Question: Given a function \(f(x)\) with domain \(D\), and a set of numbers \(X \subseteq D\). If there exists a positive number \(M\) such that \(|f(x)| \leqslant M\) holds for every \(x \in X\), the function \(f(x)\) is said to be what on \(X\)?

Choices:
- Bounded
- Unbounded
- Monotonic
- Periodic

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Let the domain of the function \(f(x)\) be \(D\), and the interval \(I \subseteq D\). If for any two points \(x_{1}\) and \(x_{2}\) in the interval \(I\), when \(x_{1} < x_{2}\), it always holds that \(f(x_{1}) < f(x_{2})\), then the function \(f(x)\) on the interval \(I\) is called?

Choices:
- monotonically decreasing
- monotonically increasing
- periodic
- bounded

College--Advanced Mathematics--Functions, Limits, Continuity


Question: The domain \(D\) of the function \(f(x)\) is symmetric about the origin. If for any \(x \in D\), it always holds that \(f(-x) = -f(x)\), then \(f(x)\) is called?

Choices:
- Odd function
- Even function
- Neither odd nor even function
- None of the above

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Let the domain of the function \(f(x)\) be \(D\). If there exists a positive number \(l\), such that for any \(x \in D\), \(x + l \in D\) holds, and \(f(x + l) = f(x)\) always holds, then \(f(x)\) is called what?

Choices:
- Periodic function
- Bounded function
- Monotonic function
- Continuous function

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Let \(\left\{x_{n}\right\}\) be a sequence. If there exists a constant \(a\), for any given \(\varepsilon>0\), there always exists a positive integer \(N\), such that for all \(n>N\), the inequality \(\left|x_{n}-a\right|<\varepsilon\) holds, then the sequence \(\left\{x_{n}\right\}\) is called?

Choices:
- Periodic
- Divergent
- Convergent to \(a\)
- Bounded

College--Advanced Mathematics--Functions, Limits, Continuity


Question: If there exists a constant $A$ such that for any given $\varepsilon>0$, there always exists $X>0$, such that when $|x|>X$, the function value $f(x)$ satisfies the inequality $|f(x)-A|<\varepsilon$, then the constant $A$ is called the limit of the function $f(x)$ as $x \rightarrow \infty$, denoted as?

Choices:
- $\lim _{x \rightarrow x_0} f(x)=A$
- $\lim _{x \rightarrow A} f(x)=x_0$
- $\lim _{x \rightarrow 0} f(x)=A$
- $\lim _{x \rightarrow \infty} f(x)=A$

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Among the following descriptions about the limits of sequences and functions, which one is correct?

Choices:
- If $\lim_{x \rightarrow \infty} x_n=a$ and $a>0$, then for all $n$, we have $x_n>0$.
- If $\lim_{x \rightarrow x_0} f(x)=A$ and $A>0$, then there must exist a constant $\delta>0$, such that when $0<|x-x_0|<\delta$, we have $f(x) > 0$.
- If $\lim_{x \rightarrow x_0} f(x)=A$, it is not necessary that there exists a constant $M>0$ and $\delta>0$, such that when $0<|x-x_0|<\delta$, we have $|f(x)| \leqslant M$.
- If a sequence $\{x_n\}$ is convergent, then its limit may not be unique, different from the uniqueness of the limit of a function.

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Let $\alpha$ and $\beta$ be infinitesimals with the same independent variable during a change process. If $\lim \frac{\beta}{\alpha^{k}} = c \neq 0$ and $k>0$, then $\beta$ is an infinitesimal of what order with respect to $\alpha$?

Choices:
- Equivalent
- Lower order
- $k$-th order
- Higher order

College--Advanced Mathematics--Functions, Limits, Continuity


Question: When \(x \rightarrow 0\), which of the following expressions for an equivalent infinitesimal is correct?

Choices:
- \(\mathrm{e}^x - 1 \sim x \ln e\)
- \(\log_a (1 + x) \sim \frac{x}{\ln a}\), where \(a > 0\) and \(a \neq 1\)
- \((1 + x)^\alpha - 1 \sim x\), where \(\alpha \neq 0\)
- \(\arcsin x \sim \frac{x^2}{2}\)

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Regarding the continuity of a function at a certain point, which of the following descriptions is correct?

Choices:
- If the function is continuous from the left at \(x_0\), it must also be continuous from the right at \(x_0\).
- If a function is continuous at \(x_0\), it is necessary that \(\lim_{x \rightarrow x_0} f(x) = f(x_0^{-})\).
- If the function is continuous from the right at \(x_0\), it must be continuous at \(x_0\).
- A function is continuous at point \(x_0\) if and only if it is continuous from both the left and the right at \(x_0\).

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Regarding the classification of discontinuity points in functions, which of the following descriptions is correct?

Choices:
- If \(f\left(x_{0}^{-}\right) = f\left(x_{0}^{+}\right)\) and \(f(x)\) is not defined at \(x=x_0\), then \(x_0\) is a removable discontinuity point.
- The first category of discontinuity points includes infinite discontinuity points and oscillating discontinuity points.
- An infinite discontinuity point means that either \(f\left(x_{0}^{-}\right)\) or \(f\left(x_{0}^{+}\right)\) is at least \(\infty\).
- An oscillating discontinuity point refers to the case where the limit value of the function at \(x_0\) is infinite.

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Regarding the properties of continuous functions on a closed interval, which of the following statements is correct?

Choices:
- If function \(f(x)\) is continuous on the closed interval \([a, b]\), then the function cannot achieve its maximum and minimum values in this interval.
- If function \(f(x)\) is continuous on the closed interval \([a, b]\), then the function may be unbounded in this interval.
- If function \(f(x)\) is continuous on the closed interval \([a, b]\) and \(f(a) = f(b)\), then there must exist \(\xi \in (a, b)\) such that \(f(\xi) = f(a)\).
- The Intermediate Value Theorem states that if function \(f(x)\) is continuous on the closed interval \([a, b]\), then for any \(C\) between \(f(a)\) and \(f(b)\), there must exist \(\xi \in (a, b)\) such that \(f(\xi) = C\).

College--Advanced Mathematics--Functions, Limits, Continuity


Question: Regarding the limit of sequences and functions, which of the following statements is correct?

Choices:
- \(\lim_{x \rightarrow 0}(1+x)^{\frac{1}{x}}\) is not equal to \(\mathrm{e}\).
- If the sequence \(\{x_n\}\) is monotonically increasing and bounded above, then \(\lim_{n \rightarrow \infty} x_n\) might not exist.
- For the sequences \(\{x_n\}\), \(\{y_n\}\), \(\{z_n\}\), if there exists \(n_0\) such that for \(n > n_0\), \(x_n \leq y_n \leq z_n\) and \(\lim_{n \rightarrow \infty} x_n = \lim_{n \rightarrow \infty} z_n = a\), then \(\lim_{n \rightarrow \infty} y_n\) must equal \(a\).
- \(\lim_{x \rightarrow 0} \frac{\sin x}{x}\) might not equal 1.

College--Advanced Mathematics--Functions, Limits, Continuity


Question: In which of the following cases is $x_{0}$ the derivative of the function $y=f(x)$?

Choices:
- The left derivative exists.
- Both the left and right derivatives exist but are not equal.
- Both the left and right derivatives exist and are equal.
- The right derivative exists.

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: There is a relationship between derivative and differential. For a univariate function, which of the following statements is correct:

Choices:
- The function being differentiable is a subset of the function being derivable
- The function being differentiable is unrelated to the function being derivable
- The function being differentiable is equivalent to the function being derivable
- The function being derivable is a subset of the function being differentiable

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: For $a>0, a \neq 1$, the derivative of the function $\left(a^{x}\right)^{\prime}=$ is:

Choices:
- $a^{x} \ln a$
- $a^{x}$
- $a$
- $\ln a$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: The derivatives of functions $u,v$ exist within their domain, and $v \neq 0$, then the derivative of $\left(\frac{u}{v}\right)^{\prime}$ is?

Choices:
- $\frac{u^{\prime} v-u v^{\prime}}{v}$
- $u^{\prime} v-u v^{\prime}$
- $\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$
- $u^{\prime}v^{\prime}$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: Let $y=f(u)$, and $u=g(x)$, and both $f(u)$ and $g(x)$ are differentiable, then what is the derivative of the composite function $y=f(g(x))$?

Choices:
- $f^{\prime}(x) \cdot g^{\prime}(x)$
- $f^{\prime}(g(x)) \cdot g^{\prime}(g(x))$
- $f(g(x)) \cdot g^{\prime}(x)$
- $f^{\prime}(g(x)) \cdot g^{\prime}(x)$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If the function $y=f(x)$ is monotonic, derivable within the interval $I_{x}$, and $f^{\prime}(x) \neq 0$, then its inverse function $x=f^{-1}(y)$ is also derivable within the interval $I_{y}=\{y \mid y=f(x), x \in I_{x}\}$, and $\left[f^{-1}(y)\right]^{\prime}$ is.

Choices:
- $f^{\prime}(x)$
- $\frac{1}{f^{\prime}(x)}$
- $f(x)$
- $\frac{1}{f(x)}$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If a function is defined by the parametric equations $\left\{\begin{array}{l}x=\varphi(t), \\ y=\psi(t)\end{array}\right.$, where $\varphi(t), \psi(t)$ are both differentiable up to the second order, and $\varphi^{\prime}(t) \neq 0$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is:

Choices:
- $\frac{\psi(t)}{\varphi^{\prime}(t)}$
- $\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}$
- $\frac{\psi(t)}{\varphi(t)}$
- $\frac{\psi^{\prime}(t)}{\varphi(t)}$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If the functions $u,v$ are both differentiable, then $\mathrm{d}(uv)$ is

Choices:
- $v \mathrm{d}u + u \mathrm{d}v$
- $\mathrm{d}u \mathrm{d}v$
- $v \mathrm{d}u + uv$
- $vu + u \mathrm{d}v$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: Given that the formula for curvature is $K=\frac{\left|y^{\prime \prime}\right|}{\left[1+\left(y^{\prime}\right)^{2}\right]^{\frac{3}{2}}}$, and let the curve $y=f(x)$ have a curvature of $K(K \neq 0)$ at point $M(x, y)$. On the normal line of the curve at point $M$, pick a point $D$ on the concave side. Use $D$ as the center and $\rho$ as the radius to draw a circle. This circle is the osculating circle of the curve at point $M$. What value should $\rho$ take so that the circle and the curve have the same tangent and curvature at point $M$, and have the same concavity nearby $M$?

Choices:
- $\sqrt{K}$
- $\frac{1}{K}$
- $K$
- $\frac{1}{K^2}$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: Let the function $f(x)$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If $f^{\prime}(x) \geq 0$ within $(a, b)$, and the equality holds only at a finite number of points, then $f(x)$ on $[a, b]$ is.

Choices:
- Monotonically increasing but not strictly
- Strictly monotonically decreasing
- Strictly monotonically increasing
- Monotonically decreasing but not strictly

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: Suppose the function $f(x)$ is differentiable at $x_{0}$, and attains an extreme value at $x_{0}$, then

Choices:
- Uncertain
- $\leq 0$
- $\geq 0$
- 0

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: Suppose the function $f(x)$ is continuous on $[a, b]$ and twice differentiable within $(a, b)$. If the curve $y=f(x)$ is concave on $[a, b]$, then on $[a, b]$,

Choices:
- $f^{\prime \prime}(x)=0$
- $f^{\prime \prime}(x)$ is indeterminate
- $f^{\prime \prime}(x)<0$
- $f^{\prime \prime}(x)>0$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If the function $f(x)$ is twice differentiable in the interval $I$, and $x_{0}$ is a point within $I$, and the point $\left(x_{0}, f\left(x_{0}\right)\right)$ is an inflection point of the curve $y=f(x)$, then $f^{\prime \prime}\left(x_{0}\right)$:

Choices:
- Indeterminate
- $\geq 0$
- 0
- $\leq 0$

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If the distance between a moving point $M$ on the curve $y=f(x)$ and a certain fixed line $L$ approaches zero as the point $M$ moves infinitely far away from the origin along the curve, then $L$ is called an asymptote of the curve $y=f(x)$. If $\lim _{x \rightarrow a^{+}} f(x)=\infty$ or $\lim _{x \rightarrow a^{-}} f(x)=\infty$, then the line $x=a$ is called

Choices:
- Uncertain
- Horizontal asymptote
- Vertical asymptote
- Oblique asymptote

College--Advanced Mathematics--Functions, Limits, Continuity


Question: If the function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $f(a)=f(b)$, then how many $\xi(a<\xi<b)$ exist at least, such that $f^{\prime}(\xi)=0$?

Choices:
- 1
- 0
- 2
- 3

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question:
If the function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then how many points at least exist within $(a, b)$ such that $\xi(a<\xi<b)$, the equation
$$
f(b)-f(a)=f^{\prime}(\xi)(b-a)
$$holds true

Choices:
- 2
- 3
- 0
- 1

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If functions $f(x)$ and $F(x)$ are continuous on $[a, b]$ and are differentiable on $(a, b)$, and for any $x \in (a, b), F^{\prime}(x) \neq 0$, then at least how many points $\xi(a<\xi<b)$ exist such that the equation $$\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f^{\prime}(\xi)}{F^{\prime}(\xi)}$$ holds true

Choices:
- 3
- 2
- 1
- 0

College--Advanced Mathematics--Differential Calculus of Single-Variable Functions


Question: If, in the interval $I$, the derivative of a differentiable function $F(x)$ is $f(x)$, then the function $F(x)$ is called what of $f(x)$ (or $f(x) \mathrm{d} x)$ over the interval $I$?

