2008高数 天津理

2008年普通高等学校招生全国统一考试理科数学（天津卷）学校:___________姓名：___________班级：___________考号：___________一、单选题1．i 是虚数单位，()=-+113i i i (A) 1- (B) 1 (C) i - (D) i2．设变量满足约束条件，则目标函数的最大值为A ．2B ．3C ．4D ．53．设函数()R x x x f ∈⎪⎭⎫⎝⎛-=,22sin π，则()x f 是 (A) 最小正周期为π的奇函数 (B) 最小正周期为π的偶函数(C) 最小正周期为2π的奇函数 (D) 最小正周期为2π的偶函数 4．设是两条直线，是两个平面，则的一个充分条件是 A ． B ． C ．D ．5．设椭圆上一点P 到其左焦点的距离为3，到右焦点的距离为1，则P 点到右准线的距离为 A ．6B ．2C ．D ．6．设集合，则的取值范围是A ．B ．C ．或D ．或7．设函数的反函数为，则A ．在其定义域上是增函数且最大值为1B ．在其定义域上是减函数且最小值为0C ．在其定义域上是减函数且最大值为1D ．在其定义域上是增函数且最小值为08．已知函数，则不等式的解集是A ．B ．C ．D ．9．已知函数是R 上的偶函数，且在区间上是增函数.令，则A ．B ．C ．D ．10．有8张卡片分别标有数字1，2，3，4，5，6，7，8，从中取出6张卡片排成3行2列，要求3行中仅有中间行的两张卡片上的数字之和为5，则不同的排法共有 A ．1344种 B ．1248种 C ．1056种 D ．960种二、填空题 11．的二项展开式中，的系数是 （用数字作答）.12．一个正方体的各顶点均在同一球的球面上，若该球的体积为，则该正方体的表面积为 .13．已知圆C 的圆心与抛物线x y 42=的焦点关于直线x y =对称，直线0234=--y x 与圆C 相交于B A ,两点，且6=AB ，则圆C 的方程为 .14．如图，在平行四边形ABCD 中，()()2,3,2,1-==BD AC ，则=⋅AC AD .15．已知数列{}n a 中，()*31,1111N n a a a n n n ∈=-=++，则=∞→nn a lim .16．设，若仅有一个常数c使得对于任意的，都有满足方程，这时，的取值的集合为.三、解答题17．（本小题满分12分）已知.（Ⅰ）求的值；（Ⅱ）求的值.18．甲、乙两个篮球运动员互不影响地在同一位置投球，命中率分别为与，且乙投球2次均未命中的概率为.（Ⅰ）求乙投球的命中率；（Ⅱ）若甲投球1次，乙投球2次，两人共命中的次数记为，求的分布列和数学期望. 19．如图，在四棱锥中，底面是矩形.已知.（Ⅰ）证明平面；（Ⅱ）求异面直线与所成的角的大小；（Ⅲ）求二面角的大小.20．已知函数，其中.（Ⅰ）若曲线在点处的切线方程为，求函数的解析式；（Ⅱ）讨论函数的单调性；（Ⅲ）若对于任意的，不等式在上恒成立，求的取值范围. 21．已知中心在原点的双曲线C的一个焦点是，一条渐近线的方程是.（Ⅰ）求双曲线C的方程；（Ⅱ）若以为斜率的直线与双曲线C相交于两个不同的点M，N，线段MN的垂直平分线与两坐标轴围成的三角形的面积为，求的取值范围.22．在数列与中，，数列的前项和满足,为与的等比中项，.（Ⅰ）求的值；（Ⅱ）求数列与的通项公式；（Ⅲ）设.证明.参考答案【答案】A【解析】()31(1)11111i i i i ii i i +-+-===----，选A ． 2．D 【解析】如图，由图象可知目标函数过点(1,0)A 时z 取得最大值，max 5z =，选D ．3．B【解析】()cos 2f x x =-是周期为π的偶函数，选B ． 4．C 【解析】A 、B 、D 直线,a b 可能平行，选C ． 5．B 【解析】由椭圆第一定义知2a =，所以24m =，椭圆方程为22111432x y e d +=⇒==所以2d =，选B ． 6．A 【解析】{|15}S x x x =-或，所以1{3185a a a <-⇒-<<-+>，选A ． 7．D【解析】1y =为减函数，由复合函数单调性知()f x 为增函数，所以()1f x -单调递增，排除B 、C ；又()1f x -的值域为()f x 的定义域，所以()1f x -最小值为0．视频 8．C 【解析】 依题意得1010{{(1)()1(1)1x x x x x x x x +<+≥++-≤++≤或所以11{{111111x x x x x x Rx ≥-<-⇒<--≤≤⇒≤∈≤≤或或，选C ．9．A【解析】52(cos)(cos )77b f f ππ=-=，52(tan )(tan )77c f f ππ=-= 因为2472πππ<<，所以2220cos sin 1tan 777πππ<<<<，所以b a c <<，选A ．10．B【解析】首先确定中间行的数字只能为1，4或2，3，共有12224C A =种排法.然后确定其余4个数字的排法数.用总数46360A =去掉不合题意的情况数：中间行数字和为5，还有一行数字和为5，有4种排法，余下两个数字有2412A =种排法.所以此时余下的这4个数字共有360412312-⨯=种方法．由乘法原理可知共有43121248⨯=种不同的排法，选B ．视频 11．40 【解析】3552155((2)r r rr r rr T C xC x --+==-，所以2r，系数为225(2)40C -=.12．24 【解析】试题分析：设正方体的外接球的半径为R ,由：343R π=，解得：R 设该正方体的边长为a ，根据223412a R ==解得2a =，所以正方体的表面积为：266424a =⨯=，所以答案为24.考点：1.求的体积公式；2.正方体的外接球；3.球的表面积和体积公式.13．22(1)10x y +-=【解析】抛物线的焦点为(1,0)，所以圆心坐标为(0,1)，2222(032)3105r --=+=，圆C 的方程为22(1)10x y +-=. 14．3【解析】令AB a =，AD b =，则(1,2)(2,0),(1,2)(3,2)a b a b a b ⎧+=⎪⇒==-⎨-+=-⎪⎩ 所以()3AD AC b a b ⋅=⋅+=. 15．76【解析】22111211111()13())33(n n n n n n n a a a a a a a a ----+-+=+++=-++++所以2173lim 11613n n a →∞=+=-.16．{}2 【解析】由已知得ca y x=，单调递减，所以当[,2]x a a ∈时，所以1122log 2{{23c a c a a aa a a --≥+≥⇒≤≤，因为有且只有一个常数c 符合题意，所以2log 23a +=，解得2a =，所以的取值的集合为{}2.17．（Ⅰ）4sin 5x =（Ⅱ）sin 23x π⎛⎫+= ⎪⎝⎭【解析】（Ⅰ）因为，所以，于是（Ⅱ）因为，故所以18．（Ⅰ）（Ⅱ）的分布列为的数学期望2E ξ= 【详解】试题分析：对于问题（I ）由题目条件并结合间接法，即可求出乙投球的命中率p ；对于问题（II ），首先列出两人共命中的次数ξ的所有可能的取值情况，再根据题目条件分别求出ξ取各个值时所对应的概率，就可得到ξ的分布列．试题解析：（I ）设“甲投球一次命中”为事件A ，“乙投球一次命中”为事件B . 由题意得221(1())(1)16P B p -=-=解得34p =或54（舍去），所以乙投球的命中率为34.（II ）由题设知（I ）知1()2P A =，1()2P A =，3()4P B =，1()4P B =， ξ可能取值为0,1,2,3故2111(0)()()()2432P P A P B B ξ==⋅=⨯=， 12(1)()()()()()P P A P B B C P B P B P A ξ==⋅+⋅⋅2113117()22444232=⨯+⨯⨯⨯=， 2139(3)()()()2432P P A P B B ξ==⋅=⨯=15(2)1(0)(1)(3)32P P P P ξξξξ==-=-=-==ξ的分布列为171590123232323232E ξ=⨯+⨯+⨯+⨯= 考点：1、概率；2、离散型随机变量及其分布列.19．（Ⅰ）证明见解析． （Ⅱ）（Ⅲ）【解析】 （Ⅰ）证明：在中，由题设可得于是.在矩形中，.又，所以平面.（Ⅱ）由题设，，所以（或其补角）是异面直线与所成的角.