From f80121edff3e015366ca1ec0867ff785c198700e Mon Sep 17 00:00:00 2001 From: wangweiye7840 Date: Tue, 5 Sep 2023 07:48:37 +0800 Subject: [PATCH] =?UTF-8?q?Revert=20"=E6=94=B6=E5=BD=952025=E5=B1=8A?= =?UTF-8?q?=E9=AB=98=E4=B8=80=E4=B8=8A=E5=AD=A6=E6=9C=9F=E5=91=A8=E6=9C=AB?= =?UTF-8?q?=E5=8D=B7=E5=8F=8A=E6=B5=8B=E9=AA=8C=E5=8D=B7=E9=A2=98=E7=9B=AE?= =?UTF-8?q?"?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit This reverts commit cc77d7af9fd078b58c264019aee349896b2cc33a. --- 题库0.3/Problems.json | 10160 ---------------------------------------- 1 file changed, 10160 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 664e1560..450290e5 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -567541,10166 +567541,6 @@ "space": "4em", "unrelated": [] }, - "022139": { - "id": "022139", - "content": "用``$\\in$''或``$\\notin$''填空:\\\\\n(1) $\\dfrac{2}{3}$\\blank{20}$\\mathbf{N}$;\\\\\n(2) $-10$\\blank{20}$\\mathbf{Z}$;\\\\\n(3) $3.14$\\blank{20}$\\mathbf{Q}$;\\\\\n(4) $3.\\dot{1}\\dot{4}$\\blank{20}$\\mathbf{Q}$;\\\\\n(5) $\\dfrac{\\pi}{3}$\\blank{20}$\\mathbf{Q}$;\\\\\n(6) $\\dfrac{\\pi}{3}$\\blank{20}$\\mathbf{R}$;\\\\\n(7) $\\sqrt{\\pi}$\\blank{20}$\\mathbf{R}$;\\\\\n(8) $0$\\blank{20}$\\varnothing$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022140": { - "id": "022140", - "content": "用描述法表示下列集合:\\\\\n(1) 所有偶数组成的集合: \\blank{150};\\\\\n(2) 被$5$除余数等于$2$的正整数的集合: \\blank{150};\\\\\n(3) 被$5$除余数等于$2$的整数的集合: \\blank{150};\\\\\n(4) 大于$1$且不大于$10$的实数组成的集合: \\blank{150};\\\\\n(5) 平面直角坐标系内$x$轴上的点的坐标组成的集合: \\blank{150};\\\\\n(6) 平面直角坐标系内第二象限内的所有点的坐标组成的集合: \\blank{150};\\\\\n(7) 平面直角坐标系内不在第二象限内的所有点的坐标组成的集合: \\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022141": { - "id": "022141", - "content": "用列举法表示下列集合:\\\\\n(1) $\\{x | \\dfrac{6}{x}\\in \\mathbf{N}, x \\in \\mathbf{Z}\\}=$\\blank{150}.\\\\\n(2) $\\{y | y=-x^2+4, x \\in \\mathbf{R}, y \\in \\mathbf{N}\\}=$\\blank{150}.\\\\\n(3) $\\{(x, y) | x+(y+1)^2=4, x \\in \\mathbf{N}$, $y \\in \\mathbf{Q}$ 且 $y<0\\}=$\\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022142": { - "id": "022142", - "content": "用区间表示下列集合:\\\\\n(1) $\\{x |-22$: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022160": { - "id": "022160", - "content": "写出下列命题的否定形式:\\\\\n(1) 不论 $k$ 取何实数, $x^2+x+k=0$ 必有实数根: \\blank{150};\\\\\n(2) 三角形中至多有一个钝角: \\blank{150};\\\\\n(3) 正 $n$($n \\geq 3$) 边形的 $n$ 个内角全相等: \\blank{150};\\\\\n(4) 张三是中科大或北大的学生: \\blank{150};\\\\\n(5) 如果 $x^2-x-2=0$, 那么 $x \\neq-1$ 且 $x \\neq-2$: \\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022161": { - "id": "022161", - "content": "请给出下列两个集合之间的包含关系, 并证明你的结论.\n$A=\\{n | n$ 是 $6$, $16$ 的正公约数 $\\}$, $B=\\{m | m$ 是 $6$, $12$ 的正公约数 $\\}$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022162": { - "id": "022162", - "content": "已知 $a$、$b\\in \\mathbf{R}$, 元素 $(1,2) \\in A \\cap B$, 且 $A=\\{(x, y) | a x-y^2+b=0\\}$, $B=\\{(x, y) | x^2-a y-b=0\\}$, 求 $a$、$b$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022163": { - "id": "022163", - "content": "设 $a$ 为实数, 且集合 $M=\\{x | a x^2+2 x+1=0\\}$ 的子集有且只有两个, 求满足上述条件 $a$ 所构成的集合.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022164": { - "id": "022164", - "content": "已知集合 $A=\\{1,2,3, m\\}$, $B=\\{m^2, 3\\}$, 满足 $A \\cup B=\\{1,2,3, m\\}$, 求实数 $m$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022165": { - "id": "022165", - "content": "设 $m \\in \\mathbf{R}$, 已知集合 $A=\\{x | x^2-3 x+2=0\\}$, $B=\\{x | m x+1=0\\}$, 且 $B \\subseteq A$, 求 $m$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022166": { - "id": "022166", - "content": "用描述法表示:\\\\\n(1) 被 $4$ 除的余数等于 $1$ 的负整数的集合:\\blank{100}\\\\\n(2) 平面直角坐标系内``不在第三象限内、也不在坐标轴上''的所有点的坐标组成的集合: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022167": { - "id": "022167", - "content": "用列举法表示:\\\\\n(1) $\\{y | y=-x^2+\\dfrac{9}{2}, x \\in \\mathbf{R}, y \\in \\mathbf{N}\\}=$\\blank{100};\\\\\n(2) $\\{(x, y) | x+(y+2)^2=5, x \\in \\mathbf{N}, y \\in \\mathbf{Q}, y<0\\}=$\\blank{100};\\\\\n(3) 若集合 $M=\\{0,3,4\\}$, $P=\\{(x, y) | x=a+b, y=a b, a,b \\in M\\}=$\\blank{100};\\\\\n(4) 平面上的两条曲线 $y=x^2+1$ 和 $y=x+6$ 的交点组成的集合\\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022168": { - "id": "022168", - "content": "因式分解:\\\\\n(1) $x^4+x^2 y^2+y^4=$\\blank{100};\\\\\n(2) $3 x^5-x^4+3 x^3-x^2-3 x+1=$\\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022169": { - "id": "022169", - "content": "给出四个命题: \\textcircled{1} 若 $x^2=y^2$, 则 $x=y$; \\textcircled{2}若 $x \\neq y$, 则 $x^2 \\neq y^2$; \\textcircled{3}若 $x^2 \\neq y^2$, 则 $x \\neq y$;\\\\\n\\textcircled{4} 若 $x \\neq y$ 且 $x \\neq-y$, 则 $x^2 \\neq y^2$, 其中真命题的序号是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022170": { - "id": "022170", - "content": "若``存在 $x \\in\\{x | 1 \\leq x \\leq 2\\}$ 使得 $3 x+1+a \\geq 0$''是真命题, 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022171": { - "id": "022171", - "content": "满足 $a, b \\in\\{-1,0,1,2\\}$, 且关于 $x$ 的方程 $a x^2+2 x+b=0$ 有实数解的有序实数对 $(a, b)$ 的个数为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022172": { - "id": "022172", - "content": "若 $A=\\{y | y=x^2, x=2 k+1, k \\in \\mathbf{Z}\\}$, $B=\\{x | x=4 k+1, k \\in \\mathbf{N}\\}$, 求证: $A \\subset B$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022173": { - "id": "022173", - "content": "已知集合 $P=\\{x |-2 \\leq x \\leq 5\\}$, $Q=\\{x | k+1 \\leq x \\leq 2 k-1\\}$, 若 $P \\cap Q=\\varnothing$, 求 $k$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022174": { - "id": "022174", - "content": "已知集合 $A=\\{x-y, x+y, x y\\}$, $B=\\{x^2+y^2, x^2-y^2, 0\\}$, 且 $A=B$, 求 $x, y$ 和集合 $A$、$B$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022175": { - "id": "022175", - "content": "用符号``$\\in$''或``$\\notin$''或``$\\subseteq$''或``$\\subset$''填空:\n(1) $\\{x | 2 x^2-3 x-2=0, x>0\\}$\\blank{20}$\\mathbf{Z}$;\\\\\n(2) $-\\sqrt{2}$\\blank{20}$\\mathbf{Q}$;\\\\\n(3) $\\sqrt{4-2 \\sqrt{3}}$\\blank{20}$\\{1,2, \\sqrt{3}-1\\}$;\\\\\n(4) $\\{(-1,1)\\}$\\blank{20}$\\{(x, y) | y=x^2\\}$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022176": { - "id": "022176", - "content": "已知全集 $U=\\{0,1,2,3,4,5,6,7,8,9\\}$, 集合 $A=\\{0,1,3,5,8\\}$, 集合 $B=\\{2,4,5,6,8\\}$, 则 $\\overline{A}\\cap \\overline{B}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022177": { - "id": "022177", - "content": "已知 $M=\\{x | y^2=x+2\\}$, $N=\\{x | y^2=-2 x+8\\}$, 则 $M \\cap N=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022178": { - "id": "022178", - "content": "集合 $A=\\{(x, y) | y=x^2\\}$, $B=\\{(x, y) | \\dfrac{y-1}{x-1}=\\dfrac{1}{2}\\}$, 则 $A \\cap B=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022179": { - "id": "022179", - "content": "集合 $A=\\{m | m=n^2+1, n \\in \\mathbf{N}, n \\geq 1\\}$, $B=\\{y | y=x^2-2 x+2, x \\in \\mathbf{N}, x \\geq 1\\}$, 探究 $A$、$B$ 之间的关系, 并证明你的结论.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022180": { - "id": "022180", - "content": "设 $A$ 是整数集的一个非空子集, 对 $k \\in A$, 若 $k-1 \\notin A$ 、且 $k+1 \\notin A$, 则称 $k$ 是 $A$ 的一个``孤立元'', 给定 $S=\\{1,2,3,4,5,6\\}$, 在 $S$ 的所有 $3$ 元子集中, 含孤立元的集合共有\\blank{50}个.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷02小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022181": { - "id": "022181", - "content": "指出陈述句 $P$ 是陈述句 $Q$ 的什么条件. (用选项回答: A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分又不必要)\\\\\n(1) $P$: $x=0$, $Q$: $x^2=0$: \\bracket{20};\\\\\n(2) $P$: $x+y+z>0$, $Q$: $x>0$ 或 $y>0$ 或 $z>0$: \\bracket{20};\\\\\n(3) $P$: $x+y+z>0$, $Q$: $x>0$ 且 $y>0$ 且 $z>0$: \\bracket{20};\\\\\n(4) $P$: $x / y>0$, $Q$: $x y>0$: \\bracket{20};\\\\\n(5) $P$: $x / y \\geq 0$, $Q$: $x y \\geq 0$: \\bracket{20};\\\\\n(6) $P$: $(x-3)^2+(y+4)^2=0$, $Q$: $(x-3)(y-4)=0$: \\bracket{20};\\\\\n(7) $P$: $x+y>0$ 且 $x y>0$, $Q$: $x>0$ 且 $y>0$: \\bracket{20};\\\\\n(8) $P$: $x+y>4$ 且 $x y>4$, $Q$: $x>2$ 且 $y>2$: \\bracket{20};\\\\\n(9) $P$: $x+y \\neq 3$, $Q$: $x \\neq 1$ 或 $y \\neq 2$: \\bracket{20};\\\\\n(10) $P$: $x+y \\neq 3$, $Q$: $x \\neq 1$ 且 $y \\neq 2$: \\bracket{20};\\\\\n(11) $P$: 函数 $y=k x+b$ 的图像通过原点 (其中 $k, b$ 为常数), $Q$: $b=0$: \\bracket{20};\\\\\n(12) $P$: $x z>y z$ 且 $z \\neq 0$, $Q$: $x>y$ 且 $z \\neq 0$: \\bracket{20};\\\\\n(13) $P$: $|x|<2$, $Q$: $x<2$: \\bracket{20};\\\\\n(14) $P$: $x^2+1=y$ 且 $x+1=y$, $Q$: $x^2=x$ 且 $x^2+1=y$: \\bracket{20}.", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022182": { - "id": "022182", - "content": "设全集 $U=\\mathbf{R}$, 集合 $A=(0,2)$, $B=(-1,1)$, 则 $A \\cap \\overline{B}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022183": { - "id": "022183", - "content": "集合 $\\{y | y=x^2-6, x \\in \\mathbf{R}\\}\\cap\\{y | y=-x^2+2 x+6, x \\in \\mathbf{R}\\}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022184": { - "id": "022184", - "content": "设全集 $U=\\{1,2,3,4,5\\}$, 若集合 $A$、$B$ 满足 $\\overline{A}\\cup \\overline{B}=\\{1,2,4,5\\}$, $\\overline{A}\\cap B=\\{5\\}$, $A \\cap \\overline{B}=\\{2\\}$, 则集合 $A=$\\blank{50}, $B=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022185": { - "id": "022185", - "content": "已知 $m$ 是实常数, 若集合 $A=\\{0,1\\}$, $B=\\{x | x^2-m x-m>0\\}$ 且 $A \\cap B=\\{1\\}$, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022186": { - "id": "022186", - "content": "已知 $m$ 是实常数, 若 $\\alpha: 2 \\leq x \\leq 4$, $\\beta: 3 m-1a$ 且 $x<-3 a\\}$ 且满足 $\\overline{A}\\cap \\overline{B}\\subset C$, 则 $\\overline{A}\\cap \\overline{B}=$\\blank{50}, 实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022189": { - "id": "022189", - "content": "已知常数 $a, b, c \\in \\mathbf{R}$, 证明``$a c<0$''是``关于 $x$ 的一元二次方程 $a x^2+b x+c=0$ 有两个不同的实数根''的充分非必要条件.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022190": { - "id": "022190", - "content": "已知 $a$ 为实常数, 集合 $A=\\{x | x^2+4 x=0\\}$, $B=\\{x | x^2+2(a+1) x+a^2-1=0, x \\in \\mathbf{R}\\}$.\\\\\n(1) 若 $A \\cup B=B$, 求实数 $a$ 的值;\\\\\n(2) 若 $A \\cap B=B$, 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022191": { - "id": "022191", - "content": "设 $m$ 为实常数, 命题甲: 关于 $x$ 的方程 $x^2+x+m=0$ 有两个不同的负根; 命题乙: 关于 $x$ 的方程 $4 x^2+x+m=0$ 无实根. 若这两个命题有且只有一个成立, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022192": { - "id": "022192", - "content": "已知 $a$ 为实常数, 集合 $A=\\{x | x^2-2 a x+a=0, x \\in \\mathbf{R}\\}$, 集合 $B=\\{x | x^2-4 x+a+5=0, x \\in \\mathbf{R}\\}$.\\\\\n(1) 若 $A=B=\\varnothing$, 求实数 $a$ 的取值范围;\\\\\n(2) 若 $A$、$B$ 中有且只有一个是 $\\varnothing$, 求实数 $a$ 的取值范围;\\\\\n(3) 若 $A$、$B$ 中至少有一个是 $\\varnothing$, 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022193": { - "id": "022193", - "content": "``$a \\in \\mathbf{R}$''是`` $a^2>0$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022194": { - "id": "022194", - "content": "``$a<0$''是``$a^2>a$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022195": { - "id": "022195", - "content": "设集合 $M=\\{x | x>2\\}$, $P=\\{x | x<3\\}$, 则``$x \\in M$ 或 $x \\in P$ \"是``$x \\in P \\cap M$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022196": { - "id": "022196", - "content": "``$|x+y|<2$ \" 是\" $|x|<1$ 且 $|y|<1''$ 的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022197": { - "id": "022197", - "content": "已知 $p$ 是 $r$ 的充分非必要条件, $s$ 是 $r$ 的必要条件, $q$ 是 $s$ 的必要条件, 那么 $p$ 是 $q$ 的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022198": { - "id": "022198", - "content": "判断命题真假:\\\\\n(1) 若 $x^2+4 x-m=0$ 有两实根, 则 $m>-5$: \\blank{50};\\\\\n(2) 若 $x+y \\neq 3$, 则 $x \\neq 1$ 或 $y \\neq 2$: \\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022199": { - "id": "022199", - "content": "设有两个命题: \\textcircled{1} 方程 $x^2-m x+2=0$ 没有实根; \\textcircled{2} 实数 $m$ 为非正数. 如果这两个命题有且只有一个是假命题, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022200": { - "id": "022200", - "content": "在直角坐标系用图形表示下列集合:\\\\\n(1) $\\{(x, y)|| x |=3$ 且 $y=x\\}$;\\\\\n(2) $\\{(x, y) | x<3$ 且 $y<2\\}$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022201": { - "id": "022201", - "content": "已知 $A=\\{y | y=x^2-6 x+10, x, y \\in \\mathbf{N}\\}$, $B=\\{y | y=-x^2-6 x+18, x \\in \\mathbf{Z}, y \\in \\mathbf{N}\\}$, 求 $A \\cap B$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022202": { - "id": "022202", - "content": "已知 $A=\\{(x, y) | y=x^2-6 x+10, x, y \\in \\mathbf{N}\\}$, $B=\\{(x, y) | y=-x^2-6 x+18, x \\in \\mathbf{Z}, y \\in \\mathbf{N}\\}$, 求 $A \\cap B$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022203": { - "id": "022203", - "content": "指出陈述句 $P$ 是陈述句 $Q$ 的什么条件. (用选项回答: A. 