Choices:
- Derivative
- Parent function
- Subfunction
- Antiderivative

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Suppose the function $f(x)$ is bounded on $[a, b]$, and by inserting any number of partition points $$a=x_{0}<x_{1}<x_{2}<\cdots<x_{n-1}<x_{n}=b \text {, }$$
the interval $[a, b]$ is divided into $n$ subintervals $\left[x_{0}, x_{1}\right],\left[x_{1}, x_{2}\right], \cdots,\left[x_{n-1}, x_{n}\right]$, with the lengths of the subintervals being
$$
\Delta x_{1}=x_{1}-x_{0}, \quad \Delta x_{2}=x_{2}-x_{1}, \quad \cdots, \quad \Delta x_{n}=x_{n}-x_{n-1} .
$$
Within each subinterval $\left[x_{i-1}, x_{i}\right]$, a point $\xi_{i}\left(x_{i-1} \leqslant \xi_{i} \leqslant x_{i}\right)$ is chosen, and the product of the function value $f\left(\xi_{i}\right)$ and the length of the subinterval $\Delta x_{i}$, i.e., $f\left(\xi_{i}\right) \Delta x_{i}(i=1,2, \cdots, n)$, is calculated, and the sum is made
$$
S=\sum_{i=1}^{n} f\left(\xi_{i}\right) \Delta x_{i} .
$$
Let $\lambda=\max \left\{\Delta x_{1}, \Delta x_{2}, \cdots, \Delta x_{n}\right\}$, if as $\lambda \rightarrow 0$, this sum's limit always exists, and is independent of the partition of the closed interval $[a, b]$ and the choice of points $\xi_{i}$, then this limit $I$ is called the function $f(x)$ on the interval $[a, b]$'s what?

Choices:
- Derivative
- Calculus
- Definite integral
- Indefinite integral

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Suppose the function $f(x)$ is bounded on the interval $[a, b]$, and any number of points are inserted arbitrarily in $[a, b]$
$$
a=x_{0}<x_{1}<x_{2}<\cdots<x_{n-1}<x_{n}=b \text {, }
$$
dividing the interval $[a, b]$ into $n$ subintervals $\left[x_{0}, x_{1}\right], \left[x_{1}, x_{2}\right], \cdots, \left[x_{n-1}, x_{n}\right]$, with the lengths of these subintervals being respectively
$$
\Delta x_{1}=x_{1}-x_{0}, \quad \Delta x_{2}=x_{2}-x_{1}, \quad \cdots, \quad \Delta x_{n}=x_{n}-x_{n-1} .
$$
In each subinterval $\left[x_{i-1}, x_{i}\right]$, choose any point $\xi_{i}\left(x_{i-1} \leqslant \xi_{i} \leqslant x_{i}\right)$, calculate the product of the function value $f\left(\xi_{i}\right)$ and the length of the subinterval $\Delta x_{i}$, $f\left(\xi_{i}\right) \Delta x_{i}(i=1,2, \cdots, n)$, and compute the sum
$$
S=\sum_{i=1}^{n} f\left(\xi_{i}\right) \Delta x_{i} .
$$
Let $\lambda=\max \left\{\Delta x_{1}, \Delta x_{2}, \cdots, \Delta x_{n}\right\}$. If as $\lambda \rightarrow 0$, this sum's limit always exists, and is independent of how the interval $[a, b]$ is divided and the choice of points $\xi_{i}$, then this limit $I$ is called the definite integral of function $f(x)$ over the interval $[a, b]$, where $f(x)dx$ is known as?

Choices:
- the integrand
- the interval of integration
- the variable of integration
- the function to be integrated

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Suppose $\alpha$ and $\beta$ are constants, then
$$
\int_{a}^{b}[\alpha f(x)+\beta g(x)] \mathrm{d} x=\alpha \int_{a}^{b} f(x) \mathrm{d} x+\beta \int_{a}^{b} g(x) \mathrm{d} x .
$$
What property of definite integrals does this illustrate?

Choices:
- Preservation of Sign
- Linearity
- Identity
- Additivity over intervals

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Given $a<c<b$, then
$$
\int_{a}^{b} f(x) \mathrm{d} x=\int_{a}^{c} f(x) \mathrm{d} x+\int_{c}^{b} f(x) \mathrm{d} x .
$$
What property of definite integration does this illustrate?

Choices:
- Identity property
- Linearity
- Preservation of sign
- Additivity over intervals

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
If on the interval $[a, b]$ $f(x) \equiv 1$, then
$$
\int_{a}^{b} 1 \mathrm{~d} x=? .
$$

Choices:
- $1$
- $b-a$
- $b+a$
- $a-b$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question: The absolute value of the integral of $f(x)$ from $a$ to $b$, $\left|\int_{a}^{b} f(x) \mathrm{d} x\right|$, is <<<Answer>>> the integral from $a$ to $b$ of the absolute value of $f(x)$, $\int_{a}^{b}|f(x)| \mathrm{d} x$.

Choices:
- $\geqslant$
- $\equiv$
- $\leqslant$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Let $M$ and $m$ be the maximum and minimum values of the function $f(x)$ in the interval $[a, b]$, respectively, then
$$
m(b-a) \<<<Answer1>>> \int_{a}^{b} f(x) \mathrm{d} x <<<Answer2>>> M(b-a) .
$$

Choices:
- $\geqslant$,  $\leqslant$
- $\leqslant$, $\geqslant$
- $\leqslant$,  $\leqslant$
- $\geqslant$, $\geqslant$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
If the function $f(x)$ is continuous on the interval $[a, b]$, then there exists at least one point $\xi$ in $[a, b]$ such that
$$
\int_{a}^{b} f(x) \mathrm{d} x=f(\xi)(b-a) \quad(a \leqslant \xi \leqslant b) .
$$
What is the above formula known as?

Choices:
- Mean value theorem for integrals
- L'Hôpital's rule
- All other options are incorrect
- Lagrange's mean value theorem

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question: According to the Mean Value Theorem for integrals, $f(\xi)=$ $\frac{1}{b-a} \int_{a}^{b} f(x) \mathrm{d} x$ is referred to as the what of the function $f(x)$ over the interval $[a, b]$?

Choices:
- Maximum value
- Minimum value
- Average value
- All other options are incorrect

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question: Regarding the choice of $u, v$ in the integral by parts formula $\int u v^{\prime} \mathrm{d} x=u v-\int u^{\prime} v \mathrm{~d} x$ (or $\left.\int u \mathrm{~d} v=u v-\int v \mathrm{~d} u\right)$, if $P_{n}(x)$ is an $n$th degree polynomial, and the form of the integrand is $P_{n}(x) \mathrm{e}^{a x}, P_{n}(x) \sin a x$, or $P_{n}(x) \cos a x$ etc., where $a$ is a non-zero constant, then what is the selection of $u, v$?

Choices:
- $u=P_{n}(x), v^{\prime}=\mathrm{(e-1)}^{a x}, \sin a x, \cos a x$
- $u=P_{n}(x), v^{\prime}=\mathrm{e}^{a x}, \sin a x, \cos a x$
- $u=P_{n-1}(x), v^{\prime}=\mathrm{e}^{a x}, \tan a x$
- $u=P_{n-1}(x), v^{\prime}=\mathrm{e}^{a x}, \sin a x, \cos a x$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question: Regarding the choice of $u, v$ in the integration by parts formula $\int u v^{\prime} \mathrm{d} x=u v-\int u^{\prime} v \mathrm{~d} x$ (or $\left.\int u \mathrm{~d} v=u v-\int v \mathrm{~d} u\right)$, where $P_{n}(x)$ is an $n$-th degree polynomial, if the form of the integrand is $P_{n}(x) \ln ^{m} x, m$ being a positive integer, $P_{n}(x) \arcsin x, P_{n}(x) \arctan x$, etc., then what should be the choice of $u, v$?

Choices:
- $u=\ln ^{(m-1)} x, \arcsin x, \arctan x$, $v^{\prime}=P_{n}(x)$
- $u=\ln ^{(m-1)} x, \arccot x$, $v^{\prime}=P_{n}(x)$
- $u=\lg ^{(m-1)} x, \arcsin x, \arctan x$, $v^{\prime}=P_{n}(x)$
- $u=\ln ^{m} x, \arcsin x, \arctan x$, $v^{\prime}=P_{n}(x)$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question: Regarding the selection of $u, v$ in the integration by parts formula $\int u v' \mathrm{d} x=uv-\int u' v \mathrm{d} x$ (or $\int u \mathrm{d} v=uv-\int v \mathrm{d} u$), let $P_{n}(x)$ be a polynomial of degree $n$. If the integrand is in the form of $\mathrm{e}^{ax} \sin bx, \mathrm{e}^{ax} \cos bx$, where $a, b$ are non-zero constants, what should be the selection of $u, v$?

Choices:
- $u=\mathrm{e}^{ax}, v'=\sin bx, \cos bx$
- $u=\sin ax, \cos ax, v'=\mathrm{e}^{bx}$
- $u=\mathrm{e}^{bx}, v'=\sin ax, \cos ax$
- $u=\tan bx, v'=\mathrm{e}^{ax}$

College--Advanced Mathematics--Integral Calculus of Single-Variable Functions


Question:
Suppose the function $z=f(x, y)$ is defined in some neighborhood of the point $(x_{0}, y_{0})$, when $y$ is fixed at $y_{0}$ and $x$ has an increment $\Delta x$ at $x_{0}$, the corresponding function increment is $f(x_{0}+\Delta x, y_{0})-f(x_{0}, y_{0})$, if

$$
\lim _{\Delta x \rightarrow 0} \frac{f(x_{0}+\Delta x, y_{0})-f(x_{0}, y_{0})}{\Delta x}
$$

exists, then this limit is called the function $z=f(x, y)$ at the point $(x_{0}, y_{0})$ with respect to $x$'s ?

Choices:
- gradient
- differential
- partial derivative
- subgradient

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: The partial derivative of the function $z=f(x, y)$ at the point $(x_{0}, y_{0})$ with respect to $x$ is denoted by?

Choices:
- $\left.\frac{\partial z}{\partial f}\right|_{\substack{x=x_{0} \\ y=y_{0}}}$
- $\left.\frac{\partial z}{\partial x}\right|_{\substack{x=x_{0} \\ y=y_{0}}}$
- $\left.\frac{\partial x}{\partial f}\right|_{\substack{x=x_{0} \\ y=y_{0}}}$
- $\left.\frac{\partial y}{\partial f}\right|_{\substack{x=x_{0} \\ y=y_{0}}}$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: The partial derivative of the function $z=f(x, y)$ at the point $(x_{0}, y_{0})$ with respect to $x$, is denoted by?

Choices:
- $f_{y}^{\prime}(x_{0}, y_{0})$
- $f_{x}^{\prime\prime}(x_{0}, y_{0})$
- $f_{x}(x_{0}, y_{0})$
- $f_{x}^{\prime}(x_{0}, y_{0})$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
Let the function $z=f(x, y)$ be defined in some neighborhood of the point $(x_{0}, y_{0})$. When $x$ is fixed at $x_{0}$ and $y$ has an increment $\Delta y$ at $x_{0}$, the corresponding function increment is $f\left(x_{0}, y_{0}+\Delta y\right)-f\left(x_{0}, y_{0}\right)$. If

$$
\lim _{\Delta y \rightarrow 0} \frac{f\left(x_{0}, y_{0} + \Delta y\right)-f\left(x_{0}, y_{0}\right)}{\Delta y}
$$

exists, then this limit is called the function $z=f(x, y)$ at point $(x_{0}, y_{0})$ with respect to $y$'s

Choices:
- derivative
- subgradient
- gradient
- partial derivative

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Suppose the function $z=f(x, y)$ is defined in some neighborhood of the point $(x_{0}, y_{0})$. When $x$ is fixed at $x_{0}$ and $y$ has an increment $\Delta y$ at $x_{0}$, the corresponding function increment is $f\left(x_{0}, y_{0}+\Delta y\right)-f\left(x_{0}, y_{0}\right)$. If the limit ? exists, then this limit is called the partial derivative of the function $z=f(x, y)$ at the point $(x_{0}, y_{0})$ with respect to $y$

Choices:
- $\n\lim _{\Delta x \rightarrow 0} \frac{f\left(x_{0}, y_{0} + \Delta y\right)-f\left(x_{0}, y_{0}\right)}{\Delta y}\n$
- $\n\lim _{\Delta y \rightarrow 0} \frac{f\left(x_{0}, y_{0} + \Delta x\right)-f\left(x_{0}, y_{0}\right)}{\Delta y}\n$
- $\n\lim _{\Delta y \rightarrow 0} \frac{f\left(x_{0}, y_{0} + \Delta y\right)-f\left(x_{0}, y_{0}\right)}{\Delta x}\n$
- $\n\lim _{\Delta y \rightarrow 0} \frac{f\left(x_{0}, y_{0} + \Delta y\right)-f\left(x_{0}, y_{0}\right)}{\Delta y}\n$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
Suppose the function $z=f(x, y)$ is defined within a neighborhood of point $(x, y)$. If the total increment $\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y)$ at point $(x, y)$ can be expressed as

$$
\Delta z=A \Delta x+B \Delta y+o(\rho),
$$

where $A$ and $B$ depend only on $x$ and $y$ and not on $\Delta x$ and $\Delta y$, with $\rho=$ $\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}$, then the function $z=f(x, y)$ at point $(x, y)$ is considered?