在中，由余弦定理得由（Ⅰ）知平面，平面，所以，因而，于是是直角三角形，故所以异面直线与所成的角的大小为.（Ⅲ）解：过点P做于H，过点H做于E，连结PE因为平面，平面，所以.又，因而平面，故HE为PE再平面ABCD内的射影.由三垂线定理可知，，从而是二面角的平面角．由题设可得，于是再中，所以二面角的大小为．20．（Ⅰ）8()9f x xx=-+（Ⅱ）见详解（Ⅲ）7(,]4【详解】本小题主要考查导数的几何意义、利用导数研究函数的单调性、解不等式等基础知识，考查运算能力、综合分析和解决问题的能力．（Ⅰ）解：2()1af x x'=-，由导数的几何意义得(2)3f '=，于是8a =-． 由切点(2,(2))P f 在直线31yx 上可得27b -+=，解得9b =．所以函数()f x 的解析式为8()9f x x x=-+． （Ⅱ）解：2()1a f x x '=-． 当0a ≤时，显然()0f x '>（0x ≠）．这时()f x 在(,0)-∞，(0,)+∞上内是增函数． 当0a >时，令()0f x '=，解得x =当x 变化时，()f x '，()f x 的变化情况如下表：所以()f x 在(,-∞，)+∞内是增函数，在(，内是减函数． 综上，当0a ≤时，()f x 在(,0)-∞，(0,)+∞上内是增函数；当0a >时，()f x 在(,-∞，)+∞内是增函数，在(，内是减函数．（Ⅲ）解：由（Ⅱ）知，()f x 在1[,1]4上的最大值为1()4f 与(1)f 的较大者，对于任意的1[,2]2a ∈，不等式()10f x ≤在1[,1]4上恒成立，当且仅当1()104(1)10f f ⎧≤⎪⎨⎪≤⎩，即39449b a b a⎧≤-⎪⎨⎪≤-⎩，对任意的1[,2]2a ∈成立．从而得74b ≤，所以满足条件的b 的取值范围是7(,]4． 21．（Ⅰ）22145x y -=（Ⅱ）55(,)((0,(,)4224-∞-⋃-⋃⋃+∞ 【解析】试题分析：(1)因为中心在原点的双曲线C 的一个焦点是F 1（一3,0)，一条渐近线的方程是，两个条件即可求出双曲线的方程.(2)依题意可得通过假设直线l 的方程，联立双曲线方程消去y ，即可得到一个关于x 的二次方程，运用韦达定理以及判别式要大于零，即可写出线段MN 的中垂线的直线方程，从而求出直线与两坐标轴的交点，即可表示出所求的三角形的面积，从而得到一个等式结合判别式的关系式，即可得到结论.试题解析：（1）设双曲线C 的方程为22221(00)x y a b a b，-=>>，由题设得229{a b b a +==，解得224{ 5.a b ==，，所以双曲线C 的方程为22145x y -=； （2）设直线l 的方程为(0)y kx m k =+≠，点11()M x y ，，22()N x y ，的坐标满足方程组22 { 1. 45y kx m x y =+-=，①②，将①式代入②式，得22()145x kx m +-=，整理得222(54)84200k x kmx m ----=，此方程有两个不等实根，于是2540k -≠， 且222(8)4(54)(420)0km k m ∆=-+-+>，整理得22540m k +->.③ 由根与系数的关系可知线段MN 的中点坐标00()x y ，满足：12024254x x km x k +==-，002554my kx m k =+=-，从而线段MN 的垂直平分线的方程为225145454m km y x k k k ⎛⎫-=-- ⎪--⎝⎭，此直线与x 轴，y 轴的交点坐标分别为29054km k ⎛⎫ ⎪-⎝⎭，，29054m k⎛⎫⎪-⎝⎭，， 由题设可得2219981·254542km m k k =--，整理得222(54)k m k-=，0k ≠， 将上式代入③式得222(54)540k k k-+->，整理得22(45)(45)0k k k --->，0k ≠，解得0k <<54k >， 所以k 的取值范围是55004224⎛⎫⎛⎛⎫⎛⎫-∞-⋃-⋃⋃+∞ ⎪ ⎪ ⎪ ⎪ ⎝⎭⎝⎭⎝⎭⎝⎭，，，． 考点：1.待定系数的应用.2.直线与圆锥曲线的位置关系.3.三角形的面积的表示方法.4.韦达定理.5.代数的运算能力.22．（Ⅰ）23a =，29b = （Ⅱ）(1)2n n n a +=，2(1)n b n =+ （Ⅲ）证明见解析. 【解析】本小题主要考查等差数列的概念、通项公式及前n 项和公式、等比数列的概念、等比中项、不等式证明、数学归纳等基础知识，考查运算能力和推理论证能力及分类讨论的思想方法．满分14分（Ⅰ）解：由题设有，，解得14b =．由题设又有，，解得11a =． （Ⅱ）解法一：由题设，，，及14b =，11a =，进一步可得，425b =，，，猜想，，*n N ∈．先证，*n N ∈．当2n ≥时，，等式成立．当时用数学归纳法证明如下：（1当n k =时，，等式成立．（2）假设2k ≥时等式成立，即，．由题设，①的两边分别减去②的两边，整理得，从而 ．这就是说，当时等式也成立．根据（1）和（2）可知，等式对任何的成立． 综上所述，等式对任何的*n N ∈都成立再用数学归纳法证明，*n N ∈． （1）当2n ≥时，，等式成立．（2）假设当2k ≥时等式成立，即，那么．这就是说，当时等式也成立．根据（1）和（2）可知，等式对任何的*n N ∈都成立．解法二：由题设1(3)n n nS n S +=+1(1)(2)n n n S n S --=+①的两边分别减去②的两边，整理得1(2)n n na n a +=+，．所以3224a a =， 4335a a =，……1(1)(1)n n n a n a +-=+，3n ≥．将以上各式左右两端分别相乘，得2(1)!(1)!6n n n a a +-=， 由（Ⅰ）并化简得2(1)(1)62n n n n n a a ++==，3n ≥． 止式对1,2n =也成立．由题设有2114n n n b b a ++=，所以221(2)(1)n n b b n n +=++，即1221(1)(2)n n b b n n +⋅=++，*n N ∈． 令2(1)n n b x n =+，则11n n x x +=，即11n n x x +=．由11x =得1n x =，1n ≥．所以21(1)nb n =+，即，1n ≥．解法三：由题设有1(3)n n nS n S +=+，*n N ∈，所以214S S =， 3225S S =，……1(1)(2)n n n S n S --=+，．将以上各式左右两端分别相乘，得112(1)45(2)n n S n S ⨯⨯⨯-=⨯⨯⨯+，化简得1(1)(2)(1)(2)236n n n n n n n S a ++++==⨯，3n ≥．由（Ⅰ），上式对1,2n =也成立．所以1(1)2n n n n n a S S -+=-=，． 上式对2n ≥时也成立． 以下同解法二，可得，1n ≥．（Ⅲ）证明：12(1)222212(1)(1)(1)23(1)(1)nn n a a a n n T b b b n +=-+-++-=--++-+．当4n k =，*k N ∈时，222222222345(42)(41)(4)(41)n T k k k k =--++-----+++．注意到2222(42)(41)(4)(41)324k k k k k ----+++=-，故(1)32(12)43242n k k T k k k +=⨯+++-=⨯-224(44)4(4)343k k k k k n n =+-=+⨯=+．当41n k =-，*k N ∈时，2222(4)34(41)(1)3(1)(2)n T k k k n n n n =+⨯-+=+++-+= 当42n k =-，*k N ∈时，22222(4)34(41)(4)3(2)(3)33n T k k k k n n n n =+⨯-+-=+-+=---．当43n k =-，*k N ∈时，222234(41)(41)3(3)(4)(2)3n T k k k n n n n =⨯-++-=+-+++=--．所以．从而3n ≥时，有222132,5,9,13,3312,6,10,14,{12,3,7,11,312,4,8,12,n n n n n T n nnn n n n+<=++<==<=+<=总之，当3n ≥时有22n T n <，即22n T n <．。

