MikaStars39/nano-eval huggingface dataset:
A U-tube has two limbs of internal diameter $6$ mm and 16 mm respectively and contains some water. Calculate the difference in water levels in the two limbs. Surface tension of water = 0.073 N/m.

Options:
A. $$
1.6 \ \mathrm{mm}
$$
B. $$
4.0 \ \mathrm{mm}
$$
C. $$
3.5 \ \mathrm{mm}
$$
D. $$
2.4 \ \mathrm{mm}
$$
E. $$
4.2 \ \mathrm{mm}
$$
F. $$
3.8 \ \mathrm{mm}
$$
G. $$
3. 1 \ \mathrm{m m}
$$
H. $$
5.0 \ \mathrm{mm}
$$
I. $$
2.6 \ \mathrm{mm}
$$
J. $$
2.8 \ \mathrm{mm}
$$
Answer: G
Find the pressure represented by a column of$$
1 0 \mathrm{~ c m ~ o f ~ w a t e r.}
$$

Options:
A. $$
9 2 0 \mathrm{N / m}^{2}
$$
B. $$
9 7 9 \mathrm{N / m}^{2}
$$
C. $$
1 0 0 1 \mathrm{N / m}^{2}
$$
D. $$
1 0 5 0 \mathrm{N / m}^{2}
$$
E. $$
9 5 5 \mathrm{N / m}^{2}
$$
F. $$
9 6 8 \mathrm{N / m}^{2}
$$
G. $$
8 7 9 \mathrm{N / m}^{2}
$$
H. $$
1 0 2 0 \mathrm{N / m}^{2}
$$
I. $$
9 8 5 \mathrm{N / m}^{2}
$$
J. $$
1 1 0 0 \mathrm{N / m}^{2}
$$
Answer: B
Assuming an isothermal process, find the pressure at an elevation of 2500 m above sea level. The mass density and atmospheric pressure at an elevation of 500 m above sea level are known to be 1.1677 $kg/m^3$ and 95480 Pa. What is the density of air at that level?

Options:
A. $$
0. 8 2 7 4 \ \mathrm{k g} / \mathrm{m}^{3}
$$
B. $$
1. 0 5 9 8 \ \mathrm{k g} / \mathrm{m}^{3}
$$
C. $$
0. 7 5 6 4 \ \mathrm{k g} / \mathrm{m}^{3}
$$
D. $$
1. 0 3 5 2 \ \mathrm{k g} / \mathrm{m}^{3}
$$
E. $$
1. 1 2 4 3 \ \mathrm{k g} / \mathrm{m}^{3}
$$
F. $$
0. 9 3 4 2 \ \mathrm{k g} / \mathrm{m}^{3}
$$
G. $$
0. 9 1 8 6 \ \mathrm{k g} / \mathrm{m}^{3}
$$
H. $$
0. 8 8 3 1 \ \mathrm{k g} / \mathrm{m}^{3}
$$
I. $$
0. 7 9 6 5 \ \mathrm{k g} / \mathrm{m}^{3}
$$
J. $$
0. 9 5 0 7 \ \mathrm{k g} / \mathrm{m}^{3}
$$
Answer: G
For the following velocity vectors determine the magnitude of velocity at A $( x=2, \; y=-\; 3, \; z=1,$ $t=2 )$ .
$$
\mathbf{v}=( 1 0 t+x y ) \mathbf{i}+(-y z-1 0 t ) \mathbf{j}+(-y z+z^{2} / 2 ) \mathbf{k}
$$

Options:
A. $$
2 1. 4 \mathrm{\ u n i t s}
$$
B. $$
2 0. 0 \mathrm{\ u n i t s}
$$
C. $$
2 2. 3 \mathrm{\ u n i t s}
$$
D. $$
1 7. 9 \mathrm{\ u n i t s}
$$
E. $$
2 5. 1 \mathrm{\ u n i t s}
$$
F. $$
1 9. 2 \mathrm{\ u n i t s}
$$
G. $$
1 8. 7 \mathrm{\ u n i t s}
$$
H. $$
2 3. 0 \mathrm{\ u n i t s}
$$
I. $$
2 4. 8 \mathrm{\ u n i t s}
$$
J. $$
1 2. 5 \mathrm{\ u n i t s}
$$
Answer: C
The velocity $u$ in a laminar boundary layer over a aupacisobw ${\frac{u} {U_{0}}}=\left[ 2 \, {\frac{y} {\delta}}-\left( {\frac{y} {\delta}} \right)^{2} \right] {\mathrm{w h e r e ~}} U_{0}$ = constant and $\delta$ = boundary layer thickness and is given by $\delta=k \sqrt{x}$ . Obtain an expression for the velocity component $V$ across the boundary layer is___.

Options:
A. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{3}} {4 \delta^{3}}-\frac{y^{2}} {2 \delta^{4}} \Biggr]
$$
B. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{3}} {3 \delta^{3}}-\frac{y} {2 \delta} \Biggr]
$$
C. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y} {\delta^{2}}-\frac{y^{2}} {2 \delta^{3}} \Biggr]
$$
D. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{2}} {3 \delta^{2}}-\frac{y^{4}} {4 \delta^{4}} \Biggr]
$$
E. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{3}} {2 \delta^{2}}-\frac{y^{2}} {3 \delta^{3}} \Biggr]
$$
F. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{2}} {4 \delta^{2}}-\frac{y^{3}} {2 \delta^{3}} \Biggr]
$$
G. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{2}} {2 \delta}-\frac{y^{3}} {4 \delta^{4}} \Biggr]
$$
H. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y^{2}} {2 \delta^{2}}-\frac{y^{3}} {3 \delta^{3}} \Biggr]
$$
I. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y} {\delta}-\frac{y^{4}} {4 \delta^{4}} \Biggr]
$$
J. $$
\frac{k U_{0}} {\sqrt{x}} \Biggl[ \frac{y} {\delta^{3}}-\frac{y^{4}} {3 \delta^{4}} \Biggr]
$$
Answer: H
Values of $\psi$ for two flows are given. Determine the corresponding values of the potential function $\varphi$.
$$
(\mathrm{i}) \ \ \psi = \mathrm{A} \ (x^2 - y^2)
$$
$$
(\mathrm{ii}) \ \ \psi = y^3 - 3x^2y
$$

Options:
A. $$
2 \mathbf{A} y^2+\mathbf{c o n s t a n t}
$$
$$
3 x^2-y^3+\mathrm{c o n s t a n t}
$$
B. $$
- \mathbf{A} (x^2 - 3y^2)+\mathbf{c o n s t a n t}
$$
$$
y^{3}-2 x^{3}+\mathrm{c o n s t a n t}
$$
C. $$
- \mathbf{A} (2xy)+\mathbf{c o n s t a n t}
$$
$$
x^3-yx^{2}+\mathrm{c o n s t a n t}
$$
D. $$
4 \mathbf{A} xy+\mathbf{c o n s t a n t}
$$
$$
x^3-y^{2} x+\mathrm{c o n s t a n t}
$$
E. $$
2 \mathbf{A} (x+y)+\mathbf{c o n s t a n t}
$$
$$
y^2 x-x^{2} y+\mathrm{c o n s t a n t}
$$
F. $$
\mathbf{A} (x^2 - 2y^2)+\mathbf{c o n s t a n t}
$$
$$
2 y^3-3 x+\mathrm{c o n s t a n t}
$$
G. $$
- \mathbf{A} (2x^2-y^2)+\mathbf{c o n s t a n t}
$$
$$
x^2y - y^3+\mathrm{c o n s t a n t}
$$
H. $$
2 \mathbf{A} x^2+\mathbf{c o n s t a n t}
$$
$$
x^3-y^3+\mathrm{c o n s t a n t}
$$
I. $$
- 2 \mathbf{A} x y+\mathbf{c o n s t a n t}
$$
$$
3 y^{2} x-x^{3}+\mathrm{c o n s t a n t}
$$
J. $$
- \mathbf{A} (x^2 + y^2)+\mathbf{c o n s t a n t}
$$
$$
y^3-x^3+\mathrm{c o n s t a n t}
$$
Answer: I
The velocity distribution in a pipe of radius $r_{0}$ is given by
$$
{\frac{u} {u_{m}}}=\left( 1-{\frac{r} {r_{0}}} \right)^{m}
$$
with $m < 1,$ Determine the momentum correction factor $\beta$ and ealculate the value of $\beta$ for $m=1 / 7$

Options:
A. $$
\frac{1} {5} \frac{( m+2 )^{3} \ ( m+1 )} {( 2 m+2 )}
$$
$$
1. 0 7
$$
B. $$
\frac{1} {4} \frac{( m+2 )^{2} \ ( m+1 )} {( 4 m+1 )}
$$
$$
1. 0 1
$$
C. $$
\frac{1} {5} \frac{( m+3 )^{2} \ ( m+2 )} {( 2 m+1 )}
$$
$$
1. 0 8
$$
D. $$
\frac{1} {6} \frac{( m+2 )^{2} \ ( m+3 )} {( 2 m+1 )}
$$
$$
1. 0 9
$$
E. $$
\frac{1} {4} \frac{( m+2 )^{2} \ ( m+1 )} {( 2 m+1 )}
$$
$$
1. 0 2
$$
F. $$
\frac{1} {3} \frac{( m+2 )^{2} \ ( m+2 )} {( 3 m+1 )}
$$
$$
1. 0 6
$$
G. $$
\frac{1} {4} \frac{( m+4 )^{2} \ ( m+1 )} {( 2 m+2 )}
$$
$$
1. 0 4
$$
H. $$
\frac{1} {3} \frac{( m+3 )^{2} \ ( m+1 )} {( 2 m+1 )}
$$
$$
1. 0 3
$$
I. $$
\frac{1} {4} \frac{( m+3 )^{2} \ ( m+1 )} {( 3 m+1 )}
$$
$$
1. 1 2
$$
J. $$
\frac{1} {2} \frac{( m+1 )^{2} \ ( m+2 )} {( 2 m+1 )}
$$
$$
1. 0 5
$$
Answer: E
For laminar flow in a pipe, the drop in pressure $\Delta p$ is a function of the pipe length $L,$ its diameter $D$ , mean velocity of flow $V$ and the dynamic viscosity $\mu$ Using Raleigh's method, develop an expression for $\Delta p$ .

