2014年高考数学文科(高考真题+模拟新题)分类：M单元 推理与证明

第一篇：2014年高考数学文科(高考真题+模拟新题)分类：M单元 推理与证明

数学

M单元 推理与证明

M1 合情推理与演绎推理

16．，[2014·福建卷] 已知集合{a，b，c}＝{0，1，2}，且下列三个关系：①a≠2；②b＝2；③c≠0有且只有一个正确，则100a＋10b＋c等于________．

16．201 [解析](i)若①正确，则②③不正确，由③不正确得c＝0，由①正确得a＝1，所以b＝2，与②不正确矛盾，故①不正确．

(ii)若②正确，则①③不正确，由①不正确得a＝2，与②正确矛盾，故②不正确．(iii)若③正确，则①②不正确，由①不正确得a＝2，由②不正确及③正确得b＝0，c＝1，故③正确．

则100a＋10b＋c＝100×2＋10×0＋1＝201.14．[2014·全国新课标卷Ⅰ] 甲、乙、丙三位同学被问到是否去过A，B，C三个城市时，甲说：我去过的城市比乙多，但没去过B城市．乙说：我没去过C城市．丙说：我们三人去过同一城市．

由此可判断乙去过的城市为________．

14．A [解析] 由甲没去过B城市，乙没去过C城市，而三人去过同一城市，可知三人去过城市A，又由甲最多去过两个城市，且去过的城市比乙多，故乙只去过A城市．

x14．[2014·陕西卷] 已知f(x)＝x≥0，若f1(x)＝f(x)，fn＋1(x)＝f(fn(x))，n∈N＋，则1＋x

f2014(x)的表达式为________．

xx14.[解析] 由题意，得f1(x)＝f(x)＝ 1＋2014x1＋x

x

1＋xxxf2(x)＝f3(x)＝，„，x1＋2x1＋3x11＋x

由此归纳推理可得f2014(x)＝x.1＋2014x

M2 直接证明与间接证明

21．、[2014·湖南卷] 已知函数f(x)＝xcos x－sin x＋1(x＞0)．

(1)求f(x)的单调区间；

111(2)记xi为f(x)的从小到大的第i(i∈N*)个零点，证明：对一切n∈N*，有x1x2xn

321．解：(1)f′(x)＝cos x－xsin x－cos x＝－xsin x.令f′(x)＝0，得x＝kπ(k∈N*)．

当x∈(2kπ，(2k＋1)π)(k∈N)时，sin x>0，此时f′(x)<0;

当x∈((2k＋1)π，(2k＋2)π)(k∈N)时，sin x<0，此时f′(x)>0.故f(x)的单调递减区间为(2kπ，(2k＋1)π)(k∈N)，单调递增区间为((2k＋1)π，(2k＋2)π)(k∈N)．

ππ(2)由(1)知，f(x)在区间(0，π)上单调递减．又f＝0，故x1＝.2

2当n∈N*时，因为

＋f(nπ)f[（n＋1）π]＝[(－1)nnπ＋1][(－1)n1(n＋1)π＋1]＜0，且函数f(x)的图像是连续不断的，所以f(x)在区间(nπ，(n＋1)π)内至少存在一个零点．又f(x)在区间(nπ，(n＋1)π)上是单调的，故

nπ＜xn＋1＜(n＋1)π.142因此，当n＝1时，＜； x1π3

1112当n＝2时，＋(4＋1)＜ x1x2π3

当n≥3时，1111114＋1＋ 2（n－1）x1x2xnπ

11151＜＜（n－2）（n－1）1×2ππ5＋1－1＋11＋„＋11 223n－2n－1

1162＝6－n－1＜＜π3π

1112综上所述，对一切n∈N*，.x1x2xn3

M3数学归纳法

sin x23．、[2014·江苏卷] 已知函数f0(x)＝(x>0)，设fn(x)为fn－1(x)的导数，n∈N*.x

πππ(1)求2f1＋f2的值； 222

πππ2(2)证明：对任意的n∈N*，等式nfn－1＋n＝ 4442

sin xcos xsin x23．解：(1)由已知，得f1(x)＝f′0(x)＝′＝－，xxx

cos xxsin ′＝ 于是f2(x)＝f1′(x)＝′－xx－sin x2cos x2sin x＋，xxx

ππ4216所以f1＝－f2＝－22πππ

πππ故2f12＝－1.222(2)证明：由已知得，xf0(x)＝sin x，等式两边分别对x求导，得f0(x)＋xf0′(x)＝cos x，π即f0(x)＋xf1(x)＝cos x＝sinx＋.2

