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@ -63777,13 +63777,13 @@
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|
},
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|
|
|
},
|
|
|
|
"002300": {
|
|
|
|
"002300": {
|
|
|
|
"id": "002300",
|
|
|
|
"id": "002300",
|
|
|
|
"content": "若两圆$(x-a)^2+y^2=1$与$x^2+(y-b)^2=1$外切, 则点$P(a,b)$的轨迹方程是\\blank{100}.\n%\\ans{$a^2+b^2=4$}",
|
|
|
|
"content": "若两圆$(x-a)^2+y^2=1$与$x^2+(y-b)^2=1$外切, 则点$P(a,b)$的轨迹方程是\\blank{100}.",
|
|
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|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
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|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$a^2+b^2=4$",
|
|
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|
"solution": "",
|
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|
"solution": "",
|
|
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|
"duration": -1,
|
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"duration": -1,
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"usages": [],
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"usages": [],
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@ -63798,13 +63798,13 @@
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},
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},
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"002301": {
|
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"002301": {
|
|
|
|
"id": "002301",
|
|
|
|
"id": "002301",
|
|
|
|
"content": "已知$A(-1,1)$, $B(-2,0)$是$\\triangle ABC$的两个顶点, 且$|AC|=\\sqrt{2}|BC|$,\n则顶点$C$的轨迹方程为\\blank{100}.\n%\\ans{$(x+3)^2+(y+1)^2=4, (x+3,y+1)\\ne \\pm(\\sqrt{2},\\sqrt{2})$}",
|
|
|
|
"content": "已知$A(-1,1)$, $B(-2,0)$是$\\triangle ABC$的两个顶点, 且$|AC|=\\sqrt{2}|BC|$,\n则顶点$C$的轨迹方程为\\blank{100}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$(x+3)^2+(y+1)^2=4$, $(x+3,y+1)\\ne \\pm(\\sqrt{2},\\sqrt{2})$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
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|
"duration": -1,
|
|
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"usages": [],
|
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"usages": [],
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|
@ -63819,13 +63819,13 @@
|
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|
},
|
|
|
|
},
|
|
|
|
"002302": {
|
|
|
|
"002302": {
|
|
|
|
"id": "002302",
|
|
|
|
"id": "002302",
|
|
|
|
"content": "与$x$轴(整个)和射线$y=-\\sqrt{3}x\\ (x<0)$都相切的圆的圆心的轨迹方程为\\blank{100}.\n%\\ans{$y=\\sqrt{3}x, \\ x>0$or$y=-\\dfrac{\\sqrt{3}}{3}x, \\ x<0$}",
|
|
|
|
"content": "与$x$轴(整个)和射线$y=-\\sqrt{3}x\\ (x<0)$都相切的圆的圆心的轨迹方程为\\blank{100}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$y=\\sqrt{3}x$, \\ x>0$或$y=-\\dfrac{\\sqrt{3}}{3}x, \\ x<0$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -63840,13 +63840,13 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002303": {
|
|
|
|
"002303": {
|
|
|
|
"id": "002303",
|
|
|
|
"id": "002303",
|
|
|
|
"content": "自$A(4,0)$引圆$O:x^2+y^2=4$的割线$ABC$, $B,C$是该割线和圆的两个交点,\n则弦$BC$中点$P$的轨迹方程为\\blank{100}.\n%\\ans{$x^2+y^2-4x=0, \\ x<1$}",
|
|
|
|
"content": "自$A(4,0)$引圆$O:x^2+y^2=4$的割线$ABC$, $B,C$是该割线和圆的两个交点,\n则弦$BC$中点$P$的轨迹方程为\\blank{100}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$x^2+y^2-4x=0, \\ x<1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -63861,13 +63861,13 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002304": {
|
|
|
|
"002304": {
|
|
|
|
"id": "002304",
|
|
|
|
"id": "002304",
|
|
|
|
"content": "一动圆被两直线$3x+y=0$, $3x-y=0$所截, 截得的弦长分别为$8$和$4$(所谓分别, 就是前者对前者, 后者对后者的意思, 英语里叫``{\\textrm respectively}''), 则动圆圆心$P$的轨迹方程为\\blank{100}.\n%\\ans{$xy+10=0$}",
|
|
|
|
"content": "一动圆被两直线$3x+y=0$, $3x-y=0$所截, 截得的弦长分别为$8$和$4$, 则动圆圆心$P$的轨迹方程为\\blank{100}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$xy+10=0$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -63882,13 +63882,13 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002305": {
|
|
|
|
"002305": {
|
|
|
|
"id": "002305",
|
|
|
|
"id": "002305",
|
|
|
|
"content": "已知点$A(-1,0)$与点$B(1,0)$, $C$是圆$x^2+y^2=1$上的动点(不能和$A,B$重合), 连接$BC$并延长到$D$, 使$|CD|=|BC|$, 求直线$AC$与直线$OD$的交点$P$的轨迹方程.\n%\\ans{$\\left(x+\\dfrac{1}{3}\\right)^2+y^2=\\dfrac{4}{9}$}",
|
|
|
|
"content": "已知点$A(-1,0)$与点$B(1,0)$, $C$是圆$x^2+y^2=1$上的动点(不能和$A,B$重合), 连接$BC$并延长到$D$, 使$|CD|=|BC|$, 求直线$AC$与直线$OD$的交点$P$的轨迹方程.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "解答题",
|
|
|
|
"genre": "解答题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$\\left(x+\\dfrac{1}{3}\\right)^2+y^2=\\dfrac{4}{9}$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -64102,7 +64102,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002314": {
|
|
|
|
"002314": {
|
|
|
|
"id": "002314",
|
|
|
|
"id": "002314",
|
|
|
|
"content": "若方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示椭圆, 则实数$k$的取值范围为\\blank{80}.\n%\\ans{$(3,4)\\cup (4,5)$}",
|
|
|
|
"content": "若方程$\\dfrac{x^2}{k-5}+\\dfrac{y^2}{3-k}=-1$表示椭圆, 则实数$k$的取值范围为\\blank{80}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"K0713002X"
|
|
|
|
"K0713002X"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
@ -64211,7 +64211,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002318": {
|
|
|
|
"002318": {
|
|
|
|
"id": "002318",
|
|
|
|
"id": "002318",
|
|
|
|
"content": "焦点在$x$轴上的椭圆上有一点$P(3,y)$, 若$P$点到两焦点的距离分别为$6.5$和$3.5$, 则此椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/25+4y^2/75=1$}",
|
|
|
|
"content": "焦点在$x$轴上的椭圆上有一点$P(3,y)$, 若$P$点到两焦点的距离分别为$6.5$和$3.5$, 则此椭圆的标准方程为\\blank{100}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"K0713004X"
|
|
|
|
"K0713004X"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
@ -64220,7 +64220,7 @@
|
|
|
|
"椭圆"
|
|
|
|
"椭圆"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$x^2/25+4y^2/75=1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -64259,13 +64259,13 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002320": {
|
|
|
|
"002320": {
|
|
|
|
"id": "002320",
|
|
|
|
"id": "002320",
|
|
|
|
"content": "三角形$ABC$的两个顶点$A,B$的坐标分别是$(-6,0),(6,0)$, $AC,BC$边所在直线的斜率之积等于$\\dfrac{4}{9}$,\n则顶点$C$的轨迹方程为\\blank{100}.\n%\\ans{$\\dfrac{x^2}{36}+\\dfrac{y^2}{16}=1,y\\neq 0$}",
|
|
|
|
"content": "三角形$ABC$的两个顶点$A,B$的坐标分别是$(-6,0),(6,0)$, $AC,BC$边所在直线的斜率之积等于$\\dfrac{4}{9}$,\n则顶点$C$的轨迹方程为\\blank{100}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第七单元"
|
|
|
|
"第七单元"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$\\dfrac{x^2}{36}+\\dfrac{y^2}{16}=1,y\\neq 0$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -64437,7 +64437,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002327": {
|
|
|
|
"002327": {
|
|
|
|
"id": "002327",
|
|
|
|
"id": "002327",
|
|
|
|
"content": "一个焦点把长轴分成长度为$7$和$1$两段的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/16+y^2/7=1$or$x^2/7+y^2/16=1$}",
|
|
|
|
"content": "一个焦点把长轴分成长度为$7$和$1$两段的椭圆的标准方程为\\blank{100}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"K0714003X",
|
|
|
|
"K0714003X",
|
|
|
|
"K0713002X"
|
|
|
|
"K0713002X"
|
|
|
|
@ -64447,7 +64447,7 @@
|
|
|
|
"椭圆"
|
|
|
|
"椭圆"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$x^2/16+y^2/7=1$or$x^2/7+y^2/16=1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -64462,7 +64462,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002328": {
|
|
|
|
"002328": {
|
|
|
|
"id": "002328",
|
|
|
|
"id": "002328",
|
|
|
|
"content": "已知长轴长与短轴长之比为$2:1$, 一条准线方程为$x+4=0$的椭圆的标准方程为\\blank{100}.\n%\\ans{$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$}",
|
|
|
|
"content": "已知长轴长与短轴长之比为$2:1$, 一条准线方程为$x+4=0$的椭圆的标准方程为\\blank{100}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"KNONE"
|
|
|
|
"KNONE"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
@ -64471,7 +64471,7 @@
|
|
|
|
"椭圆"
|
|
|
|
"椭圆"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -64486,7 +64486,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"002329": {
|
|
|
|
"002329": {
|
|
|
|
"id": "002329",
|
|
|
|
"id": "002329",
|
|
|
|
"content": "以直线$3x+4y-12=0$和两轴的交点之一作为顶点, 另一交点作为焦点的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/25+y^2/9=1$or$x^2/16+y^2/25=1$}",
|
|
|
|
"content": "以直线$3x+4y-12=0$和两轴的交点之一作为顶点, 另一交点作为焦点的椭圆的标准方程为\\blank{100}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
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|
"K0713003X"
|
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"K0713003X"
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|
|
],
|
|
|
|
],
|
|
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@ -64523,7 +64523,7 @@
|
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},
|
|
|
|
},
|
|
|