Choices:
- Differentiable
- Non-integrable
- Integrable
- Non-differentiable

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
Suppose the function $z=f(x, y)$ is defined within a certain neighborhood of the point $(x, y)$. If at the point $(x, y)$, the total increment $\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y)$ can be expressed as

?

where $A$ and $B$ do not depend on $\Delta x$ and $\Delta y$, but only on $x$ and $y$, and $\rho=$ $\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}$, then the function $z=f(x, y)$ is said to be differentiable at the point $(x, y)$

Choices:
- $$\Delta z=Ax+B \Delta y+o(\rho),$$
- $$\Delta z=A \Delta x+B \Delta y+o(\rho),$$
- $$\Delta z=A \Delta x+By+o(\rho),$$
- $$\Delta z=Ax+By+o(\rho),$$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
Assume the function $z=f(x, y)$ is defined within some neighborhood of point $(x, y)$, if the total increment $\Delta z=f(x+\Delta x, y+\Delta y)-f(x, y)$ at point $(x, y)$ can be expressed as

$$
\Delta z=A \Delta x+B \Delta y+o(\rho),
$$

where $A$ and $B$ do not depend on $\Delta x$ and $\Delta y$, but only on $x$ and $y$, and $\rho=$ $\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}$, then $A \Delta x+B \Delta y$ is called the what of function $z=f(x, y)$ at point $(x, y)$, denoted as $\mathrm{d} z$?

Choices:
- total differential
- directional derivative
- derivative
- subdifferential

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: If a function is differentiable at every point within region $D$, what is this function considered to be within $D$?

Choices:
- Integrable
- Differentiable
- Bounded
- Has Extremes

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: If the functions $u=\varphi(t)$ and $v=\psi(t)$ are both derivable at point $t$, and the function $z=f(u, v)$ has continuous partial derivatives at the corresponding point $(u, v)$, then is the composite function $z=f[\varphi(t), \psi(t)]$ derivable at point $t$?

Choices:
- Not necessarily derivable
- Definitely not derivable
- Cannot determine
- Derivable

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
If the functions $u=\varphi(t)$ and $v=$ $\psi(t)$ are both derivable at point $t$, and the function $z=f(u, v)$ has continuous partial derivatives at the corresponding point $(u, v)$, then the composite function $z=f[\varphi(t), \psi(t)]$ is derivable at point $t$, and it follows that
<<<Answer>>>

Choices:
-
$$
\frac{\mathrm{d} z}{\mathrm{~d} t}=\frac{\partial z}{\partial u} \frac{\mathrm{d} u}{\mathrm{~d} t}\times\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} t}
$$
-
$$
\frac{\mathrm{d} z}{\mathrm{~d} t}=\frac{\partial z}{\partial u} \frac{\mathrm{d} u}{\mathrm{~d} t}-\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} t}
$$
-
$$
\frac{\mathrm{d} z}{\mathrm{~d} t}=\frac{\partial z}{\partial u} \frac{\mathrm{d} u}{\mathrm{~d} t}\cdot\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} t}
$$
-
$$
\frac{\mathrm{d} z}{\mathrm{~d} t}=\frac{\partial z}{\partial u} \frac{\mathrm{d} u}{\mathrm{~d} t}+\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} t}
$$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: If the functions $u=\varphi(x, y)$ and $v=\psi(x, y)$ both possess partial derivatives with respect to $x$ and $y$ at the point $(x, y)$, and the function $z=f(u, v)$ has continuous partial derivatives at the corresponding point $(u, v)$, then what are the two partial derivatives of the composite function $z=f[\varphi(x, y), \psi(x, y)]$ at the point $(x, y)$?

Choices:
- Neither exists
- Only the one with respect to x exists
- Both exist
- Only the one with respect to y exists

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: If the function $u=\varphi(x, y)$ has partial derivatives with respect to $x$ and $y$ at the point $(x, y)$, the function $v=\psi(y)$ is differentiable at point $y$, and the function $z=f(u, v)$ has continuous partial derivatives at the corresponding point $(u, v)$, then the composite function $z=f[\varphi(x, y), \psi(y)]$ has two partial derivatives at point $(x, y)$, and they are?

Choices:
-
$$
\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}, \quad \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}\cdot\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} y}
$$
-
$$
\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}, \quad \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}\times\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} y}
$$
-
$$
\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} +\frac{\partial u}{\partial x}, \quad \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} y}
$$
-
$$
\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial x}, \quad \frac{\partial z}{\partial y}=\frac{\partial z}{\partial u} \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v} \frac{\mathrm{d} v}{\mathrm{~d} y}
$$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Assume the function $F(x, y)$ has continuous partial derivatives in some neighborhood of the point $P\left(x_{0}, y_{0}\right)$, and $F\left(x_{0}, y_{0}\right)=0, F_{y}^{\prime}\left(x_{0}, y_{0}\right) \neq 0$. Then, the equation $F(x, y)=0$ uniquely determines a continuous function $y=f(x)$ with a continuous derivative in some neighborhood of the point $\left(x_{0}, y_{0}\right)$, which satisfies the condition $y_{0}=f\left(x_{0}\right)$. What is the derivative of $y$ with respect to $x$?

Choices:
-
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{F_{y}^{\prime}}{F_{x}^{\prime}} .
$$
-
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{F_{x}^{\prime}}{F_{y}^{\prime}} .
$$
-
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{F_{x}^{\prime}}{F_{y}^{\prime\prime}} .
$$
-
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{F_{x}^{\prime\prime}}{F_{y}^{\prime}} .
$$

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Let the domain of function $f(x, y)$ be $D, P_{0}\left(x_{0}, y_{0}\right)$ be an interior point of $D$. If there exists some neighborhood $U\left(P_{0}\right) \subseteq D$ of $P_{0}$, such that for any point $(x, y)$ within the neighborhood and different from $P_{0}$, it holds that $f(x, y)<f\left(x_{0}, y_{0}\right)$, then the function $f(x, y)$ is said to have what at the point $\left(x_{0}, y_{0}\right)$? $f\left(x_{0}, y_{0}\right)$

Choices:
- A local maximum
- A local minimum
- A global minimum
- A global maximum

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Let the domain of the function $f(x, y)$ be $D$, and $P_{0}(x_{0}, y_{0})$ be an interior point of $D$. If there exists a neighborhood $U(P_{0}) \subseteq D$ of $P_{0}$ such that for any point $(x, y)$ within this neighborhood, different from $P_{0}$, it holds that $f(x, y)<f(x_{0}, y_{0})$, then the point $(x_{0}, y_{0})$ is called what of the function $f(x, y)$?

Choices:
- Maximum point
- Minimum point
- Local minimum point
- Local maximum point

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Suppose the domain of the function $f(x, y)$ is $D, P_{0}(x_{0}, y_{0})$ is an interior point of $D$. If there exists a neighborhood $U(P_{0}) \subseteq D$ of $P_{0}$, such that for any point $(x, y)$ in this neighborhood different from $P_{0}$, it holds that $f(x, y)>f(x_{0}, y_{0})$, then the point $(x_{0}, y_{0})$ is called the ? of the function $f(x, y)$

Choices:
- maximum point
- local maximum point
- local minimum point
- minimum point

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Given a function $f(x, y)$ with domain $D, P_{0}(x_{0}, y_{0})$ is an interior point of $D$. If there exists a neighborhood $U(P_{0}) \subseteq D$ around $P_{0}$ such that for any point $(x, y)$ different from $P_{0}$ within the neighborhood, it holds that $f(x, y)>f(x_{0}, y_{0})$, then the function $f(x, y)$ is said to have what at the point $(x_{0}, y_{0})$? $f(x_{0}, y_{0})$

Choices:
- Local minimum
- Maximum
- Local maximum
- Minimum

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Points $(x_{0}, y_{0})$ that satisfy $f_{x}^{\prime}(x, y)=0, f_{y}^{\prime}(x, y)=0$ simultaneously are called ____ of the function $z=f(x, y)$.

Choices:
- maximum points
- local maximum points
- local minimum points
- stationary points

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question:
$$
L(x, y)=f(x, y)+\lambda \varphi(x, y),
$$

where $\lambda$ is a parameter. Calculate the first-order partial derivatives with respect to $x$ and $y$, and set them to zero, then solve the system of equations together with $\varphi(x, y)=0$:

$$
\left\{\begin{array}{l}
f_{x}^{\prime}(x, y)+\lambda \varphi_{x}^{\prime}(x, y)=0 \\
f_{y}^{\prime}(x, y)+\lambda \varphi_{y}^{\prime}(x, y)=0 \\
\varphi(x, y)=0
\end{array}\right.
$$

Solve the system of equations for $x, y$ and $\lambda$. The $(x, y)$ obtained in this way represents all possible ? of the function $f(x, y)$ under the additional condition $\varphi(x, y)=0$.

Choices:
- extreme points
- type 1 discontinuities
- continuous points
- type 2 discontinuities

College--Advanced Mathematics--Differential Calculus of Multivariable Functions


Question: Which of the following is a property of double integration?

Choices:
- None of the above
-
Let $M$ and $m$ be the maximum and minimum values of $f(x, y)$ over the closed region $D$, respectively, and let $\sigma$ be the area of $D$, then we have

$$
m \sigma < \iint_{D} f(x, y) \mathrm{d} \sigma < M \sigma \text{.}
$$
-
$$
\iint_{D} [\alpha f(x, y) + \beta g(x, y)] \mathrm{d} \sigma = \alpha\beta \iint_{D} f(x, y) \mathrm{d} \sigma + \alpha\beta \iint_{D} g(x, y) \mathrm{d} \sigma.
$$
-
$$
\iint_{D} [\alpha f(x, y) + \beta g(x, y)] \mathrm{d} \sigma = \alpha \iint_{D} f(x, y) \mathrm{d} \sigma + \beta \iint_{D} g(x, y) \mathrm{d} \sigma.
$$


Question: When the integration region $D$ is symmetric about the $y$-axis, which of the following statements is correct:

Choices:
- If $f(x, y)$ is an odd function with respect to $y$, that is $f(-x, y)=-f(x, y)$, then $\iint_{D} f(x, y) \mathrm{d} \sigma=0$.
- If $f(x, y)$ is an odd function with respect to $y$, that is $f(x,-y)=-f(x, y)$, then $\iint_{D} f(x, y) \mathrm{d} \sigma=0$.
- If $f(x, y)$ is an odd function with respect to $y$, that is $f(x,-y)=f(x, y)$, then $\iint_{D} f(x, y) \mathrm{d} \sigma=2 \iint_{D_{2}} f(x, y) \mathrm{d} \sigma$, where $D_{2}$ is the part of $D$ in the half-plane $y \geqslant 0$.
- $\iint_{D} f(x, y) \mathrm{d} \sigma=\iint_{D} f(y, x) \mathrm{d} \sigma$.


Question: Let $f(x, y)$ be a continuous function on a bounded closed region $D$. When the region of integration $D$ is symmetric about the line $y=x$, which of the following statements is correct:

Choices:
- $f(x,-y)=-f(x, y)$
- If $f(x, y)$ is an odd function with respect to $y$, i.e., $f(x,-y)=f(x, y)$, then $\iint_{D} f(x, y) \mathrm{d} \sigma=2 \iint_{D_{2}} f(x, y) \mathrm{d} \sigma$, where $D_{2}$ is the part of $D$ in the half-plane $y \geqslant 0$.
- $f(x,-y)=f(x, y)$
- '$\iint_{D_{1}} f(x, y) \mathrm{d} \sigma=\iint_{D_{2}} f(y, x) \mathrm{d} \sigma$, where $D_{1}$ and $D_{2}$ are respectively the parts of $D$ located above and below the line $y=x$


Question:
Suppose the integral region $D$ can be represented by the inequalities
$$
\varphi_{1}(x) \leqslant y \leqslant \varphi_{2}(x), \quad a \leqslant x \leqslant b
$$

where the functions $\varphi_{1}(x), \varphi_{2}(x)$ are continuous on the interval $[a, b]$, we call this type of region an $\mathrm{X}$ type region. The boundaries of such a region can be represented by $x=a, x=b$ and $y=\varphi_{1}(x)$, $y=\varphi_{2}(x)$, where the two boundary curves $y=\varphi_{1}(x), y=\varphi_{2}(x)$ are functions of $x$. Transforming the double integral into a double integral that integrates over $y$ first and then over $x$ can be transformed into$$
\iint_{D} f(x, y) \mathrm{d} \sigma=$$?

