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2018 高三数学文科 4 月月考试卷(山西大学附中有答案) 5 西大学附中 )+(x2+1,2- )= (1,- ), 所以 x1+x2= -2 ,1+2= (2- ) ………① 又,, 两式相减得 3(x1+x2)(x1-x2)+ 4(1+2)(1-2)=0………..② 以①式代入可得 AB 的斜率= 为定值; 试题分析解(I) , ∵ ,由 ,∴ 在 上单调递增。 由 ,∴ 在 上单调递减。 ∴ 的单调递减区间为 ,单调递增区间为 。 (II) , 恒成立 当 时, 取得最大值 。∴ ,∴ (III)若 的图象与 的图象恰有四个不同得交点,即 有四个不 同的根,亦即 有四个不同的根。 令, 则 当 x 变化时, 、 的变化情况如下表 x 的符号 +-+- 的单调性 由表格知 , 画出草图和验证 可知,当 时, 与 恰有四个不同的交点。 ∴当 时, 的图象与 的图象恰有四个不同的交点。 22【解析】 试题分析设点 P 的坐标为(x,),则有 消去参数 α ,可得 由于 α ∈[0,π ],∴≥0,故点 P 的轨迹是上半圆 ∵曲线 c ,即 ,即 ρ sinθ -ρ csθ =10,故曲线 c 的直角坐标方程 x-+10=0.(2)如图 所示由题意可得点 Q 在直线 x-+10=0 上,点 P 在半圆上,半圆的圆 心 c(1,0)到直线 x-+10=0 的距离等于 .即|PQ|的最小值为 -1. 23 解(Ⅰ) 2分 5分 (Ⅱ)由(Ⅰ)可知, 在 为单调增函数 且 7分 当 时, ; 当 时, ; 当 时, 综上所述 10 分 5

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