MikaStars39/nano-eval huggingface dataset:
Let $U=\{u_{1}, u_{2}, u_{3}\}$ and $V=\{v_{1}, v_{2}, v_{3}, v_{4}\}$, and let
$$
R=\left\{ \begin{matrix} {{{0.2}}} & {{{0.7}}} & {{{0}}} & {{{0.1}}} \\ {{{0}}} & {{{1}}} & {{{0.3}}} & {{{0.2}}} \\ {{{0.4}}} & {{{0.5}}} & {{{0}}} & {{{0.6}}} \\ \end{matrix} \right\}, \quad R \in \mathcal{P}\left( \begin{matrix} {{{U \times V}}} \\ \end{matrix} \right)
$$
$$
A=\{\, u_{1}, u_{2}\,\}, \quad B=\frac{0.1}{u_{1}} + \frac{0.3}{u_{2}} + \frac{0.7}{u_{3}}
$$
What are $T_{R}(A)$ and $T_{R}(B)$?

Options:
A. $T_{R}(A)=(0.2,0,0.3,0.2),T_{R}(B)=(0.4,1,0.5,0.6)$
B. $T_{R}(A)=(0.4,1,0.5,0.2),T_{R}(B)=(0.2,0.7,0.3,0.6)$
C. $T_{R}(A)=(0.2,0.7,0,0.1),T_{R}(B)=(0.4,0.5,0.1,0.6)$
D. $T_{R}(A)=(0.2,0.7,0.5,0.2),T_{R}(B)=(0.4,1,0.3,0.6)$
E. $T_{R}(A)=(0.2,1,0.7,0.2),T_{R}(B)=(0,0.5,0,0.6)$
F. $T_{R}(A)=(0.2,1,0.3,0),T_{R}(B)=(0.4,0.7,0,0.6)$
G. $T_{R}(A)=(0.2,1,0.3,0.2),T_{R}(B)=(0.4,0.5,0.3,0.6)$
H. $T_{R}(A)=(0.2,1,0.3,0.1),T_{R}(B)=(0.4,0,0.5,0.6)$
I. $T_{R}(A)=(0.4,1,0.3,0.6),T_{R}(B)=(0.2,0.7,0,0.1)$
J. $T_{R}(A)=(0,1,0.3,0.2),T_{R}(B)=(0.2,0.5,0.7,0.6)$
Answer: G
Let $\quad{\underline{{{2}}}}=\int_{1}^{2} {\frac{x-1} {x}}+\int_{2}^{3} {\frac{3-x} {x}}, \quad{\underline{{{3}}}}=\int_{2}^{3} {\frac{x-2} {x}}+\int_{3}^{4} {\frac{4-x} {x}}$ . What are $\quad{\underline{{{2}}}}+\quad{\underline{{{3}}}}$ and $\quad{\underline{{{2}}}}-\quad{\underline{{{3}}}}$?

Options:
A. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-4}{3}&4<z\leqslant6\\\\\frac{6-z}{3}&6<z<8\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{2-z}{3}&-2\leqslant z<0\\\\\frac{4+z}{3}&-4<z<-2\end{cases}$$
B. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-3}{3}&3<z\leqslant5\\\\\frac{8-z}{3}&5<z<7\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{z-2}{3}&-3\leqslant z<-1\\\\\frac{4+z}{3}&-4<z<-2\end{cases}$$
C. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-3}{2}&3<z\leqslant5\\\\\frac{6-z}{2}&5<z<7\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{z-1}{2}&-1<z\leqslant1\\\\\frac{3-z}{2}&-3<z<-1\end{cases}$$
D. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-2}{3}&2<z\leqslant4\\\\\frac{5-z}{3}&4<z<6\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{1+z}{3}&-3\leqslant z<-1\\\\\frac{4-z}{3}&-1<z<1\end{cases}$$
E. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-4}{2}&4<z\leqslant6\\\\\frac{8-z}{2}&6<z<8\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{3-z}{2}&-2<z\leqslant0\\\\\frac{5+z}{2}&-5<z<-2\end{cases}$$
F. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-2}{2}&2<z\leqslant4\\\\\frac{5-z}{2}&4<z<6\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{1-z}{2}&-1\leqslant z<1\\\\\frac{4+z}{2}&-4<z<-1\end{cases}$$
G. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-2}{2}&2<z\leqslant4\\\\\frac{7-z}{2}&4<z<6\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{1+z}{2}&-2\leqslant z\leqslant0\\\\\frac{3-z}{2}&-3<z<-2\end{cases}$$
H. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-3}{2}&3<z\leqslant5\\\\\frac{7-z}{2}&5<z<7\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{1-z}{2}&-1\leqslant z<1\\\\\frac{3+z}{2}&-3<z<-1\end{cases}$$
I. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-3}{3}&3<z\leqslant5\\\\\frac{9-z}{3}&5<z<7\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{z-1}{3}&-2<z\leqslant1\\\\\frac{6-z}{3}&-6<z<-2\end{cases}$$
J. $$(\underline{2}+\underline{3})(z)=\begin{cases}\frac{z-2}{2}&2<z\leqslant4\\\\\frac{6-z}{2}&4<z<6\end{cases},\quad(\underline{2}-\underline{3})(z)=\begin{cases}\frac{2-z}{2}&-1<z\leqslant1\\\\\frac{5+z}{2}&-5<z<-1\end{cases}$$
Answer: H
In a certain type of food, a specific flavoring agent is added, and its content per unit of food is $x \text{ g}$. Through statistical investigations, a preference function is obtained as follows:
$$
f(x) = \begin{cases}
\frac{x}{2} \mathrm{e}^{\left(1-\frac{x}{10}\right)} & \quad 0 \leq x \leq 100 \\
0 & \quad \text{otherwise}
\end{cases}
$$
If there are no specific restrictions on $x$, the optimal amount to add should be the maximum value of $f(x)$, namely
$$
x^{*} = 10 \text{ (g)}
$$
However, as $x$ increases, production costs rise, leading to higher prices, which in turn reduce profits. Therefore, $x$ needs to be limited. Suppose the restriction is a set $\mathrm{F}$:
$$
A(x) = \begin{cases}
1 & \quad \text{when } 0 \leq x \leq 1 \\
\frac{1}{1+(x-1)^{2}} & \quad \text{when } 1 < x \leq 100
\end{cases}
$$
How can the optimal amount be determined using the symmetric type $\mathrm{F}$ planning method?

Options:
A. $$ 0.9(g) $$
B. $$ 12.4(g) $$
C. $$ 3.3(g) $$
D. $$ 18.5(g) $$
E. $$ 9.8(g) $$
F. $$ 7.2(g) $$
G. $$
2.1(g)
$$
H. $$ 15.0(g) $$
I. $$ 5.5(g) $$
J. $$ 42.0(g) $$
Answer: G
The relative humidity of air equals the ratio of the partial pressure of water in the air to the equilibrium vapor pressure of water at the same temperature, times 100%.. If the relative humidity of the air is 58% and its temnper-ature is 68 ℉,how many molecules of water are present in a room measuring 12 $\mathrm{f t} \times1 0 \mathrm{~ f t} \times8 \mathrm{~ f t}?$

Options:
A. $$
5 \times1 0^{2 4}
$$
B. $$
1 \times1 0^{2 4}
$$
C. $$
7 \times1 0^{2 5}
$$
D. $$
3 \times1 0^{2 3}
$$
E. $$
8 \times1 0^{2 5}
$$
F. $$
9 \times1 0^{2 4}
$$
G. $$
8 \times1 0^{2 3}
$$
H. $$
2 \times1 0^{2 3}
$$
I. $$
4 \times1 0^{2 4}
$$
J. $$
6 \times1 0^{2 5}
$$
Answer: F
A 2.200-g sample of quinone ($C_6H_4O_2$) is burned in a bomb calorimeter whose total heat capacity is 7.854 kJ/°C. The temperature of the calorimeter increas- es from 23.44 °C to 30.57 C. What is the heat of combus- tion per gram of quinone?

Options:
A. $$
- 2. 9 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
B. $$
- 3. 0 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
C. $$
- 2. 9 3 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
D. $$
- 2. 7 8 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
E. $$
- 2. 5 8 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
F. $$
- 2. 6 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
G. $$
- 2. 7 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
H. $$
- 2. 7 5 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
I. $$
- 2. 8 0 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
J. $$
- 2. 8 5 \times1 0^{3} \mathrm{~ k J / m o l ~ C_6 H_4 O_2}
$$
Answer: H
The heat of combustion of ethanol, $C_2H_5OH(l)$, is -1367 kJ/mol. A batch of Sauvignon Blanc wine contains 10.6% ethanol by mass. Assuming the density of the wine to be $1. 0 \ \mathrm{g / m L}$ . what caloric content does the alcohol (ethanol) in a 6-oz glass of wine (177 mL) have?