2008年普通高等学校招生全国统一考试（天津卷）理科综合能力测试本试卷分第Ⅰ卷（选择题）和第Ⅱ卷两部分，共300分，考试用时150分钟。

第Ⅰ卷l 至5页，第Ⅱ卷6至15页．考试结束后，将本试卷和答题卡一并交回。

祝各位考生考试顺利！第I 卷注意事项：1.答第I卷前，考生务必将自己的姓名、准考兮、科目涂写在答题卡上，并在规定位置粘贴考试用条形码。

2．每小题选出答案后，用铅笔把答题卡上对应题目的答案标号涂黑。

如需改动，用橡皮擦干净后，再选涂其他答案标号。

答在试卷上的无效。

3．本卷共21题，每题6分．共126分。

在每题列出的四个选项中，只有一项是最符合题目要求的。

以下数据可供解题时参考：相对原子质最：H 1C12O 16 S 32 Cl 35.5 K 39 Br 801.下列关于蛋白质和氨基酸的叙述，正确的是A.具有生物催化作用的酶都是由氨基酸组成的B.高等动物能合成生命活动所需的20种氨基酸C.细胞中氨基酸种类和数量相同的蛋白质是同一种蛋白质D.在胚胎发育过程中，基因选择性表达，细胞会产生新的蛋白质2.下列有关生物膜的叙述．正确的是A.大鼠脾细胞与免造血千细胞的细胞膜能够发生融合B.用蛋白酶处理生物膜可改变其组成，不改变其通透性C.在生长激素的合成和分泌过程中，生物膜只发生结构上的联系D.兴奋在神经纤维上传导和在神经元间传递时，生物膜发生的变化是相同的3.下列有关特异性免疫的叙述，正确的是A.当抗原侵入宿主细胞时．细胞免疫才开始发挥作用B.效应B细胞的产生，需要T细胞和抗原的共同刺激C.在体液免疫过程中．每个效应B细胞只分泌一种特异性抗体D.当同一种抗原再次进入机体时．产生的效应B细胞均来自记忆细胞4.为获得纯合高蔓抗病番茄植株，采用了下图所示的方法；图中两对相对性状独立遗传。

据图分析，不正确．．．的是 A .过程①的自交代数越多，纯合高蔓抗病植株的比例越高 B .过程②可以取任一植株的适宜花药作培养材料 C .过程③包括脱分化和再分化两个过程 D .图中筛选过程不改变抗病基因频率5.为研究人工生态系统中大草履虫和栉毛虫间捕食关系的影响因素，设计两组实验： 实验一：在培养液中依次加入草履虫和栉毛虫，得到种群数量变化曲线（见甲图）； 实验二：在培养液中先加入沉渣作隐蔽场所，再同时加入大草履虫和栉毛虫，得到种群数量变化曲线（见乙图）。

2008年高考真题精品解析2008年普通高等学校招生全国统一考试（天津卷）（理科） 测试题 2019.91,i 是虚数单位，()=-+113i i i(A) 1-(B) 1 (C) i - (D) i2,设变量y x ,满足约束条件⎪⎩⎪⎨⎧≥+≤+≥-1210y x y x y x ，则目标函数y x z +=5的最大值为 (A) 2 (B) 3 (C) 4 (D) 53,设函数()Rx x x f ∈⎪⎭⎫ ⎝⎛-=,22sin π，则()x f 是(A) 最小正周期为π的奇函数 (B) 最小正周期为π的偶函数(C) 最小正周期为2π的奇函数 (D) 最小正周期为2π的偶函数 4,设b a ,是两条直线，βα,是两个平面，则b a ⊥的一个充分条件是(A) βαβα⊥⊥,//,b a (B) βαβα//,,⊥⊥b a (C) βαβα//,,⊥⊂b a (D) βαβα⊥⊂,//,b a5,设椭圆()1112222>=-+m m y m x 上一点P 到其左焦点的距离为3，到右焦点的距离为1，则P 点到右准线的距离为(A) 6 (B) 2 (C) 21(D) 7726,设集合{}{}RT S a x a x T x x S =+<<=>-= ,8|,32|，则a 的取值范围是(A) 13-<<-a (B) 13-≤≤-a (C) 3-≤a 或1-≥a (D) 3-<a 或1->a 7,设函数()()1011<≤-=x xx f 的反函数为()x f1-，则(A) ()x f 1-在其定义域上是增函数且最大值为1 (B) ()x f1-在其定义域上是减函数且最小值为0 (C) ()x f 1-在其定义域上是减函数且最大值为1(D) ()x f1-在其定义域上是增函数且最小值为08,已知函数()⎩⎨⎧≥-<+-=0101x x x x x f ，则不等式()()111≤+++x f x x 的解集是 (A) {}121|-≤≤-x x (B) {}1|≤x x(C) {}12|-≤x x (D) {}1212|-≤≤--x x9,已知函数()x f 是R 上的偶函数，且在区间[)+∞,0上是增函数.令⎪⎭⎫⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=75tan,75cos ,72sin πππf c f b f a ，则(A) c a b <<(B) a b c << (C) a c b << (D) c b a <<10,有8张卡片分别标有数字1，2，3，4，5，6，7，8，从中取出6张卡片排成3行2列，要求3行中仅有中间行的两张卡片上的数字之和为5，则不同的排法共有(A) 1344种(B) 1248种(C) 1056种 (D) 960种测试题答案1, 解析：，选A2,解析：如图，由图象可知目标函数y x z +=5过点时取得最大值，，选D ．3, 解析：是周期为的偶函数，选B ．()31(1)11111i i i i ii i i +-+-===----(1,0)A z max 5z =()cos 2f x x =-π4, 解析：A 、B 、D 直线可能平行，选C ．5, 解析：由椭圆第一定义知，所以，椭圆方程为所以，选B ．6, 解析：，所以，选A ．7, 解析：为减函数，由复合函数单调性知为增函数，所以单调递增，排除B 、C ；又的值域为的定义域，所以最小值为0．8, 解析：依题意得 所以，选C ．9, 解析：，因为，所以，所以，选A ．10, 解析：首先确定中间行的数字只能为1，4或2，3，共有种排法.然后确定其余4个数字的排法数.用总数去掉不合题意的情况数：中间行数字和为5，还有一行数字和为5，有4种排法，余下两个数字有种排法.所以此时余下的这4个数字共有种方法．由乘法原理可知共有种不同的排法，选B ．,a b 2a =24m =22111432x y e d +=⇒==2d ={|15}S x x x =<->或13185a a a <-⎧⇒-<<-⎨+>⎩1y =()f x 1()f x -1()f x -()f x 1()f x -11010(1)()(1)1x x x x x x x x +<+⎧⎧⎨⎨++-++⎩≥≤⎩≤或11111111x x x x x x R x ⎧≥-≤≤⇒≤∈≤≤<-⎧⎪⇒<--⎨⎨⎪⎩⎩或或5(cos)(c 2os )77b f f ππ=-=5(tan )(t 2an )77c f f ππ=-=2472πππ<<220cos sin 1tan 7772πππ<<<<b a c <<12224C A =46360A =2412A =360412312-⨯=31248412⨯=。

