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录入T20260205新题

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3
工具v2/文本文件/新题收录列表.txt
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@ -415,3 +415,6 @@
20240307-163014 高一下学期统考复习卷04
024679:024697
20240307-163632 高一下学期统考复习卷05
024698:024717

468
题库0.3/Problems.json
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@ -94743,7 +94743,9 @@
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"content": "若定义在$\\{t|t\\ne -1\\}$上的函数 $y=f(x)$满足对任意$x\\ne -1$, 均成立$f(\\dfrac{1-x}{1+x})=x$, 则 $f(x)=$\\blank{50}.",
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"content": "判断函数 $f(x)=\\begin{cases}x^2-2 x+5, & x \\geq 0,\\\\x^2+2 x+5, & x<0\\end{cases}$ 的奇偶性, 它是\\blank{50}函数.",
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"content": "若函数 $f(x)=x+\\dfrac{4}{x}$($1 \\leq x \\leq 3$) 的最大值为 $M$, 最小值为 $m$. 则 $M-m=$\\blank{50}.",
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"content": "若定义在$[1,+\\infty)$上的函数 $y=f(x)$ 满足: 对任意$x\\ge 0$, 总成立 $f(\\sqrt{x}+1)=x+2 \\sqrt{x}-1$, 则函数 $f(x)=$\\blank{50}.",
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"content": "已知函数 $f(x)=a^x$($a>0$, $a \\neq 1$) 在区间 $[-1,2]$ 上的最大值为 $8$, 最小值为 $m$, 若函数 $g(x)=(3-10 m) \\sqrt{x}$ 在 $[0,+\\infty)$ 上是严格减函数,则 $a=$\\blank{50}.",
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"content": "如果单调函数 $f(x)$ 的图像经过点 $A(1,1)$、$B(4,3)$, 那么不等式 $|f(x)-2| \\leq 1$ 的解集是\\blank{50}.",
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"content": "函数 $f(x)=\\sqrt{4-3 x-x^2}$ 的单调增区间是\\blank{50}.",
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"content": "已知函数 $f(x)$ 满足: \\textcircled{1} 对任意的实数 $x$ 都有 $f(x)=3 f(x+1)$; \\textcircled{2} 当 $0 \\leq x \\leq 1$ 时, $f(x)=x(4-2 x)$, 则 $f(-1.5)=$\\blank{50}.",
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"content": "设函数 $f(x)=\\begin{cases}3^x-a, & x<1,\\\\2(x-a)(x-2 a), & x \\geq 1.\\end{cases}$ 若 $f(x)$ 恰有 $2$ 个零点, 则实数 $a$ 的取值范围是\\blank{50}.",
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"content": "若函数 $f(x)=\\sqrt{x^2-5 x+6}$ 的定义域是 $F$, $g(x)=\\sqrt{x-2}+ \\sqrt{x-3}$ 的定义域是 $G$, 则 $F$ 和 $G$ 的关系是\\bracket{20}.\n\\fourch{$G \\subset F$}{$F \\subset G$}{$F=G$}{$F \\cap G=\\varnothing$}",
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"content": "下列各组函数中, 表示同一函数的是\\bracket{20}.\n\\twoch{$y=x+1$ 和 $y=\\dfrac{x^2-1}{x+1}$}{$y=x^0$ 和 $y=1$}{$f(x)=x^2$ 和 $g(x)=(x+1)^2$}{$f(x)=\\dfrac{(\\sqrt{x})^2}{x}$ 和 $g(x)=\\dfrac{x}{(\\sqrt{x})^2}$}",
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"content": "若 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 且在 $\\mathbf{R}$ 上是严格减函数,若 $f(2-a)+f(4-a)<0$, 则 $a$ 的取值范围为\\bracket{20}.\n\\fourch{$a<1$}{$a<3$}{$a>1$}{$a>3$}",
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"024711": {