充分不必要 B. 必要不充分 C. 充分必要 D. 既不充分又不必要)\\\\\n(1) $P$: $x+y+z>3$, $Q$: $x>1$ 且 $y>1$ 且 $z>1$: \\bracket{20};\\\\\n(2) $P$: $x+y>0$ 且 $x y>0$, $Q$: $x>0$ 且 $y>0$: \\bracket{20};\\\\\n(3) $P$: $x+y \\neq 3$, $Q$: $x \\neq 1$ 且 $y \\neq 2$: \\bracket{20};\\\\\n(4) $P$: $x z>y z$ 且 $z \\neq 0$, $Q$: $x>y$ 且 $z \\neq 0$: \\bracket{20};\\\\\n(5) $P$: $|x|<2$, $Q$: $x>-2$: \\bracket{20}.", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022204": { - "id": "022204", - "content": "设全集 $U=\\mathbf{R}$, 集合 $A=(0,2)$, $B=(-1,1]$, 则 $A \\cap \\overline{B}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022205": { - "id": "022205", - "content": "集合 $\\{y | y=x^2-5, x \\in \\mathbf{R}\\}\\cap\\{y | y=-x^2+2 x+5, x \\in \\mathbf{R}\\}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022206": { - "id": "022206", - "content": "已知 $m$ 是实常数, 若 $\\alpha: 2 \\leq x \\leq 4$, $\\beta: 3 m-12\\}$, $P=\\{x | x<3\\}$, 则``$x \\in M$ 或 $x \\in P$ \"是``$x \\in P \\cap M$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022209": { - "id": "022209", - "content": "``$|x+y|<4$ \"是``$|x|<1$ 且 $|y|<2$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022210": { - "id": "022210", - "content": "设 $m$ 为实常数, 命题甲: 关于 $x$ 的方程 $x^2+2 x+m-1=0$ 有个正根和一个负根; 命题乙: $m$ 满足不等式 $4 m+1 \\leq 0$. 若这两个命题有且只有一个成立, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷03小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022211": { - "id": "022211", - "content": "已知 $a$、$b$、$c$ 是实数, 下列三个命题:\\\\\n\\textcircled{1} 若 $(a-2) c=(b-2) c$, 则 $a=b$;\\\\\n\\textcircled{2} $\\dfrac{a}{c}=\\dfrac{b}{c}$, 则 $a=b$;\\\\\n\\textcircled{3} 若 $ab^2$; \\textcircled{2} $a^3>b^3$; \\textcircled{3} $\\dfrac{1}{a}<\\dfrac{1}{b}$; \\textcircled{4} $\\dfrac{a}{b}>1$; \\textcircled{5} $\\dfrac{1}{a-b}>\\dfrac{1}{a}$; \\textcircled{6} $|a|>-b$. 其中正确项的序号是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022218": { - "id": "022218", - "content": "若 $a>b>0$, $c1$ 且 $b>1$}{$a>1$ 或 $b>1$}{$a>0$ 且 $b>2$}{$a>0$ 或 $b>2$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022220": { - "id": "022220", - "content": "已知互不相等的正数 $a$、$b$、$c$, 满足 $a^2+c^2=2 b c$, 则下列不等式中可能成立的是\\bracket{20}.\n\\fourch{$a>b>c$}{$b>a>c$}{$b>c>a$}{$c>a>b$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022221": { - "id": "022221", - "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x$ 的方程 $|x-1|=a-1$ 的解集.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022222": { - "id": "022222", - "content": "已知关于 $x$ 的方程 $x^2-2 k x+k^2-k-1=0$ 有两个不相等的实数根 $x_1$、$x_2$.\\\\\n(1) 求实数 $k$ 的取值范围;\\\\\n(2) 若 $x_1-3 x_2=2$, 求实数 $k$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022223": { - "id": "022223", - "content": "某超市在国庆节前进行某个商品降价促销活动, 拟分两次降价. 有两种降价方案: 甲方案是第一次打 $a$ 折销售, 第二次打 $b$ 折销售; 乙方案是两次都打 $\\dfrac{a+b}{2}$ 折销售. 试问: 经过两次降价后哪种方案的商品价格更低? 请说明理由.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022224": { - "id": "022224", - "content": "设 $a_1$、$b_1$、$c_1$、$a_2$、$b_2$、$c_2$ 均为非零实数, 方程 $a_1 x^2+b_1 x+c_1=0$ 和方程 $a_2 x^2+b_2 x+c_2=0$ 的实数解集分别为 $M$ 和 $N$, 则``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022225": { - "id": "022225", - "content": "设常数 $m \\in \\mathbf{R}$, $A=\\{(x, y) | y=-x^2+m x-1, x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | x+y=3, x \\in \\mathbf{R}\\}$, 且 $A \\cap B$ 的子集有两个, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022226": { - "id": "022226", - "content": "设 $x$ 是实数. 求证: $|x-1|,|x+1|$ 中至少有一个大于等于$1$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022227": { - "id": "022227", - "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x$ 的方程 $\\dfrac{x}{x+1}=\\dfrac{1}{x+a}$ 的解集.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022228": { - "id": "022228", - "content": "设常数 $a \\in \\mathbf{R}$, 求关于 $x, y$ 的方程组 $\\begin{cases}a x+2 y=1+a,\\\\2 x+2(a-1) y=3\\end{cases}$ 的解集.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022229": { - "id": "022229", - "content": "已知关于 $x$ 的方程 $m x+4=3 x-n$($m$、$n \\in \\mathbf{R}$).\\\\\n(1) 求方程的解集 $A$;\\\\\n(2)若 $n=4$ 时方程仅有正整数解, 求整数 $m$ 的所有取值组成的集合 $B$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022230": { - "id": "022230", - "content": "设实数 $a$、$b$、$c$ 满足 $\\begin{cases}b+c=3 a^2-4 a+6,\\\\c-b=a^2-4 a+4,\\end{cases}$ 试确定 $a$、$b$、$c$ 的大小关系, 并说明理由.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022231": { - "id": "022231", - "content": "已知 $A=\\{x | x+a>0, x \\in \\mathbf{R}\\}$, $B=\\{x | b x<1, x \\in \\mathbf{R}\\}$, 其中 $a, b$ 均为实常数, 且 $b \\neq 0$.\\\\\n(1) 若 $A \\cap B=\\{x | 2b$, $cb d$: \\blank{20};\\\\\n(5) 若 $a>b>0$, $c>d>0$, 则 $\\dfrac{a}{c}>\\dfrac{b}{d}$: \\blank{20};\n(6) 若 $\\sqrt[3]{a}>\\sqrt[3]{b}$, 则 $a \\geq b$: \\blank{20};\\\\\n(7) 若 $\\dfrac{c}{a}=\\dfrac{c}{b}$, 则 $a=b \\neq 0$: \\blank{20};\\\\ (8) 若 $a=b$, 则 $\\sqrt[n]{a}=\\sqrt[n]{b}$, 其中 $n$ 是正整数: \\blank{20}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022233": { - "id": "022233", - "content": "集合 $A=\\{x | x^2+p x+q=0\\}$, $B=\\{x | x^2-x+r=0\\}$, 且 $A \\cap B=\\{-1\\}$, $A \\cup B=\\{-1,2\\}$, 则 $p=$\\blank{50}, $q=$\\blank{50}, $r=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022234": { - "id": "022234", - "content": "已知集合 $E=\\{x | x^2-3 x+2=0\\}$, 集合 $F=\\{x | x^2-a x+(a-1)=0\\}$, 若 $F$ 不是 $E$ 的真子集, 则实数 $a$ 所有值的集合是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022235": { - "id": "022235", - "content": "设 $k \\in \\mathbf{R}$, 若关于 $x$ 与 $y$ 的二元一次方程组 $\\begin{cases}3 x-y=1,\\\\k x-y=2\\end{cases}$ 的解集为空集, 则 $k=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022236": { - "id": "022236", - "content": "解关于 $x$ 的不等式: $m(x+2) \\leq x+m$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022237": { - "id": "022237", - "content": "设常数 $b \\in \\mathbf{R}$, 若存在两个不同的实数 $x_1, x_2$ 使得 $\\begin{cases}x_1+x_2=2 b,\\\\x_1^2+x_2^2=2 b^2-b-1\\end{cases}$ 成立, 求 $b$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷04小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022238": { - "id": "022238", - "content": "已知 $A, B, C$ 都是全集 $U$ 的子集. 判断下列命题是否为真(T or F):\\\\\n(1) 任一集合至少有两个不同的子集: \\blank{20};\\\\\n(2) 空集是任何非空集合的子集: \\blank{20};\\\\\n(3) 如果一个集合 $A$ 与全集 $U$ 的每一个子集的交集都是空集, 那么 $A$ 是空集: \\blank{20};\\\\\n(4) 如果 $U$ 不是空集, 那么 $U$ 的子集中必定存在两个集合 $S, T$, 使得 $S \\cap T=\\varnothing$: \\blank{20};\\\\\n(5) 已知 $A \\subseteq U$, $B \\subseteq U$. `` $A \\cap B=\\varnothing$''是``$A \\subseteq \\overline{B}$''的充要条件: \\blank{20}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022239": { - "id": "022239", - "content": "用列举法表示下列集合:\\\\\n(1) $\\{(x, y) |(x+1)^2+(y-1)^2 \\leq 4, x, y \\in \\mathbf{Z}\\}=$ \\blank{100};\\\\\n(2) $\\{x | \\dfrac{10}{x-4}\\in \\mathbf{Z}, x \\in \\mathbf{N}\\}=$\\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022240": { - "id": "022240", - "content": "设 $a$ 是实常数, 集合 $A=\\{x | x^2-5 x+4<0\\}$, 集合 $B=\\{x | a \\cdot x=16\\}$, 且 $B \\subseteq A$, 则 $a$ 的所有可能取值组成的集合为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022241": { - "id": "022241", - "content": "设 $a$ 是实常数, 集合 $A=\\{y | y=a x^2+x+a, x \\in \\mathbf{R}\\}$. 求 $a$ 的取值范围, 使得 $A \\cap(0,+\\infty) \\neq \\varnothing$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022242": { - "id": "022242", - "content": "写出下列陈述句的否定形式:\\\\\n(1) $a>1$ 且 $b<1$: \\blank{150};\\\\\n(2) $a=b=c$: \\blank{150};\\\\\n(3) 所有三角形都是直角三角形: \\blank{150};\\\\\n(4) 存在正整数 $n$, 使得 $n^2$ 是素数: \\blank{150};\\\\\n(5) 恰好有两个实数的平方等于$2$: \\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022243": { - "id": "022243", - "content": "判断下列陈述句 $\\alpha$ 与 $\\beta$ 中, $\\alpha$ 是 $\\beta$ 的什么条件?(A: $\\alpha$ 是 $\\beta$ 的充分非必要条件, B: $\\alpha$ 是 $\\beta$ 的必要非充分条件, C: $\\alpha$ 是 $\\beta$ 的充要条件, D: $\\alpha$ 是 $\\beta$ 的既不充分又不必要条件). \\\\\n(1) $\\alpha$: $m$ 是正整数, $\\beta$: $m$ 是有理数: \\bracket{20};\\\\ (2) $\\alpha$: 自然数 $x$ 的末位数字是 $0$, $\\beta$: 自然数 $x$ 是 $5$ 的倍数: \\bracket{20};\\\\\n(3) $\\alpha$: $x+y+z>0$, $\\beta$: $x, y, z$ 不都小于零: \\bracket{20};\\\\\n(4) $\\alpha$: $x^2+3 x+2 \\neq 0$, $\\beta: x \\neq-1$ 或 $x \\neq-2$: \\bracket{20};\\\\\n(5) $\\alpha$: 正整数 $k$ 是 $3$ 的倍数, $\\beta$: 正整数 $k$ 的末位数字是 $3$, $6$ 或 $9$: \\bracket{20}.", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022244": { - "id": "022244", - "content": "已知 $a, b, c$ 是三个实数. 证明:``三个实数 $a, b, c$ 中任意两数之和大于第三数''是``这三个实数 $a, b, c$ 中任意两数之差的绝对值小于第三数\"的充分必要条件.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022245": { - "id": "022245", - "content": "用反证法证明:对任意的 $x \\in \\mathbf{R}$, 关于 $x$ 的方程 $x^2-5 x+m=0$ 与 $2 x^2+x+6-m=0$ 至少有一个方程有实根.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022246": { - "id": "022246", - "content": "判断下列命题是否为真(T or F):\\\\\n(1) 若 $a>b$, $c=d$, 则 $a c>b d$: \\blank{20};\\\\\n(2) 若 $\\dfrac{a}{c^2}<\\dfrac{b}{c^2}$, 则 $ab$, 则 $a c^2>b c^2$: \\blank{20};\\\\\n(5) 若 $a>b$, $cb d$: \\blank{20};\\\\\n(6) 若 $a=b$, 则 $\\sqrt[n]{a}=\\sqrt[n]{b}$, 其中 $n$ 是正整数: \\blank{20};\\\\\n(7) 若 $a>b>0$, $c>d>0$, 则 $\\dfrac{a}{c}>\\dfrac{b}{d}$: \\blank{20};\\\\\n(8) 若 $a>b$, $c \\geq d$, 则 $a+c>b+d$: \\blank{20};\\\\\n(9) 若 $\\sqrt[3]{a}>\\sqrt[3]{b}$, 则 $a \\geq b$: \\blank{20};\\\\ \n(10) 若 $a b^2 \\geq 0$, 则 $a \\geq 0$: \\blank{20}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022247": { - "id": "022247", - "content": "设 $k \\in \\mathbf{R}$, 若关于 $x$ 与 $y$ 的二元一次方程组 $\\begin{cases}3 x-y=1,\\\\k x-y=2\\end{cases}$ 的解集为空集, 则 $k=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022248": { - "id": "022248", - "content": "设常数 $a, b \\in \\mathbf{R}$, 比较以下各组中两数的大小:\\\\\n(1) $-(a+1)^2$ 与 $-2 a^2-3 a-4$\\\\\n(2) $a^2+a b+b^2$ 与 $0$", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022249": { - "id": "022249", - "content": "写出下列一元二次不等式的解集:\\\\\n(1) $x^2+2 x-15>0$: \\blank{100};\\\\\n(2) $x^2+4 x-45 \\geq 0$: \\blank{100};\\\\\n(3) $-3 x^2+2 x-4 \\geq 0$: \\blank{100};\\\\\n(4) $x^2+12 x+36 \\leq 0$: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022250": { - "id": "022250", - "content": "若集合 $A=\\{x | a x^2-2 x-1=0\\}$ 只有一个元素, 则实数 $a=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022251": { - "id": "022251", - "content": "已知一元二次方程 $2 x^2+3 x-4=0$ 的两根为 $x_1$ 与 $x_2$, 求下列各式的值:\\\\\n(1) $x_1^2+x_2^2$;\\\\\n(2) $x_1^3-x_2^3$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022252": { - "id": "022252", - "content": "设关于 $x$ 的二次不等式 $a x^2+b x+c>0$ 的解集为 $(-1,3)$. 求二次不等式 $c x^2-b x+a \\leq 0$ 的解集.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022253": { - "id": "022253", - "content": "设 $a$ 是实常数, 已知 $M=(0,+\\infty)$, 二次不等式 $x^2+x-a \\leq 0$ 的解集为 $S$.\\\\\n(1) 若 $S \\subseteq M$, 求 $a$ 的取值范围.\\\\\n(2) 若 $S \\cap M=\\varnothing$, 求 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022254": { - "id": "022254", - "content": "设 $a$ 是实常数, 求下列关于 $x$ 的二次不等式的解集:\\\\\n(1) $x^2-a x+1 \\leq 0$\\\\\n(2) $a x^2-2(a+1) x+4>0$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022255": { - "id": "022255", - "content": "在下列不等式的后面空格内填写该不等式的解集:\\\\\n(1) $\\dfrac{2 x+3}{x-2}<0$: \\blank{100};\\\\\n(2) $\\dfrac{2 x+3}{4 x+5}\\leq 0$: \\blank{100};\\\\\n(3) $\\dfrac{x^2}{x-1}\\geq 0$: \\blank{100};\\\\\n(4) $\\dfrac{1}{x}<1$: \\blank{100};\\\\\n(5) $3 x+5+\\dfrac{1}{x-4}>2 x+\\dfrac{1}{x-4}+3$: \\blank{100};\\\\\n(6) $\\dfrac{4}{x-1}\\leq x-1$: \\blank{100};\\\\\n(7) $\\dfrac{2}{x+1}\\geq \\dfrac{x+1}{2}$: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022256": { - "id": "022256", - "content": "``$x>1$''是``$\\dfrac{1}{x}<1$''成立的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022257": { - "id": "022257", - "content": "若对一切实数 $x$, 均有 $\\dfrac{2 x^2+x-2}{x^2+2 x+2}>k$, 则实数 $k$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022258": { - "id": "022258", - "content": "若关于 $x$ 的不等式 $-80$: \\blank{100};\\\\\n(4) $\\dfrac{2 x-3}{x+1}<0$: \\blank{100};\\\\\n(5) $\\dfrac{2-x}{x+6}\\geq 0$: \\blank{100};\\\\\n(6) $\\dfrac{x+5}{(x-1)^2}\\geq 2$: \\blank{100};\\\\\n(7) $\\dfrac{1}{x}\\leq 2$: \\blank{100};\\\\\n(8) $x^2+5+\\dfrac{1}{x-4}\\geq 7 x+\\dfrac{1}{x-4}-7$: \\blank{100};", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022261": { - "id": "022261", - "content": "若关于 $x$ 的不等式 $x^2+a x-2<0$ 的解集为 $(-2,1)$, 则实数 $a=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022262": { - "id": "022262", - "content": "设 $a_1$、$b_1$、$c_1$、$a_2$、$b_2$、$c_2$ 均为非零实数, 不等式 $a_1 x^2+b_1 x+c_1>0$ 和方程 $a_2 x^2+b_2 x+c_2>0$ 的解集分别为 $M$ 和 $N$, 则``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022263": { - "id": "022263", - "content": "已知关于$x$的不等式 $x^2-a x-2 a^2<0$($a \\in \\mathbf{R}$), 若$x=1$ 不是该不等式的解, 则 $m$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022264": { - "id": "022264", - "content": "已知关于 $x$ 的不等式 $m x^2-m x+1 \\geq 0$ 在 $\\mathbf{R}$ 上恒成立, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022265": { - "id": "022265", - "content": "求关于 $x$ 的不等式 $\\dfrac{2 x-a-1}{x-2 a}\\geq 1$(其中 $a \\in \\mathbf{R}$) 的解集.