MikaStars39/nano-eval huggingface dataset:
The plane is flying along the axis of y with constant speed $v$. When the plane is found when it reaches $(0, 1)$, then a missile is launched from $(-1, 0)$ on the axis of $x$ to fly to the plane. If the missile direction always points to the plane and the velocity is $2 v$. The differential square satisfied by the trajectory of the missile is () and the initial condition is ().

Options:
A. $$
\left\{\begin{array} {l} {{{3x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{1+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=1}}} \\ \end{array} \right.
$$
B. $$
\left\{\begin{array} {l} {{{2x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{1+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=1}}} \\ \end{array} \right.
$$
C. $$\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{3+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=1}}} \\ \end{array} \right.
$$
D. $$\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {3} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=5}}} \\ \end{array} \right.
$$
E. $$\left\{\begin{array} {l} {{{2x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=3}}} \\ \end{array} \right.
$$
F. $$\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=2}}} \\ \end{array} \right.
$$
G. $$\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=1}}} \\ \end{array} \right.
$$
H. $$\left\{\begin{array} {l} {{{2x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{3} {4} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=6}}} \\ \end{array} \right.
$$
I. $$\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{5} {3} \sqrt{2+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=5}}} \\ \end{array} \right.
$$
J. $$
\left\{\begin{array} {l} {{{x \frac{\mathrm{d}^{2} y} {\mathrm{d} x^{2}}+\frac{1} {2} \sqrt{1+\left( \frac{\mathrm{d} y} {\mathrm{d} x} \right)^{2}}=0}}} \\ {{{y (-1 )=0, y^{\prime} (-1 )=1}}} \\ \end{array} \right.
$$
Answer: J
Let the function $y=f \left(x \right)$ satisfy the differential equation $y \prime+y={\frac{\mathrm{e}^{-x} \operatorname{c o s} x} {2 \sqrt{\operatorname{s i n} x}}}$, and $f \left(\pi\right)=0$, find the curve $y=f \left(x \right) \left(x \geqslant0 \right)$ The volume of the body of rotation obtained by rotating about the $x$ axis once is ().

Options:
A. $$ \frac{5\pi} {3 ( 5-\mathrm{e}^{-2\pi} )} $$
B. $$ \frac{2\pi} {5 ( 2-\mathrm{e}^{-\pi} )} $$
C. $$ \frac{2\pi} {5 ( 1-\mathrm{e}^{-\pi} )} $$
D. $$ \frac{3\pi} {3 ( 4-\mathrm{e}^{-2\pi} )} $$
E. $$ \frac{5\pi} {3 ( 5-\mathrm{e}^{-\pi} )} $$
F. $$ \frac{3\pi} {3 ( 5-\mathrm{e}^{-\pi} )} $$
G. $$ \frac{2\pi} {5 ( 2-\mathrm{e}^{-2\pi} )} $$
H. $$ \frac{\pi} {5 ( 1-\mathrm{e}^{-2 \pi} )} $$
I. $$ \frac{3\pi} {5 ( 2-\mathrm{e}^{-\pi} )} $$
J. $$ \frac{\pi} {5 ( 2-\mathrm{e}^{-\pi} )} $$
Answer: H
Do integral equation $y^{\prime} \left( x \right)+y \left( x \right)=\frac{\left(-x \right)^{n-1}} {3^{n} \mathrm{e}^{x}}$general solution for the (), Where $n$is any positive integer.

Options:
A. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n} \left( \frac{x} {5} \right)^{n-1}+C \right]$,$C$ is any constant
B. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n} \left( \frac{x} {3} \right)^{n-1}+C \right]$,$C$ is any constant
C. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n-2} \left( \frac{x} {3} \right)^{n}+C \right]$,$C$ is any constant
D. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n} \left( \frac{x} {3} \right)^{n+1}+C \right]$,$C$ is any constant
E. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n} \left( \frac{x} {3} \right)^{n}+C \right]$,$C$ is any constant
F. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {2n} \left( \frac{x} {3} \right)^{n}+C \right]$,$C$ is any constant
G. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {n+2} \left( \frac{x} {3} \right)^{n}+C \right]$,$C$ is any constant
H. $y \left( x \right)=\mathrm{e}^{-x} \left[ \frac{\left(-1 \right)^{n-1}} {3n} \left( \frac{x} {3} \right)^{n}+C \right]$,$C$ is any constant
Answer: E
Let the derivative of the function $f \left(x \right)$ on the domain $I$ be greater than zero , if for any $x_{0} \in I$, Curve $y=f \left( x \right)$ points $(x_ {0}, f (x_ {0}))$ at the tangent and linear $x=x_{0}$ and $x$ axis formed by the area of the region of constant $4$, And $f \left(0 \right)=2$, the expression for $f \left(x \right)$ is ().

Options:
A. $$\frac{7} {4-3x^3}, x \in I$$
B. $$\frac{8} {4-2x^3}, x \in I$$
C. $$\frac{4} {4-x}, x \in I$$
D. $$\frac{8} {4-x^3}, x \in I$$
E. $$\frac{2} {4-x}, x \in I$$
F. $$\frac{8} {4-x}, x \in I$$
G. $$\frac{8} {4-3x}, x \in I$$
H. $$\frac{8} {4-5x}, x \in I$$
I. $$\frac{6} {5-x^3}, x \in I$$
J. $$\frac{8} {4-x^2}, x \in I$$
Answer: F
Let the hot water temperature in the thermos bottle be $t$, the indoor temperature be $T_{0}$, and $t$ be time (in hours). According to Newton's law of cooling: hot water temperature drop rate and $T - T_ {0}$ is proportional to the set again $T_ {0} = 2 0 ^ {\circ} \mathrm {C}$ when $t = 0$, $T=1 0 0^{\circ} \mathrm{C}$, and the temperature in the bottle after 24 hours is $5 0^{\circ} \mathrm{C}$. How many hours later is the temperature in the bottle $9 5^{\circ} \mathrm{C}$ ?

Options:
A. $1.58$ hours
B. $2.28$ hours
C. $2.89$ hours
D. $2.53$ hours
E. $4.89$ hours
F. $2.19$ hours
G. $3.29$ hours
H. $3.16$ hours
I. $3.46$ hours
J. $3.39$ hours
Answer: A
A product produced by a manufacturer is sold in two markets at the same time with the selling price of $p_{1}$and $p_{2}$, the sales volume of $q_{1}$and $q_{2}$, and the demand function of
$q_ {1} = 2, 4-0. 2 p_ {1}$ and $q_ {2} = 1 0 0. 0 5 p_ {2}$ total cost function is $C = 35 + 40(q_ ({1} + q_ {2})$. How does the manufacturer determine the selling price in both markets to maximize its total profit? What is the maximum total profit?