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"002330": {
|
|
|
|
"002330": {
|
|
|
|
"id": "002330",
|
|
|
|
"id": "002330",
|
|
|
|
"content": "过点$P(3,0)$, 且长轴长是短轴长的三倍的椭圆的标准方程为\\blank{100}.\n%\\ans{$x^2/9+y^2=1$or$x^2/9+y^2/81=1$}",
|
|
|
|
"content": "过点$P(3,0)$, 且长轴长是短轴长的三倍的椭圆的标准方程为\\blank{100}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"K0714003X",
|
|
|
|
"K0714003X",
|
|
|
|
"K0713004X"
|
|
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|
"K0713004X"
|
|
|
|
@ -64533,7 +64533,7 @@
|
|
|
|
"椭圆"
|
|
|
|
"椭圆"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$x^2/9+y^2=1$or$x^2/9+y^2/81=1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
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|
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|
"usages": [],
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|
@ -64644,7 +64644,7 @@
|
|
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|
},
|
|
|
|
},
|
|
|
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"002335": {
|
|
|
|
"002335": {
|
|
|
|
"id": "002335",
|
|
|
|
"id": "002335",
|
|
|
|
"content": "以椭圆的两个焦点为直径端点的圆交椭圆于四个点, 若顺次连接这四个点及两个焦点恰好组成一个正六边形, 则椭圆的离心率为\\blank{50}.\n%\\ans{$\\sqrt{3}-1$}",
|
|
|
|
"content": "以椭圆的两个焦点为直径端点的圆交椭圆于四个点, 若顺次连接这四个点及两个焦点恰好组成一个正六边形, 则椭圆的离心率为\\blank{50}.",
|
|
|
|
"objs": [
|
|
|
|
"objs": [
|
|
|
|
"K0714005X",
|
|
|
|
"K0714005X",
|
|
|
|
"K0713004X"
|
|
|
|
"K0713004X"
|
|
|
|
@ -64654,7 +64654,7 @@
|
|
|
|
"椭圆"
|
|
|
|
"椭圆"
|
|
|
|
],
|
|
|
|
],
|
|
|
|
"genre": "填空题",
|
|
|
|
"genre": "填空题",
|
|
|
|
"ans": "",
|
|
|
|
"ans": "$\\sqrt{3}-1$",
|
|
|
|
"solution": "",
|
|
|
|
"solution": "",
|
|
|
|
"duration": -1,
|
|
|
|
"duration": -1,
|
|
|
|
"usages": [],
|
|
|
|
"usages": [],
|
|
|
|
@ -335088,7 +335088,7 @@
|
|
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|
},
|
|
|
|
},
|
|
|
|
"013711": {
|
|
|
|
"013711": {
|
|
|
|
"id": "013711",
|
|
|
|
"id": "013711",
|
|
|
|
"content": "已知集合$M=\\{1,2,3, \\cdots, 10\\}$, 集合$A \\subseteq M$, 定义$M(A)$为$A$中元素的最小值, 当$A$取谝$M$的所有非空子集时, 对应的$M(A)$的和记为$S_{10}$, 则$S_{10}=$\\blank{50}.\n%02",
|
|
|
|
"content": "已知集合$M=\\{1,2,3, \\cdots, 10\\}$, 集合$A \\subseteq M$, 定义$M(A)$为$A$中元素的最小值, 当$A$取谝$M$的所有非空子集时, 对应的$M(A)$的和记为$S_{10}$, 则$S_{10}=$\\blank{50}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第八单元"
|
|
|
|
"第八单元"
|
|
|
|
@ -335511,7 +335511,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013731": {
|
|
|
|
"013731": {
|
|
|
|
"id": "013731",
|
|
|
|
"id": "013731",
|
|
|
|
"content": "$a>b>0$, 那么当代数式$a^2+\\dfrac{16}{b(a-b)}$取得最小值时点$P(a, b)$的坐标为\\blank{50}.\n%03",
|
|
|
|
"content": "$a>b>0$, 那么当代数式$a^2+\\dfrac{16}{b(a-b)}$取得最小值时点$P(a, b)$的坐标为\\blank{50}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第一单元"
|
|
|
|
"第一单元"
|
|
|
|
@ -335958,7 +335958,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013752": {
|
|
|
|
"013752": {
|
|
|
|
"id": "013752",
|
|
|
|
"id": "013752",
|
|
|
|
"content": "已知$a, b, \\in (0,+\\infty)$, 且$a \\neq b$, $n$是正整数. 求证: $(a+b)(a^n+b^n)<2(a^{n+1}+b^{n+1})$.\n%04",
|
|
|
|
"content": "已知$a, b, \\in (0,+\\infty)$, 且$a \\neq b$, $n$是正整数. 求证: $(a+b)(a^n+b^n)<2(a^{n+1}+b^{n+1})$.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第一单元"
|
|
|
|
"第一单元"
|
|
|
|
@ -336087,7 +336087,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013758": {
|
|
|
|
"013758": {
|
|
|
|
"id": "013758",
|
|
|
|
"id": "013758",
|
|
|
|
"content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{x \\cdot y | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.\n%05",
|
|
|
|
"content": "设$a, b$是两个实数, $A=\\{(x, y) | x=n, y=n a+b, n \\in \\mathbf{Z}\\}$, $B=\\{(x, y) | x=m, \\ y=3(m^2+5),\\ m \\in \\mathbf{Z}\\}$, $C=\\{x \\cdot y | x^2+y^2 \\leq 144\\}$, 讨论是否存在$a, b$使得$A \\cap B \\neq \\varnothing$且$(a, b) \\in C$.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第一单元"
|
|
|
|
"第一单元"
|
|
|
|
@ -336571,7 +336571,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013781": {
|
|
|
|
"013781": {
|
|
|
|
"id": "013781",
|
|
|
|
"id": "013781",
|
|
|
|
"content": "定义在$[0,1]$上的函数$f(x)$满足$f(0)=0$, $f(x)+f(1-x)=1$, $f(\\dfrac{x}{5})=\\dfrac{1}{2} f(x)$, 且当$0 \\leq x_1<x_2 \\leq 1$时, $f(x_1) \\leq f(x_2)$. 则$f(\\dfrac{1}{2020})=$\\blank{50}.\n%06",
|
|
|
|
"content": "定义在$[0,1]$上的函数$f(x)$满足$f(0)=0$, $f(x)+f(1-x)=1$, $f(\\dfrac{x}{5})=\\dfrac{1}{2} f(x)$, 且当$0 \\leq x_1<x_2 \\leq 1$时, $f(x_1) \\leq f(x_2)$. 则$f(\\dfrac{1}{2020})=$\\blank{50}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第二单元"
|
|
|
|
"第二单元"
|
|
|
|
@ -336970,7 +336970,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013800": {
|
|
|
|
"013800": {
|
|
|
|
"id": "013800",
|
|
|
|
"id": "013800",
|
|
|
|
"content": "已知函数$f(x)=4^x-2^x$, 实数$s, t$满足$f(s)+f(t)=0$, $a=2^s+2^t$, $b=2^{s+t}$.\\\\\n(1) 当函数$f(x)$的定义域为$[-1,1]$时, 求函数$f(x)$的值域;\\\\\n(2) 求函数关系式$b=g(a)$, 并求函数$g(a)$的定义域$D$;\\\\\n(3) 在 (2) 的结论中, 对任意$x_1 \\in D$, 都存在$x_2 \\in[-1,1]$, 使得$g(x_1)=f(x_2)+m$成立, 求实数$m$的取值范围.\n%07",
|
|
|
|
"content": "已知函数$f(x)=4^x-2^x$, 实数$s, t$满足$f(s)+f(t)=0$, $a=2^s+2^t$, $b=2^{s+t}$.\\\\\n(1) 当函数$f(x)$的定义域为$[-1,1]$时, 求函数$f(x)$的值域;\\\\\n(2) 求函数关系式$b=g(a)$, 并求函数$g(a)$的定义域$D$;\\\\\n(3) 在 (2) 的结论中, 对任意$x_1 \\in D$, 都存在$x_2 \\in[-1,1]$, 使得$g(x_1)=f(x_2)+m$成立, 求实数$m$的取值范围.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第二单元"
|
|
|
|
"第二单元"
|
|
|
|
@ -337375,7 +337375,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013819": {
|
|
|
|
"013819": {
|
|
|
|
"id": "013819",
|
|
|
|
"id": "013819",
|
|
|
|
"content": "对于实数$a$和$b$, 定义运算``$\\ast$'', $a \\ast b=\\begin{cases}a^2-a b, & a \\leq b, \\\\ b^2-a b, & a>b.\\end{cases}$ 设$f(x)=(2 x-1) \\ast (x-1)$, 且关于$x$的方程为$f(x)=m$($m \\in \\mathbf{R}$)恰有三个互不相等的实数根$x_1, x_2, x_3$, 则$x_1 x_2 x_3$的取值范围是\\blank{50}.\n%08",
|
|
|
|
"content": "对于实数$a$和$b$, 定义运算``$\\ast$'', $a \\ast b=\\begin{cases}a^2-a b, & a \\leq b, \\\\ b^2-a b, & a>b.\\end{cases}$ 设$f(x)=(2 x-1) \\ast (x-1)$, 且关于$x$的方程为$f(x)=m$($m \\in \\mathbf{R}$)恰有三个互不相等的实数根$x_1, x_2, x_3$, 则$x_1 x_2 x_3$的取值范围是\\blank{50}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第二单元"
|
|
|
|
"第二单元"
|
|
|
|
@ -337755,7 +337755,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013837": {
|
|
|
|
"013837": {
|
|
|
|
"id": "013837",
|
|
|
|
"id": "013837",
|
|
|
|
"content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(\\dfrac{1}{3},+\\infty)$, 使得$\\{y | y=f(x), x \\in[a, b]\\}=[m a, m b]$, 则实数$m$的取值范围是\\blank{50}.\n%09",
|
|
|
|
"content": "已知$f(x)=4-\\dfrac{1}{x}$, 若存在区间$[a, b] \\subseteq(\\dfrac{1}{3},+\\infty)$, 使得$\\{y | y=f(x), x \\in[a, b]\\}=[m a, m b]$, 则实数$m$的取值范围是\\blank{50}.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第二单元"
|
|
|
|
"第二单元"
|
|
|
|
@ -337903,7 +337903,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013844": {
|
|
|
|
"013844": {
|
|
|
|
"id": "013844",
|
|
|
|
"id": "013844",
|
|
|
|
"content": "已知函数$f(x)=x^2+(x-1)|x-a|$, 是否存在实数$a$, 使不等式$f(x) \\geq 2 x-3$对一切实数$x$恒成立? 若存在, 求出$a$的取值范围, 若不存在, 请说明理由.\n%10",
|
|
|
|
"content": "已知函数$f(x)=x^2+(x-1)|x-a|$, 是否存在实数$a$, 使不等式$f(x) \\geq 2 x-3$对一切实数$x$恒成立? 若存在, 求出$a$的取值范围, 若不存在, 请说明理由.",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第二单元"
|
|
|
|
"第二单元"
|
|
|
|
@ -338218,7 +338218,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013859": {
|
|
|
|
"013859": {
|
|
|
|
"id": "013859",
|
|
|
|
"id": "013859",
|
|
|
|
"content": "如图, 位于$A$处的信息中心获悉: 在其正东方向相距$40$海里的$B$处有一艘渔船遭遇不测, 在原地等待营救. 信息中心立即把消息告知在其南偏西$30^{\\circ}$、相距$20$海里的$C$处的搜救船只, 现搜救船只朝北偏东$\\theta$的方向即沿直线$CB$前往$B$处救援, 求$\\cos \\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\draw (-120:2) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\draw [dashed] (0,-2) coordinate (S) -- (0,2);\n\\draw [dashed] (0,0) -- (-1,0);\n\\draw (A) pic [draw, \"$30^\\circ$\", angle eccentricity = 1.8] {angle = C--A--S};\n\\draw (A)--(C) node [midway, above left] {$20$};\n\\draw (A)--(B) node [midway, above] {$40$};\n\\draw [->] (0.7,1) -- (1.5,1) node [right] {东};\n\\draw [->] (1,0.7) -- (1,1.5) node [above] {北};\n\\end{tikzpicture}\n\\end{center}\n%11",
|
|
|
|
"content": "如图, 位于$A$处的信息中心获悉: 在其正东方向相距$40$海里的$B$处有一艘渔船遭遇不测, 在原地等待营救. 信息中心立即把消息告知在其南偏西$30^{\\circ}$、相距$20$海里的$C$处的搜救船只, 现搜救船只朝北偏东$\\theta$的方向即沿直线$CB$前往$B$处救援, 求$\\cos \\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$A$} coordinate (A);\n\\draw (4,0) node [right] {$B$} coordinate (B);\n\\draw (-120:2) node [below] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--cycle;\n\\draw [dashed] (0,-2) coordinate (S) -- (0,2);\n\\draw [dashed] (0,0) -- (-1,0);\n\\draw (A) pic [draw, \"$30^\\circ$\", angle eccentricity = 1.8] {angle = C--A--S};\n\\draw (A)--(C) node [midway, above left] {$20$};\n\\draw (A)--(B) node [midway, above] {$40$};\n\\draw [->] (0.7,1) -- (1.5,1) node [right] {东};\n\\draw [->] (1,0.7) -- (1,1.5) node [above] {北};\n\\end{tikzpicture}\n\\end{center}",
