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www7727occ:2014-2015学年湖南省郴州市高一(上)期末数学试卷(解析版)

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2014-2015 学年湖南省郴州市高一(上)期末数学试卷 一、选择题(每小题 4 分,共 40 分) 1. (4.00 分)若集合 A={x|﹣2≤x<1},B={x|0<x≤2},则 A∩B=( A.{x|﹣2≤x≤2} B.{x|﹣2≤x<0} C.{x|0<x<1} D.{x|1<x≤2} 2. (4.00 分)下列函数中,在 R 上单调递减的是( A.y=|x| B.y=log2x C.y=x D.y=( )x 的定义域为( ) ) ) 3. (4.00 分)函数 f(x)= A.[1,2)∪(2,+∞) B. (1,+∞) C.[1,2) D.[1,+∞) ,c=( )﹣1,则( ) 4. (4.00 分)设 a=2﹣1,b=log3 A.a>b>c B.b<a<c C.a<b<c D.b<c<a 5. (4.00 分)已知直线 m、n 与平面 α,β,给出下列三个命题: ①若 m∥α,n∥α,则 m∥n; ②若 m∥α,n⊥α,则 n⊥m; ③若 m⊥α,m∥β,则 α⊥β. 其中真命题的个数是( A.0 B.1 C.2 ) D.3 ) 6. (4.00 分)三视图如图的几何体的全面积是( A. B. C. D. 7. (4.00 分)已知正三棱锥的侧棱长是底面边长的 2 倍,则侧棱与底面所成角的 余弦值等于( A. B. ) C. D. 8. (4.00 分)函数 f(x)=log3x﹣8+2x 的零点一定位于区间( A. (5,6) B. (3,4) C. (2,3) D. (1,2) ) 9. (4.00 分)设点 P 是 Z 轴上一点,且点 P 到 M(1,0,2)与点 N(1,﹣3,1) 的距离相等,则点 P 的坐标是( ) A. (﹣3,﹣3,0) B. (0,0,3) C. (0,﹣3,﹣3) D. (0,0,﹣3) 10. (4.00 分)设 r>0,两圆(x﹣1)2+(y+3)2=r2 与 x2+y2=16 可能( A.相离 B.相交 ) C.内切或内含或相交 D.外切或外离 二、填空题(每小题 4 分,共 20 分) 11. (4.00 分)如果轴截面为正方形的圆柱的侧面积是 4π,那么圆柱的体积等 于 . . . 12. (4.00 分) 已知点 A (1, 2) 、 B (3, 1) , 则线段 AB 的垂直平分线的方程是 13. (4.00 分)已知函 f(x)= ,则 f(f( ) )= 14. (4.00 分)若 f(x)为奇函数,当 x<0 时,f(x)=x2+ax,且 f(3)=6,则 实数 a 的值为 15. (4.00 分)过点 . 的直线 l 将圆(x﹣2)2+y2=8 分成两段弧,当劣弧 . 所对的圆心角最小时,直线 l 的斜率 k= 三、解答题 16. (6.00 分)求经过直线 l1:3x+2y﹣5=0,l2:3x﹣2y﹣1=0 的交点且平行于直 线 2x+y﹣5=0 的直线方程. 17. (8.00 分)已知函数 f(x)= (Ⅰ)判断函数 f(x)的奇偶性 (Ⅱ)判断 f(x)在(﹣∞,+∞)上的单调性,并用定义证明. 18. (8.00 分)如图,已知等腰直角三角形 RBC,其中∠RBC=90°,RB=BC=2,点 A,D 分别是 RB,RC 的中点,现将△RAD 沿着边 AD 折起到△PAD 位置,使 PA (a>1) ⊥AB,连结 PB,PC (Ⅰ)求证:BC⊥PB (Ⅱ)求 PC 与平面 ABCD 所成角的余弦值. 19. (9.00 分)如图,有一块矩形空地,要在这块空地上辟一个内接四边形为绿 地,使其四个顶点分别落在矩形的四条边上,已知 AB=a ( a > 2 ) , BC=2 ,且 AE=AH=CF=CG,设 AE=x,绿地面积为 y. (1)写出 y 关于 x 的函数关系式,并指出这个函数的定义域. (2)当 AE 为何值时,绿地面积最大? 20. (9.00 分)已知圆方程 x2+y2﹣2x﹣4y+m=0. (1)若圆与直线 x+2y﹣4=0 相交于 M,N 两点,且 OM⊥ON(O 为坐标原点) 求 m 的值; (2)在(1)的条件下,求以 MN 为直径的圆的方程. 2014-2015 学年湖南省郴州市高一(上)期末数学试卷 参考答案与试题解析 一、选择题(每小题 4 分,共 40 分) 1. (4.00 分)若集合 A={x|﹣2≤x<1},B={x|0<x≤2},则 A∩B=( A.{x|﹣2≤x≤2} B.{x|﹣2≤x<0} C.{x|0<x<1} D.{x|1<x≤2} 【解答】解:∵A={x|﹣2≤x<1},B={x|0<x≤2}, ∴A∩B={x|0<x<1}. 故选:C. ) 2. (4.00 分)下列函数中,在 R 上单调递减的是( A.y=|x| B.y=log2x C.y=x D.y=( )x ) 【解答】解:y=|x|在(﹣∞,0]上为减函数,在[0,+∞)上为增函数,不满足 条件. y=log2x 在[0,+∞)上为增函数,不满足条件. y=x 在 R 上为增函数,不满足条件. y=( )x 在 R 上单调递减,满足条件. 故选:D. 3. (4.00 分)函数 f(x)= A.[1,2)∪(2,+∞) 【解答】解:由题意 故选:A. 的定义域为( ) B. (1,+∞) C.[1,2) D.[1,+∞) 解得 x∈[1,2)∪(2,+∝) 4. (4.00 分)设 a=2﹣1,b=log3 ,c=( )﹣1,则( ) A.a>b>c B.b<a<c C.a<b<c D.b<c<a 【解答】解:a=2 1= ,b=log3 ﹣ <0,c=( ) 1= , ﹣ 所以 b<a<c, 故选:B. 5. (4.00 分)已知直线 m、n 与平面 α,β,给出下列三个命题: ①若 m∥α,n∥α,则 m∥n; ②若 m∥

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