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[
{
"type": "text",
"text": "024年普通高等学校招生全国统一考试",
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"text": "数学",
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{
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"text": "本试卷共4页22小题满分150分。考试用时120分钟。",
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{
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"text": "意事项1答题前请务必将自已的姓名、准考证号用黑色字迹的签字笔或钢笔分别填写在试题卷和答题卡上。用2B铅笔将试卷类型A填涂在答题卡相应位置上。将条形码横贴在答题卡右上角“条形码粘贴处”。2.作答选择题时选出每小题答案后用2B铅笔把答题卡上对应题目选项的答案信息点涂黑如需改动用橡皮擦干净后再选涂其他答案答案不能答在试卷上。3.非选择题必须用黑色字迹钢笔或签字笔作答答案必须写在答题卡各题目指定区域内相应位置上如需改动先划掉原来的答案然后再写上新的答案不准使用铅笔和涂改液。不按以上要求作答的答案无效。4.考生必须保持答题卡的整洁。考试结束后,将试卷和答题卡一并交回。",
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{
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"text": "一、选择题本大题共8小题每小题5分共计40分.每小题给出的四个选项中,只有一个选项是正确的,请把正确的选项填涂在答题卡相应的位置上.",
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{
"type": "text",
"text": "1.已知集合A={x|5<x<5}B={3,1,0,2,3}则A∩B= ",
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},
{
"type": "text",
"text": "A.(-1.0) B.(2.3} C.(-3,-1.0) D.(-1,0,2 2.三 =1+i则z= ",
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},
{
"type": "text",
"text": "A.-1-i B.-1+i C.1-i D.1+i3.已知向量a=(0,1)b=(2,x)若b⊥(b4a)则x=",
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},
{
"type": "text",
"text": "A.-2 B.-1 C.1 D.2 4.已知cos(α+β)=mtanαtanβ=2则cos(α−β)= ",
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},
{
"type": "text",
"text": "A.-3m B.\\a\\$ c. m D.3m ,已知圆柱和圆锥的底面半径相等,侧面积相等,且它们的高均为√,则圆锥的体积为 ",
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},
{
"type": "text",
"text": "",
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},
{
"type": "text",
"text": "A.2√\\}\\$ \\$B.3√\\n}\\$ \\$C.6\\\\$ D.9\\\\$ ",
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},
{
"type": "text",
"text": "6.已知函数f(x)= -x-2ax-a,x<0₂在R上单调递增则a的取值范围是le+ln(x+1).x≥0",
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},
{
"type": "text",
"text": "A.(-20,0] B. [-1.0] C. [-1.1] D.[0,+) ",
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},
{
"type": "text",
"text": "7.当xe[0.2n]时曲线y=sinx与y=2sin(3x-)的交点个数为 ",
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},
{
"type": "text",
"text": "A.3 B.4 C.6 D.8 ",
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},
{
"type": "text",
"text": "8.已知函数f(x)的定义域为Rf(x)>f(x1)+f(x2)且当x<3时f(x)=x则下列结论中一定正确的是A.f(10)>100 B.f(20)>1000 C.f(10)<1000D.f(20)<10000二、选择题本大题共3小题每小题6分共计18分.每小题给出的四个选项中有多项符合题目要求。全部选对得6分选对但不全的得部分分有选错的得0分。",
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{
"type": "text",
"text": "",
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},
{
"type": "text",
"text": "9.为了解推动出口后的亩收入单位万元情况从该种植区抽取样本得到推动出口后亩收入的样本均值2.1,样本方差=0.01已知该种植区以往的亩收入X服从正态分布N(1.8,0.1²)假设推动出口后的亩收入Y服从正态分布(x,)若随机变量Z服从正态分布N(μσ)则P(Z<+)≈0.8413",
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},
{
"type": "text",
"text": "A.P(X>2)>0.2 B.P(X>2)<0.5 \nC.P(Y>2)>0.5 D.P(Y>2)<0.8",
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},
{
"type": "text",
"text": "10.设函数f(x)=(x1)(x4),则",
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{
"type": "text",
"text": "A.x=3是f(x)的极小值点 B.当0<x<1时f(x)<f(x)C.当1<x<2时-4<f(2x-1)<0 D.当-1<x<0时f(2x)>f(x)",