2008年普通高等学校招生全国统一考试（天津卷）数学（理工类）本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分，共150分，考试用时120分钟。

第Ⅰ卷1至2页，第Ⅱ卷3至10页。

考试结束后，将本试卷和答题卡一并交回。

祝各位考生考试顺利！第Ⅰ卷注意事项：1.答第Ⅰ卷前，考生务必将自己的姓名、准考证号、科目涂写在答题卡上，并在规定位置粘贴考试用条形码。

2.每小题选出答案后，用铅笔把答题卡上对应题目的答案标号涂黑。

如需改动，用橡皮擦干净后，再选涂其他答案标号。

答在试卷上的无效。

3.本卷共10小题，每小题5分，共50分。

参考公式：·如果时间A ，B 互斥，那么·球的表面积公式P （A+B ）=P （A ）+P （B ）24S R π=.·如果事件A ，B 相互独立，那么 其中R 表示球的半径.P （A·B ）=P （A ）·P （B ）一、选择题：在每小题给出的四个选项中只有一项是符合题目要求的.（1）i 是虚数单位，()=-+113i i i(A) 1- (B) 1 (C) i - (D) i（2）设变量y x ,满足约束条件⎪⎩⎪⎨⎧≥+≤+≥-1210y x y x y x ，则目标函数y x z +=5的最大值为(A) 2 (B) 3 (C) 4 (D) 5 （3）设函数()R x x x f ∈⎪⎭⎫⎝⎛-=,22sin π，则()x f 是 (A) 最小正周期为π的奇函数 (B) 最小正周期为π的偶函数(C) 最小正周期为2π的奇函数 (D) 最小正周期为2π的偶函数（4）设b a ,是两条直线，βα,是两个平面，则b a ⊥的一个充分条件是(A) βαβα⊥⊥,//,b a (B) βαβα//,,⊥⊥b a (C) βαβα//,,⊥⊂b a (D) βαβα⊥⊂,//,b a（5）设椭圆()1112222>=-+m m ymx 上一点P 到其左焦点的距离为3，到右焦点的距离为1，则P 点到右准线的距离为(A) 6 (B) 2 (C)21 (D)772（6）设集合{}{}R T S a x a x T x x S =+<<=>-= ,8|,32|，则a 的取值范围是(A) 13-<<-a (B) 13-≤≤-a (C) 3-≤a 或1-≥a (D) 3-<a 或1->a （7）设函数()()1011<≤-=x xx f 的反函数为()x f1-，则(A) ()x f 1-在其定义域上是增函数且最大值为1 (B) ()x f 1-在其定义域上是减函数且最小值为0 (C) ()x f 1-在其定义域上是减函数且最大值为1 (D) ()x f1-在其定义域上是增函数且最小值为0（8）已知函数()⎩⎨⎧≥-<+-=0101x x x x x f ，则不等式()()111≤+++x f x x 的解集是(A) {}121|-≤≤-x x (B) {}1|≤x x(C) {}12|-≤x x (D) {}1212|-≤≤--x x（9）已知函数()x f 是R 上的偶函数，且在区间[)+∞,0上是增函数.令⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=75tan ,75cos ,72sin πππf c f b f a ，则(A) c a b << (B) a b c << (C) a c b << (D) c b a <<（10）有8张卡片分别标有数字1，2，3，4，5，6，7，8，从中取出6张卡片排成3行2列，要求3行中仅有中间行的两张卡片上的数字之和为5，则不同的排法共有(A) 1344种 (B) 1248种 (C) 1056种 (D) 960种第Ⅱ卷注意事项： 1．答卷前将密封线内的项目填写清楚。