|
|
|
|
"objs": [],
|
|
|
|
"objs": [],
|
|
|
|
"tags": [
|
|
|
|
"tags": [
|
|
|
|
"第三单元"
|
|
|
|
"第三单元"
|
|
|
|
@ -338535,7 +338535,7 @@
|
|
|
|
},
|
|
|
|
},
|
|
|
|
"013874": {
|
|
|
|
"013874": {
|
|
|
|
"id": "013874",
|
|
|
|
"id": "013874",
|
|
|
|
"content": "已知定义在区间$[-\\dfrac{\\pi}{2}, \\pi]$上的函数$y=f(x)$的图像关于直线$x=\\dfrac{\\pi}{4}$对称, 当$x \\geq \\dfrac{\\pi}{4}$时, 函数解析式为$f(x)=\\sin x$.\\\\\n(1) 求$y=f(x)$的函数表达式;\\\\\n(2) 如果关于$x$的方程$f(x)=a$有解, 那么将方程在$a$取某一确定值时所求得的所有解的和记为$M_a$, 求$M_a$的所有可能取值及相应的$a$的取值范围.\n%12",
|
|
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"content": "已知定义在区间$[-\\dfrac{\\pi}{2}, \\pi]$上的函数$y=f(x)$的图像关于直线$x=\\dfrac{\\pi}{4}$对称, 当$x \\geq \\dfrac{\\pi}{4}$时, 函数解析式为$f(x)=\\sin x$.\\\\\n(1) 求$y=f(x)$的函数表达式;\\\\\n(2) 如果关于$x$的方程$f(x)=a$有解, 那么将方程在$a$取某一确定值时所求得的所有解的和记为$M_a$, 求$M_a$的所有可能取值及相应的$a$的取值范围.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第三单元"
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"第三单元"
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@ -338810,7 +338810,7 @@
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},
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},
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"013887": {
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"013887": {
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"id": "013887",
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"id": "013887",
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"content": "已知点$A(2,0)$, 点$P$是以原点$O$为圆心、$1$为半径的圆上的任意一点, 将点$P$绕点$O$逆时针旋转$90^{\\circ}$得点$Q$, 线段$AP$的中点为$M$, 求$|MQ|$的最大值与最小值.\n%13",
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"content": "已知点$A(2,0)$, 点$P$是以原点$O$为圆心、$1$为半径的圆上的任意一点, 将点$P$绕点$O$逆时针旋转$90^{\\circ}$得点$Q$, 线段$AP$的中点为$M$, 求$|MQ|$的最大值与最小值.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第三单元"
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"第三单元"
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@ -339041,7 +339041,7 @@
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},
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},
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"013898": {
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"013898": {
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"id": "013898",
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"id": "013898",
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"content": "正方形$ABCD$的边长为$2$, 对角线$AC$、$BD$相交于点$O$, 动点$P$满足$|\\overrightarrow{OP}|=\\dfrac{\\sqrt{2}}{2}$, 若$\\overrightarrow{AP}=m \\overrightarrow{AB}+n \\overrightarrow{AD}$, 其中$m$、$n \\in \\mathbf{R}$, 则$\\dfrac{2 m+1}{2 n+2}$的最大值是\\blank{50}.\n%14",
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"content": "正方形$ABCD$的边长为$2$, 对角线$AC$、$BD$相交于点$O$, 动点$P$满足$|\\overrightarrow{OP}|=\\dfrac{\\sqrt{2}}{2}$, 若$\\overrightarrow{AP}=m \\overrightarrow{AB}+n \\overrightarrow{AD}$, 其中$m$、$n \\in \\mathbf{R}$, 则$\\dfrac{2 m+1}{2 n+2}$的最大值是\\blank{50}.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第五单元"
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"第五单元"
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@ -339295,7 +339295,7 @@
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},
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},
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"013910": {
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"013910": {
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"id": "013910",
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"id": "013910",
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"content": "已知平面上三个不同的单位向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}=\\dfrac{1}{2}$, 若$\\overrightarrow {e}$为平面内的任意单位向量, 则$|\\overrightarrow {a} \\cdot \\overrightarrow {e}|+2|\\overrightarrow {b} \\cdot \\overrightarrow {e}|+3|\\overrightarrow {c} \\cdot \\overrightarrow {e}|$的最大值为\\blank{50}.\n%15",
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"content": "已知平面上三个不同的单位向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$满足$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {b} \\cdot \\overrightarrow {c}=\\dfrac{1}{2}$, 若$\\overrightarrow {e}$为平面内的任意单位向量, 则$|\\overrightarrow {a} \\cdot \\overrightarrow {e}|+2|\\overrightarrow {b} \\cdot \\overrightarrow {e}|+3|\\overrightarrow {c} \\cdot \\overrightarrow {e}|$的最大值为\\blank{50}.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第五单元"
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"第五单元"
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@ -339443,7 +339443,7 @@
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},
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},
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"013917": {
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"013917": {
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"id": "013917",
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"id": "013917",
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"content": "已知$\\triangle ABC$中, $|\\overrightarrow{BC}|=3 \\sqrt{2}$, $|\\overrightarrow{CA}|=4$, $|\\overrightarrow{AB}|=2 \\sqrt{3}$, $PQ$是以$A$为圆心, $2$为半径的圆的直径.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale =0.6]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw ({3*sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\path [name path = c1] (B) ++ ({2*sqrt(3)},0) arc (0:70:{2*sqrt(3)});\n\\path [name path = c2] (C) ++ (-4,0) arc (180:125:4);\n\\path [name intersections = {of = c1 and c2, by = A}];\n\\draw (A)--(B) (A) node [above] {$A$} --(C) (B)--(C);\n\\draw (A) circle (2);\n\\draw (A) ++ (220:2) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (40:2) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\draw [->] (B)--(P);\n\\draw [->] (C)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值、最小值, 并指出取最大值、 最小值时向量$\\overrightarrow{PQ}$的方向.\n%16",
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"content": "已知$\\triangle ABC$中, $|\\overrightarrow{BC}|=3 \\sqrt{2}$, $|\\overrightarrow{CA}|=4$, $|\\overrightarrow{AB}|=2 \\sqrt{3}$, $PQ$是以$A$为圆心, $2$为半径的圆的直径.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale =0.6]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw ({3*sqrt(2)},0) node [right] {$C$} coordinate (C);\n\\path [name path = c1] (B) ++ ({2*sqrt(3)},0) arc (0:70:{2*sqrt(3)});\n\\path [name path = c2] (C) ++ (-4,0) arc (180:125:4);\n\\path [name intersections = {of = c1 and c2, by = A}];\n\\draw (A)--(B) (A) node [above] {$A$} --(C) (B)--(C);\n\\draw (A) circle (2);\n\\draw (A) ++ (220:2) node [left] {$P$} coordinate (P);\n\\draw (A) ++ (40:2) node [right] {$Q$} coordinate (Q);\n\\draw (P)--(Q);\n\\draw [->] (B)--(P);\n\\draw [->] (C)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值、最小值, 并指出取最大值、 最小值时向量$\\overrightarrow{PQ}$的方向.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第五单元"
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"第五单元"
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@ -339716,7 +339716,7 @@
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},
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},
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"013930": {
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"013930": {
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"id": "013930",
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"id": "013930",
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"content": "已知定义在$\\mathbf{R}$上的函数$f(x)$是奇函数, 且满足$f(x)=f(x+3)$, $f(-2)=-3$. 若数列$\\{a_n\\}$中, $a_1=-1$, 且前$n$项和$S_n$满足$\\dfrac{S_n}{n}=2 \\times \\dfrac{a_n}{n}+1$, 则$f(a_5)+f(a_6)=$\\blank{50}.\n%17",
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"content": "已知定义在$\\mathbf{R}$上的函数$f(x)$是奇函数, 且满足$f(x)=f(x+3)$, $f(-2)=-3$. 若数列$\\{a_n\\}$中, $a_1=-1$, 且前$n$项和$S_n$满足$\\dfrac{S_n}{n}=2 \\times \\dfrac{a_n}{n}+1$, 则$f(a_5)+f(a_6)=$\\blank{50}.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第四单元"
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"第四单元"
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@ -339989,7 +339989,7 @@
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},
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},
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"013943": {
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"013943": {
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"id": "013943",