Options:
A. When $p_{1}=8 5, \, \, \, p_{2}=1 1 5$, the maximum profit is 595
B. When $p_{1}=9 0, \, \, \, p_{2}=1 2 0$, the maximum profit is 605
C. When $p_{1}=8 0, \, \, \, p_{2}=1 1 0$, the maximum profit is 610
D. When $p_{1}=7 5, \, \, \, p_{2}=1 3 0$, the maximum profit is 595
E. When $p_{1}=9 0, \, \, \, p_{2}=1 5 0$, the maximum profit is 650
F. When $p_{1}=8 5, \, \, \, p_{2}=1 2 5$, the maximum profit is 625
G. When $p_{1}=8 0, \, \, \, p_{2}=1 2 0$, the maximum profit is 605
H. When $p_{1}=8 5, \, \, \, p_{2}=1 3 0$, the maximum profit is 625
I. When $p_{1}=7 5, \, \, \, p_{2}=1 0 5$, the maximum profit is 575
J. When $p_{1}=7 0, \, \, \, p_{2}=1 1 0$, the maximum profit is 555
Answer: G
Consider a series circuit consisting of a resistor $R=1 0 \; \Omega$, an inductor $L=2 \mathrm{~ H}$ , and a power supply voltage $E=2 0 \operatorname{s i n} 5 t$ V After the switch $S$ is closed, current flows through the circuit. Find the function relationship between the current $i$ and $t$ is ().

Options:
A. $$i \left( t \right)=\mathrm{e}^{-5 t^2}+\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{\pi} {4} \right)$$
B. $$i \left( t \right)=\mathrm{e}^{-5 t}-\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{\pi} {4} \right)$$
C. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{3\pi} {4} \right)$$
D. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t^2-\frac{\pi} {4} \right)$$
E. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t^2-\frac{3\pi} {4} \right)$$
F. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t+\frac{3\pi} {4} \right)$$
G. $$i \left( t \right)=\mathrm{2e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{\pi} {4} \right)$$
H. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{\pi} {4} \right)$$
I. $$i \left( t \right)=\mathrm{e}^{-5 t}+2\sqrt{2} \operatorname{s i n} \left( 5 t-\frac{\pi} {4} \right)$$
J. $$i \left( t \right)=\mathrm{e}^{-5 t}+\sqrt{2} \operatorname{s i n} \left( 5 t+\frac{\pi} {4} \right)$$
Answer: H
Assume that the continuous random variables $$x_{1}, x_{2}$$ are independent of each other, and the variances all exist $$x_{1}$$, the probability densities of $$x_{2}$$ are $$f_{1}( x), f_{2}(x)$$, the probability density of random variable $$Y_{1}$$ is $$f_{Y_{1}} ( y )=\frac{1} {2} ( f_ {1} ( y )+f_{2} ( y ) )$$, random variable $$\mathrm{Y}_{2}=\mathrm{~ ( x_{1}+x_{2} ) ~} / 2$$, then ().

Options:
A. EY1=EY2,DY1=DY2
B. EY1>EY2, DY1=DY2
C. EY1<ELY2, DY1=DY2
D. EY1>EY2, DY1<DY2
E. EY1<EY2, DY1<DY2
F. EY1=EY2,DY1<DY1<DY2
G. EY1>EY2, DY1>DY2
H. EY1<DY2, DY1=DY2
I. EY1=EY2, DY1<DY2
J. EY1=EY2, DY1>DY2
Answer: J
Given a population $\mathrm{X} \sim \mathrm{N} \; \; ( \mu, \; \; \sigma^{2} )$, where $\sigma^{2}$ is known, and given a random sample $\mathbf{X}_{1}$, $\mathbf{x}_{2}, \ldots, \mathbf{x}_{n}$, to test the population mean $\mu$, let $\mathbf{H}_{0}$ : $\mu=\mu_{0}$, $\mathbf{H}_{1}$ : $\mu \neq \mu_{0}$, then ( ).

Options:
A. If the null hypothesis $\mathrm{H}_{0}$ is accepted at a significance level $\alpha = 0.01$, then it must also be accepted at $\alpha = 0.05$.
B. If the null hypothesis $\mathrm{H}_{0}$ is accepted at a significance level $\alpha = 0.10$, then it must also be accepted at $\alpha = 0.05$.
C. If the null hypothesis $\mathrm{H}_{0}$ is rejected at a significance level $\alpha = 0.05$, then it must also be rejected at $\alpha = 0.10$.
D. If the null hypothesis $\mathrm{H}_{0}$ is accepted at a significance level $\alpha = 0.05$, then it must be rejected at $\alpha = 0.01$.
E. If the null hypothesis $\mathrm{H}_{0}$ is rejected at a significance level $\alpha = 0.05$, then it must be accepted at $\alpha = 0.01$.
F. If the null hypothesis $\mathrm{H}_{0}$ is rejected at a significance level $\alpha = 0.01$, then it must be accepted at $\alpha = 0.05$.
G. If the null hypothesis $\mathrm{H}_{0}$ is rejected at a significance level $\alpha = 0.05$, then it must also be rejected at $\alpha = 0.01$.
H. If the null hypothesis $\mathrm{H}_{0}$ is accepted at a significance level $\alpha = 0.05$, then it must also be accepted at $\alpha = 0.01$.
Answer: H
Let X₁, X₂,..., $X_m$ be a simple random sample from a binomial distribution population B(n,p), with X and $S^{2}$ respectively being the sample mean and sample variance. If $X + kS^{2}$ is an unbiased estimator of $np^2$, then k = ___.

Options:
A. 3
B. -6
C. 0
D. 4
E. 2
F. -4
G. -2
H. -1
I. 5
J. -7
Answer: H
Given that a>0, for any two numbers x and y, where 0 < x < a and 0 < y < a, the probability of $xy < a^2/4$ is ____.

Options:
A. P(A)=0.5842
B. P(A)=0.5556
C. P(A)=0.5877
D. P(A)=0.5723
E. P(A)=0.6094
F. P(A)=0.5149
G. P(A)=0.5841
H. P(A)=0.5966
I. P(A)=0.6312
J. P(A)=0.6235
Answer: H
A certain gas station replenishes its fuel supply once a week, and if the weekly sales volume (in kiloliters) at this station is a random variable, its density function is
$$
p ( x )=\left\{\begin{aligned} {{0. 0 5 ( 1-\frac{x} {1 0 0} )^{4},}} & {{{} 0 < x < 1 0 0,}} \\ 0, & others \\ \end{aligned} \right.
$$
How large must the storage tank at this gas station be to keep the probability of running out of fuel within a week below 5%?

Options:
A. $k \ge 38.500$
B. $k \ge 46.982$
C. $k \ge 42.865$
D. $k \ge 45.072$
E. $k \ge 57.215$
F. $k \ge 41.057$
G. $k \ge 51.303$
H. $k \ge 34.276$
I. $k \ge 52.893$
J. $k \ge 48.109$
Answer: D
Let the cumulative distribution function of the continuous random variable \(X\) be \(F(x)\), and assume the expected value exists. What is the correct expected value of the equation?