2008年普通高等学校招生全国统一考试（天津卷）数学（理工类）本试卷分第I 卷（选择题）和第II 卷（非选择题）两部分，共150分．考试用时120分钟．第I 卷1至2页，第II 卷3至10页．考试结束后，将本试卷和答题卡一并交回．祝各位考生考试顺利！第I 卷注意事项： 1．答第I 卷前，考生务必将自己的姓名、准考号、科目涂写在答题卡上．并在规定位置粘贴考试用条形码． 2．每小题选出答案后，用铅笔把答题卡上对应题目的答案标号涂黑．如需改动，用橡皮擦干净后，再选涂其他答案标号．答在试卷上的无效． 3．本卷共10小题，每小题5分，共50分． 参考公式： 如果事件A B ，互斥，那么球的表面积公式24πS R =()()()P A B P A P B +=+球的体积公式34π3V R =如果事件A B ，相互独立，那么 其中R 表示球的半径()()()P A B P A P B =一、选择题：在每小题给出的四个选项中，只有一项是符合题目要求的．1．i 是虚数单位，3i (i 1)i 1+=-（ ） A ．1-B ．1C ．i -D ．i2．设变量x y ，满足约束条件012 1.x y x y x y -⎧⎪+⎨⎪+⎩≥，≤，≥则目标函数5z x y =+的最大值为（ ）A ．2B ．3C ．4D ．53．设函数()sin 22f x x x π⎛⎫=-∈ ⎪⎝⎭R ，，则()f x 是（ ） A ．最小正周期为π的奇函数B ．最小正周期为π的偶函数C ．最小正周期为π2的奇函数 D ．最小正周期为π2的偶函数 4．设a b ，是两条直线，αβ，是两个平面，则a b ⊥的一个充分条件是（ ） A ．a b αβαβ⊥⊥，∥， B ．a b αβαβ⊥⊥，，∥ C ．a b αβαβ⊂⊥，，∥D ．a b αβαβ⊂⊥，∥，5．设椭圆22221(1)1x y m m m +=>-上一点P 到其左焦点的距离为3，到右焦点的距离为1，则P 到右准线的距离为（ ） A ．6B ．2C ．12D6．设集合{}23S x x =->，{}8T x a x a =<<+，S T =R ，则a 的取值范围是（ ）A ．31a -<<-B ．31a --≤≤C ．3a -≤或1a -≥D ．3a <-或1a >-7．设函数()1)f x x =<≤的反函数为1()f x -，则（ D ）A ．1()f x -在其定义域上是增函数且最大值为1B ．1()f x -在其定义域上是减函数且最小值为0C ．1()f x -在其定义域上是减函数且最大值为1D ．1()f x -在其定义域上是增函数且最小值为0 8．已知函数10()10x x f x x x -+<⎧=⎨-⎩，，，≥，则不等式(1)(1)1x x f x +++≤的解集是（ ）A．{}11x x -≤B ．{}1x x ≤C．{}1x xD．{}11x x ≤9．已知函数()f x 是定义在R 上的偶函数，且在区间[)0+，∞上是增函数．令2sin 7a f π⎛⎫= ⎪⎝⎭，5cos 7b f π⎛⎫= ⎪⎝⎭，5tan 7c f π⎛⎫= ⎪⎝⎭，则（ ）A ．b a c <<B ．c b a <<C ．b c a <<D ．a b c <<10．有8张卡片分别标有数字1，2，3，4，5，6，7，8，从中取出6张卡片排成3行2列，要求3行中仅有．．中间行的两张卡片上的数字之和为5，则不同的排法共有（ ） A ．1344种 B ．1248种C ．1056种D ．960种2008年普通高等学校招生全国统一考试（天津卷）数学（理工类）第Ⅱ卷注意事项：1．答卷前将密封线内的项目填写清楚． 2．用钢笔或圆珠笔直接答在试卷上． 3．本卷共12小题，共100分．二、填空题：本大题共6小题，每小题4分，共24分．把答案填在题中横线上．11．5x ⎛- ⎝的二项展开式中2x 的系数是 （用数字作答）． 12．一个正方体的各顶点均在同一球的球面上，若该球的体积为，则该正方体的表面积为 ．13．已知圆C 的圆心与抛物线24y x =的焦点关于直线y x =对称，直线4320x y --=与圆C 相交于A B ，两点，且6AB =，则圆C 的方程为 ．14．如图，在平行四边形ABCD 中，(12)AC =，，(32)BD =-，，。

2008年普通高等学校招生全国统一考试文科综合能力测试（天津卷）(地理)第Ⅰ卷分析图1中天津的文化景观，回答1－2题。

甲－奥运体育场乙－海河上的桥丙－南开大学丁－石油钻井平台图11、由自然条件决定其主要功能的一组文化景观是A.甲乙B. 乙丙C. 乙丁D.丙丁2、在天津的下列产业部门中，能充分合理利用丁图中设施所开采资源的是A. 化学工业B. 电力工业C.交通运输业D.汽车工业读某区域部分地理信息图（图2），回答3－4题。

图23、甲区域气候一年分干、湿两季，据图判断其形成的主要因素是A.纬度位置B.地势C.大气环流D.洋流4、乙海域有一大范围渔场，若用洋流剖面示意图来解释其成因，应选A B C D香港地形以山地丘陵为主，有著名的天然良港，结合图文信息（图3）回答5－6题。

图35、香港建设的有利自然条件是A. 地平坡缓B.岛多浪小C.滩阔岸直D.河多沙厚6、填海造陆对香港的影响有A. 港区行船更加通畅B. 经济活动远离了海岸C. 海洋生态得以维护D.利于沿海功能区布局读冀北某地等高线地形图（图4），回答7－8题。

图47、为把铝土矿石运到火车站，计划修一条公路，合理的选线是a、b、c、d中的A. aB. bC.cD.d8、图中有四座小水泥厂，原料主要来自采石场，产品主要外运，若在环境整治中只保留一座，应保留A. 甲B.乙C.丙D.丁读从宇宙空间拍摄的地球和月球照片（图5），回答9－10题。

图59、专家认为，地球也像月球一样，曾遭受许多陨石撞击，而目前陆地表面却很少发现明显的陨石坑。

探究其原因，主要是A.月球阻挡B.地质作用改造C. 地表布满岩石D.地表水体覆盖10、在白天，图中地球上空大片浓密的云层，对其覆盖地区的影响是利于A. 地表水分蒸发B. 近地面空气对流C.大气对地面的保温D.紫外线辐射到达地面图6中X、Y分别为晨昏圈与纬线圈的切点。

据图回答11－12题。

图611、X、Y两点情况相同的是A. 太阳所在方向B.所在时区C. 正午太阳高度D.白昼长短12、太阳直射点正向什么方向运动？能确定的是A. 向北B.向南C.向东D.向西第Ⅱ卷36、（34分）结合甲、乙两区域图（图9），回答问题。

