| {"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\\)?", "options": ["Bounded", "Unbounded", "Monotonic", "Periodic"], "answer": "A", "topic": "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?", "options": ["monotonically decreasing", "monotonically increasing", "periodic", "bounded"], "answer": "B", "topic": "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?", "options": ["Odd function", "Even function", "Neither odd nor even function", "None of the above"], "answer": "A", "topic": "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?", "options": ["Periodic function", "Bounded function", "Monotonic function", "Continuous function"], "answer": "A", "topic": "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?", "options": ["Periodic", "Divergent", "Convergent to \\(a\\)", "Bounded"], "answer": "C", "topic": "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?", "options": ["$\\lim _{x \\rightarrow x_0} f(x)=A$", "$\\lim _{x \\rightarrow A} f(x)=x_0$", "$\\lim _{x \\rightarrow 0} f(x)=A$", "$\\lim _{x \\rightarrow \\infty} f(x)=A$"], "answer": "D", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "Among the following descriptions about the limits of sequences and functions, which one is correct?", "options": ["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."], "answer": "B", "topic": "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 $\\alpha$?", "options": ["Equivalent", "Lower order", "$k$-th order", "Higher order"], "answer": "C", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "When \\(x \\rightarrow 0\\), which of the following expressions for an equivalent infinitesimal is correct?", "options": ["\\(\\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}\\)"], "answer": "B", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "Regarding the continuity of a function at a certain point, which of the following descriptions is correct?", "options": ["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\\)."], "answer": "D", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "Regarding the classification of discontinuity points in functions, which of the following descriptions is correct?", "options": ["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."], "answer": "A", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "Regarding the properties of continuous functions on a closed interval, which of the following statements is correct?", "options": ["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\\)."], "answer": "D", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "Regarding the limit of sequences and functions, which of the following statements is correct?", "options": ["\\(\\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."], "answer": "C", "topic": "College--Advanced Mathematics--Functions, Limits, Continuity"} |
| {"question": "In which of the following cases is $x_{0}$ the derivative of the function $y=f(x)$?", "options": ["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."], "answer": "C", "topic": "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:", "options": ["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"], "answer": "C", "topic": "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:", "options": ["$a^{x} \\ln a$", "$a^{x}$", "$a$", "$\\ln a$"], "answer": "A", "topic": "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?", "options": ["$\\frac{u^{\\prime} v-u v^{\\prime}}{v}$", "$u^{\\prime} v-u v^{\\prime}$", "$\\frac{u^{\\prime} v-u v^{\\prime}}{v^{2}}$", "$u^{\\prime}v^{\\prime}$"], "answer": "C", "topic": "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 $y=f(g(x))$?", "options": ["$f^{\\prime}(x) \\cdot g^{\\prime}(x)$", "$f^{\\prime}(g(x)) \\cdot g^{\\prime}(g(x))$", "$f(g(x)) \\cdot g^{\\prime}(x)$", "$f^{\\prime}(g(x)) \\cdot g^{\\prime}(x)$"], "answer": "D", "topic": "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.", "options": ["$f^{\\prime}(x)$", "$\\frac{1}{f^{\\prime}(x)}$", "$f(x)$", "$\\frac{1}{f(x)}$"], "answer": "B", "topic": "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:", "options": ["$\\frac{\\psi(t)}{\\varphi^{\\prime}(t)}$", "$\\frac{\\psi^{\\prime}(t)}{\\varphi^{\\prime}(t)}$", "$\\frac{\\psi(t)}{\\varphi(t)}$", "$\\frac{\\psi^{\\prime}(t)}{\\varphi(t)}$"], "answer": "B", "topic": "College--Advanced Mathematics--Differential Calculus of Single-Variable Functions"} |
| {"question": "If the functions $u,v$ are both differentiable, then $\\mathrm{d}(uv)$ is", "options": ["$v \\mathrm{d}u + u \\mathrm{d}v$", "$\\mathrm{d}u \\mathrm{d}v$", "$v \\mathrm{d}u + uv$", "$vu + u \\mathrm{d}v$"], "answer": "A", "topic": "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 $M$?", "options": ["$\\sqrt{K}$", "$\\frac{1}{K}$", "$K$", "$\\frac{1}{K^2}$"], "answer": "B", "topic": "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.", "options": ["Monotonically increasing but not strictly", "Strictly monotonically decreasing", "Strictly monotonically increasing", "Monotonically decreasing but not strictly"], "answer": "B", "topic": "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", "options": ["Uncertain", "$\\leq 0$", "$\\geq 0$", "0"], "answer": "D", "topic": "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 $[a, b]$,", "options": ["$f^{\\prime \\prime}(x)=0$", "$f^{\\prime \\prime}(x)$ is indeterminate", "$f^{\\prime \\prime}(x)<0$", "$f^{\\prime \\prime}(x)>0$"], "answer": "D", "topic": "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 $f^{\\prime \\prime}\\left(x_{0}\\right)$:", "options": ["Indeterminate", "$\\geq 0$", "0", "$\\leq 0$"], "answer": "C", "topic": "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", "options": ["Uncertain", "Horizontal asymptote", "Vertical asymptote", "Oblique asymptote"], "answer": "C", "topic": "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 $f^{\\prime}(\\xi)=0$?", "options": ["1", "0", "2", "3"], "answer": "A", "topic": "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\n$$\nf(b)-f(a)=f^{\\prime}(\\xi)(b-a)\n$$holds true", "options": ["2", "3", "0", "1"], "answer": "D", "topic": "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", "options": ["3", "2", "1", "0"], "answer": "C", "topic": "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 $I$?", "options": ["Derivative", "Parent function", "Subfunction", "Antiderivative"], "answer": "D", "topic": "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 $$a=x_{0}<x_{1}<x_{2}<\\cdots<x_{n-1}<x_{n}=b \\text {, }$$\nthe 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\n$$\n\\Delta x_{1}=x_{1}-x_{0}, \\quad \\Delta x_{2}=x_{2}-x_{1}, \\quad \\cdots, \\quad \\Delta x_{n}=x_{n}-x_{n-1} .\n$$\nWithin 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\n$$\nS=\\sum_{i=1}^{n} f\\left(\\xi_{i}\\right) \\Delta x_{i} .\n$$\nLet $\\lambda=\\max \\left\\{\\Delta x_{1}, \\Delta x_{2}, \\cdots, \\Delta x_{n}\\right\\}$, 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?", "options": ["Derivative", "Calculus", "Definite integral", "Indefinite integral"], "answer": "C", "topic": "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 $[a, b]$\n$$\na=x_{0}<x_{1}<x_{2}<\\cdots<x_{n-1}<x_{n}=b \\text {, }\n$$\ndividing 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\n$$\n\\Delta x_{1}=x_{1}-x_{0}, \\quad \\Delta x_{2}=x_{2}-x_{1}, \\quad \\cdots, \\quad \\Delta x_{n}=x_{n}-x_{n-1} .\n$$\nIn 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\n$$\nS=\\sum_{i=1}^{n} f\\left(\\xi_{i}\\right) \\Delta x_{i} .\n$$\nLet $\\lambda=\\max \\left\\{\\Delta x_{1}, \\Delta x_{2}, \\cdots, \\Delta x_{n}\\right\\}$. 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?", "options": ["the integrand", "the interval of integration", "the variable of integration", "the function to be integrated"], "answer": "A", "topic": "College--Advanced Mathematics--Integral Calculus of Single-Variable Functions"} |
| {"question": "Suppose $\\alpha$ and $\\beta$ are constants, then\n$$\n\\int_{a}^{b}[\\alpha f(x)+\\beta g(x)] \\mathrm{d} x=\\alpha \\int_{a}^{b} f(x) \\mathrm{d} x+\\beta \\int_{a}^{b} g(x) \\mathrm{d} x .\n$$\nWhat property of definite integrals does this illustrate?", "options": ["Preservation of Sign", "Linearity", "Identity", "Additivity over intervals"], "answer": "B", "topic": "College--Advanced Mathematics--Integral Calculus of Single-Variable Functions"} |
| {"question": "Given $a<c<b$, then\n$$\n\\int_{a}^{b} f(x) \\mathrm{d} x=\\int_{a}^{c} f(x) \\mathrm{d} x+\\int_{c}^{b} f(x) \\mathrm{d} x .\n$$\nWhat property of definite integration does this illustrate?", "options": ["Identity property", "Linearity", "Preservation of sign", "Additivity over intervals"], "answer": "D", "topic": "College--Advanced Mathematics--Integral Calculus of Single-Variable Functions"} |
| {"question": "If on the interval $[a, b]$ $f(x) \\equiv 1$, then\n$$\n\\int_{a}^{b} 1 \\mathrm{~d} x=? .\n$$", "options": ["$1$", "$b-a$", "$b+a$", "$a-b$"], "answer": "B", "topic": "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 $f(x)$, $\\int_{a}^{b}|f(x)| \\mathrm{d} x$.", "options": ["$\\geqslant$", "$\\equiv$", "$\\leqslant$", "$=$"], "answer": "C", "topic": "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\n$$\nm(b-a) \\<<<Answer1>>> \\int_{a}^{b} f(x) \\mathrm{d} x <<<Answer2>>> M(b-a) .\n$$", "options": ["$\\geqslant$, $\\leqslant$", "$\\leqslant$, $\\geqslant$", "$\\leqslant$, $\\leqslant$", "$\\geqslant$, $\\geqslant$"], "answer": "C", "topic": "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\n$$\n\\int_{a}^{b} f(x) \\mathrm{d} x=f(\\xi)(b-a) \\quad(a \\leqslant \\xi \\leqslant b) .\n$$\nWhat is the above formula known as?", "options": ["Mean value theorem for integrals", "L'Hôpital's rule", "All other options are incorrect", "Lagrange's mean value theorem"], "answer": "A", "topic": "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 $[a, b]$?", "options": ["Maximum value", "Minimum value", "Average value", "All other options are incorrect"], "answer": "C", "topic": "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 $u, v$?", "options": ["$u=P_{n}(x), v^{\\prime}=\\mathrm{(e-1)}^{a x}, \\sin a x, \\cos a x$", "$u=P_{n}(x), v^{\\prime}=\\mathrm{e}^{a x}, \\sin a x, \\cos a x$", "$u=P_{n-1}(x), v^{\\prime}=\\mathrm{e}^{a x}, \\tan a x$", "$u=P_{n-1}(x), v^{\\prime}=\\mathrm{e}^{a x}, \\sin a x, \\cos a x$"], "answer": "B", "topic": "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 $u, v$?", "options": ["$u=\\ln ^{(m-1)} x, \\arcsin x, \\arctan x$, $v^{\\prime}=P_{n}(x)$", "$u=\\ln ^{(m-1)} x, \\arccot x$, $v^{\\prime}=P_{n}(x)$", "$u=\\lg ^{(m-1)} x, \\arcsin x, \\arctan x$, $v^{\\prime}=P_{n}(x)$", "$u=\\ln ^{m} x, \\arcsin x, \\arctan x$, $v^{\\prime}=P_{n}(x)$"], "answer": "D", "topic": "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 $u, v$?", "options": ["$u=\\mathrm{e}^{ax}, v'=\\sin bx, \\cos bx$", "$u=\\sin ax, \\cos ax, v'=\\mathrm{e}^{bx}$", "$u=\\mathrm{e}^{bx}, v'=\\sin ax, \\cos ax$", "$u=\\tan bx, v'=\\mathrm{e}^{ax}$"], "answer": "A", "topic": "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\n\n$$\n\\lim _{\\Delta x \\rightarrow 0} \\frac{f(x_{0}+\\Delta x, y_{0})-f(x_{0}, y_{0})}{\\Delta x}\n$$\n\nexists, then this limit is called the function $z=f(x, y)$ at the point $(x_{0}, y_{0})$ with respect to $x$'s ?", "options": ["gradient", "differential", "partial derivative", "subgradient"], "answer": "C", "topic": "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?", "options": ["$\\left.\\frac{\\partial z}{\\partial f}\\right|_{\\substack{x=x_{0} \\\\ y=y_{0}}}$", "$\\left.\\frac{\\partial z}{\\partial x}\\right|_{\\substack{x=x_{0} \\\\ y=y_{0}}}$", "$\\left.\\frac{\\partial x}{\\partial f}\\right|_{\\substack{x=x_{0} \\\\ y=y_{0}}}$", "$\\left.\\frac{\\partial y}{\\partial f}\\right|_{\\substack{x=x_{0} \\\\ y=y_{0}}}$"], "answer": "B", "topic": "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?", "options": ["$f_{y}^{\\prime}(x_{0}, y_{0})$", "$f_{x}^{\\prime\\prime}(x_{0}, y_{0})$", "$f_{x}(x_{0}, y_{0})$", "$f_{x}^{\\prime}(x_{0}, y_{0})$"], "answer": "D", "topic": "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\n\n$$\n\\lim _{\\Delta y \\rightarrow 0} \\frac{f\\left(x_{0}, y_{0} + \\Delta y\\right)-f\\left(x_{0}, y_{0}\\right)}{\\Delta y}\n$$\n\nexists, then this limit is called the function $z=f(x, y)$ at point $(x_{0}, y_{0})$ with respect to $y$'s", "options": ["derivative", "subgradient", "gradient", "partial derivative"], "answer": "D", "topic": "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 $y$", "options": ["$\\n\\lim _{\\Delta x \\rightarrow 0} \\frac{f\\left(x_{0}, y_{0} + \\Delta y\\right)-f\\left(x_{0}, y_{0}\\right)}{\\Delta y}\\n$", "$\\n\\lim _{\\Delta y \\rightarrow 0} \\frac{f\\left(x_{0}, y_{0} + \\Delta x\\right)-f\\left(x_{0}, y_{0}\\right)}{\\Delta y}\\n$", "$\\n\\lim _{\\Delta y \\rightarrow 0} \\frac{f\\left(x_{0}, y_{0} + \\Delta y\\right)-f\\left(x_{0}, y_{0}\\right)}{\\Delta x}\\n$", "$\\n\\lim _{\\Delta y \\rightarrow 0} \\frac{f\\left(x_{0}, y_{0} + \\Delta y\\right)-f\\left(x_{0}, y_{0}\\right)}{\\Delta y}\\n$"], "answer": "D", "topic": "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\n\n$$\n\\Delta z=A \\Delta x+B \\Delta y+o(\\rho),\n$$\n\nwhere $A$ 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?", "options": ["Differentiable", "Non-integrable", "Integrable", "Non-differentiable"], "answer": "A", "topic": "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\n\n?