类似可得

2f1(x)＋xf2(x)＝－sin x＝sin(x＋π)，3π3f2(x)＋xf3(x)＝－cos x＝sinx＋，2

4f3(x)＋xf4(x)＝sin x＝sin(x＋2π)．

nπ下面用数学归纳法证明等式nfn－1(x)＋xfn(x)＝sinx＋对所有的n∈N*都成立． 2

(i)当n＝1时，由上可知等式成立．

kπ(ii)假设当n＝k时等式成立，即kfk－1(x)＋xfk(x)＝sinx.2

因为[kfk－1(x)＋xfk(x)]′＝kfk－1′(x)＋fk(x)＋xfk′(x)＝(k＋1)fk(x)＋xfk＋1(x)，sinx＋kπ′＝cosx＋kπ·x＋kπ′＝sinx＋（k＋1）π，2222

（k＋1）π所以(k＋1)fk(x)＋xfk＋1(x)＝sinx＋2，因此当n＝k＋1时，等式也成立．

nπ综合(i)(ii)可知，等式nfn－1(x)＋xfn(x)＝sinx＋对所有的n∈N*都成立． 2

πππππnπ令x＝nfn－1＋fn＝sin＋(n∈N*)，424444

πππ所以nfn－1＋fn＝444(n∈N*)．

M4单元综合5．[2014·湖南长郡中学月考] 记Sk＝1k＋2k＋3k＋„＋nk，当k＝1，2，3，„时，观察

111111111111下列等式：S1＝n2＋n，S2＝n3＋2＋n，S34＋3＋2，S4＝n5n4＋3－n，2232642452330

115S56＋5＋n4＋An2，„由此可以推测A＝____________． 6212

11155．－ [解析] 根据所给等式可知，各等式右边的各项系数之和为1，所以＋126212

1A＝1，解得A＝－12

6．[2014·日照一中月考] 二维空间中圆的一维测度(周长)l＝2πr，二维测度(面积)S＝π

4r2，观察发现S′＝l；三维空间中球的二维测度(表面积)S＝4πr2，三维测度(体积)V＝πr3，3

观察发现V′＝S.已知四维空间中“超球”的三维测度V＝8πr3，猜想其四维测度W＝________.6．2πr4 [解析] 因为W′＝8πr3，所以W＝2πr4.7．[2014·甘肃天水一中期末] 观察下列等式：

(1＋1)＝2×1；

(2＋1)(2＋2)＝22×1×3；

(3＋1)(3＋2)(3＋3)＝23×1×3×5.照此规律，第n个等式为________________________________________________________________________．

7．(n＋1)(n＋2)(n＋3)„(n＋n)＝2n×1×3×5ׄ×(2n－1)

[解析] 观察等式规律可知第n个等式为(n＋1)(n＋2)(n＋3)„(n＋n)＝2n×1×3×5ׄ×(2n－1)．

8．[2014·南昌调研] 已知整数对的序列为(1，1)，(1，2)，(2，1)，(1，3)，(2，2)，(3，1)，(1，4)，(2，3)，(3，2)，(4，1)，(1，5)，(2，4)，„，则第57个数对是________．

8．(2，10)[解析] 由题意，发现所给序数列有如下规律：

(1，1)的和为2，共1个；

(1，2)，(2，1)的和为3，共2个；

(1，3)，(2，2)，(3，1)的和为4，共3个；

(1，4)，(2，3)，(3，2)，(4，1)的和为5，共4个；

(1，5)，(2，4)，(3，3)，(4，2)，(5，1)的和为6，共5个．

由此可知，当数对中两个数字之和为n时，有n－1个数对．易知第57个数对中两数之和为12，且是两数之和为12的数对中的第2个数对，故为(2，10)．

9．[2014·福州模拟] 已知点A(x1，ax1)，B(x2，ax2)是函数y＝ax(a>1)的图像上任意不同的两点，依据图像可知，线段AB总是位于A，B两点之间函数图像的上方，因此有结论ax1＋ax2x1＋x2>a成立．运用类比的思想方法可知，若点A(x1，sin x1)，B(x2，sin x2)是函数y22

