山东省烟台市期末学业水平诊断2025届高三数学答案.docx

高三数学答案（第1页，共5页）

2024～2025学年度第一学期期末学业水平诊断

高三数学试题参考答案

一、选择题

DCADBCBB

二、选择题

9.BC10.ACD11.BCD

三、填空题

12.如x?2，答案不唯一）13.14.

四、解答题

15.解：因为由正弦定理得················2分

化简得ac=a2+c2?b2，由余弦定理cosB=···········4分

又因为，所以B=.···························································6分

（2）由正弦定理，所以a=3sinA，c=3sinC，又所以A+C=所以，

所以ΔABC的周长l=2+································8分

=2+4=2+4sin·····························10分

又△ABC为锐角三角形，0A，0?A，即，·········11分

所以所以3sin

所以周长.·······························································13分

16.解：（1）f(1)=1，因为点(1,1)在ax?by+1=0上，所以a?b+1=0.···········2分

又·························································3分

高三数学答案（第2页，共5页）

所以联立0，·······················································4分

解得a=1,b=2或a=?··················································6分

（2）f(x)的定义域为，············8分

令g(x)=ax2+2(a?1)x+a，x∈(0,+∞).

①当a≤0时，f′(x)0，函数f(x)在(0,+∞)上单调递减；························10分

②当a0时，Δ=4(a?1)2?4a2=4(1?2a)，

当a≥时，Δ≤0，g(x)≥0，即f′(x)≥0，f(x)在(0,+∞)上单增；·········12分

当0a时，Δ0，g(x)=ax2+2(a?1)x+a=0有正根x1,x2=1?2a.当，或x∈时，f′0，f单增；

当时，f′0，f单减.··············14分

综上，当a≤0时，函数f(x)在(0,+∞)上单调递减；

当0a时，f(x)在(0,)和(,+∞)上单调递增，在单减；当a≥在单增.··15分

17.解：（1）连接AC，取BC中点O，连接AO,OP，

在直角梯形ABCD中，由已知可得AD=3，AC=2，·································2分

又AB=BC=2，O为等边ΔABC的边BC中点，所以AO丄BC.····················3分

又因为△PBC为等边三角形，O为BC中点，所以OP丄BC.···························4分

因为AO∩OP=O，AO?平面AOP，PO?平面AOP，

所以BC丄平面AOP，·········································5分

又因为AP?平面AOP，所以AP丄BC.····························