Options:
A. $$E( X )=\int_{-x}^{0} [ 1-F ( x ) ] \mathrm{d} x-\int_{0}^{-x} F ( x ) \mathrm{d} x$$
B. $$E( X )=\int_{0}^{x} [ 1-F ( -x ) ] \mathrm{d} x-\int_{-x}^{0} F ( x ) \mathrm{d} x$$
C. $$E( X )=\int_{0}^{x} [ 1-F ( x ) ] \mathrm{d} x+\int_{-x}^{0} F ( x ) \mathrm{d} x$$
D. $$E( X )=\int_{0}^{x} [ 1+F ( x ) ] \mathrm{d} x-\int_{x}^{0} F ( x ) \mathrm{d} x$$
E. $$E( X )=\int_{x}^{0} [ 1-F ( x ) ] \mathrm{d} x-\int_{-1}^{0} F ( x ) \mathrm{d} x$$
F. $$E( X )=\int_{x}^{1} [ 1-F ( x ) ] \mathrm{d} x-\int_{0}^{-x} F ( x ) \mathrm{d} x$$
G. $$E( X )=\int_{0}^{-x} [ 1+F ( x ) ] \mathrm{d} x-\int_{-x}^{0} F ( x ) \mathrm{d} x$$
H. $$E( X )=\int_{0}^{x} [ 1-F ( x ) ] \mathrm{d} x-\int_{-x}^{0} F ( x ) \mathrm{d} x$$
I. $$
E ( X )=\int_{0}^{-x} [ 1-F ( x ) ] \mathrm{d} x-\int_{-x}^{0} F ( x ) \mathrm{d} x
$$
J. $$E( X )=\int_{x}^{0} [ 1-F ( x ) ] \mathrm{d} x-\int_{0}^{-x} F ( x ) \mathrm{d} x$$
Answer: I
The inflection point of the curve $\mathrm{y}=\mathrm{x s i n x}+2 \mathrm{c o s x} \ \ (-\pi/ 2 < \mathrm{x} < 2 \pi)$ is ()?

Options:
A. $$( 0,2 )$$
B. $$(- \pi,- 2 )$$
C. $$( -3\pi, 2 )$$
D. $$( -\pi, 1 )$$
E. $$(-3 \pi/ 2,3 \pi/ 2 )$$
F. $$( -2\pi, 2 )$$
G. $$( -2\pi, 3 )$$
H. $$( \pi/ 4,\pi/ 2 )$$
I. $$( -\pi/2, \pi/2 )$$
J. $$( -\pi, 2 )$$
Answer: E
Let the domain of the function f (x) be $( 0, \ +x )$
and satisfy $2 f \left( x \right)+x^{2} f \left( {\frac{1} {x}} \right)={\frac{x^{2}+2 x} {\sqrt{1+x^{2}}}}$ , then what is the expression for f(x)?The curves $\mathrm{y=f( x )} \; \; \;, \; \; \mathrm{y=1 / 2} \;, \; \; \; \mathrm{y=\frac{\sqrt{3}} {2} \, }$ and the Y-axis what is the volume of the rotation of the graph around which they revolve around the X-axis?

Options:
A. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!+\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{4} $$
B. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!-\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{2} $$
C. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!+\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{2} $$
D. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!-\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{12} $$
E. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!+\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{6} $$
F. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!-\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{4} $$
G. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!-\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{3} $$
H. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!+\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{12} $$
I. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!+\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{3} $$
J. $$
f ( x ) \!=\! {\frac{x} {\sqrt{1 \!-\! x^{2}}}} \;, \; \; \mathrm{x} \! \in\! \; \; ( 0 \;, \; \;+\infty)
$$,
$$V=\frac{\pi ^{2} }{6} $$
Answer: E
Let f (x) have a second continuous derivative, $\begin{array} {r c l r} {\mathbf{f}{( 0 )}} & {{=\mathbf{f}^{\prime}}} & {{( 0 )}} & {{=0 \,,}} & {{\mathbf{f}^{\prime\prime}}} & {{( \mathbf{x} )}} & {{> 0}} \\ \end{array}$, and let $\mathbf{u}=\mathbf{u} ( \mathbf{x} )$ be the intercept of the tangent line of the curve y=f(x) at point (x, f (x)) on the X-axis, then what is the value of $\operatorname* {l i m}_{x \to0} {\frac{x} {u ( x )}}$?

Options:
A. $$ -\frac{1}{2} $$
B. $$ 3 $$
C. $$- \frac{3}{2} $$
D. $$ 1 $$
E. $$ -1 $$
F. $$ \frac{3}{2} $$
G. $$ 0 $$
H. $$ \frac{1}{2} $$
I. $$ -2 $$
J. $$ 2 $$
Answer: J
Given that $\mathbf{y}=f( \mathbf{x} )$ satisfies $\mathbf{y}^{\prime\prime}+2 \mathbf{y}^{\prime}+5 f \ \ \mathbf{( x )} \ =0$, and $f \ \ ( 0 ) \ \ =1 \,, \ \ f^{\prime} \ \ ( 0 ) \ \ =\ -1 \,$.
We set $a_{n}=\int_{n \pi}^{+\infty} f \left( x \right) \! \mathrm{d} x$ , so what is the value of $\sum_{n=1}^{\infty} a_{n}$?

Options:
A. $$
\frac{1} {5 \left( e^{\pi}+1 \right)}
$$
B. $$
\frac{1} {4 \left( e^{\pi}-1 \right)}
$$
C. $$
\frac{1} { \left( e^{\pi}-1 \right)}
$$
D. $$
\frac{1} {3 \left( e^{\pi}-1 \right)}
$$
E. $$
\frac{1} {3 \left( e^{\pi}+1 \right)}
$$
F. $$
\frac{1} {2 \left( e^{\pi}+1 \right)}
$$
G. $$
\frac{1} {1 \left( e^{\pi}+1 \right)}
$$
H. $$
\frac{1} {4 \left( e^{\pi}+1 \right)}
$$
I. $$
\frac{1} {5 \left( e^{\pi}-1 \right)}
$$
J. $$
\frac{1} {2\left( e^{\pi}-1 \right)}
$$
Answer: I
Let $3 ( \alpha_{1}-\alpha)+2 ( \alpha_{2}+\alpha)=5 ( \alpha_{3}+\alpha),$ , and we know that $\alpha_{1}={\left( \begin{array} {l} {{2}} \\ {{5}} \\ {{1}} \\ {{3}} \\ \end{array} \right)}, \alpha_{2}={\left( \begin{array} {l} {{1 0}} \\ {{1}} \\ {{5}} \\ {{1 0}} \\ \end{array} \right)}, \alpha_{3}={\left( \begin{array} {l} {{4}} \\ {{1}} \\ {{-1}} \\ {{1}} \\ \end{array} \right)}$ , then what is the value of $\alpha$?