＝sin x(x∈(0，π))的图像上任意不同的两点，则类似地有________________成立．

9.sin x1＋sin x2x1＋x2

sin x1＋sin x2x1＋x2总是位于A，B两点之间函数图像的下方，所以有

第二篇：2013高考数学_(真题+模拟新题分类)_推理与证明_理

推理与证明

M1 合情推理与演绎推理

15．B13，J3，M1[2013·福建卷] 当x∈R，|x|<1时，有如下表达式： 12n

1＋x＋x＋„＋x.1－x

11121n1

1两边同时积分得：∫01dx0xdx＋∫0xdx＋„＋∫0xdx0，222221－x从而得到如下等式：

1n＋1111211311×＋＋＋„＋＋„＝ln 2.22232n＋12请根据以上材料所蕴含的数学思想方法，计算：

1111212131n1n＋1CCn×＋n×＋„＋n×＝__________．

22232n＋12

0n

13n＋1n0122nn

15.[解析](1＋x)＝Cn＋Cnx＋Cnx＋„＋Cnx，－1n＋12

11121n1012nn

两边同时积分得Cn∫01dx＋Cn∫0xdx＋Cn∫xdx＋„＋Cn∫0xdx＝∫(1＋x)dx，***n1n＋113n＋10

得Cn×Cn×＋n×Cn＝－1.22232n＋12n＋1

214．M1[2013·湖北卷] 古希腊毕达哥拉斯学派的数学家研究过各种多边形数，如三角形n（n＋1）121

数1，3，6，10，„，第n个三角形数为＝n，记第n个k边形数为N(n，k)(k≥3)，222以下列出了部分k边形数中第n个数的表达式：

121

三角形数 N(n，3)＝＋n，22正方形数 N(n，4)＝n，321

五边形数 N(n，5)＝－n，22

六边形数 N(n，6)＝2n－n，„„

可以推测N(n，k)的表达式，由此计算N(10，24)＝________.11k－22

14．1 000 [解析] 观察得k每增加1，n项系数增加n项系数减少，N(n，k)＝

222n2

n＋(4－k)N(10，24)＝1 000.20，0

ln x，x≥1.

＋

2-1-

①若a>0，b>0，则ln(a)＝blna；

＋＋＋

②若a>0，b>0，则ln(ab)＝lna＋lnb；

＋a＋＋

③若a>0，b>0，则ln≥lna－lnb；

b

＋b＋

④若a>0，b>0，则ln(a＋b)≤lna＋lnb＋ln 2.其中的真命题有________．(写出所有真命题的编号)

b＋bb

16．①③④ [解析] ①中，当a≥1时，∵b>0，∴a≥1，ln(a)＝ln a＝bln a＝bln＋b＋b＋

a；当00，∴0

＋＋＋

②中，当01时，左边＝ln(ab)＝0，右边＝lna＋lnb＝ln a＋0＝ln a>0，∴②不成立；

aa＋＋

≤1，即a≤b时，左边＝0，右边＝lna－lnb≤0bba

时，左边＝lnln a－ln b>0，若a>b>1时，右边＝ln a－ln b，左边≥右边成立；若0

1ba

时，右边＝0, 左边≥右边成立；若a>1>b>0，左边＝ln＝ln a－ln b>ln a，右边＝ln a，b左边≥右边成立，∴③正确；

④中，若0

＋

＋

＋＋＋

(a＋b)＝0，右边＝ln＋a＋ln＋b＋ln 2＝ln 2>0，左边≤

a＋b，2

(a＋b)－ln 2＝ln(a＋b)－ln 2＝a＋ba＋ba＋ba＋b又∵≤a或≤b，a，b至少有1个大于1，∴lnln a或lnln b，即

2222有ln

＋

(a＋b)－ln 2＝ln(a＋b)－ln 2＝ln

a＋b＋＋

≤lna＋lnb，∴④正确． 2

14．M1[2013·陕西卷] 观察下列等式： 2

1＝1 22

1－2＝－3 222

1－2＋3＝6 2222

1－2＋3－4＝－10 „„

照此规律，第n个等式可为________． 14．1－2＋3－4＋„＋(－1)

n＋12

n＝(－1)

n＋1

n（n＋1）

[解析] 结合已知所给几项的特2

点，可知式子左边共n项，且正负交错，奇数项为正，偶数项为负，右边的绝对值为左边底

2222n＋12n

数的和，系数和最后一项正负保持一致，故表达式为1－2＋3－4＋„＋(－1)n＝(－1)