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"id": "013943",
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"content": "已知$f(x)=\\dfrac{1}{4} x^2+a x+b$, 且关于$x$的不等式$f(x)<0$的解集为$(-2,0)$. 各项均为正数的数列$\\{a_n\\}$的前$n$项和为$S_n$, 点列$(a_n, S_n)$($n \\in \\mathbf{N}$, $n\\ge 1$)在函数$y=f(x)$的图像上.\\\\\n(1) 求函数$y=f(x)$的解析式;\\\\\n(2) 若$b_n=k^{\\frac{a_n}{2}}$($k>0$), 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{2 b_n-1}{b_n+2}$的值;\\\\\n(3) 令$c_n=\\begin{cases}a_n, n \\text {为奇数}, \\\\ c_{\\frac n 2}, n\\text{为偶数},\\end{cases}$ 求数列$\\{c_n\\}$的前$2012$项中满足$c_m=6$的所有项数之和.\n%18",
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"content": "已知$f(x)=\\dfrac{1}{4} x^2+a x+b$, 且关于$x$的不等式$f(x)<0$的解集为$(-2,0)$. 各项均为正数的数列$\\{a_n\\}$的前$n$项和为$S_n$, 点列$(a_n, S_n)$($n \\in \\mathbf{N}$, $n\\ge 1$)在函数$y=f(x)$的图像上.\\\\\n(1) 求函数$y=f(x)$的解析式;\\\\\n(2) 若$b_n=k^{\\frac{a_n}{2}}$($k>0$), 求$\\displaystyle\\lim_{n\\to\\infty} \\dfrac{2 b_n-1}{b_n+2}$的值;\\\\\n(3) 令$c_n=\\begin{cases}a_n, n \\text {为奇数}, \\\\ c_{\\frac n 2}, n\\text{为偶数},\\end{cases}$ 求数列$\\{c_n\\}$的前$2012$项中满足$c_m=6$的所有项数之和.",
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"objs": [],
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"objs": [],
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"tags": [
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|
"tags": [
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|
"第四单元"
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"第四单元"
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@ -340221,7 +340221,7 @@
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},
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},
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"013954": {
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"013954": {
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"id": "013954",
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"id": "013954",
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"content": "设数列$\\{a_n\\}$的通项公式为$a_n=p n+q$($n \\in \\mathbf{N}$, $n\\ge 1$, $p>0$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n \\geq m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$, 求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_m\\}$的前$2 m$项和;\\\\\n(3) 是否存在$(p, q)$, 使得$b_m=3 m+2$($n \\in \\mathbf{N}$)? 如果存在, 求所有的$(p, q)$所构成的集合; 如果不存在, 请说明理由.\n%19",
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"content": "设数列$\\{a_n\\}$的通项公式为$a_n=p n+q$($n \\in \\mathbf{N}$, $n\\ge 1$, $p>0$). 数列$\\{b_n\\}$定义如下: 对于正整数$m$, $b_m$是使得不等式$a_n \\geq m$成立的所有$n$中的最小值.\\\\\n(1) 若$p=\\dfrac{1}{2}$, $q=-\\dfrac{1}{3}$, 求$b_3$;\\\\\n(2) 若$p=2$, $q=-1$, 求数列$\\{b_m\\}$的前$2 m$项和;\\\\\n(3) 是否存在$(p, q)$, 使得$b_m=3 m+2$($n \\in \\mathbf{N}$)? 如果存在, 求所有的$(p, q)$所构成的集合; 如果不存在, 请说明理由.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第四单元"
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"第四单元"
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@ -340348,7 +340348,7 @@
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},
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},
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"013960": {
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"013960": {
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"id": "013960",
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"id": "013960",
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"content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_2=5$, $\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+S_{n-2}=2S_{n-1}+2^{n-1}$($n \\geq 3$).\\\\\n(1) 试求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=\\dfrac{2^{n-1}}{a_n \\cdot a_{n+1}}$, $T_n$是数列$\\{b_n\\}$的前$n$项和, 证明: $T_n<\\dfrac{1}{6}$;\\\\ \n(3) 证明: 对任意给定的$m \\in(0, \\dfrac{1}{6})$, 均存在$n_0 \\in \\mathbf{N}$, 使得当$n>n_0$时, (2)中的$T_n>m$恒成立.\n%20",
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|
"content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_2=5$, $\\{a_n\\}$的前$n$项和为$S_n$, 且满足$S_n+S_{n-2}=2S_{n-1}+2^{n-1}$($n \\geq 3$).\\\\\n(1) 试求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 令$b_n=\\dfrac{2^{n-1}}{a_n \\cdot a_{n+1}}$, $T_n$是数列$\\{b_n\\}$的前$n$项和, 证明: $T_n<\\dfrac{1}{6}$;\\\\ \n(3) 证明: 对任意给定的$m \\in(0, \\dfrac{1}{6})$, 均存在$n_0 \\in \\mathbf{N}$, 使得当$n>n_0$时, (2)中的$T_n>m$恒成立.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第四单元"
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"第四单元"
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@ -340687,7 +340687,7 @@
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},
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},
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"013976": {
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"013976": {
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"id": "013976",
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"id": "013976",
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"content": "如图, 为了保护河上古桥$OA$, 规划建一座新桥$BC$, 同时设立一个圆形保护区. 规划要求: 新桥$BC$与河岸$AB$垂直; 保护区的边界为圆心$M$在线段$OA$上并与$BC$相切的圆. 且古桥两端$O$和$A$到该圆上任意一点的距离均不少于$80 \\text{m}$. 经测量, 点$A$位于点$O$正北方向$60 \\text{m}$处, 点$C$位于点$O$正东方向$170 \\text{m}$处($OC$为河岸), $\\tan \\angle BCO=\\dfrac{4}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (3.4,0) node [below right] {$C$} coordinate (C);\n\\draw (0,1.2) node [above left] {$A$} coordinate (A);\n\\draw (C) ++ ({90+atan(3/4)}:3) coordinate (T);\n\\draw ($(C)!(A)!(T)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(B)--(C) (A)--(O)--(C);\\\n\\draw (O) --++ (0,-0.5) coordinate (O1) (C) --++ (0,-0.5) coordinate (C1);\n\\draw [<->] ($(O)!0.5!(O1)$) -- ($(C)!0.5!(C1)$) node [midway, fill = white] {$170\\text{m}$};\n\\draw (O) --++ (-0.5,0) coordinate (O2) (A) --++ (-0.5,0) coordinate (A2);\n\\draw [<->] ($(O)!0.5!(O2)$) -- ($(A)!0.5!(A2)$) node [midway, fill = white, rotate = 90] {$60\\text{m}$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {东};\n\\draw [->] (A) -- ($(O)!1.6!(A)$) node [left] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 求新桥$BC$的长;\\\\\n(2) 当$OM$多长时, 圆形保护区的面积最大?\n%21",
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"content": "如图, 为了保护河上古桥$OA$, 规划建一座新桥$BC$, 同时设立一个圆形保护区. 规划要求: 新桥$BC$与河岸$AB$垂直; 保护区的边界为圆心$M$在线段$OA$上并与$BC$相切的圆. 且古桥两端$O$和$A$到该圆上任意一点的距离均不少于$80 \\text{m}$. 经测量, 点$A$位于点$O$正北方向$60 \\text{m}$处, 点$C$位于点$O$正东方向$170 \\text{m}$处($OC$为河岸), $\\tan \\angle BCO=\\dfrac{4}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (3.4,0) node [below right] {$C$} coordinate (C);\n\\draw (0,1.2) node [above left] {$A$} coordinate (A);\n\\draw (C) ++ ({90+atan(3/4)}:3) coordinate (T);\n\\draw ($(C)!(A)!(T)$) node [above] {$B$} coordinate (B);\n\\draw (A)--(B)--(C) (A)--(O)--(C);\\\n\\draw (O) --++ (0,-0.5) coordinate (O1) (C) --++ (0,-0.5) coordinate (C1);\n\\draw [<->] ($(O)!0.5!(O1)$) -- ($(C)!0.5!(C1)$) node [midway, fill = white] {$170\\text{m}$};\n\\draw (O) --++ (-0.5,0) coordinate (O2) (A) --++ (-0.5,0) coordinate (A2);\n\\draw [<->] ($(O)!0.5!(O2)$) -- ($(A)!0.5!(A2)$) node [midway, fill = white, rotate = 90] {$60\\text{m}$};\n\\draw [->] (C) -- ($(O)!1.3!(C)$) node [below] {东};\n\\draw [->] (A) -- ($(O)!1.6!(A)$) node [left] {北};\n\\end{tikzpicture}\n\\end{center}\n(1) 求新桥$BC$的长;\\\\\n(2) 当$OM$多长时, 圆形保护区的面积最大?",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第七单元"
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"第七单元"
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@ -341002,7 +341002,7 @@
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},
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},
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"013991": {
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"013991": {
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"id": "013991",
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"id": "013991",
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"content": "如图, 已知椭圆$C_1$与$C_2$的中心是坐标原点$O$, 长轴均为$MN$且在$x$轴上, 短轴长分别为$2 m$, $2 n$($m>n$), 过原点且不与$x$轴重合的直线$l$与$C_1, C_2$的四个交点按纵坐标从大到小依次为$A$、$B$、$C$、$D$. 记$\\lambda=\\dfrac{m}{n}$, $\\triangle BDM$和$\\triangle ABN$的面积分别为$S_1$和$S_2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\path [name path = elli1, draw] (0,0) ellipse (3 and 2);\n\\path [name path = elli2, draw] (0,0) ellipse (3 and 1);\n\\draw (-3,0) node [below left] {$M$} coordinate (M);\n\\draw (3,0) node [below right] {$N$} coordinate (N);\n\\path [name path = line, draw] (-2,-2.5) -- (2,2.5);\n\\path [name intersections = {of = line and elli1, by = {A,D}}];\n\\path [name intersections = {of = line and elli2, by = {B,C}}];\n\\draw (D) node [below] {$D$}--(M)--(B) node [above] {$B$};\n\\draw (C) node [below] {$C$} (A) node [above] {$A$} -- (N)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 设直线$l: y=k x$($k>0$), 若$S_1=3S_2$, 证明: $B$、$C$是线段$AD$的四等分点;\\\\\n(2) 当直线$l$与$y$轴重合时, 若$S_1=\\lambda S_2$, 求$\\lambda$的值;\\\\\n(3) 当$\\lambda$变化时, 是否存在与坐标轴不重合的直线$l$, 使得$S_1=\\lambda S_2$? 并说明理由.\n%22",