Options:
A. $$
1. 3 \times1 0^{2} \; \mathrm{C a l}
$$
B. $$
4. 0 \times1 0^{2} \; \mathrm{C a l}
$$
C. $$
1. 5 \times1 0^{2} \; \mathrm{C a l}
$$
D. $$
3. 0 \times1 0^{2} \; \mathrm{C a l}
$$
E. $$
1. 8 \times1 0^{2} \; \mathrm{C a l}
$$
F. $$
2. 5 \times1 0^{2} \; \mathrm{C a l}
$$
G. $$
2. 2 \times1 0^{2} \; \mathrm{C a l}
$$
H. $$
1. 0 \times1 0^{3} \; \mathrm{C a l}
$$
I. $$
1. 2 \times1 0^{2} \; \mathrm{C a l}
$$
J. $$
2. 0 \times1 0^{2} \; \mathrm{C a l}
$$
Answer: A
The sun supplies about 1.0 kilowatt of energy for each square meter of surface area (1.0 $\mathrm{k W / m^{2}}$ where a watt $= 1 \, \mathrm{J} / \mathrm{s} )$ . Plants produce the equivalent of about 0.20 $\mathrm{g}$ of sucrose ($C_{\mathrm{12}}$$H_{\mathrm{22}}$$O_{\mathrm{11}}$) per hour per square meter. Assuming that the sucrose is produced as follows, calculate the percentage of sunlight used to produce sucrose.
$$
1 2 \ \mathrm{C O}_{2} ( g ) \ +\ 1 1 \ \mathrm{H}_{2} \mathrm{O} ( l ) \ \longrightarrow\ \mathrm{C_{1 2} H}_{2 2} \mathrm{O}_{1 1} \ +\ 1 2 \ \mathrm{O}_{2} ( g )~ \Delta H=5 6 4 5 \, \mathrm{k J}
$$

Options:
A. $$
0.44\%
$$
B. $$
0.87\%
$$
C. $$
0.052\%
$$
D. $$
0.61\%
$$
E. $$
0.28\%
$$
F. $$
0.092\%
$$
G. $$
0.19\%
$$
H. $$
0.75\%
$$
I. $$
0.15\%
$$
J. $$
0.033\%
$$
Answer: F
Let $S^{n}$ be the unit sphere in $\mathbb{R}^{n+1}$ , and let $r_{i} : S^{n} \to S^{n}$ be the reflection in the ith axis; that is, the map
$$
( x_{0}, \ldots, x_{n} ) \mapsto( x_{0}, \ldots, x_{i-1},-x_{i}, x_{i+1}, \ldots, x_{n} )
$$
What is the degree of $r_{i}$?

Options:
A. $$
1^{n}
$$
B. $$
(n+1)^{0}
$$
C. $$
(-1 )^{n+1}
$$
D. $$
(-1)^{i+1}
$$
E. $$
(1)^{n+1}
$$
F. $$
(2)^{n+1}
$$
G. $$
n^{(-1)}
$$
H. $$
(-1)^{n}
$$
I. $$
0^{n+1}
$$
J. $$
(-1)^{i}
$$
Answer: C
How many roots of
$$
p ( z )=z^{7}-2 z^{5}+6 z^{3}-z+1=0
$$
are there in the unit disc in $| z | < 1$?

Options:
A. 6
B. 0
C. 2
D. 9
E. 4
F. 7
G. 1
H. 5
I. 8
J. 3
Answer: J
Let $f : [ 0, 1 ] \to\mathbb{R}$ be in the Sobolev space $H^{1} ( [ 0, 1 ] )$ ; that is, functions $f$ such that both $f$ and its derivative are $L^{2}$ -integrable.
What is the limit of $\operatorname* {l i m}_{n \to\infty} \left( n \int_{0}^{1} f ( x ) e^{-2 \pi i n x} d x \right)$?

Options:
A. $$
\frac{f(1) + f(0)}{\pi i}
$$
B. $$
\frac{f(0) - f(1)}{\pi i}
$$
C. $$
\frac{2(f(1) - f(0))}{\pi i}
$$
D. $$
\frac{f(1) + f(0)}{2 \pi i}
$$
E. $$
\frac{f(1) - f(0)}{\pi i}
$$
F. $$
\frac{f(1) - f(0)}{2 \pi i}
$$
G. $$
\frac{f(1) - f(0)}{4 \pi i}
$$
H. $$
\frac{f(1) + f(0)}{4 \pi i}
$$
I. $$
\frac{f(1) - f(0)}{2 \pi}
$$
J. $$
\frac{f(0) - f(1)}{2 \pi i}
$$
Answer: F
Let $C$ be the smooth projective curve over $\mathbb{C}$ with affine equation $y^{2}=f ( x )$ , where $f \in\mathbb{C} [ x ]$ is a square-free monic polynomial of degree $d=2 n$ .
What is the deficit of $C$?

Options:
A. $n-1$
B. $n+1$
C. $2n-1$
D. $n^{2}-1$
E. $3n-1$
F. $n+0.5$
G. $n-3$
H. $2n$
I. $n-4$
J. $n-2$
Answer: A
Let $C$ be a smooth projective curve over $\mathbb{C}$ and $\omega_{C}^{\otimes2}$ the square of its canonical sheaf.\
Suppose $g ( C ) \, \geq\, 2$ and $s \, \in\, \Gamma( C, \omega_{C}^{\otimes2} )$ is a section with simple zeros. What is the genus of $\Sigma=\{x^{2}=s \}$ in the total space of the line bundle $\omega_{C}$? i.e. the curve defined by the 2-valued 1-form $\sqrt{s}$ .

Options:
A. $$
g ( \Sigma)=4 g+0.
$$
B. $$
g ( \Sigma)=3 g+2.
$$
C. $$
g ( \Sigma)=5 g-4.
$$
D. $$
g ( \Sigma)=3 g-5.
$$
E. $$
g ( \Sigma)=4 g+1.
$$
F. $$
g ( \Sigma)=4 g-5.
$$
G. $$
g ( \Sigma)=5 g-2.
$$
H. $$
g ( \Sigma)=4 g-3.
$$
I. $$
g ( \Sigma)=2 g-3.
$$
J. $$
g ( \Sigma)=6 g-1.
$$
Answer: H
Consider the algebra $M_{2} ( k )$ of $2 \times2$ matrices over a field $k$ . Recall that an idempotent in an algebra is an element $e$ such that $e^{2}=e$ .\
In case $k=\mathbb{F}_{p}$ is the prime field with $p$ elements.What is the number of idempotents in $M_{2} ( k )$ ?(Count 0 and 1 in.)

Options:
A. $$
3p^{2}-p
$$
B. $$
p^{3}+p-2
$$
C. $$
p^{2}+2p-1
$$
D. $$
p^{3}+3p
$$
E. $$
p^{2}+2p+1
$$
F. $$
p^{2}+3p+2
$$
```
G. $$
2p^{2}+p
$$
H. $$
2p^{3}+p
$$
I. $$
p^{2}-p+3
$$
J. $$
p^{2}+p+2
$$
Answer: J
Let $Y \subset\mathbb{P}^{n}$ be a closed subvariety of dimension $k$ and degree $a$ , and
$$
Z :=v_{d} ( Y ) \subset\mathbb{P}^{N}
$$
where $v_{d} : \mathbb{P}^{n} \to\mathbb{P}^{N}$ is the Veronese map of degree $d$ . What is the degree of $Z$ ?

Options:
A. $$
d^{k}
a$$
B. $$a^{k}d$$
C. $$d^{k}a^{d}$$
D. $$d^{k+1}a$$
E. $$d^{k}a^{2}$$
F. $$d^{k-2}a$$
G. $$d^{k-1}a$$
H. $$2d^{k}a$$
I. $$d^{k} \cdot a$$
J. $$d^{k}a^{1/2}$$
Answer: A
Evaluate the value is ().
$$
\int_{-\infty}^{\infty} {\frac{\operatorname{s i n}^{2} x} {x^{2}}} d x .
$$

Options:
A. $$\pi^2$$
B. $$
\pi
$$
C. $$1$$
D. $$2\pi$$
E. $$4$$
F. $$3\pi$$
G. $$\displaystyle\frac{\pi}{2}$$
H. $$\infty$$
I. $$\displaystyle\frac{\pi^2}{2}$$
J. $$0$$
Answer: B
Let $S \subset\mathbb{P}^{3}$ be a smooth algebraic surface of degree $d$ , and $S^{*} \subset\mathbb{P}^{3 *}$ the dual surface, that is, the locus of tangent planes to $S$ .\
Assuming that a general tangent plane to $S$ is tangent at only one point (this is true in characteristic O). What is the degree of $S^{*}$ ?

Options:
A. $$ (d+1)^{2}d $$
B. $$ d(d-2)^{2} $$
C. $$ d^{3}-(d-1)^{2} $$
D. $$
(d-1)^{2}d
$$
E. $$ (d-1)d^{2} $$
F. $$ (d+1)^{3} $$
G. $$ d^{2}(d-2) $$
H. $$ d(d-1)^{2} $$
I. $$ (d-1)^{3} $$
J. $$ (d+1)^{2}(d-1) $$
Answer: D
Let $p$ be an odd prime.