＋1

n（n＋1）

M2 直接证明与间接证明

20．M2，D2，D3，D5[2013·北京卷] 已知{an}是由非负整数组成的无穷数列，该数列前n项的最大值记为An，第n项之后各项an＋1，an＋2，„的最小值记为Bn，dn＝An－Bn.-2-

(1)若{an}为2，1，4，3，2，1，4，3，„，是一个周期为4的数列(即对任意n∈N，an

＋4＝an)，写出d1，d2，d3，d4的值；

(2)设d是非负整数，证明：dn＝－d(n＝1，2，3，„)的充分必要条件为{an}是公差为d的等差数列；

(3)证明：若a1＝2，dn＝1(n＝1，2，3，„)，则{an}的项只能是1或者2，且有无穷多项为1.20．解：(1)d1＝d2＝1，d3＝d4＝3.(2)(充分性)因为{an}是公差为d的等差数列，且d≥0，所以a1≤a2≤„≤an≤„.因此An＝an，Bn＝an＋1，dn＝an－an＋1＝－d(n＝1，2，3，„)．

(必要性)因为dn＝－d≤0(n＝1，2，3，„)．所以An＝Bn＋dn≤Bn.又因为an≤An，an＋1≥Bn，所以an≤an＋1.于是，An＝an，Bn＝an＋1.因此an＋1－an＝Bn－An＝－dn＝d，即{an}是公差为d的等差数列．

(3)因为a1＝2，d1＝1，所以A1＝a1＝2，B1＝A1－d1＝1.故对任意n≥1，an≥B1＝1.假设{an}(n≥2)中存在大于2的项． 设m为满足am>2的最小正整数，则m≥2，并且对任意1≤k2，于是，Bm＝Am－dm>2－1＝1，Bm－1＝min{am，Bm}>1.故dm－1＝Am－1－Bm－1<2－1＝1，与dm－1＝1矛盾．

所以对于任意n≥1，有an≤2，即非负整数列{an}的各项只能为1或2.因为对任意n≥1，an≤2＝a1，所以An＝2.故Bn＝An－dn＝2－1＝1.因此对于任意正整数n，存在m满足m>n，且am＝1，即数列{an}有无穷多项为1.M3 数学归纳法

M4 单元综合1111

1．[2013·黄山质检] 已知n为正偶数，用数学归纳法证明1－＋－＋„＋234n＋1

1112(＋„＋)时，若已假设n＝k(k≥2为偶数)时命题为真，则还需要用归纳假设n＋2n＋42n

再证n＝()时等式成立()

A．k＋1B．k＋2 C．2k＋2D．2(k＋2)

1．B [解析] 根据数学归纳法的步骤可知，则n＝k(k≥2为偶数)下一个偶数为k＋2，故答案为B.2．[2013·石景山期末] 在整数集Z中，被5除所得余数为k的所有整数组成一个“类”，记为[k]，即[k]＝{5n＋k|n∈Z}，k＝0，1，2，3，4.给出如下四个结论：

*

①2 013∈[3]；②－2∈[2]；③Z＝[0]∪[1]∪[2]∪[3]∪[4]；④整数a，b属于同一“类”的充要条件是a－b∈[0]．

其中，正确结论的个数为()

A．1B．2C．3D．

42．C [解析] 因为2 013＝402×5＋3，所以2 013∈[3]，①正确．－2＝－1×5＋3，－2∈[3]，所以②不正确．因为整数集中的数被5除的余数可以且只可以分成五类，所以③正确．整数a，b属于同一“类”，则整数a，b被5除的余数相同，从而a－b被5除的余数为0，反之也成立，故整数a，b属于同一“类”的充要条件是a－b∈[0]，故④正确．所以正确的结论个数为3，选C.223344