Options:
A. $$
\frac{\mu D} {L} \cdot\mathrm{f n} \left( {\frac{L^2} {D}} \right)
$$
B. $$
\frac{\mu V} {D^2} \cdot\mathrm{f n} \left( {\frac{D L} {V}} \right)
$$
C. $$
\frac{\mu V} {L} \cdot\mathrm{f n} \left( {\frac{D} {L}} \right)
$$
D. $$
\frac{V^2} {\mu L} \cdot\mathrm{f n} \left( {\frac{D} {L}} \right)
$$
E. $$
\frac{\mu V D} {L} \cdot\mathrm{f n} \left( {\frac{L} {D}} \right)
$$
F. $$
\frac{\mu} {V L} \cdot\mathrm{f n} \left( {\frac{D^2} {L}} \right)
$$
G. $$
\frac{\mu V^2} {D} \cdot\mathrm{f n} \left( {\frac{L} {D}} \right)
$$
H. $$
\frac{\mu} {D} \cdot\mathrm{f n} \left( {\frac{L} {D^2}} \right)
$$
I. $$
\frac{\mu^2 V} {L} \cdot\mathrm{f n} \left( {\frac{L} {D}} \right)
$$
J. $$
\frac{\mu L} {V} \cdot\mathrm{f n} \left( {\frac{L} {D}} \right)
$$
Answer: C
Young's modulus for a spider's thread is about $0. 2 \times1 0^{1 0}$ Pa, and the thread breaks when it undergoes a strain of about 50% [Kohler and Vollrath (1995)]. Calculate the strain that steel undergoes when it breaks is (). (Assume that a linear relationship between stress and strain holds until it breaks.) Compare the breaking strain to the spider's thread.

Options:
A. 0.36%
B. 0.05%
C. 0.38%
D. 0.29%
E. 0.08%
F. 0.75%
G. 0.35%
H. 5.00%
I. 0.15%
J. 0.25%
Answer: J
Consider laminar viscous flow in the following situation, which models flow in the bronchi or a network of branching blood vessels. A vessel of radius $R$ connects to $N$ smaller vessels, each of radius $xR$.
What is the relationship between total cross-sectional area of the smaller vessels and that of the larger vessel if the pressure gradient is the same in both sets of vessels ?

Options:
A. $$
R \cdot N
$$
B. $$
N^2
$$
C. $$
\frac{1}{\sqrt{N}}
$$
D. $$
N^{\frac{1}{3}}
$$
E. $$
\frac{1}{N}
$$
F. $$
R \times c^N
$$
G. $$
c^2 N
$$
H. $$
N^c
$$
I. $$
\sqrt{N}
$$
J. $$
N \times c
$$
Answer: I
The radioactive nucleus 64 $C u$ decays independently by three different paths. The relative decay rates of these three modes are in the ratio 2:2:1. The half-life is 12.8 $h$ . Calculate the total decay rate $b$ , and the three partial decay rates $b_{1}, b_{2}$ , and $b_{3}$ is _______.

Options:
A. $$
0.0 5 4 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=6. 0 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=3. 0 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
B. $$
0.0 5 6 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=6. 3 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=3. 2 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
C. $$
0.0 5 7 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=5. 8 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=2. 9 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
D. $$
0.0 5 8 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=6. 2 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=3. 1 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
E. $$
0.0 4 9 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=5. 5 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=2. 7 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
F. $$
0.0 5 3 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=5. 9 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=2. 9 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
G. $$
0.0 4 4 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=5. 6 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=2. 8 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
H. $$
0.0 5 1 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=6. 1 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=3. 1 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
I. $$
0.0 4 5 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=5. 0 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=2. 5 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
J. $$
0.0 6 0 \, \mathrm{h r}^{-1}
$$
$$
b_{1}=b_{2}=7. 0 \times1 0^{-6} \; \mathrm{s}^{-1}, b_{3}=3. 5 \times1 0^{-6} \; \mathrm{s}^{-1}.
$$
Answer: A
Ice and water coerist at 273K. To melt 1 mol of ice at this temperature, 6000 J are needed. Calculate the entropy difference and the ratio of the number of microstates for 1 mol of ice and 1 mol of water at this temperature is(). (Do not worry about any volume changes of the ice and water.)

Options:
A. $$
1 0^{6. 9 \times1 0^{2 3}}
$$
B. $$
1 0^{7. 1 \times1 0^{2 3}}
$$
C. $$
1 0^{6. 8 \times1 0^{2 3}}
$$
D. $$
1 0^{6. 2 \times1 0^{2 3}}
$$
E. $$
1 0^{5. 9 \times1 0^{2 3}}
$$
F. $$
1 0^{8. 3 \times1 0^{2 3}}
$$
G. $$
1 0^{7. 4 \times1 0^{2 3}}
$$
H. $$
1 0^{8. 1 \times1 0^{2 3}}
$$
I. $$
1 0^{6. 5 \times1 0^{2 3}}
$$
J. $$
1 0^{5. 8 \times1 0^{2 3}}
$$
Answer: A
If the osmotic pressure in human blood is 7.7 atm at $3 7^{\circ} \mathrm{C},$ what is the solute concentration assuming that $\sigma=1?$ What would be the osmotic pressure at $4 \, {}^{\circ} \mathrm{C} \,?$

Options:
A. $$
0. 3 0 2 \mathrm{~ m o l ~} 1^{-1}
$$
$$
6. 8 8 \mathrm{\ at m}
$$
B. $$
0. 3 5 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 1 0 \mathrm{\ at m}
$$
C. $$
0. 3 1 5 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 2 2 \mathrm{\ at m}
$$
D. $$
0. 2 5 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 9 2 \mathrm{\ at m}
$$
E. $$
0. 3 7 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
8. 1 5 \mathrm{\ at m}
$$
F. $$
0. 2 8 5 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 4 5 \mathrm{\ at m}
$$
G. $$
0. 2 7 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 6 8 \mathrm{\ at m}
$$
H. $$
0. 2 9 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
6. 7 5 \mathrm{\ at m}
$$
I. $$
0. 4 0 2 \mathrm{~ m o l ~} 1^{-1}
$$
$$
8. 0 0 \mathrm{\ at m}
$$
J. $$
0. 3 3 0 \mathrm{~ m o l ~} 1^{-1}
$$
$$
7. 0 5 \mathrm{\ at m}
$$
Answer: A
Radiation from the nearby star Alpha Centauri is observed to be reduced in wavelength (after correction for Earth's orbital motion) by a factor of 0.999933. What is the recession velocity of Alpha Centauri relative to the Sun?

Options:
A. $$
- 1 8. 2 ~ \mathrm{k m / s e c}
$$
B. $$
+ 0 4. 2 ~ \mathrm{k m / s e c}
$$
C. $$
- 2 5. 4 ~ \mathrm{k m / s e c}
$$
D. $$
- 0 8. 7 ~ \mathrm{k m / s e c}
$$
E. $$
+ 1 5. 7 ~ \mathrm{k m / s e c}
$$
F. $$
- 1 0. 6 ~ \mathrm{k m / s e c}
$$
G. $$
+ 2 0. 9 ~ \mathrm{k m / s e c}
$$
H. $$
- 0 5. 3 ~ \mathrm{k m / s e c}
$$
I. $$
- 2 0. 1 ~ \mathrm{k m / s e c}
$$
J. $$
+ 1 9. 8 ~ \mathrm{k m / s e c}
$$
Answer: I
A space-based telescope can achieve a diffraction-limited angular resolution of 0.05" for red light (wavelength 700 nm). What would the resolution of the instrument be in the infrared, at $3. 5 \, \mu\mathrm{m}$ , and in the ultraviolet, at 140 nm?

Options:
A. $$
0. 4 0 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 1 5 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
B. $$
0. 3 0 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 3 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
C. $$
0. 3 5 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 0 7 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
D. $$
0. 6 0 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 0 1 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
E. $$
0. 2 8 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 2 5 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
F. $$
0. 2 9 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 3 5 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
G. $$
0. 1 2 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 0 9 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
H. $$
0. 1 5 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 0 5 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
I. $$
0. 2 5 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 01 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
J. $$
0. 5 0 \mathrm{~ s e c o n d s ~ o f ~ a r c},
$$
$$
0. 0 2 \mathrm{~ s e c o n d s ~ o f ~ a r c}
$$
Answer: I
$\mathrm{A}$ star lying 20 pc from the Sun has proper motion of 0.50/yr. What is the star's transverse velocity?

