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苏锡常镇四市2012届高三教学调研测试(一)数学试题

苏锡常镇四市2012届高三教学调研测试(一)数学试题
苏锡常镇四市2012届高三教学调研测试(一)数学试题

苏锡常镇四市2012届高三教学调研测试(一)

1.已知集合{}3,2,1=A ,集合{}4,3=B ,则=B A .

2.已知复数i z 21-=(i 为虚数单位),则=2z .

3.已知命题:p 直线a ,b 相交,命题:q 直线a ,b 异面,则p ?是q 的 条件.

4.某公司为了改善职工的出行条件,随机抽取100名职工,调查了他们的居住地与公司间的距离d (单位:千米).由其数据绘制的频率分布直方图如图所示,则样本中职工居住地与公司间的距离不超过4千米的人数为 .

5.如图,给出一个算法的伪代码,已知输出值为3,则输入值=x .

Read x

If 0≥x Then

13)(2--←x x x f

Else

)5(log )(2+←x x f

End If

Print )(x f

6.已知角α(πα20<≤)的终边过点)3

2cos ,32(sin

ππP ,则=α .

7.写出一个满足1)()()(-+=y f x f xy f (x ,0>y )的函数=)(x f .

8.已知点M 与双曲线19

162

2=-y x 的左,右焦点的距离之比为3:2,则点M 的轨迹方程为 .

9.先后投掷一颗质地均匀的骰子两次,得到其向上的点数分别为m ,n ,设向量),(n m =,

5<的概率为 .

10.等差数列{}n a 中,已知158≥a ,139≤a ,则12a 的取值范围是 .

11.已知a ,b 为正实数,函数x bx ax x f 2)(3++=在[]1,0上的最大值为4,则)(x f 在[]0,1-上的最小值为 .

12.如图,已知二次函数c bx ax y ++=2(a ,b ,c 为实数,0≠a )的图象过点)2,(t C ,且与x 轴交于A ,B 两点,若BC AC ⊥,则a 的值为 .

13.设)(n u 表示正整数n 的个位数,)()(2n u n u a n -=,则数列{}n a 的前2012项和等于 .

14.将函数3322-++-=x x y ([]2,0∈x )的图象绕坐标原点逆时针旋转θ(θ为锐角),若所得曲线仍是一个函数的图象,则θ的最大值为 .

二.解答题:本大题共6小题,共90分。

15.在ABC ?中,角A ,B ,C 的对边分别为a ,b ,c ,向量)sin ,2

cos 2(C C -=,)sin 2,2

(cos

C C =,且.⊥ (1)求角C 的大小; (2)若2222c b a +=,求A tan 的值.

16.如图1所示,在ABC Rt ?中,6=AC ,3=BC ,?=∠90ABC ,CD 为ACB ∠的平分线,点E 在线段AC 上,4=CE .如图2所示,将BCD ?沿CD 折起,使得平面⊥BCD 平面ACD ,连结AB ,设点F 是AB 的中点.

(1)求证:⊥DE 平面BCD ;

(2)若//EF 平面BDG ,其中G 为直线AC 与平面BDG 的交点,求三棱锥DEG B -的体积.

17.如图,两个圆形飞轮通过皮带传动,大飞轮1O 的半径为r 2(r 为常数),小飞轮2O 的半径为r ,r O O 421=.在大飞轮的边缘上有两个点A ,B ,满足31π

=∠A BO ,在小飞轮的

边缘上有点C .设大飞轮逆时针旋转一圈,传动开始时,点B ,C 在水平直线21O O 上.

(1)求点A 到达最高点时A ,C 间的距离;

(2)求点B ,C 在传动过程中高度差的最大值.

18.如图,已知椭圆125

100:2

2=+y x E 的上顶点为A ,直线4-=y 交椭圆E 于点B ,C (点B 在点C 的左侧),点P 在椭圆E 上.

(1)若点P 的坐标为)4,6(,求四边形ABCP 的面积;

(2)若四边形ABCP 为梯形,求点P 的坐标;

(3)若BC n BA m BP ?+?=(m ,n 为实数),求n m +的最大值.

19.数列{}n a 中,11=a ,22=a .数列{}n b 满足n n n n a a b )1(1-+=+,.+∈N n

(1)若数列{}n a 是等差数列,求数列{}n b 的前6项和6S ;

(2)若数列{}n b 是公差为2的等差数列,求数列{}n a 的通项公式;

(3)若0122=--n n b b ,n n n b b 2

6212=

++,+∈N n ,求数列{}n a 的前n 2项的和.2n T

20.若斜率为k 的两条平行直线l ,m 经过曲线C 的端点或与曲线C 相切,且曲线C 上的所有点都在l ,m 之间(也可在直线l ,m 上),则把l ,m 间的距离称为曲线C 在“k 方向上的宽度”,记为).(k d

(1)若曲线)21(12:2≤≤--=x x y C ,求)1(-d ;

(2)已知2>k ,若曲线)21(:3≤≤--=x x x y C ,求关于k 的函数关系式).(k d

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