\n\nwhere $A$ 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 $(x, y)$", "options": ["$$\\Delta z=Ax+B \\Delta y+o(\\rho),$$", "$$\\Delta z=A \\Delta x+B \\Delta y+o(\\rho),$$", "$$\\Delta z=A \\Delta x+By+o(\\rho),$$", "$$\\Delta z=Ax+By+o(\\rho),$$"], "answer": "B", "topic": "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\n\n$$\n\\Delta z=A \\Delta x+B \\Delta y+o(\\rho),\n$$\n\nwhere $A$ 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 $\\mathrm{d} z$?", "options": ["total differential", "directional derivative", "derivative", "subdifferential"], "answer": "A", "topic": "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 $D$?", "options": ["Integrable", "Differentiable", "Bounded", "Has Extremes"], "answer": "B", "topic": "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 $t$?", "options": ["Not necessarily derivable", "Definitely not derivable", "Cannot determine", "Derivable"], "answer": "D", "topic": "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\n<<<Answer>>>", "options": ["$$\n\\frac{\\mathrm{d} z}{\\mathrm{~d} t}=\\frac{\\partial z}{\\partial u} \\frac{\\mathrm{d} u}{\\mathrm{~d} t}\\times\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} t}\n$$", "$$\n\\frac{\\mathrm{d} z}{\\mathrm{~d} t}=\\frac{\\partial z}{\\partial u} \\frac{\\mathrm{d} u}{\\mathrm{~d} t}-\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} t}\n$$", "$$\n\\frac{\\mathrm{d} z}{\\mathrm{~d} t}=\\frac{\\partial z}{\\partial u} \\frac{\\mathrm{d} u}{\\mathrm{~d} t}\\cdot\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} t}\n$$", "$$\n\\frac{\\mathrm{d} z}{\\mathrm{~d} t}=\\frac{\\partial z}{\\partial u} \\frac{\\mathrm{d} u}{\\mathrm{~d} t}+\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} t}\n$$"], "answer": "D", "topic": "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 $(x, y)$?", "options": ["Neither exists", "Only the one with respect to x exists", "Both exist", "Only the one with respect to y exists"], "answer": "C", "topic": "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?", "options": ["$$\n\\frac{\\partial z}{\\partial x}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x}, \\quad \\frac{\\partial z}{\\partial y}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial y}\\cdot\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} y}\n$$", "$$\n\\frac{\\partial z}{\\partial x}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x}, \\quad \\frac{\\partial z}{\\partial y}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial y}\\times\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} y}\n$$", "$$\n\\frac{\\partial z}{\\partial x}=\\frac{\\partial z}{\\partial u} +\\frac{\\partial u}{\\partial x}, \\quad \\frac{\\partial z}{\\partial y}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial y}+\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} y}\n$$", "$$\n\\frac{\\partial z}{\\partial x}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial x}, \\quad \\frac{\\partial z}{\\partial y}=\\frac{\\partial z}{\\partial u} \\frac{\\partial u}{\\partial y}+\\frac{\\partial z}{\\partial v} \\frac{\\mathrm{d} v}{\\mathrm{~d} y}\n$$"], "answer": "D", "topic": "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 $x$?", "options": ["$$\n\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=-\\frac{F_{y}^{\\prime}}{F_{x}^{\\prime}} .\n$$", "$$\n\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=-\\frac{F_{x}^{\\prime}}{F_{y}^{\\prime}} .\n$$", "$$\n\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=-\\frac{F_{x}^{\\prime}}{F_{y}^{\\prime\\prime}} .\n$$", "$$\n\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=-\\frac{F_{x}^{\\prime\\prime}}{F_{y}^{\\prime}} .\n$$"], "answer": "B", "topic": "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 $\\left(x_{0}, y_{0}\\right)$? $f\\left(x_{0}, y_{0}\\right)$", "options": ["A local maximum", "A local minimum", "A global minimum", "A global maximum"], "answer": "A", "topic": "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 $f(x, y)$?", "options": ["Maximum point", "Minimum point", "Local minimum point", "Local maximum point"], "answer": "D", "topic": "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 $f(x, y)$", "options": ["maximum point", "local maximum point", "local minimum point", "minimum point"], "answer": "C", "topic": "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 $(x_{0}, y_{0})$? $f(x_{0}, y_{0})$", "options": ["Local minimum", "Maximum", "Local maximum", "Minimum"], "answer": "A", "topic": "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 $z=f(x, y)$.", "options": ["maximum points", "local maximum points", "local minimum points", "stationary points"], "answer": "D", "topic": "College--Advanced Mathematics--Differential Calculus of Multivariable Functions"} |
| {"question": "$$\nL(x, y)=f(x, y)+\\lambda \\varphi(x, y),\n$$\n\nwhere $\\lambda$ 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 $\\varphi(x, y)=0$:\n\n$$\n\\left\\{\\begin{array}{l}\nf_{x}^{\\prime}(x, y)+\\lambda \\varphi_{x}^{\\prime}(x, y)=0 \\\\\nf_{y}^{\\prime}(x, y)+\\lambda \\varphi_{y}^{\\prime}(x, y)=0 \\\\\n\\varphi(x, y)=0\n\\end{array}\\right.\n$$\n\nSolve 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 $\\varphi(x, y)=0$.", "options": ["extreme points", "type 1 discontinuities", "continuous points", "type 2 discontinuities"], "answer": "A", "topic": "College--Advanced Mathematics--Differential Calculus of Multivariable Functions"} |
| {"question": "Which of the following is a property of double integration?", "options": ["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\n\n$$\nm \\sigma < \\iint_{D} f(x, y) \\mathrm{d} \\sigma < M \\sigma \\text{.}\n$$", "$$\n\\iint_{D} [\\alpha f(x, y) + \\beta g(x, y)] \\mathrm{d} \\sigma = \\alpha\\beta \\iint_{D} f(x, y) \\mathrm{d} \\sigma + \\alpha\\beta \\iint_{D} g(x, y) \\mathrm{d} \\sigma. \n$$", "$$\n\\iint_{D} [\\alpha f(x, y) + \\beta g(x, y)] \\mathrm{d} \\sigma = \\alpha \\iint_{D} f(x, y) \\mathrm{d} \\sigma + \\beta \\iint_{D} g(x, y) \\mathrm{d} \\sigma. \n$$"], "answer": "D", "topic": "College--Advanced Mathematics--Double Integrals"} |
| {"question": "When the integration region $D$ is symmetric about the $y$-axis, which of the following statements is correct:", "options": ["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 $y \\geqslant 0$.", "$\\iint_{D} f(x, y) \\mathrm{d} \\sigma=\\iint_{D} f(y, x) \\mathrm{d} \\sigma$."], "answer": "B", "topic": "College--Advanced Mathematics--Double Integrals"} |
| {"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:", "options": ["$f(x,-y)=-f(x, y)$", "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 $y=x$"], "answer": "D", "topic": "College--Advanced Mathematics--Double Integrals"} |
| {"question": "Suppose the integral region $D$ can be represented by the inequalities\n$$\n\\varphi_{1}(x) \\leqslant y \\leqslant \\varphi_{2}(x), \\quad a \\leqslant x \\leqslant b\n$$\n\nwhere 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$$\n\\iint_{D} f(x, y) \\mathrm{d} \\sigma=$$?", "options": ["$$\n\\iint_{D} f(x, y) \\mathrm{d} \\sigma=\\int_{a}^{b}\\left[\\int_{\\varphi_{2}(x)}^{\\varphi_{1}(x)} f(x, y) \\mathrm{d} y\\right] \\mathrm{d} x .\n$$", "$$ \\int_{b}^{a}\\left[\\int_{\\varphi_{2}(x)}^{\\varphi_{1}(x)} f(x, y) \\mathrm{d} y\\right] \\mathrm{d} x .\n$$", "$$ \\int_{a}^{b}\\left[\\int_{\\varphi_{1}(x)}^{\\varphi_{2}(x)} f(x, y) \\mathrm{d} y\\right] \\mathrm{d} x .\n$$", "$$ \\int_{b}^{a}\\left[\\int_{\\varphi_{1}(x)}^{\\varphi_{2}(x)} f(x, y) \\mathrm{d} y\\right] \\mathrm{d} x .\n$$"], "answer": "C", "topic": "College--Advanced Mathematics--Double Integrals"} |
| {"question": "The Cartesian coordinates $(x, y)$ and polar coordinates $(r, \\theta)$ of point $M$ are related by $\\left\\{\\begin{array}{l}x=r \\cos \\theta, \\\\ y=r \\sin \\theta .\\end{array}\\right.$\nWhen calculating double integrals in polar coordinates, the formula is\n\n$$\n\\iint_{D} f(x, y) \\mathrm{d} \\sigma=\n$$ _______", "options": ["$$\\iint_{D} f(r \\sin \\theta, r \\sin \\theta) r \\mathrm{~d} r \\mathrm{~d} \\theta$$", "$$\\iint_{D} f(r \\cos \\theta, -\\sin \\theta) r \\mathrm{~d} r \\mathrm{~d} \\theta$$", "$$\\iint_{D} f(\\cos \\theta, \\sin \\theta) r \\mathrm{~d} r \\mathrm{~d} \\theta$$", "$$\\iint_{D} f(r \\sin \\theta, r \\cos \\theta) r \\mathrm{~d} r \\mathrm{~d} \\theta$$"], "answer": "D", "topic": "College--Advanced Mathematics--Double Integrals"} |
| {"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?", "options": ["Partial differential equation", "Non-separable variable differential equation.", "Separable variable differential equation.", "None of the above"], "answer": "C", "topic": "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?", "options": ["Average", "Sum", "Derive", "Integrate"], "answer": "D", "topic": "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", "options": ["homogeneous equation", "singular equation", "non-homogeneous equation", "linear equation"], "answer": "A", "topic": "College--Advanced Mathematics--Ordinary Differential Equations"} |
| {"question": "The solution method for the homogeneous equation $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=\\varphi\\left(\\frac{y}{x}\\right)$:\n\nPerform 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\n<<<Answer>>>\n\nAfter solving by the method of separation of variables, substitute back $u=\\frac{y}{x}$ and then solve for $y$.", "options": ["$$x \\frac{\\mathrm{d} u}{\\mathrm{~d} x}=\\varphi(u)+u .$$", "$$x \\frac{\\mathrm{d} u}{\\mathrm{~d} x}=\\varphi(u)/u .$$", "$$x \\frac{\\mathrm{d} u}{\\mathrm{~d} x}=\\varphi(u)-u .$$", "$$x \\frac{\\mathrm{d} u}{\\mathrm{~d} x}=\\varphi(u)*u .$$"], "answer": "C", "topic": "College--Advanced Mathematics--Ordinary Differential Equations"} |
| {"question": "The solution of the homogeneous equation $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=\\varphi\\left(\\frac{y}{x}\\right)$:\n\nPerform 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\n$$\nx \\frac{\\mathrm{d} u}{\\mathrm{~d} x}=\\varphi(u)-u .\n$$\n\nSolve it using () and then substitute back $u=\\frac{y}{x}$ and solve for $y$ to find the solution.", "options": ["Method of Separation of Variables", "General Constant Method", "General Variable Method", "Separation of Constants Method"], "answer": "A", "topic": "College--Advanced Mathematics--Ordinary Differential Equations"} |
| {"question": "The equation $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}+p(x) y=q(x)$ is called a()", "options": ["first-order linear differential equation.", "first-order nonlinear differential equation.", "second-order nonlinear differential equation.", "second-order linear differential equation."], "answer": "A", "topic": "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 $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}+p(x) y=q(x)$.", "options": ["$q(x)>0$", "$q(x)\\neq0$", "$q(x)=0$", "$q(x)<0$"], "answer": "C", "topic": "College--Advanced Mathematics--Ordinary Differential Equations"} |
| {"question": "() can generally be divided into the following three categories:\n\n(1) $y''=f(x, y')$ type: Let $y'=p$, then $y''=p'$, the equation is transformed into $p'=f(x, p)$.\n\n(2) $y''=f(y, y')$ type: Let $y'=p$, then $y''=p \\frac{dp}{dy}$, the equation is transformed into $p \\frac{dp}{dy}=f(y, p)$.\n\n(3) $y^{(n)}=f(x)$ type: Perform $n$ indefinite integrations on $f(x)$.", "options": ["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"], "answer": "D", "topic": "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 $y=C_{1} e^{r_{1} x}+C_{2} e^{r_{2} x}$;", "options": ["distinct real roots", "equal real roots", "conjugate complex roots", "none of the above"], "answer": "A", "topic": "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 $y=\\left(C_{1}+C_{2} x\\right) \\mathrm{e}^{r_{1} x}$", "options": ["equal real roots", "unequal real roots", "conjugate complex roots", "none of the above"], "answer": "A", "topic": "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\n\n$$\ny^{*}=x^{k} R_{m}(x) \\mathrm{e}^{\\lambda x}\n$$\n\nwhere $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\\lambda$ is not a root of the characteristic equation, k=()", "options": ["0", "1", "2", "3"], "answer": "A", "topic": "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\n\n$$\ny^{*}=x^{k} R_{m}(x) \\mathrm{e}^{\\lambda x}\n$$\n\nwhere $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\\lambda$ is a root of the characteristic equation, k equals", "options": ["0", "2", "3", "1"], "answer": "D", "topic": "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\n\n$$\ny^{*}=x^{k} R_{m}(x) \\mathrm{e}^{\\lambda x}\n$$\n\nwhere $R_{m}(x)$ is a polynomial of the same degree as $P_{m}(x)$. When $\\lambda$ is a repeated root of the characteristic equation, k equals ()", "options": ["2", "0", "1", "3"], "answer": "A", "topic": "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=()", "options": ["2", "0", "3", "1"], "answer": "D", "topic": "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\n\n$$\ny^{*}=x^{k} \\mathrm{e}^{\\lambda x}[R_{m}^{(1)}(x) \\cos \\omega x+R_{m}^{(2)}(x) \\sin \\omega x]$$\n\nwhere $R_{m}^{(1)}(x), R_{m}^{(2)}(x)$ 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", "options": ["0", "1", "2", "3"], "answer": "A", "topic": "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.", "options": ["$y=C_{1} y_{1}(x) * C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x) ^ C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x)+C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x) / C_{2} y_{2}(x)$"], "answer": "C", "topic": "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.", "options": ["$y=C_{1} y_{1}(x)+C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x) / C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x) * C_{2} y_{2}(x)$", "$y=C_{1} y_{1}(x) ^ C_{2} y_{2}(x)$"], "answer": "A", "topic": "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 $y^{\\prime \\prime}+p(x) y^{\\prime}+q(x) y=f(x)$.", "options": ["$y=Y(x)/y^{*}(x)$", "$y=Y(x)-y^{*}(x)$", "$y=Y(x)*y^{*}(x)$", "$y=Y(x)+y^{*}(x)$"], "answer": "D", "topic": "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 $y^{\\prime \\prime}+p(x) y^{\\prime}+q(x) y=0$.", "options": ["$y_{1}(x)/y_{2}(x)$", "$y_{1}(x)-y_{2}(x)$", "$y_{1}(x)+y_{2}(x)$", "$y_{1}(x)*y_{2}(x)$"], "answer": "B", "topic": "College--Advanced Mathematics--Ordinary Differential Equations"} |