Options:
A. $\alpha={\left( \begin{array} {l} {{1}} \\ {{2}} \\ {{4}} \\ {{4}} \\ \end{array} \right)}$
B. $\alpha={\left( \begin{array} {l} {{1}} \\ {{2}} \\ {{2}} \\ {{2}} \\ \end{array} \right)}$
C. $\alpha={\left( \begin{array} {l} {{1}} \\ {{4}} \\ {{1}} \\ {{4}} \\ \end{array} \right)}$
D. $\alpha={\left( \begin{array} {l} {{1}} \\ {{4}} \\ {{4}} \\ {{4}} \\ \end{array} \right)}$
E. $\alpha={\left( \begin{array} {l} {{1}} \\ {{1}} \\ {{2}} \\ {{2}} \\ \end{array} \right)}$
F. $\alpha={\left( \begin{array} {l} {{1}} \\ {{1}} \\ {{1}} \\ {{2}} \\ \end{array} \right)}$
G. $\alpha={\left( \begin{array} {l} {{1}} \\ {{1}} \\ {{3}} \\ {{3}} \\ \end{array} \right)}$
H. $\alpha={\left( \begin{array} {l} {{1}} \\ {{2}} \\ {{3}} \\ {{4}} \\ \end{array} \right)}$
I. $\alpha={\left( \begin{array} {l} {{1}} \\ {{4}} \\ {{3}} \\ {{4}} \\ \end{array} \right)}$
J. $\alpha={\left( \begin{array} {l} {{1}} \\ {{3}} \\ {{3}} \\ {{3}} \\ \end{array} \right)}$
Answer: H
What are the values of the following double integrals?
$$\iint_{| x |+| y | \leq1} | x y | d x d y$$

Options:
A. $$\frac{4}{3} $$
B. $$\frac{1}{2} $$
C. $$\frac{2}{3} $$
D. $$\frac{1}{6} $$
E. $$1 $$
F. $$\frac{5}{6} $$
G. $$\frac{7}{6} $$
H. $$\frac{1}{3} $$
I. $$\frac{3}{2} $$
J. $$\frac{5}{3} $$
Answer: C
When $a$ takes what value, the following system of linear equations has infinitely many solutions?
$$ \left\{\begin{matrix} {{a x_{1}+x_{2}+x_{3}=1}} \\ {{x_{1}+a x_{2}+x_{3}=a}} \\ {{x_{1}+x_{2}+a x_{3}=a^{2}}} \\ \end{matrix} \right. $$

Options:
A. $$
a=3
$$
B. $$
a=4
$$
C. $$
a=-4
$$
D. $$
a=2
$$
E. $$
a=1
$$
F. $$
a=5
$$
G. $$
a=-1
$$
H. $$
a=-3
$$
I. $$
a=-2
$$
J. $$
a=0
$$
Answer: E
In a vector space $F_{n} [ x ]$ , define a linear transformation $\tau$ as: for any $\tau( f ( x ) )=x f^{\prime} ( x )-f ( x )$ , where $f^{\prime} ( x )$ represents the derivative of $f ( x )$ . What are $\mathrm{K e r} \tau$ and $\mathrm{I m} \tau$ ?

Options:
A. $$
\mathrm{K e r} \, \tau\!=\! L ( x ^{3}), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n} )
$$
B. $$
\mathrm{K e r} \, \tau\!=\! L ( x^{2} ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n} )
$$
C. $$
\mathrm{K e r} \, \tau\!=\! L ( x^{5} ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n} )
$$
D. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n} )
$$
E. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n-5} )
$$
F. $$
\mathrm{K e r} \, \tau\!=\! L ( x^{4} ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n} )
$$
G. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n-2} )
$$
H. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n-3} )
$$
I. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n-4} )
$$
J. $$
\mathrm{K e r} \, \tau\!=\! L ( x ), \, \mathrm{I m} \, \tau\!=\! L ( 1, \, \, x^{2} \,, \, \, \, \cdots\, \,, x^{n-1} )
$$
Answer: D
What is the result of expanding the determinant $D \!=\! \left| \begin{array} {c c c c} {{{1}}} & {{{0}}} & {{{-1}}} & {{{-1}}} \\ {{{0}}} & {{{-1}}} & {{{-1}}} & {{{1}}} \\ {{{-1}}} & {{{-1}}} & {{{1}}} & {{{0}}} \\ {{{a}}} & {{{b}}} & {{{c}}} & {{{d}}} \\ \end{array} \right|$ by the fourth row?

Options:
A. $$
- 2 a+b-2 c-d
$$
B. $$
- 3 a+2b-2 c-d
$$
C. $$
- a+b-2 c-d
$$
D. $$
- 3 a-b-2 c-d
$$
E. $$
- 3 a+b-2 c-d
$$
F. $$
- 3 a+b+2 c-d
$$
G. $$
- 3 a+b- c-d
$$
H. $$
- 3 a+b-2 c-2d
$$
I. $$
3 a+b-2 c-d
$$
J. $$
- 3 a+b-2 c+d
$$
Answer: E
Let $A$ be a square matrix of third order, and $\mathrm{d e t} A={\frac{1} {2}}$ , what is the value of $\operatorname* {d e t} \! \left( {\frac{1} {3}} A^{-1} \!-\! 1 0 A^{*} \right) \!$ ?

Options:
A. $$
2 \times( \frac{1 4} {7} )^{3}
$$
B. $$
- 2 \times( \frac{1 3} {3} )^{3}
$$
C. $$
- 2 \times( \frac{1 4} {5} )^{3}
$$
D. $$
2 \times( \frac{1 4} {3} )^{3}
$$
E. $$
- 2 \times( \frac{1 4} {9} )^{3}
$$
F. $$
2 \times( \frac{1 3} {3} )^{3}
$$
G. $$
2 \times( \frac{1 4} {5} )^{3}
$$
H. $$
- 2 \times( \frac{1 4} {3} )^{3}
$$
I. $$
2 \times( \frac{1 4} {9} )^{3}
$$
J. $$
- 2 \times( \frac{1 4} {7} )^{3}
$$
Answer: H
Let the matrix $A$ be similar to $B$ , where $A \!=\! \left( \! \! \begin{array} {c c c} {{{-2}}} & {{{0}}} & {{{0}}} \\ {{{2}}} & {{{a}}} & {{{2}}} \\ {{{3}}} & {{{1}}} & {{{1}}} \\ \end{array} \! \! \right)$ , $B {=} \left( \begin{matrix} {{{-1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{2}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{b}}} \\ \end{matrix} \right)$ .
What are the values of $a$ and $b$ respectively?

Options:
A. $$
a=2, \quad b=-2
$$
B. $$
a=-2, \quad b=-2
$$
C. $$
a=-1, \quad b=-2
$$
D. $$
a=0, \quad b=-1
$$
E. $$
a=0, \quad b=-2
$$
F. $$
a=1, \quad b=-2
$$
G. $$
a=0, \quad b=0
$$
H. $$
a=0, \quad b=3
$$
I. $$
a=0, \quad b=1
$$
J. $$
a=0, \quad b=2
$$
Answer: E
Set $f (x) = x ^ {3} + (1 + t) x ^ {2} + 2 x + 2 u$ and $g (x) = x ^ {3} + t x+ u$ the greatest common factor is a quadratic polynomial, How many possible values of $t$ and $u$ are there?

