河北省石家庄市高三下学期一模考试理科数学试题及答案.doc

2015年石家庄市高中毕业班第一次模拟考试 高三数学(理科答案)一、 选择题（a卷）1-5 cbacd 6-10 badcb 11-12ba一、选择题（b卷）1-5 dbadc 6-10 bacdb 11-12ba二、 填空题13 14 815 16 三、 解答题（阅卷时发现的正确解答，请教师参阅此评分标准酌情给分）17解：（1）解法1 ，即，又数列为以1为首项，公比为的等比数列，2分，整理得，得4分，6分解法2： ，整理得，得2分 ，即，又数列为以1为首项，公比为2的等比数列，4分，6分（2）8分 得10分 整理得：12分18解：（）三个电子元件能正常工作分别记为事件，则. 依题意，集成电路e需要维修有两种情形：3个元件都不能正常工作，概率为； 2分3个元件中的2个不能正常工作，概率为 5分所以,集成电路e需要维修的概率为 6分（）设为维修集成电路的个数，则，而， 9分的分布列为：0100200 10分或. 12分19解：(1) 证明一连接交于点,在平面中做交于, 因为平面，平面 平面，-2 因为,-4 证明二在棱上取点，使得,-2连接交于点, 所以， 因为平面，平面所以平面-4 (2)取上一点使得连结,则为正方形.过作平面,垂足为.连结.，所以和都是等边三角形，因此,所以,即点为正方形对角线的交点,-7（或取的中点,连结,则为正方形.连接交于点，连接， -7）以坐标原点，分别以的方向为轴，轴,轴的正方向建立如图所示的空间直角坐标系.设棱的长为 ，则 , -9,-10-11解得t=-1220解：（1）由题意可知圆心到的距离等于到直线的距离，由抛物线的定义可知，圆心的轨迹方程：.4分（2）设，直线pb的方程为：，又圆心（1，0）到pb的距离为1，整理得：， 6分同理可得：，所以，可知是方程的两根，所以：8分依题意，即，则，因为，所以：，10分所以，当时上式取得等号，所以面积最小值为8.12分解二：（2）设，直线pb：与圆d相切，则，整理得：，6分,8分依题意那么，由韦达定理得：，则，10分所以当时上式取得等号，所以面积最小值为8.12分21. 解：（1）由，得因为在区间上单调递增，则在上恒成立，2分即在上恒成立，设，则，所以在上单调递减，故，所以4分（2）解法一：而故欲证 ，只需证6分即证成立8分设，则令得，列表如下：极小值 10分 ， 即 当时，12分解法二：对于任意两个不相等的正数、有 8分 而 10分故： ， 即 当时，12分22. 证明：（1）连结，为的直径，为的直径, , ，,为弧中点，,，3分，。5分 （2）由（1）知，,7分 ,由（1）知， 10分23.解：（1）曲线的普通方程为，2分 曲线的普通方程为. 4分 （2）法一：由曲线：，可得其参数方程为，所以点坐标为，由题意可知.因此 6分.所以当时，有最大值28，8分因此的最大值为. 10分法二：设点坐标为，则，由题意可知.因此 6分.所以当时，有最大值28，8分因此的最大值为. 10分24.解：（1）：因为函数定义域为，所以0恒成立，2分设函数，则不大于函数的最小值，又，即的最小值为4，所以.5分（2）：由（1）知，所以 6分 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