| {"question": "If a non-homogeneous linear differential equation is in the form\n\n$$\ny''+p(x)y'+q(x)y=f_{1}(x)+f_{2}(x),\n$$\n\nand $y_{1}^{*}(x)$ 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 $y''+p(x)y'$ $+q(x)y=f_{1}(x)+f_{2}(x)$.", "options": ["$y_{1}^{*}(x)-y_{2}^{*}(x)$", "$y_{1}^{*}(x)/y_{2}^{*}(x)$", "$y_{1}^{*}(x)+y_{2}^{*}(x)$", "$y_{1}^{*}(x)*y_{2}^{*}(x)$"], "answer": "C", "topic": "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 $\\sum_{n=1}^{\\infty} a_{n}$ .", "options": ["Conditionally convergent", "Convergent", "Absolutely convergent", "Divergent"], "answer": "B", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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>>>;", "options": ["Conditionally Convergent", "Convergent", "All other options are incorrect", "Divergent"], "answer": "B", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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 $\\sum_{n=1}^{\\infty} a_{n}$ .", "options": ["Converges", "Absolutely converges", "Diverges", "Conditionally converges"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_{n}}=\\rho$ exists (including $\\rho=\\infty$), then $\\rho<1$, series.", "options": ["diverges", "conditionally converges", "absolutely converges", "can't be sure"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_{n}}=\\rho$ exists $($ including $\\rho=\\infty)$, and $\\rho>1$, then the series is.", "options": ["Conditionally convergent", "Convergent", "Divergent", "Absolutely convergent"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_{n}}=\\rho$ exists (including $\\rho=\\infty)$, then $\\rho=1$, the series.", "options": ["diverges", "converges", "inconclusive", "not divergent"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\sqrt[n]{a_{n}}=\\rho$ exists (including $\\rho=\\infty$), then $\\rho<1$, the series", "options": ["absolutely converges", "conditionally converges", "can't be sure", "diverges"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\sqrt[n]{a_{n}}=\\rho$ exists (including $\\rho=\\infty$), then $\\rho>1$, the series is.", "options": ["absolutely convergent", "convergent", "conditionally convergent", "divergent"], "answer": "D", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If $\\lim _{n \\rightarrow \\infty} \\sqrt[n]{a_{n}}=\\rho$ exists (including $\\rho=\\infty$), then $\\rho=1$, the series is.", "options": ["indefinite", "divergent", "conditionally convergent", "convergent"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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 $\\sum_{n=1}^{\\infty} a_{n}$.", "options": ["Conditionally convergent", "Divergent", "Absolutely convergent", "Convergent but not absolutely convergent"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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", "options": ["Absolutely convergent", "Divergent", "Conditionally convergent", "Convergent"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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>>>.", "options": ["Absolutely convergent", "Convergent", "Conditionally convergent", "Divergent"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If the series $\\sum_{n=1}^{\\infty} a_{n}$ is absolutely convergent, then the series $\\sum_{n=1}^{\\infty} a_{n}$ necessarily <<<Answer>>>.", "options": ["converges", "diverges", "conditionally converges", "absolutely converges"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If the alternating series $\\sum_{n=1}^{\\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ satisfies the conditions:\n(1) $a_{n} \\geqslant a_{n+1} (n=1,2,3, \\cdots)$;\n(2) $\\lim _{n \\rightarrow \\infty} a_{n}=0$,\nthen the series.", "options": ["diverges", "conditionally converges", "converges", "absolutely converges"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If the alternating series $\\sum_{n=1}^{\\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ meets the conditions:\n(1) $a_{n} \\geqslant a_{n+1}(n=1,2,3, \\cdots)$;\n(2) $\\lim _{n \\rightarrow \\infty} a_{n}=0$,\nthen the series converges, and its sum $s$ <<<Answer>>> $a_{1}$.", "options": ["$\\leq$", "$\\le$", "$\\leqslant$", "$\\neq$"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "If the alternating series $\\sum_{n=1}^{\\bar{n}}(-1)^{n-1} a_{n}$ $(a_{n}>0)$ satisfies the conditions:\n(1) $a_{n} \\geqslant a_{n+1}(n=1,2,3, \\cdots)$;\n(2) $\\lim _{n \\rightarrow \\infty} a_{n}=0$,\nthen the series converges, and its sum $s$ <<<Answer>>> $a_{1}$.", "options": ["$\\leqslant$", "$\\neq$", "$\\leq$", "$\\le$"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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>>>.", "options": ["$\\frac{1}{l}$", "$l$", "$-l$", "$\\frac{-1}{l}$"], "answer": "A", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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>>>.", "options": ["$-l$", "$\\frac{-1}{l}$", "$\\frac{1}{l}$", "$l$"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\mathrm{e}^{x}$ =.", "options": ["$\\sum{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{1}{n !} x^{n}(-\\infty<x<+\\infty)$"], "answer": "D", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "Infinite series $a^{x}$ =.", "options": ["$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n)!} x^{2n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} x^{2n+1}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{1}{n!} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n!} x^{n}(-\\infty<x<+\\infty)$"], "answer": "D", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\sin x$ =.", "options": ["$\\sum{n=0}^{\\infty} \\frac{1}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\cos x$ =.", "options": ["$\\sum{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$", "$\\sum{n=0}^{\\infty} \\frac{1}{n !} x^{n}(-\\infty<x<+\\infty)$"], "answer": "B", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\frac{1}{1+x}$ =.", "options": ["$\\sum{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty}(-1)^{n} x^{n}(-1<x<1)$", "$\\sum{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$"], "answer": "C", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\ln (1+x)$ equals.", "options": ["$\\sum_{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1) !} x^{2n+1}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n) !} x^{2n}(-\\infty<x<+\\infty)$", "$\\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1}}{n} x^{n}(-\\infty<x<+\\infty)$"], "answer": "D", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\frac{1}{1+x^{2}}$ equals.", "options": ["$\\sum_{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{1}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$"], "answer": "B", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"question": "The infinite series $\\arctan x=$ .", "options": ["$\\sum_{n=0}^{\\infty} \\frac{(\\ln a)^{n}}{n !} x^{n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2 n+1} x^{2 n+1}(-1 \\leqslant x \\leqslant 1 \\text { )}$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n) !} x^{2 n}(-\\infty<x<+\\infty)$", "$\\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}(-\\infty<x<+\\infty)$"], "answer": "B", "topic": "College--Advanced Mathematics--Infinite Series"} |
| {"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 $\\iiint_{\\Omega} f(x, y, z) \\mathrm{d} v=\\lim _{\\lambda \\rightarrow 0} \\sum_{i=1}^{n} f(\\xi_{i}, \\eta_{i}, \\zeta_{i}) \\Delta v_{i}, where____is called the integrand?", "options": ["$\\Omega$", "$\\mathrm{d} v$", "$f(x, y, z)$", "$ f(x, y, z) \\mathrm{d} v$"], "answer": "C", "topic": "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 $L$:\n\\int_{L} P \\mathrm{~d} x+Q \\mathrm{~d} y=\\int_{L}(P \\cos \\alpha+Q \\cos \\beta) \\mathrm{d} s,\nwhere $\\alpha(x, y), \\beta(x, y)$ are the angles of the direction of the ______ at the point $(x, y)$ on the directed curve arc $L$.", "options": ["tangent", "normal", "normal vector", "tangent vector"], "answer": "D", "topic": "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\n\\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,\nwhere $\\hat{L}$ is the positively oriented boundary curve of $D$.", "options": ["second derivatives", "first-order continuous partial derivatives", "second-order continuous partial derivatives", "first-order derivatives"], "answer": "B", "topic": "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", "options": ["$\\iint_{\\Sigma} f(x, y, z) \\mathrm{d} S=\\iint_{D_{xy}} f(x, y, z(x, y)) \\sqrt{1+z_{x}^{\\prime}(x, y)+z_{y}^{\\prime}(x, y)} \\mathrm{~d} x \\mathrm{~d} y$ .", "$\\iint_{\\Sigma} f(x, y, z) \\mathrm{d} S=\\iint_{D_{xy}} f(x, y, z(x, y)) \\sqrt{2+z_{x}^{\\prime}(x, y)+z_{y}^{\\prime}(x, y)} \\mathrm{~d} x \\mathrm{~d} y$ .", "$\\iint_{\\Sigma} f(x, y, z) \\mathrm{d} S=\\iint_{D_{xy}} f(x, y, z(x, y)) \\sqrt{z_{x}^{\\prime}(x, y)+z_{y}^{\\prime}(x, y)} \\mathrm{~d} x \\mathrm{~d} y$ .", "$\\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 $."], "answer": "D", "topic": "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 \n\\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.", "options": ["Defined and continuous", "Twice differentiable", "Continuous", "First-order differentiable"], "answer": "A", "topic": "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)?", "options": ["Becomes $\\frac{1}{x}$ times the original", "Becomes $1+x$ times the original", "Remains the same", "Becomes $x$ times the original"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "What is the relationship between a determinant and its transpose determinant $|\\boldsymbol{A}|$ and $\\left|\\boldsymbol{A}^{\\top}\\right|$?", "options": ["$|\\boldsymbol{A}| = 2\\left|\\boldsymbol{A}^{\\top}\\right|$", "$|\\boldsymbol{A}| = \\frac{1}{\\left|\\boldsymbol{A}^{\\top}\\right|}$", "$|\\boldsymbol{A}| = \\left|\\boldsymbol{A}^{\\top}\\right|$", "$|\\boldsymbol{A}| = \\left|\\boldsymbol{A}^{\\top}\\right|$ - 1"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "What change occurs to a determinant after swapping two rows (or columns)?", "options": ["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)"], "answer": "A", "topic": "College--Linear Algebra--Determinants"} |
| {"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?", "options": ["$k^{n}$ times", "Becomes $\\frac{1}{k}$ of the original", "Unchanged", "Becomes $k$ times the original"], "answer": "D", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "If in a determinant, two rows (or columns) are in proportion k, what is the value of the determinant?", "options": ["1", "k", "0", "$\\frac{1}{k}$"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "What is the relationship between the following three matrices?\n$$\nD_1=\\left|\\begin{array}{cccc}\na_{11} & a_{12} & \\cdots & a_{1 n} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{i 1}+a_{i 1}^{\\prime} & a_{i 2}+a_{i 2}^{\\prime} & \\cdots & a_{i n}+a_{i n}^{\\prime} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{n 1} & a_{n 2} & \\cdots & a_{n n}\n\\end{array}\\right|,\n$$\n\n$$\nD_2=\\left|\\begin{array}{cccc}\na_{11} & a_{12} & \\cdots & a_{1 n} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{n 1} & a_{i 2} & \\cdots & a_{i n} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{n 1} & a_{n 2} & \\cdots & a_{n n}\n\\end{array}\\right|\nD_3 = \\left|\\begin{array}{cccc}\na_{11} & a_{12} & \\cdots & a_{1 n} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{i 1}^{\\prime} & a_{22}^{\\prime} & \\cdots & a_{i n}^{\\prime} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\na_{n 1} & a_{n 2} & \\cdots & a_{n n}\n\\end{array}\\right| .\n$$", "options": ["$D_1 = D_2 * D_3$", "$D_1 = D_2 + D_3$", "No relation", "$D_1 = \\frac{D_2}{D_3}$"], "answer": "B", "topic": "College--Linear Algebra--Determinants"} |