Options:
A. $$7
$$
B. $$6
$$
C. $$5
$$
D. $$10
$$
E. $$8
$$
F. $$9
$$
G. $$4
$$
H. $$2
$$
I. $$3
$$
J. $$1
$$
Answer: G
Let$$
\begin{array} {l c r} {{{\alpha_{1} {=} ( 1, 0,-1 ),}}} & {{{\alpha_{2} {=} (-1, 1, 0 ),}}} & {{{\alpha_{3} {=} ( 1, 2, 3 ),}}} \\ {{{\beta_{1} {=} ( 0, 1, 1 ),}}} & {{{\beta_{2} {=} ( 1, 0, 1 ),}}} & {{{\beta_{3} {=} ( 1, 1, 1 )}.}} \\ \end{array}
$$ What is the transition matrix from basis $\{\alpha_{1}, \, \alpha_{2}, \, \alpha_{s} \}$ to basis $\{\beta_{1}, \beta_{2}, \beta_{3} \}$ in $R^{3}$ ?

Options:
A. $$
\frac{1} {6} \! \left( \! \! \begin{array} {c c c} {{{0}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
B. $$
\frac{1} {5} \! \left( \! \! \begin{array} {c c c} {{{1}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
C. $$
\frac{1} {4} \! \left( \! \! \begin{array} {c c c} {{{0}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
D. $$
\frac{1} {2} \! \left( \! \! \begin{array} {c c c} {{{0}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
E. $$
\frac{1} {2} \! \left( \! \! \begin{array} {c c c} {{{1}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
F. $$
\frac{1} {4} \! \left( \! \! \begin{array} {c c c} {{{1}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
G. $$
\frac{1} {3} \! \left( \! \! \begin{array} {c c c} {{{0}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
H. $$
\frac{1} {5} \! \left( \! \! \begin{array} {c c c} {{{0}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
I. $$
\frac{1} {6} \! \left( \! \! \begin{array} {c c c} {{{1}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
J. $$
\frac{1} {3} \! \left( \! \! \begin{array} {c c c} {{{1}}} & {{{0}}} & {{{3}}} \\ {{{2}}} & {{{-4}}} & {{{0}}} \\ {{{2}}} & {{{2}}} & {{{3}}} \\ \end{array} \! \! \right)
$$
Answer: A
$f ( x_{1}, x_{2} )=5 x_{1}+5 x_{2}-4 x_{1} x_{2}$ .Under the condition $\left\| X \right\|=1$, what is the maximum value of the quadratic form $f$ and the unit vector at which this maximum is attained?

Options:
A. $$8
$$
B. $$6
$$
C. $$2
$$
D. $$5
$$
E. $$10
$$
F. $$7
$$
G. $$4
$$
H. $$1
$$
I. $$9
$$
J. $$3
$$
Answer: F
Let $A=\left( \begin{matrix} {{{0}}} & {{{-1}}} & {{{}}} & {{{4}}} \\ {{{-1}}} & {{{}}} & {{{3}}} & {{{}}} & {{{a}}} \\ {{{4}}} & {{{}}} & {{{a}}} & {{{}}} & {{{0}}} \\ \end{matrix} \right)$ be such that there exists an orthogonal matrix $Q$ for which $Q^T A Q = \Lambda$ results in a diagonal matrix. If the first column of $Q$ is ${\frac{1} {\sqrt{6}}} ( 1, 2, 1 ) T$ , what are the values in the diagonal matrix $\Lambda$ ?

Options:
A. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{1}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
B. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{3}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
C. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{3}} \\ \end{matrix} \right)
$$
D. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{1}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
E. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{1}} & {{5}} \\ \end{matrix} \right)
$$
F. $$
\left( \begin{matrix} {{4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
G. $$
\left( \begin{matrix} {{-4}} & {{1}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
H. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{1}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
I. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{1}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
J. $$
\left( \begin{matrix} {{-4}} & {{0}} & {{0}} \\ {{0}} & {{2}} & {{0}} \\ {{0}} & {{0}} & {{5}} \\ \end{matrix} \right)
$$
Answer: J
Let the fourth-order square matrix $A$ satisfy condition $| 3 E+A \models0, A A T=2 E, | A|<0$ , then what is an eigenvalue of the adjugate matrix $A^{*}$ of matrix $A$?

Options:
A. $$
\frac{1}{4}
$$
B. $$
\frac{7}{3}
$$
C. $$
\frac{1}{3}
$$
D. $$
\frac{5}{4}
$$
E. $$
\frac{4}{7}
$$
F. $$
\frac{3}{4}
$$
G. $$
\frac{4}{3}
$$
H. $$
\frac{2}{3}
$$
I. $$
\frac{4}{9}
$$
J. $$
\frac{4}{13}
$$
Answer: G
Set $A=\left( \begin{matrix} {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{1}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right), B=P^{-1} A P$, $P$ for third-order invertible matrix, the $B_{2 0 0 4} - 2 A_ {2}$ value is how much?

Options:
A. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{1}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
B. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{1}}} & {{{-1}}} \\ \end{matrix} \right)
$$
C. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
D. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
E. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)
$$
F. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
G. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-3}}} \\ \end{matrix} \right)
$$
H. $$
\left( \begin{matrix} {{{3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{-1}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
I. $$
\left( \begin{matrix} {{{-3}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
J. $$
\left( \begin{matrix} {{{3}}} & {{{1}}} & {{{0}}} \\ {{{0}}} & {{{3}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)
$$
Answer: F
Let $A=\left( \begin{matrix} {{{1}}} & {{{-1}}} & {{{1}}} \\ {{{2}}} & {{{4}}} & {{{a}}} \\ {{{-3}}} & {{{-3}}} & {{{5}}} \\ \end{matrix} \right)$ and the eigenvalue of $A$ be $\lambda_{1}=6\ , \lambda_{2, 3}=2$, what is the value of $a$ ?

Options:
A. $$
- 6
$$
B. $$
2
$$
C. $$
- 1
$$
D. $$
0
$$
E. $$
8
$$
F. $$
- 8
$$
G. $$
- 4
$$
H. $$
6
$$
I. $$
4
$$
J. $$
- 2
$$
Answer: J
Set $A=\left( \begin{matrix} {{{1}}} & {{{2}}} & {{{3}}} \\ {{{-1}}} & {{{g}}} & {{{2}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ \end{matrix} \right)$, and the eigenvalues of $A$ are 1,2,3, then what is the value of $g$ ?

