20.解:(1) 第r + 1项项系数为,第r项系数为,第r + 2项系数为
∴ 展开式中系数最大的项为
18.解:(1)
(2)
当在R上递增,满足题意;
当
∴ , ∴
∴ 综上,a的取值范围是.
19.解法一:
(1) 过O作OF⊥BC于F,连接O1F,
∵OO1⊥面AC,∴BC⊥O1F,
∴∠O1FO是二面角O1-BC-D的平面角,······················· 3分
∵OB = 2,∠OBF = 60°,∴OF =.
在Rt△O1OF中,tan∠O1FO =
∴∠O1FO=60° 即二面角O1-BC-D的大小为60°·············································· 6分
(2) 在△O1AC中,OE是△O1AC的中位线,∴OE∥O1C
∴OE∥O1BC,∵BC⊥面O1OF,∴面O1BC⊥面O1OF,交线O1F.
过O作OH⊥O1F于H,则OH是点O到面O1BC的距离,································· 10分
∴OH = ∴点E到面O1BC的距离等于····················································· 12分
解法二:
(1) ∵OO1⊥平面AC,
∴OO1⊥OA,OO1⊥OB,又OA⊥OB,······················· 2分
建立如图所示的空间直角坐标系(如图)
∵底面ABCD是边长为4,∠DAB = 60°的菱形,
∴OA = 2,OB = 2,
则A(2,0,0),B(0,2,0),C(-2,0,0),O1(0,0,3)·········· 3分
设平面O1BC的法向量为=(x,y,z),则⊥,⊥,
∴,则z = 2,则x=-,y = 3,
∴=(-,3,2),而平面AC的法向量=(0,0,3)··························· 5分
∴ cos<,>=,
设O1-BC-D的平面角为α, ∴cosα=∴α=60°.
故二面角O1-BC-D为60°.············································································· 6分
(2) 设点E到平面O1BC的距离为d,
∵E是O1A的中点,∴=(-,0,),············································· 9分
则d=
∴点E到面O1BC的距离等于.··································································· 12分
17.解:(1) 法一:设、两项技术指标达标的概率分别为、
由题意得: ······················································ 3分
解得:或,∴.
即,一个零件经过检测为合格品的概率为.·············································· 6分
法二:
(2) 任意抽出5个零件进行检查,其中至多3个零件是合格品的概率为
············································································· 13分
21.(本小题满分12分)
已知函数.
(1) 若函数的图象在点P(1,)处的切线的倾斜角为,求实数a的值;
(2) 设的导函数是,在 (1) 的条件下,若,求的最小值.
(3) 若存在,使,求a的取值范围.
(命题人:周 静 审题人:赵文丽)
∴ a的取值范围为
20.(本小题满分12分)
已知的展开式中,某一项的系数是它前一项系数的2倍,而等于它后一项的系数的.
(1) 求该展开式中二项式系数最大的项;
(2) 求展开式中系数最大的项.
19.(本小题满分12分)
如图,直四棱柱ABCD-A1B1C1D1的高为3,底面是边长为4且∠DAB = 60°的菱形,ACBD = O,A1C1B1D1 = O1,E是O1A的中点.
(1) 求二面角O1-BC-D的大小;
(2) 求点E到平面O1BC的距离.
18.(本小题满分13分)
已知函数.
(1) 若在x = 0处取得极值为 – 2,求a、b的值;
(2) 若在上是增函数,求实数a的取值范围.
17.(本小题满分13分)
某工厂在试验阶段大量生产一种零件.这种零件有、两项技术指标需要检测,设各项技术指标达标与否互不影响.若有且仅有一项技术指标达标的概率为,至少一项技术指标达标的概率为.按质量检验规定:两项技术指标都达标的零件为合格品.
(1) 求一个零件经过检测为合格品的概率是多少?
(2) 任意依次抽出5个零件进行检测,求其中至多3个零件是合格品的概率是多少?
16.(本小题满分13分)
设集合,若,求实数a的取值范围.
15. 将右图中编有号的五个区域染色,有五种颜色可供选择,要求有公共边的两个区域不能同色,则不同的涂色方法总数为________________(用数字作答).
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