2023-2024学年湖北省云学名校新高考联盟高一（下）联考数学试卷（5月份）（含答案）.doc

第1页共8页20232024学年湖北省云学名校新高考联盟高一下联考数学试卷5月份一单选题本题共8小题每小题5分共40分在每小题给出的选项中只有一项是符合题目要求的已知集合119860119909311986011986113211986011986132119861119860132119860119861119860119861119862中角119860119861119862所对的边分别为119886119887119888已知1198872119861120587611988811990011990411986035则11988625285281562156已知两条不同的直线119898119899两个不同的平面120572120573则下列说法正确的是若120572120573119898120572119899120573则119898119899若119898120572119899119898则119899120572若120572120573120572120573119899119899119898则119898120573若120572120573119899119898120572119898120573则119898119899在平面直角坐标系中119874为坐标原点1198722311987331则1198881199001199042119872119874119873566536516653365已知11990901199100且41199091199101则1199092119910119909119910的最小值为624246已知向量1198981119904119894119899120572119899119888119900119904120572212058721205721205872则下列命题中不正确的是存在120572使得119898119899当1198981198993时1199051198861198991205722当119898与119899垂直时11990511988611989912057222119898与119899可能平行若1198911199091141199091对任意实数119886119887则1198861198870是1198911198861198911198871成立的充分且必要条件充分不必要条件必要不充分条件既不充分也不必要条件第2页共8页若函数119891119909119886011990911989911988611199091198991119886119899111990911988611989911988600有119899个不同的零点11990911199092119909119899则119891119909119886011990911990911199091199092119909119909119899已知11989211990911990911990942存在实数119886119887119888119905满足1198921198861198921198871198921198881199051198868816与实数119905有关二多选题本题共3小题共18分在每小题给出的选项中有多项符合题目要求设11991111199112是复数则下列说法正确的是若1199111为纯虚数则11991112若11991111199112则1199111211991122若11991111199112则11991111199112若11991111199112119877则11991111199112下列说法正确的非零向量119886119887若119886与119887共线则119886119886119887119887非零向量119886119887满足119886119887119886119887则119886119887在119860119861119862中若1198601198611198601198611198601198621198601198621198611198620且11986011986111986011986111986011986211986011986212则119860119861119862为等边三角形已知单位向量119886119887119888满足2119886311988741198880则11988611988714在长方体1198601198611198621198631198601119861111986211198631中119860119860141198601198611198611198622动点119873在线段1198601119862上不含端点119872在线段119860119861上则存在点119873使得119861119873平面11986011198601198631198631存在点119873使得119860119862119861119873119873119861119873119863的最小值为2303119872119873的最小值为455三填空题本题共3小题每小题5分共15分已知119894是虚数单位复数1199114311989432119894则复数119911的虚部为文翁千载一时珍醉卧襟花听暗吟表达了对李时珍学识渊博才华横溢的赞叹李时珍是湖北省蕲春县人明代著名医药学家他历经27个寒暑三易其稿完成了192万字的巨著本草纲目被后世尊为药圣为纪念李时珍人们在美丽的蕲春县独山修建了一座雕像如图所示某数学学习小组为测第3页共8页量雕像的高度在地面上选取共线的三点119860119861119862分别测得雕像顶的仰角为604530且11986011986111986111986267610米则雕像高为米用平行于棱锥底面的平面去截棱锥把底面和截面之间的那部分多面体叫做棱台在正三棱台119860119861119862119860111986111198621中侧棱119860119860141198601198616119860111986112则侧棱1198601198601与底面119860119861119862所成角的正弦值为该三棱台的体积为四解答题本题共5小题共77分解答应写出文字说明证明过程或演算步骤本小题13分已知向量11988621198881199001199041199091199041198941198991199091198881199001199041199091198873119904119894119899119909119904119894119899119909119888119900119904119909且函数119891119909119886119887119898在119909119877时的最大值为231求常数119898的值2当1199090120587时求函数119891119909的单调递增区间本小题15分已知复数11991132119894119894为虚数单位1求11991111198942若1199115119894119903119888119900119904120579119894119904119894119899120579其中119903012057902120587求119903120579的值3若119911213且1199111199112是纯虚数求1199112本小题15分如图所示圆内接四边形119860119861119862119863中119860119861311986011986323119862为圆周上一动点11986111986211986312058731求四边形119860119861119862119863周长的最大值2若11986111986211986211986312求119860119862的长本小题17分在四棱锥119875119860119861119862119863中平面119875119860119863平面119860119861119862119863119864为119860119863边上一点119865为119875119861中点1198611198621198601198631198601198611198611198621198751198602119860119863411986011986411198751198632311986011986111986212058731求四棱锥119875119860119861119862119863的体积第4页共8页2证明119860119865平面1198751198621198643证明平面119875119860119861平面119875119861119862本小题17分如图设119874119909119874119910是平面内相交成1205720面坐标系119909119874119910为120572仿射坐标系在120572仿射坐标系中若11987411987511990911989011199101198902则记1198741198751199091199101在120572仿射坐标系中若119886119898119899求119886若1198861311988731且119886与119887的夹角为1205873求1198881199001199041205722如上图所示在1205873仿射坐标系中119861119862分别在119909轴119910轴正半轴上1198611198621119874119863719119874119862119864119865分别为119861119863119861119862中点求119874119864119874119865的最大值第5页共8页答案11986311986111986311986211986311986311986011986011986011986211986111986211986111986211986310201635223解1由题意11988611988723119904119894119899119909119888119900119904119909sin2119909cos2119909311990411989411989921199091198881199001199042119909211990411989411989921199091205876又119891119909211990411989411989921199091205876119898119909119877的最大值为23所以119891119909119898119886119909211989823解得11989832由1得1198911199092119904119894119899211990912058763令120587221198961205872119909120587612058722119896120587119896119885解得12058761198961205871199091205873119896120587119896119885又1199090120587故119891119909的单调递增区间为01205873和51205876120587解1依题意11991111198942119894所以1199111119894221252119911511989432119894511989432119894511989451198945119894131311989426121211989422222211989422cos7120587411989411990411989411989971205874第6页共8页所以11990322120579712058743设1199112119886119887119894119886119887119877则11991121