Choices:
-
$$
\iint_{D} f(x, y) \mathrm{d} \sigma=\int_{a}^{b}\left[\int_{\varphi_{2}(x)}^{\varphi_{1}(x)} f(x, y) \mathrm{d} y\right] \mathrm{d} x .
$$
-
$$ \int_{b}^{a}\left[\int_{\varphi_{2}(x)}^{\varphi_{1}(x)} f(x, y) \mathrm{d} y\right] \mathrm{d} x .
$$
-
$$ \int_{a}^{b}\left[\int_{\varphi_{1}(x)}^{\varphi_{2}(x)} f(x, y) \mathrm{d} y\right] \mathrm{d} x .
$$
-
$$ \int_{b}^{a}\left[\int_{\varphi_{1}(x)}^{\varphi_{2}(x)} f(x, y) \mathrm{d} y\right] \mathrm{d} x .
$$


Question:
The Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ of point $M$ are related by $\left\{\begin{array}{l}x=r \cos \theta, \\ y=r \sin \theta .\end{array}\right.$
When calculating double integrals in polar coordinates, the formula is

$$
\iint_{D} f(x, y) \mathrm{d} \sigma=
$$ _______

Choices:
- $$\iint_{D} f(r \sin \theta, r \sin \theta) r \mathrm{~d} r \mathrm{~d} \theta$$
- $$\iint_{D} f(r \cos \theta, -\sin \theta) r \mathrm{~d} r \mathrm{~d} \theta$$
- $$\iint_{D} f(\cos \theta, \sin \theta) r \mathrm{~d} r \mathrm{~d} \theta$$
- $$\iint_{D} f(r \sin \theta, r \cos \theta) r \mathrm{~d} r \mathrm{~d} \theta$$


Question: If a first-order differential equation can be written in the form $g(y) \mathrm{d} y=f(x) \mathrm{d} x$, that is, the differential equation can be written such that one side contains only a function of $y$ and $\mathrm{d} y$, and the other side contains only a function of $x$ and $\mathrm{d} x$, what is the original equation called?

Choices:
- Partial differential equation
- Non-separable variable differential equation.
- Separable variable differential equation.
- None of the above

College--Advanced Mathematics--Ordinary Differential Equations


Question: Solution method for separable differential equations: After writing the original equation in the form $g(y) \mathrm{d} y=f(x) \mathrm{d} x$, what should be done simultaneously to both sides of the equation?

Choices:
- Average
- Sum
- Derive
- Integrate

College--Advanced Mathematics--Ordinary Differential Equations


Question: If a first-order differential equation can be converted into the form $\frac{\mathrm{d} y}{\mathrm{~d} x}=\varphi\left(\frac{y}{x}\right)$, then this equation is called

Choices:
- homogeneous equation
- singular equation
- non-homogeneous equation
- linear equation

College--Advanced Mathematics--Ordinary Differential Equations


Question:
The solution method for the homogeneous equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\varphi\left(\frac{y}{x}\right)$:

Perform the transformation $u=\frac{y}{x}$, then $y=u x, \frac{\mathrm{d} y}{\mathrm{~d} x}=x \frac{\mathrm{d} u}{\mathrm{~d} x}+u$. Thus, the original equation can be transformed to
<<<Answer>>>

After solving by the method of separation of variables, substitute back $u=\frac{y}{x}$ and then solve for $y$.

Choices:
- $$x \frac{\mathrm{d} u}{\mathrm{~d} x}=\varphi(u)+u .$$
- $$x \frac{\mathrm{d} u}{\mathrm{~d} x}=\varphi(u)/u .$$
- $$x \frac{\mathrm{d} u}{\mathrm{~d} x}=\varphi(u)-u .$$
- $$x \frac{\mathrm{d} u}{\mathrm{~d} x}=\varphi(u)*u .$$

College--Advanced Mathematics--Ordinary Differential Equations


Question:
The solution of the homogeneous equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\varphi\left(\frac{y}{x}\right)$:

Perform the transformation $u=\frac{y}{x}$, then $y=u x, \frac{\mathrm{d} y}{\mathrm{~d} x}=x \frac{\mathrm{d} u}{\mathrm{~d} x}+u$. Hence, the original equation can be transformed into
$$
x \frac{\mathrm{d} u}{\mathrm{~d} x}=\varphi(u)-u .
$$

Solve it using () and then substitute back $u=\frac{y}{x}$ and solve for $y$ to find the solution.

Choices:
- Method of Separation of Variables
- General Constant Method
- General Variable Method
- Separation of Constants Method

College--Advanced Mathematics--Ordinary Differential Equations


Question: The equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+p(x) y=q(x)$ is called a()

Choices:
- first-order linear differential equation.
- first-order nonlinear differential equation.
- second-order nonlinear differential equation.
- second-order linear differential equation.

College--Advanced Mathematics--Ordinary Differential Equations


Question: If (), then $\frac{\mathrm{d} y}{\mathrm{~d} x}+p(x) y=0$ is the homogeneous linear equation corresponding to $\frac{\mathrm{d} y}{\mathrm{~d} x}+p(x) y=q(x)$.

Choices:
- $q(x)>0$
- $q(x)\neq0$
- $q(x)=0$
- $q(x)<0$

College--Advanced Mathematics--Ordinary Differential Equations


Question:
() can generally be divided into the following three categories:

(1) $y''=f(x, y')$ type: Let $y'=p$, then $y''=p'$, the equation is transformed into $p'=f(x, p)$.

(2) $y''=f(y, y')$ type: Let $y'=p$, then $y''=p \frac{dp}{dy}$, the equation is transformed into $p \frac{dp}{dy}=f(y, p)$.

(3) $y^{(n)}=f(x)$ type: Perform $n$ indefinite integrations on $f(x)$.

Choices:
- Lower order differential equations that can be reduced
- Higher order differential equations that cannot be reduced
- Lower order differential equations that cannot be reduced
- Higher order differential equations that can be reduced

College--Advanced Mathematics--Ordinary Differential Equations


Question: Find the roots $r_{1}$ and $r_{2}$ of the characteristic equation $r^{2}+p r+q=0$, then write the general solution of the homogenous equation based on the conditions of $r_{1}$ and $r_{2}$.\n\n- If $r_{1}$ and $r_{2}$ are (), then $y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}$;

Choices:
- distinct real roots
- equal real roots
- conjugate complex roots
- none of the above

College--Advanced Mathematics--Ordinary Differential Equations


Question: Find the roots $r_{1}$ and $r_{2}$ of the characteristic equation $r^{2}+p r+q=0$, then write the general solution of the homogeneous equation based on the conditions of $r_{1}$ and $r_{2}$.\n\n- If $r_{1}$ and $r_{2}$ are (), then $y=\left(C_{1}+C_{2} x\right) \mathrm{e}^{r_{1} x}$

Choices:
- equal real roots
- unequal real roots
- conjugate complex roots
- none of the above

College--Advanced Mathematics--Ordinary Differential Equations


Question:
When $f(x)=\mathrm{e}^{\lambda x} P_{m}(x)$, where $\lambda$ is a constant, $P_{m}(x)$ is a polynomial in $x$ of degree $m$, then the differential equation $y^{\prime \prime}+p y^{\prime}+q y=f(x)$ has a particular solution of the form

$$
y^{*}=x^{k} R_{m}(x) \mathrm{e}^{\lambda x}
$$

where $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\lambda$ is not a root of the characteristic equation, k=()

Choices:
- 0
- 1
- 2
- 3

College--Advanced Mathematics--Ordinary Differential Equations


Question:
When $f(x)=\mathrm{e}^{\lambda x} P_{m}(x)$, where $\lambda$ is a constant, $P_{m}(x)$ is a polynomial of $x$ of degree $m$, then $y^{\prime \prime}+p y^{\prime}+q y=f(x)$ has a particular solution of the form

$$
y^{*}=x^{k} R_{m}(x) \mathrm{e}^{\lambda x}
$$

where $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\lambda$ is a root of the characteristic equation, k equals

Choices:
- 0
- 2
- 3
- 1

College--Advanced Mathematics--Ordinary Differential Equations


Question:
When $f(x)=\mathrm{e}^{\lambda x} P_{m}(x)$, where $\lambda$ is a constant, and $P_{m}(x)$ is an $m$-degree polynomial of $x$, $y^{\prime \prime}+p y^{\prime}+q y=f(x)$ has a particular solution of the form

$$
y^{*}=x^{k} R_{m}(x) \mathrm{e}^{\lambda x}
$$

where $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\lambda$ is a repeated root of the characteristic equation, k equals ()

Choices:
- 2
- 0
- 1
- 3

College--Advanced Mathematics--Ordinary Differential Equations


Question: Given $f(x)=\mathrm{e}^{\lambda x}\left[P_{t}(x) \cos \omega x+Q_{n}(x) \sin \omega x\right]$, where $\lambda, \omega$ are constants, $\omega \neq 0, P_{l}(x), and Q_{n}(x)$ are the polynomials of degree $l$ and $n$ respectively, with only one allowed to be zero at a time, $y^{\prime \prime}+py^{\prime}+qy=f(x)$ has a particular solution of the form $$y^{*}=x^{k} \mathrm{e}^{\lambda x}\left[R_{m}^{(1)}(x) \cos \omega x+R_{m}^{(2)} \sin \omega x\right]$$ where $R_{m}^{(1)}(x), R_{m}^{(2)}(x)$ are polynomials of degree $m=\max\{l, n\}$. When $\lambda+\omega \mathrm{i}$ (or $\lambda-\omega \mathrm{i})$ is not a root of the characteristic equation, $k=0$; when $\lambda+\omega \mathrm{i}$ (or $\lambda-\omega \mathrm{i})$ is a single root of the characteristic equation, k=()

Choices:
- 2
- 0
- 3
- 1

College--Advanced Mathematics--Ordinary Differential Equations


Question:
Given $f(x)=\mathrm{e}^{\lambda x}[P_{t}(x) \cos \omega x+Q_{n}(x) \sin \omega x]$, where $\lambda, \omega$ are constants, $\omega \neq 0$, $P_{l}(x)$ and $Q_{n}(x)$ are polynomials of degree $l$ and $n$ respectively, and only one of them can be zero, the differential equation $y''+py'+qy=f(x)$ has a particular solution of the form

$$
y^{*}=x^{k} \mathrm{e}^{\lambda x}[R_{m}^{(1)}(x) \cos \omega x+R_{m}^{(2)}(x) \sin \omega x]$$

where $R_{m}^{(1)}(x), R_{m}^{(2)}(x)$ are polynomials of degree $m$ with $m=\max\{l, n\}$. When $\lambda+\omega \mathrm{i}$ (or $\lambda-\omega \mathrm{i})$ is not a root of the characteristic equation, k equals

Choices:
- 0
- 1
- 2
- 3

College--Advanced Mathematics--Ordinary Differential Equations


Question: If $y_{1}(x)$ and $y_{2}(x)$ are two solutions of the second order homogeneous linear equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$, then () is also a solution of the equation, where $C_{1}, C_{2}$ are arbitrary constants.

Choices:
- $y=C_{1} y_{1}(x) * C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x) ^ C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x)+C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x) / C_{2} y_{2}(x)$

College--Advanced Mathematics--Ordinary Differential Equations


Question: If functions $y_{1}(x)$ and $y_{2}(x)$ are two linearly independent solutions of the second-order homogeneous linear equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$, then <<<Answer>>> is the general solution of the equation, where $C_{1}, C_{2}$ are arbitrary constants.

Choices:
- $y=C_{1} y_{1}(x)+C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x) / C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x) * C_{2} y_{2}(x)$
- $y=C_{1} y_{1}(x) ^ C_{2} y_{2}(x)$

College--Advanced Mathematics--Ordinary Differential Equations


Question: Let $y^{*}(x)$ be a particular solution of the second order nonhomogeneous linear equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)$. $Y(x)$ is the general solution of the corresponding homogeneous equation, then <<<Answer>>> is the general solution of the second order nonhomogeneous linear equation $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=f(x)$.

Choices:
- $y=Y(x)/y^{*}(x)$
- $y=Y(x)-y^{*}(x)$
- $y=Y(x)*y^{*}(x)$
- $y=Y(x)+y^{*}(x)$

College--Advanced Mathematics--Ordinary Differential Equations


Question: If functions $y_{1}(x), y_{2}(x)$ are both solutions to the second-order nonhomogeneous linear differential equation $y^{\prime \prime}+p(x) y^{\prime}+$ $q(x) y=f(x)$, then () is a solution to $y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$.

Choices:
- $y_{1}(x)/y_{2}(x)$
- $y_{1}(x)-y_{2}(x)$
- $y_{1}(x)+y_{2}(x)$
- $y_{1}(x)*y_{2}(x)$

College--Advanced Mathematics--Ordinary Differential Equations


Question:
If a non-homogeneous linear differential equation is in the form

$$
y''+p(x)y'+q(x)y=f_{1}(x)+f_{2}(x),
$$

and $y_{1}^{*}(x)$ and $y_{2}^{*}(x)$ are particular solutions of the equations $y''+p(x)y'+q(x)y=f_{1}(x)$ and $y''+$ $p(x)y'+q(x)y=f_{2}(x)$ respectively, then () is also a particular solution of the equation $y''+p(x)y'$ $+q(x)y=f_{1}(x)+f_{2}(x)$.