| {"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)?", "options": ["Becomes $\\frac{1}{k}$ times the original", "Becomes $(k+1)$ times the original", "Remains unchanged", "Becomes k times the original"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"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?", "options": ["$\\lambda^{n}|\\boldsymbol{A}|$", "$\\lambda^{2}|\\boldsymbol{A}|$", "$\\lambda|\\boldsymbol{A}|$", "$|\\boldsymbol{A}|$"], "answer": "A", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "If $\\boldsymbol{A}, \\boldsymbol{B}$ are both $n$th order matrices, what is the determinant $|\\boldsymbol{A B}|$ equal to?", "options": ["$|\\boldsymbol{A}| / |\\boldsymbol{B}|$", "$|\\boldsymbol{A}| + |\\boldsymbol{B}|$", "$|\\boldsymbol{A}| \\cdot |\\boldsymbol{B}|$", "$|\\boldsymbol{A} - \\boldsymbol{B}|$"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"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 $\\boldsymbol{A}$?", "options": ["$|\\boldsymbol{A}|^{n}$", "$|\\boldsymbol{A}|^{n+1}$", "$|\\boldsymbol{A}|^{n-1}$", "$|\\boldsymbol{A}|$"], "answer": "C", "topic": "College--Linear Algebra--Determinants"} |
| {"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?", "options": ["$|\\boldsymbol{A}|$", "$-|\\boldsymbol{A}|$", "$|\\boldsymbol{A}|^{2}$", "$\\frac{1}{|\\boldsymbol{A}|}$"], "answer": "D", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "If matrices $\\boldsymbol{A}, \\boldsymbol{B}$ are similar, what is the relationship between their determinants $|\\boldsymbol{A}|$ and $|\\boldsymbol{B}|$?", "options": ["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., $|\\boldsymbol{A}|=|\\boldsymbol{B}|$"], "answer": "D", "topic": "College--Linear Algebra--Determinants"} |
| {"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?", "options": ["$\\sum_{i=1}^{n} \\lambda_{i}$", "1", "$\\max(\\lambda_{i})$", "$\\prod_{i=1}^{n} \\lambda_{i}$"], "answer": "D", "topic": "College--Linear Algebra--Determinants"} |
| {"question": "Which type of elementary row operations does not include?", "options": ["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"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "How to define the relationship between elementary column operations and row operations of a matrix?", "options": ["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."], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "Under what condition is the matrix $\\boldsymbol{A}$ called an invertible matrix?", "options": ["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 $\\boldsymbol{A B}=\\boldsymbol{I}$"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the inverse matrix of matrix $\\boldsymbol{A}$ denoted as?", "options": ["$\\boldsymbol{A}^{-1}$", "$\\boldsymbol{A}^{T}$", "$\\boldsymbol{A}^{*}$", "$\\boldsymbol{A}^{+}$"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "How is each element of the adjugate matrix $\\boldsymbol{A}^{*}$ obtained?", "options": ["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 $\\boldsymbol{A}$. "], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "If matrix $\\boldsymbol{A}$ is invertible, what is the value of its determinant $|\\boldsymbol{A}|$?", "options": ["Not equal to 0", "Equal to 0", "Equal to 1", "Cannot be determined"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "If the determinant $|\\boldsymbol{A}| \\neq 0$, how can the inverse matrix $\\boldsymbol{A}^{-1}$ of the matrix $\\boldsymbol{A}$ be represented?", "options": ["$\\boldsymbol{A}^{-1}=\\frac{1}{|\\boldsymbol{A}|} \\boldsymbol{A}^{T}$", "$\\boldsymbol{A}^{-1}=\\frac{1}{|\\boldsymbol{A}|} \\boldsymbol{A}^{-1}$", "$\\boldsymbol{A}^{-1}=\\frac{1}{|\\boldsymbol{A}|} \\boldsymbol{A}$", "$\\boldsymbol{A}^{-1}=\\frac{1}{|\\boldsymbol{A}|} \\boldsymbol{A}^{*}$"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "How is the rank of matrix $\\boldsymbol{A}$ defined?", "options": ["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}."], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the rank of the zero matrix?", "options": ["1", "The number of rows in the matrix", "The number of columns in the matrix", "0"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the relationship between the row rank and the column rank of a matrix?", "options": ["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."], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is a full row rank matrix?", "options": ["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."], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is a full column rank matrix?", "options": ["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."], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{B}$?", "options": ["$r(\\boldsymbol{A})<r(\\boldsymbol{B})$", "$r(\\boldsymbol{A})=r(\\boldsymbol{B})$", "$r(\\boldsymbol{A})>r(\\boldsymbol{B})$", "$r(\\boldsymbol{A})=2r(\\boldsymbol{B})$"], "answer": "B", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the range of the rank $r(\\boldsymbol{A})$ of the matrix $\\boldsymbol{A}_{m \\times n}$?", "options": ["$0 \\leqslant r(\\boldsymbol{A}) \\leqslant m \\times n$", "$0 \\leqslant r(\\boldsymbol{A}) \\leqslant \\max \\{m, n\\}$", "$0 \\leqslant r(\\boldsymbol{A}) \\leqslant m + n$", "$0 \\leqslant r(\\boldsymbol{A}) \\leqslant \\min \\{m, n\\}$"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the relationship between the rank of matrix $\\boldsymbol{A}$ and the rank of its transpose $\\boldsymbol{A}^{\\mathrm{T}}$?", "options": ["$r(\\boldsymbol{A}^{\\mathrm{T}}) > r(\\boldsymbol{A})$", "$r(\\boldsymbol{A}^{\\mathrm{T}}) < r(\\boldsymbol{A})$", "$r(\\boldsymbol{A}^{\\mathrm{T}})=2r(\\boldsymbol{A})$", "$r(\\boldsymbol{A}^{\\mathrm{T}})=r(\\boldsymbol{A})$"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"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?", "options": ["$r(\\boldsymbol{A}) < r\\left(\\boldsymbol{A}^{\\mathrm{T}} \\boldsymbol{A}\\right)$", "$r(\\boldsymbol{A})=r\\left(\\boldsymbol{A}^{\\mathrm{T}} \\boldsymbol{A}\\right)=r\\left(\\boldsymbol{A} \\boldsymbol{A}^{\\mathrm{T}}\\right)$", "$r(\\boldsymbol{A}) > r\\left(\\boldsymbol{A} \\boldsymbol{A}^{\\mathrm{T}}\\right)$", "$r(\\boldsymbol{A})=2r\\left(\\boldsymbol{A}^{\\mathrm{T}} \\boldsymbol{A}\\right)$"], "answer": "B", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the relationship between the rank of matrix $\\boldsymbol{A}+\\boldsymbol{B}$ and the ranks of matrices $\\boldsymbol{A}$ and $\\boldsymbol{B}$?", "options": ["$r(\\boldsymbol{A}+\\boldsymbol{B}) = r(\\boldsymbol{A})+r(\\boldsymbol{B})$", "$r(\\boldsymbol{A}+\\boldsymbol{B}) > r(\\boldsymbol{A})+r(\\boldsymbol{B})$", "$r(\\boldsymbol{A}+\\boldsymbol{B}) \\leqslant r(\\boldsymbol{A})+r(\\boldsymbol{B})$", "$r(\\boldsymbol{A}+\\boldsymbol{B}) = r(\\boldsymbol{A})-r(\\boldsymbol{B})$"], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the relationship between the rank of matrix $\\boldsymbol{A} \\boldsymbol{B}$ and the ranks of matrices $\\boldsymbol{A}$ and $\\boldsymbol{B}$?", "options": ["$r(\\boldsymbol{A} \\boldsymbol{B}) \\leqslant \\min\\{r(\\boldsymbol{A}), r(\\boldsymbol{B})\\}$", "$r(\\boldsymbol{A} \\boldsymbol{B}) = r(\\boldsymbol{A}) \\times r(\\boldsymbol{B})$", "$r(\\boldsymbol{A} \\boldsymbol{B}) > \\max\\{r(\\boldsymbol{A}), r(\\boldsymbol{B})\\}$", "$r(\\boldsymbol{A} \\boldsymbol{B}) = r(\\boldsymbol{A}) + r(\\boldsymbol{B})$"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "If $\\boldsymbol{\\alpha}, \\boldsymbol{\\beta}$ are non-zero column vectors, what is the rank of the matrix $\\boldsymbol{\\alpha} \\boldsymbol{\\beta}^{\\mathrm{T}}$?", "options": ["0", "$\\min\\{r(\\boldsymbol{\\alpha}), r(\\boldsymbol{\\beta})\\}$", "1", "2"], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{A}$?", "options": ["$r(\\boldsymbol{P A Q})=r(\\boldsymbol{A})$", "$r(\\boldsymbol{P A Q})=2r(\\boldsymbol{A})$", "$r(\\boldsymbol{P A Q}) > r(\\boldsymbol{A})$", "$r(\\boldsymbol{P A Q}) < r(\\boldsymbol{A})$"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{B}$?", "options": ["$n$", "$m + l$", "$m + n$", "$n + l$"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the necessary and sufficient condition for the matrix equation $\\boldsymbol{A} \\boldsymbol{X}=\\boldsymbol{B}$ to have a solution?", "options": ["$r(\\boldsymbol{A}) < r(\\boldsymbol{A}, \\boldsymbol{B})$", "$r(\\boldsymbol{A}) > r(\\boldsymbol{A}, \\boldsymbol{B})$", "$r(\\boldsymbol{A})=r(\\boldsymbol{B})$", "$r(\\boldsymbol{A})=r(\\boldsymbol{A}, \\boldsymbol{B})$"], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{A}^{*}$?", "options": ["$r\\left(\\boldsymbol{A}^{*}\\right)=1$", "$r\\left(\\boldsymbol{A}^{*}\\right)=0$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n-1$"], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{A}^{*}$?", "options": ["$r\\left(\\boldsymbol{A}^{*}\\right)=0$", "$r\\left(\\boldsymbol{A}^{*}\\right)=1$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n-1$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n$"], "answer": "B", "topic": "College--Linear Algebra--Matrices"} |
| {"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 $\\boldsymbol{A}^{*}$?", "options": ["$r\\left(\\boldsymbol{A}^{*}\\right)=0$", "$r\\left(\\boldsymbol{A}^{*}\\right)=1$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n$", "$r\\left(\\boldsymbol{A}^{*}\\right)=n-1$"], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "How to define the linear dependence of a set of vectors?", "options": ["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"], "answer": "B", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "When is a vector $\\boldsymbol{\\alpha}$ linearly dependent?", "options": ["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 $\\boldsymbol{\\alpha} \\neq \\boldsymbol{0}$"], "answer": "B", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "Is a nonzero vector $\\boldsymbol{\\alpha}$ linearly independent?", "options": ["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 $\\boldsymbol{\\alpha} \\neq \\boldsymbol{0}$"], "answer": "C", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "What is the necessary and sufficient condition for two vectors $\\boldsymbol{\\alpha}_1, \\boldsymbol{\\alpha}_2$ to be linearly dependent?", "options": ["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"], "answer": "A", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "If two vectors $\\boldsymbol{\\alpha}_{1}$ and $\\boldsymbol{\\alpha}_{2}$ are linearly dependent, what is their geometric relationship?", "options": ["They are independent", "They are coplanar but not necessarily collinear", "They are perpendicular", "They are collinear"], "answer": "D", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "If three three-dimensional vectors $\\boldsymbol{\\alpha}_{1}, \\boldsymbol{\\alpha}_{2}, \\boldsymbol{\\alpha}_{3}$ are linearly dependent, what geometric relationship do they have?", "options": ["They are mutually perpendicular", "They are coplanar", "They form a triangle", "They are linearly independent"], "answer": "B", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "What is the condition for the vector $\\boldsymbol{\\beta}$ to be linearly represented by the set of vectors $\\boldsymbol{\\alpha}_{1}, \\boldsymbol{\\alpha}_{2}, \\cdots, \\boldsymbol{\\alpha}_{l}$?", "options": ["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 $k_{1} \\boldsymbol{\\alpha}_{1}+k_{2} \\boldsymbol{\\alpha}_{2}+\\cdots+k_{l} \\boldsymbol{\\alpha}_{l}=\\boldsymbol{\\beta}$."], "answer": "D", "topic": "College--Linear Algebra--Vectors"} |
| {"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 $\\boldsymbol{A x}=\\boldsymbol{\\beta}$?", "options": ["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."], "answer": "B", "topic": "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 $\\boldsymbol{\\alpha}_{1}, \\boldsymbol{\\alpha}_{2}, ..., \\boldsymbol{\\alpha}_{l}$?", "options": ["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."], "answer": "A", "topic": "College--Linear Algebra--Vectors"} |
| {"question": "What is the necessary and sufficient condition for the system of linear equations $Ax=b$ to have no solution?", "options": ["$r(A)<r(A, b)$", "$r(A)>r(A, b)$", "$r(A)=r(A, b)$", "$r(A)=r(A, b)=n$"], "answer": "A", "topic": "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?", "options": ["$r(A)=r(A, b)<n$", "$r(A)>r(A, b)$", "$r(A)<r(A, b)$", "$r(A)=r(A, b)=n$"], "answer": "D", "topic": "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?", "options": ["$r(A)=r(A, b)=n$", "$r(A)=r(A, b)<n$", "$r(A)<r(A, b)$", "$r(A)>r(A, b)$"], "answer": "B", "topic": "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?", "options": ["$r(A)>n$", "$r(A)=n$", "$r(A)=r(A, b)$", "$r(A)<n$"], "answer": "D", "topic": "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?", "options": ["$r(A)<r(A, b)$", "$r(A)=r(A, b)$", "$r(A)=n$", "$r(A)>r(A, b)$"], "answer": "B", "topic": "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?", "options": ["$r(A)>r(A, B)$", "$r(A)<r(A, B)$", "$r(A)=r(A, B)$", "$r(A, B)<n$"], "answer": "C", "topic": "College--Linear Algebra--Systems of Linear Equations"} |
| {"question": "Given the matrix equation $AB=C$, what is the maximum possible value of $r(C)$?", "options": ["$\\min \\{r(A), r(B)\\}$", "$r(A)+r(B)$", "$r(A) - r(B)$", "$r(A) \\cdot r(B)$"], "answer": "A", "topic": "College--Linear Algebra--Systems of Linear Equations"} |