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷05小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022266": { - "id": "022266", - "content": "如果 $a^2>b^2$, 那么下列不等式中正确的是\\bracket{20}.\n\\fourch{$a>0>b$}{$a>b>0$}{$|a|>|b|$}{$a>|b|$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022267": { - "id": "022267", - "content": "如果 $aa b$}{$\\dfrac{1}{b^2}<\\dfrac{1}{a^2}$}{$\\dfrac{1}{a}<\\dfrac{1}{b}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022268": { - "id": "022268", - "content": "如果 $a<0b^2$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022269": { - "id": "022269", - "content": "若 $x\\dfrac{1}{x}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022270": { - "id": "022270", - "content": "以下几个关系中正确的是\\bracket{20}.\n\\fourch{$0 \\in \\varnothing$}{$\\varnothing \\in\\{x | x^2+1=0\\}$}{$\\varnothing \\subset\\{0\\}$}{$(A\\cap B) \\subset(A\\cup B)$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022271": { - "id": "022271", - "content": "满足关系式 $\\{a_1, a_2\\}\\subset A \\subset\\{a_1, a_2, a_3, a_4, a_5, a_6\\}$ 的集合 $A$ 的个数是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022272": { - "id": "022272", - "content": "已知 $A=\\{x |-3k+1$ 且 $x<2 k-1\\}$, 若 $P \\cap Q \\neq \\varnothing$, 求实数 $k$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022279": { - "id": "022279", - "content": "设全集 $U=\\mathbf{N}$, $A=\\{x | x$ 为正偶数 $\\}, B=\\{x | x$ 是 $5$ 的倍数 $\\}$, 则 $B \\cap \\overline{A}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022280": { - "id": "022280", - "content": "填写``A''(充分非必要)、``B''(必要非充分)、``C''(充要)、``D''(既非充分又非必要)之一:\\\\\n(1) ``$|x+y|<2$'' 是``$|x|<1$ 且 $|y|<1$''的\\bracket{20}条件;\\\\\n(2) ``$|x|<3$''是``$x^2<9$''的\\bracket{20}条件;\\\\\n(3) ``$x^2+y^2>0$'' 是``$x \\neq 0$''的\\bracket{20}条件;\\\\\n(4) ``$x$ 不是 2 的倍数''是``$x$ 不是 6 的倍数''的\\bracket{20}条件;\\\\\n(5) ``$x+y \\neq 3$''是``$x \\neq 1$ 且 $y \\neq 2$''的\\bracket{20}条件.", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022281": { - "id": "022281", - "content": "已知 $m$ 是实常数, 若 $\\alpha: 1 \\leq x \\leq 5$, $\\beta: 3 m-1p^2-q^2$;\\\\\n(2) $(x-a)(x-2 a)<0$($a<0$).", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022286": { - "id": "022286", - "content": "若关于 $x$ 的不等式 $a x^2-(a+1) x+1>0$ 的解集为 $\\varnothing$, 求实数 $a$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷06", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022287": { - "id": "022287", - "content": "不等式 $|\\dfrac{3 x-9}{2}| \\leq 6$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022288": { - "id": "022288", - "content": "不等式 $|x|+|x^2-4|<8$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022289": { - "id": "022289", - "content": "$x \\in \\mathbf{R}$, 代数式 $3 x^2+\\dfrac{1}{2 x^2}$ 的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022290": { - "id": "022290", - "content": "$x<0$, 代数式 $\\dfrac{x^2+2 x+4}{x}$ 的最大值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022291": { - "id": "022291", - "content": "若 $a>0$, $b>0$, 且 $2 a+3 b=1$, 则 $a b$ 的最大值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022292": { - "id": "022292", - "content": "若 $0k$, 则实数 $k$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022296": { - "id": "022296", - "content": "使不等式 $|x-4|+|x-3|>a$ 的解集为 $\\mathbf{R}$ 的实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022297": { - "id": "022297", - "content": "$x$ 为实数, 且 $|x-5|+|x-3|-1$ 时, $\\dfrac{(x+5)(x+2)}{x+1}$ 的最小值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022299": { - "id": "022299", - "content": "设 $p: x^2-x-2<0$, $q: \\dfrac{1+x}{|x-2|}<0$, 则 $p$ 是 $q$ 的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既不充分也不必要}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022300": { - "id": "022300", - "content": "下列命题中正确的是\\bracket{20}.\n\\twoch{$x+\\dfrac{1}{x}$ 的最小值是 $2$}{$\\dfrac{x^2+2}{\\sqrt{x^2+1}}$ 的最小值是 $2$}{$\\dfrac{x^2+5}{\\sqrt{x^2+4}}$ 的最小值是 $2$}{$2-3 x-\\dfrac{4}{x}$ 的最小值是 $2$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022301": { - "id": "022301", - "content": "若$a b>0$, 则\\textcircled{1} $|a+b|>|a|$; \\textcircled{2} $|a+b|<|b|$; \\textcircled{3} $|a+b|<|a-b|$; \\textcircled{4} $|a+b|>|a-b|$ 四个式中正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{4}}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022302": { - "id": "022302", - "content": "给出以下四个式子: \\textcircled{1} $x^2+4>3 x$, \\textcircled{2} $\\dfrac{y}{x}+\\dfrac{x}{y}\\geq 2$, \\textcircled{3} $x^2+y^2 \\geq 2(x-y-1)$, \\textcircled{4} $x^7+y^7 \\geq x^4 y^3+x^3 y^4$, 其中恒成立的个数是\\bracket{20}.\n\\fourch{0}{1}{2}{4}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022303": { - "id": "022303", - "content": "解下列不等式(组):\\\\\n(1) $3 \\triangleleft x-2 |<5$;\\\\\n(2) $\\begin{cases}\\dfrac{x^2+x-6}{2 x-1}>2\\\\x^2(3 x-2)(x-5)<0\\end{cases}$;\\\\\n(3) $|x-1|+|x+2| \\leq 4$;\\\\\n(4) $|x^2+2 x| \\leq x+1$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022304": { - "id": "022304", - "content": "不等式组 $\\begin{cases}4 x^2-27 x+18>0\\\\x^2-6 x+8 \\leq 0\\end{cases}$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022305": { - "id": "022305", - "content": "不等式 $\\dfrac{2 x+1}{x-1}\\geq 1$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022306": { - "id": "022306", - "content": "设 $a, b \\in \\mathbf{R}$, 若关于 $x$ 的不等式 $(4 a-3 b) x \\geq 2 b-a$ 的解集为 $(-\\infty, \\dfrac{4}{9}]$, 则 $a x \\leq b$ 的解集为为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022307": { - "id": "022307", - "content": "设 $k \\in \\mathbf{R}$, 若关于 $x$ 的不等式 $2 k x^2+k x+1 \\geq 0$ 对一切实数均成立, 则 $k$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022308": { - "id": "022308", - "content": "若集合 $A=\\{x | x^2+(a-3) x+a<0\\}$ 非空, 且 $A \\subset (0,+\\infty)$, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022309": { - "id": "022309", - "content": "设 $a \\in \\mathbf{R}$, 若不等式组 $\\begin{cases}a x^2-x-2 \\leq 0,\\\\x^2-x \\geq a(1-x)\\end{cases}$ 的解集为 $\\mathbf{R}$, 则 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022310": { - "id": "022310", - "content": "甲乙两位同学分别解``$x, y$ 为正实数, 求 $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})$ 的最小值''的过程如下:\\\\\n甲: $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y}) \\geq 2 \\sqrt{x y}\\cdot 2 \\sqrt{\\dfrac{2}{x y}}=4 \\sqrt{2}$, 所以$(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})_{\\min}=4 \\sqrt{2}$.\\\\\n乙: $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})=3+\\dfrac{y}{x}+\\dfrac{2 x}{y}\\geq 3+2 \\sqrt{\\dfrac{y}{x}\\cdot \\dfrac{2 x}{y}}=3+2 \\sqrt{2}$,\n当 $y=\\sqrt{2}x$ 时, $(x+y)(\\dfrac{1}{x}+\\dfrac{2}{y})_{\\text{min}}=3+2 \\sqrt{2}$.\\\\\n试判断谁的结论有错? 为什么错?", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022311": { - "id": "022311", - "content": "已知 $a, b, c$ 为正数, $a+b+c=1$, 求证:\\\\\n(1) $\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c}\\geq 9$;\\\\\n(2) $a b+b c+c a \\leq \\dfrac{1}{3}$;\\\\\n(3) $a^2+b^2+c^2 \\geq \\dfrac{1}{3}$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022312": { - "id": "022312", - "content": "已知 $f(x)=x^2+a x+b$($a, b \\in \\mathbf{R}$), 求证: $|f(1)|+2|f(2)|+|f(3)| \\geq 2$. (提示: $|f(1)|+2|f(2)|+|f(3)| \\geq|f(1)-2 f(2)+f(3)|$)", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022313": { - "id": "022313", - "content": "不等式 $-3 x^2+7 x-10<0$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022314": { - "id": "022314", - "content": "不等式 $\\dfrac{x-4}{3-x}\\geq 0$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022315": { - "id": "022315", - "content": "不等式 $\\dfrac{x^2+3 x-4}{x^2+2 x+3}\\geq 1$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022316": { - "id": "022316", - "content": "不等式 $|x^2+2 x-3|2$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022318": { - "id": "022318", - "content": "若 $x$、$y>0$, 且 $x+y=1$, 则 $\\dfrac{1}{x}+\\dfrac{2}{y}$ 的最小值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022319": { - "id": "022319", - "content": "设 $a$ 为常数, 若关于 $x$ 的不等式 $a x^2-(a+1) x+1<0$ 解集为空, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022320": { - "id": "022320", - "content": "设 $a$ 为常数, 若关于 $x$ 的不等式 $a x^2-(a+1) x+1<0$ 有实数解, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022321": { - "id": "022321", - "content": "已知集合 $A=\\{x | \\dfrac{a x}{x-1}<1\\}$.\\\\\n(1) 若 $A=\\{x | x<1$ 或 $x>2\\}$, 求实数 $a$ 的值;\\\\\n(2) 已知集合 $B=\\{x | x^2-5 x+6 \\leq 0\\}$, 若 $A \\cap B=B$ , 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷07小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022322": { - "id": "022322", - "content": "设 $a>0$, 在\\textcircled{1} $a^{2 n}\\cdot a^n=a^{3 n}$; \\textcircled{2} $2^3 \\cdot 3^2=6^5$; \\textcircled{3} $a^2 \\cdot a^2=5^a$; \\textcircled{4} $(-a)^2 \\cdot(-a)^3=a^5$; \\textcircled{5} $2^3 \\cdot 3^3=6^3$ 中, \n正确的是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022323": { - "id": "022323", - "content": "$\\dfrac{\\sqrt[3]{x \\sqrt{x}}}{x}=$\\blank{50}.(用分数指数幂表示)", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022324": { - "id": "022324", - "content": "已知$(5 x-\\dfrac{1}{2})^{-\\frac{3}{4}}+(x-10)^0$ 有意义, 则 $x$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022325": { - "id": "022325", - "content": "若$\\log _{(a-5)}(9-a)$ 有意义, 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022326": { - "id": "022326", - "content": "化简: $(a^{\\frac{2}{3}}b^{\\frac{1}{2}})(-3 a^{\\frac{1}{2}}b^{\\frac{1}{3}}) \\div(\\dfrac{1}{3}a^{\\frac{1}{6}}b^{\\frac{5}{6}})=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022327": { - "id": "022327", - "content": "已知 $a>0$ 且 $a \\neq 1, M$, $N \\in \\mathbf{R}$. ``$M=N$''是``$\\log _a M=\\log _a N$''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022328": { - "id": "022328", - "content": "已知自然常数 $\\mathrm{e} \\approx 2.71828 \\cdots$, 若 $\\mathrm{e}^\\alpha=4$, $\\beta=\\ln 2$, 则 $\\mathrm{e}^{2 \\alpha+\\beta}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022329": { - "id": "022329", - "content": "方程 $\\log _2 x^2=-2$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022330": { - "id": "022330", - "content": "计算下列各式的值(不能使用计算器).\\\\\n(1) $\\log _535+2 \\log _{\\frac{1}{2}}\\sqrt{2}-\\log _5 \\dfrac{1}{50}-\\log _514=$\\blank{50}.\\\\\n(2) $\\dfrac{\\log _5 \\sqrt{2}\\times \\log _79}{\\log _5 \\dfrac{1}{3}\\times \\log _7 \\sqrt[3]{4}}=$\\blank{50}.\\\\\n(3) $(\\log _43+\\log _83) \\times \\dfrac{\\log _72}{\\log _73}=$\\blank{50}.\\\\\n(4) $2^{\\log _4(\\sqrt{3}-2)^2}=$\\blank{50}.\\\\\n(5) $\\log _6^23+\\dfrac{\\log _618}{\\log _26}=$\\blank{50}.\\\\\n(6) $\\log _2 \\dfrac{\\sqrt{2}}{2}-\\log _3 \\dfrac{\\sqrt{3}}{3}+(\\sqrt{7})^{\\frac{1}{2}\\log _714-\\frac{1}{2}}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022331": { - "id": "022331", - "content": "用 $a, b$ 的不含对数的式子表示下列各式的值.\\\\\n(1) 若 $\\log _53=a$, 则 $\\log _{45}9=$\\blank{50}.\\\\\n(2) 若 $\\log _{18}6=b$, 则 $\\log _{12}27=$\\blank{50}.\\\\\n(3) 若 $\\log _35=a$, $\\log _25=b$, 则 $\\log _65=$\\blank{50}.\\\\\n(4) 若 $\\log _754=a$, $\\log _772=b$, 则 $\\log _712=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022332": { - "id": "022332", - "content": "若方程 $(\\lg x)^2+(\\lg 7+\\lg 5) \\lg x+\\lg 7 \\cdot \\lg 5=0$ 的两根为 $\\alpha, \\beta$, 则 $\\alpha \\beta=\\bracket{20}$\\bracket{20}.