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"content": "如图, 已知椭圆$C_1$与$C_2$的中心是坐标原点$O$, 长轴均为$MN$且在$x$轴上, 短轴长分别为$2 m$, $2 n$($m>n$), 过原点且不与$x$轴重合的直线$l$与$C_1, C_2$的四个交点按纵坐标从大到小依次为$A$、$B$、$C$、$D$. 记$\\lambda=\\dfrac{m}{n}$, $\\triangle BDM$和$\\triangle ABN$的面积分别为$S_1$和$S_2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\path [name path = elli1, draw] (0,0) ellipse (3 and 2);\n\\path [name path = elli2, draw] (0,0) ellipse (3 and 1);\n\\draw (-3,0) node [below left] {$M$} coordinate (M);\n\\draw (3,0) node [below right] {$N$} coordinate (N);\n\\path [name path = line, draw] (-2,-2.5) -- (2,2.5);\n\\path [name intersections = {of = line and elli1, by = {A,D}}];\n\\path [name intersections = {of = line and elli2, by = {B,C}}];\n\\draw (D) node [below] {$D$}--(M)--(B) node [above] {$B$};\n\\draw (C) node [below] {$C$} (A) node [above] {$A$} -- (N)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 设直线$l: y=k x$($k>0$), 若$S_1=3S_2$, 证明: $B$、$C$是线段$AD$的四等分点;\\\\\n(2) 当直线$l$与$y$轴重合时, 若$S_1=\\lambda S_2$, 求$\\lambda$的值;\\\\\n(3) 当$\\lambda$变化时, 是否存在与坐标轴不重合的直线$l$, 使得$S_1=\\lambda S_2$? 并说明理由.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第七单元"
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"第七单元"
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@ -341275,7 +341275,7 @@
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},
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},
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"014004": {
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"014004": {
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"id": "014004",
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"id": "014004",
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"content": "设椭圆$\\Gamma$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 点$O$为坐标原点, 点$A$的坐标为$(a, 0)$, 点$B$的坐标为$(0, b)$, 点$M$在线段$AB$上, 满足$|BM|=2|MA|$, 直线$OM$的斜率为$\\dfrac{\\sqrt{5}}{10}$.\\\\\n(1) 若$a=\\lambda b$, 求$\\lambda$的值;\\\\\n(2) 设点$C$的坐标为$(0,-b)$, $N$为线段$AC$的中点, 点$N$关于直线$AB$的对称点的纵坐标为$\\dfrac{7}{2}$, 求椭圆$\\Gamma$的方程.\n%23",
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"content": "设椭圆$\\Gamma$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 点$O$为坐标原点, 点$A$的坐标为$(a, 0)$, 点$B$的坐标为$(0, b)$, 点$M$在线段$AB$上, 满足$|BM|=2|MA|$, 直线$OM$的斜率为$\\dfrac{\\sqrt{5}}{10}$.\\\\\n(1) 若$a=\\lambda b$, 求$\\lambda$的值;\\\\\n(2) 设点$C$的坐标为$(0,-b)$, $N$为线段$AC$的中点, 点$N$关于直线$AB$的对称点的纵坐标为$\\dfrac{7}{2}$, 求椭圆$\\Gamma$的方程.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第七单元"
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"第七单元"
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@ -341506,7 +341506,7 @@
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},
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},
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"014015": {
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"014015": {
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"id": "014015",
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"id": "014015",
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"content": "(1) 设$P$是圆$F_1: x^2+y^2+2 x-15=0$上的动点, $F(1,0)$, 且线段$PF$的垂直平分线交直线$PF_1$于点$Q$, 求点$Q$的轨迹$C$的方程$f(x, y)=0$;\\\\\n(2) 我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(I) 已知``盾圆$D$''的方程为$y^2= \\begin{cases}4 x, & 0 \\leq x \\leq 3, \\\\ -12(x-4), & 3<x \\leq 4.\\end{cases}$ 设``盾圆$D$''上的任意一点$M$到$F(1,0)$的距离为$d_1$, $M$到直线$l: x=3$的距离为$d_2$. 求证: $d_1+d_2$为定值;\\\\\n(II) 由抛物线弧$E_1: y^2=4 x$($0 \\leq x \\leq \\dfrac{2}{3}$)与(1)中轨迹$C$上的曲线弧$E_2$: $f(x, y)=0$($\\dfrac{2}{3} \\leq x \\leq 2$)所合成的封闭曲线为``盾圆$E$''. 设``盾圆$E$''上的两点$A$、$B$关于$x$轴对称, $O$为坐标原点. 求$\\triangle OAB$面积的最大值.\n%24",
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"content": "(1) 设$P$是圆$F_1: x^2+y^2+2 x-15=0$上的动点, $F(1,0)$, 且线段$PF$的垂直平分线交直线$PF_1$于点$Q$, 求点$Q$的轨迹$C$的方程$f(x, y)=0$;\\\\\n(2) 我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(I) 已知``盾圆$D$''的方程为$y^2= \\begin{cases}4 x, & 0 \\leq x \\leq 3, \\\\ -12(x-4), & 3<x \\leq 4.\\end{cases}$ 设``盾圆$D$''上的任意一点$M$到$F(1,0)$的距离为$d_1$, $M$到直线$l: x=3$的距离为$d_2$. 求证: $d_1+d_2$为定值;\\\\\n(II) 由抛物线弧$E_1: y^2=4 x$($0 \\leq x \\leq \\dfrac{2}{3}$)与(1)中轨迹$C$上的曲线弧$E_2$: $f(x, y)=0$($\\dfrac{2}{3} \\leq x \\leq 2$)所合成的封闭曲线为``盾圆$E$''. 设``盾圆$E$''上的两点$A$、$B$关于$x$轴对称, $O$为坐标原点. 求$\\triangle OAB$面积的最大值.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第七单元"
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"第七单元"
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@ -341654,7 +341654,7 @@
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},
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},
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"014022": {
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"014022": {
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"id": "014022",
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"id": "014022",
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"content": "在平面直角坐标系$xOy$中, 已知椭圆$C: \\dfrac{x^2}{3}+y^2=1$. 斜率为$k$($k>0$)且不过原点的直线$l$交椭圆$C$于$A$、$B$两点, 线段$AB$的中点为$E$, 射线$OE$交椭圆$C$于点$G$, 交直线$x=-3$于点$D(-3, m)$.\\\\\n(1) 求$mk$的值;\\\\\n(2) 若$|OG|^2=|OD| \\cdot|OE|$, 求证: 直线$l$过定点;\\\\\n(3) 在(2)的条件下, 判断点$B$与$G$能否关于$x$轴对称? 若能, 求出此时点$B$的坐标; 若不能, 请说明理由.\n%25",
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"content": "在平面直角坐标系$xOy$中, 已知椭圆$C: \\dfrac{x^2}{3}+y^2=1$. 斜率为$k$($k>0$)且不过原点的直线$l$交椭圆$C$于$A$、$B$两点, 线段$AB$的中点为$E$, 射线$OE$交椭圆$C$于点$G$, 交直线$x=-3$于点$D(-3, m)$.\\\\\n(1) 求$mk$的值;\\\\\n(2) 若$|OG|^2=|OD| \\cdot|OE|$, 求证: 直线$l$过定点;\\\\\n(3) 在(2)的条件下, 判断点$B$与$G$能否关于$x$轴对称? 若能, 求出此时点$B$的坐标; 若不能, 请说明理由.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第七单元"
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"第七单元"
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@ -341990,7 +341990,7 @@
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},
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},
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"014038": {
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"014038": {
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"id": "014038",
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"id": "014038",
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"content": "将半径都为$1$的$4$个钢球完全装入形状为正四面体的容器里, 这个正四面体的高的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}+2 \\sqrt{6}}{3}$}{$2+\\dfrac{2 \\sqrt{6}}{3}$}{$4+\\dfrac{2 \\sqrt{6}}{3}$}{$\\dfrac{4 \\sqrt{3}+2 \\sqrt{6}}{3}$}\n%26",
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"content": "将半径都为$1$的$4$个钢球完全装入形状为正四面体的容器里, 这个正四面体的高的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{3}+2 \\sqrt{6}}{3}$}{$2+\\dfrac{2 \\sqrt{6}}{3}$}{$4+\\dfrac{2 \\sqrt{6}}{3}$}{$\\dfrac{4 \\sqrt{3}+2 \\sqrt{6}}{3}$}",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第六单元"
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"第六单元"
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@ -342242,7 +342242,7 @@
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},
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},
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"014050": {
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"014050": {
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"id": "014050",
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"id": "014050",
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"content": "已知棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为侧面$BB_1C_1C$的中心, $F$在棱$AD$上运动, 正方体表面上有一点$P$满足$\\overrightarrow{D_1P}=\\lambda \\overrightarrow{D_1F}+\\mu \\overrightarrow{D_1E}$($\\lambda \\geq 0$, $\\mu \\geq 0$), 则所有满足条件的$P$点构成图形的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.4!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (F)--(D1);\n\\draw (B)--(C1);\n\\filldraw (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n%27",
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"content": "已知棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $E$为侧面$BB_1C_1C$的中心, $F$在棱$AD$上运动, 正方体表面上有一点$P$满足$\\overrightarrow{D_1P}=\\lambda \\overrightarrow{D_1F}+\\mu \\overrightarrow{D_1E}$($\\lambda \\geq 0$, $\\mu \\geq 0$), 则所有满足条件的$P$点构成图形的面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(C1)$) node [right] {$E$} coordinate (E);\n\\draw ($(A)!0.4!(D)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (F)--(D1);\n\\draw (B)--(C1);\n\\filldraw (E) circle (0.03);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第六单元"