| {"question": "What is the fundamental system of solutions of a homogeneous system of linear equations?", "options": ["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."], "answer": "B", "topic": "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?", "options": ["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 $\\boldsymbol{x}=\\boldsymbol{\\xi}_{1}+\\boldsymbol{\\xi}_{2}+\\cdots+\\boldsymbol{\\xi}_{t}$."], "answer": "C", "topic": "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?", "options": ["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."], "answer": "D", "topic": "College--Linear Algebra--Systems of Linear Equations"} |
| {"question": "What is the solution space of a system of $n$ homogeneous linear equations?", "options": ["$S=\\{\\boldsymbol{x} \\mid \\boldsymbol{A x}=\\mathbf{1}\\}$ 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."], "answer": "D", "topic": "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?", "options": ["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."], "answer": "C", "topic": "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?", "options": ["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 $\\boldsymbol{A x}=\\mathbf{0}$."], "answer": "D", "topic": "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 $\\boldsymbol{A x}=\\mathbf{0}$?", "options": ["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 $\\boldsymbol{A x}=\\mathbf{0}$."], "answer": "D", "topic": "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 $\\boldsymbol{A x}=\\boldsymbol{b}$?", "options": ["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."], "answer": "A", "topic": "College--Linear Algebra--Systems of Linear Equations"} |
| {"question": "What does the general solution of a non-homogeneous linear system consist of?", "options": ["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."], "answer": "B", "topic": "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?", "options": ["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"], "answer": "A", "topic": "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 $\\boldsymbol{A x}=\\boldsymbol{b}$?", "options": ["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 $\\boldsymbol{b}$"], "answer": "B", "topic": "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 $\\boldsymbol{A x}=\\boldsymbol{b}$?", "options": ["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"], "answer": "D", "topic": "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?", "options": ["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"], "answer": "A", "topic": "College--Linear Algebra--Systems of Linear Equations"} |
| {"question": "What are the eigenvalues and eigenvectors of a matrix?", "options": ["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 $\\boldsymbol{A}$."], "answer": "A", "topic": "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?", "options": ["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."], "answer": "D", "topic": "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?", "options": ["$\\lambda^{k}$ 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 $\\boldsymbol{A}$."], "answer": "D", "topic": "College--Linear Algebra--Eigenvalues and Eigenvectors"} |
| {"question": "When a matrix has multiple eigenvalues, are the corresponding eigenvectors linearly independent?", "options": ["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."], "answer": "B", "topic": "College--Linear Algebra--Eigenvalues and Eigenvectors"} |
| {"question": "Is a linear combination of eigenvectors belonging to the same eigenvalue also an eigenvector of that eigenvalue?", "options": ["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."], "answer": "B", "topic": "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 $\\lambda$?", "options": ["At most $k$.", "Only one.", "At most $k+1$.", "At most $k-1$."], "answer": "A", "topic": "College--Linear Algebra--Eigenvalues and Eigenvectors"} |
| {"question": "How are two matrices $\\boldsymbol{A}$ and $\\boldsymbol{B}$ defined to be similar?", "options": ["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."], "answer": "A", "topic": "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?", "options": ["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."], "answer": "A", "topic": "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?", "options": ["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."], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the necessary and sufficient condition for the $n$-th order matrix $\\boldsymbol{A}$ to be similar to a diagonal matrix?", "options": ["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."], "answer": "B", "topic": "College--Linear Algebra--Eigenvalues and Eigenvectors"} |
| {"question": "What are the necessary conditions for two matrices to be similar?", "options": ["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."], "answer": "A", "topic": "College--Linear Algebra--Eigenvalues and Eigenvectors"} |
| {"question": "What is a quadratic form?", "options": ["A quadratic form is any polynomial function whose highest order term is 2.", "A quadratic form is a quadratic homogeneous function with $n$ variables $x_{1}, x_{2}, \\cdots, x_{n}$, which can be expressed in the form $f=\\boldsymbol{x}^{\\mathrm{T}} \\boldsymbol{A x}$, where $\\boldsymbol{A}$ is a symmetric matrix.", "A quadratic form is a system of linear equations, which contains $n$ variables $x_{1}, x_{2}, \\cdots, x_{n}$.", "A quadratic form is a system of inequalities, which contains $n$ variables $x_{1}, x_{2}, \\cdots, x_{n}$."], "answer": "B", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What are the characteristics of matrix $\\boldsymbol{A}$ in the quadratic form $f=\\boldsymbol{x}^{\\mathrm{T}} \\boldsymbol{A x}$?", "options": ["Matrix $\\boldsymbol{A}$ is a diagonal matrix.", "Matrix $\\boldsymbol{A}$ is a skew-symmetric matrix, i.e., $\\boldsymbol{A} = -\\boldsymbol{A}^{\\top}$.", "Matrix $\\boldsymbol{A}$ is an upper triangular matrix.", "Matrix $\\boldsymbol{A}$ is a symmetric matrix, i.e., $\\boldsymbol{A} = \\boldsymbol{A}^{\\top}$."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "How is the rank of a quadratic form defined?", "options": ["The rank of a quadratic form is defined as the number of eigenvalues of its corresponding symmetric matrix $\\boldsymbol{A}$.", "The rank of a quadratic form is defined as the rank of its corresponding symmetric matrix $\\boldsymbol{A}$.", "The rank of a quadratic form is defined as the number of non-zero elements in its corresponding symmetric matrix $\\boldsymbol{A}$.", "The rank of a quadratic form is defined as the value of the determinant of its corresponding symmetric matrix $\\boldsymbol{A}$."], "answer": "B", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the standard form of a quadratic form?", "options": ["The standard form of a quadratic form refers to the sum of the coefficients of each term after the quadratic form is expanded.", "If there exists an invertible linear transformation that makes the quadratic form contain only square terms, then this form containing only square terms is called the standard form of a quadratic form.", "The standard form of a quadratic form refers to the quadratic form being decomposable into the product of two linear forms.", "The standard form of a quadratic form refers to the form of the quadratic form containing cross terms."], "answer": "B", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the canonical form of a quadratic form?", "options": ["The canonical form of a quadratic form refers to its representation as a complete square form.", "The canonical form of a quadratic form means that the coefficients of all variables in the quadratic form are $1$.", "If the coefficients of a quadratic form in its standard form can only be $1$, $-1$, or $0$, then this standard form is called the canonical form.", "The canonical form of a quadratic form means that the coefficients of all variables in the quadratic form are equal."], "answer": "C", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What transformation can be used to reduce the quadratic form $\\boldsymbol{x}^{\\mathrm{T}} \\boldsymbol{A x}$ to its standard form?", "options": ["Using a linear transformation $\\boldsymbol{x}=\\boldsymbol{A y}$.", "Using an orthogonal transformation $\\boldsymbol{x}=\\boldsymbol{P y}$.", "Using an oblique transformation $\\boldsymbol{x}=\\boldsymbol{B y}$.", "Using a similarity transformation $\\boldsymbol{x}=\\boldsymbol{A}^{-1} \\boldsymbol{y}$."], "answer": "B", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "In the process of turning a real symmetric matrix $\\boldsymbol{A}$ into a diagonal matrix through orthogonal transformation, what are the elements of the diagonal matrix?", "options": ["The value of the trace of the real symmetric matrix $\\boldsymbol{A}$.", "The value of the determinant of the real symmetric matrix $\\boldsymbol{A}$.", "The eigenvectors of the real symmetric matrix $\\boldsymbol{A}$.", "The eigenvalues of the real symmetric matrix $\\boldsymbol{A}$."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What method might we need to use when orthogonally diagonalizing a real symmetric matrix $\\boldsymbol{A}$ if there are repeated eigenvalues?", "options": ["Jacobi method.", "Gaussian elimination.", "Schmidt orthogonalization.", "Power method."], "answer": "C", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What should the column vectors of the orthogonal matrix $\\boldsymbol{P}$ consist of?", "options": ["The column vectors of the original matrix $\\boldsymbol{A}$.", "The row vectors of the matrix $\\boldsymbol{A}$.", "The column vectors of the identity matrix $\\boldsymbol{E}$.", "Vectors after orthonormalization of eigenvectors."], "answer": "D", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "How does the orthogonal matrix $\\boldsymbol{P}$ operate with the real symmetric matrix $\\boldsymbol{A}$ to obtain a diagonal matrix?", "options": ["Through $\\boldsymbol{P} \\boldsymbol{A} - \\boldsymbol{P}$.", "Through $\\boldsymbol{P} \\boldsymbol{A} \\boldsymbol{P}}$.", "Through $\\boldsymbol{P} + \\boldsymbol{A}$.", "Through $\\boldsymbol{P}^{-1} \\boldsymbol{A P}$."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the congruence of matrices?", "options": ["If there exists an invertible matrix $\\boldsymbol{C}$ such that $\\boldsymbol{B}=\\boldsymbol{C} \\boldsymbol{A}$, then matrix $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are said to be congruent.", "If there exists an invertible matrix $\\boldsymbol{C}$ such that $\\boldsymbol{B}=\\boldsymbol{C}^{\\mathrm{T}} \\boldsymbol{A C}$, then matrix $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are said to be congruent.", "If there exists an invertible matrix $\\boldsymbol{C}$ such that $\\boldsymbol{B}=\\boldsymbol{C} \\boldsymbol{A C}$, then matrix $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are said to be congruent.", "If there exists an invertible matrix $\\boldsymbol{C}$ such that $\\boldsymbol{B}=\\boldsymbol{A C}$, then matrix $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are said to be congruent."], "answer": "B", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What property of quadratic forms does the inertia theorem tell us about?", "options": ["The total sum of the coefficients in the standard form of a quadratic form is invariant, not changing with transformation.", "The product of the coefficients in the standard form of a quadratic form is invariant, not changing with transformation.", "The number of zero coefficients in the standard form of a quadratic form is invariant, not changing with transformation.", "The number of positive and negative coefficients (i.e., the positive and negative inertia indices) in the standard form of a quadratic form is invariant, not changing with transformation."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "If the quadratic form $f=\\boldsymbol{x}^{\\top} \\boldsymbol{A x}$ changes to $f=k_{1} y_{1}^{2}+\\cdots+k_{r} y_{r}^{2}$ and $f=\\lambda_{1} z_{1}^{2}+\\cdots+\\lambda_{r} z_{r}^{2}$, does the number of positive coefficients remain the same in these two forms?", "options": ["No, because the sign of the coefficients might change.", "No, the number of positive coefficients is different.", "Yes, but only when the transformation is orthogonal.", "Yes, the number of positive coefficients is the same."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the number of positive coefficients in the standard form of a quadratic form called?", "options": ["Called the positive inertia index of the quadratic form.", "Called the rank of the quadratic form.", "Called the trace of the quadratic form.", "Called the determinant of the quadratic form."], "answer": "A", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the term for the number of negative coefficients in the standard form of a quadratic form?", "options": ["Called the rank of the quadratic form.", "Called the trace of the quadratic form.", "Called the negative inertia index of the quadratic form.", "Called the determinant of the quadratic form."], "answer": "C", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the necessary and sufficient condition for two real symmetric matrices to be congruent?", "options": ["They have the same rank and the same number of positive and negative inertia indexes.", "They have the same trace and the same determinant.", "They have the same eigenvalues.", "They have the same eigenvectors."], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is a positive definite quadratic form?", "options": ["If for every non-zero vector $\\boldsymbol{x}$, it holds that $f(\\boldsymbol{x})=x^{\\top} A x>0$, then the quadratic form $f$ is called positive definite, and the corresponding symmetric matrix $\\boldsymbol{A}$ is also called positive definite.", "If for every non-zero vector $\\boldsymbol{x}$, it holds that $f(\\boldsymbol{x})=x^{\\top} A x\\geq0$, then the quadratic form $f$ is called positive definite.", "If for every non-zero vector $\\boldsymbol{x}$, it holds that $f(\\boldsymbol{x})=x^{\\top} A x=0$, then the quadratic form $f$ is called positive definite.", "If for every non-zero vector $\\boldsymbol{x}$, it holds that $f(\\boldsymbol{x})=x^{\\top} A x<0$, then the quadratic form $f$ is called positive definite."], "answer": "A", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the necessary and sufficient condition for a quadratic form $f=\\boldsymbol{x}^{\\mathrm{T}} \\boldsymbol{A} \\boldsymbol{x}$ to be positive definite?", "options": ["The standard form has at least one of its $n$ coefficients equal to zero.", "All $n$ coefficients of its standard form are non-positive.", "All $n$ coefficients of its standard form are negative.", "All $n$ coefficients of its standard form are positive, i.e., all $n$ coefficients of its canonical form are 1, that is, its positive inertia index is equal to $n$."], "answer": "D", "topic": "College--Linear Algebra--Quadratic Forms"} |
