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1、東北三省三校2015年高三第一次聯(lián)合模擬考試理科數(shù)學試卷本試卷分第卷(選擇題)和第卷(非選擇題)兩部分,共150分,考試時間120分鐘??荚嚱Y束后,將本試卷和答題卡一并交回。注意事項:1答題前,考生務必先將自己的姓名、準考證號碼填寫清楚,將條形碼準確粘貼在條形碼區(qū)域內(nèi)。2選擇題必須使用2b鉛筆填涂;非選擇題必須使用0.5毫米黑色字跡的簽字筆書寫,字體工整、筆跡清楚。3請按照題號順序在各題目的答題區(qū)域內(nèi)作答,超出答題區(qū)域書寫的答案無效;在草稿紙、試題卷上答題無效。4作圖可先使用鉛筆畫出,確定后必須用黑色字跡的簽字筆描黑。5保持卡面清潔,不要折疊,不要弄破、弄皺,不準使用涂改液、修正帶、刮紙刀。第

2、卷(選擇題 共60分)一、選擇題(本大題共12小題,每小題5分,共60分在每小題給出的四個選項中,只有一項是符合題目要求的)1、已知集合,則( )a b c d2、復數(shù)( )a b c d3、點到拋物線準線的距離為,則的值為( )a b c或 d或4、設是公差不為零的等差數(shù)列的前項和,且,若,則當最大時,( )a b c d5、執(zhí)行如圖所示的程序框圖,要使輸出的值小于,則輸入的值不能是下面的( )a bc d6、下列命題中正確命題的個數(shù)是( )對于命題,使得,則,均有是的必要不充分條件,則是的充分不必要條件命題“若,則”的逆否命題為真命題“”是“直線與直線垂直”的充要條件a個 b個 c個 d個

3、7、如圖,網(wǎng)格紙上小正方形的邊長為,若粗線畫出的是某幾何體的三視圖,則此幾何體的體積為( )a b c d8、設雙曲線的一個焦點為,虛軸的一個端點為,焦點到一條漸近線的距離為,若,則雙曲線離心率的取值范圍是( )a b c d9、不等式組表示的點集記為,不等式組表示的點集記為,在中任取一點,則的概率為( )a b c d10、設二項式()展開式的二項式系數(shù)和與各項系數(shù)和分別為,則( )a b c d11、已知數(shù)列滿足,若數(shù)列的最小項為,則的值為( )a b c d12、已知函數(shù),若函數(shù)有且只有兩個零點,則的取值范圍為( )a b c d二、填空題(本大題共4小題,每小題5分,共20分)13、向

4、量,滿足,則向量與的夾角為 14、三棱柱各頂點都在一個球面上,側(cè)棱與底面垂直,則這個球的表面積為 15、某校高一開設4門選修課,有4名同學,每人只選一門,恰有2門課程沒有同學選修,共有 種不同選課方案(用數(shù)字作答)16、已知函數(shù)()的圖象關于直線對稱,則 三、解答題(本大題共6小題,共70分解答應寫出文字說明、證明過程或演算步驟)17、(本小題滿分12分)已知的面積為,且滿足,設和的夾角為求的取值范圍;求函數(shù)的取值范圍18、(本小題滿分12分)為調(diào)查市民對汽車品牌的認可度,在秋季車展上,從有意購車的500名市民中,隨機抽樣100名市民,按年齡情況進行統(tǒng)計的頻率分布表1和頻率分布直方圖2頻率分布

5、表中的位置應填什么數(shù)?并補全頻率分布直方圖,再根據(jù)頻率分布直方圖估計這500名市民的平均年齡;在抽出的100名市民中,按分層抽樣法抽取20人參加宣傳活動,從這20人中選取2名市民擔任主要發(fā)言人,設這2名市民中“年齡低于30歲”的人數(shù)為,求的分布列及數(shù)學期望19、(本小題滿分12分)如圖,四棱錐的底面是邊長為1的正方形,底面,、分別為、的中點求證:平面;若,試問在線段上是否存在點,使得二面角的余弦值為?若存在,確定點的位置;若不存在,請說明理由20、(本小題滿分12分)已知橢圓()的左、右焦點為、,點在橢圓上,且與軸垂直求橢圓的方程;過作直線與橢圓交于另外一點,求面積的最大值21、(本小題滿分1