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"第六单元"
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@ -342368,7 +342368,7 @@
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},
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},
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"014056": {
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"014056": {
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"id": "014056",
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"id": "014056",
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"content": "如图, 正四棱锥$P-ABCD$中, $B_1$为$PB$的中点, $D_1$为$PD$的中点, 则棱锥$A-B_1CD_1$与$P-ABCD$的体积之比为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (1,0,-1) node [right] {$D$} coordinate (D);\n\\draw (-1,0,-1) node [below] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [above left] {$B_1$} coordinate (B_1);\n\\draw ($(P)!0.5!(D)$) node [above right] {$D_1$} coordinate (D_1);\n\\draw (P)--(B)(P)--(C)(P)--(D)(B)--(C)--(D);\n\\draw [dashed] (P)--(A)(B)--(A)--(D);\n\\draw (C)--(B_1)(C)--(D_1);\n\\draw [dashed] (C)--(A)--(B_1)(D_1)--(A)(D_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}\n%28",
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"content": "如图, 正四棱锥$P-ABCD$中, $B_1$为$PB$的中点, $D_1$为$PD$的中点, 则棱锥$A-B_1CD_1$与$P-ABCD$的体积之比为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0,1) node [left] {$B$} coordinate (B);\n\\draw (1,0,1) node [right] {$C$} coordinate (C);\n\\draw (1,0,-1) node [right] {$D$} coordinate (D);\n\\draw (-1,0,-1) node [below] {$A$} coordinate (A);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(P)$) node [above left] {$B_1$} coordinate (B_1);\n\\draw ($(P)!0.5!(D)$) node [above right] {$D_1$} coordinate (D_1);\n\\draw (P)--(B)(P)--(C)(P)--(D)(B)--(C)--(D);\n\\draw [dashed] (P)--(A)(B)--(A)--(D);\n\\draw (C)--(B_1)(C)--(D_1);\n\\draw [dashed] (C)--(A)--(B_1)(D_1)--(A)(D_1)--(B_1);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第六单元"
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"第六单元"
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@ -342641,7 +342641,7 @@
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},
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},
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"014069": {
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"014069": {
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"id": "014069",
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"id": "014069",
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"content": "已知关于$z$的方程$z^2-(4+\\mathrm{i}) z+4-m \\mathrm{i}=0$($m \\in \\mathbf{R}$)有实根$\\lambda$.\\\\\n(1) 分别求实数根$\\lambda$以及相应利$m$的值;\\\\\n(2) 在(1)的条件下, 若$M=\\{(x, y) |$存在$b, n \\in \\mathbf{R}$, 使得$(m-n \\mathrm{i})(1-b \\mathrm{i})=x+y \\mathrm{i}, \\ x, y \\in \\mathbf{R}\\}$, 是否存在$t, \\alpha \\in \\mathbf{R}$, 满足$(t \\cos \\alpha, \\sqrt{7} \\sin \\alpha) \\in M$, 若存在, 求出$t$的取值范围, 若不存在, 请说明理由.\n%29",
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"content": "已知关于$z$的方程$z^2-(4+\\mathrm{i}) z+4-m \\mathrm{i}=0$($m \\in \\mathbf{R}$)有实根$\\lambda$.\\\\\n(1) 分别求实数根$\\lambda$以及相应利$m$的值;\\\\\n(2) 在(1)的条件下, 若$M=\\{(x, y) |$存在$b, n \\in \\mathbf{R}$, 使得$(m-n \\mathrm{i})(1-b \\mathrm{i})=x+y \\mathrm{i}, \\ x, y \\in \\mathbf{R}\\}$, 是否存在$t, \\alpha \\in \\mathbf{R}$, 满足$(t \\cos \\alpha, \\sqrt{7} \\sin \\alpha) \\in M$, 若存在, 求出$t$的取值范围, 若不存在, 请说明理由.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第五单元"
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"第五单元"
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@ -342872,7 +342872,7 @@
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},
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},
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"014080": {
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"014080": {
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"id": "014080",
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"id": "014080",
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"content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n$、$m$是正整数, 且$m \\leq n)$的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的性质$\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$, 是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能, 写出推广的形式, 并给出证明; 若不能, 说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.\n%30",
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"content": "规定$\\mathrm{C}_x^m=\\dfrac{x(x-1) \\cdots(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}$, $m$是正整数, 且$\\mathrm{C}_x^0=1$, 这是组合数$\\mathrm{C}_n^m$($n$、$m$是正整数, 且$m \\leq n)$的一种推广.\\\\\n(1) 求$\\mathrm{C}_{-15}^5$的值;\\\\\n(2) 组合数的性质$\\mathrm{C}_n^m=\\mathrm{C}_n^{n-m}$; $\\mathrm{C}_n^m+\\mathrm{C}_n^{m-1}=\\mathrm{C}_{n+1}^m$, 是否都能推广到$\\mathrm{C}_x^m$($x \\in \\mathbf{R}$, $m$是正整数)的情形? 若能, 写出推广的形式, 并给出证明; 若不能, 说明理由;\\\\\n(3) 已知组合数$\\mathrm{C}_n^m$是正整数, 证明: 当$x \\in \\mathbf{Z}$, $m$是正整数时, $\\mathrm{C}_x^m \\in \\mathbf{Z}$.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第八单元"
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"第八单元"
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@ -343167,7 +343167,7 @@
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},
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},
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"014094": {
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"014094": {
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"id": "014094",
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"id": "014094",
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"content": "某人有$5$把钥匙, 其中有$k$把是房门钥匙, 但忘记了开房门的是哪一把. 于是, 他逐把不重复地试开, 问:\\\\\n(1) 若$k=1$, 则恰好第三次打开房门锁的概率是多少?\\\\\n(2) 若$k=1$, 则三次内打开的概率是多少?\\\\\n(3) 若$k=2$, 则三次内打开的概率是多少?\n%31",
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"content": "某人有$5$把钥匙, 其中有$k$把是房门钥匙, 但忘记了开房门的是哪一把. 于是, 他逐把不重复地试开, 问:\\\\\n(1) 若$k=1$, 则恰好第三次打开房门锁的概率是多少?\\\\\n(2) 若$k=1$, 则三次内打开的概率是多少?\\\\\n(3) 若$k=2$, 则三次内打开的概率是多少?",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第八单元"
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"第八单元"
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@ -343719,7 +343719,7 @@
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},
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},
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"014122": {
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"014122": {
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"id": "014122",
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"id": "014122",
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"content": "设集合$P_1=\\{x | x^2+a x+1>0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a, b \\in \\mathbf{R}$. 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}\n%02",
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"content": "设集合$P_1=\\{x | x^2+a x+1>0\\}$, $P_2=\\{x | x^2+a x+2>0\\}$, $Q_1=\\{x | x^2+x+b>0\\}$, $Q_2=\\{x | x^2+2 x+b>0\\}$, 其中$a, b \\in \\mathbf{R}$. 下列说法正确的是\\bracket{20}.\n\\onech{对任意$a$, $P_1$是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{对任意$a$, $P_1$是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 对任意$b$, $Q_1$不是$Q_2$的子集}{存在$a$, 使得$P_1$不是$P_2$的子集; 存在$b$, 使得$Q_1$是$Q_2$的子集}",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "选择题",
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"genre": "选择题",
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@ -344137,7 +344137,7 @@
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},
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},
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"014144": {
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"014144": {
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"id": "014144",
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"id": "014144",
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"content": "对于给定的正数$a$, 存在最大的实数$M$, 使得关于$x$的不等式$|x^2-2 a x| \\leq 5$对一切$x \\in[0, M]$都成立, 试将$M$表示为$a$的函数.\n%03",
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"content": "对于给定的正数$a$, 存在最大的实数$M$, 使得关于$x$的不等式$|x^2-2 a x| \\leq 5$对一切$x \\in[0, M]$都成立, 试将$M$表示为$a$的函数.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -344536,7 +344536,7 @@
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},
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},