What is the order of $G L_{2} ( \mathbb{F}_{p} )?$

Options:
A. $$(p^{2}+1)(p^{2}-p)$$
B. $$
(p^{2}-1)(p^{2}-p)
$$
C. $$(p-1)(p^{2}+p+1)$$
D. $$(p^{2}+p)(p^{2}-p-1)$$
E. $$(p^{2}+1)(p^{2}+p)$$
F. $$(p^{2}-1)(p^{2}-p-1)$$
G. $$(p^{2}+1)(p^{2}-p+1)$$
H. $$(p^{2}-p)(p^{2}-p+1)$$
I. $$(p^{2}-1)(p^{2}+1)$$
J. $$(p^{2}-1)(p^{2}+p)$$
Answer: B
Use following formula to estimate the dependence of $\tau$ and $L_{c}$ on $M$ for stars with a given composition is().
$$
\tau=\frac{6} {5} \, \frac{k T M} {A m_{u} L} \,, \qquad\theta_{\mathrm{D}} \, \ll\, T \ll\, T_{g} \
$$

Options:
A. $$
\tau\sim M^{-1. 5}, \; L_{c} \sim M^{3. 5}
$$
B. $$
\tau\sim M^{-3. 0}, \; L_{c} \sim M^{1. 5}
$$
C. $$
\tau\sim M^{-2. 0}, \; L_{c} \sim M^{4. 0}
$$
D. $$
\tau\sim M^{-0. 5}, \; L_{c} \sim M^{2. 0}
$$
E. $$
\tau\sim M^{2. 5}, \; L_{c} \sim M^{-0. 5}
$$
F. $$
\tau\sim M^{0. 5}, \; L_{c} \sim M^{3. 0}
$$
G. $$
\tau\sim M^{3. 5}, \; L_{c} \sim M^{-1. 0}
$$
H. $$
\tau\sim M^{1. 0}, \; L_{c} \sim M^{5. 0}
$$
I. $$
\tau\sim M^{-2. 5}, \; L_{c} \sim M^{4. 5}
$$
J. $$
\tau\sim M^{0}, \; L_{c} \sim M^{2. 5}
$$
Answer: I
Estimate an upper imit to the thermal neutrino cooling rate from a neutron star of radius 10 km (Bludman and Ruderman, 1975) is(). Assume that the star radiates electron neutrinos and antineutrinos like a Fermi-Dirac blackbody" at temperature $T$ and chemical potential $\mu_{\nu}=\mu_{\bar{\nu}}=0$ . For sufficiently high $T$ ail the luminosities calculated above exceed this ra nis discrepancy.

Options:
A. $$
8 \times 10^{40} \, T_{9}^{2} \mathrm{~ erg ~ s^{-1} ~}
$$
B. $$
1 \times 10^{45} \, T_{8}^{4} \mathrm{~ erg ~ s^{-1} ~}
$$
C. $$
5 \times 10^{43} \, T_{9}^{3} \mathrm{~ erg ~ s^{-1} ~}
$$
D. $$
7 \times 10^{43} \, T_{9}^{2} \mathrm{~ erg ~ s^{-1} ~}
$$
E. $$
2 \times 10^{40} \, T_{9}^{6} \mathrm{~ erg ~ s^{-1} ~}
$$
F. $$
9 \times 10^{44} \, T_{6}^{3} \mathrm{~ erg ~ s^{-1} ~}
$$
G. $$
7 \times 10^{41} \, T_{9}^{4} \mathrm{~ erg ~ s^{-1} ~}
$$
H. $$
6 \times1 0^{4 4} \, T_{9}^{4} \mathrm{~ e r g ~ s^{-1} ~}
$$
I. $$
4 \times 10^{43} \, T_{10}^{4} \mathrm{~ erg ~ s^{-1} ~}
$$
J. $$
3 \times 10^{42} \, T_{7}^{5} \mathrm{~ erg ~ s^{-1} ~}
$$
Answer: H
Estimate the entropy of the sun is(). Assume it consists of completely ionized hydrogen, with a mean density of $1 \ \mathrm{g} \ \mathrm{c m}^{-3}$ and a mean temperature of $1 0^{6} \ \ \mathrm{K}$ .

Options:
A. $$
3 \times1 0^{5 7} \ k
$$
B. $$
4 \times1 0^{5 7} \ k
$$
C. $$
5 \times6 0^{5 9} \ k
$$
D. $$
7 \times1 0^{5 7} \ k
$$
E. $$
6 \times2 0^{5 8} \ k
$$
F. $$
1 \times1 0^{5 8} \ k
$$
G. $$
2 \times1 0^{5 8} \ k
$$
H. $$
1 \times5 0^{5 8} \ k
$$
I. $$
9 \times1 0^{5 8} \ k
$$
J. $$
8 \times1 0^{5 8} \ k
$$
Answer: G
Estimate the characteristic temperature of matter behind the shock front. What happens to the nuclei after passing through a shock?

Options:
A. $$
\frac{3} {2} k T < \frac{2} {3} m_{B} u_{s}^{2} \sim\ 1 8 \ \mathrm{M e V} > \mathrm{electromagnetic ~ absorption}
$$
B. $$
\frac{3} {2} k T > \frac{1} {3} m_{B} u_{s}^{2} \sim\ 2 0 \ \mathrm{M e V} = \mathrm{atomic ~ binding ~ energy}
$$
C. $$
\frac{3} {2} k T \leq\frac{1} {2} m_{B} u_{s}^{2} \sim\ 1 2 \ \mathrm{M e V} < \mathrm{ionization ~ potential ~ value}
$$
D. $$
\frac{3} {2} k T = \frac{1} {2} m_{B} u_{s}^{2} \sim\ 1 5 \ \mathrm{M e V} < \mathrm{thermal ~ diffusion ~ rate}
$$
E. $$
\frac{2} {3} k T \geq\frac{3} {2} m_{B} u_{s}^{2} \sim\ 1 7 \ \mathrm{M e V} < \mathrm{nuclear ~ fusion ~ energy}
$$
F. $$
\frac{1} {2} k T = \frac{2} {3} m_{B} u_{s}^{2} \sim\ 1 0 \ \mathrm{M e V} > \mathrm{chemical ~ reaction ~ energy}
$$
G. $$
\frac{1} {2} k T > \frac{3} {2} m_{B} u_{s}^{2} \sim\ 1 4 \ \mathrm{M e V} = \mathrm{photonic ~ emission ~ rate}
$$
H. $$
\frac{1} {3} k T \leq\frac{3} {2} m_{B} u_{s}^{2} \sim\ 2 5 \ \mathrm{M e V} \leq \mathrm{gamma ~ ray ~ energy}
$$
I. $$
\frac{3} {2} k T \leq\frac{1} {2} m_{B} u_{s}^{2} \sim\ 1 3 \ \mathrm{M e V} > \mathrm{n u c l e a r ~ b i n d i n g ~ e n e r g y}
$$
J. $$
\frac{3} {2} k T \geq\frac{1} {2} m_{B} u_{s}^{2} \sim\ 2 1 \ \mathrm{M e V} = \mathrm{plasma ~ conductivity ~ rate}
$$
Answer: I
Find the redshift factor, $G M / R c^{2}$ , at the onset of instability.

Options:
A. $$
0. 4 5 6 7 \bigg( \frac{V_{\odot}} {V} \bigg)^{6 / 8}
$$
B. $$
0. 9 0 7 1 \bigg( \frac{G_{\odot}} {G} \bigg)^{4 / 6}
$$
C. $$
0. 1 8 3 4 \bigg( \frac{P_{\odot}} {P} \bigg)^{5 / 7}
$$
D. $$
0. 5 4 2 9 \bigg( \frac{L_{\odot}} {L} \bigg)^{1 / 4}
$$
E. $$
0. 2 9 8 1 \bigg( \frac{D_{\odot}} {D} \bigg)^{7 / 9}
$$
F. $$
0. 7 3 1 8 \bigg( \frac{R_{\odot}} {R} \bigg)^{1 / 3}
$$
G. $$
0. 6 5 9 2 \bigg( \frac{S_{\odot}} {S} \bigg)^{2 / 3}
$$
H. $$
0. 3 2 4 8 \bigg( \frac{K_{\odot}} {K} \bigg)^{3 / 5}
$$
I. $$
0. 8 0 1 5 \bigg( \frac{T_{\odot}} {T} \bigg)^{1 / 5}
$$
J. $$
0. 6 2 9 5 \bigg( \frac{M_{\odot}} {M} \bigg)^{1 / 2}
$$
Answer: J
Evaluate the integral is(J=\int_{\mathcal{V}}(r^2+2)\:\nabla\cdot\left(\frac{\hat{\mathbf{r}}}{r^2}\right)\:d\tau, where \mathcal{V} is a sphere of radius R centered at the origin.) J is ().

Options:
A. $$2\pi R^2$$
B. $$5\pi$$
C. $$0$$
D. $$12\pi R$$
E. $$3\pi R^4$$
F. $$6\pi R$$
G. $$\pi R^2$$
H. $$-4\pi R$$
I. $$8\pi$$
J. $$4\pi R^3$$
Answer: I
Check Stokes thorem using the function $\mathbf{v}=a y \, {\hat{\mathbf{x}}}+b x \, {\hat{\mathbf{y}}} \, ( a$ and $b$ uewms ndthecirular ahofrdad $R$ cenedathogih $x$ yplae. Anwr $\pi R^{2} ( b-a ) ]$

Options:
A. $$
2 \pi R^{2} ( b+a )
$$
B. $$
\frac{1} {2} \pi R^{2} ( b-a )
$$
C. $$
\frac{1}{2} \pi R ( b+a )
$$
D. $$
\pi R^{3} ( b-a )
$$
E. $$
\pi R ( b-a )
$$
F. $$
\pi R^{2} ( b-a-b )
$$
G. $$
\frac{1}{4} \pi R^{2} ( a-b )
$$
H. $$
\frac{1}{3} \pi R^{2} ( b-a )
$$
I. $$
\pi^2 R ( b-a )
$$
J. $$
\pi R^{2} ( a+b )
$$
Answer: B
An inverted hemispherical bowl of radius $R$ carries a uniform surface charge desis $\sigma$ . Find the potential difference between the north pole" and the center.