Options:
A. $$
4 4. 5 \ \mathrm{k m / s e c}
$$
B. $$
5 2. 4 \ \mathrm{k m / s e c}
$$
C. $$
4 7. 5 \ \mathrm{k m / s e c}
$$
D. $$
5 0. 1 \ \mathrm{k m / s e c}
$$
E. $$
5 1. 3 \ \mathrm{k m / s e c}
$$
F. $$
4 3. 9 \ \mathrm{k m / s e c}
$$
G. $$
4 6. 8 \ \mathrm{k m / s e c}
$$
H. $$
3 7. 6 \ \mathrm{k m / s e c}
$$
I. $$
4 9. 2 \ \mathrm{k m / s e c}
$$
J. $$
3 8. 2 \ \mathrm{k m / s e c}
$$
Answer: C
The telescope best suited to observing dark dust clouds is().

Options:
A. $$
\mathrm{a\ m i c r o w a v e ~ t e l e s c o p e}
$$
B. $$
\mathrm{a\ u l t r a v i o l e t ~ t e l e s c o p e}
$$
C. $$
\mathrm{a\ r a d i o ~ t e l e s c o p e}
$$
D. $$
\mathrm{a\ r e f l e c t i n g ~ t e l e s c o p e}
$$
E. $$
\mathrm{a\ p h o t o n ~ s e n s o r ~ t e l e s c o p e}
$$
F. $$
\mathrm{a\ t h e r m a l ~ i n f r a r e d ~ t e l e s c o p e}
$$
G. $$
\mathrm{a\ g a m m a ~ r a y ~ t e l e s c o p e}
$$
H. $$
\mathrm{a\ r e f r a c t i n g ~ t e l e s c o p e}
$$
I. $$
\mathrm{a\ n e a r - i n f r a r e d ~ t e l e s c o p e}
$$
J. $$
\mathrm{a\ f i b e r - o p t i c ~ t e l e s c o p e}
$$
Answer: C
A beam of light shining through a dense molecular cloud is diminished in intensity by a factor of 2 for every $5$ pc it travels. How many magnitudes is the light from a background star dimmed if the total thickness of the cloud is 60 pc?

Options:
A. $$
7. 9 8
$$
B. $$
6. 2 8
$$
C. $$
9. 0 3
$$
D. $$
5. 7 2
$$
E. $$
7. 4 5
$$
F. $$
6. 9 3
$$
G. $$
10. 0 4
$$
H. $$
8. 1 9
$$
I. $$
10. 3 7
$$
J. $$
8. 5 6
$$
Answer: C
A typical open cluster will dissolve in about the same amount of time as the time since ().

Options:
A. flowering plants became dominant
B. tectonic plates started slow drift
C. major ice ages repeatedly ended
D. humans domesticated wild sheep
E. oceans formed from extensive rainfall
F. hominids began using basic tools
G. mammals first evolved on Earth
H. dinosaurs walked on Earth
I. continents formed distinct landmasses
J. forests first appeared worldwide
Answer: H
After the core of a Sun-like star starts to fuse helium on the horizontal branch, the core becomes().

Options:
A. $$
\operatorname{b r i g h t e r}
$$
B. $$
\operatorname{l e s s \, d e n s e}
$$
C. $$
\operatorname{c o o l e r}
$$
D. $$
\operatorname{s m a l l e r}
$$
E. $$
\operatorname{l a r g e r}
$$
F. $$
\operatorname{dimmer with time}
$$
G. $$
\operatorname{m o r e \, s t a b l e}
$$
H. $$
\operatorname{l e s s \, s t a b l e}
$$
I. $$
\operatorname{m o r e \, d e n s e}
$$
J. $$
\operatorname{h o t t e r}
$$
Answer: J
Use the radius-luminosity-temperature relation to calculate the radius of a 12,000-K (twice the temperature of the Sun) 0.0004-solar-luminosity white dwarf.

Options:
A. $$
3 0 0 0 \, \mathrm{k m}
$$
B. $$
4 0 0 0 \, \mathrm{k m}
$$
C. $$
1 5 0 0 \, \mathrm{k m}
$$
D. $$
4 5 0 0 \, \mathrm{k m}
$$
E. $$
2 4 0 0 \, \mathrm{k m}
$$
F. $$
3 6 0 0 \, \mathrm{k m}
$$
G. $$
5 0 0 0 \, \mathrm{k m}
$$
H. $$
3 2 3 0 \, \mathrm{k m}
$$
I. $$
2 8 0 0 \, \mathrm{k m}
$$
J. $$
3 8 5 0 \, \mathrm{k m}
$$
Answer: H
An observable supernova should occur in our Galaxy about once every().

Options:
A. $$
\mathrm{q u a r t e r}
$$
B. $$
\mathrm{w e e k l y}
$$
C. $$
\mathrm{d a i l y}
$$
D. $$
\mathrm{m o n t h l y}
$$
E. $$
\mathrm{d e c a d e s }
$$
F. $$
\mathrm{a n n u a l l y}
$$
G. $$
\mathrm{c e n t u r y}
$$
H. $$
\mathrm{m i l l e n n i u m}
$$
I. $$
\mathrm{d e c a d e}
$$
J. $$
\mathrm{b i e n n i u m}
$$
Answer: G
Using the Sun's current luminosity, estimate its total energy output, assuming a $1 0^{1 0}$ year main-sequence lifetime. How does this compare with the energy released by a typical supernova?

Options:
A. about twenty times less than the energy from fusion reactions
B. approximately half of the energy produced by a galaxy each year
C. about the same as the energy emitted by a black hole merge
D. similar to the energy emitted by gamma-ray bursts in the early universe
E. equal to the energy absorbed by the Sun during its formation phase
F. slightly greater than the total energy output of all visible stars combined
G. about one-tenth of the energy it gives off in the form of neutrinos
H. comparable to the energy released during planetary formation processes
I. roughly double the energy stored in the Earth's magnetic field
J. marginally less than the energy generated by cosmic ray collisions
Answer: G
Calculate the surface gravitational acceleration and escape speed of a 1.4-solar-mass neutron star with a radius of 10 km.

Options:
A. $$
8. 5 \times1 0^{1 0} \, \mathrm{m} / \mathrm{s}^{2}
$$
B. $$
2. 8 \times1 0^{1 0} \, \mathrm{m} / \mathrm{s}^{2}
$$
C. $$
3. 5 \times1 0^{1 1} \, \mathrm{m} / \mathrm{s}^{2}
$$
D. $$
6. 1 \times1 0^{1 2} \, \mathrm{m} / \mathrm{s}^{2}
$$
E. $$
2. 3 \times1 0^{1 1} \, \mathrm{m} / \mathrm{s}^{2}
$$
F. $$
5. 8 \times1 0^{1 2} \, \mathrm{m} / \mathrm{s}^{2}
$$
G. $$
4. 2 \times1 0^{1 3} \, \mathrm{m} / \mathrm{s}^{2}
$$
H. $$
1. 0 \times1 0^{1 3} \, \mathrm{m} / \mathrm{s}^{2}
$$
I. $$
1. 9 \times1 0^{1 2} \, \mathrm{m} / \mathrm{s}^{2}
$$
J. $$
7. 9 \times1 0^{9} \, \mathrm{m} / \mathrm{s}^{2}
$$
Answer: I
Set $F$: $A = \int_{x} \mathrm{e}^{-x^{2}} / x$, what are the cut sets $A_{\frac{1}{e}}$ and $A_{1}$ respectively?

Options:
A. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{1,2}\\\end{array}\right];A_{1}=\mid2\}
$$
B. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{1,-1}\\\end{array}\right];A_{1}=\mid1\}
$$
C. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-1,1}\\\end{array}\right];A_{1}=\mid0\}
$$
D. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-4,4}\\\end{array}\right];A_{1}=\mid3\}
$$
E. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{2,-2}\\\end{array}\right];A_{1}=\mid-1\}
$$
F. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-1,0}\\\end{array}\right];A_{1}=\mid-2\}
$$
G. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-0.5,0.5}\\\end{array}\right];A_{1}=\mid0\}
$$
H. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-3,3}\\\end{array}\right];A_{1}=\mid0\}
$$
I. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{-2,2}\\\end{array}\right];A_{1}=\mid1\}
$$
J. $$
A_{\frac{1}{e}}=\left[\begin{array}{c}{0,1}\\\end{array}\right];A_{1}=\mid-1\}
$$
Answer: C
Let $\mathrm{F}$ be the set
$$
A = {\frac{0.6}{u_{1}}} + {\frac{0.8}{u_{2}}} + {\frac{1}{u_{3}}} + {\frac{0.8}{u_{4}}} + {\frac{0.6}{u_{5}}} + {\frac{0.2}{u_{6}}}, \qquad B = {\frac{0.4}{u_{1}}} + {\frac{0.6}{u_{2}}} + {\frac{0.5}{u_{3}}} + {\frac{1}{u_{4}}} + {\frac{0.8}{u_{5}}} + {\frac{0.3}{u_{6}}}
$$
Calculate $N(A, B)$ using Hamming nearness degree.