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"014165": {
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"014165": {
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"id": "014165",
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"id": "014165",
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"content": "冬奥会期间, 冰墩墩成热销商品, 一家冰墩墩生产公司为加大生产, 计划租地建造临时仓库储存货物, 若记仓库到车站的距离为$x$(单位: $\\text{km}$), 经过市场调查了解到: 每月土地占地费$y_1$(单位: 万元) 与$x+1$成反比, 每月库存货物费$y_2$(单位: 万元) 与$4 x+1$成正比; 若在距离车站$5 \\text{km}$处建仓库, 则$y_1$与$y_2$分别为$12.5$万元和 7 万元. 记两项费用之和为$w$. 问这家公司应该把仓库建在距离车站多少千米处, 才能使两项费用之和最小? 并求出最小值.\n%04",
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"content": "冬奥会期间, 冰墩墩成热销商品, 一家冰墩墩生产公司为加大生产, 计划租地建造临时仓库储存货物, 若记仓库到车站的距离为$x$(单位: $\\text{km}$), 经过市场调查了解到: 每月土地占地费$y_1$(单位: 万元) 与$x+1$成反比, 每月库存货物费$y_2$(单位: 万元) 与$4 x+1$成正比; 若在距离车站$5 \\text{km}$处建仓库, 则$y_1$与$y_2$分别为$12.5$万元和 7 万元. 记两项费用之和为$w$. 问这家公司应该把仓库建在距离车站多少千米处, 才能使两项费用之和最小? 并求出最小值.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -344897,7 +344897,7 @@
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},
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},
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"014184": {
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"014184": {
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"id": "014184",
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"id": "014184",
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"content": "我们知道当$a>0$时, $a^{m+n}=a^m \\cdot a^n$对一切$m, n \\in \\mathbf{R}$都成立. 学生小贤在进一步研究指数幂的性质时, 发现有这么一个等式$2^{1+1}=2^1+2^1$, 带着好奇, 他进一步对$2^{m+n}=2^m+2^n$进行深入研究.\\\\\n(1) 当$m=2$时, 求$n$的值;\\\\\n(2) 当$m \\leq 0$时, 求证: $n$是不存在的.\n%05",
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"content": "我们知道当$a>0$时, $a^{m+n}=a^m \\cdot a^n$对一切$m, n \\in \\mathbf{R}$都成立. 学生小贤在进一步研究指数幂的性质时, 发现有这么一个等式$2^{1+1}=2^1+2^1$, 带着好奇, 他进一步对$2^{m+n}=2^m+2^n$进行深入研究.\\\\\n(1) 当$m=2$时, 求$n$的值;\\\\\n(2) 当$m \\leq 0$时, 求证: $n$是不存在的.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -345258,7 +345258,7 @@
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},
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},
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"014203": {
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"014203": {
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"id": "014203",
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"id": "014203",
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"content": "求满足方程$12^x+5^x=13^x$的实数$x$的值.\n%08",
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"content": "求满足方程$12^x+5^x=13^x$的实数$x$的值.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -345638,7 +345638,7 @@
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},
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},
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"014223": {
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"014223": {
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"id": "014223",
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"id": "014223",
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"content": "设$\\alpha, \\beta$均为锐角, 且满足$3 \\sin ^2 \\alpha+2 \\sin ^2 \\beta=1$, $2 \\sin 2 \\beta-3 \\sin 2 \\alpha=0$. 求证: $\\alpha+2 \\beta=\\dfrac{\\pi}{2}$.\n%09",
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"content": "设$\\alpha, \\beta$均为锐角, 且满足$3 \\sin ^2 \\alpha+2 \\sin ^2 \\beta=1$, $2 \\sin 2 \\beta-3 \\sin 2 \\alpha=0$. 求证: $\\alpha+2 \\beta=\\dfrac{\\pi}{2}$.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -346018,7 +346018,7 @@
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},
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},
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"014243": {
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"014243": {
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"id": "014243",
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"id": "014243",
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"content": "在临港滴水湖畔拟建造一个四边形的露营基地, 如图$ABCD$所示. 为考虑露营客人娱乐休闲的需求, 在四边形$ABCD$区域中, 将$\\triangle ABD$区域设立成花卉观赏区, $\\triangle BCD$区域设立成烧烤区, 边$AB$、$BC$、$CD$、$DA$修建成观赏步道, 边$BD$修建隔离防护栏. 其中$CD=100$米, $BC=200$米, $\\angle A=\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw (0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0) node [left] {$D$} coordinate (D);\n\\draw (-80:2) node [right] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(3)}!30:(D)$) coordinate (O);\n\\draw ($(O)!1!100:(D)$) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (D) -- (B);\n\\draw (barycentric cs:A=1,B=1,D=1) node {花卉观赏区};\n\\draw (barycentric cs:B=1,C=1,D=1) node {烧烤区};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果烧烤区是一个占地面积为$9600$平方米的钝角三角形, 那么需要修建多长的隔离防护栏(精确到$0.1$米)?\\\\\n(2) 考虑到烧烤区的安全性, 在规划四边形$ABCD$区域时, 首先保证烧烤区的占地面积最大时, 再使得花卉观赏区的面积尽可能大, 则应如何设计观赏步道?\n%10",
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"content": "在临港滴水湖畔拟建造一个四边形的露营基地, 如图$ABCD$所示. 为考虑露营客人娱乐休闲的需求, 在四边形$ABCD$区域中, 将$\\triangle ABD$区域设立成花卉观赏区, $\\triangle BCD$区域设立成烧烤区, 边$AB$、$BC$、$CD$、$DA$修建成观赏步道, 边$BD$修建隔离防护栏. 其中$CD=100$米, $BC=200$米, $\\angle A=\\dfrac{\\pi}3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.7]\n\\draw (0,0) node [right] {$C$} coordinate (C);\n\\draw (-1,0) node [left] {$D$} coordinate (D);\n\\draw (-80:2) node [right] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(3)}!30:(D)$) coordinate (O);\n\\draw ($(O)!1!100:(D)$) node [left] {$A$} coordinate (A);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (D) -- (B);\n\\draw (barycentric cs:A=1,B=1,D=1) node {花卉观赏区};\n\\draw (barycentric cs:B=1,C=1,D=1) node {烧烤区};\n\\end{tikzpicture}\n\\end{center}\n(1) 如果烧烤区是一个占地面积为$9600$平方米的钝角三角形, 那么需要修建多长的隔离防护栏(精确到$0.1$米)?\\\\\n(2) 考虑到烧烤区的安全性, 在规划四边形$ABCD$区域时, 首先保证烧烤区的占地面积最大时, 再使得花卉观赏区的面积尽可能大, 则应如何设计观赏步道?",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -346455,7 +346455,7 @@
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},
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},
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"014266": {
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"014266": {
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"id": "014266",
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"id": "014266",
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"content": "设$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$), 函数$y=f(x)$的最小正周期为$\\pi$, 且直线$x=-\\dfrac{\\pi}{2}$是其图像的一条对称轴.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{4}$个单位, 再将所得的图像上每一点的纵坐标不变, 横坐标伸长为原来的$2$倍后得到函数$y=g(x)$的图像, 设常数$\\lambda \\in \\mathbf{R}$, $n$为正整数, 且函数$y=f(x)+\\lambda g(x)$在区间$(0, n \\pi)$内恰有$2023$个零点, 求常数$\\lambda$与$n$的值.\n%11",
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"content": "设$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$), 函数$y=f(x)$的最小正周期为$\\pi$, 且直线$x=-\\dfrac{\\pi}{2}$是其图像的一条对称轴.\\\\\n(1) 求函数$y=f(x)$的表达式;\\\\\n(2) 将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{4}$个单位, 再将所得的图像上每一点的纵坐标不变, 横坐标伸长为原来的$2$倍后得到函数$y=g(x)$的图像, 设常数$\\lambda \\in \\mathbf{R}$, $n$为正整数, 且函数$y=f(x)+\\lambda g(x)$在区间$(0, n \\pi)$内恰有$2023$个零点, 求常数$\\lambda$与$n$的值.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -347215,7 +347215,7 @@
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},
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},
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"014306": {
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"014306": {
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"id": "014306",
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"id": "014306",
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"content": "已知$a \\in \\mathbf{R}$, 若对任意$x_1, x_2 \\in[1,+\\infty)$, 当$x_1<x_2$时, 恒有$a \\ln \\dfrac{x_2}{x_1}<2(x_2-x_1)$成立, 求$a$的取值范围.\n%07",
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"content": "已知$a \\in \\mathbf{R}$, 若对任意$x_1, x_2 \\in[1,+\\infty)$, 当$x_1<x_2$时, 恒有$a \\ln \\dfrac{x_2}{x_1}<2(x_2-x_1)$成立, 求$a$的取值范围.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -347557,7 +347557,7 @@
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},
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},
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"014324": {
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"014324": {
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"id": "014324",
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"id": "014324",
