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2018-2018 学年高二年级 6 月份数学月考试题(文科)答案 一、选择题 D.B.B.C.B 二、填空题 13. 2 14. 能 15.0.32,36,0.18 16. ①④ 9 1 7 21. 解:设容器的高为 x ,容器的体积为 V, 则 V ? x(90 ? 2x)(48 ? 2x) , 0 ? x ? 24 , 即 V ? 4 x3 ? 276 x2 ? 4320 x , C.B.B.B.C D.B ∴ V ? ? 12x2 ? 552x ? 4320 ? 12( x ?10)( x ? 36) , x f '( x) f ( x) (0,10) + 10 0 极大值 (10,24) - 17.(1)每个工厂在某一天停电的概率为 P= ( )5 ? 1 7 1 16807 ∴当 x=10,V 有极大值 V(10)=1960 ,∵定义域端点取不到, ∴极大值就是最大 值, ∴该容器的高为 10cm 时,容器的容积最大,最大容积是 1960. 22.解:(I) f '( x) =3 x2 -2 x -1 若 f '( x) =0,则 x ==- , x =1 当 x 变化时, f '( x) , f ( x) 变化情况如下表: (-∞,- x 1 ) 3 1 3 (没有算出最后结果,也可以) ………………6 分 (2)至少有两个工厂选择同一天停电的概率 5 A7 14287 P ? 1? 5 ? ……………………………………………………12 分 7 16807 18.解:设全班男生 x 名,则女生(36-x)名,选的 2 名都是男生的概率为 2 2 Cx x( x ? 1) C36 (36 ? x)(35 ? x) ?x ? ? 2 2 C36 36 ? 35 选的的 2 名都是女生的概率为 C36 36 ? 35 - 0 x( x ? 1) (36 ? x)(35 ? x) 1 ? 依题意的 36 ? 35 + 36 ? 35 2 解得 x=15 或 x=21 即男生 15 名,女生 21 名;或男生 21 名,女生 15 名,相差 6 名 1 n! 1 1 ( n ? k ? 1)(n ? k ? 2) ? ? ? k k n k !(n ? k )! n k! nk 19.证明: (1) 1 n ? k ?1 n ? k ? 2 1 1 ? ? ? ? ? k! n n n k! k Cn ? 1 2 ? Cn (2) (1 ? )n ? Cn0 ? Cn 1 3 (- , 1) - 1 3 1 0 极小 值 (1,+ ∞) + f '( x) + f ( x) 极大值 1 3 2 ?1 ……………6 分 ∴ f ( x) 的极大值是 f (? ) ? 5 ? a ,极小值是 f (1) ? a ? 1 27 (II)函数 f ( x) ? x3 ? x2 ? x ? a ? ( x ? 1)2 ( x ? 1) ? a ? 1 由此可知,取足够大的正数时,有 f ( x) >0,取足够小的负数时有 f ( x) <0, 1 1 1 n 1 ? ? Cn 2 n n n nn 1 1 1 1 1 1 1 ? ? ? ? ? ? 1? ? ? ? 1! 2! 3! n 1? 2 2 ? 3 3 ? 4 1 1 1 1 1 1 ? 1?1? ? ? ? ? ? ? 2? ? 3 2 2 3 n ?1 n n ? 1 n ? (n ? 1) 所以曲线 y = f ( x) 与 x 轴至少有一个交点 结合 f ( x) 的单调性可知: 当 f ( x) 的极大值 5 5 ? a <0,即 a ? (??, ? ) 时,它的极小值也小于 0,因此 27 27 曲线 y = f ( x) 与 x 轴仅有一个交点,它在(1,+∞)上。 当 f ( x) 的极小值 a -1>0 即 a ? (1,+∞)时,它的极大值也大于 0,因此曲 线 y = f ( x) 与 x 轴仅有一个交点,它在(-∞,- )上。 ∴当 a ? (??, ? 5 ) ∪(1,+∞)时,曲线 y = f ( x) 与 x 轴仅有一个交点。 27 1 3

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