\n\\fourch{$\\lg 7 \\cdot \\lg 5$}{$\\lg 35$}{$35$}{$\\dfrac{1}{35}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022333": { - "id": "022333", - "content": "若 $a$、$b$、$c$ 都是正数, 且 $3^a=4^b=6^c$, 则 $\\bracket{20}$\\bracket{20}.\n\\fourch{$\\dfrac{1}{c}=\\dfrac{1}{a}+\\dfrac{1}{b}$}{$\\dfrac{2}{c}=\\dfrac{2}{a}+\\dfrac{1}{b}$}{$\\dfrac{1}{c}=\\dfrac{2}{a}+\\dfrac{2}{b}$}{$\\dfrac{2}{c}=\\dfrac{1}{a}+\\dfrac{2}{b}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022334": { - "id": "022334", - "content": "已知 $x=\\log _23$,求 $\\dfrac{2^{3 x}-2^{-3 x}}{2^x-2^{-x}}$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022335": { - "id": "022335", - "content": "已知 $a>0$, 若 $0 \\leq r \\leq 8$, $r \\in \\mathbf{N}$, 式子 $(\\sqrt{a})^{8-r}\\cdot(\\dfrac{1}{\\sqrt[4]{a}})^r$ 能化为关于 $a$ 的整数指数幂, 求 $r$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022336": { - "id": "022336", - "content": "已知 $x_1$、$x_2$ 是方程 $x^2-m x+3=0$ 的两个根, 且 $\\dfrac{\\ln (x_1+x_2)}{\\ln x_1+\\ln x_2}=2$, 求 $m$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022337": { - "id": "022337", - "content": "已知 $\\log _a(x^2+1)+\\log _a(y^2+4)=\\log _a 8+\\log _a x+\\log _a y$, 求 $\\log _8(x y)$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022338": { - "id": "022338", - "content": "已知 $a>0$ 且 $a \\neq 1$, $x>y>0$, 下列四个等式中, 恒成立的是\\blank{50}.\\\\\n\\textcircled{1} $\\log _a x \\log _a y=\\log _a(x+y)$;\\\\\n\\textcircled{2} $\\log _a x+\\log _a y=\\log _a(x+y)$;\\\\\n\\textcircled{3} $\\log _a \\dfrac{x}{y}=\\log _a x-\\log _a y$;\\\\\n\\textcircled{4} $\\log _a(x-y)=\\dfrac{\\log _a x}{\\log _a y}$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022339": { - "id": "022339", - "content": "将 $a^{2 k}=N$($a>0$, $a \\neq 1$) 转化成对数式, 其中错误的是\\blank{50}.\\\\\n\\textcircled{1} $k=\\log _{a^2}N$\\\\\n\\textcircled{2} $2=\\log _{a^k}N$\\\\\n\\textcircled{3} $k=\\dfrac{1}{2}\\log _a N$\\\\\n\\textcircled{4} $k=\\log _a \\dfrac{N}{2}$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022340": { - "id": "022340", - "content": "设 $x$、$y>1$, 且 $2 \\log _x y-2 \\log _y x+3=0$, 求 $T=x^2-4 y^2$ 的最小值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022341": { - "id": "022341", - "content": "设 $a$、$b$、$c$ 是直角三角形的三边长, 其中 $c$ 为斜边, 且 $c+b \\neq 1$, $c-b \\neq 1$, 求证: $\\log _{(c+b)}a+\\log _{(c-b)}a=2 \\log _{(c+b)}a \\cdot \\log _{(c-b)}a$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022342": { - "id": "022342", - "content": "若实数 $a$、$b$ 满足 $a^2+b^2=1$, 则 $a b$ 的最小值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022343": { - "id": "022343", - "content": "已知正数 $x, y$ 满足 $x^3+y^3=2$, 则 $x+y$ 的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022344": { - "id": "022344", - "content": "若直角三角形的斜边为 1 , 则此三角形内切圆半径的最大值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022345": { - "id": "022345", - "content": "使 $(\\sqrt{x}+\\sqrt{y}) \\leq a \\sqrt{x+y}$, $x>0$, $y>0$ 恒成立的 $a$ 的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022346": { - "id": "022346", - "content": "若 $a, b>0$, $2 a+b=1$, 则 $S=2 \\sqrt{a b}-4 a^2-b^2$ 的的最大值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022347": { - "id": "022347", - "content": "若 $a$、$b>0$, 满足 $a+b+1=a b$, 则 $3 a+2 b$ 的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022348": { - "id": "022348", - "content": "$0b>1$, 若 $\\log _a b+\\log _b a=\\dfrac{10}{3}$, $a^b=b^a$, 求由数对 $(a, b)$ 构成的集合.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷08小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022358": { - "id": "022358", - "content": "写出下列函数的定义域:\\\\\n(1) $y=x^{-\\frac{4}{5}}$: \\blank{100};\\\\\n(2) $y=x^{-\\frac{3}{2}}$: \\blank{100};\\\\\n(3) $y=x^{\\frac{7}{4}}$: \\blank{100};\\\\\n(4) $y=x^{\\frac{2}{3}}$: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022359": { - "id": "022359", - "content": "设 $m \\in \\mathbf{Z}$, 若幂函数 $y=x^{-m^2+2 m}$ 的定义域为 $\\mathbf{R}$, 则 $m$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022360": { - "id": "022360", - "content": "若幂函数 $f(x)=x^{m^2+2 m-3}$($m \\in Z$) 的图像关于 $y$ 轴对称, 且在区间 $(0,+\\infty)$ 上是减函数, 则 $m=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022361": { - "id": "022361", - "content": "设 $m \\in \\mathbf{R}$, 若 $(3-2 m)^{\\frac{1}{2}}>(m+1)^{\\frac{1}{2}}$, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022362": { - "id": "022362", - "content": "已知 $x>y>1$, 且 $0a^{\\frac{1}{y}}$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022363": { - "id": "022363", - "content": "有下列说法: \\textcircled{1} 函数 $y=x^{\\frac{2}{3}}$ 在定义域上是增函数; \\textcircled{2} 函数 $y=x^{\\frac{5}{3}}$ 在定义域上是增函数; \\textcircled{3} 函数 $y=x^{-\\frac{1}{3}}$在定义域上是减函数; \\textcircled{4} 函数 $y=x^{-\\frac{3}{2}}$ 在定义域上是减函数. 其中正确的有\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022364": { - "id": "022364", - "content": "设常数 $a \\in \\mathbf{R}$, 则``$b=2$''是``函数 $y=(2 b^2-3 b-1) x^a$ 为幂函数''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022365": { - "id": "022365", - "content": "若点 $A(a, b)$ 在幂函数 $y=x^n$($n \\in \\mathbf{Q}$) 的图像上, 那么下列结论中不能成立的是 \\bracket{20}.\n\\fourch{$\\begin{cases}a>0,\\\\b>0\\end{cases}$}{$\\begin{cases}a>0,\\\\b<0\\end{cases}$}{$\\begin{cases}a<0,\\\\b<0\\end{cases}$}{$\\begin{cases}a<0,\\\\b>0\\end{cases}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022366": { - "id": "022366", - "content": "若集合 $A=\\{y | y=x^{\\frac{1}{3}},-1 \\leq x \\leq 1\\}$, $B=\\{y | y=2-\\dfrac{1}{x}, 0=latex,scale = 0.5]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:2, samples = 100] plot (\\x,\\x*\\x) node [right] {$c_1$};\n\\draw [domain = 0:2, samples = 100] plot (\\x*\\x,\\x) node [right] {$c_2$};\n\\draw [domain = 0.0625:4, samples = 100] plot (\\x, {1/sqrt(\\x)}) node [right] {$c_3$};\n\\draw [domain = 0.5:4, samples = 100] plot (\\x, {1/\\x/\\x}) node [right] {$c_4$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$-2,-\\dfrac 12,\\dfrac 12,2$}{$2,\\dfrac 12,-\\dfrac 12,-2$}{$-\\dfrac 12,-2,2,\\dfrac 12$}{$2,\\dfrac 12,-2,-\\dfrac 12$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022369": { - "id": "022369", - "content": "下列函数的图像为(A)、(B)、(C)、(D)之一, 试将正确的字母标号填在相应函数后面的横线上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-3^(1/4)}:{3^(1/4)}] plot (\\x*\\x*\\x,\\x*\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(A)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-3^(1/5)}:{3^(1/5)}] plot (\\x*\\x*\\x,\\x*\\x*\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(B)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale =0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {0}:{3^(1/3)}] plot (\\x*\\x,\\x*\\x*\\x);\n\\draw (0,-3) node [below] {(C)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {3^(-3/2)}:3] plot (\\x,{\\x^(-2/3)}) plot (-\\x,{\\x^(-2/3)});\n\\draw (0,-3) node [below] {(D)};\n\\end{tikzpicture}\n\\end{center}\n(1) $y=x^\\frac 32$\\blank{50}; (2) $y=x^\\frac 43$\\blank{50}; (3) $y=x^\\frac 53$\\blank{50}; (4) $y=x^{-\\frac 23}$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022370": { - "id": "022370", - "content": "试求下列函数的定义域、值域, 并画出它的大致图像:\\\\\n(1) $y=(x+2)^{-2}$\\\\\n(2) $y=(x+2)^{-\\frac{2}{3}}-3$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022371": { - "id": "022371", - "content": "已知函数 $y=\\dfrac{2 x+1-a}{x+a}$ 在 $[1,+\\infty)$ 上是严格增函数, 并且函数值恒不为负, 求出所有符合条件的 $a$ 取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022372": { - "id": "022372", - "content": "若 $(a+1)^{-\\frac{2}{3}}>(3-2 a)^{-\\frac{2}{3}}$, 则 $a$ 取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022373": { - "id": "022373", - "content": "若 $(a+1)^{-3}>(3-2 a)^{-3}$, 则 $a$ 取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022374": { - "id": "022374", - "content": "函数 $f(x)=\\dfrac{x+3}{x-1}$ 的图像关于对称.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022375": { - "id": "022375", - "content": "已知幂函数$y=x^{\\frac qp}$($p,q$都是正整数, $p,q$互质)的图像如图所示, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [dashed] (-1,-1) -- (2,2);\n\\draw [domain = 0:3, samples = 400, very thick] plot (\\x,{\\x^(3/8)}) plot (-\\x,{\\x^(3/8)});\n\\draw [dashed] (1,0) node [below] {$1$} -- (1,1) -- (0,1) node [left] {$1$};\n\\end{tikzpicture}\n\\end{center}\n\\twoch{$p,q$均为奇数}{$p$是奇数, $q$是偶数, 且$0<\\dfrac qp<1$}{$p$是偶数, $q$是奇数}{$p$是奇数, $q$是偶数, 且$\\dfrac qp>1$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022376": { - "id": "022376", - "content": "作出函数 $y=\\dfrac{x^2+4 x+6}{x^2+4 x+4}$ 的大致图像, 并根据图像判断 $x$ 分别取 $-\\pi$ 和 $-\\dfrac{\\sqrt{2}}{2}$ 时, 函数值 $y_1$ 和 $y_2$ 的大小.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022377": { - "id": "022377", - "content": "若 $\\lg a, \\lg b$ 是方程 $2 x^2-4 x+1=0$ 的两个根, 则 $(\\lg \\dfrac{a}{b})^2=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022378": { - "id": "022378", - "content": "设 $\\lg x+\\lg y=2 \\lg (x-2 y)$, 则 $\\log _{\\sqrt{2}}\\dfrac{x}{y}$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022379": { - "id": "022379", - "content": "$2^{\\log _4(\\sqrt{3}-2)^2}+\\lg (\\sqrt{3+\\sqrt{5}}+\\sqrt{3-\\sqrt{5}})=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022380": { - "id": "022380", - "content": "已知 $\\log _35=a$, $\\log _25=b$, 用 $a, b$ 表示 $\\log _{18}5=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022381": { - "id": "022381", - "content": "若关于 $x$ 的不等式 $a x^2+b x+c<0$ 的解集为 $(-\\infty,-2) \\cup$($3,+\\infty$), 则 $c x^2-b x+a>0$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022382": { - "id": "022382", - "content": "已知不等式 $x^2-|2 x-4| \\geq p$ 对任意实数 $x$ 恒成立, 求实数 $p$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022383": { - "id": "022383", - "content": "已知集合 $A=\\{x | \\dfrac{a x}{x-2}>1\\}$.\\\\\n(1) 是否存在实数 $a$ 使得集合 $A=\\{x | x<1$, 或 $x>2\\}$? 若存在, 求出 $a$ 的值; 若不存在, 说明理由;\\\\\n(2) 若集合 $B=\\{x|| x-1 |<3\\}$, 且 $A \\cup B=B$, 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022384": { - "id": "022384", - "content": "已知 $\\alpha: x<3 m-1$ 或 $x>-m+1$, $\\beta: x<2$ 或 $x \\geq 4$. 若 $\\alpha$ 是 $\\beta$ 的必要条件, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022385": { - "id": "022385", - "content": "幂函数 $y=x^{\\frac{1}{2}}$ 的图像可由函数 $y=1+\\sqrt{x-1}$ 的图像向\\blank{20}(填``左''或``右'') 平移一个单位, 再向\\blank{20}(填``上''或``下'') 平移一个单位得到.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022386": { - "id": "022386", - "content": "若不等式 $|x+m|-|x-1| \\leq 4$ 对于任意实数 $x$ 恒成立, 则 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022387": { - "id": "022387", - "content": "$(\\log _43+\\log _83)(\\log _32+\\log _92)+\\log _{\\frac{1}{2}}\\sqrt[4]{32}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022388": { - "id": "022388", - "content": "设 $\\lg x+\\lg y=\\lg (x-2 y)^2$, 则 $\\log _{\\sqrt{2}}\\dfrac{x}{y}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022389": { - "id": "022389", - "content": "若设 $m \\in \\mathbf{Z}$, 若幂函数 $y=x^{\\frac{m-1}{m-4}}$ 的定义域为 $\\{x | x \\neq 0\\}$, 则 $m$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022390": { - "id": "022390", - "content": "若幂函数 $y=(m^2-3 m+3) x^{m^2-2 m+2}$ 的图像关于原点中心对称, 则 $m=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022391": { - "id": "022391", - "content": "已知 $x, y, z \\in$($0,+\\infty$), 且 $x y z(x+y+z)=1$, 求 $(x+y)(y+z)$ 的最小值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022392": { - "id": "022392", - "content": "已知 $(a+1)^{-\\frac{2}{3}}>(3-4 a)^{-\\frac{2}{3}}$, 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022393": { - "id": "022393", - "content": "已知存在 $x \\in[-2,2]$ 使得不等式 $x^2-|2 x-4| \\geq p$ 成立, 求实数 $p$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷09小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022394": { - "id": "022394", - "content": "写出下列函数的定义域 (用集合形式表示):\\\\\n(1) $y=2^{\\sqrt{-x^2}}$: \\blank{150};\\\\\n(2) $y=\\dfrac{1}{4^{x-5}-2}$: \\blank{150};\\\\\n(3) $y=\\dfrac{1}{\\log _2 x}$: \\blank{150};\\\\\n(4) $y=\\sqrt{\\log _{0.8}x-1}$: \\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022395": { - "id": "022395", - "content": "不使用计算器, 比较下列各数的大小:\\\\\n(1) $(\\dfrac{1}{3})^4$\\blank{20}$3^{0.01}$;\\\\\n(2) $\\log _2 \\pi$\\blank{20}$\\log _2 \\mathrm{e}$;\\\\\n(3) $\\log _{0.5}3$\\blank{20}$\\log _24$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022396": { - "id": "022396", - "content": "设常数 $a>0$ 且 $a \\neq 1$. 