198862119887213即11988621198872131199111199112321198941198861198871198943119886211988731198872119886119894由1199111199112是纯虚数则有311988621198870311988721198860由1198862119887213311988621198870311988721198860解得11988621198873或11988621198873所以119911223119894或1199112231198941解连接119861119863因为1198611198621198631205873所以11986311986011986121205873在119860119861119863中由余弦定理得119861119863211986011986121198601198632211986011986111986011986311988811990011990421205873可得11986111986321在119861119862119863中由余弦定理得119861119863211986111986221198621198632211986111986211986211986311988811990011990412058731198611198622119862119863211986111986211986211986311986111986211986211986323119861119862119862119863所以1198611198621198621198632213119861119862119862119863因为11986111986211986211986311986111986211986211986322当且仅当11986111986211986211986321时等号成立所以119861119862119862119863221341198611198621198621198632119861119862119862119863221所以周长的最大值为221332解依题意得11986111986211986211986312设1198611198621199091198621198632119909在119861119862119863中由余弦定理得11986111986321198611198622119862119863221198611198621198621198631198881199001199041205873可得11986111986321所以2121199092211990921199092119909解得11990927所以119861119862711986211986327可得119862119863211986111986221198611198632所以1198631198611198621205872在119860119861119863中由正弦定理119861119863sin21205873119860119863sin119860119861119863所以sin119860119861119863119860119863sin21205873119861119863217则cos119860119861119862cos1198601198611198631205872sin119860119861119863217第7页共8页在119860119861119862中由余弦定理得1198601198622119860119861211986111986222119860119861119861119862cos11986011986111986216所以1198601198624解11198751198602119875119863231198601198634119860119863211987511986021198751198632119875119860119875119863且1198751198601198631205873又119875119860211986011986411198751198601198631205873由余弦定理得1198751198644122123119875119860211986011986421198751198642119875119864119860119863又平面119875119860119863平面119860119861119862119863平面119875119860119863平面119860119861119862119863119860119863119875119864平面119875119860119863119875119864平面119860119861119862119863连接119860119862119860119861119861119862211986011986111986212058731198601198622119860119861119862为等边三角形119861119862119860119863119860119861119862120587311986311986011986121205873119863119860119862120587311986011986221198621198634119860119862119862119863119860119862119863为直角三角形119878四边形1198601198611198621198631198781198601198611198621198781198601198621198633422122233311988111987511986011986111986211986313119875119864119878四边形1198601198611198621198631333332取119875119862中点119872119865为119875119861中点119865119872为119875119861119862中位线119865119872119861119862且119865119872121198611198621又119860119864119861119862且119860119864121198611198621119865119872119860119864且119865119872119860119864四边形119860119865119872119864为平行四边形119860119865119872119864又119872119864平面119875119862119864119860119865平面119875119862119864119860119865平面1198751198621198643由2得四边形119860119865119872119864为平行四边形1198621198641198751198643119872为119875119862的中点119872119864119875119862又119860119865119872119864第8页共8页119860119865119875119862在119875119860119861中119860119861119875119860119865为119875119861中点119860119865119875119861119875119861平面119875119861119862119875119862平面119875119861119862119875119861119875119862119875119860119865平面119875119861119862又119860119865平面119875119860119861平面119875119860119861平面119875119861119862解1因为11988611989811989911988611989811989011198991198902所以1198862119898119890111989911989022119898211989012211989811989911989011198902119899211989022119898221198981198991198881199001199041205721198992所以119886119898221198981198991198881199001199041205721198992因为1198861311988731所以119886122131198881199001199041205723210611988811990011990412057211988732231119888119900119904120572121061198881199001199041205721198861198871198901311989023119890111989023119890123119890221011989011198902610119888119900119904120572因为119886与119887的夹角为1205873所以cos120587311988611988711988611988761011988811990011990412057210611988811990011990412057212解得119888119900119904120572172依题意设1198611198980119862011989911989801198990则119861119874119862120587311986111986211198741198637191198741198620719119899因为119865为119861119862中点所以1198741198651211987411986212119874119861121198981198901121198991198902因为119864为119861119863中点所以11987411986412119874119863121198741198611211989811989017381198991198902所以1198741198641198741198651211989811989011211989911989021211989811989017381198991198902141198982119890127761198992119890227761198981198991411989811989911989011198902因为119890121198902211198901119890211cos120587312则1198741198641198741198651411989827761198992776119898119899141198981198991214119898277611989921376119898119899在119874119861119862中依据余弦定理得119898211989921198981198991所以119898119899119898211989921代入上式得11987411986411987411986551911989828191198992137611951198982811989921376设11989821198992111989811989911990511989821411990511989921199050则1119905119898211411990511989921令111990511411990558得3211990521211990550解得119905158119905214舍所以38119898235119899215119898281198992403则119874119864119874119865119511989828119899213761194031376121228