Options:
A. 0.72
B. 0.78
C. 0.74
D. 0.88
E. 0.68
F. 0.82
G. 0.85
H. 0.70
I. 0.76
J. 0.80
Answer: I
Let
$$
R_{1}=\left\{ \begin{matrix} {{{0.1}}} & {{{0}}} & {{{0.8}}} \\ {{{0.9}}} & {{{0.5}}} & {{{0}}} \\ {{{0}}} & {{{0.4}}} & {{{0.3}}} \\ \end{matrix} \right\}, \quad R_{2}=\left\{ \begin{matrix} {{{0.7}}} & {{{0.2}}} & {{{0.4}}} \\ {{{0.3}}} & {{{0.1}}} & {{{0.6}}} \\ {{{1}}} & {{{0.5}}} & {{{0.2}}} \\ \end{matrix} \right\}
$$
What is $R_{1} \cup R_{2}$?

Options:
A. $$
R_1\cup R_2=\left\{\begin{matrix}0.1&0.2&0.8\\0.3&0.5&0.6\\1&0.4&0.3\end{matrix}\right\}
$$
B. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0&0.8\\0.3&0.5&0\\1&0.4&0.3\end{matrix}\right\}
$$
C. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0.2&0.8\\0.3&0.1&0.6\\0&0.5&0.2\end{matrix}\right\}
$$
D. $$
R_1\cup R_2=\left\{\begin{matrix}0.1&0&0.8\\0.9&0.1&0.6\\1&0.5&0.2\end{matrix}\right\}
$$
E. $$
R_1\cup R_2=\left\{\begin{matrix}0.1&0.2&0.4\\0.9&0.5&0.6\\1&0.4&0.2\end{matrix}\right\}
$$
F. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0.2&0.4\\0.3&0.5&0.6\\0&0.4&0.3\end{matrix}\right\}
$$
G. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0.2&0.8\\0.9&0.5&0.6\\1&0.5&0.3\end{matrix}\right\}
$$
H. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0.2&0.4\\0.9&0.5&0.6\\1&0.4&0.2\end{matrix}\right\}
$$
I. $$
R_1\cup R_2=\left\{\begin{matrix}0.1&0.2&0.4\\0.9&0.1&0.6\\1&0.5&0.2\end{matrix}\right\}
$$
J. $$
R_1\cup R_2=\left\{\begin{matrix}0.7&0&0.4\\0.3&0.5&0.6\\1&0.4&0.2\end{matrix}\right\}
$$
Answer: G
$U=V$ represents the real number field, $R \! \in\! {\mathcal{F}} ( U \! \times\! V )$ ,
$$
R \left( u \,, v \right)=\frac{1} {1+k \left( u-v \right)^{2}} \quad\left( k > 1 \right)
$$
What is $f ( u ) ( v )$?

Options:
A. $$
\frac{k}{1+\left(u-v\right)^{2}}
$$
B. $$
\frac{1}{1+k\left(v-u\right)^{3}}
$$
C. $$
\frac{1+k}{1+\left(u-v\right)^{2}}
$$
D. $$
k \cdot \frac{1}{1+\left(u-v\right)}
$$
E. $$
\frac{1}{1+k\left(u+v\right)^{2}}
$$
F. $$
\frac{1}{1-k\left(u-v\right)^{2}}
$$
G. $$
\frac{1}{k+\left(u-v\right)^{2}}
$$
H. $$
\frac{1}{1+k\left(u-v\right)^{2}}
$$
I. $$
\frac{\left(u-v\right)^{2}}{1+k}
$$
J. $$
\frac{1}{1+k\left(v-u\right)}
$$
Answer: H
Solve the equation
$$
(x_{1} \wedge 0.6) \lor (x_{2} \wedge 0.7) \lor (x_{3} \wedge 0.5) \lor (x_{4} \wedge 0.3) = 0.5.
$$

Options:
A. $$X=([0,0.5],[0.1,1],[0.5,0.6],[0,1])U([0.5,0.6],[0,1],[0.7,1],[0.4,1])U([0,0.6],[0.4,0.5],[0.5,1],[0,1])$$
B. $$X=([0.5,0.7],[0,1],[0.5,1],[0.2,1])U([0.7,1],[0.5,0.7],[0,1],[0.1,0.5])U([0.6,1],[0.7,1],[0,1],[0.5,1])$$
C. $$X=([0.5,1],[0.6,1],[0,0.5],[0.3,1])U([0,0.5],[0.4,1],[0.5,1],[0.6,1])U([0,0.5],[0.6,1],[0.5,1],[0,0.3])$$
D. $$X=([0.4,0.5],[0,1],[0.5,0.7],[0.8,1])U([0.6,1],[0.3,0.5],[0,1],[0,0.5])U([0.5,0.7],[0,1],[0,0.4],[0.5,1])$$
E. $$X=([0.5,1],[0,0.7],[0.3,1],[0.6,1])U([0.3,0.5],[0.5,1],[0,1],[0,0.6])U([0.5,0.6],[0,1],[0,0.5],[0,0.4])$$
F. $$X=([0.5,0.6],[0,0.3],[0.6,1],[0,0.5])U([0.3,0.5],[0.4,1],[0.5,0.6],[0.7,1])U([0,1],[0.6,1],[0.5,0.7],[0,1])$$
G. $$X=([0.7,1],[0,1],[0.3,0.5],[0.1,1])U([0.3,0.5],[0.5,1],[0,1],[0.4,0.6])U([0.5,0.6],[0,1],[0,0.4],[0.5,1])$$
H. $$X=([0.5,1],[0.5,0.7],[0,0.5],[0,1])U([0.5,1],[0.5,0.6],[0,1],[0.3,1])U([0.5,1],[0.3,0.5],[0,0.4],[0.5,1])$$
I. $$
X=(0.5,[0,0.5],[0,1],[0,1])U([0,0.5],0.5,[0,1],[0,1])U([0,0.5],[0,0.5],[0.5,1],[0,1])
$$
J. $$X=([0.6,1],[0,0.5],[0,1],[0.5,1])U([0.5,0.6],[0,0.5],[0.7,1],[0,1])U([0.5,0.6],[0,1],[0.7,1],[0.5,1])$$
Answer: I
What is the simplified form of the following $\mathrm{F}$ logical functions?
$$
\begin{array} {l}
{{f_{1}=y z+x \bar{y} z+x \bar{x} y \bar{y} \bar{z}+\bar{x} \bar{y} z \bar{z};}} \\
{{f_{2}=x_{1} x_{3} \bar{x}_{1}+x_{1} \bar{x}_{1} \ \bar{x}_{3} \ \bar{x}_{4}+x_{1} \bar{x}_{1} \ \bar{x}_{3} \ {x}_{4};}} \\
{{f_{3}=x_{2} x_{3}+x_{1} \bar{x}_{2} x_{3}+x_{1} \ \bar{x}_{1} \ {x}_{2} \ \bar{x}_{2} \ \bar{x}_{3}+x_{1} \ \bar{x}_{1} \ \bar{x}_{2} \ {x}_{3} \ \bar{x}_{3};}} \end{array}
$$

Options:
A. $$f_{1}=yz+x\bar{y}z+\bar{x}x\bar{y}y+\bar{z}z, f_{2}=x_{1}\bar{x}_{1}, f_{3}=x_{2}x_{3}+x_{1}\bar{x}_{2}x_{3}+x_{1}\bar{x}_{1}x_{2}\bar{x}_{2},$$
B. $$f_{1}=\bar{x}z+y\bar{z}+x\bar{y}z+\bar{x}\bar{z}, f_{2}=\bar{x}_{1}x_{3}, f_{3}=x_{3}\bar{x}_{3}+\bar{x}_{2}x_{1}x_{2}+\bar{x}_{1}x_{3},$$
C. $$f_{1}=y\bar{z}+x\bar{y}z+\bar{x}y\bar{y}+\bar{z}, f_{2}=x_{1}\bar{x}_{3}, f_{3}=x_{2}x_{3}+x_{1}x_{3}+x_{2}\bar{x}_{3},$$
D. $$f_{1}=z\bar{y}+x\bar{y}y+\bar{x}\bar{y}z\bar{z}+x\bar{z}, f_{2}=\bar{x}_{1}\bar{x}_{4}, f_{3}=x_{3}x_{2}+\bar{x}_{1}x_{3}+\bar{x}_{2},$$
E. $$f_{1}=y\bar{z}+x\bar{x}z+\bar{y}\bar{x}+z\bar{z}, f_{2}=\bar{x}_{1}x_{4}, f_{3}=x_{3}x_{2}+x_{2}x_{1}+x_{3}\bar{x}_{1},$$
F. $$f_{1}=yz+x\bar{z}y+x\bar{x}+z, f_{2}=x_{1}x_{3}, f_{3}=\bar{x}_{2}x_{3}+x_{1}x_{2}+\bar{x}_{1}\bar{x}_{3},$$
G. $$f_{1}=xz+y\bar{y}z+x\bar{z}+y, f_{2}=x_{1}\bar{x}_{4}, f_{3}=\bar{x}_{1}x_{3}+x_{2}\bar{x}_{3}+\bar{x}_{2}x_{1},$$
H. $$f_{1}=xz+y\bar{z}x+\bar{x}z\bar{y}+x\bar{z}, f_{2}=\bar{x}_{1}x_{3}, f_{3}=x_{3}x_{2}+\bar{x}_{1}x_{2}x_{3}+\bar{x}_{2}\bar{x}_{3},$$
I. $$f_{1}=yz+\bar{x}x\bar{z}+y\bar{y}+z, f_{2}=x_{1}x_{3}, f_{3}=x_{3}x_{2}+\bar{x}_{2}x_{1}+\bar{x}_{1}\bar{x}_{3},$$
J. $$f_{1}=z\bar{y}+x\bar{y}z+x\bar{x}y+\bar{z}, f_{2}=x_{1}x_{4}, f_{3}=x_{2}\bar{x}_{3}+x_{1}x_{2}x_{3}+\bar{x}_{2}\bar{x}_{3},$$
Answer: A
It is known that the defect rate of a certain product is 0.01. Now, suppose 100 items are randomly selected for inspection. Let $A$ represent "approximately 2 defective products are found among the selected items." What is the probability of $A$? Where
$$
A={\frac{0.6}{1}}+{\frac{1}{2}}+{\frac{0.5}{3}}
$$

