and $\Psi$ ?\
(b) What is the degrees of $\Phi$ and $\Psi$ ?

Options:
A. (a)$\dim\Phi=4,\dim\Psi=6$.\
(b)$\Phi=3,\Psi=15$.
B. (a)$\dim\Phi=4,\dim\Psi=9$.\
(b)$\Phi=2,\Psi=11$.
C. (a)$\dim\Phi=4,\dim\Psi=8$.\
(b)$\Phi=5,\Psi=13$.
D. (a)$\dim\Phi=6,\dim\Psi=4$.\
(b)$\Phi=7,\Psi=18$.
E. (a)$\dim\Phi=5,\dim\Psi=6$.\
(b)$\Phi=6,\Psi=13$.
F. (a)$\dim\Phi=3,\dim\Psi=5$.\
(b)$\Phi=4,\Psi=16$.
G. (a)$\dim\Phi=2,\dim\Psi=7$.\
(b)$\Phi=4,\Psi=12$.
H. (a)$\dim\Phi=5,\dim\Psi=5$.\
(b)$\Phi=6,\Psi=17$.
I. (a)$\dim\Phi=5,\dim\Psi=7$.\
(b)$\Phi=2,\Psi=14$.
J. (a)$\dim\Phi=3,\dim\Psi=8$.\
(b)$\Phi=5,\Psi=14$.
Answer: A
Establish an upper limit to the surface temperature $T_{s}$ of the Crab pulsar by assuming that all of the rotational energy is dissipated as thermal blackbody radiation from the surface, what is the upper limit?

Options:
A. $$
T_{s} \leq1 \times1 0^{8} \mathrm{~ K}
$$
B. $$
T_{s} \leq3 \times1 0^{6} \mathrm{~ K}
$$
C. $$
T_{s} \leq9 \times1 0^{7} \mathrm{~ K}
$$
D. $$
T_{s} \leq2 \times1 0^{7} \mathrm{~ K}
$$
E. $$
T_{s} \leq5 \times1 0^{7} \mathrm{~ K}
$$
F. $$
T_{s} \leq7 \times1 0^{6} \mathrm{~ K}
$$
G. $$
T_{s} \leq8 \times1 0^{7} \mathrm{~ K}
$$
H. $$
T_{s} \leq4 \times1 0^{7} \mathrm{~ K}
$$
I. $$
T_{s} \leq6 \times1 0^{6} \mathrm{~ K}
$$
J. $$
T_{s} \leq3 \times1 0^{7} \mathrm{~ K}
$$
Answer: J
The imaging proportional counter on the Einstein Observatory has a sensitivity of 2.5 X $1 0^{-5}$ photons $\mathrm{c m^{-2} ~ s^{-1}}$ between 0.2-3.0 keV. Assuming a neutron star of radius 10 km at a distance of 100 pc, what upper limit can the satellite put on $T_{e}?$

Options:
A. $$
T_{e} \leq1. 0 \times1 0^{6} \mathrm{~ K}
$$
B. $$
T_{e} \leq2. 0 \times1 0^{5} \mathrm{~ K}
$$
C. $$
T_{e} \leq0. 2 \times1 0^{6} \mathrm{~ K}
$$
D. $$
T_{e} \leq2. 5 \times1 0^{6} \mathrm{~ K}
$$
E. $$
T_{e} \leq1. 5 \times1 0^{7} \mathrm{~ K}
$$
F. $$
T_{e} \leq0. 5 \times1 0^{5} \mathrm{~ K}
$$
G. $$
T_{e} \leq5. 0 \times1 0^{4} \mathrm{~ K}
$$
H. $$
T_{e} \leq7. 0 \times1 0^{5} \mathrm{~ K}
$$
I. $$
T_{e} \leq8. 0 \times1 0^{4} \mathrm{~ K}
$$
J. $$
T_{e} \leq9. 0 \times1 0^{5} \mathrm{~ K}
$$
Answer: C
Evaluate $\rho( r ) / \rho_{\infty}=n ( r ) / n_{\infty}$ at the event horizon is().