Options:
A. $$1
$$
B. $$2
$$
C. $$-2
$$
D. $$-4
$$
E. $$3
$$
F. $$4
$$
G. $$0
$$
H. $$-1
$$
I. $$5
$$
J. $$-3
$$
Answer: F
If we know that matrix $A=\left( \begin{matrix} {{{2}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{1}}} \\ {{{0}}} & {{{1}}} & {{{i}}} \\ \end{matrix} \right)$ is similar to matrix $B=\left( \begin{matrix} {{{2}}} & {{{0}}} & {{{0}}} \\ {{{0}}} & {{{j}}} & {{{0}}} \\ {{{0}}} & {{{0}}} & {{{-1}}} \\ \end{matrix} \right)$ , what are the values of the constants $i$and $j$?

Options:
A. $$
i=0, j=-1
$$
B. $$
i=0, j=2
$$
C. $$
i=0, j=0
$$
D. $$
i=0, j=-2
$$
E. $$
i=-1, j=0
$$
F. $$
i=0, j=3
$$
G. $$
i=-1, j=1
$$
H. $$
i=0, j=1
$$
I. $$
i=0, j=-3
$$
J. $$
i=1, j=1
$$
Answer: H
The eigenvalues of the 3x3 matrix $A$ are known to be 1, 2, and 3. Therefore, what are the respective eigenvalues of $A^{3}-5A^{2}+7A$ and $A$ ?

Options:
A. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$6,3,2$$
B. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$3,3,2$$
C. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$2,3,2$$
D. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!1, \varphi (3 ) \!=\!3$$
$$6,3,2$$
E. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$1,3,2$$
F. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$4,3,2$$
G. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!1$$
$$6,3,2$$
H. $$\varphi ( 1 ) \!=\!1, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$6,3,2$$
I. $$\varphi ( 1 ) \!=\!3, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!2$$
$$6,3,2$$
J. $$\varphi ( 1 ) \!=\!2, \varphi (2 ) \!=\!2, \varphi (3 ) \!=\!3$$
$$6,3,2$$
Answer: A
Set $\alpha_{{}_{1}}=( 1, 1, 1 ) \tau\,, \alpha_{{}_{2}}=( 0, 2, 5 ) \tau\,, \alpha_{{}_{3}}=( 1, 3, 6 ) \tau\, ; \beta=( 1, 5, 1 1 ) \tau$ .
So what is the dimension of a set of bases of a vector space generated by the set of vectors $\alpha_{1}, \alpha_{2}, \alpha_{3}$ ? And what are the coordinates of the vector $\beta$ with respect to this basis?

Options:
A. $$4; (0, 2)
$$
B. $$2; (1, 2)
$$
C. $$1; (0, 2)
$$
D. $$5; (1, 2)
$$
E. $$1; (1, 2)
$$
F. $$5; (0, 2)
$$
G. $$2; (0, 2)
$$
H. $$3; (0, 2)
$$
I. $$3; (1, 2)
$$
J. $$4; (1, 2)
$$
Answer: B
Set matrix $A=\left( \begin{matrix} {{{2}}} & {{{1}}} & {{{3}}} & {{{4}}} \\ {{{0}}} & {{{2}}} & {{{1}}} & {{{3}}} \\ {{{0}}} & {{{0}}} & {{{2}}} & {{{1}}} \\ {{{0}}} & {{{0}}} & {{{0}}} & {{{2}}} \\ \end{matrix} \right), B=\left( \begin{matrix} {{{2}}} & {{{1}}} & {{{2}}} & {{{6}}} \\ {{{0}}} & {{{3}}} & {{{3}}} & {{{4}}} \\ {{{0}}} & {{{0}}} & {{{4}}} & {{{5}}} \\ {{{0}}} & {{{0}}} & {{{0}}} & {{{5}}} \\ \end{matrix} \right)$ . Then what is the value of $\left| ( E-A^{-1} B ) \tau A \tau\right|$?

Options:
A. $$( -1, 0,-1 ) \tau
$$
B. $$ ( -1, 1, 3 ) \tau~
$$
C. $$ ( 4, 1, 2 ) \tau~
$$
D. $$1
$$
E. $$ ( 1, 1, 2 ) \tau~
$$
F. $$ ( 0, 1, 2 ) \tau~
$$
G. $$ ( 3, 1, 2 ) \tau~
$$
H. $$0
$$
I. $$ ( 0, 2, 2 ) \tau~
$$
J. $$ ( 2, 1, 2 ) \tau~
$$
Answer: H
Let matrix $A=\left( \begin{matrix} {{{1}}} & {{{2}}} & {{{-1}}} & {{{\lambda}}} \\ {{{2}}} & {{{5}}} & {{{\lambda}}} & {{{-1}}} \\ {{{1}}} & {{{1}}} & {{{-6}}} & {{{1 0}}} \\ \end{matrix} \right)$ , and $R(A)=3$. What is the value of $\lambda$ ?

Options:
A. $$\lambda=0
$$
B. $$\lambda=-1
$$
C. $$\lambda=1
$$
D. $$\lambda\neq3
$$
E. $$\lambda\neq2
$$
F. $$\lambda\neq0
$$
G. $$\lambda=2
$$
H. $$\lambda\neq-1
$$
I. $$\lambda\neq1
$$
J. $$\lambda=3
$$
Answer: D
Known $A$ for third-order phalanx, and $\left| A \right|=-2$, then what are the values of $\left| ( 2 A )^{-1} \right|$ , $A^{*}$ and $\left| A^{*}-\frac{1} {2} A^{-1} \right|$ respectively?

Options:
A. $$-\frac{1} {4} ;4;\frac{12 5} {1 6}
$$
B. $$-\frac{1} {20} ;4;\frac{12 5} {1 6}
$$
C. $$-\frac{1} {1 8} ;4;\frac{12 5} {1 6}
$$
D. $$-\frac{1} {1 6} ;4;\frac{12 5} {1 6}
$$
E. $$-\frac{1} {1 0} ;4;\frac{12 5} {1 6}
$$
F. $$-\frac{1} {6} ;4;\frac{12 5} {1 6}
$$
G. $$-\frac{1} {1 2} ;4;\frac{12 5} {1 6}
$$
H. $$-\frac{1} {2} ;4;\frac{12 5} {1 6}
$$
I. $$-\frac{1} {1 4} ;4;\frac{12 5} {1 6}
$$
J. $$-\frac{1} {8} ;4;\frac{12 5} {1 6}
$$
Answer: D
We know that the matrix $\alpha=\left( \begin{matrix} {{{1}}} & {{{2}}} & {{{3}}} \\ \end{matrix} \right), \ \ \beta=\left( \begin{matrix} {{{1}}} & {{{\frac{1} {2}}}} & {{{\frac{1} {3}}}} \\ \end{matrix} \right)$. Let $A=\alpha\tau\beta$ , then what is the value of $A^n$ (where $n$ is a positive integer)?

