2017年东北三省三校高三第二次联合模拟考试理科数学试题及答案.doc

二模理科数学参考答案题号123456789101112答案adcbcbdaabcc13 14 15 1617（）解：当时， 2分又 4分 数列是首项为，公比为的等比数列， 6分（）， 8分所以 10分12分18（）解：第三组的频率是0.1502=0.3；第四组的频率是0.1002=0.2；第五组的频率是0.0502=0.1 3分（）由题意可知，在分层抽样的过程中第三组应抽到60.5=3个，而第三组共有1000.3=30个，所以甲乙两产品同时被选中的概率为 7分第四组共有x个产品被购买，所以x的取值为0,1,2；所以x的分布列为 10分 12分19（）证明: 连结 3分（）于是7分()如图建立直角坐标系，设平面的法向量为 9分 11分所以，直线与平面所成角的正弦值为 12分20（）设，则 4分（）设直线：，将直线代入到中得，所以5分又因为所以8分或 10分所以恒过定点 12分21 （）令 2分有两实根不妨记为极小极大所以，有两个极值点 ，一个极大值点一个极小值点 4分（），由韦达定理得 6分，所以 7分()因为，所以 8分又因为当时，不等式恒成立所以，原问题对一切恒成立法一、设（）设，当时，所以，当时，所以，所以在上单调递增，又因为所以当时， ，当时， 所以在上递减，递增，所以 10分所以当时， ，当时， 所以在上递减，递增，所以所以 12分法二不妨设，当时，所以在上单调递增，所以在上单调递增， ，所以当时成立10分当时得当时所以在上单调递减，所以在上单调递减，与条件矛盾，同理时亦如此综上 12分22 （）5分（）为切线又因为为切线 10分23（）， 5分（）设，则点到直线的距离 8分当且仅当，即（）时取等 10分24解：（）由柯西不等式得， 所以的取值范围是 5分（）同理， 7分若不等式对一切实数恒成立，则，解集为 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