Options:
A. 0.432
B. 0.361
C. 0.678
D. 0.392
E. 0.538
F. 0.316
G. 0.524
H. 0.284
I. 0.274
J. 0.486
Answer: A
Clausthalite is a mineral composed of lead selenide (PbSe). The mineral adopts a NaCl-type structure. The density of PbSe at 25 °C is $8. 2 7 \, \mathrm{g / c m}^{3}.$ What is the edge length of the PbSe unit cell?

Options:
A. $5.78\AA$
B. $6.50\AA$
C. $5.90\AA$
D. $5.85\AA$
E. $6.13\AA$
F. $6.30\AA$
G. $6.40\AA$
H. $6.05\AA$
I. $6.25\AA$
J. $5.72\AA$
Answer: E
What is the kinetic energy in joules of an 850-lb motorcycle moving at 66 mph?

Options:
A. $$
1. 7 \times1 0^{5} \mathrm{~ J}
$$
B. $$
8. 8 \times1 0^{5} \mathrm{~ J}
$$
C. $$
2. 5 \times1 0^{5} \mathrm{~ J}
$$
D. $$
5. 3 \times1 0^{5} \mathrm{~ J}
$$
E. $$
7. 6 \times1 0^{5} \mathrm{~ J}
$$
F. $$
6. 1 \times1 0^{5} \mathrm{~ J}
$$
G. $$
4. 9 \times1 0^{5} \mathrm{~ J}
$$
H. $$
9. 4 \times1 0^{5} \mathrm{~ J}
$$
I. $$
3. 2 \times1 0^{5} \mathrm{~ J}
$$
J. $$
1. 0 \times1 0^{5} \mathrm{~ J}
$$
Answer: A
$\mathbf{A}$ particular chip snack food is composed of 12% protein, 14% fat, and the rest carbohydrate. What percentage of the calorie content of this food is fat?

Options:
A. $$16\%$$
B. $$25\%$$
C. $$18\%$$
D. $$29\%$$
E. $$
27\%
$$
F. $$30\%$$
G. $$22\%$$
H. $$20\%$$
I. $$12\%$$
J. $$24\%$$
Answer: E
When a 0.235-g sample of benzoic acid is combusted in a bomb calorimeter, the temperature rises 1.642 °C. When a 0.265-g sample of caffeine, $\mathrm{C_{8} H_{1 0} O_{2} N_{4}},$ is burned, the temperature rises 1.525 °C. Using the value 26.38 $\mathrm{k J / g}$ for the heat of combustion of benzoic acid, calculate the heat of combustion per mole of caffeine at constant volume.

Options:
A. $$
3. 6 8 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
B. $$
4. 1 5 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
C. $$
3. 7 4 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
D. $$
3. 5 6 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
E. $$
4. 0 2 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
F. $$
3. 8 1 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
G. $$
3. 8 9 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
H. $$
3. 7 8 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
I. $$
3. 7 2 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
J. $$
3. 9 1 \, \mathrm{k J} /^{\circ} \, \mathrm{C}
$$
Answer: H
Let $V, W$ be complex vector spaces of dimensions $m \geq n \geq 2$, respectively. Let ${\mathbb{P}\text{Hom}(V, W) \cong \mathbb{P}^{mn-1} }$be the projective space of nonzero linear maps $\phi: V \to W$modulo scalars. Further, let $\Phi \subset \mathbb{P}\text{Hom}(V, W)$ be the subset of those linear maps $\phi$which do not have full rank $n$ . $\Phi$is an irreducible subvariety of $\mathbb{P}^{mn-1}$ .What is its dimensionality?

Options:
A. $$
mn-n^2-1
$$
B. $$
m^2-nm+m-1
$$
C. $$
mn-m^2+n-2
$$
D. $$
m^2-n^2-2
$$
E. $$
mn-m+n-3
$$
F. $$
mn-n+m-4
$$
G. $$
mn-n-3m+1
$$
H. $$
m^2-nm+n+3
$$
I. $$
mn-m+n-2
$$
J. $$
m^2-n+2
$$
Answer: I
Let $M$ be a connected closed $4$ -manifold such that $\pi_{1} ( M )$ is perfect; that is, does not have any non-trivial abelian quotients. What is the possible cohomology groups $H^{*} ( M, \mathbb{Z} )$ ?

Options:
A. $\mathbb{Z},0,A,0,\mathbb{Z}$.\
where $A$ is free of fnite rank.
B. $\mathbb{Z},0,J,0,\mathbb{Z}_2$.\
where $J$ is finitely generated.
C. $\mathbb{Z},0,0,0,\mathbb{Z}$.\
where no extra conditions apply.
D. $\mathbb{Z},0,F,0,0$.\
where $F$ is finite cyclic.
E. $\mathbb{Z},0,C,0,2\mathbb{Z}$.\
where $C$ contains torsion elements.
F. $\mathbb{Z},\mathbb{Z},G,0,\mathbb{Z}$.\
where $G$ is not finitely generated.
G. $0,0,H,0,\mathbb{Z}$.\
where $H$ is free of finite rank.
H. $\mathbb{Z},0,B,0,\mathbb{Z}$.\
where $B$ is not free of finite rank.
I. $2\mathbb{Z},0,I,0,\mathbb{Z}$.\
where $I$ has infinite rank.
J. $\mathbb{Z},0,D,D,\mathbb{Z}$.\
where $D$ is torsion free.
Answer: A
Let $\mathbb{P}^{N}$ be the space of nonzero homogeneous polynomials of degree $d$ in $n+1$ variables over a field $K$ , modulo multiplication by nonzero scalars, and let $U \subset\mathbb{P}^{N}$ be the subset of irreducible polynomials $F$ such that the zero locus $V ( F ) \subset\mathbb{P}^{n}$ is smooth.\
What is the dimension of the complement $D=\mathbb{P}^{N} \setminus U?$

Options:
A. It follows that $D$ is reducible of dimension $N-1.$
B. It follows that $D$ is again irreducible of dimension $N-1.$
C. It follows that $D$ is closed with dimension $N+3.$
D. It follows that $D$ is dense in dimension $N$ minus one.
E. It follows that $D$ is fully open of dimension $N$ minus one.
F. It follows that $D$ has singularities of dimension $N+2.$
G. It follows that $D$ is compact of dimension $N$ minus 1.
H. It follows that $D$ is non-empty of dimension $N+1.$
I. It follows that $D$ is irreducible of degree $N$ minus 1.
J. It follows that $D$ is disconnected of dimension $N-1.$
Answer: B
Let $C \subset\mathbb{P}^{2}$ be a smooth plane curve of degree $d$.
Let $K_{C}$ be the canonical bundle of $C$ . For what integer $n$ is it the case that $K_{C} \cong\mathcal{O}_{C} ( n )$ ?

Options:
A. $$ 3d-2 $$
B. $$ 4-d $$
C. $$ 2d-2 $$
D. $$ d-2 $$
E. $$ d+1 $$
F. $$ d-3 $$
G. $$ d^2-3 $$
H. $$ 2-d $$
I. $$ 3-d $$
J. $$ 2d-3 $$
Answer: F
The integer 8871870642308873326043363 is the $1 3^{t h}$ power of an integer $n$ . What is $n$ ?

Options:
A. 115
B. 104
C. 127
D. 83
E. 45
F. 96
G. 139
H. 37
I. 72
J. 58
Answer: D
Let $U$ the be upper half plane $U=\{( x, y ) \in\mathbb{R}^{2} | y > 0 \}$ and introduce the Poincaré metric
$$
g=y^{-2} ( d x \otimes d x+d y \otimes d y ).
$$
What is the geodesic equation?