Options:
A. $$
\frac{\sigma R} {4 \epsilon_{0}} ( 2\sqrt{2}-2 ).
$$
B. $$
\frac{\sigma R} {\epsilon_{0}} ( 2\sqrt{2}-1 ).
$$
C. $$
\frac{3\sigma R} {2 \epsilon_{0}} ( 1-\sqrt{2} ).
$$
D. $$
\frac{\sigma R^2} {2 \epsilon_{0}} ( \sqrt{2}-1 ).
$$
E. $$
\frac{2\sigma R} {3 \epsilon_{0}} ( \sqrt{2}+1 ).
$$
F. $$
\frac{\sigma R} {3 \epsilon_{0}} ( \sqrt{3}-1 ).
$$
G. $$
\frac{2 \sigma R} {\epsilon_{0}} ( \sqrt{2}+1 ).
$$
H. $$
\frac{\sigma R} {2 \epsilon_{0}} ( 1-\sqrt{2} ).
$$
I. $$
\frac{\sigma R} {2 \epsilon_{0}} ( 3-\sqrt{2} ).
$$
J. $$
\frac{\sigma R} {2 \epsilon_{0}} ( \sqrt{2}-1 ).
$$
Answer: J
When a positron is annihilated by combination with an electron, two photons of equal energy result. What is the wavelength of these photons?

Options:
A. $$
7. 2 6 3 \times1 0^{-3} \mathrm{~ n m}
$$
B. $$
1. 2 0 7 \times1 0^{-3} \mathrm{~ n m}
$$
C. $$
5. 5 9 4 \times1 0^{-3} \mathrm{~ n m}
$$
D. $$
6. 9 8 1 \times1 0^{-3} \mathrm{~ n m}
$$
E. $$
2. 4 2 6 \times1 0^{-3} \mathrm{~ n m}
$$
F. $$
8. 5 3 8 \times1 0^{-3} \mathrm{~ n m}
$$
G. $$
4. 0 1 9 \times1 0^{-3} \mathrm{~ n m}
$$
H. $$
9. 4 5 5 \times1 0^{-3} \mathrm{~ n m}
$$
I. $$
3. 1 0 5 \times1 0^{-3} \mathrm{~ n m}
$$
J. $$
3. 7 4 2 \times1 0^{-3} \mathrm{~ n m}
$$
Answer: E
The density of the fat tristearin is 0.95 g cm-3. Calculate the change in molar Gibbs energy of tristearin when a deep-sea creature is brought to the surface $p=1. 0$ atm) from a depth of 1.5 km. The hydrostatic pressure, take the mean density of water to be 1.03 g cm-3 is ().

Options:
A. $$
- 1 2 k J m o l^{-1}
$$
B. $$
- 1 4 k J m o l^{-1}
$$
C. $$
+ 1 6 k J m o l^{-1}
$$
D. $$
+ 1 4 k J m o l^{-1}
$$
E. $$
- 1 6 k J m o l^{-1}
$$
F. $$
+ 1 3 k J m o l^{-1}
$$
G. $$
+ 1 2 k J m o l^{-1}
$$
H. $$
+ 2 0 k J m o l^{-1}
$$
I. $$
+ 1 0 k J m o l^{-1}
$$
J. $$
+ 1 8 k J m o l^{-1}
$$
Answer: D
The standard reaction Gibbs energies (at pH= 7) for the hydrolysis of glucose-1-phosphate, glucose-6-phosphate, and glucose-3-phosphate are-21, -14, and -9.2 kJ mol. The equilibrium constants for the hydrolyses at 37°C is ().

Options:
A. $$
4. 1 \times1 0^{2};1. 8 \times1 0^{3};{9 0}
$$
B. $$
5. 6 \times1 0^{0};7. 1 \times1 0^{2};{1 5}
$$
C. $$
7. 4 \times1 0^{3};6. 5 \times1 0^{1};{2 1}
$$
D. $$
2. 7 \times1 0^{3};4. 9 \times1 0^{1};{3 0}
$$
E. $$
3. 0 \times1 0^{2};5. 3 \times1 0^{0};{8 8}
$$
F. $$
3. 5 \times1 0^{3};2. 3 \times1 0^{2};{3 6}
$$
G. $$
2. 3 \times1 0^{1};8. 0 \times1 0^{0};{6 5}
$$
H. $$
6. 8 \times1 0^{0};4. 0 \times1 0^{2};{4 4}
$$
I. $$
9. 1 \times1 0^{1};3. 6 \times1 0^{0};{7 2}
$$
J. $$
1. 2 \times1 0^{1};9. 8 \times1 0^{3};{5 4}
$$
Answer: F
Calculate the standard biological Gibbs energy for the reaction.
$$
\mathrm{P y r u v a t e^{-} ( a q )+N A D H ( a q )+H^{+} ( a q ) \to F a c t a t e^{-} ( a q )+N A D^{+} ( a q )}
$$
at 310 K given that $\Delta_{r} G^{\ominus }=-6 6. 6$ kJ mol1. (NAD is the oxidized form of nicotinamide dinucleotide.) This reaction occurs in muscle cells deprived of oxygen during strenuous exercise and can lead to cramp.

Options:
A. $$
{-3 0. 2 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
B. $$
{-6 3. 5 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
C. $$
{-2 5. 1 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
D. $$
{-1 8. 9 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
E. $$
{-4 2. 7 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
F. $$
{-5 0. 4 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
G. $$
{-1 7. 7 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
H. $$
{-3 5. 6 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
I. $$
{-7 2. 8 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
J. $$
{-2 9. 9 \, \mathrm{k J} \, \mathrm{m o l}^{-1}}
$$
Answer: C
Estimate the pH of a solution of 0.50 M HCl(aq). The mean activity coefficient at this concentration is 0.769. What is a more reliable value of the pH?

Options:
A. $$
9. 5 4
$$
$$
{9. 7 6}
$$
B. $$
8. 1 1
$$
$$
{8. 3 8}
$$
C. $$
6. 9 0
$$
$$
{6. 1 2}
$$
D. $$
2. 5 6
$$
$$
{2. 7 8}
$$
E. $$
0. 3 0
$$
$$
{0. 4 2}
$$
F. $$
3. 6 4
$$
$$
{3. 8 7}
$$
G. $$
5. 8 9
$$
$$
{5. 0 1}
$$
H. $$
4. 7 8
$$
$$
{4. 9 9}
$$
I. $$
7. 2 5
$$
$$
{7. 4 3}
$$
J. $$
1. 2 3
$$
$$
{1. 4 5}
$$
Answer: E
The permanganate ion is a common oxidizing agent. What is the standard potential of the $\mathrm{M n O_{4}^{-}, H^{+} / M n^{2+}}$ couple at $\mathrm{p H}=6. 0 0$?

Options:
A. $$
{+0. 8 5 \mathrm{~ V}}
$$
B. $$
{-0. 3 2 \mathrm{~ V}}
$$
C. $$
{+1. 3 3 \mathrm{~ V}}
$$
D. $$
{+1. 1 2 \mathrm{~ V}}
$$
E. $$
{+1. 4 8 \mathrm{~ V}}
$$
F. $$
{+0. 5 9 \mathrm{~ V}}
$$
G. $$
{+0. 7 6 \mathrm{~ V}}
$$
H. $$
{+0. 9 4 \mathrm{~ V}}
$$
I. $$
{+0. 2 5 \mathrm{~ V}}
$$
J. $$
{+0. 1 0 \mathrm{~ V}}
$$
Answer: H
The net potential energy between two adjacent ions, $E_{N},$ that $i s$
$$
E_{N}=-{\frac{A} {r}}+{\frac{B} {r^{n}}}
$$
The bonding energy $E_{0}$ in terms of the parameters $A$, $B$ is ().

Options:
A. $$
-\frac{A} {\left( \frac{n B} {A} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{n B} {A} \right)^{n / ( 1-n )}}
$$
B. $$
-\frac{A} {\left( \frac{A} {n B} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{A} {n B} \right)^{n / ( 1-n )}}
$$
C. $$
-\frac{A} {\left( \frac{n A} {B} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{n A} {B} \right)^{n / ( 1-n )}}
$$
D. $$
-\frac{B} {\left( \frac{B} {n A} \right)^{1 / ( 1-n )}}+\frac{A} {\left( \frac{B} {n A} \right)^{n / ( 1-n )}}
$$
E. $$
-\frac{A} {\left( \frac{B} {A n} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{B} {A n} \right)^{n / ( 1-n )}}
$$
F. $$
-\frac{B} {\left( \frac{n A} {B} \right)^{1 / ( 1-n )}}+\frac{A} {\left( \frac{n A} {B} \right)^{n / ( 1-n )}}
$$
G. $$
-\frac{B} {\left( \frac{A} {n B} \right)^{1 / ( 1-n )}}+\frac{A} {\left( \frac{A} {n B} \right)^{n / ( 1-n )}}
$$
H. $$
-\frac{B} {\left( \frac{A} {A n} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{A} {A n} \right)^{n / ( 1-n )}}
$$
I. $$
-\frac{A} {\left( \frac{B} {n A} \right)^{1 / ( 1-n )}}+\frac{B} {\left( \frac{B} {n A} \right)^{n / ( 1-n )}}
$$
J. $$
-\frac{A} {\left( \frac{A} {B n} \right)^{1 / ( 1-n )}}+\frac{A} {\left( \frac{A} {B n} \right)^{n / ( 1-n )}}
$$
Answer: B
Magnesium (Mg) has an HCP crystal structure and a density of $1.7 4g/cm^{3}$. What is the volume of its unit cell in cubic centimeters?

