后期复习参考资料选择
题
快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题
: 1.在中,角,,的对边分别是,,.若,则() (A) (B) (C) (D) 2.【理】设为正整数,二项式的展开式中含有的项,则的最小值为() (A) (B) (C) (D) 3.已知函数.则“”是“恒成立”的() (A)充分而不必要条件 (B)必要而不充分条件 (C)充要条件 (D)既不充分又不必要条件 4.已知函数,其中,且.若在区间上的最大值与最小值互为相反数,则() (A) (B) (C) (D) 5.设等比数列的公比为,前项和,则的取值范围是() (A) (B) (C) (D) 6.【理】设,则函数的最大值是() (A) (B) (C) (D) 7.设函数,集合,且.在直角坐标系中,集合所
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示的区域的面积为() (A) (B) (C) (D) 8.已知函数是定义在上的增函数,当时,.若,其中,则() (A) (B) (C) (D) 9.设,函数在区间上有两个不同的零点,则的取值范围是() (A) (B) (C) (D) 10.平面上的点的坐标满足,且时,称点为“有理点”.设是给定的正实数,则圆上的有理点的个数() (A)最多有个 (B)最多有个 (C)最多有个 (D)可以有无穷多个二、填空题:11.函数为偶函数的充分必要条件是______.12.若存在,使得成立,则的取值范围是______.13.已知直线经过椭圆的一个顶点和一个焦点,那么这个椭圆的离心率为______.14.设集合中元素的最大值和最小值分别为,则______.15.设数列的通项公式为.数列定义如下:对任意,是数列中不大于的项的个数,则_______;数列的前项和_______.16.已知函数,其中实数,均随机选自区间.则方程有实根的概率为______.17.已知曲线的方程是.给出下列三个结论:①曲线C关于原点对称;②曲线关于直线对称;③曲线C所围成的区域的面积大于.其中,所有正确结论的序号是_____.18.已知数列的前项和为.,,则_____;使得成立的的最小值是_____.19.在矩形中,.为矩形所在平面内一点,.则______.20.已知实数序列满足:任何连续5项之和均为负数,且任何连续9项之和均为正数,则的最大值是_____.三、解答题:21.在△中,已知,.(Ⅰ)求的值;(Ⅱ)若,求△的面积.22.如图,已知半圆及点,为半圆周上任意一点,以为一边作等边△.设,其中.(Ⅰ)将边的长表示为的函数;(Ⅱ)求四边形面积的最大值.23.【理】如图,在直四棱柱中,于,,是棱上一点.(Ⅰ)如果过,,的平面与底面交于直线,求证:;(Ⅱ)当是棱中点时,求证:;(Ⅲ)设二面角的平面角为,当时,求的长.24.如图,在四棱锥中,底面为正方形,底面,.过点的平面与棱分别交于点(三点均不在棱的端点处).(Ⅰ)求证:平面平面;(Ⅱ)求证:不可能与平面平行;(Ⅲ)若平面,试确定点位置,并
证明
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:.25.将各项均为正数的数列排成如图所示的三角形数阵,表示数阵中第行第列的数.已知数列为等比数列,且从第行开始,各行均构成公差为的等差数列,,,.(Ⅰ)求数列的通项公式;(Ⅱ)求的值;(Ⅲ)是否在该数阵中,说明理由.26.已知函数,其中.(Ⅰ)当时,求曲线在点处的切线方程;(Ⅱ)求函数的单调区间.27.已知函数,.对任意的,都有.(Ⅰ)求实数的取值范围;(Ⅱ)若,证明:.28.已知函数,其中.(Ⅰ)记的导函数为,求在内的单调区间;(Ⅱ)若在内恰有一个极大值和一个极小值,求的取值范围.29.已知集合,其中.将中所有不同值的个数记为.(Ⅰ)设集合,,求,;(Ⅱ)设集合,证明:;(Ⅲ)求的最小值.30.设是由个实数组成的有序数组,满足:①,;②;③,.(Ⅰ)当时,写出满足题设条件的全部;(Ⅱ)设,其中,求的取值集合;(Ⅲ)给定正整数,求的个数.后期复习参考资料选择题:1.B;2.C;3.B;4.D;5.D;6.C;7.B;8.C;9.D;10.B.提示:1.由,得中为锐角三角形.由正弦定理,得,即.所以,解得,从而.选B.2..令,所以时,取得最小值6.选C.3.恒成立所以必有;反之不一定,如.选B.6..令,并由,得.易知是该函数的极大值点.因为,,,所以.选C.7.因为,所以,即.因为,所以,即,所以或在直角坐标系中,集合所表示的区域如图所示,所以,其面积为,选B.8.依题意有,从而,即,所以.若,得,所以,矛盾!若,得,所以,这与是上的增函数矛盾!所以.所以,得;所以,得;所以,得.因为,且,从而,.所以,选C.9.设的两个零点分别是,则.因为,且,由均值不等式得:.选D.10.设上的任意两个相异的有理点为,其中.