英语第I卷第一部分：英语知识运用(共两节，满分45分)1. We’ll have a picnic in the park this Sunday _____ it rains or it’s very cold.A. sinceB. ifC. unlessD. until2. My brother is really ____. He often works in his office far into the night.A. open-mindedB. hard-workingC. self-confidentD. warm-hearted3. ---- I just can’t stop worrying about the result of the job interview.----_____. There’s nothing you can do now but wait.A. RelaxB. Go aheadC. Go for itD. Good luck4. _____ their hats into the air, the fans of the winning team let out loud shouts of victory.A. To throwB. ThrownC. ThrowingD. Being thrown5. To know more about the British Museum, you can use the Internet to go to the library, or _______.A. neitherB. someC. allD. both6. She ______ have left school, for her bike is still here.A. can’tB. wouldn’tC. shouldn’tD. needn’t7. The meal over, the managers went back to the meeting room to ______ their discussion.A. put awayB. take downC. look overD. carry on8. It was along the Mississippi River _______ Mark Twain spent much of his childhood.A. howB. whichC. thatD. where9. ----How much do I owe you for lunch?----______. It’s nothing.A. You’re welcomeB. Forget itC. With pleasureD. That’s right10. Many Chinese universities provided scholarships for students ______ financial aid.A. in favour ofB. in honour ofC. in face ofD. in need of11. Most air pollution is caused by the burning of ____ like coal, gas and oil.A. fuelsB. articlesC. goodsD. products12. The last time we had great fun was _____ we were visiting the Water Park.A. whereB. howC. whenD. why13. Her shoes ______ her dress; they look very well together.A. suitB. fitC. compareD. match14. He _____ football regularly for many years when he was young.A. was playingB. playedC. has playedD. had played15. At the railway station, the mother waved goodbye to her daughter until the train was _______.A. out f sightB. out f reachC. out f orderD. out f place第二节：完形填空（共20小题；每小题1.5分，满分30分）Jenna, a popular girl from Westwood Middle School, had graduated first in her class and was ready for new 16 in high school.17, high school was different. In the first week, Jenna went to tryouts(选拔赛)for cheerleaders(拉拉队队员). She was competing against very talented girls, and she knew it would be 18 for her to be selected. Two hours later, the 19 read a list of the girls for a second tryout. Her heart 20 as the list ended without her name. Feeling 21 , she walked home carrying her schoolbag full of homework.Arriving home, she started with math. She had always been a good math student, but now she was 22 . Shemoved on to English and history, and was 23 to find that she didn’t have any trouble with those subjects. Feeling better, she decided not to 24 math for the time being.The nest day Jenna went to see Mrs. Biden about being on the school 25 . Mrs. Biden wasn’t as 26 as Jenna. “I’m sorry, but we have enough 27 for the newspaper already. Co me back next year and we’ll talk then.” Jenna smiled 28 and left. “Why is high school so29 ?” she sighed.Later in 30 class, Jenna devoted herself to figuring out the problems that had given her so much 31 . By the end of class, she understood how t o get them right. As she gathered her books, Jenna decided she’d continue to try to 32 at her new school. She wasn’t sure if she’d succeed, but she knew she had to33 . High school was just as her mom had said: “You will feel like a small fish in a big pond 34 a big fish in a small pond. The challenge is to become the 35 fish you can be.”16. A. processes B. decision C. challenges D. exercises17. A. Therefore B. However C. Otherwise D. Besides18. A. difficult B. easy C. boring D. interesting19. A. editor B. boss C. candidate D. judge20. A. jumped B. sank C. stopped D. raced21. A. strange B. happy C. awful D. lonely22. A. struggling B. improving C. working D. complaining23. A. ashamed B. disappointed C. shocked D. relieved24. A. put up B. prepare for C. worry about D. give up25. A. committee B. newspaper C. radio D. team26. A. enthusiastic B. artistic C. sympathetic D. realistic27. A. speakers B. readers C. cheerleaders D. writers28. A. widely B. weakly C. excitedly D. brightly29. A. similar B. ordinary C. different D. familiar30. A. physics B. history C. English D. math31. A. pleasure B. hope C. trouble D. sorrow32. A. fit in B. look out C. stay up D. get around33. A. swim B. try C. ask D. escape34. A. in return for B. in case of C. in terms of D. instead of35. A. slimmest B. smallest C. best D. gentlest第二部分：阅读理解（共20小题;每小题2分，满分40分）ASandra Cisneros was born in Chicago in 1954 to a Mexican American family. As the only girl in a family of seven children, she often felt like she had “seven fathers,” because her six brothers, as well as her father, tried to control her. Feeling shy and unimportant, she retreated(躲避) into books. Despite her love of reading, she did not do well in elementary school because she was too shy to participate.In high school, with the encouragement of one particular teacher, Cisneros improved her grades and worked for the school literary magazine. Her father encouraged her to go to college because her thought it would be a good way for her to find a husband. Cisneros did attend college, but instead of searching for a husband, she found a teacher who helped her join the famous graduate writing program at the University of Iowa. At the university’s Writers’ Workshop, however, she felt lonely----a Mexican American from a poor neighborhood among students from wealthy families. The feeling of being so different helped Cisneros find her “Creative voice.”“It was not until this moment when I considered myself truly different that my writing acquired a voice. I knew I was a Mexican woman, but I didn’t think it had anything to do with why I felt so much imbalance in my life, but it hadeverything to do with it! That’s when I decided I would write about something my classmates couldn’t write about.”Cisneros published her first work, The House on Mango Street, when she was twenty-nine. The book tells about a young Mexican American girl growing up in a Spanish-speaking area in Chicago, much like the neighborhoods in which Cisneros lived as a child. The book won an award in 1985 and has been used in classes from high school through graduate school le vel. Since then, Cisneros has published several books of poetry, a children’s book, and a short-story collection.36. Which of the following is TRUE about Cisneros in her childhood?A. She had seven brothers.B. She felt herself a nobody.C. She was too shy to go to school.D. She did not have any good teachers.37. The graduate program gave Cisneros a chance to _____.A. work for a school magazineB. run away from her familyC. make a lot of friendsD. develop her writing style38. According to Cisneros, what played the decisive role in her success?A. Her early years in college.B. Her training in the Workshop.C. Her feeling of being different.D. Her childhood experience.39. What do we learn about The House on Mango Street?A. It is quite popular among students.B. It is the only book ever written by Cisneros.C. It wasn’t success as it was written in Spanish.D. It won an award when Cisneros was twenty-nine.BI love charity(慈善) shops and so do lots of other people in Britain because you find quite a few of them on every high street. The charity shop is a British institution, selling everything from clothes to electric goods, all at very good prices. You can get things you won’t find in the shops anymore. The thing I like best about them is that your money is going to a good cause and not into the pockets of profit-driven companies, and you are not damaging the planet, but finding a new home for unwanted goods.The first charity shop was opened in 1947 by Oxfam. The famous charity’s appeal to aid postwar Greece had been so successful it had been flooded with donations(捐赠物). They decided to set up a shop to sell some of these donations to raise money for that appeal. Now there are over 7,000 charity shops in the UK. My favourite charity shop in my hometown is the Red Cross shop, where I always find children’s books, all 10 or 20 pence each.Most of the people working in the charity shops are volunteers, although there is often a manager who gets paid. Over 90% of the goods in the charity shops are donated by the public. Every morning you see bags of unwanted items outside the front of shops, although they don’t encourage this, rather ask people to bring things in when the shop is open.The shops have very low running costs: all profits go to charity work. Charity shops raise more than ￡110 million a year, funding（资助）medical research, overseas aid, supporting sick and poor children, homeless and disabled people, and much more. What better place to spend your money? You get something special for a very good price and a good moral sense. You provide funds to a good cause and tread lightly on the environment.40. The author loves the charity shop mainly because of _______.A. its convenient locationB. its great variety of goodsC. its spirit of goodwillD. its nice shopping environment41. The first charity shop in the UK was set up to ____.A. sell cheap productsB. deal with unwanted thingsC. raise money for patientsD. help a foreign country42. Which of the following is TRUE about charity shops?A. The operating costs are very low.B. The staff are usually well paid.C. 90% of the donations are second-hand.D. They are open twenty-four hours a day.43. Which of the following may be the best title for the passage?A. What to Buy a Charity Shops.B. Charity Shop: Its Origin & Development.C. Charity Shop: Where You Buy to Donate.D. The Public’s Concern about Charity Shops.CMichael Fish may soon be replaced as a weather forecaster by something truly fishier---the shark(鲨鱼).Research by a British biology student suggests that sharks could be used to predict storms.Lauren Smith, 24, is close to completing her study on shark’s ability to sense pressure.If her studies prove the theory, scientists may be able to monitor the behaviour of sharks to predict bad weather.Miss Smith had previously studied the behaviour of lemon sharks in the Bahamas.She then used their close relatives, lesser spotted dogfish, for further research at Aberdeen University.Her work---thought to be the first of its kind to test the pressure theory ---- resulted from the observation that juvenile blacktip sharks off Florida moved into deeper water ahead of a violent storm in 2001.Miss Smith said: “I’ve always been crazy about traveling and diving and this led me to an interest in sharks.”“I was delighted to have been able to research in the area for my degree. I know there’s so much more we need to understand ---- but it certainly opens the way to more research.”It has been discovered that a shark senses pressure using hair cells in its balance system.At the Bimini Shark Lab in the Bahamas, Miss Smith fixed hi-tech sensors to sharks to record pressure and temperature, while also tracking them using GPS (Global Positioning System) technology.In Aberdeen, she was able to study the effects of tidal(潮汐的) and temperature changes on dogfish----none of which were harmed. She also used a special lab which can mimic(模拟) oceanic pressure changes caused by weather fronts.She is due to complete her study and graduate later this year. She says she will be looking for a job which will give her the chance to enrich her experience of shark research.44.The passage is most probably taken from _____.A. a short-story collectionB. a popular science magazineC. a research paperD. a personal diary45. What do we learn from the first four paragraph of the passage?A. Sharks may be used to predict bad weather.B. Sharks’ behaviour can be controlled.C. Michael Fish is not qualified for his job.D. Lauren Smith will become a weather forecaster.46. Lauren Smith conducted her research by _______.A. removing hair cells from a shark’s balance systemB. measuring the air pressure of weather frontsC. recording sharks’ body temperatureD. monitoring sharks’ reaction to weather changes47. What is the passage mainly about?A. A popular way of forecasting weather.B. A new research effort in predicting storms.C. Biologists’ interest in the secrets of sharks.D. Lauren Smith’s devotion to scientific resear ch.DWe can achieve knowledge either actively or passively（被动地）. We achieve it actively by direct experience, by testing and proving an idea, or by reasoning.We achieve knowledge passively by being told by someone else. Most of the learning that takes place in the classroom and the kind that happens when we watch TV or read newspapers or magazines is passive. Conditioned as we are to passive learning, it’s not surprising that we depend on it in our everyday communication with friends and co-workers.Unfortunately, passive learning has a serious problem. It makes us tend to accept what we are told even when it is little more than hearsay and rumor(谣言).Did you ever play the game Rumor? It begins when one person writes down a message but doesn’t show it to anyone. Then the person whispers it, word for word, to another person. That person, in turn, whispers it to still another, and so on, through all the people playing the game. The last person writes down the message word for word as he or she hears it. Then the two written statements are compared. Typically, the original message has changed.That’s what happens in daily life. The simple fact that peo ple repeat a story in their own words changes the story. Then, too, most people listen imperfectly. And many enjoy adding their own creative touch to a story, trying to improve on it, stamping（打上标记）it with their own personal style. Yet those who hear it think they know.This process is also found among scholars and authors: A statement of opinion by one writer may be re-stated as fact by another, who may in turn be quoted by yet another; and this process may continue, unless it occurs to someone to question the facts on which the original writer based his opinion or to challenge the interpretation he placed upon those facts.48. According to the passage, passive learning may occur in _______.A. doing a medical experimentB. solving a math problemC. visiting an exhibitionD. doing scientific reasoning49. The underlined word “it” in Paragraph 2 refers to _____.A. active learningB. knowledgeC. communicationD. passive learning50. The author mentions the game Rumor to show that _____.A. a message may be changed when being passed onB. a message should be delivered in different waysC. people may have problems with their sense of hearingD. people tend not to believe in what they know as rumor51. What can we infer from the passage?A. Active learning is less important.B. Passive learning may not be reliable.C. Active learning occurs more frequently.D. Passive learning is not found among scholars.EAs kids, my friends and I spent a lot of time out in the woods. “The woods” was our part-time address, destination, purpose, and excuse. If I went to a friend’s house and found him not at home, his mother might say, “Oh, he’s out in the woods, ” with a tone(语气) of airy acceptance. It’s similar to the tone people sometimes use nowadays to tell me that someone I’m looking for is on the golf course or at the gym, or even “away from his desk.” For us ten-year-olds, “being out in the woods” was just an excuse to d o whatever we feel like for a while.We sometimes told ourselves that what we were doing in the woods was exploring(探索). Exploring was a more popular idea back then than it is today. History seemed to be mostly about explorers. Our explorations, though, seemed to have less system than the historic kind: something usually came up along the way. Say we stayed in the woods, throwing rocks, shooting frogs, picking blackberries, digging in what we were briefly persuaded was an Italian burial mound.Often we got “lost” and had to climb a tree to find out where we were. If you read a story in which someone does that successfully, be skeptical: the topmost branches are usually too skinny to hold weight, and we could never climb high enough to see anything except other trees. There were four or five trees that we visited regularly----tall beeches, easy to climb and comfortable to sit in.It was in a tree, too, that our days of fooling around in the woods came to an end. By then some of us has reached seventh grade and had begun the rough ride of adolescence（青春期）. In March, the month when we usually took to the woods again after winter, two friends and I set out to go exploring. We climbed a tree, and all of a sudden it occurred to all three of us at the same time that were really were rather big to be up in a tree. Soon there would be the spring dances on Friday evenings in the high school cafeteria.52. The author and his fiends were often out in the woods to _______.A. spend their free timeB. play gold and other sportsC. avoid doing their schoolworkD. keep away from their parents53. What can we infer from Paragraph 2?A. The activities in the woods were well planned.B. Human history is not the result of exploration.C. Exploration should be a systematic activity.D. The author explored in the woods aimlessly.54. The underlined word “skeptical” in Paragraph 3 is closest in meaning to ______.A. calmB. doubtfulC. seriousD. optimistic55. How does the author feel about his childhood?A. Happy but short.B. Lonely but memorable.C. Boring and meaningless.D. Long and unforgettable.第三部分：写作第一节：短文改错（共10小题；每小题1分，满分10分）Last summer I go to America and studied at a language 56.school. I had many wonderful experience, but I also 57.had a sad one. One day, the school held party, where 58.I invited to talk about Tianjin. After that they asked me a lot of 59.Things about China. But I couldn’t explain them with English 60.Clearly. I felt sadly. I learnt a lesson from this experience. I 61.have already studied English for eight years, I can’t use it 62.very good. I must work hard to improve my spoken English 63.so that I will not be able to communicate freely with foreigners. 64.I hope I can be a bridge between China and others countries 65.in the future.第二节：书面表达（满分25分）66.假设2008年2月12日是你父亲的生日，下面三幅图描绘的是你给父亲买完礼物后乘坐地铁回家时经历的一件事。