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"content": "设$f(x)=\\begin{cases}|x+2|, & x<0, \\\\ x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$. 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围是\\bracket{20}.\n\\fourch{$(-2, \\dfrac{1}{2})$}{$(-6, \\dfrac{1}{2})$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac{1}{2},+\\infty)$}\n \n%12",
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"content": "设$f(x)=\\begin{cases}|x+2|, & x<0, \\\\ x^2-4 x+2, & x \\geq 0,\\end{cases}$ $g(x)=k x+1$. 若函数$y=f(x)-g(x)$的图像经过四个象限, 则实数$k$的取值范围是\\bracket{20}.\n\\fourch{$(-2, \\dfrac{1}{2})$}{$(-6, \\dfrac{1}{2})$}{$(-2,+\\infty)$}{$(-\\infty,-6) \\cup(\\dfrac{1}{2},+\\infty)$}",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "选择题",
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"genre": "选择题",
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@ -347899,7 +347899,7 @@
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},
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},
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"014342": {
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"014342": {
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"id": "014342",
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"id": "014342",
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"content": "如图, 已知$\\triangle ABC$的三边长$|AB|=8$, $|BC|=7$, $|AC|=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:3) node [below] {$C$} coordinate (C);\n\\draw (-120:8) node [left] {$B$} coordinate (B);\n\\draw (10:3) node [right] {$Q$} coordinate (Q);\n\\draw (190:3) node [left] {$P$} coordinate (P);\n\\draw (A) circle (3);\n\\draw (P)--(Q)--(C)--(B)--cycle (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 圆$A$的半径为$3$, 设$PQ$是圆$A$的一条直径, 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值和最小值.\n%17",
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"content": "如图, 已知$\\triangle ABC$的三边长$|AB|=8$, $|BC|=7$, $|AC|=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0) node [above] {$A$} coordinate (A);\n\\draw (-60:3) node [below] {$C$} coordinate (C);\n\\draw (-120:8) node [left] {$B$} coordinate (B);\n\\draw (10:3) node [right] {$Q$} coordinate (Q);\n\\draw (190:3) node [left] {$P$} coordinate (P);\n\\draw (A) circle (3);\n\\draw (P)--(Q)--(C)--(B)--cycle (B)--(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$;\\\\\n(2) 圆$A$的半径为$3$, 设$PQ$是圆$A$的一条直径, 求$\\overrightarrow{BP} \\cdot \\overrightarrow{CQ}$的最大值和最小值.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -348241,7 +348241,7 @@
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},
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},
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"014360": {
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"014360": {
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"id": "014360",
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"id": "014360",
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"content": "如图$1$是由矩形$ADEB$, Rt$\\triangle ABC$和菱形$BFGC$组成的一个平面图形, 其中$AB=1$, $BE=BF=2$, $\\angle FBC=60^{\\circ}$, 将其沿$AB, BC$折起使得$BE$与$BF$重合, 连结$DG$, 如图$2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (-2,0) node [below] {$E$} coordinate (E);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$A$} coordinate (A);\n\\draw (-2,1) node [above] {$D$} coordinate (D);\n\\draw (-60:2) node [below] {$F$} coordinate (F);\n\\draw (F) ++ (2,0) node [below] {$G$} coordinate (G);\n\\draw (E)--(C)(D)--(A)--(C)(D)--(E)(A)--(B)--(F)--(G)(C)--(G);\n\\draw (0.5,-2.5) node {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(120:0.5cm)}]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (B) ++ (1,{sqrt(3)},0) node [below right] {$E$($F$)} coordinate (E);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(G)--(D)--cycle (D)--(E)--(G) (E)--(B);\n\\draw [dashed] (A)--(C);\n\\draw (0.875,-0.75) node {图$2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 图$2$中的$A, C, G, D$四点共面, 且平面$ABC \\perp$平面$BCGE$;\\\\\n(2) 求图$2$中的二面角$B-CG-A$的大小.\n%18",
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"content": "如图$1$是由矩形$ADEB$, Rt$\\triangle ABC$和菱形$BFGC$组成的一个平面图形, 其中$AB=1$, $BE=BF=2$, $\\angle FBC=60^{\\circ}$, 将其沿$AB, BC$折起使得$BE$与$BF$重合, 连结$DG$, 如图$2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (-2,0) node [below] {$E$} coordinate (E);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$A$} coordinate (A);\n\\draw (-2,1) node [above] {$D$} coordinate (D);\n\\draw (-60:2) node [below] {$F$} coordinate (F);\n\\draw (F) ++ (2,0) node [below] {$G$} coordinate (G);\n\\draw (E)--(C)(D)--(A)--(C)(D)--(E)(A)--(B)--(F)--(G)(C)--(G);\n\\draw (0.5,-2.5) node {图$1$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex, z = {(120:0.5cm)}]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,0,1) node [left] {$A$} coordinate (A);\n\\draw (B) ++ (1,{sqrt(3)},0) node [below right] {$E$($F$)} coordinate (E);\n\\draw (C) ++ (1,{sqrt(3)},0) node [above] {$G$} coordinate (G);\n\\draw (A) ++ (1,{sqrt(3)},0) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(G)--(D)--cycle (D)--(E)--(G) (E)--(B);\n\\draw [dashed] (A)--(C);\n\\draw (0.875,-0.75) node {图$2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 图$2$中的$A, C, G, D$四点共面, 且平面$ABC \\perp$平面$BCGE$;\\\\\n(2) 求图$2$中的二面角$B-CG-A$的大小.",
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"objs": [],
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"objs": [],
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"tags": [],
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"tags": [],
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"genre": "解答题",
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"genre": "解答题",
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@ -383337,7 +383337,7 @@
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},
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},
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"030012": {
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"030012": {
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"id": "030012",
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"id": "030012",
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"content": "已知角$\\alpha$终边上一点$P$与$x$轴的距离和与轴的距离之比为$4:3$, 且$\\cos \\alpha <0$.求$\\sin \\alpha$和$\\tan \\alpha$.\n%",
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"content": "已知角$\\alpha$终边上一点$P$与$x$轴的距离和与轴的距离之比为$4:3$, 且$\\cos \\alpha <0$.求$\\sin \\alpha$和$\\tan \\alpha$.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第三单元"
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"第三单元"
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@ -383358,7 +383358,7 @@
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},
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},
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"030013": {
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"030013": {
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"id": "030013",
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"id": "030013",
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"content": "若函数$f(x)=\\log_2(x+1)+a$的图像经过点$(1,4)$, 则实数$a=$\\blank{50}.\n%010927",
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"content": "若函数$f(x)=\\log_2(x+1)+a$的图像经过点$(1,4)$, 则实数$a=$\\blank{50}.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第二单元"
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"第二单元"
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@ -383393,7 +383393,7 @@
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},
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},
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"030014": {
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"030014": {
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"id": "030014",
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"id": "030014",
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"content": "将函数$y=\\sqrt{3}\\sin 2x-\\cos 2x$的图像向左平移$m$($m>0$)个单位, 所得图像对应的函数为偶函数, 则$m$的最小值为\\blank{50}.\n%010931",
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"content": "将函数$y=\\sqrt{3}\\sin 2x-\\cos 2x$的图像向左平移$m$($m>0$)个单位, 所得图像对应的函数为偶函数, 则$m$的最小值为\\blank{50}.",
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"objs": [],
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"objs": [],
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"tags": [
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"tags": [
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"第三单元"
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"第三单元"
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