若无论 $a$ 取何值, 函数 $y=a^{x-3}+3$ 图像恒经过一个定点, 则此定点的坐标是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022397": { - "id": "022397", - "content": "设 $a>0$ 且 $a \\neq 1$. 若无论 $a$ 为何值, 函数 $y=\\log _a(x-4)-3$ 的图像总经过一个定点, 则该定点的坐标是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022398": { - "id": "022398", - "content": "设 $a>0$ 且 $a \\neq 1$. 函数 $y=\\log _a x$ 在区间 $[a, 2 a]$ 上的最大值与最小值之差为 $2$, 则 $a=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022399": { - "id": "022399", - "content": "不等式$(\\dfrac{1}{4})^x<2^{-x^2+2 x-3}$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022400": { - "id": "022400", - "content": "不等式$\\log _{\\frac{1}{2}}(3 x-1)>2$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022401": { - "id": "022401", - "content": "若指数函数 $y=(a^2-1)^x$ 的值在区间 ($-\\infty, 0$) 上恒大于 $1$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022402": { - "id": "022402", - "content": "设 $a>0$ 且 $a \\neq 1$. 若 $y=\\log _a x$ 的值在区间 $[2,+\\infty)$ 上恒大于 1 , 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022403": { - "id": "022403", - "content": "函数 $y=\\lg (k x^2-4 x+k+3)$ 的定义域为 $\\mathbf{R}$, 则实数 $k$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022404": { - "id": "022404", - "content": "已知函数\\textcircled{1} $y=3^x-1$, \\textcircled{2} $y=3^{x-1}$, \\textcircled{3} $y=-3^x$, \\textcircled{4} $y=-3^{-x}$, \\textcircled{5} $y=(\\dfrac{1}{3})^x$, \\textcircled{6} $y=(\\dfrac{1}{3})^{-x}$. 将 $y=3^x$ 的图像向右平移 $1$ 个单位得到\\blank{50}的图像; 将 $y=3^x$ 的图像向下平移 $1$ 个单位得到\\blank{50}的图像; $y=3^x$ 的图像与\\blank{50}的图像关于 $x$ 轴对称, 与\\blank{50}的图像关于 $y$ 轴对称, 与\\blank{50}的图像关于原点对称, 与\\blank{50}的图像完全相同. (用序号作答)", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022405": { - "id": "022405", - "content": "某种细菌在培养过程中, 每 20 分钟分裂一次 (一个分裂为两个), 经过 3 小时, 这种细菌由 1 个可繁殖成\\bracket{20}.\n\\fourch{18 个}{256 个}{512 个}{1024 个}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022406": { - "id": "022406", - "content": "设 $a, b$ 均为正数且均不等于 $1$, 若 $\\log _a \\dfrac{1}{3}>\\log _b \\dfrac{1}{3}>0$, 则 $a, b$ 的关系是\\bracket{20}.\n\\fourch{$10$ 且 $a \\neq 1$) 的图像有两个公共点, 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022415": { - "id": "022415", - "content": "已知函数 $y=\\begin{cases}|\\log _2 x|,& 08 .\\end{cases}$ 若当 $x$ 分别取互不相等的实数 $a, b, c$ 时, 函数值均相等, 则 $a b c$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022416": { - "id": "022416", - "content": "设 $a>0$ 且 $a \\neq 1$, 比较 $1+\\log _a 5$ 与 $\\log _a(a+1)$ 的大小.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022417": { - "id": "022417", - "content": "关于 $x$ 的方程 $3^{2 x+1}+(m+1)(3^{x+1}-1)-(m-3) 3^x=0$ 有两个不同的实数解, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022418": { - "id": "022418", - "content": "已知函数 $y=\\lg (2+x)+\\lg (2-x)$.\\\\\n(1) 求函数的定义域;\\\\\n(2) 设 $m \\in \\mathbf{R}$, 若关于 $x$ 的不等式 $\\lg (2+x)+\\lg (2-x)-m>0$ 有解, 求实数 $m$ 的取值范围.\n %2025届高一上学期周末卷12", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022419": { - "id": "022419", - "content": "若 $f(x)$ 的定义域为 $[0,10]$, 则函数 $F(x)=f(x+2)+f(x-2)$ 的定义域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022420": { - "id": "022420", - "content": "设函数 $f(x)$ 的定义域为 $(0,6)$, $g(x)$ 的定义域为 $[2,7]$, 若 $f(x)>g(x)$ 的解集是 $(3,5)$, 则 $f(x) \\leq g(x)$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022421": { - "id": "022421", - "content": "已知 $A=\\{x | y=\\lg (x^2-x-2)\\}$, $B=\\{y | y=(\\dfrac{1}{2})^x\\}$, 则 $A \\cap B=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022422": { - "id": "022422", - "content": "函数 $f(x)=\\begin{cases}x^2,& x \\geq 4,\\\\f(x+3) & x<4\\end{cases}$ 则 $f(2)=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022423": { - "id": "022423", - "content": "若函数 $y=f(x)$ 对一切 $x \\in \\mathbf{R}$ 均成立 $f(2 x+1)=x^2$, 则 $f(x)$ 的解析式为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022424": { - "id": "022424", - "content": "若函数 $f(x)=\\sqrt{2 x^2+3}$, 函数 $g(x)=f(\\sqrt{x})$, 则函数 $y=g(x)$ 的解析式为\\blank{50}, 定义域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022425": { - "id": "022425", - "content": "函数 $f(x)=\\dfrac{-5 x+3}{x-3}$, $x \\in(-\\infty, 2) \\cup[4,+\\infty)$ 的值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022426": { - "id": "022426", - "content": "``$f(0)=0$''是``函数 $y=f(x)$ 为奇函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022427": { - "id": "022427", - "content": "下列各组函数中, 表示同一函数的是\\bracket{20}.\n\\twoch{$y=1$, $y=\\dfrac{x}{x}$}{$y=\\sqrt{x-1}\\cdot \\sqrt{x+1}$, $y=\\sqrt{x^2-1}$}{$y=x$, $y=\\sqrt[3]{x^3}$}{$y=|x|$, $y=(\\sqrt{x})^2$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022428": { - "id": "022428", - "content": "若 $a, b \\in \\mathbf{R}$, 则``$b^2<4 a c$''是``函数 $y=\\dfrac{1}{a x^2+b x+c}$ 的定义域为 $\\mathbf{R}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022429": { - "id": "022429", - "content": "若定义在 $\\mathbf{R}$ 上的函数 $f(x)$、$g(x)$ 均为奇函数, 设 $F(x)=a f(x)+b g(x)+1$, 若 $F(-2)=10$, 则 $F(2)$ 的值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022430": { - "id": "022430", - "content": "已知函数 $f(x)$ 为定义在 $\\mathbf{R}$ 上的函数, 则命题:``存在 $x_0 \\in \\mathbf{R}$, 使 $f(-x_0) \\neq f(x_0)$ 且 $f(-x_0) \\neq-f(x_0)$''是命题``$f(x)$ 为非奇非偶函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022431": { - "id": "022431", - "content": "设函数 $f(x)=2^x$, $g(x)=4^x$, 且 $f[g(x)]=g[f(x)]$, 则 $x=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022432": { - "id": "022432", - "content": "满足解析式为 $y=2 x^2+1$, 且值域为 $\\{5,19\\}$ 的函数有\\bracket{20}.\n\\fourch{4 个}{6 个}{8 个}{9 个}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022433": { - "id": "022433", - "content": "函数 $f(x)=\\lg (x^2-1)$ 的定义域为\\blank{50}, 值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022434": { - "id": "022434", - "content": "设 $f(x)$ 是定义在 $\\mathbf{R}$ 上的奇函数, 当 $x>0$ 时, $f(x)=2^x-3$, 则不等式 $f(x)<-5$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022435": { - "id": "022435", - "content": "已知函数 $y=\\log _a(2-a x)$ 在 $[0,1]$ 上严格递减, 则实数 $a$ 的取值范围是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{2}, 1) \\cup(1,2)$}{$(0, \\dfrac{1}{2}) \\cup(1,2)$}{$(1,2)$}{$(0, \\dfrac{1}{2}) \\cup$($2,+\\infty$)}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022436": { - "id": "022436", - "content": "判断下列函数的奇偶性, 并说明理由:\\\\\n(1) $f(x)=x^2+x$;\\\\\n(2) $f(x)=\\begin{cases}x(1-x), & x<0,\\\\x(1+x),& x>0;\\end{cases}$\\\\\n(3) $f(x)=\\dfrac{x(a^x-1)}{a^x+1}$($a>0$, $a \\neq 1$);\\\\\n(4) $f(x)=x^n-x^{-n}$($n \\in \\mathbf{Z}$).", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022437": { - "id": "022437", - "content": "已知定义域为 $[0,1]$ 上的函数 $f(x)=1-|1-2 x|$ 和 $g(x)=(x-1)^2$ 且 $F(x)=\\begin{cases}g(x), & f(x) \\geq g(x),\\\\f(x),& f(x)0$ 时, $f(x)=\\sqrt{x+1}+1$, 则 $f(x)=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022445": { - "id": "022445", - "content": "函数 $y=\\log _{0.5}(-x^2+8 x-7)$ 的单调递减区间是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022446": { - "id": "022446", - "content": "函数 $f(x)$ 在 $\\mathbf{R}$ 上严格递减, 且为奇函数. 若 $f(1)=-1$, 则满足 $-1 \\leq f(x-2) \\leq 1$ 的 $x$ 的取值范围是\\bracket{20}.\n \\fourch{$[-2,2]$}{$[-1,1]$}{$[0,4]$}{$[1,3]$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022447": { - "id": "022447", - "content": "已知定义在区间 $(-1,1)$ 上的函数 $f(x)=\\dfrac{a x+b}{1+x^2}$ 为奇函数, 且 $f(\\dfrac{1}{2})=\\dfrac{2}{5}$.\\\\\n (1) 求实数 $a$、$b$ 的值;\\\\\n (2) 证明: 函数 $f(x)$ 在区间 $(-1,1)$ 上是严格增函数;\\\\\n (3) 解关于 $t$ 的不等式 $f(t-1)+f(t)<0$.\n %2025届高一上学期周末卷13", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022448": { - "id": "022448", - "content": "函数 $y=\\log _{(2 x-1)}\\sqrt{3 x-2}$ 的定义域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022449": { - "id": "022449", - "content": "函数 $y=x^2+4 x-12$ 的单调递减区间为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022450": { - "id": "022450", - "content": "函数 $y=x^2+4 x-12$ 的值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022451": { - "id": "022451", - "content": "函数 $y=\\log _5(x^2+4 x-12)$ 的递增区间为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022452": { - "id": "022452", - "content": "函数 $y=\\log _5(x^2+4 x-12)$ 的值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022453": { - "id": "022453", - "content": "函数 $f(x)=a x+b$($a, b \\in \\mathbf{R}$) 为奇函数, 则 $a, b$ 满足的条件为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022454": { - "id": "022454", - "content": "函数 $f(x)=\\dfrac{3 x+2}{x-1}$ 递减区间是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022455": { - "id": "022455", - "content": "函数 $f(x)=2 x^2+3 x$, $x \\in[-2,2]$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022456": { - "id": "022456", - "content": "函数 $f(x)=\\dfrac{2 x-3}{2 x+1}$, $x \\in(-1,-\\dfrac{1}{2})$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022457": { - "id": "022457", - "content": "若函数 $f(x)=(a+1) x^2-(2 a-3) x+1$ 在 $[-1,+\\infty)$ 上单调递增, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022458": { - "id": "022458", - "content": "定义在 $(-2,2)$ 上的偶函数 $f(x)$ 在 $(-2,0]$ 上严格递增, 若 $f(a)>f(2 a-1)$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022459": { - "id": "022459", - "content": "下列函数中, 在区间 $(-\\infty, 0)$ 上是减函数的是\\bracket{20}.\n\\fourch{$y=1-x^2$}{$y=x^2+2 x$}{$y=\\dfrac{x+2}{x+1}$}{$y=\\dfrac{x}{x-1}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022460": { - "id": "022460", - "content": "下列函数中, 是奇函数的是\\bracket{20}.\n\\fourch{$f(x)=x^2+x$}{$g(x)=1-x^2$}{$h(x)=2 x+1$}{$m(x)=\\lg (\\sqrt{1+x^2}-x)$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022461": { - "id": "022461", - "content": "下列命题中, 正确的是\\bracket{20} .\n\\onech{若函数 $f(x)$ 在 $(-\\infty, 0]$ 上严格递增, 在 ($0,+\\infty$) 上也严格递增, 则它在 ($-\\infty$, $+\\infty$) 上严格递增}{若函数 $f(x)$ 在 ($-\\infty, 2]$ 上严格递增, 在 $[2,+\\infty$) 上也严格递增, 则它在 ($-\\infty$, $+\\infty$) 上严格递增}{若 $f(x)$ 在 $D$ 上严格递增, $g(x)$ 在 $D$ 上严格递减, 则 $F(x)=f(x)+g(x)$ 在 $D$ 上严格递减}{若 $f(x)$ 在 $D$ 上严格递增, $g(x)$ 在 $D$ 上严格递增, 则 $G(x)=f(x) \\cdot g(x)$ 在 $D$ 上严格递增}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022462": { - "id": "022462", - "content": "画出下列函数的图像:\\\\\n(1) $y=|x^2-2 x-3|$;\\\\\n(2) $y=3^{|x-1|}-2$;\\\\\n(3) $y=|\\log _2(|x|-1)|$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022463": { - "id": "022463", - "content": "已知 $m \\in \\mathbf{R}$, 求函数 $f(x)=x^2+2 x+3$ 在 $[m, 0]$ 上的值域.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022464": { - "id": "022464", - "content": "已知 $a$ 为实常数, 设 $f(x)=\\begin{cases}(x-a)^2,& x \\leq 0,\\\\x+\\dfrac{4}{x}+3 a,& x>0 .\\end{cases}$ 若 $f(0)$ 是 $f(x)$ 的最小值, 求 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022465": { - "id": "022465", - "content": "已知函数 $f(x)=x^2+|x+a|$, $x \\in \\mathbf{R}$, $a$ 为实常数.\\\\\n(1) 求证: 函数 $f(x)$ 不是奇函数;\\\\\n(2) 若函数 $f(x)$ 是偶函数, 求实数 $a$ 的值;", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022466": { - "id": "022466", - "content": "已知函数 $f(x)$ 的定义域是 $x \\neq 0$ 的一切实数, 对定义域内的任意 $x_1, x_2$ 都有 $f(x_1 \\cdot x_2)=f(x_1)+f(x_2)$, 且当 $x>1$ 时 $f(x)>0$, $f(2)=1$.\\\\\n(1) 求证: $f(x)$ 是偶函数;\\\\\n(2) 求证: $f(x)$ 在 ($0,+\\infty$) 上是增函数;\\\\\n(3) 解不等式 $f(2 x^2-1)<2$.