| {"question": "What is the necessary and sufficient condition for a symmetric matrix $A$ to be positive definite?", "options": ["All eigenvalues of matrix $A$ are positive.", "All eigenvalues of matrix $A$ are non-positive.", "All eigenvalues of matrix $A$ are zero.", "At least one eigenvalue of matrix $A$ is positive."], "answer": "A", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the Hurwitz theorem?", "options": ["A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that all the leading principal minors of $A$ are negative.", "A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that any leading principal minor of $A$ is positive.", "A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that all the leading principal minors of $A$ are positive.", "A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that at least one of the leading principal minors of $A$ is positive."], "answer": "C", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "What is the necessary and sufficient condition for the symmetric matrix $A$ to be negative definite?", "options": ["All leading principal minors of the symmetric matrix $A$ are positive.", "The odd-order leading principal minors of the symmetric matrix $A$ are negative, while the even-order leading principal minors are positive.", "The even-order leading principal minors of the symmetric matrix $A$ are negative, while the odd-order leading principal minors are positive.", "All leading principal minors of the symmetric matrix $A$ are negative."], "answer": "B", "topic": "College--Linear Algebra--Matrices"} |
| {"question": "$P(\\varnothing)=$", "options": ["1", "0", "0.5", "-1"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$P(\\bar{A})=$", "options": ["$1-P(A)$", "$1$", "$P(A)-1$", "$P(A)$"], "answer": "A", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "When $A_{1}, A_{2}, \\cdots, A_{n}$ are mutually exclusive events, $P\\left(A_{1} \\cup A_{2} \\cup \\cdots \\cup A_{n}\\right)=$", "options": ["$\\prod_{i=1}^{n} P\\left(A_{i}\\right)$", "$\\sum_{i=1}^{n} P\\left(A_{i}\\right)$", "$P(A_n)$", "$P(A_1)$"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$P(A \\cup B)=$", "options": ["$P(A)-P(B)$", "$P(A B)$", "$P(A)+P(B)$", "$P(A)+P(B)-P(A B)$"], "answer": "D", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "When $P(A)>0$, $P(A B)=$", "options": ["$ P(A)$", "$P(B \\mid A) P(A)$", "$ P(B)$", "$P(B \\mid A) $"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$B_{1}, B_{2}, \\cdots, B_{n}$ are a partition of the entire sample space, and $P\\left(B_{i}\\right)>0, i=1,2, \\cdots, n$, $P\\left(A \\mid B_{1}\\right) P\\left(B_{1}\\right)+P\\left(A \\mid B_{2}\\right) P\\left(B_{2}\\right)+\\cdots$ $+P\\left(A \\mid B_{n}\\right) P\\left(B_{n}\\right)=$<<<Answer>>>", "options": ["$P(A)-P(B)$", "$P(A)$", "$P(B)$", "$P(AB)$"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$P(A)=P\\left(A \\mid B_{1}\\right) P\\left(B_{1}\\right)+P\\left(A \\mid B_{2}\\right) P\\left(B_{2}\\right)+\\cdots$ $+P\\left(A \\mid B_{n}\\right) P\\left(B_{n}\\right)$, where $B_{1}, B_{2}, \\cdots, B_{n}$ are a partition of the entire sample space, and $P\\left(B_{i}\\right)>0, i=1,2, \\cdots, n$, what is this formula called?", "options": ["Multiplication Formula", "Addition Formula", "Bayes' Formula", "Law of Total Probability"], "answer": "D", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$B_{1}, B_{2}, \\cdots, B_{n}$ are partitions of the entire sample space, and $P\\left(B_{i}\\right)>0, i=1,2, \\cdots, n$, $\\frac{P\\left(A \\mid B_{i}\\right) P\\left(B_{i}\\right)}{\\sum_{j=1}^{n} P\\left(A \\mid B_{j}\\right) P\\left(B_{j}\\right)}=$", "options": ["$P\\left(B_{i} \\mid A\\right)$", "$P(AB)$", "$P(A)$", "$P(B_i)$"], "answer": "A", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$P(B_{i} | A)=\\frac{P(A | B_{i}) P(B_{i})}{\\sum_{j=1}^{n} P(A | B_{j}) P(B_{j})}, i=1,2, \\cdots, n$, where $B_{1}, B_{2}, \\cdots, B_{n}$ are a partition of the entire sample space, and $P(A)>0, P(B_{i})>0$, $i=1,2, \\cdots, n$, what is this formula called?", "options": ["Formula of Total Probability", "Bayes' Theorem", "Multiplication Rule", "Addition Rule"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "$A \\cup B=$", "options": ["$B$", "$A$", "$B \\cap A$", "$B \\cup A$"], "answer": "D", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$A \\cap B=$", "options": ["$B \\cup A$", "$B$", "$B \\cap A$", "$A$"], "answer": "C", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$A \\cup(B \\cup C)=$", "options": ["$(A \\cap B) \\cap C$", "$(A \\cup B) \\cup C$", "$B \\cap A$", "$B \\cup A$"], "answer": "B", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$A \\cap(B \\cap C)=$", "options": ["$(A \\cup B) \\cup C$", "$B \\cap A$", "$B \\cup A$", "$(A \\cap B) \\cap C$"], "answer": "D", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$A \\cup(B \\cap C)=$", "options": ["$(A \\cap B) \\cap C$", "$(A \\cap B) \\cup(A \\cap C)$", "$(A \\cup B) \\cup C$", "$(A \\cup B) \\cap(A \\cup C)$"], "answer": "D", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$A \\cap(B \\cup C)=$", "options": ["$(A \\cup B) \\cap(A \\cup C)$", "$(A \\cap B) \\cap C$", "$(A \\cup B) \\cup C$", "$(A \\cap B) \\cup(A \\cap C)$"], "answer": "D", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "$\\overline{A \\cup B}=$", "options": ["$(A \\cup B) \\cap (A \\cup C)$", "$\\overline A \\cup \\overline B$", "$\\overline A \\cap \\overline B$", "$(A \\cap B) \\cup(A \\cap C)$"], "answer": "C", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "The complement of $A \\cap B$", "options": ["$(A \\cap B) \\cup(A \\cap C)$", "$\\overline{A} \\cap \\overline{B}$", "$(A \\cup B) \\cap(A \\cup C)$", "$\\overline{A} \\cup \\overline{B}$"], "answer": "D", "topic": "College--Discrete Mathematics--Set Theory"} |
| {"question": "What are the classical probability model and the geometric probability model?", "options": ["Geometric probability model", "Unequal probability model", "Equal probability model", "Classical probability model"], "answer": "C", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "What is the difference between the classical model and the geometric model in that the sample space of the classical model contains only what elements?", "options": ["One", "A finite number", "An infinite number", "Equally likely"], "answer": "B", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "The difference between the classical probability model and the geometric probability model lies in, what elements does the sample space of the geometric probability model contain only?", "options": ["A finite number", "One", "Equally likely", "An infinite number"], "answer": "D", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Probability calculation in the classical model: If the total number of elementary events in the sample space is $n$, and event $A$ contains $k_{A}$ of these elementary events, then the probability of event $A$ occurring is $P(A)=$", "options": ["$\\frac{n}{k_{A}}$", "$n$", "$k_{A}$", "$\\frac{k_{A}}{n}$"], "answer": "D", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Probability calculation in geometric models: Let the sample space be $\\Omega, D_{A}$ is a measurable subregion of $\\Omega$. Picking a point randomly from $\\Omega$, denote the event $A$ as \"the point falls within the region $D_{A}$\", then the probability of event $A$ happening is $P(A)=$, where $\\mu_{A}, \\mu_{\\Omega}$ are the geometric measures (like length, area, volume, etc.) of $D_{A}$ and $\\Omega$ respectively=", "options": ["$\\mu_{\\Omega}$", "$\\mu_{A}$", "$\\frac{\\mu_{\\Omega}}{\\mu_{A}}$", "$\\frac{\\mu_{A}}{\\mu_{\\Omega}}$"], "answer": "D", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Under the same conditions, independently and repeatedly conducting $n$ random experiments, this type of experiment is called.", "options": ["Repeated experiment", "Random experiment", "Independent repeated experiment", "Independent experiment"], "answer": "C", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Under the same conditions, if an experiment is repeated $n$ times independently and randomly, and the experiment only has two possible outcomes: occurrence or non-occurrence, then we call this series of independent repeated random experiments <<<Answer1>>>", "options": ["independent repeated experiments", "Bernoulli trial", "$n$ fold Bernoulli trial", "random experiment"], "answer": "C", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Given in a trial, the probability of event $A$ occurring is $p(0<p<1)$, then in $n$ repeated Bernoulli trials, the probability of event $A$ occurring exactly $k$ times is", "options": ["$\\mathrm{C}_{n}^{k} p^{k}(k=0,1, \\cdots, n)$", "$\\mathrm{C}_{n}^{k} (k=0,1, \\cdots, n)$", "$\\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}(k=0,1, \\cdots, n)$", "$\\mathrm{C}_{n}^{k} (1-p)^{n-k}(k=0,1, \\cdots, n)$"], "answer": "C", "topic": "College--Probability and Statistics--Random Events and Probability"} |
| {"question": "Let the sample space of a random experiment be $S. X=X(e)$ is a real-valued single-valued function defined on the sample space $S$. Call $X=X(e)$ as", "options": ["random variable", "random experiment", "random constant", "sample space"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "Let $X$ be a random variable, $x$ be any real number, the function $F(x)=P\\{X \\leqslant x\\},-\\infty<x<+\\infty$ is called the", "options": ["distribution function", "probability density function", "random function", "sine function"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "A random variable that can take on a finite number of values or countably infinite many values is called", "options": ["integer random variable", "discrete random variable", "decimal random variable", "continuous random variable"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "A random variable that can take either a finite number of values or countably infinite many values is called", "options": ["the distribution law of continuous random variable $X$", "the distribution law of discrete random variable $Y$", "the distribution law of discrete random variable $X$", "the distribution law of continuous random variable $Y$"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "For the distribution function $F(x)$ of a random variable $X$, if there exists a non-negative integrable function $f(x)$, such that for any real number $x$, $F(x)=\\int_{-\\infty}^{x} f(t) \\mathrm{d} t$, then $X$ is referred to as", "options": ["The distribution law of a discrete random variable $X$", "A continuous random variable", "The distribution law of a continuous random variable $X$", "A discrete random variable"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "For the distribution function $F(x)$ of a random variable $X$, if there exists a non-negative integrable function $f(x)$, such that for any real number $x$, $F(x)=\\int_{-\\infty}^{x} f(t) \\mathrm{d} t$, then $f(x)$ is called the <<<Answer1>>> of $X$, also known as the probability density <<<Answer2>>>.", "options": ["continuous random variable", "discrete random variable", "probability density function", "distribution function"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "The