2015广东高考理科数学试卷

需证明(m?1)3?(m?1)2需证明m?1?em设g(m)?em?m?1?g?(m)?em?1令g?(m)?0,m?0?g(m)在(-?,0)上单调递减，在(0，+?)上单调递增?g(m)?g(0)?0?em?m?1?0?em?m?1命题得证.

20.（本小题满分14分）

22已知过原点的动直线l与圆C1:x+y-6x+5=0相交于不同的两点A，B.

（1）求圆C1的圆心坐标；

（2）求线段AB的中点M的轨迹C的方程；

（3）是否存在实数k，使得直线L:y=k(x-4)与曲线C只有一个交点？若存在，求出k的取值范围；若不存在，说明理由. 【解析】

（1）由题意知：圆C1方程为：(x?3)2?y2?4 ∴圆C1的圆心坐标为（3,0）

（2）由图可知，令M(x1,y1),|OM|?x12?y12,|C1M|?(x1?3)2?y12

|OC1|2?|OM|2?|C1M|2?32?x12?y12?(x1?3)2?y12 39?(x1?)2?y12?24∵直线L与圆C1交于A、B两点 ∴直线L与圆C1的距离：0?d?2

?0?(x1?3)2?y12?4?0?(x1?3)2?5??x1?3339?轨迹C的方程为：(x?)2?y2?2493?(x1?)2?442

5x?(,3]339（3）∵直线L：y?k(x?4)与曲线(x?)2?y2?仅有1个交点

24?y?k(x?4)5?联立方程：?，x?(,3] 32923(x?)?y??24?5得：(k2?1)x2?(8k2?3)x?16k2?0，在区间(,3]有且仅有1个解

342当?=(8k2?3)2-64k（k2+1）=0时，k??

3125此时，x??(,3]，仅有一个交点，符合题意。

53当??0时，令g(x)?(k2?1)x2?(8k2?3)x?16k2 5则有：g()g(3)?0

3解得：k?[?2525,] 7725254,]或k?? 773∴k的取值范围为：k?[?21.（本小题满分14分）

数列{an}满足：a1+2a2+（1）求a3的值；

+nan=4-n+2,n?N*. n-12（2）求数列{an}的前n项和Tn； （3）令b1=a1,bn=满足Sn<2+2lnn. 【解析】

（1）由题意知：a1?2a2?Tn-111+(1+++n231证明：数列{an}的前n项和Sn+)an(n?2)，

n?nan?4?n?2 2n?1

2?2213?2当n=3时，a1?2a2+3a3=4?22

3?22?233a3=4?2?(4?1)?2241a3=4当n=2时，a1?2a2=4?（2）

a1?2a2??a1?2a2?(n+1)an?1??nan?4?n?22n?1n?3?nan+(n+1)an?1?4?n2

n?2n?3?n2n?122n?4?n?3?

2nn?1?n21?an?1?()n2 1?an?()n?12∴{an}是首相为1，公比为

1的等边数列 211?()n2?2?(1)n?1?2?1 ∴Tn?n?11221?2（3）由（2）得：Tn?2?1 n?12111?Sn?(2?n?1)(1??)

22n111已知不等式：??ln(1?n)

23nx设f(x)?ln(1?x)?,x?0

1?x

x2?f?(x)??0,f(x)在(0,+?)单调递增1?xx?f(x)?ln(1?x)??f(0)?01?xx?ln(1?x)?在(0,+?)上恒成立1?x1令x=,n 1ln(1?)?ln(1?n)?lnn?lnn?ln(n?1)??ln2?ln1?lnnn1111ln(1?)????n23n?1111?lnn????23n?111111?Sn?(2?n?1)(1??)?2(1??)?2(1?ln2)?2?2ln222n2n

To conquer our fears, we must go past them. To fear is natural; to go past them is heroic. Fear is a normal part of life, symbolizing that there are new, extraordinary things to come across and face. To be able to see

these things before they happen is a sign of wisdom. Fear is nothing out of the ordinary. To admit that we are fearing is to prove that we value our life. And men should pay attention to the call of fears. To truly be a leader or someone who is heroic and brave, we should focus on getting past these fears. And it is frightful for people to go through life ignoring fears, because they may often forget their heart and step into life without responsibility.