\n %2025届高一上学期周末卷13小测", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022467": { - "id": "022467", - "content": "函数 $y=2^{-x}-2^x$ 的奇偶性为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022468": { - "id": "022468", - "content": "函数 $y=\\sqrt{1-2^x}$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022469": { - "id": "022469", - "content": "已知 $a \\neq 0$, 函数 $y=a^{x+1}-2$ 的图像都通过同一定点, 该点坐标是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022470": { - "id": "022470", - "content": "函数 $y=2^{x^2+4 x-5}$ 的严格增区间为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022471": { - "id": "022471", - "content": "函数 $y=\\log _{0.5}(x^2+4 x+5)$ 的值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022472": { - "id": "022472", - "content": "函数 $f(x)=\\log _a x$ 在区间 $[a, 2 a]$ 上的最大值与最小值的差为 $\\dfrac{1}{2}$, 则 $a=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022473": { - "id": "022473", - "content": "已知函数 $f(x)$ 是奇函数且定义域为 $\\mathbf{R}$, 若 $x>0$ 时, $f(x)=x^2-2 x$, 则 $f(x)=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022474": { - "id": "022474", - "content": "画出下列函数的图像:\\\\\n(1) $y=x^2-2|x|+1$;\\\\\n(2) $y=3^{|x|+1}+1$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022475": { - "id": "022475", - "content": "设 $m>0$, 若二次函数 $f(x)=-x^2+m x+1+n$, $x \\in[-1,1]$ 的最大值是 $9$ , 最小值是 $6$, 求实数 $m$、$n$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷10", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022476": { - "id": "022476", - "content": "若函数 $f(x)=x^{m^2-2 m-3}$($m \\in \\mathbf{Z}$) 为偶函数, 且在区间 ($0,+\\infty$) 上是减函数, 则 $f(x)=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022477": { - "id": "022477", - "content": "已知 $(a+1)^{-2}>(3-2 a)^{-2}$, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022478": { - "id": "022478", - "content": "函数 $y=\\log _{\\frac{1}{2}}\\sqrt{-x^2+2 x+1}$ 的定义域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022479": { - "id": "022479", - "content": "函数 $y=x+\\sqrt{2 x-1}$ 的值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022480": { - "id": "022480", - "content": "函数 $y=\\dfrac{x^2+3}{1-x}$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022481": { - "id": "022481", - "content": "函数 $y=\\dfrac{2 x}{1+x^2}$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022482": { - "id": "022482", - "content": "函数 $y=\\log _3 \\dfrac{x+2}{2 x+1}$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022483": { - "id": "022483", - "content": "函数 $y=2 x^2+\\dfrac{4}{x^2+1}x \\in[1,7]$ 的最小值是\\blank{50}, 此时 $x=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022484": { - "id": "022484", - "content": "已知函数 $f(x)=x+\\dfrac{a}{x}$ 在区间 $[3,+\\infty)$ 上为增函数, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022485": { - "id": "022485", - "content": "定义在 $(0,+\\infty)$ 上的函数 $f(x)$ 是增函数, 如果 $f(x^2-2 a x)$ 在 $x \\in[2,4]$ 上是增函数, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022486": { - "id": "022486", - "content": "若 $f(x)=\\begin{cases}x+\\dfrac{a-2}{x},& x>1,\\\\(1-a) x-1,& x \\leq 1\\end{cases}$ 是 $\\mathbf{R}$ 上的增函数, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022487": { - "id": "022487", - "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的偶函数, 且在 $[0,+\\infty)$ 上是严格增函数, 若 $f(a+1) \\leq f(4)$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022488": { - "id": "022488", - "content": "已知函数 $f(x)$ 在 $\\mathbf{R}$ 上是严格增函数且 $f(3)=0$, 则函数 $g(x)=-|f(x)|+3$ 的单调增区间是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022489": { - "id": "022489", - "content": "``$m>0$''是``函数 $f(x)=|x(m x+2)|$ 在区间 ($0,+\\infty$) 上为增函数''的条件.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022490": { - "id": "022490", - "content": "若直线 $y=2 a$ 与函数 $y=|a^x-1|$($a>0$ 且 $a \\neq 1$) 的图像有两个公共点, 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022491": { - "id": "022491", - "content": "下列函数中, 在区间 ($-\\infty, 0$) 上是减函数的是\\bracket{20}.\n\\fourch{$y=1-x^2$}{$y=x^2+2 x$}{$y=\\dfrac{x+2}{x+1}$}{$y=\\dfrac{x}{x-1}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022492": { - "id": "022492", - "content": "下列函数既是奇函数, 又在区间 $[-1,1]$ 上是增函数的是\\bracket{20}.\n\\twoch{$f(x)=|x+1|$}{$f(x)=\\dfrac{x}{|x|}$}{$f(x)=\\begin{cases}x^2,& x \\geq 0,\\\\-x^2, & x<0\\end{cases}$}{$f(x)=-\\dfrac{2}{x}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022493": { - "id": "022493", - "content": "函数 $f(x)=-x^2+4 x$ 在 $[m, n]$ 上的值域为 $[-5,4]$, 则 $m+n$ 的取值所组成的集合为\\bracket{20}.\n\\fourch{$[0,6]$}{$[-1,5]$}{$[1,5]$}{$[1,7]$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022494": { - "id": "022494", - "content": "已知函数 $f(x)=|x|+\\dfrac{m}{x}-3$($m \\in \\mathbf{R}$, $x \\neq 0$). 判断函数 $y=f(x)$ 的奇偶性, 并说明理由.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022495": { - "id": "022495", - "content": "设 $f(x)=a x^2+b x+1,$($a$、$b \\in \\mathbf{R}$).\\\\\n(1) 若 $f(-1)=0$, 且对任意实数 $x$, 均有 $f(x) \\geq 0$, 求 $a$、$b$ 的值;\\\\\n(2) 在 (1) 的条件下, $g(x)=f(x)-k x$ 在 $[-2,2]$ 上是单调函数, 求实数 $k$ 的取值范围;", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022496": { - "id": "022496", - "content": "函数 $f(x)=(a-2) x^2+2(a-2) x-4$($a \\in \\mathbf{R}$).\\\\\n(1) 若 $x \\in \\mathbf{R}$, $f(x)<0$ 恒成立, 求 $a$ 的取值范围;\\\\\n(2) 当 $x \\in[1,3]$ 时, $f(x)<0$ 恒成立, 求 $a$ 的取值范围;\\\\\n(3) 当 $x \\in[1,3]$ 时, $f(x)<0$ 有解, 求 $a$ 的取值范围;\\\\\n(4) 当 $x \\in[1,3]$ 时, $f(x)=0$ 有解, 求 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022497": { - "id": "022497", - "content": "对于函数 $y=f(x)$($x \\in D$), 如果存在实数 $a$、$b$ ($a \\neq 0$, 且 $a=1$、$b=0$ 不同时成立), 使得 $f(x)=f(a x+b)$ 对 $x \\in D$ 恒成立, 则称函数 $f(x)$ 为``$(a, b)$ 映像函数''.\\\\\n(1) 判断函数 $f(x)=x^2-2$ 是否是``$(a, b)$ 映像函数'', 如果是, 请求出相应的 $a$、$b$ 的值, 若不是, 请说明理由;\\\\\n(2) 已知函数 $y=f(x)$ 是定义在 $[0,+\\infty)$ 上的``$(2,1)$ 映像函数'', 且当 $x \\in[0,1)$ 时, $f(x)=2^x$. 求函数 $y=f(x)$ ($x \\in[3,7)$) 的解析式.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022498": { - "id": "022498", - "content": "函数 $y=x+\\sqrt{3 x-1}$ 的定义域为\\blank{50}, 值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022499": { - "id": "022499", - "content": "函数 $y=\\sqrt{\\lg (-x^2+6 x+7)}$ 的定义域为\\blank{50}, 值域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022500": { - "id": "022500", - "content": "函数 $y=2 x+\\dfrac{4}{x+1}x \\in[1,7]$ 的最小值是\\blank{50}, 此时 $x=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022501": { - "id": "022501", - "content": "已知函数 $f(x)=x+\\dfrac{a}{x}$ 在区间 $[2,+\\infty)$ 上为增函数, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022502": { - "id": "022502", - "content": "已知函数 $y=f(x)$ 是定义在 $\\mathbf{R}$ 上的偶函数, 且在 $[0,+\\infty)$ 上是增函数, 若 $f(a+1) \\leq f(4)$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022503": { - "id": "022503", - "content": "已知函数 $f(x)$ 在 $\\mathbf{R}$ 上是增函数且 $f(3)=0$, 则函数 $g(x)=-|f(x)|+3$ 的单调增区间是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022504": { - "id": "022504", - "content": "下列函数中, 在区间 ($1,+\\infty$) 上是减函数的有 (写出全部正确的序号)\\blank{50}.\\\\\n\\textcircled{1} $y=1-x^2$; \\textcircled{2} $y=x^2+2 x$; \\textcircled{3} $y=\\dfrac{x+2}{x+1}$; \\textcircled{4} $y=\\dfrac{x}{x-1}$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022505": { - "id": "022505", - "content": "若 $f(x)=\\begin{cases}x+\\dfrac{a-2}{x},& x>2,\\\\(1-a) x-1,& x \\leq 2\\end{cases}$ 是 $\\mathbf{R}$ 上的增函数, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022506": { - "id": "022506", - "content": "函数 $f(x)=(a-1) x^2+2(a-1) x-4$,($a \\in \\mathbf{R}$).\\\\\n(1) 若当 $x \\in \\mathbf{R}$时, $f(x)<0$ 恒成立, 求 $a$ 的取值范围;\\\\\n(2) 若当 $x \\in[1,4]$ 时, $f(x)<0$ 恒成立, 求 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷14小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022507": { - "id": "022507", - "content": "已知函数 $y=x^3-2 x^2+8 x^{\\frac{5}{3}}-1$ 在区间 $(0,1)$ 上有一个零点, 由二分法得这个零点约是\\blank{50}. (结果精确到$0.1$)", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022508": { - "id": "022508", - "content": "若关于 $x$ 的不等式 $x+|x-2 c|>1$ 的解集为 $\\mathbf{R}$, 则实数 $c$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022509": { - "id": "022509", - "content": "(1) 已知 $f(x)$ 是定义域为 $[1,5]$ 上的连续函数, 则``$f(1) f(5)>0$''是``$f(x)$ 不存在零点''的\\blank{50}条件;\\\\\n(2) 已知 $f(x)$ 是定义域为 $[1,5]$ 上的函数, 则``$f(1) f(5)<0$''是``$f(x)$ 存在零点''的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022510": { - "id": "022510", - "content": "若函数 $f(x)=x^2-a x+2$ 恰有两个不同零点, 且这两个零点\\\\\n(1) 都在区间 $(1,+\\infty)$ 中, 则实数 $a$ 的取值范围为\\blank{50};\\\\\n(2) 分别在区间 $(-\\infty, 1)$ 和 $(1,+\\infty)$ 中, 则实数 $a$ 的取值范围为\\blank{50};\\\\\n(3) 都在区间 $(1,4)$ 中, 则实数 $a$ 的取值范围为\\blank{50};\\\\\n(4) 分别在区间 $(-\\infty, 1)$ 和 $(3,+\\infty)$ 中, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022511": { - "id": "022511", - "content": "若函数 $f(x)=4^x+(m-3) \\cdot 2^x+m$ 有两个不相等的零点, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022512": { - "id": "022512", - "content": "若对任意 $x \\in \\mathbf{R}$, 都有 $4^x a+(3 a-1) 2^x+a>0$ 成立, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022513": { - "id": "022513", - "content": "若存在 $x \\in \\mathbf{R}$, 使得 $4^x a+(3 a-1) 2^x+a>0$ 成立, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022514": { - "id": "022514", - "content": "已知不等式 $3 x^2<\\log _a x$($a>0$, $a \\neq 1$) 在 $x \\in(0, \\dfrac{1}{3})$ 上恒成立, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022515": { - "id": "022515", - "content": "设 $n \\in \\mathbf{Z}$, 且函数 $y=10 x^3-10 x+1$ 在区间 $[n, n+2]$ 上有零点, 则 $n$ 的所有可能值为\\blank{50}. (注: 三次函数至多有三个零点)", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022516": { - "id": "022516", - "content": "若奇函数 $y=f(x)$, $x \\in \\mathbf{R}$ 有反函数 $y=f^{-1}(x)$, 则下列各点中, 一定在函数 $y=f^{-1}(x)$ 的图像上的点有\\blank{50}.\\\\\n\\textcircled{1} $(-f(a), a)$; \\textcircled{2} $(-f(a),-a)$; \\textcircled{3} $(-a,-f(a))$; \\textcircled{4} $(a, f^{-1}(a))$.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022517": { - "id": "022517", - "content": "已知函数 $y=\\dfrac{x-5}{2 x+m}$ 的图像关于直线 $y=x$ 对称, 则 $m=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022518": { - "id": "022518", - "content": "已知 $y=f(x)$, $x \\in D$ 的值域为 $A$, 它有反函数 $y=f^{-1}(x)$, $x \\in A$. 设命题``方程 $f(x)=0$ 有解 $x=a$, 且 $f(x)>x$ 对一切 $x \\in D$ 成立''是真命题, 根据上述命题, 其反函数所具有的类似的结论为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022519": { - "id": "022519", - "content": "函数 $y=x^2-2 x$, $x<0$ 的反函数为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022520": { - "id": "022520", - "content": "已知 $f(x)=9^x$, 则方程 $f(x)=f[f^{-1}(3^x+6)]$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022521": { - "id": "022521", - "content": "用函数的观点解关于 $x$ 的不等式: $\\dfrac{x+2}{x-1}-a<0$($a \\in \\mathbf{R}$).", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022522": { - "id": "022522", - "content": "判断方程 $2 x^3+3 x-2=50$ 是否有整数解, 并证明你的结论.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022523": { - "id": "022523", - "content": "已知三次函数 $f(x)=a x^3+b x^2-b x-a$($a \\neq 0$, $a,b \\in \\mathbf{R}$).\\\\\n(1) 求证: $x=1$ 是 $f(x)$ 的零点;\\\\\n(2) 如果 $x_0$ 是 $f(x)$ 的零点, 求证: $\\dfrac{1}{x_0}$ 也是 $f(x)$ 的零点.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022524": { - "id": "022524", - "content": "对于函数 $f(x)$, 若在定义域内存在实数 $x_0$, 满足 $f(-x_0)=-f(x_0)$, 则称 $f(x)$ 是``M 类函数''.\\\\\n(1) 已知函数 $f(x)=x(1-|x|)$, 试判断 $f(x)$ 是否是``$M$ 类函数''? 并说明理由;\\\\\n(2) 已知函数 $f(x)=2^x+m$ 是定义在 $[-1,1]$ 上的``$M$ 类函数'', 求实数 $m$ 的最小值;\\\\\n(3) 已知函数 $f(x)=\\begin{cases}\\log _2(x^2-2 m x),& x \\geq 2, \\\\-3,& x<2\\end{cases}$ 是``$M$ 类函数'', 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022525": { - "id": "022525", - "content": "若关于 $x$ 的方程 $\\lg (-x^2+3 x-m)=\\lg (3-x)$ 恰有一个实数解, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022526": { - "id": "022526", - "content": "若函数 $f(x)=x^2-a x+2$ 有两个不同零点, 分别在区间 ($-\\infty, 2$) 和 ($2,+\\infty$) 中, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022527": { - "id": "022527", - "content": "函数 $y=x^2-2 x$, $x<0$ 的反函数为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022528": { - "id": "022528", - "content": "函数 $f(x)=\\ln (x+1)-\\dfrac{2}{x}$ 的一个零点所在的区间是\\bracket{20}.\n\\fourch{$(0,1)$}{$(1,2)$}{$(2,3)$}{$(3,4)$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022529": { - "id": "022529", - "content": "若函数 $f(x)=x^2+x+m$ 在区间 $[-2,4]$ 内有两个不同的零点, 则实数 $m$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022530": { - "id": "022530", - "content": "已知函数 $y=\\dfrac{x}{3}+m$ 与函数 $y=n x-6$ 互为反函数, 则 $m=$\\blank{50}, $n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022531": { - "id": "022531", - "content": "若函数 $f(x)=x^2-2 a x+3$ 在区间 $[1,2)$ 上存在反函数, 则实数 $a$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022532": { - "id": "022532", - "content": "用函数的观点解关于 $x$ 的不等式: $\\dfrac{x-3}{x+2}-a<0$($a \\in \\mathbf{R}$).", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022533": { - "id": "022533", - "content": "已知函数 $f(x)=\\dfrac{2 x+1}{x+a}$($x \\neq-a$, $a \\neq \\dfrac{1}{2}$).