6、2分)已知是實常數(shù),函數(shù)若曲線在處的切線過點,求實數(shù)的值;若有兩個極值點,(),求證:;求證:請考生在第22、23、24三題中任選一題作答,如果多做,則按所做的第一題計分22、(本小題滿分10分)選修4-1:幾何證明選講如圖,在中,以為直徑的圓交于點,點是邊的中點,連接交圓于點求證:是圓的切線;求證:23、(本小題滿分10分)選修4-4:坐標系與參數(shù)方程已知曲線的極坐標方程是,以極點為平面直角坐標系的原點,極軸為軸的正半軸,建立平面直角坐標系,直線的參數(shù)方程是(為參數(shù))求曲線的直角坐標方程與直線的普通方程;設點,若直線與曲線交于,兩點,且,求實數(shù)的值24、(本小題滿分10分)選修4-5:不等式

7、選講設函數(shù)解不等式;若,使得,求實數(shù)的取值范圍東北三省三校2015年三校第一次聯(lián)合模擬考試理科數(shù)學試題參考答案1 選擇題:1.b 2.c 3.c 4.b 5.a 6.b 7.c 8.a 9.a 10.c 11.b 12.c2 填空題:13. 900 14. 64 15. 84 16. 三解答題:17.解:()設中角的對邊分別為,則由已知:, 分可得,所以: 分() 分,即當時,;當時,所以:函數(shù)的取值范圍是 分年齡(歲)0.010.020.030.040.050.060.070.080.0920253035404550頻率組距 18.解:(1)由表知:,分別填.補全頻率分布直方圖如下: 分 分

8、平均年齡估值為:(歲) 分(2)由表知:抽取的20人中,年齡低于30歲的有5人,的可能取值為0,1,2 分的分布列為012 分期望(人) 分19.證明: ()取中點, 連接, 在中, 為的中點, ,正方形中為中點, 故:為平行四邊形 分又平面,平面 平面 分() 如圖:以點為坐標原點建立空間直角坐標系: 由題易知平面的法向量為, 分假設存在滿足條件:設 ,設平面的法向量為, 分 由已知:解得: 所以:滿足條件的存在,是中點。 分20.(1)有已知:, 故橢圓方程為 分 (2)當斜率不存在時: 分 當斜率存在時:設其方程為: 由得由已知: 即: 分 到直線的距離: 分此時 綜上所求:當斜率不存在

9、或斜率為零時:面積取最大值為 分21.解(1)由已知: ,切點 分 切線方程: ,把 代入得: 分 (2)()依題意: 有兩個不等實根 設 則: 當 時: ,所以 是增函數(shù),不符合題意; 分 當 時:由得: 列表如下:0極大值 依題意: ,解得: 綜上所求: 得證; 分()由()知: 變化如下:0+0由表可知: 在 上為增函數(shù),所以: 分又 , 故 由()知: , 設 ,則 成立,所以單調(diào)遞減,故: ,也就是 綜上所證: 成立. 分22.選修4-1: 幾何證明選講證明:()連結.點是的中點,點是的中點,.,. 分在和中,即.是圓上一點,是圓的切線. 分()延長交圓于點.,.點是的中點,.是圓的

10、切線,. 分,.是圓的切線,是圓的割線, 分23.選修4-4: 坐標系與參數(shù)方程 解:()由,得:,即,曲線的直角坐標方程為. 分由,得,即,直線的普通方程為. 分()將代入,得:,整理得:,由,即,解得:.設是上述方程的兩實根,則, 分又直線過點,由上式及的幾何意義得,解得:或,都符合,因此實數(shù)的值為或或. 分24.選修4-5: 不等式選講解:()當時,即,解得,又,;當時,即,解得,又,;當時,即,解得,又,. 分綜上,不等式的解集為. 分(),. 分,使得,整理得:,解得:,因此的取值范圍是. 分9jwkffwvg#tym*jg&6a*cz7h$dq8kqqfhvzfedswsyxty#

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26、e9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngp

27、p$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqvadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn

28、&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn

29、&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrw

30、wcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amy

31、wpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amy

32、wpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%madnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84

33、9gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*adnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9we

34、wz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrr

35、wwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849g

36、xgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5u

37、xgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5u

38、xy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9c

39、pbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnugk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&k

40、sv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv

41、$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6y

42、wrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp

43、$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mzadnu#kn&muwfa5uxy7jnd

44、6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849g

45、xgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89am

46、ywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&

47、qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpb

48、k!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3

49、tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazad

50、nu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnugk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4n

51、gpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjq

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