Options:
A. $$ y^{\prime\prime}-\frac{1} {y} [ 3 ( x^{\prime} )^{2}+2 ( y^{\prime} )^{2} ]=0. $$
B. $$ y^{\prime\prime}-\frac{1} {y} [ ( x^{\prime} )^{2}+2 ( y^{\prime} )^{2} ]=0. $$
C. $$ y^{\prime\prime}-\frac{3} {y} [ ( x^{\prime} )^{2}+ ( y^{\prime} )^{2} ]=0. $$
D. $$ y^{\prime\prime}+\frac{1} {y} [ ( x^{\prime} )^{2}+3 ( y^{\prime} )^{2} ]=0. $$
E. $$ y^{\prime\prime}+\frac{1} {y} [ 2 ( x^{\prime} )^{2}+ ( y^{\prime} )^{2} ]=0. $$
F. $$ y^{\prime\prime}+\frac{1} {y} [ 3 ( x^{\prime} )^{2}+ ( y^{\prime} )^{2} ]=0. $$
G. $$ y^{\prime\prime}+\frac{2} {y} [ ( x^{\prime} )^{2}+ ( y^{\prime} )^{2} ]=0. $$
H. $$
y^{\prime\prime}-\frac{1} {y} [ ( x^{\prime} )^{2}+3 ( y^{\prime} )^{2} ]=0.
$$
I. $$ y^{\prime\prime}-\frac{2} {y} [ 3 ( x^{\prime} )^{2}+ ( y^{\prime} )^{2} ]=0. $$
J. $$ y^{\prime\prime}-\frac{1} {y} [ 2 ( x^{\prime} )^{2}+2 ( y^{\prime} )^{2} ]=0. $$
Answer: H
Let $M=U ( n ) / O ( n )$ for $n \geq1$ , where $U ( n )$ is the group of $n \times n$ unitary matrices and $O ( n )$ is the group of $n \times n$ orthogonal matrices. $M$ is a real manifold called the $Lagrangian\ Grassmannian$.
What is the dimension of $M$ ?

Options:
A. $$
\operatorname{d i m} M={\frac{n^{2} - n} {2}}
$$
B. $$
\operatorname{d i m} M={\frac{n ( n-1 )} {2}}
$$
C. $$
\operatorname{d i m} M={2n ( n+1 )}
$$
D. $$
\operatorname{d i m} M={2n^{2} - n}
$$
E. $$
\operatorname{d i m} M={n^{2} - 1}
$$
F. $$
\operatorname{d i m} M={n ( n-1 )}
$$
G. $$
\operatorname{d i m} M={n^{2} / 2}
$$
H. $$
\operatorname{d i m} M={\frac{n ( n+1 )} {2}}
$$
I. $$
\operatorname{d i m} M={3n ( n-1 )}
$$
J. $$
\operatorname{d i m} M={n^{2} + n}
$$
Answer: H
Let $p$ be a prime number and denote by $\mathbb{F}_{p}$ the field with $p$ elements. Consider the groups $G=\mathrm{G L}_{2} ( \mathbb{F}_{p} )$ . and $G^{\prime}=\mathrm{P G L}_{2} ( \mathbb{F}_{p} )=G / \mathbb{F}_{p}^{\times}$ , where the group of units in $\mathbb{F}_{p}$ embeds diagonally into $G$ .\
What is the value of $| G |$ and $| G^{\prime} |$ ?

Options:
A. $$
|G|=p(p-1)^2(p),|G^{\prime}|=p(p-1)(p+2)
$$
B. $$
|G|=p^2(p-1)^2,|G^{\prime}|=p(p-1)(p-1)
$$
C. $$
|G|=p(p-1)^2(p+2),|G^{\prime}|=p(p-1)(p)
$$
D. $$
|G|=p(p+1)^2(p-1),|G^{\prime}|=p(p)(p+1)
$$
E. $$
|G|=p^2(p-1)(p+1),|G^{\prime}|=p(p-2)(p+1)
$$
F. $$
|G|=p(p-2)^2(p+1),|G^{\prime}|=p(p-1)(p+3)
$$
G. $$
|G|=p(p-1)^2(p-1),|G^{\prime}|=p(p-1)(p-2)
$$
H. $$
|G|=p(p-1)^3(p+1),|G^{\prime}|=p(p-1)(p+3)
$$
I. $$
|G|=p(p-1)^2(p+1),|G^{\prime}|=p(p-1)(p+1)
$$
J. $$
|G|=p(p-1)^2(p+1),|G^{\prime}|=p(p+1)(p-1)
$$
Answer: I
Let $V$ and $W$ be complex vector spaces of dimensions $m$ and $n$ respec- tively and $A \subset V$ a subspace of dimension $l$ . Let $\mathbb{P H o m} ( V, W ) \cong\mathbb{P}^{m n-1}$ be the projective space of nonzero linear maps $\phi: V \to W$ mod scalars, and for any integer $k \leq l$ let
$$
\Psi_{k}=\{\phi: V \to W : \, \mathrm{r a n k} ( \phi|_{A} ) \leq k \} \subset\mathbb{P}^{m n-1}.
$$
$\Psi_{k}$ is an irreducible subvariety of $\mathbb{P}^{m n-1}$ . What its dimension?

Options:
A. $$
k ( n+m-k )-1
$$
B. $$
k ( n-m+k )-1
$$
C. $$
m ( n+k-m )-1
$$
D. $$
l ( m+n-k )-1
$$
E. $$
n ( k+m-l )-1
$$
F. $$
k ( n+m+l )-2
$$
G. $$
k ( m+n-k+l )-1
$$
H. $$
m ( l+n-k )-1
$$
I. $$
l ( n+m-l )-1
$$
J. $$
k ( l+n-m )-1
$$
Answer: A
Evaluate the integral is ()
$$
\int_{0}^{\infty} \frac{\sqrt{t}} {( 1+t )^{2}} d t.
$$

Options:
A. $$ \frac{\pi}{2} $$
B. $$ 3 \pi $$
C. $$
\pi
$$
D. $$ \frac{\sqrt{\pi}}{2} $$
E. $$ e $$
F. $$ 2 \pi $$
G. $$ 1 $$
H. $$ \frac{1}{2} $$
I. $$ \frac{\pi}{4} $$
J. $$ \ln 2 $$
Answer: C
(based on Ostriker, Richstone, and Thuan 1974) Suppose that high-mass stars( $M > 8 M_{\odot}$ , produce pulsars, while moderate-mass stars $( 4-8 M_{\odot} )$ ) explode com- pletely (Arnett and Schramm, 1973), each explosion producing from the processed core about 1.4 $M_{\odot}$ of elements close to iron in atomic number.
Use the Salpeter birth rate function and $T_{0}=1 2 \times1 0^{9}$ yr to compute the total density of iron returned to the interstellar medium by such explosions is().

Options:
A. $$
4 \times1 0^{-2} M_{\odot} \mathrm{\ p c}^{-3}.
$$
B. $$
3 \times1 0^{-2} M_{\odot} \mathrm{\ p c}^{-3}.
$$
C. $$
8 \times1 0^{-5} M_{\odot} \mathrm{\ p c}^{-3}.
$$
D. $$
2 \times1 0^{-3} M_{\odot} \mathrm{\ p c}^{-3}.
$$
E. $$
1 \times1 0^{-1} M_{\odot} \mathrm{\ p c}^{-3}.
$$
F. $$
9 \times1 0^{-3} M_{\odot} \mathrm{\ p c}^{-3}.
$$
G. $$
5 \times1 0^{-2} M_{\odot} \mathrm{\ p c}^{-3}.
$$
H. $$
6 \times1 0^{-4} M_{\odot} \mathrm{\ p c}^{-3}.
$$
I. $$
7 \times1 0^{-3} M_{\odot} \mathrm{\ p c}^{-3}.
$$
J. $$
5 \times1 0^{-4} M_{\odot} \mathrm{\ p c}^{-3}.
$$
Answer: D
The spinup of the crust is communicated to the charged particles in the interior by the magnetic field which threads them. Any distortion in the magnetic field due, for example, to differential rotation, generates magnetic "sound waves" or Alfven waves, which travel with a speed
$$
v_{\bf A} \sim \left( \frac{P_{B}} {\rho} \right)^{1 / 2}, \qquad P_{B} \sim\frac{B^{2}} {8 \pi}
$$
If differential rotation in the crust-charged component is smoothed out in the time $\tau_{\mathrm{A}}$ for an Alfven wave to traverse the neutron star, estimate $\tau_{\mathrm{A}}$ for a typical pulsar is().

Options:
A. $$
\tau_{\mathrm{A}} \sim9 0 \mathrm{~ s}
$$
B. $$
\tau_{\mathrm{A}} \sim1 0 0 \mathrm{~ s}
$$
C. $$
\tau_{\mathrm{A}} \sim6 0 \mathrm{~ s}
$$
D. $$
\tau_{\mathrm{A}} \sim7 5 \mathrm{~ s}
$$
E. $$
\tau_{\mathrm{A}} \sim3 5 \mathrm{~ s}
$$
F. $$
\tau_{\mathrm{A}} \sim4 0 \mathrm{~ s}
$$
G. $$
\tau_{\mathrm{A}} \sim8 0 \mathrm{~ s}
$$
H. $$
\tau_{\mathrm{A}} \sim2 5 \mathrm{~ s}
$$
I. $$
\tau_{\mathrm{A}} \sim5 0 \mathrm{~ s}
$$
J. $$
\tau_{\mathrm{A}} \sim2 0 \mathrm{~ s}
$$
Answer: I
Construct a density by dimensional analysis out of $c$ , $G$ , and $h$ . Evaluate numerically this "Planck density'' is() at which quantum gravitational effects would become important.