\\\\\n(1) 求 $f(x)$ 的反函数 $f^{-1}(x)$;\\\\\n(2) 设 $f(x)=f^{-1}(x)$, 求实数 $a$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷15小测", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022534": { - "id": "022534", - "content": "在等差数列 $\\{a_n\\}$ 中, 已知 $a_1=\\dfrac{5}{6}$, $d=-\\dfrac{1}{6}$, 若 $a_n=-\\dfrac{3}{2}$, 则 $n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022535": { - "id": "022535", - "content": "等差数列 $\\{a_n\\}$ 中, 若 $a_6+a_9+a_{12}+a_{15}=20$, 则 $S_{20}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022536": { - "id": "022536", - "content": "若 3 与 $x$ 的等比中项为 $3 x$, 则 $x$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022537": { - "id": "022537", - "content": "$S_n$ 是等差数列 $\\{a_n\\}$ 的前 $n$ 项和, 满足 $S_{10}=100$, $S_{20}=110$, 则 $S_{40}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022538": { - "id": "022538", - "content": "在等比数列 $\\{a_n\\}$ 中, 若 $a_3 \\cdot a_7 \\cdot a_{17}\\cdot a_{21}=25$, 则 $a_5 \\cdot a_{19}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022539": { - "id": "022539", - "content": "在等差数列 $\\{a_n\\}$ 中, 满足 $3 a_4=7 a_7$, 且 $a_1>0, S_n$ 是数列 $\\{a_n\\}$ 前 $n$ 项的和. 若 $S_n$ 取得最大值, 则 $n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022540": { - "id": "022540", - "content": "在等比数列 $\\{a_n\\}$ 中, 首项 $a_1<0$, 则 $\\{a_n\\}$ 是严格递增数列的一个充要条件是公比 $q$ 满足\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022541": { - "id": "022541", - "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和是 $S_n=2^n+n$, 则 $\\{a_n\\}$ 的通项公式是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022542": { - "id": "022542", - "content": "若关于 $x$ 的方程 $x^2-x+a=0$ 和 $x^2-x+b=0$($a \\neq b$) 的四个根可组成首项为 $\\dfrac{1}{4}$ 的等差数列, 则 $a+b$ 的值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022543": { - "id": "022543", - "content": "已知等差数列 $\\{a_n\\}$ 的前 $n$ 项之和为 $S_n$, $a_{10}=30$, $a_{20}=50$, 若 $S_k=242$, 则 $k=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022544": { - "id": "022544", - "content": "已知等差数列 $\\{a_n\\}$ 的前 $n$ 项之和为 $S_n$, 若 $S_{15}$ 为一确定常数, 则下列各式也为确定常数的是\\bracket{20}.\n\\fourch{$a_2+a_{13}$}{$a_2 \\cdot a_{13}$}{$a_1+a_8+a_{15}$}{$a_1 \\cdot a_8 \\cdot a_{15}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022545": { - "id": "022545", - "content": "若实数 $a$、$b$、$c$、$d$、$e$ 成等比, 且 $a=-81$, $e=-1$, 则 $c=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022546": { - "id": "022546", - "content": "两个数列 $\\{a_n\\}$ 与 $\\{b_n\\}$ 都成等差, 前 $n$ 项之和分别为 $S_n$ 和 $T_n$, 且 $\\dfrac{S_n}{T_n}=\\dfrac{2 n+1}{3 n-2}$, 则 $\\dfrac{a_9}{b_9}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022547": { - "id": "022547", - "content": "设 $\\{a_n\\}$ 是由正数组成的等比数列, 且 $a_1 a_2 \\cdots a_7 a_8=16$, 则 $a_4+a_5$ 的最小值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022548": { - "id": "022548", - "content": "已知 $\\{a_n\\},\\{b_n\\}$ 均是等差数列, $c_n=a_n \\cdot b_n$, $\\{c_n\\}$ 的前三项是 $7,9,9$, 则 $c_{10}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022549": { - "id": "022549", - "content": "等差数列 $\\{a_n\\}$ 中, $S_n$ 为前 $n$ 项和, 且 $S_6S_8$, 给出下列命题: \\textcircled{1} 数列 $\\{a_n\\}$ 中前 $7$ 项是递增的, 从第 $8$ 项开始递减; \\textcircled{2} $S_9$ 一定小于 $S_6$; \\textcircled{3} $a_1$ 是 $\\{a_n\\}$ 各项中的最大的; \\textcircled{4} $S_7$ 不一定是 $\\{S_n\\}$ 中最大项. 其中正确的序号是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022550": { - "id": "022550", - "content": "在等差数列 $\\{a_n\\}$ 中, $a_4=10$, 且 $a_3, a_6, a_{10}$ 成等比数列, 求数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022551": { - "id": "022551", - "content": "等差数列 $\\{a_n\\}$ 中, $S_{12}>0$, $S_{13}<0$, 则 $S_1, S_2, S_3, \\cdots, S_n$ 中最大项为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022552": { - "id": "022552", - "content": "有固定项的数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n=2 n^2+n$, 现从中抽取某一项(不包括首项、末项)后, 余下的项的平均值是 79 .\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项 $a_n$;\\\\\n(2) 求这个数列的项数, 并回答抽取的是第几项?", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022553": { - "id": "022553", - "content": "数列 $\\{a_n\\}$ 中, $a_1=8$, $a_4=2$, 且满足 $a_{n+2}-2 a_{n+1}+a_n=0$,\\\\\n(1) 求数列 $\\{a_n\\}$ 的通项公式;\\\\\n(2) 记 $b_n=\\dfrac{1}{n(12-a_n)}$, $T_n=b_1+b_2+\\cdots \\cdots+b_n$, 是否存在最大的整数 $m$, 使得任取的正整数 $n$, 有 $T_n>\\dfrac{m}{32}$ 恒成立? 若存在, 求最大的正整数$m$; 若不存在, 说明理由.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷16", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022554": { - "id": "022554", - "content": "(1) 实数 $a$、$b$、$c$ 满足 $b^2=a c$ 是 $a$、$b$、$c$ 成等比数列的\\blank{50}条件.\\\\\n(2) 实数 $a$、$b$、$c$ 满足 $b=\\sqrt{a c}$ 是 $b$ 为 $a$、$c$ 的等比中项的\\blank{50}条件.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022555": { - "id": "022555", - "content": "在等比数列 $\\{a_n\\}$ 中.\\\\\n(1) 若 $a_8=32$, $a_{11}=4$, 则 $a_3=$\\blank{50}.\\\\\n(2) 若 $a_1+a_2=324$, $a_3+a_4=36$, 则 $a_5+a_6=$\\blank{50}.\\\\\n(3) 若 $a_3 \\cdot a_7 \\cdot a_{17}\\cdot a_{21}=9$, 则 $a_5 \\cdot a_{19}=$\\blank{50}.\\\\\n(4) 若公比 $q=2$, $S_4=1$, 则 $S_8=$\\blank{50}.\\\\\n(5) 若 $a_1=2$, $a_n=162$, $S_n=242$, 则 $q=$\\blank{50}, $n=$\\blank{50}.\\\\\n(6) 若 $S_5=10$, $S_{10}=40$, 则 $S_{20}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022556": { - "id": "022556", - "content": "若实数 $a$、$b$、$c$、$d$、$e$ 成等比, 且 $a=-81$, $e=-1$, 则 $c=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022557": { - "id": "022557", - "content": "由正数组成的等比数列 $\\{a_n\\}$ 中公比为 2 , 且满足 $a_1 \\cdot a_2 \\cdot a_3 \\cdots a_{30}=2^{30}$, 那么 $a_3 \\cdot a_6 \\cdot a_9 \\cdots a_{30}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022558": { - "id": "022558", - "content": "设 $n$ 是正整数, 求和: $2+2^4+\\cdots+2^{3 n+4}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022559": { - "id": "022559", - "content": "已知数列 $\\{a_n\\}$ 满足 $a_1=3$, $a_n a_{n+1}=(\\dfrac{1}{2})^n$, 则此数列前 $2 n$ 项之和为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022560": { - "id": "022560", - "content": "数列 $\\{a_n\\}$ 的前 $n$ 项和 $S_n$ 满足: $S_n=\\begin{cases}3 n+1, &n \\leq 5,\\\\ n^2, & n \\geq 6,\\end{cases}$ 则数列 $\\{a_n\\}$ 的通项公式为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022561": { - "id": "022561", - "content": "已知公比为 $q$($00$, 且 $S_n$ 满足条件 $\\lg (S_n+1)=n$, 则数列 $\\{a_n\\}$ 是\\bracket{20}.\n\\twoch{等比数列而非等差数列}{等差数列而非等比数列}{既是等比数列又是等差数列}{既非等比数列又非等差数列}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022565": { - "id": "022565", - "content": "等比数列 $\\{a_n\\}$ 的首项 $a_1=2$, 公比 $q=3$, 若数列 $\\{a_n\\}$ 从第 $m$ 项至第 $n$($m\\dfrac{a+d}{2}$}{$\\sqrt{b c}=\\dfrac{a+d}{2}$}{$\\sqrt{b c}<\\dfrac{a+d}{2}$}{不能确定}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022567": { - "id": "022567", - "content": "数列 $\\{a_n\\}$、$\\{b_n\\}$ 满足条件: $a_1=2$, $a_{n+1}=\\dfrac{3 a_n+1}{a_n+3}$, $b_n=\\dfrac{a_n-1}{a_n+1}$, 则 $b_n$ 是\\bracket{20}数列.\n\\twoch{等差但不是等比}{等比但不是等差}{既是等差又是等比}{既不是等差又不是等比}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022568": { - "id": "022568", - "content": "已知数列的通项公式为 $a_n=-n^2+7 n+8$.\\\\\n(1) $\\dfrac{45}{4}$ 是否是数列中的项?\\\\ \n(2) 求数列 $\\{a_n\\}$ 的最大项的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022569": { - "id": "022569", - "content": "如图, 在半径为 $r$ 的圆内作内接正六边形, 再作正六边形的内切圆, 又在此内切圆内作内接正六边形, 如此无限继续下去, 设为前 $n$ 个圆的面积之和为 $S_n$, 则 $\\displaystyle\\lim _{n \\rightarrow+\\infty}S_n$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) circle (1);\n\\draw (1,0) --++ (120:1) --++ (180:1) --++ (240:1) --++ (300:1) --++ (0:1) --++ (60:1);\n\\draw (0,0) circle ({sqrt(3)/2});\n\\draw ({sqrt(3)/2},0) --++ (120:{sqrt(3)/2}) --++ (180:{sqrt(3)/2}) --++ (240:{sqrt(3)/2}) --++ (300:{sqrt(3)/2}) --++ (0:{sqrt(3)/2}) --++ (60:{sqrt(3)/2});\n\\draw (0,0) circle (0.75);\n\\draw (0,0) node {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022570": { - "id": "022570", - "content": "已知数列 $\\{a_n\\}$ 的通项公式 $a_n=(n^2-5 n+4) \\cdot 0.9^n$. \\\\\n(1) 数列中有多少项是负数?\\\\\n(2) 是否存在自然数 $\\mathbf{N}$, 使得对于任意自然数 $n$, 都有 $a_n \\leq a_N$ 成立?", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022571": { - "id": "022571", - "content": "求数列 $1,1+a, 1+a+a^2, 1+a+a^2+a^3, \\cdots, 1+a+\\cdots+a^{n-1}, \\cdots$ 的前 $n$ 项和 $S_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022572": { - "id": "022572", - "content": "已知数列 $\\{a_n\\}$, $\\{b_n\\}$ 中, $a_n=2 n-1$, $b_n=3^{n-1}$. 若 $c_n=a_n \\cdot b_n$, 求数列 $\\{c_n\\}$ 的前 $n$ 项和 $S_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022573": { - "id": "022573", - "content": "等差数列 $\\{a_n\\}$ 的第二项为 $8$, 前十项的和为 $185$, 从数列 $\\{a_n\\}$ 中, 依次取出第 $2$ 项、第 $4$ 项、第 $8$ 项、 第 $2^n$ 项按原来的顺序排成一个新数列 $\\{b_n\\}$, 求数列 $\\{b_n\\}$ 前项和公式 $S_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷17", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022574": { - "id": "022574", - "content": "已知数列 $\\{a_n\\}$ 的通项公式为 $a_n=n^2-2 n$, 则 $80$ 为数列 $\\{a_n\\}$ 中的第\\blank{50}项.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022575": { - "id": "022575", - "content": "等差数列 $\\{a_n\\}$ 中, $a_1=3$, $a_3+a_5=18$, 则数列 $\\{a_n\\}$ 的通项公式 $a_{12}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022576": { - "id": "022576", - "content": "等比数列 $\\{a_n\\}$ 中, $a_4=2$, $a_7=16$, 则数列 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022577": { - "id": "022577", - "content": "某等差数列共有 $10$ 项, 其中奇数项之和为 $15$, 偶数项之和为 $30$, 则其公差是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022578": { - "id": "022578", - "content": "在等比数列 $\\{a_n\\}$ 中, $a_5=3$, 则 $\\log _3 a_1+\\log _3 a_2+\\cdots+\\log _3 a_9$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022579": { - "id": "022579", - "content": "已知无穷等比数列 $\\{a_n\\}$ 中, 公比 $q=\\dfrac{1}{3}$, $\\displaystyle\\lim _{n\\rightarrow+\\infty}(a_1+a_2+\\cdots+a_n)=\\dfrac{1}{2}$, 则首项 $a_1$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022580": { - "id": "022580", - "content": "已知等比数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且 $S_n=3 \\cdot 2^n+a$, 则 $a_1+a_9=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022581": { - "id": "022581", - "content": "已知数列 $\\{a_n\\}$ 的通项公式 $a_n=\\begin{cases}-4 n+34,& 1 \\leq n \\leq 3,\\\\6 n-4, & n \\geq 4.\\end{cases}$ 则此数列的前 $100$ 项和 $S_{100}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022582": { - "id": "022582", - "content": "无穷等差数列 $\\{a_n\\}$ 的各项均为正数, 若 $a_2-2 a_8^2+3 a_{10}=0$, 则数列 $\\{a_n\\}$ 的公差 $d$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022583": { - "id": "022583", - "content": "已知 $\\{a_n\\}$、$\\{b_n\\}$ 均为等差数列, $c_n=a_n b_n$ ($n$ 为正整数), 若数列 $\\{c_n\\}$ 的前三项依次为 $7,9,9$, 则 $c_{10}$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022584": { - "id": "022584", - "content": "已知数列 $\\{a_n\\}$ 的前 $n$ 项和为 $S_n$, 且 $S_n+n a_n=1$($n \\geq 1$), 则 $\\{a_n\\}$ 的通项公式 $a_n=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022585": { - "id": "022585", - "content": "``数列 $\\{a_n\\}$ 为等比数列''是``数列 $\\{a_n^2\\}$ 为等比数列''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022586": { - "id": "022586", - "content": "已知 $\\{a_n\\}$ 是等比数列, $a_2=2$, $a_3=1$, 则 $a_1 a_2+a_2 a_3+\\cdots+a_n a_{n+1}$ 等于\\bracket{20}.\n\\fourch{$16(1-4^{-n})$}{$16(1-2^{-n})$}{$\\dfrac{32}{3}(1-4^{-n})$}{$\\dfrac{32}{3}(1-2^{-n})$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022587": { - "id": "022587", - "content": "关于问题``函数 $f(x)=\\dfrac{x-1}{2 x-11}$ 的最值, 以及数列 $a_n=\\dfrac{n-1}{2 n-11}$ 的最值'', 下列说法正确的是\\bracket{20}.\n\\onech{函数 $f(x)$ 既有最大值又有最小值, 数列 $\\{a_n\\}$ 既有最大值又有最小值}{函数 $f(x)$ 既无最大值又无最小值, 数列 $\\{a_n\\}$ 既有最大值又有最小值}{函数 $f(x)$ 既有最大值又有最小值, 数列 $\\{a_n\\}$ 有最大值无最小值}{函数 $f(x)$ 既无最大值又无最小值, 数列 $\\{a_n\\}$ 无最大值有最小值}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022588": { - "id": "022588", - "content": "已知等比数列 $\\{a_n\\}$ 中, $a_2=3 $, $a_4=27$, 求数列 $\\{a_n\\}$ 的通项公式及其前 $10$ 项和 $S_{10}$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022589": { - "id": "022589", - "content": "已知数列 $\\{a_n\\}$ 为等差数列, $S_n$ 为其前 $n$ 项和, $S_7=7$, $S_{15}=75$. 设 $T_n$ 为数列 $\\{\\dfrac{S_n}{n}\\}$ 的前 $n$ 项和, 求 $T_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022590": { - "id": "022590", - "content": "已知 $\\{a_n\\}$、$\\{b_n\\}$ 均为正项数列, $a_1=3$, $a_2=6$, $\\{b_n\\}$ 为等差数列, 且对任意正整数 $n$, 恒有 $a_n=b_n \\cdot b_{n+1}$ 成立. 求数列 $\\{b_n\\}$ 的通项公式.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022591": { - "id": "022591", - "content": "设 $k \\in \\mathbf{R}$, 数列 $\\{a_n\\}$ 满足 $a_n=2 n+k$, 数列 $\\{b_n\\}$ 的通项公式为 $b_n=3^n$.\\\\\n(1) 若 $k=-113$, 设 $c_n=\\dfrac{a_n}{b_n}$, 求数列 $\\{c_n\\}$ 最大项及相应的序数;\\\\\n(2) 若 $k=1$, 求数列 $\\{a_n \\cdot b_n\\}$ 的前 $n$ 项和 $T_n$;\\\\\n(3) 若 $k=1$, 设 $d_n=\\begin{cases}a_n,& n \\text{为奇数}, \\\\b_n,& n \\text{为偶数}.