3．[2013·汕头期末] 已知2＋＝3＋3 4＋＝，33881515aa

6＋＝(a，t均为正实数)，类比以上等式，可推测a，t的值，则a－t＝________． tt

3．－29 [解析] 类比等式可推测a＝6，t＝35，则a－t＝－29.x

4．[2013·福州期末] 已知点A(x1，ax1)，B(x2，ax2)是函数y＝a(a>1)的图像上任意不同两点，依据图像可知，线段AB总是位于A、B两点之间函数图像的上方，因此有结论

x1＋x2

ax1＋ax2

>a2成立．运用类比思想方法可知，若点A(x1，sin x1)，B(x2，sin x2)是函数y2

＝sin x(x∈(0，π))的图像上的不同两点，则类似地有________成立．

sin x1＋sin x2x1＋x24.[解析] 函数y＝sin x在x∈(0，π)的图像上任意不同两

sin x1＋sin x2

点A，B，依据图像可知，线段AB总是位于A，B两点之间函数图像的下方，所以

x1＋x2

[规律解读] 类比推理中的结论要注意问题在变化之后的不同，要“求同存异”才能够正确解决问题．

5．[2013·云南师大附中月考] 我们把平面内与直线垂直的非零向量称为直线的法向量，在平面直角坐标系中，利用求动点轨迹方程的方法，可以求出过点A(－3，4)，且法向量为n＝(1，－2)的直线(点法式)方程为1×(x＋3)＋(－2)×(y－4)＝0，化简得x－2y＋11＝0.类比以上方法，在空间直角坐标系中，经过点A(1，2，3)，且法向量为n＝(－1，－2，1)的平面(点法式)方程为________．

5．x＋2y－z－2＝0 [解析] 设B(x，y，z)为平面内的任一点，类比得平面的方程为(－1)×(x－1)＋(－2)×(y－2)＋1×(z－3)＝0，即x＋2y－z－2＝0.*

6．[2013·黄山质检] 已知数列{an}满足a1＝1，an＝logn(n＋1)(n≥2，n∈N)．定义：

*

使乘积a1·a2·„·ak为正整数的k(k∈N)叫作“简易数”．则在[1，2 012]内所有“简易数”的和为________．

lg（n＋1）

6．2 036 [解析] ∵an＝logn(n＋1)＝，lg n

lg 3lg 4lg（k＋1）lg（k＋1）

∴a1·a2·„·ak·＝＝log2(k＋1)，则“简

lg 2lg 3lg klg 2

nn

易数”k使log2(k＋1)为整数，即满足2＝k＋1，所以k＝2－1，则在[1，2 012]内所有“简

2（1－2）1210

易数”的和为2－1＋2－1＋„＋2－1＝－10＝1 023×2－10＝2 036.1－2

若

第三篇：2014年高考文科数学真题解析分类：M单元 推理与证明(纯word可编辑)

数学

M单元 推理与证明

M1 合情推理与演绎推理

16．，[2014·福建卷] 已知集合{a，b，c}＝{0，1，2}，且下列三个关系：①a≠2；②b＝2；③c≠0有且只有一个正确，则100a＋10b＋c等于________．

16．201 [解析](i)若①正确，则②③不正确，由③不正确得c＝0，由①正确得a＝1，所以b＝2，与②不正确矛盾，故①不正确．

(ii)若②正确，则①③不正确，由①不正确得a＝2，与②正确矛盾，故②不正确．(iii)若③正确，则①②不正确，由①不正确得a＝2，由②不正确及③正确得b＝0，c＝1，故③正确．

则100a＋10b＋c＝100×2＋10×0＋1＝201.14．[2014·全国新课标卷Ⅰ] 甲、乙、丙三位同学被问到是否去过A，B，C三个城市时，甲说：我去过的城市比乙多，但没去过B城市．乙说：我没去过C城市．丙说：我们三人去过同一城市．