distribution function of a continuous random variable is <<<Answer>>> continuous", "options": ["Not necessarily", "Under certain conditions", "Definitely not", "Inevitably"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "The probability density function for a continuous random variable is <<<Answer>>> continuous", "options": ["under certain conditions", "definitely not", "necessarily", "not necessarily"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "The probability mass function of a random variable $X$ is $P\\{X=k\\}=p^{k}(1-p)^{1-k}, 0<p<1, k=0,1$. If $X$ can only take the values 0 and 1, what distribution does this law of distribution belong to?", "options": ["Hypergeometric distribution law", "$(0-1)$ distribution law", "Binomial distribution law", "Geometric distribution law"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "The $(0-1)$ distribution is also known as", "options": ["Geometric distribution", "Binomial distribution", "Two-point distribution", "Hypergeometric distribution"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "$P\\{X=k\\}=\\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \\cdots, n$. What distribution's distribution law is this?", "options": ["Binomial distribution law", "$(0-1)$ distribution law", "Geometric distribution law", "Hypergeometric distribution law"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "In an $n$ repeated Bernoulli trial, the probability of event $A$ occurring is $p$, the random variable $X$ represents the number of times event $A$ occurs, then", "options": ["$X \\sim b(p, n)$", "$X \\sim a(n, p)$", "$X \\sim b(n)$", "$X \\sim b(n, p)$"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "What is the relationship between the binomial distribution $b(1, p)$ and the $(0-1)$ distribution when $n=1$?", "options": ["Different", "Linearly related", "Unrelated", "Identical"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "$P\\{X=k\\}=\\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \\cdots, n$. What is the distribution law of?", "options": ["Hypergeometric distribution law", "$(0-1)$ distribution law", "Geometric distribution law", "Binomial distribution law"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "$P\\{X=k\\}=\\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \\cdots, n$. What is the distribution law of?", "options": ["$(0-2)$ distribution", "Binomial distribution", "Hypergeometric distribution", "Geometric distribution"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "What distribution does the law of distribution $P\\{X=k\\}=\\frac{\\mathrm{C}_{M}^{k} \\mathrm{C}_{N-k}^{n-k}}{\\mathrm{C}_{N}^{n}}, k=0,1, \\cdots, l$, where $l=\\min \\{n, M\\}, n \\leqslant N$ belong to?", "options": ["Hypergeometric distribution", "$(0-1)$ distribution", "Geometric distribution", "Binomial distribution"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "In product sampling inspection (without replacement), assuming there are $M$ defective items among $N$ products, and $n$ products are randomly drawn, if the number of found defective items is denoted by $X$, then what distribution does $X$ follow with parameters $N, M, n$?", "options": ["Geometric distribution", "Hypergeometric distribution", "$(0-2)$ distribution", "Binomial distribution"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "$P\\{X=k\\}=\\frac{\\lambda^{k} \\mathrm{e}^{-\\lambda}}{k !}, k=0,1,2, \\cdots$, where $\\lambda>0$ is a constant. Which distribution's distribution law is this?", "options": ["Geometric distribution's distribution law", "Binomial distribution's distribution law", "Poisson distribution's distribution law", "$(0-1)$ distribution's distribution law"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "If the probability density of a continuous random variable $X$ is\n$f(x)= \\begin{cases}\\frac{1}{b-a}, & a<x<b, \\\\ 0, & \\text { otherwise, }\\end{cases}$then, $X$ is said to follow what distribution in the interval $(a, b)$?", "options": ["Exponential distribution", "Gamma distribution", "Uniform distribution", "Normal distribution"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "If the probability density of a continuous random variable $X$ is\n\n$$\nf(x)= \\begin{cases}\\lambda \\mathrm{e}^{-\\lambda x}, & x>0, \\\\ 0, & \\text { otherwise, }\\end{cases}\n$$\n\nwhere $\\lambda>0$ is a constant, then $X$ is said to follow a distribution with parameter $\\lambda$ of what kind?", "options": ["Gamma distribution", "Uniform distribution", "Normal distribution", "Exponential distribution"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "A random variable $X$ following an exponential distribution has the following property:\nFor any $s, t>0$, it holds that\n$$\nP\\{X>s+t \\mid X>s\\}=P\\{X>t\\} .\n$$\nWhat is this property called?", "options": ["Memorylessness", "Distributiveness", "Exponentiality", "With memory"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "If the probability density of the continuous random variable $X$ is\n\n$$\nf(x)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} \\mathrm{e}^{-\\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}}, \\quad-\\infty<x<+\\infty,\n$$\n\nwhere $\\mu, \\sigma(\\sigma>0)$ are constants, then $X$ is said to follow a distribution with parameters $\\mu, \\sigma$ of what type?", "options": ["Uniform distribution", "Exponential distribution", "Normal distribution", "Gamma distribution"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "How to find the probability density of $Y=g(X)$ given that the random variable $X$ has a probability density of $f(x)$ through definition?", "options": ["$F_{Y}(y)=\\int_{f(x) \\leqslant y} g(x) \\mathrm{d} x $", "$F_{Y}(y)=\\int_{g(x) \\leqslant y} f(x) \\mathrm{d} x $", "$g(x)$", "$f(x)$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "Suppose random variable $X$ has probability density $f_{X}(x),-\\infty<x<+\\infty$, and suppose the function $g(x)$ is differentiable everywhere and always has $g^{\\prime}(x)>0$ (or always has $\\left.g^{\\prime}(x)<0\\right)$, then $Y=g(X)$ is a continuous random variable, whose probability density is <<<Answer>>>> where $\\alpha=\\min \\{g(-\\infty), g(+\\infty)\\}, \\beta=\\max \\{g(-\\infty), g(+\\infty)\\}, h(y)$ is the inverse function of $g(x)$.", "options": ["$f_{X}[h(y)]$", "Answer:$f_{Y}(y)= \\begin{cases}f_{X}[h(y)]\\left|h^{\\prime}(y)\\right|, & \\alpha<y<\\beta, \\\\ 0, & \\text { others, }\\end{cases}$", "0", "Answer:$f_{Y}(y)= \\begin{cases}f_{X}[h(y)]\\left|h^{\\prime}(y)\\right|, & \\alpha>y>\\beta, \\\\ 0, & \\text { others, }\\end{cases}$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Random Variables"} |
| {"question": "In general, suppose $E$ is a random experiment with sample space $S$. If $X=X(e)$ and $Y=Y(e)$ are random variables defined on $S$, the vector $(X, Y)$ formed by them is called what?", "options": ["One-dimensional random vector", "Three-dimensional random vector", "Two-dimensional random vector", "Four-dimensional random vector"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let $(X, Y)$ be a two-dimensional random variable, for any real numbers $x, y$, the bivariate function:\n\n$$\nF(x, y)=P\\{(X \\leqslant x) \\cap(Y \\leqslant y)\\} \\xlongequal{\\text { denoted by }} P\\{X \\leqslant x, Y \\leqslant y\\}\n$$\n\nis called <<<Answer1>>>, or known as <<<Answer2>>>.", "options": ["The sine function of the two-dimensional random variable $(X, Y)$", "The probability density function of the two-dimensional random variable $(X, Y)$", "The random function of the two-dimensional random variable $(X, Y)$", "The distribution function of the two-dimensional random variable $(X, Y)$"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "If a two-dimensional random variable $(X, Y)$ can take a finite number of pairs or a countably infinite number of pairs, what is $(X, Y)$ called?", "options": ["One-dimensional discrete random variable", "One-dimensional continuous random variable", "Two-dimensional continuous random variable", "Two-dimensional discrete random variable"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let the two-dimensional discrete random variable $(X, Y)$ take all possible values $(x_{i}, y_{j}), i, j=1,2, \\cdots$, and let $P\\{X=x_{i}, Y=y_{j}\\}=$ $p_{ij}, i, j=1,2, \\cdots$. According to the definition of probability,\n$$\np_{ij} \\geq 0, \\quad \\sum_{i=1}^{\\infty} \\sum_{j=1}^{\\infty} p_{ij}=1\n$$\nWhat do we call $P\\{X=x_{i}, Y=y_{j}\\}=p_{ij}, i, j=1,2, \\cdots$?", "options": ["The distribution law of the two-dimensional continuous random variable $(X, Y)$", "The distribution law of the two-dimensional discrete random variable $(X, Y)$", "The distribution law of the one-dimensional discrete random variable X", "The distribution law of the one-dimensional discrete random variable Y"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "For the distribution function $F(x, y)$ of a two-dimensional random variable $(X, Y)$, if there exists a non-negative integrable function $f(x, y)$, such that for any $x, y$, it holds that\n$$\nF(x, y)=\\int_{-\\infty}^{y} \\int_{-\\infty}^{x} f(u, v) \\mathrm{d} u \\mathrm{~d} v,\n$$\nthen what is $(X, Y)$ called?", "options": ["Two-dimensional continuous random variable", "Two-dimensional discrete random variable", "One-dimensional discrete random variable", "One-dimensional continuous random variable"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "For the distribution function $F(x, y)$ of a two-dimensional random variable $(X, Y)$, if there exists a non-negative integrable function $f(x, y)$, such that for any $x, y$, it holds that\n$$\nF(x, y)=\\int_{-\\infty}^{y} \\int_{-\\infty}^{x} f(u, v) \\mathrm{d} u \\mathrm{~d} v,\n$$\nthen what is the function $f(x, y)$ called?", "options": ["The distribution function of the two-dimensional random variable $(X, Y)$", "The distribution function of the one-dimensional random variable $(X, Y)$", "The probability density of the two-dimensional random variable $(X, Y)$", "The probability density of the one-dimensional random variable $(X, Y)$"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The two-dimensional random variable $(X, Y)$, as a whole, has a distribution function $F(x, y)$. Both $X$ and $Y$ are random variables, each with its own distribution function, denoted as $F_{X}(x), F_{Y}(y)$, respectively called the two-dimensional random variable $(X, Y)$'s <<<Answer1>>> and <<<Answer2>>>", "options": ["marginal distribution function with respect to $Y$, marginal distribution function with respect to $X$", "distribution function with respect to $Y$, distribution function with respect to $X$", "marginal distribution function with respect to $X$, marginal distribution function with respect to $Y$", "distribution function with respect to $X$, distribution function with respect to $Y$"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The marginal distribution function can be determined by the <<<Answer>>> of $(X, Y)$.\n$$\nF_{X}(x)=P\\{X \\leqslant x\\}=P\\{X \\leqslant x, Y<+\\infty\\}=F(x,+\\infty) .\n$$", "options": ["probability density function $f(x, y)$", "random function", "sine function", "distribution function $F(x, y)$"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The two-dimensional random variable $(X, Y)$ as a whole has a distribution function $F(x, y)$. Both $X$ and $Y$ are random variables and have their own distribution functions, which are denoted as $F_{X}(x), F_{Y}(y)$, respectively referred to as the two-dimensional random variable $(X, Y)$'s <<<Answer1>>> and <<<\nAnswer2>>>", "options": ["distribution law with respect to $Y$, distribution law with respect to $X$", "marginal distribution law with respect to $X$, marginal distribution law with respect to $Y$", "distribution law with respect to $X$, distribution law with respect to $Y$", "marginal distribution law with respect to $Y$, marginal distribution law with respect to $X$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "For the continuous random variables $(X, Y)$, suppose its probability density is $f(x, y)$, define\n$$\nf_{X}(x)=\\int_{-\\infty}^{+\\infty} f(x, y) \\mathrm{d} y, \\quad f_{Y}(y)=\\int_{-\\infty}^{+\\infty} f(x, y) \\mathrm{d} x,\n$$\nrespectively, $f_{X}(x), f_{Y}(y)$ are called the <<<Answer1>>> and <<<Answer2>>>", "options": ["probability density with respect to $Y$, probability density with respect to $X$", "marginal probability density with respect to $X$, marginal probability density with respect to $Y$", "marginal probability density with respect to $Y$, marginal probability density with respect to $X$", "probability density with respect to $X$, probability density with respect to $Y$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let $(X, Y)$ be a two-dimensional discrete random variable. For a fixed $j$, if $P{Y = y_i} > 0$, then \n$$\nP\\left\\{X=x_{i} \\mid Y=y_{j}\\right\\}=\\frac{P\\left\\{X=x_{i}, Y=y_{j}\\right\\}}{P\\left\\{Y=y_{j}\\right\\}}=\\frac{p_{i j}}{p_{j}}, i=1,3, \\cdots\n$$\nis called what under the condition of $Y=y_{j}$?", "options": ["The conditional distribution law of random variable $X$", "The conditional distribution law of random variable $(X, Y)$", "The conditional distribution law of random variable $Y$", "None of the above"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "For a fixed $i$, if $P\\left\\{X=x_{i}\\right\\}>0$, then\n$$\nP\\left\\{Y=y_{j} \\mid X=x_{i}\\right\\}=\\frac{P\\left\\{X=x_{i}, Y=y_{j}\\right\\}}{P\\left\\{X=x_{i}\\right\\}}=\\frac{p_{ij}}{p_{i} .