Options:
A. $$
5. 1 \times1 0^{1 0} \big( T_{\infty} / 1 0^{4} \big)^{-1 / 2}
$$
B. $$
1. 8 \times1 0^{1 1} \big( T_{\infty} / 1 0^{4} \big)^{-3 / 4}
$$
C. $$
4. 3 \times1 0^{1 1} \big( T_{\infty} / 1 0^{4} \big)^{-3 / 1}
$$
D. $$
7. 4 \times1 0^{1 0} \big( T_{\infty} / 1 0^{4} \big)^{-4 / 2}
$$
E. $$
6. 0 \times1 0^{1 2} \big( T_{\infty} / 1 0^{4} \big)^{-2 / 3}
$$
F. $$
3. 7 \times1 0^{1 1} \big( T_{\infty} / 1 0^{4} \big)^{-3 / 2}
$$
G. $$
2. 5 \times1 0^{1 2} \big( T_{\infty} / 1 0^{4} \big)^{-2 / 2}
$$
H. $$
8. 6 \times1 0^{1 3} \big( T_{\infty} / 1 0^{4} \big)^{-5 / 2}
$$
I. $$
3. 9 \times1 0^{1 3} \big( T_{\infty} / 1 0^{4} \big)^{-4 / 3}
$$
J. $$
9. 2 \times1 0^{1 2} \big( T_{\infty} / 1 0^{4} \big)^{-6 / 3}
$$
Answer: F
Estimate the binding energy of ${\bf a} \, \sim\, 1 0 \, M_{\odot}$ star is(), withradus $R \sim5 R_{\odot}$ . surrounding a collapsing degenerate core, and com ne initial shock energy.

Options:
A. $E_{B} \sim G M^{2} / R \sim1 0^{4 9}$ erg prior to core bounce.
B. $E_{B} \sim G M^{2} / R \sim1 0^{4 8}$ erg for an accretion disk.
C. $E_{B} \sim G M^{2} / R \sim1 0^{4 7}$ erg from a binary merger.
D. $E_{B} \sim G M^{3} / R \sim1 0^{5 1}$ erg during core inflation.
E. $E_{B} \sim G M / R^{2} \sim1 0^{5 0}$ erg in post-bounce scenario.
F. $E_{B} \sim G M^{2} R \sim1 0^{5 0}$ erg beyond critical density.
G. $E_{B} \sim G M^{2} / R \sim1 0^{5 0}$ erg following core bounce.
H. $E_{B} \sim M^{2} / R \sim1 0^{5 0}$ erg after neutron capture.
I. $E_{B} \sim G R^{2} / M \sim1 0^{5 0}$ erg with a static shell.
J. $E_{B} \sim G M^{2} / R \sim1 0^{5 2}$ erg at maximum spin rate.
Answer: G
Problem 2.52 We know that the charge on a conductor moves to the surface, but determining how it is distributed there is not straightforward. A famous example where the surface charge density can be explicitly calculated is the ellipsoid:
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.
$$
where $Q$ is the total charge. By choosing appropriate values for $a$, $b$, and $c$, obtain (from Eq. 2.57): (a) the net (both sides) surface charge density $\sigma(r)$ on a circular disk of radius $R$ is ().

Options:
A. $$
\frac{Q} {3 \pi R} \frac{1} {\sqrt{R^{2}+r^{2}}}.
$$
B. $$
\frac{Q} {2 \pi R} \frac{1} {\sqrt{R+r^{2}}}.
$$
C. $$
\frac{Q} {4 \pi R} \frac{1} {\sqrt{R-r^{2}}}.
$$
D. $$
\frac{Q} {2 \pi R} \frac{1} {\sqrt{R^{2}-r^{2}}}.
$$
E. $$
\frac{Q} {2 \pi} \frac{1} {\sqrt{R^{2}-r^{2}}}.
$$
F. $$
\frac{Q} {2 \pi R^2} \frac{1} {\sqrt{R^{2}+r^{2}}}.
$$
G. $$
\frac{Q} {\pi R} \frac{1} {\sqrt{R^{2}-r}}.
$$
H. $$
\frac{Q} {2 \pi R^2} \frac{1} {\sqrt{R^{2}-r}}.
$$
I. $$
\frac{Q} {5 \pi R} \frac{1} {\sqrt{R-r^{2}}}.
$$
J. $$
\frac{Q} {6 \pi R^2} \frac{1} {\sqrt{R^{2}+r}}.
$$
Answer: D
Evaluate the following integrals:$\int _{\mathrm{all~space}}( r^{2}+ \mathbf{r} \cdot \mathbf{a} + a^{2}) \delta ^{3}( \mathbf{r} - \mathbf{a} ) d\tau$,where a is a fixed vector and $a$ is its magnitude.