则有,①整理得.等式左边是有理数,右边是无理数形式,从而.将代入①,整理得,所以点关于直线对称.由点的任意性,得圆上的有理点的个数最多有个,选B.二、填空题:11.;12.;13.;14.;15.,;16.;17.①③;18.3,238;19.0;20.12.提示:12..由在上递减,得.14.由得,,当时,取得最大值;又,当时,取得最小值.所以.15.由,得,即,所以.因为,所以是公比为的等比数列,所以.16.依题意,方程有实根,从而.如图,因为实数,随机选自区间,所以正方形的面积是.满足的点构成的区域为曲边梯形,其面积为.所以方程有实根的概率为.18.,,,,,;易知数列的周期为,且一个周期之内四项之和为.所以经过个周期,共个项之和为,所以,所以最小的.19.如图,设交于点,则为的中点,连.因为.同理可得.因为,所以.因为,又,所以,从而.20.若,可构造如下的排列方式:从“行”的角度看:每行9个数之和为正,从而数表中所有数之和为正;从“列”的角度看:每列5个数之和为负,从而数表中所有数之和为负,矛盾!从而这串数字最多有12项.考虑如下一串数字:显然满足题目要求,所以的最大值是12.三、解答题:21.(Ⅰ)由,得.所以,即.因为,所以,所以.(Ⅱ)由,,得,.所以.由正弦定理得,所以.所以△的面积.22.(Ⅰ)在△中,由余弦定理得:,所以.(Ⅱ)四边形的面积.又,则当时,四边形的面积最大值.23.(Ⅰ)因为是棱柱,所以是平行四边形.所以.因为平面,平面,所以平面.因为平面平面,所以,所以.(Ⅱ)因为于,如图建立空间直角坐标系.因为,且,所以,,,.因为是棱中点,所以.设,所以,.所以.所以.(Ⅲ)设,,平面的法向量为,又因为,,所以.因为,所以,令,则,所以.设,所以,.设平面的法向量为,所以.因为,所以,令,则,所以.又因为,所以,即.解得或.所以点或.所以或.24.(Ⅰ)因为平面,所以,因为为正方形,所以,所以平面.所以平面平面.(Ⅱ)假设平面,因为,平面.所以平面.而平面,所以平面平面,这显然矛盾!所以假设不成立,即与平面不可能平行.(Ⅲ)连接,.因为平面,所以.因为为正方形,所以,因为,所以,所以是的中点.因为平面,所以,.显然△△,所以.又因为,,所以△△,所以.所以△△,所以,在△中,有,所以.25.(Ⅰ)设的公比为,记数阵中第行第列的数为.依题意,,,所以,.因为,,解得,,所以数列的通项公式为.(Ⅱ)由(Ⅰ)可得.因为,所以,即.(Ⅲ)假设存在,使得.因为第行最小的数是,最大的数是,从而有.①当时,,当时,,于是不等式①无正整数解,从而不在该数阵中.26.(Ⅰ)当时,,.由于,,所以曲线在点处的切线方程是.(Ⅱ)的定义域为,且.所以.因为,所以.①当时,恒成立,所以的单调递减区间为,无单调递增区间.②当时,令,即,解得,或,且.与的变化情况如下表: ↘ ↗ ↘所以,的单调减区间为和;单调增区间为.27.(Ⅰ)设.则,且.若,则当时,,故在上为增函数,所以,即,与题设矛盾.当时,,故在上为减函数,所以,即,符合题意.综上,实数的取值范围是.(Ⅱ)在(Ⅰ)中,令,得时,必有.设,在上面不等式中取,则有.所以所以.28.(Ⅰ),所以.令,,得,或.由,,得,或;由,,得.所以在内的单调递减区间为和;递增区间为.(Ⅱ)由(Ⅰ)得在处取得极小值,在处取得极大值.因为,,,,所以.①若,则,从而在内单调递增,所以在内无极值,显然与题设矛盾!从而.②若,当时,在内至多有一个极值点,矛盾;当时,在内至少有三个极值点,矛盾!于是.反之,当,且时,在和内各有一个极值点.所以,在内恰有一个极大值和一个极小值的充要条件是所以,的取值范围是29.(Ⅰ)由得.由得.(Ⅱ)因为共有项,所以.又集合,任取,,①当时,不妨设,则,即.②当时,.因此,当且仅当时,.即所有的值两两不同,所以.(Ⅲ)不妨设,可得,故中至少有个不同的数,即.事实上,设成等差数列,考虑,根据等差数列的性质,当时,;当时,;因此每个或等于中的一个,或等于中的一个.故对这样的,,所以的最小值为.30.(Ⅰ),,,,,共个.(Ⅱ)首先证明,且.在③中,令,得.由①得.由②得.在③中,令,得,从而.由①得.考虑,即,,此时为最大值.现交换与,使得,此时.现将逐项前移,直至.在前移过程中,显然不变,这一过程称为1次移位.继续交换与,使得,此时.现将逐项前移,直至.在前移过程中,显然不变,执行第2次移位.依此类推,每次移位的值依次递减.经过有限次移位,一定可以调整为,交替出现.注意到为奇数,所以为最小值.所以,的取值集合为.(Ⅲ)由条件①、②可知,有序数组中,有个,个.显然,从中选个,其余为的种数共有种.下面我们考虑这样的数组中有多少个不满足条件③,记该数为.如果不满足条件③,则一定存在最小的正整数,使得(ⅰ);(ⅱ).将统统改变符号,这一对应为:,从而将变为个,个组成的有序数组.反之,任何一个个,个组成的有序数组.由于多于的个数,所以一定存在最小的正整数,使得.