Choices:
- $y_{1}^{*}(x)-y_{2}^{*}(x)$
- $y_{1}^{*}(x)/y_{2}^{*}(x)$
- $y_{1}^{*}(x)+y_{2}^{*}(x)$
- $y_{1}^{*}(x)*y_{2}^{*}(x)$

College--Advanced Mathematics--Ordinary Differential Equations


Question: Suppose $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ are both series with positive terms, and $a_{n} \leqslant b_{n}(n$ $=1,2, \cdots)$. If $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ .

Choices:
- Conditionally convergent
- Convergent
- Absolutely convergent
- Divergent


Question: Suppose both $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ are positive series, and if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=l(0 \leqslant l<+\infty)$, and the series $\sum_{n=1}^{\infty} b_{n}$ converges, then the series $\sum_{n=1}^{\infty} a_{n}$<<<Answer>>>;

Choices:
- Conditionally Convergent
- Convergent
- All other options are incorrect
- Divergent


Question: Given $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ are both series with positive terms, if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=l>0$ or $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=+\infty$, and the series $\sum_{n=1}^{\infty} b_{n}$ diverges, then the series $\sum_{n=1}^{\infty} a_{n}$ .

Choices:
- Converges
- Absolutely converges
- Diverges
- Conditionally converges


Question: If $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\rho$ exists (including $\rho=\infty$), then $\rho<1$, series.

Choices:
- diverges
- conditionally converges
- absolutely converges
- can't be sure


Question: If $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\rho$ exists $($ including $\rho=\infty)$, and $\rho>1$, then the series is.

Choices:
- Conditionally convergent
- Convergent
- Divergent
- Absolutely convergent


Question: If $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\rho$ exists (including $\rho=\infty)$, then $\rho=1$, the series.

Choices:
- diverges
- converges
- inconclusive
- not divergent


Question: If $\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=\rho$ exists (including $\rho=\infty$), then $\rho<1$, the series

Choices:
- absolutely converges
- conditionally converges
- can't be sure
- diverges


Question: If $\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=\rho$ exists (including $\rho=\infty$), then $\rho>1$, the series is.

Choices:
- absolutely convergent
- convergent
- conditionally convergent
- divergent


Question: If $\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=\rho$ exists (including $\rho=\infty$), then $\rho=1$, the series is.

Choices:
- indefinite
- divergent
- conditionally convergent
- convergent


Question: If $\lim _{n \rightarrow \infty} n a_{n}=l>0$ (or $\lim _{n \rightarrow \infty} n a_{n}=+\infty$), then the series $\sum_{n=1}^{\infty} a_{n}$ diverges; if $p>1$, and $\lim _{n \rightarrow \infty} n^{p} a_{n}=l(0 \leqslant l<+\infty)$, then the series $\sum_{n=1}^{\infty} a_{n}$.

Choices:
- Conditionally convergent
- Divergent
- Absolutely convergent
- Convergent but not absolutely convergent


Question: If the series $\sum_{n=1}^{\infty} a_{n}$ converges when the absolute values of its terms form a positive series $\sum_{n=1}^{\infty}\left|a_{n}\right|$, then the series $\sum_{n=1}^{\infty} a_{n}$ is called

Choices:
- Absolutely convergent
- Divergent
- Conditionally convergent
- Convergent


Question: If the series $\sum_{n=1}^{\infty} a_{n}$ converges, but the series $\sum_{n=1}^{\infty}\left|a_{n}\right|$ diverges, then the series $\sum_{n=1}^{\infty} a_{n}$ is said to be <<<Answer>>>.

Choices:
- Absolutely convergent
- Convergent
- Conditionally convergent
- Divergent


Question: If the series $\sum_{n=1}^{\infty} a_{n}$ is absolutely convergent, then the series $\sum_{n=1}^{\infty} a_{n}$ necessarily <<<Answer>>>.

Choices:
- converges
- diverges
- conditionally converges
- absolutely converges


Question:
If the alternating series $\sum_{n=1}^{\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ satisfies the conditions:
(1) $a_{n} \geqslant a_{n+1} (n=1,2,3, \cdots)$;
(2) $\lim _{n \rightarrow \infty} a_{n}=0$,
then the series.

Choices:
- diverges
- conditionally converges
- converges
- absolutely converges


Question:
If the alternating series $\sum_{n=1}^{\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ meets the conditions:
(1) $a_{n} \geqslant a_{n+1}(n=1,2,3, \cdots)$;
(2) $\lim _{n \rightarrow \infty} a_{n}=0$,
then the series converges, and its sum $s$ <<<Answer>>> $a_{1}$.

Choices:
- $\leq$
- $\le$
- $\leqslant$
- $\neq$


Question:
If the alternating series $\sum_{n=1}^{\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ satisfies the conditions:
(1) $a_{n} \geqslant a_{n+1}(n=1,2,3, \cdots)$;
(2) $\lim _{n \rightarrow \infty} a_{n}=0$,
then the series converges, and its sum $s$ <<<Answer>>> $a_{1}$.

Choices:
- $\leqslant$
- $\neq$
- $\leq$
- $\le$


Question: If $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=l, 0<l<+\infty$, then the radius of convergence $R$ equals <<<Answer>>>.

Choices:
- $\frac{1}{l}$
- $l$
- $-l$
- $\frac{-1}{l}$


Question: If $\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=l, 0<l<+\infty$, then the radius of convergence $R$ equals <<<Answer>>>.

Choices:
- $-l$
- $\frac{-1}{l}$
- $\frac{1}{l}$
- $l$


Question: The infinite series $\mathrm{e}^{x}$ =.

Choices:
- $\sum{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{1}{n !} x^{n}(-\infty<x<+\infty)$


Question: Infinite series $a^{x}$ =.

Choices:
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n)!} x^{2n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{1}{n!} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(\ln a)^{n}}{n!} x^{n}(-\infty<x<+\infty)$


Question: The infinite series $\sin x$ =.

Choices:
- $\sum{n=0}^{\infty} \frac{1}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$


Question: The infinite series $\cos x$ =.

Choices:
- $\sum{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$
- $\sum{n=0}^{\infty} \frac{1}{n !} x^{n}(-\infty<x<+\infty)$


Question: The infinite series $\frac{1}{1+x}$ =.

Choices:
- $\sum{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty}(-1)^{n} x^{n}(-1<x<1)$
- $\sum{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$


Question: The infinite series $\ln (1+x)$ equals.

Choices:
- $\sum_{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1) !} x^{2n+1}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n) !} x^{2n}(-\infty<x<+\infty)$
- $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^{n}(-\infty<x<+\infty)$


Question: The infinite series $\frac{1}{1+x^{2}}$ equals.

Choices:
- $\sum_{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{1}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$


Question: The infinite series $\arctan x=$ .

Choices:
- $\sum_{n=0}^{\infty} \frac{(\ln a)^{n}}{n !} x^{n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1} x^{2 n+1}(-1 \leqslant x \leqslant 1 \text { )}$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\infty<x<+\infty)$
- $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\infty<x<+\infty)$


Question: Let $f(x, y, z)$ be a bounded function on a bounded closed region $\Omega$ in space. Divide $\Omega$ into $n$ small closed regions $\Delta v{1}, \Delta v{2}, \cdots, \Delta v{n}$, where $\Delta v{i}$ represents the $i$-th small region and also its volume. Choose a point $(\xi{i}, \eta{i}, \zeta{i})$ in each $\Delta v{i}$, make the product $f(\xi{i}, \eta{i}, \zeta{i}) \Delta v{i} (i=1,2, \cdots, n)$, and take the sum $\sum{i=1}^{n} f(\xi{i}, \eta{i}, \zeta{i}) \Delta v{i}$. If, as the maximum diameter among the small closed regions $\lambda \rightarrow 0$, this sum's limit always exists and is independent of the division of the closed region $\Omega$ and the choice of points $(\xi{i}, \eta{i}, \zeta{i})$, then this limit is called the triple integral of the function $f(x, y, z)$ over the closed region $\Omega$, denoted by $\iiint_{n} f(x, y, z) \mathrm{d} v$, that is $\iiint_{\Omega} f(x, y, z) \mathrm{d} v=\lim _{\lambda \rightarrow 0} \sum_{i=1}^{n} f(\xi_{i}, \eta_{i}, \zeta_{i}) \Delta v_{i}, where____is called the integrand?

Choices:
- $\Omega$
- $\mathrm{d} v$
- $f(x, y, z)$
- $ f(x, y, z) \mathrm{d} v$

College--Advanced Mathematics--Integral Calculus of Multivariable Functions


Question:
There is the following relation between two types of curve integrals on the plane curve arc $L$:
\int_{L} P \mathrm{~d} x+Q \mathrm{~d} y=\int_{L}(P \cos \alpha+Q \cos \beta) \mathrm{d} s,
where $\alpha(x, y), \beta(x, y)$ are the angles of the direction of the ______ at the point $(x, y)$ on the directed curve arc $L$.

Choices:
- tangent
- normal
- normal vector
- tangent vector

College--Advanced Mathematics--Integral Calculus of Multivariable Functions


Question:
Let the closed region $D$ be enclosed by the piecewise smooth curve $L$. If the functions $P(x, y)$ and $Q(x, y)$ have _____ on $D$, then
\iint_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) \mathrm{d} x \mathrm{~d} y=\oint_{L} P \mathrm{~d} x+Q \mathrm{~d} y,
where $\hat{L}$ is the positively oriented boundary curve of $D$.

Choices:
- second derivatives
- first-order continuous partial derivatives
- second-order continuous partial derivatives
- first-order derivatives

College--Advanced Mathematics--Integral Calculus of Multivariable Functions


Question: If the integral surface $\Sigma$ is given by the equation $z=$ $z(x, y)$, and the projection region of $\Sigma$ on the $xOy$ plane is $D{xy}$, the function $z=z(x, y)$ has first-order continuous partial derivatives on $D{xy}$, and the integrand $f(x, y, z)$ is continuous on $\Sigma$, then

Choices:
- $\iint_{\Sigma} f(x, y, z) \mathrm{d} S=\iint_{D_{xy}} f(x, y, z(x, y)) \sqrt{1+z_{x}^{\prime}(x, y)+z_{y}^{\prime}(x, y)} \mathrm{~d} x \mathrm{~d} y$ .
- $\iint_{\Sigma} f(x, y, z) \mathrm{d} S=\iint_{D_{xy}} f(x, y, z(x, y)) \sqrt{2+z_{x}^{\prime}(x, y)+z_{y}^{\prime}(x, y)} \mathrm{~d} x \mathrm{~d} y$ .
- $\iint_{\Sigma} f(x, y, z) \mathrm{d} S=\iint_{D_{xy}} f(x, y, z(x, y)) \sqrt{z_{x}^{\prime}(x, y)+z_{y}^{\prime}(x, y)} \mathrm{~d} x \mathrm{~d} y$ .
- $\iint_{\Sigma} f(x, y, z) \mathrm{d} S=\iint_{D_{xy}} f(x, y, z(x, y)) \sqrt{1+\left[z_{x}^{\prime}(x, y)\right]^{2}+\left[z_{y}^{\prime}(x, y)\right]^{2}} \mathrm{~d} x \mathrm{~d} y $.

College--Advanced Mathematics--Integral Calculus of Multivariable Functions


Question:
Suppose $f(x, y)$ is defined and continuous on the curve arc $L$, (2) The situation where curve $L$ is represented by $y=\psi(x)\left(x{0} \leqslant x \leqslant x{1}\right)$: This case can be viewed as a special parametric equation: $x=x, y=\psi(x)\left(x{0} \leqslant x \leqslant x{1}\right)$, thus
\int_{L} f(x, y) \mathrm{d} s=\int_{x_{0}}^{x_{1}} f(x, \psi(x)) \sqrt{1+\left[\psi^{\prime}(x)\right]^{2}} \mathrm{~d} x.

Choices:
- Defined and continuous
- Twice differentiable
- Continuous
- First-order differentiable

College--Advanced Mathematics--Integral Calculus of Multivariable Functions


Question: What happens to the determinant if a row (column) has each of its elements multiplied by the same number and then added to the corresponding elements of another row (column)?

Choices:
- Becomes $\frac{1}{x}$ times the original
- Becomes $1+x$ times the original
- Remains the same
- Becomes $x$ times the original


Question: What is the relationship between a determinant and its transpose determinant $|\boldsymbol{A}|$ and $\left|\boldsymbol{A}^{\top}\right|$?

Choices:
- $|\boldsymbol{A}| = 2\left|\boldsymbol{A}^{\top}\right|$
- $|\boldsymbol{A}| = \frac{1}{\left|\boldsymbol{A}^{\top}\right|}$
- $|\boldsymbol{A}| = \left|\boldsymbol{A}^{\top}\right|$
- $|\boldsymbol{A}| = \left|\boldsymbol{A}^{\top}\right|$ - 1


Question: What change occurs to a determinant after swapping two rows (or columns)?

Choices:
- Changes sign
- Remains unchanged
- Becomes $k$ times the original, where $k$ is the sum of the swapped rows (or columns)
- Becomes $\frac{1}{k}$ times the original, where $k$ is the sum of the swapped rows (or columns)


Question: If all elements in a certain row (or column) of a determinant are multiplied by the same number $k$, then the resultant new determinant is the original determinant multiplied by what?