2008年高考真题精品解析2008年普通高等学校招生全国统一考试（天津卷）（理科） 测试题 2019.91,已知⎪⎭⎫⎝⎛3∈=⎪⎭⎫ ⎝⎛-4,2,1024cos πππx x . （Ⅰ）求x sin 的值；（Ⅱ）求⎪⎭⎫ ⎝⎛+32sin πx 的值.2,甲、乙两个篮球运动员互不影响地在同一位置投球，命中率分别为21与p ，且乙投球2次均未命中的概率为161. （Ⅰ）求乙投球的命中率p ；（Ⅱ）若甲投球1次，乙投球2次，两人共命中的次数记为ξ，求ξ的分布列和数学期望.3,如图，在四棱锥ABCD P -中，底面ABCD 是矩形.已知60,22,2,2,3=∠====PAB PD PA AD AB .（Ⅰ）证明⊥AD 平面PAB ；（Ⅱ）求异面直线PC 与AD 所成的角的大小； （Ⅲ）求二面角A BD P --的大小.4,已知函数()()0≠++=x b x ax x f ，其中R b a ∈,.（Ⅰ）若曲线()x f y =在点()()2,2f P 处的切线方程为13+=x y ，求函数()x f的解析式；（Ⅱ）讨论函数()x f 的单调性；（Ⅲ）若对于任意的⎥⎦⎤⎢⎣⎡∈2,21a ，不等式()10≤x f 在⎥⎦⎤⎢⎣⎡1,41上恒成立，求b 的取值范围.5,已知中心在原点的双曲线C 的一个焦点是()0,31-F ，一条渐近线的方程是025=-y x .（Ⅰ）求双曲线C 的方程；（Ⅱ）若以()0≠k k 为斜率的直线l 与双曲线C 相交于两个不同的点M ，N ，线段MN 的垂直平分线与两坐标轴围成的三角形的面积为281，求k 的取值范围.6,在数列{}n a 与{}n b 中，4,111==b a ，数列{}n a 的前n 项和n S 满足()031=+-+n n S n nS ,12+n a 为n b 与1+n b 的等比中项，*N n ∈.（Ⅰ）求22,b a 的值；（Ⅱ）求数列{}n a 与{}n b 的通项公式；（Ⅲ）设()()()*,1112121N n b b b T n a a a n n∈-++-+-= .证明3,22≥<n n T n .7,52⎪⎪⎭⎫⎝⎛-x x 的二项展开式中，2x 的系数是 （用数字作答）. 8,一个正方体的各定点均在同一球的球面上，若该球的体积为π34，则该正方体的表面积为 .9,已知圆C 的圆心与抛物线x y 42=的焦点关于直线x y =对称.直线0234=--y x 与圆C 相交于B A ,两点，且6=AB ,则圆C 的方程为 .10,如图，在平行四边形ABCD 中，()()2,3,2,1-==， 则=⋅AC AD .测试题答案1, 解：（Ⅰ）因为⎪⎭⎫⎝⎛∈43,2ππx ，所以⎪⎭⎫ ⎝⎛∈-2,44πππx ，于是 10274cos 14sin 2=⎪⎭⎫ ⎝⎛--=⎪⎭⎫ ⎝⎛-ππx x 54221022210274sin 4cos 4cos 4sin 44sin sin =⨯+⨯=⎪⎭⎫ ⎝⎛-+⎪⎭⎫ ⎝⎛-=⎪⎪⎭⎫ ⎝⎛+⎪⎭⎫ ⎝⎛-=ππππππx x x x（Ⅱ）因为⎪⎭⎫⎝⎛∈43,2ππx ，故53541sin 1cos 22-=⎪⎭⎫⎝⎛--=--=x x 2571cos 22cos ,2524cos sin 22sin 2-=-=-==x x x x x所以5037243sin 2cos 3cos 2sin 32sin +-=+=⎪⎭⎫ ⎝⎛+πππx x x2, 解：（Ⅰ）设“甲投球一次命中”为事件A ，“乙投球一次命中”为事件B 由题意得()()()1611122=-=-p B P解得43=p 或45（舍去），所以乙投球的命中率为43（Ⅱ）由题设和（Ⅰ）知()()()()41,43,21,21====B P B P A P A Pξ可能的取值为0，1，2，3，故()()()321412102=⎪⎭⎫ ⎝⎛⨯=⋅==B B P A P P ξ()()()()()()32721414324121321412112212=⨯⨯⨯+⎪⎭⎫ ⎝⎛⨯==⎪⎭⎫ ⎝⎛⨯=+⋅==A P B P B P C B B P A P P ξ()()()329432132=⎪⎭⎫ ⎝⎛⨯=⋅==B B P A P P ξ()()()()321531012==-=-=-==ξξξξP P P Pξ的分布列为ξ的数学期望232933215232713210=⨯+⨯+⨯+⨯=ξE3,解：（Ⅰ）证明：在PAD ∆中，由题设22,2==PD PA 可得222PD AD PA =+于是PA AD ⊥.在矩形ABCD 中，AB AD ⊥.又A AB PA = ， 所以⊥AD 平面PAB .（Ⅱ）证明：由题设，AD BC //，所以PCB ∠（或其补角）是异面直线PC 与AD 所成的角.在PAB ∆中，由余弦定理得由（Ⅰ）知⊥AD 平面PAB ，⊂PB 平面PAB ，所以PB AD ⊥，因而PB BC ⊥，于是PBC ∆是直角三角形，故27tan ==BC PB PCB所以异面直线PC 与AD 所成的角的大小为27arctan. （Ⅲ）解：过点P 做AB PH ⊥于H ，过点H 做BD HE ⊥于E ，连结PE因为⊥AD 平面PAB ，⊂PH 平面PAB ，所以PH AD ⊥.又A AB AD = ，因而⊥PH 平面ABCD ，故HE 为PE 再平面ABCD 内的射影.由三垂线定理可知，PE BD ⊥，从而PEH ∠是二面角A BD P --的平面角。