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},
{
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"text": "11.造型可以做成关丽的丝带将其看作图中曲线C的一部分.已知C过坐标原点O且C上的点满足横坐标大于-2到点F(2,0)的距离与到定直线x=a(a<0)的距离之积为4则",
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},
{
"type": "image",
"img_path": "images/59086cc384346487ce511d49671d1292dc0c811a3ee51527fda22ad5320566cc.jpg",
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{
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"text": "A.a=-2 \nB.点(2√2,0)在C上 \nC.C在第一象限的点的纵坐标的最大值为1 \nD.当点(x3)在C上时≤x+2",
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{
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"text": "三、填空题本大题共3小题每小题5分共计15分",
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{
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"text": "12.设双曲线C =>0,b>0)的左右焦点分别为FF2过作平行于y轴的直 }\\$ 线交C于A、B两点若|FA=13|AB|=10则C的离心率为_ 13.若曲线在点(0,1)处的切线也是曲线y=ln(x+1)+a的切线则a=_ ",
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},
{
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"text": "14.甲乙两人各有四张卡片每张卡片上标有一个数字假的卡片上分别标有数字1357乙的卡片上分别标有数字2468.两人进行四轮比赛在每轮比赛中两人各自从自己持有的卡片中随机选一张并比较所选卡片上的数字大小数字大的人得1分数字小的人得0分然后各白弃置此轮所选的卡片弃置的卡片在此后的轮次中不能使用则四轮比赛后甲的总得分不小于2的概率为",
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{
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"text": "四、解答题本题共6小题共70分。解答应写出文字说明、证明过程或演算步骤。15.13分",
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},
{
"type": "text",
"text": "记△ABC的内角ABC的对边分别为abc已知sinC=√coB。",
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},
{
"type": "text",
"text": "^{+b²-c{=√2ab. ",
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},
{
"type": "text",
"text": "1求B \n2若△ABC的面积为3+√求c.",
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},
{
"type": "text",
"text": "16.15分",
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},
{
"type": "text",
"text": "=1(a>b>0)上两点.\\$b{r}\\$ \n1求C的离心率 \n2若过P的直线I交C于另一点B且△ABP的面积为9求I的方程.",
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},
{
"type": "text",
"text": "17.15分",
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},
{
"type": "text",
"text": "如图四棱锥P-ABCD中PA⊥底面ABCDPA=AC=2BC=1AB=√.1若AD⊥PB证明AD//平面PBC(2)若AD⊥DC且二面角A-CP-D的正弦值为√2\\$7求AD.",
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},
{
"type": "image",
"img_path": "images/68816d467f0086c289a436af04fad25fde27512b22554b1094715129ecce776e.jpg",
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{
"type": "text",
"text": "18.17分",
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},
{
"type": "text",
"text": "已知函数f(x)=lnx +αx+b(x1).2-x \n1若b=0且f(x)≥0求a的最小值 \n2证明曲线y=f(x)是中心对称图形; \n3若f(x)>2当且仅当1<x<2求b的取值范围.",
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},
{
"type": "text",
"text": "19.17分",
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},
{
"type": "text",
"text": "设m为正整数数列aq2\"2是公差不为0的等差数列若从中删去两项a 和a(i<j)后剩余的4m项可被平均分为m组且每组的4个数都能构成等差数列则称数 列aq2dm2是(i,j)—可分数列. ",
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},
{
"type": "text",
"text": "(1)写出所有的(i)1≤i<j≤6使得数列aq2\"a是(i,j)一可分数列2当m≥3时证明数列aqq2m2是(2,13)—可分数列:(3从124m+2中一次任取两个数i和j(i<)记数列aq2q2是(,)可分数列的概率为P证明P_",
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}
]