Options:
A. $$ \left( \begin{matrix} {{{{{-{\bf3}}_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} &- {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{{\bf3}_{n}}}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
B. $$ \left( \begin{matrix} {{{{{{\bf3}}_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{{\bf3}_{n}}}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
C. $$ \left( \begin{matrix} {{{{{-{\bf3}}_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{{\bf3}_{n}}}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
D. $$ \left( \begin{matrix} {{{{{{\bf3}}_{n-1}}}}} & {{{{\frac{{\bf3}_{n-1}}{2} }}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} &- {{{{\frac{{\bf3}_{n}}{2} }}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)$$
E. $$ \left( \begin{matrix} {{{{{{\bf3}}_{n-1}}}}} & {{{{\frac{{\bf3}_{n-1}}{2} }}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{\frac{{\bf3}_{n}}{2} }}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)$$
F. $$ \left( \begin{matrix} {{{{{{\bf3}}_{n-1}}}}} & {{{{\frac{{\bf3}_{n-1}}{2} }}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & -{{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{\frac{{\bf3}_{n}}{2} }}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)$$
G. $$ \left( \begin{matrix} {{{{{-{\bf3}}_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{{\bf3}_{n}}}}} & -{{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
H. $$ \left( \begin{matrix} {{{{{-{\bf3}}_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & -{{{{{\bf3}_{n}}}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
I. $$ \left( \begin{matrix} {{{{{{\bf3}}_{n-1}}}}} & {{{{\frac{{\bf3}_{n-1}}{2} }}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} &- {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{\frac{{\bf3}_{n}}{2} }}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)$$
J. $$ \left( \begin{matrix} {{{{{-{\bf3}}_{n-1}}}}} &- {{{{{\bf3}_{n-1}}}}} & {{{{{\bf3}_{n-2}}}}} \\ {{{{{\bf2} \cdot3_{n-1}}}}} & {{{{{\bf3}_{n-1}}}}} & {{{{{\bf2} \cdot3_{n-1}}}}} \\ {{{{{\bf3}_{n}}}}} & {{{{{\bf3}_{n}}}}} & {{{{\bf3}_{n-1}}}} \\ \end{matrix} \right)
$$
Answer: E
What lies at the heart of a database system?

Options:
A. Database transaction log
B. Data dictionary
C. Information storage unit
D. Information management protocol
E. Data retrieval interface
F. Database query engine
G. Database
H. Database management system
I. Database administrator
J. Data processing unit
Answer: H
In SQL, the user can directly manipulate ().

Options:
A. Data structures and arrays
B. Basic table
C. Full-text indexes and searches
D. Temporary files and folders
E. Basic tables and views
F. Relational schemas and models
G. View
H. External links and sources
I. Stored procedures and functions
J. Basic table or view
Answer: E
Why not use the relational model directly during the database design phase, instead of designing with ODL or E-R diagrams first?

Options:
A. Because the relational model is too simple to express complex inter-entity relationships.
B. Because ODL and E-R diagrams can better support transaction processing.
C. Because the relational model cannot represent many-to-many relationships between entities.
D. Because the relational model cannot represent the attributes of entities.
E. ODL and E-R diagrams are easier for programmers to understand.
F. ODL and E-R diagrams are more suitable for data security and integrity control.
G. Because ODL and E-R diagrams are better suited for physical data storage.
H. Because ODL and E-R diagrams can be more easily converted to other non-relational database models.
I. Relationship model has only a single concept - relations, the lack of like ODL or e-r diagram of auxiliary concept.
J. Because the relational model can only represent one-to-one and one-to-many relationships.
Answer: I
Given the relation $\mathbf{R} \{\mathbf{a} \,, \, \ \mathbf{b} \,, \ \mathbf{c} \}$ with the primary key being $\{\mathbf{a}$ , $\mathbf{b} \}$, the corresponding SQL statement would be _.

Options:
A. create table testl(a char(10),b char(10),c char(10),foreign key(a,b))
B. create table testl(a char(10)primary key,b char(10),c char(10))
C. create table testl(a char(10) primary key,b char(10) primary key, c char(10))
D. create table testl(a char(10),b char(10),c char(10),primary key(a,b,c))
E. create table testl(a char(10),b char(10),c char(10),unique key(a,b))
F. create table testl(a char(10),b char(10),c char(10),constraint pk_testl primary key(a,b))
G. create table testl(char(10),b char(10)primary key,c char(10))
H. create table testl(a char(10),b char(10),c char(10),primary key(a,b))
I. create table testl(a char(10)primary key,b char(10),c char(10),primary key(b))
J. create table testl(a char(10), b char(10), c char(10), key(a,b))
Answer: H
There is a relational pattern Teacher(Tno, Tname, Title, Dno,Dname, Cno, Cname, Credit), which satisfies the function dependency set: {Tno→Tname, Tno→Title, Tno→Dno, Dno→Dname, Tno→Cno, Cno→Cname, Cno→Credit}. The following I~IV gives the primary keys of this relationship pattern, and V ~ VI gives the decomposition scheme of this relationship pattern:
I. Tno
II. Tno, Dno
III. Tno, Cno
IV. Tno, Dno, Cno
V. (Tno, Tname, Title, Dno, Cno), (Dno, Dname), (Cno, Cname, Credit)
VI. (Tno, Tname, Title), (Dno, Dname), (Cno, Cname, Credit)
VII. (Tno, Tname, Title, Dno), (Dno, Dname, Cno), (Cno, Cname, Credit)
The above is the main code of the relational pattern Teacher, and its decomposition satisfies 3NF is ().

Options:
A. only III
B. only VI
C. only I
D. only II
E. Only II and VII
F. Only III and VI
G. Only IV and VII
H. only IV
I. Only I and V
J. only V
Answer: G
Database recovery is based on the use of dumped redundant data. These dumped redundant packets refer to ( ).

Options:
A. Data dictionaries, applications, log files, audit files
B. Redundant databases, application data, log files
C. Data dictionary, application, audit file, database backup copy
D. Redundant databases, audit trails, log backups
E. Backup copies of log files and databases
F. Data dictionary, application logs, database backup
G. Database log files, audit records, application data
H. Audit trails, database snapshots, log archives
I. Data dictionary, application, database backup copy
J. Backup data, application logs and dictionary files
Answer: E
In the architecture of the distributed database system, the data storage site is determined according to the partitioning strategy, which is the ( ) of the distributed database system.

Options:
A. Server allocation model
B. Fragment mode
C. Global structure design
D. Distribution mode
E. Data array system
F. Shard arrangement model
G. Network configuration mode
H. Intra-local mode
I. Component organization pattern
J. Local conceptual mode
Answer: D
A shopping mall wants to find out the attribute characteristics of customers who are inclined to buy HDTV based on shopping records, and the data mining method that can meet this demand is ( ).

Options:
A. Regression analysis
B. Cluster analysis
C. trait analysis
D. web page mining
E. Association analysis
F. Configuration synthesis analysis
G. Variation and deviation analysis
H. Classification analysis
I. Attribute comparison assessment
J. Time series analysis
Answer: H
The data model is composed of three parts: (), (), and ().

