1 . 正方体
中,
,
分别在
上,且
,
,则下列正确的有( )个
①
,②
,③
,④点
到平面
距离为1
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/564a9f661332d97d09e82dfd6f499d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0963c102bfc6da34a4e8c4e16d503065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac240e81e4dbf3097c843b75235ec661.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40e89457e4eafe86d2c6f404f967c753.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661df691bacf67f20557fef29c1af1eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6469878a955cc09fac22ba5aea3fb962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028856d5101687dd8eaf130846489cfd.png)
A.1 | B.2 | C.3 | D.4 |
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解题方法
2 . 已知函数
若
,且
,则下列关系式一定成立的为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/353372712f9b302d01c9eff8edb1d335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c71152396d26b0c146e6a3c042bbdae9.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
3 . 在
中,角
所对的边分别为
.若
,且边
上的中线
长为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b05e85a0a24292779fa5e5e37358ee28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa9afeb51c4f3b08fea4641d3ce364b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/734281a7115f8eaf345e2587f774bbec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
A.![]() | B.![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
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解题方法
4 .
中,
,当
时,
的最小值为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ca90e8a784f990c4097eec9219908d.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6060d9a82ed5405a1ea8cd824448b6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3ede869e508a8c8bda34a16782f863.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9baf78638bf6a9798800efc59248d92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c54fccc669a643c76daccc13562e265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ca90e8a784f990c4097eec9219908d.png)
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名校
5 . 对于数列
,如果存在正整数
,当任意正整数
时均有
,则称
为
的“
项递增相伴数列”.若
可取任意的正整数,则称
为
的“无限递增相伴数列”.
(1)已知
,请写出一个数列
的“无限递增相伴数列
”,并说明理由?
(2)若
满足
,其中
是首项
的等差数列,当
为
的“无限递增相伴数列”时,求
的通项公式:
(3)已知等差数列
和正整数等比数列
满足:
,其中k是正整数,求证:存在正整数k,使得
为
的“2024项递增相伴数列”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe74b815af88e4056e62e18414a0f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b3388bf956dc7be8efe787af3f5e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c10b985b5dd226a844ada49bab1b3bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267c99ff3f6386113dbaa7b1e49612da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6d2c17b1c0e71877c295cbfe05adc63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)已知等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db7c81e956379f426859fe4b8c0bddac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
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名校
解题方法
6 . 如图,直线
与直线
,分别与抛物线
交于点A,B和点C,D(A,D在x轴同侧).当
经过T的焦点F且垂直于x轴时,
.
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
,求四边形ABCD的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16660ffd67194f17709d0b35f85ba095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25445686787e27c15ce3cbe20bbf2ea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430cd4dfec1c0932fe44320a3ef85171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a983df57f48039f3c03303a8ed2fb543.png)
(2)线段AC与BD交于点H,线段AB与CD的中点分别为M,N
①求证:M,H,N三点共线;
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782f7729242188e0a9fbb12d3984512a.png)
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7 . 已知函数
.
(1)当
时,求函数
在
处的切线方程;
(2)若函数至多一个零点,求a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c107523e11ad70647d2494e82cd5fd1.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)若函数至多一个零点,求a的取值范围.
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名校
解题方法
8 . 已知
,则下列说法中错误的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92dd8ad6dc4ac1a14dd42c4b5bf9ccb8.png)
A.![]() |
B.![]() ![]() |
C.![]() ![]() |
D.当![]() ![]() |
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解题方法
9 . 已知函数
,
.
(1)求
的极值点以及极值、最值点以及最值;
(2)设
,其中
,若存在唯一的整数
,使得
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3c82a47d1b9a0e4694643325bf3f48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f417f76e2e7eb5231d8e90fb85c5b17.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffd5dfcac1e93e91962a5efc18d43947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49394e68ffeca8ef55bfde18c7ef0d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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10 . 在2023年杭州亚运会最后两个竞技项目男女马拉松比赛中,中国选手何杰以2小时13分02秒夺得男子组冠军,这是中国队亚运史上首枚男子马拉松金牌.人类长跑运动一般分为两个阶段,第一阶段为前1小时的稳定阶段,第二阶段为疲劳阶段.小明想通过数学建模的方式研究运动员的运动时长与其剩余体力的关系.通过查找资料,小明得知:一位60kg的复健马拉松运动员进行4小时长跑训练,稳定阶段平均速度为30km/h,该阶段每千克体重消耗体力
(
表示该阶段所用时间),疲劳阶段由于体力消耗过大,在原有基础上随时间变大,速度降低,比例系数为
.同时,疲劳阶段速度降低,体力得到一定恢复,该阶段每千克体重消耗体力
,(
表示该阶段所用时间).同时,根据比赛现场的环境,其他运动员的平均配速,以及比赛策略等各方面因素,产生上下5%~10%的速度浮动,其对于运动员的体力影响也更为复杂.已知该运动员初始体力为
,请帮助小明补充完善数学建模的过程:
(1)对于数学建模,我们需要给出合理假设.
假设一:假设该运动员稳定阶段作速度为
的匀速运动;疲劳阶段做
的减速运动
假设二:_________________
(2)提出问题一:该运动员剩余体力Q关于时间t有何关系?请写出函数
;
提出问题二:该运动员在4小时内何时体力达到最低值,最低值为多少?
(3)总结运用:请根据以上计算结论,给出一定的实际建议.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f0ede408390464cffb0308ce938f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c7eb49a823f757461cd5260757b088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eff35e4e3cdc188643c46265591575c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166636b1f3567f864d7321534afed858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd84a8f95166367063218ee03ffd5a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71dba305f8b63d590f0233275eaa3f10.png)
(1)对于数学建模,我们需要给出合理假设.
假设一:假设该运动员稳定阶段作速度为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5ee1c3f44dc7187d75effa7133fa678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0f7cb632d784a4c00b291cadab83f8d.png)
假设二:_________________
(2)提出问题一:该运动员剩余体力Q关于时间t有何关系?请写出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2feeb7462a45a01b9b9530248604063e.png)
提出问题二:该运动员在4小时内何时体力达到最低值,最低值为多少?
(3)总结运用:请根据以上计算结论,给出一定的实际建议.
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