Options:
A. $$
1. 3 9 \times1 0^{-2 8} \mathrm{~ m^3 / u n i t ~ c e l l}
$$
B. $$ 5. 6 4 \times 10^{-27} \mathrm{~ m^3 / unit ~ cell} $$
C. $$ 7. 8 5 \times 10^{-29} \mathrm{~ m^3 / unit ~ cell} $$
D. $$ 2. 5 1 \times 10^{-28} \mathrm{~ m^3 / unit ~ cell} $$
E. $$ 9. 2 8 \times 10^{-28} \mathrm{~ m^3 / unit ~ cell} $$
F. $$ 3. 0 8 \times 10^{-30} \mathrm{~ m^3 / unit ~ cell} $$
G. $$ 8. 4 9 \times 10^{-30} \mathrm{~ m^3 / unit ~ cell} $$
H. $$ 6. 1 7 \times 10^{-27} \mathrm{~ m^3 / unit ~ cell} $$
I. $$ 4. 7 3 \times 10^{-29} \mathrm{~ m^3 / unit ~ cell} $$
J. $$ 5. 3 5 \times 10^{-28} \mathrm{~ m^3 / unit ~ cell} $$
Answer: A
Given that the equilibrium number of vacancies at 850*C (1123 K) is 4.7 x 1022 m-3. The atomic weight and density (at 850*C) for $N i$ are, respectively, 58.69 g/mol and 8.80 g/cm. The energy for vacancy formation in nickel (Ni) is ().

Options:
A. $$
1. 4 0 \; \mathrm{e V} / \mathrm{a t o m}
$$
B. $$ 5. 2 9 \; \mathrm{e V} / \mathrm{a t o m} $$
C. $$ 1. 2 3 \; \mathrm{e V} / \mathrm{a t o m} $$
D. $$ 3. 3 9 \; \mathrm{e V} / \mathrm{a t o m} $$
E. $$ 2. 1 6 \; \mathrm{e V} / \mathrm{a t o m} $$
F. $$ 0. 9 8 \; \mathrm{e V} / \mathrm{a t o m} $$
G. $$ 2. 5 6 \; \mathrm{e V} / \mathrm{a t o m} $$
H. $$ 3. 0 7 \; \mathrm{e V} / \mathrm{a t o m} $$
I. $$ 2. 8 4 \; \mathrm{e V} / \mathrm{a t o m} $$
J. $$ 4. 4 1 \; \mathrm{e V} / \mathrm{a t o m} $$
Answer: A
What is the ASTM grain size number if 25 grains per square inch are measured at a magnification of 75x?

Options:
A. 4.8
B. 6.5
C. 3.8
D. 6.1
E. 4.0
F. 5.0
G. 4.2
H. 3.5
I. 5.5
J. 3.2
Answer: A
Carbon is allowed to diffuse through $a$ steel plate 10 mm thick. The concentrations of carbon at the
two faces are 0.85 and 0.40 kg C/cm$^{3}$ Fe, which are maintained constant. If the preexponential and activation energy are $5.0\times10^{-7}m^2/s$ and 77,000 J/mol, respectively, the temperature at which the diffusion flux is 6.3 $\times$ $10^{- 10}$ $kg/ m^{2}\cdot s$ is().

Options:
A. $$
6 1 1^{\circ} \mathrm{C}
$$
B. $$
6 5 4^{\circ} \mathrm{C}
$$
C. $$
5 7 1^{\circ} \mathrm{C}
$$
D. $$
6 9 2^{\circ} \mathrm{C}
$$
E. $$
7 1 0^{\circ} \mathrm{C}
$$
F. $$
5 9 8^{\circ} \mathrm{C}
$$
G. $$
7 4 0^{\circ} \mathrm{C}
$$
H. $$
6 8 0^{\circ} \mathrm{C}
$$
I. $$
7 2 5^{\circ} \mathrm{C}
$$
J. $$
5 8 3^{\circ} \mathrm{C}
$$
Answer: A
What is the diffusion coefficient for coper in aluminum at 600°C?Preexponential and activation energy values for this system are $6. 5 \times1 0^{-5} m^2 / s$ and 136,000 J/mol, respectively.

Options:
A. $$
6. 8 \times1 0^{-1 4} \mathrm{~ m^{2} / s}
$$
B. $$
5. 4 \times1 0^{-1 6} \mathrm{~ m^{2} / s}
$$
C. $$
9. 5 \times1 0^{-1 5} \mathrm{~ m^{2} / s}
$$
D. $$
7. 1 \times1 0^{-1 8} \mathrm{~ m^{2} / s}
$$
E. $$
3. 5 \times1 0^{-1 7} \mathrm{~ m^{2} / s}
$$
F. $$
2. 3 \times1 0^{-1 2} \mathrm{~ m^{2} / s}
$$
G. $$
2. 9 \times1 0^{-2 0} \mathrm{~ m^{2} / s}
$$
H. $$
8. 6 \times1 0^{-1 9} \mathrm{~ m^{2} / s}
$$
I. $$
1. 2 \times1 0^{-1 1} \mathrm{~ m^{2} / s}
$$
J. $$
4. 7 \times1 0^{-1 3} \mathrm{~ m^{2} / s}
$$
Answer: J
A large tower is to be supported by a series of steel wires; it is estimated that the load on each wire will be $13,300 \ N(3000 Ib_{f})$ . The minimum required wire diameter is (), assuming a factor of safety of 2.0 and a yield strength of 860 MPa (125,000 psi) for the steel.

Options:
A. $$
6. 3 \, {\mathrm{m m}} \ \ ( 0. 2 5 \, {\mathrm{i n}}. )
$$
B. $$
5. 9 \, {\mathrm{m m}} \ \ ( 0. 2 3 \, {\mathrm{i n}}. )
$$
C. $$
7. 5 \, {\mathrm{m m}} \ \ ( 0. 3 0 \, {\mathrm{i n}}. )
$$
D. $$
6. 8 \, {\mathrm{m m}} \ \ ( 0. 2 7 \, {\mathrm{i n}}. )
$$
E. $$
7. 2 \, {\mathrm{m m}} \ \ ( 0. 2 9 \, {\mathrm{i n}}. )
$$
F. $$
5. 5 \, {\mathrm{m m}} \ \ ( 0. 2 2 \, {\mathrm{i n}}. )
$$
G. $$
6. 5 \, {\mathrm{m m}} \ \ ( 0. 2 6 \, {\mathrm{i n}}. )
$$
H. $$
6. 1 \, {\mathrm{m m}} \ \ ( 0. 2 4 \, {\mathrm{i n}}. )
$$
I. $$
7. 0 \, {\mathrm{m m}} \ \ ( 0. 2 8 \, {\mathrm{i n}}. )
$$
J. $$
5. 2 \, {\mathrm{m m}} \ \ ( 0. 2 0 \, {\mathrm{i n}}. )
$$
Answer: A
A clindrca rodobrass originaly 1.2m .40 in in dameteris tobecol wokredby draing the circular cross section will be maintained during deformation. $A$ cold-worked yield strength in excess of 380 MPa (55,000 psi) and a ductiliy fat least 156EL are desired. Furthermore, the final diameter must be 7.6 mm (0.30 in.). Explain how this may be accomplished.is().

Options:
A. $$6. 5 2 \ \mathrm{m m} \ ( 0. 2 8 5 \ \mathrm{i n} )
$$
B. $$7. 7 8 \ \mathrm{m m} \ ( 0. 3 0 1 \ \mathrm{i n} )
$$
C. $$8. 0 5 \ \mathrm{m m} \ ( 0. 3 1 7 \ \mathrm{i n} )
$$
D. $$7. 2 0 \ \mathrm{m m} \ ( 0. 2 6 7 \ \mathrm{i n} )
$$
E. $$7. 5 0 \ \mathrm{m m} \ ( 0. 3 1 1 \ \mathrm{i n} )
$$
F. $$6. 6 3 \ \mathrm{m m} \ ( 0. 2 9 9 \ \mathrm{i n} )
$$
G. $$8. 9 0 \ \mathrm{m m} \ ( 0. 3 5 1 \ \mathrm{i n} )
$$
H. $$8. 1 9 \ \mathrm{m m} \ ( 0. 3 4 4 \ \mathrm{i n} )
$$
I. $$4. 5 9 \ \mathrm{m m} \ ( 0. 2 7 0 \ \mathrm{i n} )
$$
J. $$9. 0 1 \ \mathrm{m m} \ ( 0. 3 2 9 \ \mathrm{i n} )
$$
Answer: G
A structural componnet is faricated from an alloy that has a plane-strain fracture toughness of 62 $M P a {\sqrt{m}}$ thas een emina thth his somon fiasaresf fo S ow he te meimwmlenghofa surface crack is 1.6 mm What is the maximum allowable surface crack length (in mm) without fracture for this same component exposed to a stress of 250 MPa and made from another alloy that has a plane strain fracture touglmess of 51 $M P a {\sqrt{m^{2}}}$ ?