Options:
A. Data types, data operations, integrity constraints
B. Data structures, data types, integrity constraints
C. Data structure, data operation, data backup
D. Data encoding, data manipulation, integrity constraints
E. Data structure, data recovery, integrity constraints
F. Data structure, data access, integrity constraints.
G. Data structure, data operation, data security
H. Data structure; Data operation; Integrity constraint
I. Data format, data operation, integrity constraints
J. Data operations, integrity constraints, data storage
Answer: H
Let the database relational schema R=(A, B, C, D, E), have the following functional dependencies: $\mathrm{A} \to\mathrm{B C}, \ \mathrm{D} \to\mathrm{E}, \ \mathrm{C} \to\mathrm{D}$ ;Which of the following factorizations of R are lossless join factorizations of R?
I. (A, B, C)(C, D, E)
II. (A, B)(A, C, D, E)
III. (A, C) (B, C, D, E)
IV. (A, B)(C, D, E)

Options:
A. Only III
B. Only I and IV
C. Only IV
D. None of them are.
E. Only I and II
F. Only I
G. Only I, II and III
H. All of the above
I. Only II
J. Only I and III
Answer: E
In SQL Server 2000, the common database backup methods include full backup, differential backup, and log backup. In order to ensure the reliability of a database, it is necessary to use these three methods to back up the database. The following statement is true ( ).

Options:
A. The cycles of these three types of backup operations must be conducted in a fixed order, and the backups must be stored on the same backup device.
B. The frequency of these three types of backup operations can vary, but they must be backed up on the same backup device.
C. The cycles for these three types of backup operations can differ, but a full backup must be performed first.
D. The periods for these three types of backup operations can vary, and they can be backed up on different backup devices.
E. The cycles of these three types of backup operations must be identical, and they must all be backed up on the same backup device.
F. The frequency of these three types of backup operations can differ, but they must be carried out in the following order: full backup -> differential backup -> log backup.
G. The frequency of these three types of backup operations must be the same, but they can be backed up on different backup devices.
H. The frequency of these three types of backup operations can differ, but a log backup must be performed first.
I. The intervals for these three types of backup operations can be different, but a differential backup must be performed first.
J. These three types of backup operations can have different cycles, but they must be backed up on different disks of the same server.
Answer: D
There are multiple departments and multiple employees in the company, each employee can only belong to one department, and a department can have multiple employees. The contact type from staff to department is ().

Options:
A. Network relationship
B. One to many
C. Many-to-many
D. Set relation
E. One to one
F. One-way dependence
G. Bidirectional dependence
H. Hierarchical relationship
I. Self-correlation
J. many-for-one
Answer: J
In the security control of the database, when the data object has what conditions to obtain authorization, the authorization subsystem is more flexible?

Options:
A. More user rights
B. The larger the range
C. The more diverse the data types
D. The finer the authorization granularity is
E. The smaller the scope
F. The more complex the data classification
G. Large constraint range
H. Longer access control lists
I. The more detailed the constraints
J. The more authorization levels
Answer: E
What is the decision-making criterion that achieves the same result as the maximum expected profit criterion for the same risk decision issue?

Options:
A. Minimum Expected Loss Criterion
B. Minimum Risk Criterion
C. Maximum Return on Investment Criterion
D. Maximum Profit Criterion
E. Minimum Cost Criterion
F. Maximum Likelihood Criterion
G. Maximum Certainty Equivalent Criterion
H. Minimum Regret Criterion
I. Maximum Utility Criterion
J. Minimum Variance Criterion
Answer: A
When using the relative permeability $\mu_{r}$ to characterize the properties of three types of magnetic materials, which of the following statements is correct?

Options:
A. Paramagnetic materials $\mu_{r} > 1$, Diamagnetic materials $\mu_{r} > 1$, Ferromagnetic materials $\mu_{r} \gg 1$
B. Paramagnetic materials $\mu_{r} > 1$, Diamagnetic materials $\mu_{r} < 1$, Ferromagnetic materials $\mu_{r} \gg 1$
C. Paramagnetic materials $\mu_{r} = 1$, Diamagnetic materials $\mu_{r} < 1$, Ferromagnetic materials $\mu_{r} \gg 1$
D. Paramagnetic materials $\mu_{r} > 0$, Diamagnetic materials $\mu_{r} < 0$, Ferromagnetic materials $\mu_{r} \gg 1$
E. Paramagnetic materials $\mu_{r} \approx 1$, Diamagnetic materials $\mu_{r} = 1$, Ferromagnetic materials $\mu_{r} \gg 1$
F. Paramagnetic materials $\mu_{r} \approx 1$, Diamagnetic materials $\mu_{r} < 1$, Ferromagnetic materials $\mu_{r} \gg 1$
G. Paramagnetic materials $\mu_{r} > 0$, Diamagnetic materials $\mu_{r} < 0$, Ferromagnetic materials $\mu_{r} > 1$
H. Paramagnetic materials $\mu_{r} > 1$, Diamagnetic materials $\mu_{r} = 1$, Ferromagnetic materials $\mu_{r} \gg 1$
I. Paramagnetic materials $\mu_{r} > 0$, Diamagnetic materials $\mu_{r} < 1$, Ferromagnetic materials $\mu_{r} \gg 1$
J. Paramagnetic materials $\mu_{r} < 1$, Diamagnetic materials $\mu_{r} > 1$, Ferromagnetic materials $\mu_{r} \gg 1$
Answer: B
Moving a positively charged body A from a distance to a position near an uncharged conductor B will cause the electric potential of conductor B to _____.

Options:
A. Decrease
B. Invert
C. Fluctuate
D. Stabilize
E. Remain unchanged
F. Nullify
G. Equalize
H. Increase
I. Be indeterminate
J. Oscillate
Answer: H
It is known that the energy band of electrons in a one-dimensional lattice can be written as $E \! \left( k \right) \!=\! \frac{\hbar^{2}} {m a^{2}} \! \left ( \frac{7} {8} \!-\! \operatorname{c o s} k a \!+\! \frac{1} {8} \! \operatorname{c o s} 2 k a \right)$ where a is Lattice constant, m is the mass of electron, find: Band gap $\Delta E$=______.

Options:
A. $\frac{ \hbar^{2}} {5m a^{2}}$
B. $\frac{ \hbar^{2}} {3m a^{2}}$
C. $\frac{\hbar^{2}} {4m a^{2}}$
D. $\frac{2 \hbar^{2}} {m a^{2}}$
E. $\frac{3 \hbar^{2}} {m a^{2}}$
F. $\frac{ \hbar^{2}} {m a^{2}}$
G. $\frac{5 \hbar^{2}} {m a^{2}}$
H. $\frac{4 \hbar^{2}} {m a^{2}}$
I. $\frac{ \hbar^{2}} {2m a^{2}}$
J. $\frac{2 \hbar^{2}} {3m a^{2}}$
Answer: D