}, j=1,2, \\cdots\n$$\nis referred to as what under the condition $X=x_{i}$?", "options": ["The conditional distribution law of random variable $Y$", "None of the above", "The conditional distribution law of random variable $(X, Y)$", "The conditional distribution law of random variable $X$"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The probability density of the two-dimensional random variable $(X, Y)$ is $f(x, y)$. The marginal probability density of $(X, Y)$ with respect to $Y$ is $f_{Y}(y)$. If for a fixed $y, f_{Y}(y)>0$, then $\\frac{f(x, y)}{f_{Y}(y)}$ is called what under the condition $Y=y$?", "options": ["Conditional probability", "None of these", "Conditional probability density", "Probability density"], "answer": "C", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The probability density of the two-dimensional random variables $(X, Y)$ is $f(x, y)$. The marginal probability density of $(X, Y)$ with respect to $Y$ is $f_{Y}(y)$. If for a fixed $y, f_{Y}(y)>0$, then $\\frac{f(x, y)}{f_{Y}(y)}$ is called the conditional probability density of $X$ given $Y=y$, denoted as", "options": ["$$f_{\\mathrm{X|Y}}(x \\mid y)=\\frac{f(x, y)}{f_{Y}(y)}$$", "$$f_{\\mathrm{X|Y}}(x \\mid y)=f_{Y}(y$$", "none of the above", "$$f_{\\mathrm{X|Y}}(x \\mid y)=f(x, y)$$"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "What conditions must a conditional probability density satisfy?", "options": ["None of the above", "$f_{X|Y}(x \\mid y)=\\frac{f(x, y)}{f_{Y}(y)} \\geq 0$", "$f_{X|Y}(x \\mid y)=\\frac{f(x, y)}{f_{Y}(y)} = 0$", "$f_{X|Y}(x \\mid y)=\\frac{f(x, y)}{f_{Y}(y)} \\leq 0$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "The conditional probability density satisfies the condition: $f_{X|Y}(x \\mid y)=\\frac{f(x, y)}{f_{Y}(y)} \\geqslant 0$;\n$$\n\\int_{-\\infty}^{+\\infty} f_{X Y}(x \\mid y) \\mathrm{d} x=\\int_{-\\infty}^{+\\infty} \\frac{f(x, y)}{f_{Y}(y)} \\mathrm{d} x=\\frac{1}{f_{Y}(y)} \\int_{-\\infty}^{+\\infty} f(x, y) \\mathrm{d} x=1 .\n$$\n$$\nF_{X|Y}(x \\mid y)=P\\{X \\leqslant x \\mid Y=y\\}=\\int_{-\\infty}^{x} \\frac{f(x, y)}{f_{Y}(y)} \\mathrm{d} x .\n$$\nWhat is $\\int_{-\\infty}^{x} f_{\\mathrm{M} Y}(x \\mid y) \\mathrm{d} x=\\int_{-\\infty}^{x} \\frac{f(x, y)}{f_{Y}(y)} \\mathrm{d} x$ under the condition $Y=y$ for $X$ called?", "options": ["Probability Law", "Conditional Probability Density", "Distribution Function", "Conditional Distribution Function"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let $X, Y$ be two independent random variables, with their distribution functions being $F_{X}(x)$ and $F_{Y}(y)$ respectively. What is the distribution function of $M=\\max\\{X, Y\\}$?", "options": ["None of the above", "$F_{\\max }(z) = F_{X}(z) F_{Y}(z)$", "$F_{\\max }(z) = F_{X}(z)$", "$F_{\\max }(z) = F_{Y}(z)$"], "answer": "B", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let $X, Y$ be two independent random variables, with distribution functions $F_{X}(x)$ and $F_{Y}(y)$ respectively. What is the distribution function of $N=\\min \\{X, Y\\}$?", "options": ["$F_{\\min }(z) =P\\{N \\leqslant z\\}=1-[1-F_{Y}(z)]$", "None of the above", "$F_{\\min }(z) =P\\{N \\leqslant z\\}=1-[1-F_{X}(z)]$", "$F_{\\min }(z) =P\\{N \\leqslant z\\}=1-[1-F_{X}(z)][1-F_{Y}(z)]$"], "answer": "D", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "When $X_{1}, X_{2}, \\cdots, X_{n}$ are mutually independent and have the same distribution function $F(x)$, what is $F_{\\max }(z)$ equal to?", "options": ["$F(z)^{n}$", "$nF(z)$", "$\\frac{F(z)}{n}$", "None of the above"], "answer": "A", "topic": "College--Probability and Statistics--Distribution of Multidimensional Random Variables"} |
| {"question": "Let the probability distribution of a discrete random variable $X$ be \n\n$$\nP\\left\\{X=x_{k}\\right\\}=p_{k}, \\quad k=1,2, \\cdots .\n$$\n\nIf the series $\\sum_{k=1}^{\\infty} x_{k} p_{k}$ is absolutely convergent, the sum of the series $\\sum_{k=1}^{\\infty} x_{k} p_{k}$ is called what of the random variable $X$?", "options": ["Standard deviation", "Variance", "Covariance", "Expected value"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "For a continuous random variable $X$ with probability density $f(x)$, if the integral $\\int_{-\\infty}^{+\\infty} x f(x) \\mathrm{d} x$ is absolutely convergent, then the value of the integral $\\int_{-\\infty}^{+\\infty} x f(x) \\mathrm{d} x$ is called what of the random variable $X$?", "options": ["Mathematical Expectation", "Covariance", "Variance", "Standard Deviation"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$X$ is a discrete random variable, its distribution law is $P\\left\\{X=x_{k}\\right\\}=p_{k}, k=1,2, \\cdots$. If $\\sum_{k=1}^{\\infty} g\\left(x_{k}\\right) p_{k}$ is absolutely convergent, then $E(Y)=$", "options": ["None of the above", "$\\sum_{k=1}^{\\infty} p_{k}$", "$\\sum_{k=1}^{\\infty} g\\left(x_{k}\\right) $", "$\\sum_{k=1}^{\\infty} g\\left(x_{k}\\right) p_{k}$"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Let $Y$ be a function of the random variable $X$: $Y=g(X)$ (where $g$ is a continuous function). If $X$ is a continuous random variable with probability density $f(x)$, and if $\\int_{-\\infty}^{+\\infty} g(x) f(x) \\mathrm{d} x$ is absolutely convergent, then $E(Y)=$", "options": ["$\\int_{-\\infty}f(x) \\mathrm{d} x$", "$\\int_{-\\infty} g(x) \\mathrm{d} x$", "$f(x)g(x)$", "$\\int_{-\\infty}^{+\\infty} g(x) f(x) \\mathrm{d} x$"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Assuming that the improper integrals mentioned below are absolutely convergent. If $(X, Y)$ has the joint probability density $f(x, y)$, then $E(X)=$", "options": ["$\\int_{-\\infty}^{+\\infty} y f_{Y}(y) \\mathrm{d} y$", "$\\int_{-\\infty}^{+\\infty} y f_{X}(x) \\mathrm{d} y$", "$\\int_{-\\infty}^{+\\infty} x f_{Y}(y) \\mathrm{d} x $", "$\\int_{-\\infty}^{+\\infty} x f_{X}(x) \\mathrm{d} x $"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Assuming that the improper integrals below are all absolutely convergent. If $(X, Y)$ is the joint probability density of $f(x, y)$, then $E(Y)=$", "options": ["$\\int_{-\\infty}^{+\\infty} x f_{Y}(y) \\mathrm{d} x $", "$\\int_{-\\infty}^{+\\infty} x f_{X}(x) \\mathrm{d} x $", "$\\int_{-\\infty}^{+\\infty} y f_{X}(x) \\mathrm{d} y$", "$\\int_{-\\infty}^{+\\infty} y f_{Y}(y) \\mathrm{d} y$"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Assuming that the improper integrals mentioned below are all absolutely convergent. If the joint probability density of $(X, Y)$ is $f(x, y)$ and $Z=g(X, Y)$, then $E(Z)=$", "options": ["$\\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} g(x, y) \\mathrm{d} x \\mathrm{~d} y$", "$\\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} f(x, y) \\mathrm{d} x \\mathrm{~d} y$", "$\\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} g(x, y) f(x, y) \\mathrm{d} x \\mathrm{~d} y$", "None of these"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "If $C$ is a constant, then $E(C)=$.", "options": ["$C^2$", "$2C$", "$C$", "$\\sqrt{C}$"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$X$ is a random variable, $C$ is a constant, then $E(CX)=$.", "options": ["$CE(X)$", "$C^2E(X)$", "$E(X)$", "$C$"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Let $X, Y$ be two random variables, then $E(X+Y)=$.", "options": ["$E(X)$", "$E(X)+E(Y)$", "$E(Y)$", "$E(X)E(Y)$"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Given that $X, Y$ are independent random variables, then $E(X Y)=$", "options": ["$E(X)+E(Y)$", "$E(X)E(Y)$", "$E(X)$", "$E(Y)$"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Let $X$ be a random variable, if $E\\left\\{[X-E(X)]^{2}\\right\\}$ exists, then $E\\left\\{[X-E(X)]^{2}\\right\\}$ is called what of $Y$?", "options": ["Covariance", "Variance", "Mathematical Expectation", "Standard Deviation"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Given $\\sigma(X)=\\sqrt{D(X)}$, it is called", "options": ["Covariance", "Standard Deviation", "Variance", "Mathematical Expectation"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "If $C$ is a constant, then $D(C)=$", "options": ["$C$", "$C^2$", "0", "$C^2D(x)$"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "If $X$ is a random variable, $C$ is a constant, then $D(CX)=$", "options": ["$CD(X)$", "$C^{2}D(X)$", "$C$", "$D(X)$"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "If $X$ is a random variable, and $C$ is a constant, then $D(X+C)=$", "options": ["$D(X)$", "$CD(X)$", "$C$", "$C^{2} D(X)$"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Let $X, Y$ be two random variables, then $D(X+Y)=$", "options": ["$D(X)+D(Y) $", "$D(X)+D(Y)+2 \\operatorname{Cov}(X, Y) $", "$\\operatorname{Cov}(X, Y) $", "$D(X)+D(Y)-2 \\operatorname{Cov}(X, Y) $"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "The necessary and sufficient condition for $D(X)=0$ is", "options": ["$P\\{X=E(X)\\}=1 $", "$P\\{X=D(X)\\}=1 $", "$P\\{X=E(X)\\}=0 $", "$P\\{X=D(X)\\}=0 $"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$E\\{[X-E(X)][Y-E(Y)]\\}$ is called the _____ of random variables $X$ and $Y$", "options": ["variance", "mathematical expectation", "standard deviation", "covariance"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$\\rho_{X Y}=\\frac{\\operatorname{Cov}(X, Y)}{\\sqrt{D(X)} \\sqrt{D(Y)}}$ is called the", "options": ["mathematical expectation", "variance", "correlation coefficient", "standard deviation"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$\\mathrm{Cov}(X, Y)=$", "options": ["$E(XY)-E(X)E(Y)$", "$D(X)$", "$E(X)E(Y)-E(XY)$", "0"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$\\operatorname{Cov}(X, X)=$", "options": ["0", "$D(X)$", "$E(X) E(Y)-E(X Y)$", "$E(X Y)-E(X) E(Y)$"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$\\operatorname{Cov}(X, a)=$", "options": ["$D(X)$", "$E(X Y)-E(X) E(Y)$", "$E(X) E(Y)-E(X Y)$", "0"], "answer": "D", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$D(X \\pm Y)=$", "options": ["$D(X)+D(Y) $", "$D(X)-D(Y) $", "$D(X)+D(Y) \\pm 2 \\operatorname{Cov}(X, Y)$", "$D(X)-D(Y) \\pm 2 \\operatorname{Cov}(X, Y)$"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$\\operatorname{Cov}(X, Y)=$", "options": ["$D(Y, X)$", "$\\operatorname{Cov}(Y, X)$", "$D(X)+D(Y) \\pm 2 \\operatorname{Cov}(X, Y)$", "$E(Y, X)$"], "answer": "B", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "When $a, b$ are constants, $\\operatorname{Cov}(a X, b Y)=$<<<Answer>>>", "options": ["$a b \\operatorname{Cov}(X, Y)$", "$\\operatorname{Cov}(X, Y)$", "$2 a b \\operatorname{Cov}(X, Y)$", "$a b$"], "answer": "A", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "$Cov(X_1 \\pm X_2, Y) = $", "options": ["$Cov(X_1, Y) - Cov(X_2, Y)$", "$Cov(X_1, Y)Cov(X_2, Y)$", "$Cov(X_1, Y) \\pm Cov(X_2, Y)$", "$Cov(X_1, Y) + Cov(X_2, Y)$"], "answer": "C", "topic": "College--Probability and Statistics--Numerical Characteristics of Random Variables"} |
| {"question": "Let the population $X$ have a distribution containing unknown parameters $\\theta_{1}, \\theta_{2}, \\cdots, \\theta_{k}$, $X_{1}, X_{2}, \\cdots, X_{n}$ are samples from $X$. Suppose the first $k$ moments of the population $X$, $\\mu_{l}=E\\left(X^{t}\\right)(l=1$, $2, \\cdots, k)$, exist, using the sample moments $A_{l}=\\frac{1}{n} \\sum_{i=1}^{n} X_{i}^{l}$ as the estimators for the corresponding population moments $\\mu_{t}$, and using continuous functions of sample moments as estimators for continuous functions of population moments, this method of estimation is called", "options": ["None of the above", "Point estimation", "Maximum likelihood estimation", "Method of moments"], "answer": "D", "topic": "College--Probability and Statistics--Mathematical Statistics"} |
| {"question": "The observed value of the method of moments estimator is called", "options": ["None of the above", "Method of moments estimate", "Point estimate", "Maximum likelihood estimate"], "answer": "B", "topic": "College--Probability and Statistics--Mathematical Statistics"} |
| {"question": "Let $X_{1}, X_{2}, \\cdots, X_{n}$ be a sample from the population $X$, $x_{1}, x_{2}, \\cdots, x_{n}$ be the sample values, $\\theta$ be the parameter to be estimated, and $\\Theta$ be the possible range of values for $\\theta$. The parameter $\\hat{\\theta}=\\hat{\\theta}\\left(x_{1}, x_{2}, \\cdots, x_{n}\\right)$ that maximizes the likelihood function $L\\left(x_{1}, x_{2}, \\cdots, x_{n} ; \\theta\\right)$ within $\\Theta$ is called the maximum likelihood estimate of the parameter $\\theta$, and the corresponding statistic $\\hat{\\theta}=\\hat{\\theta}\\left(X_{1}, X_{2}, \\cdots, X_{n}\\right)$ is called the maximum likelihood estimator of the parameter $\\theta$. This estimation method is called", "options": ["Maximum likelihood estimation", "Method of moments", "Neither", "Point estimation"], "answer": "A", "topic": "College--Probability and Statistics--Mathematical Statistics"} |
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