Options:
A. $$
1. 0 8 \ \mathrm{m m}
$$
B. $$
1. 1 7 \ \mathrm{m m}
$$
C. $$
1. 3 4 \ \mathrm{m m}
$$
D. $$
1. 3 1 \ \mathrm{m m}
$$
E. $$
1. 2 5 \ \mathrm{m m}
$$
F. $$
1. 0 7 \ \mathrm{m m}
$$
G. $$
1. 4 7 \ \mathrm{m m}
$$
H. $$
2. 0 0 \ \mathrm{m m}
$$
I. $$
1. 0 9 \ \mathrm{m m}
$$
J. $$
1. 5 2 \ \mathrm{m m}
$$
Answer: A
Compute the atomic packing factor for the rock salt crystal structure; what is the packing factor? Where $r_{C} / r_{A}=0. 4 1 4.$

Options:
A. 0.758
B. 0.512
C. 0.846
D. 0.793
E. 0.617
F. 0.823
G. 0.684
H. 0.903
I. 0.670
J. 0.738
Answer: D
A three-point bending test is performed on a spinel (MgAl2O4) specimen having a rectangular cross section of height $d=3. 8$ mm (0.15 in.) and widh $b=9$ mm (0.35 in.); the distance between support points is 25 mm (1.0 in).
Compute the flexural strength if the load at fracture is 350 N (80 lby).

Options:
A. $$
1 1 0 \mathrm{~ M P a ~} \ ( 1 6, 0 0 0 \mathrm{~ p s i )}
$$
B. $$
1 4 0 \mathrm{~ M P a ~} \ ( 2 0, 0 0 0 \mathrm{~ p s i )}
$$
C. $$
1 3 0 \mathrm{~ M P a ~} \ ( 1 9, 5 0 0 \mathrm{~ p s i )}
$$
D. $$
9 0 M \mathrm{~ P a ~} \ ( 1 3, 0 0 0 \mathrm{~ p s i )}
$$
E. $$
9 5 M \mathrm{~ P a ~} \ ( 1 3, 8 0 0 \mathrm{~ p s i )}
$$
F. $$
1 0 0 \mathrm{~ M P a ~} \ ( 1 4, 5 0 0 \mathrm{~ p s i )}
$$
G. $$
7 5 M \mathrm{~ P a ~} \ ( 1 1, 5 0 0 \mathrm{~ p s i )}
$$
H. $$
8 0 M \mathrm{~ P a ~} \ ( 1 2, 0 0 0 \mathrm{~ p s i )}
$$
I. $$
1 2 0 \mathrm{~ M P a ~} \ ( 1 7, 4 0 0 \mathrm{~ p s i )}
$$
J. $$
1 0 1 \mathrm{~ M P a ~} \ ( 1 5, 2 0 0 \mathrm{~ p s i )}
$$
Answer: J
The modulus of elasticity for titanium carbide (TiC) having 5 vol% porosity is 310 GPa ($45 \times1 0^{6}$
psi).
The modulus of elasticity for the nonporous material is ( ).

Options:
A. $$
2 8 3 \ \mathrm{G P a} \ \ \left( 4 1. 0 \times1 0^{6} \ \mathrm{p s i} \right)
$$
B. $$
3 9 9 \ \mathrm{G P a} \ \ \left( 5 8. 2 \times1 0^{6} \ \mathrm{p s i} \right)
$$
C. $$
1 7 2 \ \mathrm{G P a} \ \ \left( 2 5. 3 \times1 0^{6} \ \mathrm{p s i} \right)
$$
D. $$
1 5 8 \ \mathrm{G P a} \ \ \left( 2 3. 2 \times1 0^{6} \ \mathrm{p s i} \right)
$$
E. $$
1 8 4 \ \mathrm{G P a} \ \ \left( 2 6. 4 \times1 0^{6} \ \mathrm{p s i} \right)
$$
F. $$
4 2 6 \ \mathrm{G P a} \ \ \left( 6 1. 9 \times1 0^{6} \ \mathrm{p s i} \right)
$$
G. $$
2 1 7 \ \mathrm{G P a} \ \ \left( 3 1. 5 \times1 0^{6} \ \mathrm{p s i} \right)
$$
H. $$
4 7 5 \ \mathrm{G P a} \ \ \left( 6 9. 0 \times1 0^{6} \ \mathrm{p s i} \right)
$$
I. $$
3 4 2 \ \mathrm{G P a} \ \ \left( 4 9. 6 \times1 0^{6} \ \mathrm{p s i} \right)
$$
J. $$
2 9 5 \ \mathrm{G P a} \ \ \left( 4 4. 8 \times1 0^{6} \ \mathrm{p s i} \right)
$$
Answer: I
For liner, freely rotating polymer molecule, the total chain length $L$ depends on the bond length between chain atoms $d$ the total number of bonds in the molecule $N$ and the angle between adjacent backbone
chain atoms $\theta$ as follows:
$$
L=N d \operatorname{s i n} \! \left( \! {\frac{\theta} {2}} \! \right)
$$
Furthermore, the average end-to-end distance for a series of polymer molecules r in Figure 14.6 is equal to
$$
r=d {\sqrt{N}}
$$
A linear polyethylene has a mumber-average molecular weight of 300,000 ghmol; compue average values of $L$ and r
for this material.

Options:
A. $$
2 3. 4 \ \mathrm{n m}
$$
B. $$
2 5. 3 \ \mathrm{n m}
$$
C. $$
1 8. 7 \ \mathrm{n m}
$$
D. $$
2 0. 0 \ \mathrm{n m}
$$
E. $$
2 2. 5 \ \mathrm{n m}
$$
F. $$
2 7. 9 \ \mathrm{n m}
$$
G. $$
2 1. 0 \ \mathrm{n m}
$$
H. $$
1 9. 5 \ \mathrm{n m}
$$
I. $$
3 0. 1 \ \mathrm{n m}
$$
J. $$
2 6. 8 \ \mathrm{n m}
$$
Answer: E
A random poly(styrene-butadiene) copolymer has a mumber-average molecular weight of 350,000 g/mol and a degree of polymerization of 500 0. Compute the fraction of styrene and butadiene repeat units in this copolymer.

Options:
A. 0.57
B. 0.50
C. 0.62
D. 0.68
E. 0.45
F. 0.82
G. 0.73
H. 0.78
I. 0.60
J. 0.54
Answer: D
The maximum and minimum thermal conductivity values for a cermet that contains 90 vol% tianium carbide (TiC) particles in a nickel matrix. Assume thermal conductivities of 27 and 67 Wlm-K for TiC and Ni, respectively is().

Options:
A. $$
2 5. 9 \, \mathrm{W / m-K}
$$
B. $$
5 2. 3 \, \mathrm{W / m-K}
$$
C. $$
3 9. 6 \, \mathrm{W / m-K}
$$
D. $$
3 5. 0 \, \mathrm{W / m-K}
$$
E. $$
5 4. 1 \, \mathrm{W / m-K}
$$
F. $$
4 1. 8 \, \mathrm{W / m-K}
$$
G. $$
6 7. 0 \, \mathrm{W / m-K}
$$
H. $$
2 8. 7 \, \mathrm{W / m-K}
$$
I. $$
4 7. 2 \, \mathrm{W / m-K}
$$
J. $$
3 0. 5 \, \mathrm{W / m-K}
$$
Answer: H
A thick steel sheet of area 100 $in^{2}$ is exposed to air near the ocean. After a one-year period it was found to experience a weight loss of 485 g due to corrosion. To what rate of corrosion, in both mpy and $mm/yr$, does this correspond?

Options:
A. $$
3 9. 7 \, {\mathrm{m p y}}
$$
B. $$
2 8. 5 \, {\mathrm{m p y}}
$$
C. $$
2 9. 3 \, {\mathrm{m p y}}
$$
D. $$
4 0. 2 \, {\mathrm{m p y}}
$$
E. $$
4 3. 0 \, {\mathrm{m p y}}
$$
F. $$
4 1. 6 \, {\mathrm{m p y}}
$$
G. $$
3 8. 9 \, {\mathrm{m p y}}
$$
H. $$
3 5. 8 \, {\mathrm{m p y}}
$$
I. $$
3 7. 4 \, {\mathrm{m p y}}
$$
J. $$
3 6. 1 \, {\mathrm{m p y}}
$$
Answer: I
A n-type semi-conductor is known to have an electron conentration of $5 \times1 0^{1 7} \, m^{-3}$ lf he electron drift velocity is 350 m/s in an electric field of 1000 V/m, calculate the conductivity of this material.

Options:
A. $$
0. 0 2 5 \ ( \Omega\mathrm{-m} )^{-1}
$$
B. $$
0. 0 2 8 \ ( \Omega\mathrm{-m} )^{-1}
$$
C. $$
0. 0 1 9 \ ( \Omega\mathrm{-m} )^{-1}
$$
D. $$
0. 0 4 0 \ ( \Omega\mathrm{-m} )^{-1}
$$
E. $$
0. 0 2 6 \ ( \Omega\mathrm{-m} )^{-1}
$$
F. $$
0. 0 2 2 \ ( \Omega\mathrm{-m} )^{-1}
$$
G. $$
0. 0 3 8 \ ( \Omega\mathrm{-m} )^{-1}
$$
H. $$
0. 0 3 1 \ ( \Omega\mathrm{-m} )^{-1}
$$
I. $$
0. 0 3 3 \ ( \Omega\mathrm{-m} )^{-1}
$$
J. $$
0. 0 3 5 \ ( \Omega\mathrm{-m} )^{-1}
$$
Answer: B
Estimate the electrical conducivity at 75°C of silicon that has been doped with $10^{22}$ $m^{-3}$ of phosphorus atoms.