Options:
A. $$
{3 a^{2}}.
$$
B. $$
{5 a^{2}}.
$$
C. $$
{2 a^{2}}.
$$
D. $$
{a^{3}}.
$$
E. $$
{4 a^{2}}.
$$
F. $$
{6 a^{2}}.
$$
G. $$
{a^{2} \times 3}.
$$
H. $$
{a^{2} - a}.
$$
I. $$
{2 a^{3}}.
$$
J. $$
{a^{2} + a}.
$$
Answer: A
Find the potential outside a charged metal sphere (charge $Q$ ,radius $R$ ) placed PaNadesnineiesisto $\mathbf{E}_{0}$ . Explain clearly where you are setting the zero of potential.

Options:
A. $$
E_{0} \left( r+\frac{R^{3}} {r^{3}} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
B. $$
E_{0} \left( r+\frac{R^{3}} {r^{2}} \right) \operatorname{c o s} \theta-\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
C. $$
-E_{0} \left( \frac{r}{2}-\frac{R^{3}} {r} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r^2}.
$$
D. $$
-E_{0} \left( r^3-\frac{R^{3}} {r^{5}} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
E. $$
-E_{0} \left( r^2-\frac{R^{3}} {r} \right) \operatorname{c o s} \theta-\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
F. $$
E_{0} \left( r-\frac{R^{3}} {r^{2}} \right) \operatorname{s i n} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
G. $$
-E_{0} \left( r-\frac{R^{3}} {r^{2}} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r}.
$$
H. $$
E_{0} \left( r-\frac{R} {r} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q} {r^2}.
$$
I. $$
-E_{0} \left( r^2-\frac{R^{3}} {r^{2}} \right) \operatorname{c o s} \theta-\frac{1} {4 \pi\epsilon_{0}} \frac{Q^2} {r}.
$$
J. $$
-E_{0} \left( r-\frac{R^{2}} {r} \right) \operatorname{c o s} \theta+\frac{1} {4 \pi\epsilon_{0}} \frac{Q^2} {r}.
$$
Answer: G
Hydrogen sulfide ($H_{2}S$) is a common and troublesome pollutant in industrial wastewaters. One way to remove $H_{2}S$ is to treat the water with chlorine, in which case the following reaction occurs:
$$
\mathrm{H_{2} S ( a q )} \,+\, \mathrm{C l_{2} ( a q )} \, \longrightarrow\, \mathrm{S ( s )} \,+\, 2 \, \mathrm{H}^{+} ( a q ) \,+\, 2 \, \mathrm{C l}^{-} ( a q )
$$
The rate of this reaction is first order in each reactant. The rate constant for the disappearance of ${H_{2} S}$ at 28°C is
$3. 5 \times1 0^{-2} \, M^{-1} \, \mathrm{s}^{-1}.$ If at a given time the concentra-tion of ${H_{2} S}$ is 2.0 $\times1 0^{-4} \, M$ and that of $Cl_{2}$ is 0.025M,what is the reat of formation of $CI^{-}$?

Options:
A. $$
3. 5 \times1 0^{-7} \ M / {\mathrm{s}}
$$
B. $$
9. 0 \times1 0^{-6} \ M / {\mathrm{s}}
$$
C. $$
2. 1 \times1 0^{-8} \ M / {\mathrm{s}}
$$
D. $$
2. 9 \times1 0^{-7} \ M / {\mathrm{s}}
$$
E. $$
4. 1 \times1 0^{-7} \ M / {\mathrm{s}}
$$
F. $$
5. 7 \times1 0^{-7} \ M / {\mathrm{s}}
$$
G. $$
1. 8 \times1 0^{-7} \ M / {\mathrm{s}}
$$
H. $$
3. 5 \times1 0^{-5} \ M / {\mathrm{s}}
$$
I. $$
4. 2 \times1 0^{-6} \ M / {\mathrm{s}}
$$
J. $$
6. 3 \times1 0^{-9} \ M / {\mathrm{s}}
$$
Answer: A
Gold is found in seawater at very low levels, about 0.05 ppb by mass. Assuming that gold is worth about $800 per troy ounce, how many liters of seawater would you have to process to obtain $1,000,000 worth of gold?
Assume the density of seawater is 1.03 g/mL and that your gold recovery process is 50% efficient.