Options:
A. $$
\rho\sim1 0^{8 1} \mathrm{~ g ~ c m}^{-3}
$$
B. $$
\rho\sim1 0^{6 5} \mathrm{~ g ~ c m}^{-3}
$$
C. $$
\rho\sim1 0^{7 5} \mathrm{~ g ~ c m}^{-3}
$$
D. $$
\rho\sim1 0^{6 2} \mathrm{~ g ~ c m}^{-3}
$$
E. $$
\rho\sim1 0^{7 0} \mathrm{~ g ~ c m}^{-3}
$$
F. $$
\rho\sim1 0^{9 4} \mathrm{~ g ~ c m}^{-3}
$$
G. $$
\rho\sim1 0^{7 8} \mathrm{~ g ~ c m}^{-3}
$$
H. $$
\rho\sim1 0^{6 8} \mathrm{~ g ~ c m}^{-3}
$$
I. $$
\rho\sim1 0^{5 2} \mathrm{~ g ~ c m}^{-3}
$$
J. $$
\rho\sim1 0^{9 0} \mathrm{~ g ~ c m}^{-3}
$$
Answer: F
Calculate the radiation efficiency is() of a thin, Newtonian accretion disk that extends inward to $r_{l}=6 G M / c^{2}$ around a central black hole. Compare your result to the efficiency of a relativistic disk around a nonrotating black hole.

Options:
A. $$
\frac{1} {2 5} ; \ 8. 5 \% \mathrm{~ vs~ 6. 0 \%}
$$
B. $$
\frac{1} {6 6} ; \ 8. 4 \% \mathrm{~ vs~ 6. 5 \%}
$$
C. $$
\frac{1} {7 8} ; \ 8. 6 \% \mathrm{~ vs~ 6. 8 \%}
$$
D. $$
\frac{1} {8 9} ; \ 7. 8 \% \mathrm{~ vs~ 5. 3 \%}
$$
E. $$
\frac{1} {1 2} ; \ 8. 3 \% \mathrm{~ vs~ 5. 7 \%}
$$
F. $$
\frac{1} {9 0} ; \ 7. 3 \% \mathrm{~ vs~ 5. 5 \%}
$$
G. $$
\frac{1} {4 5} ; \ 7. 1 \% \mathrm{~ vs~ 4. 9 \%}
$$
H. $$
\frac{1} {5 6} ; \ 9. 1 \% \mathrm{~ vs~ 5. 4 \%}
$$
I. $$
\frac{1} {3 6} ; \ 8. 0 \% \mathrm{~ vs~ 4. 8 \%}
$$
J. $$
\frac{1} {2 0} ; \ 9. 7 \% \mathrm{~ vs~ 6. 2 \%}
$$
Answer: E
What is the mean energy of the neutrino emitted following electron capture? [Assume conditions consistent with Eq.]
$$
I \simeq\int_{Q}^{\mu_{e}} \! d E_{e} \, E_{e}^{2} \left( E_{e}-Q \right)^{2} \simeq\int_{Q}^{\mu_{e}} \! d E_{e} \, E_{e}^{4}=\frac{1} {5} \mu_{e}^{5}
$$

Options:
A. $$
\langle E_{\nu} \rangle={\frac{7} {9}} \mu_{e}={\frac{1 4} {9}} \langle E_{e} \rangle
$$
B. $$
\langle E_{\nu} \rangle={\frac{4} {5}} \mu_{e}={\frac{7} {9}} \langle E_{e} \rangle
$$
C. $$
\langle E_{\nu} \rangle={\frac{2} {3}} \mu_{e}={\frac{8} {9}} \langle E_{e} \rangle
$$
D. $$
\langle E_{\nu} \rangle={\frac{5} {6}} \mu_{e}={\frac{1 0} {9}} \langle E_{e} \rangle
$$
E. $$
\langle E_{\nu} \rangle={\frac{1} {3}} \mu_{e}={\frac{1 5} {9}} \langle E_{e} \rangle
$$
F. $$
\langle E_{\nu} \rangle={\frac{5} {9}} \mu_{e}={\frac{1 6} {9}} \langle E_{e} \rangle
$$
G. $$
\langle E_{\nu} \rangle={\frac{5} {8}} \mu_{e}={\frac{1 2} {9}} \langle E_{e} \rangle
$$
H. $$
\langle E_{\nu} \rangle={\frac{1} {2}} \mu_{e}={\frac{1 3} {9}} \langle E_{e} \rangle
$$
I. $$
\langle E_{\nu} \rangle={\frac{3} {5}} \mu_{e}={\frac{1 7} {9}} \langle E_{e} \rangle
$$
J. $$
\langle E_{\nu} \rangle={\frac{3} {4}} \mu_{e}={\frac{1 1} {9}} \langle E_{e} \rangle
$$
Answer: D
Construct a vector function that has zero divergence and zero curl everywhere.

Options:
A. $$
( \operatorname{s i n} x ) ( \operatorname{c o s h} z ) \, \hat{\mathbf{x}}-( \operatorname{s i n h} z ) ( \operatorname{s i n} y ) \, \hat{\mathbf{y}} ;
$$
B. $$
( \operatorname{c o s} x ) ( \operatorname{s i n h} y ) \, \hat{\mathbf{x}}-( \operatorname{c o s} y ) ( \operatorname{s i n h} z ) \, \hat{\mathbf{y}} ;
$$
C. $$
( \operatorname{t a n} x ) ( \operatorname{c o s h} y ) \, \hat{\mathbf{x}}-( \operatorname{s i n} y ) (\operatorname{t a n h} z ) \, \hat{\mathbf{y}} ;
$$
D. $$
( \operatorname{s i n} z ) ( \operatorname{c o s h} x ) \, \hat{\mathbf{x}}-( \operatorname{c o s} y ) ( \operatorname{s i n h} x ) \, \hat{\mathbf{y}} ;
$$
E. $$
( \operatorname{c o s} z ) ( \operatorname{s i n h} y ) \, \hat{\mathbf{x}}-( \operatorname{s i n} x ) ( \operatorname{c o s h} y ) \, \hat{\mathbf{y}} ;
$$
F. $$
( \operatorname{c o s} x ) (\operatorname{s i n h} z ) \, \hat{\mathbf{x}}-( \operatorname{c o s} y ) ( \operatorname{c o s h} x ) \, \hat{\mathbf{y}} ;
$$
G. $$
( \operatorname{t a n} y ) ( \operatorname{s i n h} x ) \, \hat{\mathbf{x}}-( \operatorname{c o s} z ) ( \operatorname{t a n h} y ) \, \hat{\mathbf{y}} ;
$$
H. $$
( \operatorname{c o s} x ) ( \operatorname{c o s h} y ) \, \hat{\mathbf{x}}-( \operatorname{s i n} y ) ( \operatorname{c o s h} z ) \, \hat{\mathbf{y}} ;
$$
I. $$
( \operatorname{s i n} x ) ( \operatorname{c o s h} y ) \, \hat{\mathbf{x}}-( \operatorname{c o s} x ) ( \operatorname{s i n h} y ) \, \hat{\mathbf{y}} ;
$$
J. $$
( \operatorname{s i n} y ) ( \operatorname{t a n h} x ) \, \hat{\mathbf{x}}-( \operatorname{c o s} x ) ( \operatorname{c o s h} z ) \, \hat{\mathbf{y}} ;
$$
Answer: I
Let $\mathbf{F}_1=x^{2}\hat{\mathbf{z}}$ and $\mathbf{F}_2=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}.$ Calculate the divergence and curl of $\mathbf{F}_{1}$ and F$_2.$ Which one can be written as the gradient of a scalar? Find a scalar potential that does the job Which one can be written as the curl of a vector? Find a suitable vector potential.

Options:
A. $$2$$
$$3$$
$$
- x {\hat{\mathbf{z}}}
$$
$$0$$
B. $$0$$
$$2$$
$$
- 2 y {\hat{\mathbf{x}}}
$$
$$0$$
C. $$2$$
$$3$$
$$
- x {\hat{\mathbf{y}}}
$$
$$0$$
D. $$1$$
$$3$$
$$
0 {\hat{\mathbf{z}}}
$$
$$0$$
E. $$1$$
$$2$$
$$
- 2 x {\hat{\mathbf{x}}}
$$
$$0$$
F. $$0$$
$$4$$
$$
- 2 y {\hat{\mathbf{y}}}
$$
$$0$$
G. $$0$$
$$3$$
$$
- 2 x {\hat{\mathbf{z}}}
$$
$$0$$
H. $$0$$
$$3$$
$$
- 3 x {\hat{\mathbf{y}}}
$$
$$0$$
I. $$0$$
$$3$$
$$
- 2 x {\hat{\mathbf{y}}}
$$
$$0$$
J. $$1$$
$$3$$
$$
- 2 x {\hat{\mathbf{y}}}
$$
$$1$$
Answer: I
Here is a fourth way of computing the energy of a uniformly charged sphere: Assemble the sphere layer by layer, each time bringing in an infinitesimal charge $d q$ from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work $d W$ does it take to build up the radius by an amount $d r$ ?