2008年普通高等学校招生全国统一考试（天津卷）理科综合能力测试第Ⅰ卷（本卷共21题，每题6分，共126分。

在每题列出的四个选项中，只有一项是最符合题目要求的。

）以下数据可供解题时参考：相对原子质量：H—1 C—12 O—16 S—32 Cl—35.5 K—39 Br—80一、选择题1．下列关于蛋白质和氨基酸的叙述，正确的是（）A．具有生物催化作用的酶都是由氨基酸组成的B．高等动物能合成生命活动所需的20种氨基酸C．细胞中氨基酸种类和数量相同的蛋白质是同一种蛋白质D．在胚胎发育过程中，基因选择性表达，细胞会产生新的蛋白质2．下列有关生物膜的叙述，正确的是（）A．大鼠脾细胞与兔造血干细胞的细胞膜能够发生融合B．用蛋白酶处理生物膜可改变其组成，不改变其通透性C．在生长激素的合成和分泌过程中，生物膜只发生结构上的联系D．兴奋在神经纤维上传导和在神经元间传递时，生物膜发生的变化是相同的3．下列有关特异性免疫的叙述，正确的是（）A．当抗原侵入宿主细胞时，细胞免疫才开始发挥作用B．效应B细胞的产生，需要T细胞和抗原的共同刺激C．在体液免疫过程中，每个效应B细胞只分泌一种特异性抗体D．当同一种抗原再次进入机体时，产生的效应B细胞均来自记忆细胞4．为获得纯合高蔓抗病番茄植株，采用了下图所示的方法：图中两对相对性状独立遗传。

据图分析，不正确的是（）A．过程①的自交代数越多，纯合高蔓抗病植株的比例越高B．过程②可以取任一植株的适宜花药作培养材料C．过程③包括脱分化和再分化两个过程D．图中筛选过程不改变抗病基因频率5．为研究人工生态系统中大草履虫和栉毛虫间捕食关系的影响因素，设计两组实验：实验一：在培养液中依次加入草履虫和栉毛虫，得到种群数量变化曲线（见甲图）；实验二：在培养液中先加入沉渣作隐蔽场所，再同时加入大草履虫和栉毛虫，得到种群数量变化曲线（见乙图）。

据实验判断，正确的是（）A．沉渣对栉毛虫的种群数量变化无影响B．大草履虫以栉毛虫为食C．培养液为上述生态系统提供能量D．大草履虫的能量除用于自身代谢外，其余部分流入栉毛虫6．下列有关ATP 的叙述，正确的是（）①人长时间剧烈运动时，骨骼肌细胞中每摩尔葡萄糖生成ATP 的量与安静时相等②若细胞内Na+浓度偏高，为维持Na+浓度的稳定，细胞消耗ATP 的量增加③人在寒冷时，肾上腺素和甲状腺激素分泌增多，细胞产生式ATP 的量增加④人在饥饿时，细胞中ATP与ADP的含量难以达到动态平衡A．①② B．②③ C．③④ D．①④7．二十世纪化学合成技术的发展对人类健康水平和生活质量的提高做出了巨大贡献。