Options:
A. $2 \times1 0^{1 1} \, \mathrm{L}$ seawater
B. $8 \times1 0^{1 2} \, \mathrm{L}$ seawater
C. $1 \times1 0^{1 2} \, \mathrm{L}$ seawater
D. $6 \times1 0^{1 0} \, \mathrm{L}$ seawater
E. $5 \times1 0^{1 3} \, \mathrm{L}$ seawater
F. $3 \times1 0^{1 4} \, \mathrm{L}$ seawater
G. $7 \times1 0^{1 3} \, \mathrm{L}$ seawater
H. $4 \times1 0^{1 0} \, \mathrm{L}$ seawater
I. $9 \times1 0^{1 1} \, \mathrm{L}$ seawater
J. $1 \times1 0^{1 4} \, \mathrm{L}$ seawater
Answer: C
$\mathsf{A}$ water-carbonating plant is available for use in the home and operates by providing carbon dioxide at 1.0 atm. The molar concentration of the $\mathrm{C O_{2}}$ in the soda water it produces is ().

Options:
A. $$
28.6mmol\ dm^{-1}
$$
B. $$
36.5mmol\ dm^{-1}
$$
C. $$
29.7mmol\ dm^{-1}
$$
D. $$
39.9mmol\ dm^{-1}
$$
E. $$
25.8mmol\ dm^{-1}
$$
F. $$
31.2mmol\ dm^{-1}
$$
G. $$
45.0mmol\ dm^{-1}
$$
H. $$
34.5mmol\ dm^{-1}
$$
I. $$
30.4mmol\ dm^{-1}
$$
J. $$
42.1mmol\ dm^{-1}
$$
Answer: H
The standard reaction enthalpy of Zn(s) + $H_{\mathrm{2}}$O(g) → ZnO(s) $+H_2(g)$ is approximately constant at +224 kJ mol$^{-1}$ from 920 K up to 1280 K. The standard reaction Gibbs energy is +33 kJ mol$^{-1}$ at 1280 K. Assuming that both quantities remain constant, the temperature is () at which the equilibrium constant becomes greater than 1.

Options:
A. $$
1. 4 0 \times1 0^{3} K
$$
B. $$
1. 8 0 \times1 0^{3} K
$$
C. $$
1. 7 0 \times1 0^{3} K
$$
D. $$
2. 0 0 \times1 0^{3} K
$$
E. $$
1. 2 5 \times1 0^{3} K
$$
F. $$
1. 3 5 \times1 0^{3} K
$$
G. $$
1. 5 0 \times1 0^{3} K
$$
H. $$
2. 5 0 \times1 0^{3} K
$$
I. $$
1. 9 0 \times1 0^{3} K
$$
J. $$
1. 6 0 \times1 0^{3} K
$$
Answer: G
Show that if the logarithm of an equilibrium constant is plotted against the reciprocal of the temperature, then the standard reaction enthalpy may be determined.

Options:
A. $$
-R \times\mathrm{s l o p e} \ -\frac{R}{2} \times\mathrm{i n t e r c e p t}
$$
B. $$
-R \div\mathrm{s l o p e} \ {a n d} \frac{R}{2} \times\mathrm{i n t e r c e p t}
$$
C. $$
R \times\mathrm{s l o p e} \ {a n d} -R \times\mathrm{i n t e r c e p t}
$$
D. $$
R \times\mathrm{s l o p e} \ {a n d} \mathrm{i n t e r c e p t} \times -R
$$
E. $$
R +\mathrm{s l o p e} \ {a n d} R \ -\mathrm{i n t e r c e p t}
$$
F. $$
2 R \times\mathrm{s l o p e} \ {a n d} -R \times\mathrm{i n t e r c e p t}
$$
G. $$
-R +\mathrm{s l o p e} \ {a n d} \frac{R}{2} \times\mathrm{i n t e r c e p t}
$$
H. $$
-R \times\mathrm{s l o p e} \ {a n d} R \times\mathrm{i n t e r c e p t}
$$
I. $$
-R \div\mathrm{s l o p e} \ {a n d} R +\mathrm{i n t e r c e p t}
$$
J. $$
2 R \times\mathrm{s l o p e} \ {a n d} \frac{R}{2} \times\mathrm{i n t e r c e p t}
$$
Answer: H