Options:
A. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{4} {5} \frac{q^{2}} {R} \right)} \,.
$$
B. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{7} {10} \frac{q^{2}} {R} \right)} \,.
$$
C. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{6} {5} \frac{q^{2}} {R} \right)} \,.
$$
D. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{3} {10} \frac{q^{2}} {R} \right)} \,.
$$
E. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{5} {10} \frac{q^{2}} {R} \right)} \,.
$$
F. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{9} {20} \frac{q^{2}} {R} \right)} \,.
$$
G. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{11} {20} \frac{q^{2}} {R} \right)} \,.
$$
H. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{1} {2} \frac{q^{2}} {R} \right)} \,.
$$
I. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{2} {5} \frac{q^{2}} {R} \right)} \,.
$$
J. $$
{\frac{1} {4 \pi\epsilon_{0}} \left( \frac{3} {5} \frac{q^{2}} {R} \right)} \,.
$$
Answer: J
The half-life of tritium (hydrogen-3) is 12.3 yr. If 56.2 mg
of tritium is released from a nuclear power plant during
the course of an accident, what mass of this nuclide will
remain after 12.3 yr? After 100 yr?

Options:
A. 14.0mg ; 0.105mg
B. 30.5mg ; 0.210mg
C. 15.0mg ; 0.175mg
D. 22.3mg ; 0.230mg
E. 28.1mg ; 0.350mg
F. 29.0mg ; 0.250mg
G. 28.1mg ; 0.201mg
H. 20.0mg ; 0.150mg
I. 27.5mg ; 0.190mg
J. 25.0mg ; 0.180mg
Answer: G
The degradation of $CF_{3}CH_{2}F$ (an HFC) by OH radicals in the troposphere is first order in each reactant and has a rate constant of $k=1. 6 \times1 0^{8} \, M^{-1} \, \mathrm{s}^{-1}$ at 4 °C. If the tropospheric concentrations of $OH$ and $CF_{3}CH_{2}F$ are 8.1 $\times1 0^{5}$ and 63 $\times1 0^{8}$ molecules $\mathrm{c m}^{-3},$ respectively, what is the rate of reaction at this temperature in $M/s$?

Options:
A. $$
2. 3 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
B. $$
2. 3 \times1 0^{-1 1} \, M / {\mathrm{s}}
$$
C. $$
4. 8 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
D. $$
9. 4 \times1 0^{-1 6} \, M / {\mathrm{s}}
$$
E. $$
8. 5 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
F. $$
6. 9 \times1 0^{-2 0} \, M / {\mathrm{s}}
$$
G. $$
7. 2 \times1 0^{-1 8} \, M / {\mathrm{s}}
$$
H. $$
5. 7 \times1 0^{-1 8} \, M / {\mathrm{s}}
$$
I. $$
3. 2 \times1 0^{-1 9} \, M / {\mathrm{s}}
$$
J. $$
1. 9 \times1 0^{-1 7} \, M / {\mathrm{s}}
$$
Answer: A
Haemoglobin, the red blood protein responsible for oxygen transport, binds about 1.34 cm$^{3}$ of oxygen per gram. Normal blood has a haemoglobin concentration of 150 g dm$^{-3}$. Haemoglobin in the lungs is about 97 per cent saturated with oxygen, but in the capillary is only about 75 per cent saturated.What volume of oxygen is given up by 100 cm$^{3}$ of blood flowing from the lungs in the capillary? Breathing air at high pressures, such as in scuba diving, results in an increased concentration of dissolved nitrogen. The Henry'slaw constant in the form $c=K p$ for the solubility of nitrogen is $0. 1 8 \mu g / ( \mathrm{g H}_{2} O \mathrm{a t m}).$

Options:
A. $$
{6.5 \mathrm{c m^{3}}}
$$
B. $$
{4.2 \mathrm{c m^{3}}}
$$
C. $$
{5 \mathrm{c m^{3}}}
$$
D. $$
{9 \mathrm{c m^{3}}}
$$
E. $$
{7.5 \mathrm{c m^{3}}}
$$
F. $$
{5.5 \mathrm{c m^{3}}}
$$
G. $$
{4.8 \mathrm{c m^{3}}}
$$
H. $$
{6 \mathrm{c m^{3}}}
$$
I. $$
{8.1 \mathrm{c m^{3}}}
$$
J. $$
{7 \mathrm{c m^{3}}}
$$
Answer: C
The equilibrium constant for the reaction $\mathrm{I_{2} ( g ) \to2}$ l(g) is 0.26 at 1000 K. What is the corresponding value of $K_{\mathrm{c}}?$ ?

Options:
A. $$
0. 0 0 1 8
$$
B. $$
0. 0 0 4 3
$$
C. $$
0. 0 0 2 9
$$
D. $$
0. 0 0 7 2
$$
E. $$
0. 0 0 5 0
$$
F. $$
0. 0 0 6 3
$$
G. $$
0. 0 0 4 8
$$
H. $$
0. 0 0 5 5
$$
I. $$
0. 0 0 3 1
$$
J. $$
0. 0 0 3 7
$$
Answer: I
The equilibrium pressure of $\mathrm{H_{2}}$ over a mixture of solid uranium and solid uranium hydride at 500 $\mathsf{K}$ is 1.04 Torr. Calculate the standard Gibbs energy of formation of $UH_{\mathrm{3}}$(s) at 500 K.

Options:
A. $$
- 4 6. 1 \mathrm{~ k J ~ m o l}^{-1}
$$
B. $$
- 4 5. 0 \mathrm{~ k J ~ m o l}^{-1}
$$
C. $$
- 4 3. 5 \mathrm{~ k J ~ m o l}^{-1}
$$
D. $$
- 4 1. 0 \mathrm{~ k J ~ m o l}^{-1}
$$
E. $$
- 3 9. 8 \mathrm{~ k J ~ m o l}^{-1}
$$
F. $$
- 4 2. 7 \mathrm{~ k J ~ m o l}^{-1}
$$
G. $$
- 4 4. 2 \mathrm{~ k J ~ m o l}^{-1}
$$
H. $$
- 4 0. 0 \mathrm{~ k J ~ m o l}^{-1}
$$
I. $$
- 3 7. 0 \mathrm{~ k J ~ m o l}^{-1}
$$
J. $$
- 3 8. 3 \mathrm{~ k J ~ m o l}^{-1}
$$
Answer: D
From the biological standard half-cell potentials $E^{\oplus} ( \mathrm{O}_{2}, \mathrm{H}^{+}, \mathrm{H}_{2} \mathrm{O} )=+0. 8 2 \mathrm{V}$ and $E^{\oplus} ( \mathrm{N A D H^{+}, H^{+}, N A D H} )=-0. 3 2 \, \mathrm{V},$ calculate the standard potential arising from the reaction in which NADH is oxidized to NAD* and the corresponding biological standard reaction Gibbs energy.

Options:
A. $$
{-4 4 0 \mathrm{~ k J ~ m o l^{-1}}}
$$
B. $$ {-2 2 0 \mathrm{~ k J ~ m o l^{-1}}} $$
C. $$ {-6 0 0 \mathrm{~ k J ~ m o l^{-1}}} $$
D. $$ {-5 5 0 \mathrm{~ k J ~ m o l^{-1}}} $$
E. $$ {+1 8 0 \mathrm{~ k J ~ m o l^{-1}}} $$
F. $$ {+4 0 0 \mathrm{~ k J ~ m o l^{-1}}} $$
G. $$ {-1 1 0 \mathrm{~ k J ~ m o l^{-1}}} $$
H. $$ {+3 5 0 \mathrm{~ k J ~ m o l^{-1}}} $$
I. $$ {-3 7 0 \mathrm{~ k J ~ m o l^{-1}}} $$
J. $$ {+2 4 0 \mathrm{~ k J ~ m o l^{-1}}} $$
Answer: A
What is the force of attraction between a $\mathrm{C a^{2+}}$ and an $\mathrm{o}^{2-}$ ion whose centers are separated by a distance of 1.25 nm?

Options:
A. $$
2. 3 4 \times1 0^{-1 0} \mathrm{~ N}
$$
B. $$
7. 0 3 \times1 0^{-1 0} \mathrm{~ N}
$$
C. $$
6. 4 5 \times1 0^{-1 0} \mathrm{~ N}
$$
D. $$
8. 9 0 \times1 0^{-1 0} \mathrm{~ N}
$$
E. $$
4. 2 2 \times1 0^{-1 0} \mathrm{~ N}
$$
F. $$
0. 3 5 \times1 0^{-1 0} \mathrm{~ N}
$$
G. $$
5. 9 1 \times1 0^{-1 0} \mathrm{~ N}
$$
H. $$
1. 5 6 \times1 0^{-1 0} \mathrm{~ N}
$$
I. $$
9. 8 7 \times1 0^{-1 0} \mathrm{~ N}
$$
J. $$
3. 7 8 \times1 0^{-1 0} \mathrm{~ N}
$$
Answer: G
Calculate $\%IC$ of the interatomic bonds for the intermetallic compound $A l_{6} M n$.

Options:
A. $$
0. 2 0 \%
$$
B. $$
0. 6 5 \%
$$
C. $$
0. 2 7 \%
$$
D. $$
0. 1 5 \%
$$
E. $$
0. 4 0 \%
$$
F. $$
0. 3 0 \%
$$
G. $$
0. 5 0 \%
$$
H. $$
0. 3 5 \%
$$
I. $$
0. 2 5 \%
$$
J. $$
0. 4 5 \%
$$
Answer: I