Choices:
- $k^{n}$ times
- Becomes $\frac{1}{k}$ of the original
- Unchanged
- Becomes $k$ times the original


Question: If in a determinant, two rows (or columns) are in proportion k, what is the value of the determinant?

Choices:
- 1
- k
- 0
- $\frac{1}{k}$


Question:
What is the relationship between the following three matrices?
$$
D_1=\left|\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
\vdots & \vdots & & \vdots \\
a_{i 1}+a_{i 1}^{\prime} & a_{i 2}+a_{i 2}^{\prime} & \cdots & a_{i n}+a_{i n}^{\prime} \\
\vdots & \vdots & & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right|,
$$

$$
D_2=\left|\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
\vdots & \vdots & & \vdots \\
a_{n 1} & a_{i 2} & \cdots & a_{i n} \\
\vdots & \vdots & & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right|
D_3 = \left|\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
\vdots & \vdots & & \vdots \\
a_{i 1}^{\prime} & a_{22}^{\prime} & \cdots & a_{i n}^{\prime} \\
\vdots & \vdots & & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right| .
$$

Choices:
- $D_1 = D_2 * D_3$
- $D_1 = D_2 + D_3$
- No relation
- $D_1 = \frac{D_2}{D_3}$


Question: What happens to a determinant if the elements of one row (or column) are multiplied by a same number k and then added to the corresponding elements of another row (or column)?

Choices:
- Becomes $\frac{1}{k}$ times the original
- Becomes $(k+1)$ times the original
- Remains unchanged
- Becomes k times the original


Question: If $\boldsymbol{A}$ is an $n$th order matrix, when each element of the matrix $\boldsymbol{A}$ is multiplied by $\lambda$, what is the determinant $|\lambda \boldsymbol{A}|$ equal to?

Choices:
- $\lambda^{n}|\boldsymbol{A}|$
- $\lambda^{2}|\boldsymbol{A}|$
- $\lambda|\boldsymbol{A}|$
- $|\boldsymbol{A}|$


Question: If $\boldsymbol{A}, \boldsymbol{B}$ are both $n$th order matrices, what is the determinant $|\boldsymbol{A B}|$ equal to?

Choices:
- $|\boldsymbol{A}| / |\boldsymbol{B}|$
- $|\boldsymbol{A}| + |\boldsymbol{B}|$
- $|\boldsymbol{A}| \cdot |\boldsymbol{B}|$
- $|\boldsymbol{A} - \boldsymbol{B}|$


Question: If $\boldsymbol{A}$ is an $n$th order matrix, then what is the determinant $\left|\boldsymbol{A}^{*}\right|$ of the adjugate matrix $\boldsymbol{A}^{*}$ of $\boldsymbol{A}$?

Choices:
- $|\boldsymbol{A}|^{n}$
- $|\boldsymbol{A}|^{n+1}$
- $|\boldsymbol{A}|^{n-1}$
- $|\boldsymbol{A}|$


Question: If $\boldsymbol{A}$ is an invertible matrix of order $n$, what is the determinant of the inverse matrix $\boldsymbol{A}^{-1}$, $\left|\boldsymbol{A}^{-1}\right|$ equal to?

Choices:
- $|\boldsymbol{A}|$
- $-|\boldsymbol{A}|$
- $|\boldsymbol{A}|^{2}$
- $\frac{1}{|\boldsymbol{A}|}$


Question: If matrices $\boldsymbol{A}, \boldsymbol{B}$ are similar, what is the relationship between their determinants $|\boldsymbol{A}|$ and $|\boldsymbol{B}|$?

Choices:
- There is no fixed relationship
- They are opposite, i.e., $|\boldsymbol{A}|=-|\boldsymbol{B}|$
- They are proportional, i.e., $|\boldsymbol{A}|=k|\boldsymbol{B}|$, where $k \neq 1$
- They are equal, i.e., $|\boldsymbol{A}|=|\boldsymbol{B}|$


Question: If an $n$-order matrix $\boldsymbol{A}$ has $n$ eigenvalues $\lambda_{i}(i=1,2, \cdots, n)$, then what is the determinant $|\boldsymbol{A}|$ equal to?

Choices:
- $\sum_{i=1}^{n} \lambda_{i}$
- 1
- $\max(\lambda_{i})$
- $\prod_{i=1}^{n} \lambda_{i}$


Question: Which type of elementary row operations does not include?

Choices:
- Swapping row $i$ with column $j$
- Multiplying row $i$ by a number $k \neq 0$
- Adding row $j$ multiplied by a number $k$ to row $i$
- Swapping two numbers within the same row


Question: How to define the relationship between elementary column operations and row operations of a matrix?

Choices:
- Elementary column operations of a matrix only include swapping two columns and multiplying by a nonzero constant.
- Elementary column operations of a matrix are completely different from elementary row operations, both in terms of objects and methods of operation.
- Elementary column operations of a matrix are similar to elementary row operations, but the objects of operation are columns instead of rows.
- Elementary column operations of a matrix are the inverse operations of elementary row operations.


Question: Under what condition is the matrix $\boldsymbol{A}$ called an invertible matrix?

Choices:
- There exists a matrix $\boldsymbol{B}$ of the same order such that $\boldsymbol{A B}=\boldsymbol{A}$
- There exists a matrix $\boldsymbol{B}$ of the same order such that $\boldsymbol{A B}=\boldsymbol{B A}$
- There exists a matrix $\boldsymbol{B}$ of the same order such that $\boldsymbol{A B}=\boldsymbol{B}$
- There exists a matrix $\boldsymbol{B}$ of the same order such that $\boldsymbol{A B}=\boldsymbol{I}$


Question: What is the inverse matrix of matrix $\boldsymbol{A}$ denoted as?

Choices:
- $\boldsymbol{A}^{-1}$
- $\boldsymbol{A}^{T}$
- $\boldsymbol{A}^{*}$
- $\boldsymbol{A}^{+}$


Question: How is each element of the adjugate matrix $\boldsymbol{A}^{*}$ obtained?

Choices:
- By taking the modulus of the elements of each row and column of matrix $\boldsymbol{A}$.
- By multiplying the elements of each row and column of matrix $\boldsymbol{A}$.
- By adding up the elements of matrix $\boldsymbol{A}$.
- From the matrix formed by the algebraic cofactors $A_{ij}$ of the elements of matrix $\boldsymbol{A}$.


Question: If matrix $\boldsymbol{A}$ is invertible, what is the value of its determinant $|\boldsymbol{A}|$?

Choices:
- Not equal to 0
- Equal to 0
- Equal to 1
- Cannot be determined


Question: If the determinant $|\boldsymbol{A}| \neq 0$, how can the inverse matrix $\boldsymbol{A}^{-1}$ of the matrix $\boldsymbol{A}$ be represented?

Choices:
- $\boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}^{T}$
- $\boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}^{-1}$
- $\boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}$
- $\boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}^{*}$


Question: How is the rank of matrix $\boldsymbol{A}$ defined?

Choices:
- The rank of matrix $\boldsymbol{A}$ refers to the total sum of elements in matrix $\boldsymbol{A}$.
- The rank of matrix $\boldsymbol{A}$ refers to the number of columns in matrix $\boldsymbol{A}$.
- The rank of matrix $\boldsymbol{A}$ refers to the order $r$ of the highest order non-zero minor in matrix $\boldsymbol{A}$.
- The rank of matrix $\boldsymbol{A}$ refers to the number of non-zero elements in matrix $\boldsymbol{A}.


Question: What is the rank of the zero matrix?

Choices:
- 1
- The number of rows in the matrix
- The number of columns in the matrix
- 0


Question: What is the relationship between the row rank and the column rank of a matrix?

Choices:
- The row rank of a matrix is equal to its column rank.
- There is no fixed relationship between the row rank and column rank of a matrix.
- The row rank of a matrix is always greater than its column rank.
- The column rank of a matrix is always greater than its row rank.


Question: What is a full row rank matrix?

Choices:
- A matrix where all elements are zero.
- A matrix with a rank of 1.
- A matrix whose rank is equal to its number of columns.
- A matrix whose rank is equal to its number of rows is called a full row rank matrix.


Question: What is a full column rank matrix?

Choices:
- A matrix whose rank is equal to its number of rows.
- A matrix where all elements are zero.
- A matrix with a rank of 1.
- A matrix whose rank is equal to the number of its columns is called a full column rank matrix.


Question: If invertible matrices $\boldsymbol{P}, \boldsymbol{Q}$ make $\boldsymbol{P A Q}=\boldsymbol{B}$, what is the relationship between the ranks of matrices $\boldsymbol{A}$ and $\boldsymbol{B}$?

Choices:
- $r(\boldsymbol{A})<r(\boldsymbol{B})$
- $r(\boldsymbol{A})=r(\boldsymbol{B})$
- $r(\boldsymbol{A})>r(\boldsymbol{B})$
- $r(\boldsymbol{A})=2r(\boldsymbol{B})$


Question: What is the range of the rank $r(\boldsymbol{A})$ of the matrix $\boldsymbol{A}_{m \times n}$?

Choices:
- $0 \leqslant r(\boldsymbol{A}) \leqslant m \times n$
- $0 \leqslant r(\boldsymbol{A}) \leqslant \max \{m, n\}$
- $0 \leqslant r(\boldsymbol{A}) \leqslant m + n$
- $0 \leqslant r(\boldsymbol{A}) \leqslant \min \{m, n\}$


Question: What is the relationship between the rank of matrix $\boldsymbol{A}$ and the rank of its transpose $\boldsymbol{A}^{\mathrm{T}}$?

Choices:
- $r(\boldsymbol{A}^{\mathrm{T}}) > r(\boldsymbol{A})$
- $r(\boldsymbol{A}^{\mathrm{T}}) < r(\boldsymbol{A})$
- $r(\boldsymbol{A}^{\mathrm{T}})=2r(\boldsymbol{A})$
- $r(\boldsymbol{A}^{\mathrm{T}})=r(\boldsymbol{A})$


Question: What is the relationship between the rank of a matrix $\boldsymbol{A}$ and its transpose $\boldsymbol{A}^{\mathrm{T}}$ multiplied together versus the rank of $\boldsymbol{A}$ alone?

Choices:
- $r(\boldsymbol{A}) < r\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}\right)$
- $r(\boldsymbol{A})=r\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}\right)=r\left(\boldsymbol{A} \boldsymbol{A}^{\mathrm{T}}\right)$
- $r(\boldsymbol{A}) > r\left(\boldsymbol{A} \boldsymbol{A}^{\mathrm{T}}\right)$
- $r(\boldsymbol{A})=2r\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}\right)$


Question: What is the relationship between the rank of matrix $\boldsymbol{A}+\boldsymbol{B}$ and the ranks of matrices $\boldsymbol{A}$ and $\boldsymbol{B}$?

Choices:
- $r(\boldsymbol{A}+\boldsymbol{B}) = r(\boldsymbol{A})+r(\boldsymbol{B})$
- $r(\boldsymbol{A}+\boldsymbol{B}) > r(\boldsymbol{A})+r(\boldsymbol{B})$
- $r(\boldsymbol{A}+\boldsymbol{B}) \leqslant r(\boldsymbol{A})+r(\boldsymbol{B})$
- $r(\boldsymbol{A}+\boldsymbol{B}) = r(\boldsymbol{A})-r(\boldsymbol{B})$


Question: What is the relationship between the rank of matrix $\boldsymbol{A} \boldsymbol{B}$ and the ranks of matrices $\boldsymbol{A}$ and $\boldsymbol{B}$?

Choices:
- $r(\boldsymbol{A} \boldsymbol{B}) \leqslant \min\{r(\boldsymbol{A}), r(\boldsymbol{B})\}$
- $r(\boldsymbol{A} \boldsymbol{B}) = r(\boldsymbol{A}) \times r(\boldsymbol{B})$
- $r(\boldsymbol{A} \boldsymbol{B}) > \max\{r(\boldsymbol{A}), r(\boldsymbol{B})\}$
- $r(\boldsymbol{A} \boldsymbol{B}) = r(\boldsymbol{A}) + r(\boldsymbol{B})$


Question: If $\boldsymbol{\alpha}, \boldsymbol{\beta}$ are non-zero column vectors, what is the rank of the matrix $\boldsymbol{\alpha} \boldsymbol{\beta}^{\mathrm{T}}$?

Choices:
- 0
- $\min\{r(\boldsymbol{\alpha}), r(\boldsymbol{\beta})\}$
- 1
- 2


Question: If $\boldsymbol{P}, \boldsymbol{Q}$ are invertible, what is the relationship between the rank of the matrix $\boldsymbol{P A Q}$ and the rank of $\boldsymbol{A}$?

Choices:
- $r(\boldsymbol{P A Q})=r(\boldsymbol{A})$
- $r(\boldsymbol{P A Q})=2r(\boldsymbol{A})$
- $r(\boldsymbol{P A Q}) > r(\boldsymbol{A})$
- $r(\boldsymbol{P A Q}) < r(\boldsymbol{A})$


Question: If $\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times l}=\boldsymbol{O}$, what is the maximum value of the sum of the ranks of $\boldsymbol{A}$ and $\boldsymbol{B}$?

Choices:
- $n$
- $m + l$
- $m + n$
- $n + l$


Question: What is the necessary and sufficient condition for the matrix equation $\boldsymbol{A} \boldsymbol{X}=\boldsymbol{B}$ to have a solution?