Options:
A. $$
1 3 8 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
B. $$
1 3 0 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
C. $$
1 2 5 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
D. $$
1 2 9 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
E. $$
1 2 8 \, ( \Omega\! \!-\! \mathrm{m} )^{-1}
$$
F. $$
1 2 0 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
G. $$
1 1 8 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
H. $$
1 1 9 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
I. $$
1 3 2 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
J. $$
1 4 5 \, ( \Omega \! \!-\! \mathrm{m} )^{-1}
$$
Answer: E
The dielectric constant for a soda-lime glass measured at very high frequencies (on the order of $1 0^{15}$ Hz) is approximately 2.3. What fraction of the dielectric constant at relatively low frequencies (1 MHz) is attributed to ionic polarization? Neglect any orientation polarization contributions

Options:
A. 0.45
B. 0.67
C. 0.83
D. 0.48
E. 0.71
F. 0.62
G. 0.54
H. 0.56
I. 0.77
J. 0.39
Answer: B
A steel wire is stretched with a stress of 70 MPa (10,000 psi) at 20°C (68°C). If the length is held constant, to what temperature must the wire be heated to reduce the stress to 17 MPa (2500 psi)?

Options:
A. $$
3 8. 6^{\circ} {\mathrm{C}} \ ( 1 0 1^{\circ} {\mathrm{F}} )
$$
B. $$
3 6. 2^{\circ} {\mathrm{C}} \ ( 9 7^{\circ} {\mathrm{F}} )
$$
C. $$
4 2. 9^{\circ} {\mathrm{C}} \ ( 1 0 9^{\circ} {\mathrm{F}} )
$$
D. $$
3 9. 8^{\circ} {\mathrm{C}} \ ( 1 0 4^{\circ} {\mathrm{F}} )
$$
E. $$
5 0. 6^{\circ} {\mathrm{C}} \ ( 1 2 3^{\circ} {\mathrm{F}} )
$$
F. $$
4 7. 2^{\circ} {\mathrm{C}} \ ( 1 1 7^{\circ} {\mathrm{F}} )
$$
G. $$
4 1. 3^{\circ} {\mathrm{C}} \ ( 1 0 5^{\circ} {\mathrm{F}} )
$$
H. $$
4 4. 0^{\circ} {\mathrm{C}} \ ( 1 1 1^{\circ} {\mathrm{F}} )
$$
I. $$
4 8. 1^{\circ} {\mathrm{C}} \ ( 1 1 8^{\circ} {\mathrm{F}} )
$$
J. $$
3 5. 5^{\circ} {\mathrm{C}} \ ( 9 5^{\circ} {\mathrm{F}} )
$$
Answer: G
Zinc selenide as a band gap of 2.58 eV. Over what range of wavelengths of visible light is it transparent?

Options:
A. $$
0. 4 8 \ \mu\mathrm{m}
$$
B. $$
2. 3 4 \ \mu\mathrm{m}
$$
C. $$
0. 6 7 \ \mu\mathrm{m}
$$
D. $$
1. 2 0 \ \mu\mathrm{m}
$$
E. $$
0. 8 2 \ \mu\mathrm{m}
$$
F. $$
1. 0 1 \ \mu\mathrm{m}
$$
G. $$
2. 7 9 \ \mu\mathrm{m}
$$
H. $$
0. 9 3 \ \mu\mathrm{m}
$$
I. $$
0. 9 8 \ \mu\mathrm{m}
$$
J. $$
1. 5 5 \ \mu\mathrm{m}
$$
Answer: A
Let $\mathbb{F}_{p}$ be the field of $p$ -elements and $\mathrm{G L}_{n} ( \mathbb{F}_{p} )$ the group of invertible $n$ by $n$ matrices. Compute the order of $\mathrm{G L}_{n} ( \mathbb{F}_{p} )$ . Compute the number of Sylow $p$ -subgroups.

Options:
A. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-3} ).
$$
$$
( p^{n}-2 ) ( p^{n-1}-1 ) \cdots( p-1 ) / ( p-1 )^{n}.
$$
B. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{3} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p-1 ) / ( p^{2}-1 )^{n}.
$$
C. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-2} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p-1 ) / ( p-1 )^{n+1}.
$$
D. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p-1 ) / ( p-1 )^{n}.
$$
E. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-2}-1 ) \cdots( p-1 ) / ( p-1 )^{n}.
$$
F. $$
( p^{n}-1 ) ( p^{n}-p+1 ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p+1 ) / ( p-1 )^{n}.
$$
G. $$
( p^{n}-1 ) ( p^{n}-p )^{2} ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p-2 ) / ( p-1 )^{n}.
$$
H. $$
( p^{n}-1 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-1 ) \cdots( p-1 ) / ( p+1 )^{n}.
$$
I. $$
( p^{n}-2 ) ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n}-1 ) \cdots( p-1 ) / ( p-1 )^{n}.
$$
J. $$
( p^{n}-1 )^{2} ( p^{n}-p ) ( p^{n}-p^{2} ) \cdots( p^{n}-p^{n-1} ).
$$
$$
( p^{n}-1 ) ( p^{n-1}-2 ) \cdots( p-1 ) / ( p-1 )^{n}.
$$
Answer: D
Let $E / F$ be a field extension. Let $A$ be an $m\times m$ matrix with entries in $E$ such that $\operatorname{t r} ( A )$ belongs to $F$ for every $n \geq2$ . What is $\operatorname{t r} ( A ^n)$ ?

Options:
A. $$
\sum_{i} m_{i} t_{i}^{n}
$$
B. $$ \sum_{k} m_{k} t_{k}^{n} $$
C. $$ \prod_{i} m_{i} t_{i}^{n} $$
D. $$ \sum_{j} m_{j} t_{j}^{m} $$
E. $$ \prod_{j} m_{j} t_{j}^{n} $$
F. $$ \sum_{i} n_{i} t_{i}^{m} $$
G. $$ \sum_{i} m_{i} n_{i}^{t} $$
H. $$ \prod_{i} n_{i} t_{i}^{m} $$
I. $$ \sum_{i} t_{i} m_{i}^{n} $$
J. $$ \sum_{i} t_{i}^n m_{i} $$
Answer: A
Let $V$ be a finite dimensional vector space with a positive definite quadratic form $( *, * )$ . Let $O ( V )$ denote the orthogonal group:
$$
O ( V )=\{g \in\operatorname{G L} ( V ) \mid( g x, g y )=( x, y ), \ \forall x, y \in V \}.
$$
For any non-zero $v \in V$ , let $s_{v}$ denote the reflection on $V$ :
$$
s_{v} ( w )=w-2 \frac{( v, w )} {( v, v )} v.
$$
If $v$ and $w$ are vectors in $V$ with $\| v \|=\| w \|$ , what is $s_{v} (v )$?

Options:
A. $$
s_{u} ( v )=v-2w
$$
B. $$
s_{u} ( v )=v+w
$$
C. $$
s_{u} ( v )=v-w
$$
D. $$
s_{u} ( v )=0
$$
E. $$
s_{u} ( v )=w-v
$$
F. $$
s_{u} ( v )=2w
$$
G. $$
s_{u} ( v )=w+2v
$$
H. $$
s_{u} ( v )=v+2w
$$
I. $$
s_{u} ( v )=w
$$
J. $$
s_{u} ( v )=2v
$$
Answer: I
The mass of a red blood cell is about 33 pg (where 1 $\mathrm{p g}=1 0^{-1 2} \mathrm{g}$,and it contains typically $3 \times1 0^{8}$ haemoglobin molecules. Each haemoglobin molecule is a tetramer of myoglobin (see preceding exercise). What fraction of the mass of the cell is due to haemoglobin?

Options:
A. $$
0 .9 7
$$
B. $$
0 .8 3
$$
C. $$
0 .7 2
$$
D. $$
0 .2 5
$$
E. $$
0 .4 6
$$
F. $$
0 .6 1
$$
G. $$
0 .1 6
$$
H. $$
0 .3 9
$$
I. $$
0 .5 8
$$
J. $$
0 .0 4
$$
Answer: A
$\mathsf{A}$ domestic water-carbonating kit uses steel cylinders of carbon dioxide of volume 250 cm$^{3}$. They weigh 1.04 kg when full and 0.74 kg when empty. What is the pressure of gas in the cylinder at 20°C?

Options:
A. $$
{2 3 0 \, \mathrm{b a r}}
$$
B. $$
{8 0 0 \, \mathrm{b a r}}
$$
C. $$
{9 9 0 \, \mathrm{b a r}}
$$
D. $$
{6 6 5 \, \mathrm{b a r}}
$$
E. $$
{5 5 0 \, \mathrm{b a r}}
$$
F. $$
{7 4 0 \, \mathrm{b a r}}
$$
G. $$
{3 5 0 \, \mathrm{b a r}}
$$
H. $$
{1 2 0 \, \mathrm{b a r}}
$$
I. $$
{4 5 0 \, \mathrm{b a r}}
$$
J. $$
{5 0 0 \, \mathrm{b a r}}
$$
Answer: D
$\mathsf{A}$ determination of the density of a gas or vapour can provide a quick estimate of its molar mass even though for practical work mass spectrometry is far more precise. The density of a gaseous compound was found to be 1.23 g dm$^{-3}$ at 330 K and 25.5 kPa. What is the molar mass of the compound?