令对应为:,从而将变为个,个组成的有序数组.因此,就是个,个组成的有序数组的个数.所以的个数是.�EMBED\*MERGEFORMAT���_1234568145.unknown_1234568401.unknown_1234568529.unknown_1234568593.unknown_1234568625.unknown_1234568657.unknown_1234568673.unknown_1234568681.unknown_1234568689.unknown_1234568697.unknown_1234568701.unknown_1234568703.unknown_1234568705.unknown_1234568706.unknown_1234568704.unknown_1234568702.unknown_1234568699.unknown_1234568700.unknown_1234568698.unknown_1234568693.unknown_1234568695.unknown_1234568696.unknown_1234568694.unknown_1234568691.unknown_1234568692.unknown_1234568690.unknown_1234568685.unknown_1234568687.unknown_1234568688.unknown_1234568686.unknown_1234568683.unknown_1234568684.unknown_1234568682.unknown_1234568677.unknown_1234568679.unknown_1234568680.unknown_1234568678.unknown_1234568675.unknown_1234568676.unknown_1234568674.unknown_1234568665.unknown_1234568669.unknown_1234568671.unknown_1234568672.unknown_1234568670.unknown_1234568667.unknown_1234568668.unknown_1234568666.unknown_1234568661.unknown_1234568663.unknown_1234568664.unknown_1234568662.unknown_1234568659.unknown_1234568660.unknown_1234568658.unknown_1234568641.unknown_1234568649.unknown_1234568653.unknown_1234568655.unknown_1234568656.unknown_1234568654.unknown_1234568651.unknown_1234568652.unknown_1234568650.unknown_1234568645.unknown_1234568647.unknown_1234568648.unknown_1234568646.unknown_1234568643.unknown_1234568644.unknown_1234568642.unknown_1234568633.unknown_1234568637.unknown_1234568639.unknown_1234568640.unknown_1234568638.unknown_1234568635.unknown_1234568636.unknown_1234568634.unknown_1234568629.unknown_1234568631.unknown_1234568632.unknown_1234568630.unknown_1234568627.unknown_1234568628.unknown_1234568626.unknown_1234568609.unknown_1234568617.unknown_1234568621.unknown_1234568623.unknown_1234568624.unknown_1234568622.unknown_1234568619.unknown_1234568620.unknown_1234568618.unknown_1234568613.unknown_1234568615.unknown_1234568616.unknown_1234568614.unknown_1234568611.unknown_1234568612.unknown_1234568610.unknown_1234568601.unknown_1234568605.unknown_1234568607.unknown_1234568608.unknown_1234568606.unknown_1234568603.unknown_1234568604.unknown_1234568602.unknown_1234568597.unknown_1234568599.unknown_1234568600.unknown_1234568598.unknown_1234568595.unknown_1234568596.unknown_1234568594.unknown_1234568561.unknown_1234568577.unknown_1234568585.unknown_1234568589.unknown_1234568591.unknown_1234568592.unknown_1234568590.unknown_1234568587.unknown_1234568588.unknown_1234568586.unknown_1234568581.unknown_1234568583.unknown_1234568584.unknown_1234568582.unknown_1234568579.unknown_1234568580.unknown_1234568578.unknown_1234568569.unknown_1234568573.unknown_1234568575.unknown_1234568576.unknown_1234568574.unknown_1234568571.unknown_1234568572.unknown_1234568570.unknown_1234568565.unknown_1234568567.unknown_1234568568.unknown_1234568566.unknown_1234568563.unknown_1234568564.unknown_1234568562.unknown_1234568545.unknown_1234568553.unknown_1234568557.unknown_1234568559.unknown_1234568560.unknown_1234568558.unknown_1234568555.unknown_1234568556.unknown_1234568554.unknown_1234568549.unknown_1234568551.unknown_1234568552.unknown_1234568550.unknown_1234568547.unknown_1234568548.unknown_1234568546.unknown_1234568537.unknown_1234568541.unknown_1234568543.unknown_1234568544.unknown_1234568542.unknown_1234568539.unknown_1234568540.unknown_1234568538.unknown_1234568533.unknown_1234568535.unknown_1234568536.unknown_1234568534.unknown_1234568531.unknown_1234568532.unknown_1234568530.unknown_1234568465.unknown_1234568497.unknown_1234568513.unknown_1234568521.unknown_1234568525.unknown_1234568527.unknown_1234568528.unknown_1234568526.unknown_1234568523.unknown_1234568524.unknown_1234568522.unknown_1234568517.unknown_1234568519.unknown_1234568520.unknown_1234568518.unknown_1234568515.unknown_1234568516.unknown_1234568514.unknown_1234568505.unknown_1234568509.unknown_1234568511.unknown_1234568512.unknown_1234568510.unknown_1234568507.unknown_1234568508.unknown_1234568506.unknown_1234568501.unknown_1234568503.unknown_1234568504.unknown_1234568502.unknown_1234568499.unknown_1234568500.unknown_1234568498.unkno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