Choices:
- $r(\boldsymbol{A}) < r(\boldsymbol{A}, \boldsymbol{B})$
- $r(\boldsymbol{A}) > r(\boldsymbol{A}, \boldsymbol{B})$
- $r(\boldsymbol{A})=r(\boldsymbol{B})$
- $r(\boldsymbol{A})=r(\boldsymbol{A}, \boldsymbol{B})$


Question: For an $n(n \geqslant 2)$ order matrix $\boldsymbol{A}$, when the rank of $\boldsymbol{A}$, $r(\boldsymbol{A})$, is equal to $n$, what is the rank of its adjugate matrix $\boldsymbol{A}^{*}$?

Choices:
- $r\left(\boldsymbol{A}^{*}\right)=1$
- $r\left(\boldsymbol{A}^{*}\right)=0$
- $r\left(\boldsymbol{A}^{*}\right)=n$
- $r\left(\boldsymbol{A}^{*}\right)=n-1$


Question: For an $n(n \geqslant 2)$ order matrix $\boldsymbol{A}$, when the rank of $\boldsymbol{A}$, $r(\boldsymbol{A})$, equals $n-1$, what is the rank of its adjugate matrix $\boldsymbol{A}^{*}$?

Choices:
- $r\left(\boldsymbol{A}^{*}\right)=0$
- $r\left(\boldsymbol{A}^{*}\right)=1$
- $r\left(\boldsymbol{A}^{*}\right)=n-1$
- $r\left(\boldsymbol{A}^{*}\right)=n$


Question: For an $n(n \geqslant 2)$ order matrix $\boldsymbol{A}$, when the rank of $\boldsymbol{A}$, $r(\boldsymbol{A})$, is less than $n-1$, what is the rank of its adjugate matrix $\boldsymbol{A}^{*}$?

Choices:
- $r\left(\boldsymbol{A}^{*}\right)=0$
- $r\left(\boldsymbol{A}^{*}\right)=1$
- $r\left(\boldsymbol{A}^{*}\right)=n$
- $r\left(\boldsymbol{A}^{*}\right)=n-1$


Question: How to define the linear dependence of a set of vectors?

Choices:
- If at least one vector in the set is the zero vector, then the set of vectors is said to be linearly dependent
- If there exist not all zero scalars $k_1, k_2, \cdots, k_m$ such that $k_1 \boldsymbol{\alpha}_1 + k_2 \boldsymbol{\alpha}_2 + \cdots + k_m \boldsymbol{\alpha}_m = \mathbf{0}$, then the set of vectors is said to be linearly dependent
- If there exist all zero scalars $k_1, k_2, \cdots, k_m$ such that $k_1 \boldsymbol{\alpha}_1 + k_2 \boldsymbol{\alpha}_2 + \cdots + k_m \boldsymbol{\alpha}_m = \mathbf{0}$, then the set of vectors is said to be linearly dependent
- If there do not exist not all zero scalars $k_1, k_2, \cdots, k_m$ such that $k_1 \boldsymbol{\alpha}_1 + k_2 \boldsymbol{\alpha}_2 + \cdots + k_m \boldsymbol{\alpha}_m = \mathbf{0}$, then the set of vectors is said to be linearly dependent


Question: When is a vector $\boldsymbol{\alpha}$ linearly dependent?

Choices:
- The vector is linearly dependent when $\boldsymbol{\alpha} = \boldsymbol{0}$
- The vector is linearly dependent when $\boldsymbol{\alpha}$ can be expressed as a linear combination of other vectors
- The vector is linearly dependent when $\boldsymbol{\alpha}$ cannot be expressed as a linear combination of other vectors
- The vector is linearly dependent when $\boldsymbol{\alpha} \neq \boldsymbol{0}$


Question: Is a nonzero vector $\boldsymbol{\alpha}$ linearly independent?

Choices:
- Yes, the vector is linearly independent when $\boldsymbol{\alpha} = \boldsymbol{0}$
- No, the vector is linearly independent when $\boldsymbol{\alpha}$ can be expressed as a linear combination of other vectors
- Yes, the vector is linearly independent when $\boldsymbol{\alpha} \neq \boldsymbol{0}$
- No, the vector is linearly dependent when $\boldsymbol{\alpha} \neq \boldsymbol{0}$


Question: What is the necessary and sufficient condition for two vectors $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2$ to be linearly dependent?

Choices:
- The necessary and sufficient condition for two vectors $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2$ to be linearly dependent is that their components are proportional
- The necessary and sufficient condition for two vectors $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2$ to be linearly dependent is that they are both zero vectors
- The necessary and sufficient condition for two vectors $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2$ to be linearly dependent is that their components are not proportional
- The necessary and sufficient condition for two vectors $\boldsymbol{\alpha}_1, \boldsymbol{\alpha}_2$ to be linearly dependent is that they are both not zero vectors


Question: If two vectors $\boldsymbol{\alpha}_{1}$ and $\boldsymbol{\alpha}_{2}$ are linearly dependent, what is their geometric relationship?

Choices:
- They are independent
- They are coplanar but not necessarily collinear
- They are perpendicular
- They are collinear


Question: If three three-dimensional vectors $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$ are linearly dependent, what geometric relationship do they have?

Choices:
- They are mutually perpendicular
- They are coplanar
- They form a triangle
- They are linearly independent


Question: What is the condition for the vector $\boldsymbol{\beta}$ to be linearly represented by the set of vectors $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{l}$?

Choices:
- The vector $\boldsymbol{\beta}$ is a zero vector of the set of vectors $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{l}$.
- There exist constants $k_{1}, k_{2}, \cdots, k_{l}$ such that $k_{1} \boldsymbol{\alpha}_{1}+k_{2} \boldsymbol{\alpha}_{2}+\cdots+k_{l} \boldsymbol{\alpha}_{l}=0$.
- Any vector in the set $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{l}$ cannot be expressed as a linear combination of the other vectors.
- There exist constants $k_{1}, k_{2}, \cdots, k_{l}$ such that $k_{1} \boldsymbol{\alpha}_{1}+k_{2} \boldsymbol{\alpha}_{2}+\cdots+k_{l} \boldsymbol{\alpha}_{l}=\boldsymbol{\beta}$.


Question: What is the relationship between saying that the vector $\boldsymbol{\beta}$ can be linearly represented by the set of vectors $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \cdots, \boldsymbol{\alpha}_{l}$ and the system of equations $\boldsymbol{A x}=\boldsymbol{\beta}$?

Choices:
- It means the system of equations $\boldsymbol{A x}=\boldsymbol{\beta}$ has no solution.
- It means the system of equations $\boldsymbol{A x}=\boldsymbol{\beta}$ has a solution, where the matrix $\boldsymbol{A}$ is formed by the set of vectors.
- It means the determinant of matrix $\boldsymbol{A}$ is zero.
- It means the matrix $\boldsymbol{A}$ is a diagonal matrix.

College--Linear Algebra--Systems of Linear Equations


Question: How to use rank to describe the condition that vector $\boldsymbol{\beta}$ can be linearly represented by the set of vectors $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{l}$?

Choices:
- The condition is that the rank of the set of vectors is equal to the rank of the extended set of vectors, i.e., $r(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{l})=r(\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, ..., \boldsymbol{\alpha}_{l}, \boldsymbol{\beta})$.
- The condition is that the rank of the set of vectors is less than the rank of the extended set of vectors.
- The condition is that the rank of the extended set of vectors is equal to 1.
- The condition is that the rank of the set of vectors is greater than the rank of the extended set of vectors.


Question: What is the necessary and sufficient condition for the system of linear equations $Ax=b$ to have no solution?

Choices:
- $r(A)<r(A, b)$
- $r(A)>r(A, b)$
- $r(A)=r(A, b)$
- $r(A)=r(A, b)=n$

College--Linear Algebra--Systems of Linear Equations


Question: What is the necessary and sufficient condition for the system of linear equations $Ax=b$ to have a unique solution?

Choices:
- $r(A)=r(A, b)<n$
- $r(A)>r(A, b)$
- $r(A)<r(A, b)$
- $r(A)=r(A, b)=n$

College--Linear Algebra--Systems of Linear Equations


Question: What is the necessary and sufficient condition for the system of linear equations $Ax=b$ to have infinitely many solutions?

Choices:
- $r(A)=r(A, b)=n$
- $r(A)=r(A, b)<n$
- $r(A)<r(A, b)$
- $r(A)>r(A, b)$

College--Linear Algebra--Systems of Linear Equations


Question: What is the necessary and sufficient condition for the homogeneous linear equation system $Ax=0$ with $n$ unknowns to have a non-trivial solution?

Choices:
- $r(A)>n$
- $r(A)=n$
- $r(A)=r(A, b)$
- $r(A)<n$

College--Linear Algebra--Systems of Linear Equations


Question: What is the necessary and sufficient condition for the system of linear equations $Ax=b$ to have a solution?

Choices:
- $r(A)<r(A, b)$
- $r(A)=r(A, b)$
- $r(A)=n$
- $r(A)>r(A, b)$

College--Linear Algebra--Systems of Linear Equations


Question: What is the necessary and sufficient condition for the matrix equation $AX=B$ to have a solution?

Choices:
- $r(A)>r(A, B)$
- $r(A)<r(A, B)$
- $r(A)=r(A, B)$
- $r(A, B)<n$

College--Linear Algebra--Systems of Linear Equations


Question: Given the matrix equation $AB=C$, what is the maximum possible value of $r(C)$?

Choices:
- $\min \{r(A), r(B)\}$
- $r(A)+r(B)$
- $r(A) - r(B)$
- $r(A) \cdot r(B)$

College--Linear Algebra--Systems of Linear Equations


Question: What is the fundamental system of solutions of a homogeneous system of linear equations?

Choices:
- The fundamental system of solutions consists of the column vectors of the coefficient matrix of the homogeneous system of linear equations.
- The fundamental system of solutions is a set of maximal independent solution vectors in the solution set of the homogeneous system of linear equations.
- The fundamental system of solutions is a set of minimal independent solution vectors in the solution set of the homogeneous system of linear equations.
- The fundamental system of solutions consists of the row vectors of the coefficient matrix of the homogeneous system of linear equations.

College--Linear Algebra--Systems of Linear Equations


Question: If $\boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{t}$ are the fundamental system of solutions for the homogeneous linear equations system $\boldsymbol{A x}=\mathbf{0}$, how can the general solution of this system be represented?

Choices:
- The general solution can be represented as $\boldsymbol{x}=\boldsymbol{\xi}_{1} \times \boldsymbol{\xi}_{2} \times \cdots \times \boldsymbol{\xi}_{t}$.
- The general solution can be represented as $\boldsymbol{x}=k_{1}^2 \boldsymbol{\xi}_{1}+k_{2}^2 \boldsymbol{\xi}_{2}+\cdots+k_{t}^2 \boldsymbol{\xi}_{t}$, where $k_{1}, k_{2}, \cdots, k_{t}$ are any real numbers.
- The general solution can be represented as $\boldsymbol{x}=k_{1} \boldsymbol{\xi}_{1}+k_{2} \boldsymbol{\xi}_{2}+\cdots+k_{t} \boldsymbol{\xi}_{t}$, where $k_{1}, k_{2}, \cdots, k_{t}$ are any real numbers.
- The general solution can be represented as $\boldsymbol{x}=\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{2}+\cdots+\boldsymbol{\xi}_{t}$.

College--Linear Algebra--Systems of Linear Equations


Question: In the general solution of a homogeneous system of linear equations, what values can the coefficients $k_1, k_2, \cdots, k_t$ take?

Choices:
- The coefficients $k_1, k_2, \cdots, k_t$ can only take integer values.
- The coefficients $k_1, k_2, \cdots, k_t$ can only take positive real values.
- The coefficients $k_1, k_2, \cdots, k_t$ can only take 0 or 1.
- The coefficients $k_1, k_2, \cdots, k_t$ can take any real values.

College--Linear Algebra--Systems of Linear Equations


Question: What is the solution space of a system of $n$ homogeneous linear equations?

Choices:
- $S=\{\boldsymbol{x} \mid \boldsymbol{A x}=\mathbf{1}\}$ is a vector space, known as the solution space of the homogeneous linear equations.
- $S=\{\boldsymbol{x} \mid \boldsymbol{A x}=\mathbf{-1}\}$ is a vector space, known as the solution space of the homogeneous linear equations.
- $S=\{\boldsymbol{x} \mid \boldsymbol{A x}=\mathbf{n}\}$ is a vector space, known as the solution space of the homogeneous linear equations, where b is any positive integer.
- $S=\{\boldsymbol{x} \mid \boldsymbol{A x}=\mathbf{0}\}$ is a vector space, known as the solution space of the homogeneous linear equations.

College--Linear Algebra--Systems of Linear Equations


Question: If $\boldsymbol{\xi}_{1}$ and $\boldsymbol{\xi}_{2}$ are both solutions to the homogeneous linear equation system $\boldsymbol{A x}=\mathbf{0}$, then is their sum $\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{2}$ also a solution to the equation system?