由此可判断乙去过的城市为________．

14．A [解析] 由甲没去过B城市，乙没去过C城市，而三人去过同一城市，可知三人去过城市A，又由甲最多去过两个城市，且去过的城市比乙多，故乙只去过A城市．

x14．[2014·陕西卷] 已知f(x)＝x≥0，若f1(x)＝f(x)，fn＋1(x)＝f(fn(x))，n∈N＋，则1＋x

f2014(x)的表达式为________．

xx14.[解析] 由题意，得f1(x)＝f(x)＝ 1＋2014x1＋x

x

1＋xxxf2(x)＝f3(x)＝，…，x1＋2x1＋3x11＋x

由此归纳推理可得f2014(x)＝x.1＋2014x

M2 直接证明与间接证明

21．、[2014·湖南卷] 已知函数f(x)＝xcos x－sin x＋1(x＞0)．

(1)求f(x)的单调区间；

111(2)记xi为f(x)的从小到大的第i(i∈N*)个零点，证明：对一切n∈N*，有x1x2xn

321．解：(1)f′(x)＝cos x－xsin x－cos x＝－xsin x.令f′(x)＝0，得x＝kπ(k∈N*)．

当x∈(2kπ，(2k＋1)π)(k∈N)时，sin x>0，此时f′(x)<0;

当x∈((2k＋1)π，(2k＋2)π)(k∈N)时，sin x<0，此时f′(x)>0.故f(x)的单调递减区间为(2kπ，(2k＋1)π)(k∈N)，单调递增区间为((2k＋1)π，(2k＋2)π)(k∈N)．

ππ(2)由(1)知，f(x)在区间(0，π)上单调递减．又f＝0，故x1＝.2

2当n∈N*时，因为

＋f(nπ)f[（n＋1）π]＝[(－1)nnπ＋1][(－1)n1(n＋1)π＋1]＜0，且函数f(x)的图像是连续不断的，所以f(x)在区间(nπ，(n＋1)π)内至少存在一个零点．又f(x)在区间(nπ，(n＋1)π)上是单调的，故

nπ＜xn＋1＜(n＋1)π.142因此，当n＝1时，＜； x1π3

1112当n＝2时，＋(4＋1)＜ x1x2π3

当n≥3时，1111114＋1＋ 2（n－1）x1x2xnπ

11151＜＜（n－2）（n－1）1×2ππ5＋1－1＋11＋…＋11 223n－2n－1

1162＝6－n－1＜＜π3π

1112综上所述，对一切n∈N*，.x1x2xn3

M3数学归纳法

sin x23．、[2014·江苏卷] 已知函数f0(x)＝(x>0)，设fn(x)为fn－1(x)的导数，n∈N*.x

πππ(1)求2f1＋f2的值； 222

πππ2(2)证明：对任意的n∈N*，等式nfn－1＋n＝ 4442

sin xcos xsin x23．解：(1)由已知，得f1(x)＝f′0(x)＝′＝－，xxx

cos xxsin ′＝ 于是f2(x)＝f1′(x)＝′－xx－sin x2cos x2sin x＋，xxx

ππ4216所以f1＝－f2＝－22πππ

πππ故2f12＝－1.222(2)证明：由已知得，xf0(x)＝sin x，等式两边分别对x求导，得f0(x)＋xf0′(x)＝cos x，π即f0(x)＋xf1(x)＝cos x＝sinx＋.2

类似可得

2f1(x)＋xf2(x)＝－sin x＝sin(x＋π)，3π3f2(x)＋xf3(x)＝－cos x＝sinx＋，2

4f3(x)＋xf4(x)＝sin x＝sin(x＋2π)．

nπ下面用数学归纳法证明等式nfn－1(x)＋xfn(x)＝sinx＋对所有的n∈N*都成立． 2

(i)当n＝1时，由上可知等式成立．

kπ(ii)假设当n＝k时等式成立，即kfk－1(x)＋xfk(x)＝sinx.2

因为[kfk－1(x)＋xfk(x)]′＝kfk－1′(x)＋fk(x)＋xfk′(x)＝(k＋1)fk(x)＋xfk＋1(x)，sinx＋kπ′＝cosx＋kπ·x＋kπ′＝sinx＋（k＋1）π，2222

（k＋1）π所以(k＋1)fk(x)＋xfk＋1(x)＝sinx＋2，因此当n＝k＋1时，等式也成立．

nπ综合(i)(ii)可知，等式nfn－1(x)＋xfn(x)＝sinx＋对所有的n∈N*都成立． 2

πππππnπ令x＝nfn－1＋fn＝sin＋(n∈N*)，424444

πππ所以nfn－1＋fn＝444

M4单元综合(n∈N*)．

第四篇：2014年高考英语(高考真题+模拟新题)分类：M单元++福建

M[2014·福建卷]