\\end{cases}$ 求数列 $\\{d_n\\}$ 的前 $n$ 项和 $S_n$.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期周末卷19", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022592": { - "id": "022592", - "content": "设全集 $U=\\mathbf{R}$, 集合 $M=\\{x | x<2\\}$, $a=1$, 有以下四种说法:\\\\\n\\textcircled{1} $a \\in M$; \\textcircled{2} $-a \\in \\overline{M}$; \\textcircled{3} $-a \\in M$; \\textcircled{4} $\\{a\\}\\subset M$, 其中正确说法的序号是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022593": { - "id": "022593", - "content": "设常数 $a \\in \\mathbf{R}$, 集合 $A=\\{a, a^2\\}$, 集合 $B=\\{2 a\\}$, 且 $A \\cup B=A$, 则 $a$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022594": { - "id": "022594", - "content": "设全集 $U=\\mathbf{R}$, 集合 $A=(0,2)$, $B=(-1,1)$, 则 $A \\cup \\overline{B}=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022595": { - "id": "022595", - "content": "陈述句``存在 $x \\in \\mathbf{R}$, 使得 $2 x+c>0$''的否定形式为\\blank{150}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022596": { - "id": "022596", - "content": "满足关系: $\\{2,4,6\\}\\subseteq A \\subset\\{1,2,3,4,5,6\\}$ 的集合 $A$ 的个数为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022597": { - "id": "022597", - "content": "设全集 $U=\\mathbf{R}$, $P=\\{y | y=3 x^2+6 x+1\\}$, $Q=\\{y | y=-x^2+8 x-13\\}$, 则 $\\overline{P}\\cap Q=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022598": { - "id": "022598", - "content": "已知 $m$ 是实常数, 若集合 $A=\\{0,1\\}$, $B=\\{x | x^2-m x-m>0, x \\in \\mathbf{R}\\}$, 且 $A \\cap B=\\{1\\}$, 则实数 $m$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022599": { - "id": "022599", - "content": "已知 $a$ 为实常数, 集合 $A=[1,4]$, $B=\\{x | x \\geq a$ 且 $x \\leq 2 a+1\\}$, 且 $A \\cap B \\neq \\varnothing$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022600": { - "id": "022600", - "content": "在下列陈述句: \\textcircled{1} $a<0$ 或 $b<0$; \\textcircled{2} $a<0$ 且 $b<1$; \\textcircled{3} $a<-1$ 且 $b<0$; \\textcircled{4} $a<0$ 或 $b<-1$ 中, 能成为``$a<0$ 且 $b<0$'' 的必要非充分条件是\\blank{50}(填正确的序号).", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022601": { - "id": "022601", - "content": "以下几个关系中正确的是\\bracket{20}.\n\\fourch{$0 \\in \\varnothing$}{$\\varnothing \\in\\{x | x^2+1=0\\}$}{$\\varnothing \\subset\\{0\\}$}{$(A\\cap B) \\subset(A\\cup B)$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022602": { - "id": "022602", - "content": "``任意 $c \\in \\mathbf{R}$ 都有 $a c=b c$''是``$a=b$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022603": { - "id": "022603", - "content": "设 $a \\in \\mathbf{R}$, $b \\in \\mathbf{R}$, 集合 $A=\\{x | x^2-x=0\\}$, $B=\\{x | x^2+a x+b=0\\}$. 若 $B \\neq \\varnothing$, $B \\subseteq A$, 求 $a, b$ 的值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷01", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022604": { - "id": "022604", - "content": "若集合 $A=\\{x | x \\geq 1$ 或 $x<-4\\}$, $B=\\{x | x \\leq-2$ 或 $x>2\\}$, $C=\\{x | a0$ 的解集是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022609": { - "id": "022609", - "content": "不等式 $\\dfrac{x-1}{x-3}\\leq 0$ 的解集为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022610": { - "id": "022610", - "content": "设常数 $a, b \\in \\mathbf{R}$, 写出等式 $|a|-|b|=|a-b|$ 成立的一个充分非必要条件: \\blank{100}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022611": { - "id": "022611", - "content": "已知 $a$ 为实常数, 集合 $A=(-\\infty,-2] \\cup[6,+\\infty)$, $B=(a-4, a+4)$. 若 $A \\cup B=\\mathbf{R}$, 则实数 $a$ 的值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022612": { - "id": "022612", - "content": "已知常数 $m \\in \\mathbf{R}$, 若关于 $x$ 的方程 $x^2-2 m x-m+1=0$ 的两实根分别为 $x_1$、$x_2$ 且满足 $x_1^2+x_2^2=4$, 则实数 $m$ 的值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022613": { - "id": "022613", - "content": "若 $x>1$, 则 $2 x+\\dfrac{1}{x-1}$ 的最小值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022614": { - "id": "022614", - "content": "已知关于 $x$ 的不等式 $a x^2+b x-2<0$ 的解集是 $(-1,2)$, 则 $a+b=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022615": { - "id": "022615", - "content": "已知关于 $x$ 的不等式 $(1-a^2) x^2+(a-1) x+1>0$ 的解集为 $\\mathbf{R}$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022616": { - "id": "022616", - "content": "已知 $a>0$, 若关于 $x$ 的不等式 $(x-a)^2-(a x)^2>0$ 的解集中的整数恰有 3 个, 则 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022617": { - "id": "022617", - "content": "已知 $a, b, c$ 均是正实数, 设 $\\max (a, b, c)$ 表示实数 $a, b, c$ 的最大值, 为了使不等式 $\\max (a+\\dfrac{1}{b}, b+\\dfrac{1}{c}, c+\\dfrac{1}{a})\\geq \\lambda$ 对一切 $a, b, c$ 恒成立, 实数 $\\lambda$ 的最大可能值为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022618": { - "id": "022618", - "content": "下列不等式中, 与不等式 $x(x-2)>0$ 解集相同的是\\bracket{20}.\n\\twoch{$x^2+\\dfrac{1}{x-3}>2 x+\\dfrac{1}{x-3}$}{$x^2+\\dfrac{1}{x-1}>2 x+\\dfrac{1}{x-1}$}{$x^2 \\cdot \\sqrt{x-1}>2 x \\cdot \\sqrt{x-1}$}{$\\dfrac{x^2}{\\sqrt{x-1}}>\\dfrac{2 x}{\\sqrt{x-1}}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022619": { - "id": "022619", - "content": "对任意实数 $x, y$, 在下列命题中, 是假命题的是\\bracket{20}.\n\\twoch{$x^2>y^2$ 是 $x>|y|$ 的必要条件}{$x^2>y^2$ 是 $|x|>y$ 的充分条件}{$x>2$ 的一个充分条件是 $x^2>4$}{$\\dfrac{1}{x}>\\dfrac{1}{y}$ 的一个充分条件是 $00$, $y>0$, 且 $x+y \\leq 4$, 则下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$\\dfrac{1}{x+y}\\leq \\dfrac{1}{4}$}{$\\dfrac{1}{x}+\\dfrac{1}{y}\\geq 1$}{$\\sqrt{x y}\\geq 2$}{$\\dfrac{1}{x y}\\geq 1$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022621": { - "id": "022621", - "content": "已知常数 $a, b \\in \\mathbf{R}$, 则关于 $x$ 的不等式 $a x^2+b x+1>0$ 的解集不可能是\\bracket{20}.\n\\fourch{$\\varnothing$}{$\\mathbf{R}$}{$(-1,1)$}{($-1,+\\infty$)}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022622": { - "id": "022622", - "content": "设 $m \\in \\mathbf{R}$,已知集合 $A=\\{m |$ 关于 $x$ 的方程 $x^2+(m+1) x+(2 m-1)=0$ 无实数解$\\}$, 集合 $B=\\{z | z=a+\\dfrac{2}{b}$, $22\\}$.\\\\\n(1) 求集合 $A$;\\\\\n(2) 判断集合 $A$ 与集合 $B$ 的关系, 并说明理由.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022623": { - "id": "022623", - "content": "现欲在某足够大的广场建造一方舱医院, 设计要求病区是面积为 $4000 \\mathrm{m}^2$ 的矩形, 且病区四周是通道, 其中西南两侧的通道宽为 $3 \\mathrm{m}$, 东北两侧的通道宽为 $2 \\mathrm{m}$. 如图所示, 问如何设计病区的边长, 才能使通道的占地面积最小? 并求出通道占地面积的最小值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (0,0) rectangle (30,20);\n\\draw (5,0) -- (5,20) (8,3) rectangle (28,18) (0,5) --++ (5,0) (0,10) --++ (5,0) (0,15) --++ (5,0);\n\\draw (2.5,2.5) node {\\small 抢救室};\n\\draw (2.5,7.5) node {\\small 治疗室};\n\\draw (2.5,12.5) node {\\small CT室};\n\\draw (2.5,17.5) node {\\small 污染区};\n\\draw (18,10.5) node {病区};\n\\draw (18,1.5) node {3米宽通道};\n\\draw (18,19) node {2米宽通道};\n\\draw (6.5,10) node [rotate = 90] {3米宽通道};\n\\draw (29,10) node [rotate = 90] {2米宽通道};\n\\end{tikzpicture}\n\\end{center}", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022624": { - "id": "022624", - "content": "已知集合 $A=\\{x | \\dfrac{a x}{x-1}<1\\}$.\\\\\n(1) 若 $A=\\{x | x<1$ 或 $x>2\\}$, 求实数 $a$ 的值;\\\\\n(2) 已知集合 $B=\\{x | x^2-5 x+6 \\leq 0\\}$, 若 $A \\cap B=B$, 求实数 $a$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022625": { - "id": "022625", - "content": "已知 $a>0$, $b>0$, 且 $a+b=1$.\\\\\n(1) 求: $\\dfrac{1}{a}+\\dfrac{1}{b}$ 的最小值;\\\\\n(2) 求 $\\sqrt{a+1}+\\sqrt{b+1}$ 的最大值;\\\\\n(3) 若不等式 $|x+m|-|x+1| \\leq \\dfrac{1}{a}+\\dfrac{1}{b}$ 对任意 $x \\in \\mathbf{R}$ 及条件中的任意 $a$、$b$ 恒成立, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷02", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022626": { - "id": "022626", - "content": "函数 $y=\\sqrt{3^x-1}$ 的定义域为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022627": { - "id": "022627", - "content": "函数 $y=x^{\\frac{4}{3}}-3$ 的单调增区间为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022628": { - "id": "022628", - "content": "不论 $a$ 取何正实数, 函数 $y=a^{x+1}-2$ 的图像都通过同一定点, 该点坐标是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022629": { - "id": "022629", - "content": "函数 $y=\\log _{\\frac{1}{2}}(9 x^2-6 x+17)$ 的值域是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022630": { - "id": "022630", - "content": "若函数 $f(x)=\\dfrac{1}{3^x-1}+m$ 是奇函数, 则 $m=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022631": { - "id": "022631", - "content": "若 $\\log _a \\dfrac{2}{5}<1$, 实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022632": { - "id": "022632", - "content": "已知 $(a+3)^{\\frac{1}{2}}>(1-3 a)^{\\frac{1}{2}}$, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022633": { - "id": "022633", - "content": "已知 $0 \\leq x \\leq 2$, 则函数 $y=4^{x-\\frac{1}{2}}-3 \\cdot 2^x+5$ 的最大值是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022634": { - "id": "022634", - "content": "若函数 $f(x)=x^2+2 x+m$ 在 $[-2,4]$ 内有两个不同的零点, 则实数 $m$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022635": { - "id": "022635", - "content": "若 $f(x)=\\dfrac{3 x-2}{x-a}$ 存在反函数, 则实数 $a$ 的取值范围是\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022636": { - "id": "022636", - "content": "已知函数 $f(x)=2^x-2^{-x}$, 对任意的 $x \\in[-2,2]$, $f(m x-2)+f(x)<0$ 恒成立, 则 $m$ 的取值范围为\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022637": { - "id": "022637", - "content": "设 $a \\in \\mathbf{R}$, 若对于任意 $x>0$ 总有 $[(a-1) x-1](x^2-a x-1) \\geq 0$ 成立, 则 $a=$\\blank{50}.", - "objs": [], - "tags": [], - "genre": "填空题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022638": { - "id": "022638", - "content": "以下各函数中, 在 $[-1,1]$ 上为严格增函数的是\\bracket{20}.\n\\fourch{$y=-\\dfrac{1}{x-1}$}{$y=\\sqrt{-x^2+x}$}{$y=x^{\\frac{1}{3}}+3 x-4$}{$y=\\sqrt{x+2}-\\sqrt{x+1}$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022639": { - "id": "022639", - "content": "设函数 $y=x^3-2 x^2-3 x^{\\frac{5}{3}}+1$, 在以下四个区间中, 不含有该函数零点的是\\bracket{20} .\n\\fourch{$[-2,1]$}{$[0,3]$}{$[2,5]$}{$[4,7]$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022640": { - "id": "022640", - "content": "设 $x \\in \\mathbf{R}$, $[x]$ 表示不超过 $x$ 的最大整数, 则``$|x-y|<1$''是``$[x]=[y]$''成立的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022641": { - "id": "022641", - "content": "已知函数 $f(x)=\\begin{cases}x-\\dfrac{1}{x},& x<0,\\\\x+\\dfrac{4}{x}-1,& x>0,\\end{cases}$ 方程 $[f(x)]^2+b f(x)+c=0$ 有五个实数根, 则下列结论不正确的是\\bracket{20}.\n\\fourch{$b<-6$}{方程有一根为 2}{$c>9$}{$3 b+c+9<0$}", - "objs": [], - "tags": [], - "genre": "选择题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "", - "unrelated": [] - }, - "022642": { - "id": "022642", - "content": "已知函数 $f(x)=\\begin{cases}-x^2+a x,& x \\leq a,\\\\x-a & x>a,\\end{cases}$ $a>0$.\\\\\n(1) 若方程 $f(x)-a=0$ 总有三解, 试求 $a$ 的取值范围;\\\\\n(2) 若 $x \\in[0,3]$, 求函数 $f(x)$ 的最大值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022643": { - "id": "022643", - "content": "已知 $a$ 为常数, 函数 $f(x)=\\log _{\\frac{1}{2}}\\dfrac{1-a x}{x-1}$.\\\\\n(1) 若函数 $y=f(x)$ 是奇函数, 求 $a$ 的值;\\\\\n(2) 当 $a=2$ 时, 判断 $f(x)$ 的单调性, 并证明.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022644": { - "id": "022644", - "content": "销售甲种商品所得利润是 $P$ 万元, 它与投入资金 $t$ 万元的关系有经验公式 $P=\\dfrac{a t}{t+1}$; 销售乙种商品所得利润是 $Q$ 万元, 它与投入资金 $t$ 万元的关系有经验公式 $Q=b t$, 其中 $a$、$b$ 为常数. 现将 3 万元资金全部投入甲、乙两种商品的销售. 若全部投入甲种商品, 所得利润为 $\\dfrac{9}{4}$ 万元; 若全部投入乙种商品, 所得利润为 1 万元, 若将 3 万元资金中的 $x$ 万元投入甲种商品的销售, 余下的投入乙种商品的销售, 则所得利润总和为 $f(x)$ 万元.\\\\\n(1) 求函数 $y=f(x)$ 的表达式, 并写出定义域;\\\\\n(2) 怎样将 3 万元资金分配给甲、乙两种商品, 才能使所得到利润总和最大, 并求最大值.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022645": { - "id": "022645", - "content": "已知函数 $y=2^x-\\dfrac{1}{2^{|x|}}$.\\\\\n(1) 若 $y=2$, 求 $x$ 的值;\\\\\n(2) 判断函数 $y=a \\cdot 2^x-\\dfrac{1}{2^{|x|}}$ 的奇偶性, 并说明理由;\\\\\n(3) 已知对任意的 $x \\in[1,2]$, 都有 $2^x \\cdot(2^{2 x}-\\dfrac{1}{2^{|2 x|}})+m y \\geq 0$ 成立, 求实数 $m$ 的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, - "022646": { - "id": "022646", - "content": "已知二次函数 $f(x)=a x^2+b x+c$ 和一次函数 $g(x)=-b x$, 其中 $a, b, c$ 满足 $a>b>c$ 且 $a+b+c=0$($a, b, c \\in \\mathbf{R}$).\\\\\n(1) 求证: 两函数的图像交于不同的两点 $A, B$;\\\\\n(2) 求 $\\dfrac{c}{a}$ 的范围;\\\\\n(3) 对于 (1) 中的 $A, B$, 求线段 $AB$ 在 $x$ 轴上的射影 $A_1B_1$ 的长的取值范围.", - "objs": [], - "tags": [], - "genre": "解答题", - "ans": "", - "solution": "", - "duration": -1, - "usages": [], - "origin": "2025届高一上学期测验卷03", - "edit": [ - "20230905\t王伟叶" - ], - "same": [], - "related": [], - "remark": "", - "space": "4em", - "unrelated": [] - }, "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