Options:
A. $$
{1 0 8 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
B. $$
{1 0 5 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
C. $$
{1 6 0 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
D. $$
{1 2 3 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
E. $$
{1 4 5 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
F. $$
{1 3 2 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
G. $$
{1 7 2 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
H. $$
{1 1 8 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
I. $$
{1 4 0 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
J. $$
{1 5 0 \ \mathrm{g} \ \mathrm{m o l}^{-1}}
$$
Answer: F
Carbon dioxide, although only a minor component of the atmosphere, plays an important role in determining the weather and the composition and temperature of the atmosphere. Calculate the difference between the molar enthalpy and the molar internal energy of carbon dioxide regarded as a real gas at 298.15 K. For this calculation treat carbon dioxide as a van der Waals gas and use the data in Table 1.5.

Options:
A. $$
2. 4 6 8 k J \ m o l^{-1}
$$
B. $$ 2. 9 5 3 k J \ m o l^{-1} $$
C. $$ 1. 9 9 6 k J \ m o l^{-1} $$
D. $$ 2. 1 1 8 k J \ m o l^{-1} $$
E. $$ 1. 8 2 0 k J \ m o l^{-1} $$
F. $$ 2. 7 9 4 k J \ m o l^{-1} $$
G. $$ 4. 1 6 7 k J \ m o l^{-1} $$
H. $$ 3. 5 8 4 k J \ m o l^{-1} $$
I. $$ 3. 0 2 1 k J \ m o l^{-1} $$
J. $$ 3. 1 5 2 k J \ m o l^{-1} $$
Answer: A
A sample of carbon dioxide that initially occupies 15.0 dm$^{3}$ at 250 K and 1.00 atm is compressed isothermally. Into what volume must the gas be compressed to reduce its entropy by 10.0 J K$^{-1}$?

Options:
A. $$
{5. 0 0 \, \mathrm{d m}^{3}}
$$
B. $$
{2. 0 0 \, \mathrm{d m}^{3}}
$$
C. $$
{6. 0 0 \, \mathrm{d m}^{3}}
$$
D. $$
{3. 5 0 \, \mathrm{d m}^{3}}
$$
E. $$
{1. 8 0 \, \mathrm{d m}^{3}}
$$
F. $$
{2. 9 1 \, \mathrm{d m}^{3}}
$$
G. $$
{4. 2 5 \, \mathrm{d m}^{3}}
$$
H. $$
{2. 5 0 \, \mathrm{d m}^{3}}
$$
I. $$
{3. 0 0 \, \mathrm{d m}^{3}}
$$
J. $$
{4. 0 0 \, \mathrm{d m}^{3}}
$$
Answer: F
A piece of solid carbon dioxide with a mass of 5.50 g is placed in a 10.0-L vessel that already contains air at 705 torr and 24°C. After the carbon dioxide has totally vaporized, what is the partial pressure of carbon diox- ide and the total pressure in the container at 24 °C?

Options:
A. 0.310atm; 1.187atm
B. 0.295atm; 1.122atm
C. 0.370atm; 1.297atm
D. 0.305atm; 1.232atm
E. 0.320atm; 1.247atm
F. 0.415atm; 1.342atm
G. 0.275atm; 1.202atm
H. 0.340atm; 1.267atm
I. 0.290atm; 1.117atm
J. 0.360atm; 1.287atm
Answer: D
Thus, $\tilde{u}=0.49543$ can have two significant figures. What is the smaller positive root of the quadratic equation $x^{2}-16 \, x+1=0$ rounded to three significant figures?

Options:
A. $$
x_{1}=8-\ \sqrt{6 2}\approx0. 0 6 2 8 \,.
$$
B. $$
x_{1}=8-\ \sqrt{6 6}\approx0. 0 6 3 0 \,.
$$
C. $$
x_{1}=8-\ \sqrt{6 7}\approx0. 0 6 3 5 \,.
$$
D. $$
x_{1}=8-\ \sqrt{7 0}\approx0. 0 6 1 8 \,.
$$
E. $$
x_{1}=8-\ \sqrt{6 9}\approx0. 0 6 2 3 \,.
$$
F. $$
x_{1}=8-\ \sqrt{6 5}\approx0. 0 6 2 9 \,.
$$
G. $$
x_{1}=8-\ \sqrt{6 3}\approx0. 0 6 2 7 \,.
$$
H. $$
x_{1}=8-\ \sqrt{6 8}\approx0. 0 6 2 4 \,.
$$
I. $$
x_{1}=8-\ \sqrt{7 1}\approx0. 0 6 2 2 \,.
$$
J. $$
x_{1}=8-\ \sqrt{6 4}\approx0. 0 6 2 5 \,.
$$
Answer: G
The function $f(x) = \ln(x - \sqrt{x^2 - 1})$, find the value of $f(30)$. If a six-digit function table is used for the square root, what is the error when finding the logarithm? If another equivalent formula is used:$\ln(x - \sqrt{x^2 - 1}) = -\ln(x + \sqrt{x^2 - 1})$,
calculate and determine the error when finding the logarithm?

Options:
A. $$
\varepsilon( f^{^{*}} ) \approx0.3\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.834\times10^{-6}
$$
B. $$
\varepsilon( f^{^{*}} ) \approx0.2\times10^{-1}
$$
$$
\varepsilon( f^{*} ) \approx0.958\times10^{-6}
$$
C. $$
\varepsilon( f^{^{*}} ) \approx0.2\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.678\times10^{-6}
$$
D. $$
\varepsilon( f^{^{*}} ) \approx0.4\times10^{-3}
$$
$$
\varepsilon( f^{*} ) \approx0.465\times10^{-6}
$$
E. $$
\varepsilon( f^{^{*}} ) \approx0.8\times10^{-3}
$$
$$
\varepsilon( f^{*} ) \approx0.182\times10^{-5}
$$
F. $$
\varepsilon( f^{^{*}} ) \approx0.5\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.526\times10^{-6}
$$
G. $$
\varepsilon( f^{^{*}} ) \approx0.7\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.123\times10^{-5}
$$
H. $$
\varepsilon( f^{^{*}} ) \approx0.9\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.731\times10^{-6}
$$
I. $$
\varepsilon( f^{^{*}} ) \approx0.1\times10^{-2}
$$
$$
\varepsilon( f^{*} ) \approx0.450\times10^{-6}
$$
J. $$
\varepsilon( f^{^{*}} ) \approx0.6\times10^{-3}
$$
$$
\varepsilon( f^{*} ) \approx0.294\times10^{-6}
$$
Answer: A
When $x = 1, -1, 2$, $f(x) = 0, -3, 4$. The quadratic interpolation polynomial for $f(x)$ is____.

Options:
A. $$
L_{2}( x)=\frac{7} {8} \, x^{2}+\frac{2} {9} \, x-\frac{6} {7}
$$
B. $$
L_{2}( x)=\frac{5} {9} \, x^{2}+\frac{1} {2} \, x-\frac{4} {5}
$$
C. $$
L_{2}( x)=\frac{3} {2} \, x^{2}+\frac{5} {6} \, x-\frac{1} {4}
$$
D. $$
L_{2}( x)=\frac{4} {7} \, x^{2}+\frac{7} {9} \, x-\frac{1} {5}
$$
E. $$
L_{2}( x)=\frac{6} {11} \, x^{2}+\frac{4} {7} \, x-\frac{3} {8}
$$
F. $$
L_{2}( x)=\frac{1} {3} \, x^{2}+\frac{7} {10} \, x-\frac{5} {6}
$$
G. $$
L_{2}( x)=\frac{3} {4} \, x^{2}+\frac{4} {5} \, x-\frac{2} {3}
$$
H. $$
L_{2}( x)=\frac{2} {5} \, x^{2}+\frac{1} {3} \, x-\frac{3} {2}
$$
I. $$
L_{2}( x)=\frac{1} {2} \, x^{2}+\frac{2} {3} \, x-\frac{5} {4}
$$
J. $$
L_{2}( x)=\frac{5} {6} \, x^{2}+\frac{3} {2} \, x-\frac{7} {3}
$$
Answer: J
Let $f(x) = 4x^3 + 2x^2 + x + 1$. Try to find a polynomial $P_2(x)$ of degree no greater than 2 on $[-1, 1]$, such that it is the best uniform approximation polynomial to $f(x)$ on $[-1, 1]$. This polynomial is ______.

Options:
A. $$
P_{2} \left( x \right)=
2 \, x^{2}+4 \, x+1
$$
B. $$
P_{2} \left( x \right)=
1 \, x^{2}+3 \, x+0
$$
C. $$
P_{2} \left( x \right)=
4 \, x^{2}+0 \, x-1
$$
D. $$
P_{2} \left( x \right)=
-1 \, x^{2}+2 \, x-2
$$
E. $$
P_{2} \left( x \right)=
2 \, x^{2}-1 \, x+3
$$
F. $$
P_{2} \left( x \right)=
1 \, x^{2}+2 \, x+3
$$
G. $$
P_{2} \left( x \right)=
0 \, x^{2}-2 \, x+4
$$
H. $$
P_{2} \left( x \right)=
3 \, x^{2}-3 \, x+1
$$
I. $$
P_{2} \left( x \right)=
3 \, x^{2}+1 \, x+4
$$
J. $$
P_{2} \left( x \right)=
0 \, x^{2}+3 \, x+2
$$
Answer: A