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"content": "函数 $f(x)$ 的定义域为 $[-1,1]$, 图像如图 1 所示; 函数 $g(x)$ 的定义域为 $[-1,2]$, 图像如图 2 所示. $A=\\{x | f(g(x))=0\\}$, $B=\\{x | g(f(x))=0\\}$, 则 $A \\cap B$ 中元素的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [above] {$-1$} (1,0) node [below] {$1$} (0,1) node [left] {$1$} (0,-1) node [right] {$-1$};\n\\draw (-1,0) -- (-0.6,-1) -- (0.6,1) -- (1,0);\n\\draw [dashed] (-0.6,-1) -- (0,-1) (0.6,1) -- (0,1);\n\\draw (0,-2) node {图 1};\n\\end{tikzpicture}\n\\hspace{5em}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,1) -- (0,0) -- (1,1) -- (2,0);\n\\draw [dashed] (-1,0) -- (-1,1) -- (1,1) -- (1,0);\n\\draw (-1,0) node [below] {$-1$} (1,0) node [below] {$1$} (2,0) node [below] {$2$};\n\\draw (0,1) node [left] {$1$};\n\\draw (0,-2) node {图 2};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{1 个}{2 个}{3 个}{4 个}",
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"content": "已知 $m$ 为实数,若 $f(x)=x^2-2 m x+m-1$ 的最小值为 $g(m)$. 求:\\\\ \n(1) $g(m)$ 的解析式;\\\\\n(2) $g(m)$ 在区间 $[0,2]$ 上的最大值和最小值.",
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"content": "设函数 $f(x)=\\dfrac{x}{\\sqrt{x}}+\\dfrac{a}{x}$, $g(x)=x-\\sqrt{x}$($a>0$), 函数 $F(x)=f(x)+g(x)$.\\\\\n(1) 若 $a=3$ 时, 画出函数 $F(x)$ 的图像, 并指出函数的单调区间;\\\\\n(2) 求 $F(x)$ 在区间 $(0,2]$ 上的最小值.",
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"content": "已知函数 $f(x)$ 和 $g(x)$ 的图像关于原点对称, 且 $f(x)=x^2+2 x$.\\\\\n(1) 求函数 $g(x)$ 的解析式;\\\\\n(2) 若 $h(x)=g(x)-\\lambda f(x)+1$ 在区间 $[-1,1]$ 上是严格增函数, 求实数 $\\lambda$ 的取值范围.",
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"content": "已知函数 $y=f(x)$ 是 $\\mathbf{R}$ 上的奇函数, $x>0$ 时, $f(x)=\\dfrac{1}{x^2+1}$.求:\\\\\n(1) $y=f(x)$ 的解析式;\\\\\n(2) $y=f(x)$ 的值域.",
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"024716": {
"id": "024716",
"content": "有轨电车给市民出行带来很大便利. 已知某条线路通车后, 电车的发车时间间隔 $t$ (单位: 分钟) 满足 $2 \\leq t \\leq 20$. 经市场调研测算,电车载客量与发车时间间隔 $t$ 相关, 当 $10 \\leq t \\leq 20$ 时电车为满载状态, 载客量为 400 人, 当 $2 \\leq t<10$ 时, 载客量会减少, 减少的人数与 $(10-t)$ 的平方成正比,且发车时间间隔为 $2$ 分钟时的载客量为 $272$ 人. 记电车载客量为 $p(t)$.\\\\\n(1) 求 $p(t)$ 的表达式, 并求当发车时间间隔为 $8$ 分钟时, 电车的载客量;\\\\\n(2) 若该线路每分钟的净收益为 $Q=\\dfrac{6 p(t)-1500}{t}-60$ (元), 问当发车时间间隔为多少时,该线路每分钟的净收益最大?",
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"024717": {
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"content": "若函数 $f(x)$、$g(x)$ 都在区间 $I$ 上有定义,对任意 $x \\in I$ 都有 $|f(x)-g(x)| \\leq 1$ 成立, 则称 $f(x)$、$g(x)$ 为区间 $I$ 上的``均分函数''.\\\\\n(1) 判断 $f(x)=4^x$、$g(x)=2^x-1$ 是否为区间 $(-\\infty, 0]$ 上的``均分函数'', 并说明理由;\\\\\n(2) 若 $f(x)=\\lg x$、$g(x)=\\lg (x+1)$ 为区间 $[m,+\\infty)$ 上的``均分函数'', 求 $m$ 的取值范围;\\\\\n(3) 若 $f(x)=x^2+\\dfrac{1}{3}$、$g(x)=k x$ 为区间 $[\\dfrac{1}{2}, 1]$ 上的``均分函数'', 求 $k$ 的取值范围.",
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"030001": {
"id": "030001",
"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",