Choices:
- Only when $\boldsymbol{\xi}_{1}=\boldsymbol{\xi}_{2}$ is it a solution.
- No, $\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{2}$ is not a solution to the equation system $\boldsymbol{A x}=\mathbf{0}$.
- Yes, $\boldsymbol{\xi}_{1}+\boldsymbol{\xi}_{2}$ is also a solution to the equation system $\boldsymbol{A x}=\mathbf{0}$.
- Only when $\boldsymbol{\xi}_{1}$ and $\boldsymbol{\xi}_{2}$ are orthogonal is it a solution.

College--Linear Algebra--Systems of Linear Equations


Question: If $\boldsymbol{\xi}$ is a solution to the homogeneous system of linear equations $\boldsymbol{A x}=\mathbf{0}$, is the vector $k \boldsymbol{\xi}$, obtained by multiplying the real number $k$ with $\boldsymbol{\xi}$, also a solution to the system?

Choices:
- Only when $k=0$ is it a solution.
- Only when $k=1$ is it a solution.
- No, $k \boldsymbol{\xi}$ is not a solution to the system of equations $\boldsymbol{A x}=\mathbf{0}$.
- Yes, $k \boldsymbol{\xi}$ is also a solution to the system of equations $\boldsymbol{A x}=\mathbf{0}$.

College--Linear Algebra--Systems of Linear Equations


Question: If $\boldsymbol{\eta}_{1}$ and $\boldsymbol{\eta}_{2}$ are both solutions to the nonhomogeneous linear equation system $\boldsymbol{A x}=\boldsymbol{b}$, then is their difference $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ a solution to the corresponding homogeneous linear equation system $\boldsymbol{A x}=\mathbf{0}$?

Choices:
- It is a solution only when $\boldsymbol{\eta}_{1}$ and $\boldsymbol{\eta}_{2}$ are linearly dependent.
- It is a solution only when $\boldsymbol{\eta}_{1}=\boldsymbol{\eta}_{2}$.
- No, $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ is not a solution to the corresponding homogeneous linear equation system $\boldsymbol{A x}=\mathbf{0}$.
- Yes, $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ is a solution to the corresponding homogeneous linear equation system $\boldsymbol{A x}=\mathbf{0}$.

College--Linear Algebra--Systems of Linear Equations


Question: Let $\boldsymbol{\eta}$ be a solution to the nonhomogeneous linear equations $\boldsymbol{A x}=\boldsymbol{b}$, and let $\boldsymbol{\xi}$ be a solution to the corresponding homogeneous linear equations $\boldsymbol{A x}=\mathbf{0}$. Is the vector $\boldsymbol{\xi}+\boldsymbol{\eta}$ also a solution to the nonhomogeneous equations $\boldsymbol{A x}=\boldsymbol{b}$?

Choices:
- Yes, $\boldsymbol{\xi}+\boldsymbol{\eta}$ is still a solution to the nonhomogeneous linear equations $\boldsymbol{A x}=\boldsymbol{b}$.
- Only when $\boldsymbol{\xi}=\boldsymbol{\eta}$ it is a solution.
- No, $\boldsymbol{\xi}+\boldsymbol{\eta}$ is not a solution to the nonhomogeneous linear equations $\boldsymbol{A x}=\boldsymbol{b}$.
- Only when $\boldsymbol{\xi}$ and $\boldsymbol{\eta}$ are orthogonal is it a solution.

College--Linear Algebra--Systems of Linear Equations


Question: What does the general solution of a non-homogeneous linear system consist of?

Choices:
- The sum of all particular solutions of the non-homogeneous linear system.
- The sum of the general solution of the corresponding homogeneous linear system and a particular solution of the non-homogeneous linear system.
- The difference between any two particular solutions of the non-homogeneous linear system.
- The sum of the general solution of the corresponding homogeneous linear system and all particular solutions of the non-homogeneous linear system.

College--Linear Algebra--Systems of Linear Equations


Question: In solving an $n$ variable linear equation system through elementary row operations, if the rank of coefficient matrix $\boldsymbol{A}$ is $r(\boldsymbol{A})=r$, how should the augmented matrix $\boldsymbol{B}=(\boldsymbol{A}, \boldsymbol{b})$ be manipulated?

Choices:
- Transform the augmented matrix $\boldsymbol{B}$ into a row-reduced echelon form matrix $\widehat{\boldsymbol{B}}$, and the rank of the simplified matrix is r
- Ignore $\boldsymbol{b}$ and only row-reduce $\boldsymbol{A}$
- Convert all non-zero elements to zero elements
- Only perform row operations on the coefficient matrix $\boldsymbol{A}$, keeping $\boldsymbol{b}$ unchanged

College--Linear Algebra--Systems of Linear Equations


Question: When $r(\boldsymbol{A})<r(\boldsymbol{A}, \boldsymbol{b})$, what is the characteristic of the system of linear equations $\boldsymbol{A x}=\boldsymbol{b}$?

Choices:
- The system has infinitely many solutions
- The system has no solution
- The system has a unique solution
- The solution of the system depends on the value of $\boldsymbol{b}$

College--Linear Algebra--Systems of Linear Equations


Question: If $r(\boldsymbol{A})=r(\boldsymbol{A}, \boldsymbol{b})$, how should the augmented matrix $\boldsymbol{B}$ be handled when solving the linear system $\boldsymbol{A x}=\boldsymbol{b}$?

Choices:
- Set all elements of the augmented matrix $\boldsymbol{B}$ to 0
- Set all elements of the augmented matrix $\boldsymbol{B}$ to 1
- Just reduce the coefficient matrix $\boldsymbol{A}$ to row echelon form, ignoring $\boldsymbol{b}$
- Reduce the augmented matrix $\boldsymbol{B}$ to row echelon form

College--Linear Algebra--Systems of Linear Equations


Question: When $r(\boldsymbol{A})=r(\boldsymbol{A}, \boldsymbol{b})=r$, how to determine the non-free variables and free variables in solving the system of linear equations $\boldsymbol{A x}=\boldsymbol{b}$, and write out the general solution?

Choices:
- View the variables corresponding to the first non-zero element of the $r$ non-zero rows as non-free variables, the rest as free variables, and write out the general solution
- View all variables as non-free variables, and write out the unique solution
- Randomly select variables as non-free and free variables, and write out a possible solution
- View all variables as free variables, and write out the general solution

College--Linear Algebra--Systems of Linear Equations


Question: What are the eigenvalues and eigenvectors of a matrix?

Choices:
- If there exists a number $\lambda$ and a non-zero $n$-dimensional column vector $\boldsymbol{\alpha}$ such that $\boldsymbol{A} \boldsymbol{\alpha}=\lambda \boldsymbol{\alpha}$, then $\lambda$ is called an eigenvalue of the matrix $\boldsymbol{A}$, and the non-zero vector $\boldsymbol{\alpha}$ is called an eigenvector of $\boldsymbol{A}$ belonging to the eigenvalue $\lambda$.
- If there exists a vector $\boldsymbol{\alpha}$ such that $\boldsymbol{A} \boldsymbol{\alpha}=\boldsymbol{\alpha}$ holds, then $\boldsymbol{\alpha}$ is called an eigenvalue of the matrix $\boldsymbol{A}$.
- If there exists a number $\lambda$ such that $\boldsymbol{A} \boldsymbol{\alpha}=\boldsymbol{\alpha}$ holds for any number $\lambda$, then $\lambda$ is called an eigenvalue of the matrix $\boldsymbol{A}$.
- If there exists a number $\lambda$ such that $\boldsymbol{A} \boldsymbol{\alpha}=\lambda \boldsymbol{\alpha}$ holds for any vector $\boldsymbol{\alpha}$, then $\lambda$ is called an eigenvector of the matrix $\boldsymbol{A}$.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: What is the relationship between the eigenvalues of a matrix and the trace and determinant of the matrix?

Choices:
- The sum of the eigenvalues is equal to the determinant of the matrix, the product of the eigenvalues is equal to the trace of the matrix.
- The sum of the eigenvalues is unrelated to the trace and determinant of the matrix.
- The product of the eigenvalues is unrelated to the trace and determinant of the matrix.
- The sum of the eigenvalues is equal to the trace of the matrix, the product of the eigenvalues is equal to the determinant of the matrix.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: If $\lambda$ is an eigenvalue of the matrix $\boldsymbol{A}$, then what are $\lambda^{k}$ and $\varphi(\lambda)$ the eigenvalues of?

Choices:
- $\lambda^{k}$ is an eigenvalue of the matrix $\boldsymbol{A}$, $\varphi(\lambda)$ is an eigenvalue of the matrix $\boldsymbol{A}$.
- $\lambda^{k}$ is an eigenvector of matrix $\boldsymbol{A}$, $\varphi(\lambda)$ is an eigenvector of the polynomial $\varphi$ applied to the matrix $\boldsymbol{A}^{k}$.
- $\lambda^{k}$ is an eigenvector of the matrix $\boldsymbol{A}^{k}$, $\varphi(\lambda)$ is an eigenvector of the matrix $\boldsymbol{A}$.
- $\lambda^{k}$ is an eigenvalue of matrix $\boldsymbol{A}^{k}$, $\varphi(\lambda)$ is an eigenvalue of the polynomial $\varphi$ applied to the matrix $\boldsymbol{A}$.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: When a matrix has multiple eigenvalues, are the corresponding eigenvectors linearly independent?

Choices:
- If the eigenvalues are equal, then the corresponding eigenvectors are linearly independent.
- If the eigenvalues are all different, then the corresponding eigenvectors are linearly independent.
- Eigenvectors are always linearly dependent.
- Eigenvectors are always linearly independent.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: Is a linear combination of eigenvectors belonging to the same eigenvalue also an eigenvector of that eigenvalue?

Choices:
- Only when the eigenvalue is 1.
- Yes, a linear combination of eigenvectors belonging to the same eigenvalue is still an eigenvector of that eigenvalue.
- No, a linear combination of eigenvectors belonging to the same eigenvalue is no longer an eigenvector of that eigenvalue.
- Only when the eigenvalue is 0.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: If $\lambda$ is a $k$-fold eigenvalue of the matrix $\boldsymbol{A}$, then what is the maximum number of linearly independent eigenvectors associated with $\lambda$?

Choices:
- At most $k$.
- Only one.
- At most $k+1$.
- At most $k-1$.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: How are two matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ defined to be similar?

Choices:
- If there exists an invertible matrix $\boldsymbol{P}$ such that $\boldsymbol{P}^{-1} \boldsymbol{A} \boldsymbol{P}=\boldsymbol{B}$, then matrix $\boldsymbol{B}$ is called similar to matrix $\boldsymbol{A}$.
- If there exists a matrix $\boldsymbol{P}$ such that $\boldsymbol{P} \boldsymbol{A} \boldsymbol{P}=\boldsymbol{B}$, then matrix $\boldsymbol{B}$ is called similar to matrix $\boldsymbol{A}$.
- If there exists a matrix $\boldsymbol{P}$ such that $\boldsymbol{P} \boldsymbol{A}=\boldsymbol{B} \boldsymbol{P}$, then matrix $\boldsymbol{B}$ is called similar to matrix $\boldsymbol{A}$.
- If matrix $\boldsymbol{A}$ and $\boldsymbol{B}$ have the same eigenvalues, then they are similar.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: When two $n$-th order matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are similar, what properties do they share?

Choices:
- If matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are similar, then they have the same characteristic polynomial and eigenvalues.
- If matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are similar, then they have the same row vectors and column vectors.
- If matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are similar, then they have the same dimensions and basis space.
- If matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are similar, then they have the same determinant and inverse matrix.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: What is the sufficient condition for an $n$-th order matrix $\boldsymbol{A}$ to be similar to a diagonal matrix?

Choices:
- A sufficient condition is all eigenvalues of matrix $\boldsymbol{A}$ are integers.
- A sufficient condition is all eigenvalues of matrix $\boldsymbol{A}$ are positive.
- A sufficient condition is matrix $\boldsymbol{A}$ is invertible.
- Sufficient conditions include the matrix $\boldsymbol{A}$ having $n$ distinct eigenvalues, or the matrix $\boldsymbol{A}$ being a real symmetric matrix.


Question: What is the necessary and sufficient condition for the $n$-th order matrix $\boldsymbol{A}$ to be similar to a diagonal matrix?

Choices:
- The necessary and sufficient condition is that the matrix $\boldsymbol{A}$ is symmetric.
- The necessary and sufficient condition is that the matrix $\boldsymbol{A}$ has $n$ linearly independent eigenvectors, and the number of linearly independent eigenvectors corresponding to each eigenvalue of matrix $\boldsymbol{A}$ is equal to the multiplicity of that eigenvalue.
- The necessary and sufficient condition is that matrix $\boldsymbol{A}$ has $n$ eigenvalues.
- The necessary and sufficient condition is that the determinant of matrix $\boldsymbol{A}$ is non-zero.

College--Linear Algebra--Eigenvalues and Eigenvectors


Question: What are the necessary conditions for two matrices to be similar?

Choices:
- If two matrices are similar, they must have the same rank, the same characteristic polynomial and eigenvalues, the same trace, and the same determinant.
- If two matrices are similar, they must have the same dimensions and determinant.
- If two matrices are similar, they must have the same inverse matrices and eigenvectors.
- If two matrices are similar, they must have the same row vectors and column vectors.

College--Linear Algebra--Eigenvalues and Eigenvectors