阅读下面短文，根据以下提示：1)汉语提示，2)首字母提示，3)语境提示，在每个空格内填入一个适当的英语单词，所填单词要求意义准确，拼写正确。

Many of us were raised with the saying “Waste not, want not.” None of us, 76.，can completely avoid waste in our lives.Any kind of waste is thoughtless.Whether we waste our potential talents, our own time, our limited natural 77.________(资源)，our money, or other people's time, each of us can become more aware and careful.The smallest good habits can ma It's a good feeling to know in our hearts we are doing ourin a world that is in serious trouble.By focusing on 80.________(节省)oil, water, paper, food, and clothing, we are playing a part 81.________ cutting down on waste.We must keep reminding 82.________(自己)that it is easier to get into something in the history of our evolution.It's time for us to 85.________no to waste so that our grandchildren's children will be able to develop well.We can't solve all the problems of waste, but we can encourage mindfulness.Waste not!

76．however考查副词。我们中的许多人在成长的过程中都知道了“不浪费，就不会匮乏”。“然而”，在生活中我们没有一个人能够完全避免浪费。

77．resources考查名词。resource是可数名词，需要使用复数形式。

78．difference考查固定短语。make a(big)difference是固定短语，意为“有(很大的)作用，有(很大的)影响”。

79．best考查固定短语。do one's best是固定短语，意为“竭尽全力，全力以赴”。

80．saving 考查固定短语和非谓语动词。focus on意为“集中于，专注于”，其中on是介词，后接v.­ing形式。

81．in 考查固定短语。play a part/role in„是固定短语，意为“在„„中发挥作用、扮演角色”。

82.ourselves 考查反身代词。我们必须一直提醒我们自己。此处“自己”指代的是前面的主语we，故用ourselves。

83．than 考查连词。根据前面的easier可知，此处需用连词than。

84．done 考查固定短语和非谓语动词。A do damage to B是固定短语，意为“A对B造成伤害”。此处do 与damage是动宾关系，故使用过去分词。

85．say考查固定短语和非谓语动词。现在到了与浪费说再见的时候了。在句型“It's time for sb to do sth.”中，需要使用不定式，故填say。

(一)[2014·福建龙岩质检]

As children, loving our parents is an important part of life.It is our parents 1.________ create us, raise us，make us who we are and keep a roof over our heads in all kinds of weather.Here are some ways to love our parents.Firstly, tell them we love them every day.A gentle “I love you” will 2.w________a cold heart.Parents brought us into this world.3.________ them，we might still wander at an unknown corner of an unknown world.Then, show 4.________(尊敬)to them and don't get angry easily because anger helps 5.________ us nor our parents.Instead, keep 6.c________and sometimes share our feelings with them.Besides，obey their requests, 7.________ will make our attitudes

better.What's more, understand that parents should be 8.f________ when they make mistakes.We should also keep company with them as much as 9.p________.Learn from them by listening to their stories as parents are the 10.________(资源)of our growth and even our teachers in one way or another.1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．who/that 2.warm 3.Without 4.respect 5.neither

6．calm 7.which 8.forgiven/forgivable 9.possible 10.resources

(二)[2014·福建泉州质检]

A boy trembled in the cold winter, wrapping his arms around himself on a bus stop bench.He wasn't 1.________(穿着)warm clothes and the temperature was －10℃.What a heartbreaking scene!But the good 2.d________ of the ordinary people who witnessed the 11-year-old Johannes were both joyous 3.________ inspiring.A woman, sitting next to the boy, discovered he was

4.________ a school trip and was told to meet his teacher at the bus stop.She selflessly

5.c________ her own coat around his shoulders.Later, 6.________ woman at first gave him her scarf, then wrapped him in her large jacket.Throughout the day, more and 7.________ people offered Johannes their gloves and even the coats off their backs.8.________(事实上), it was a hidden camera experiment by Norwegian charity SOS Children's Village as part of their winter campaign to gather donations to send much-needed coats and blankets to 9.h________ Syrian children get through the winter.Synne Ronning, the information head of the organization, also noted that the child was a 10.v________ who was never in any danger during the filming.1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．wearing 2.deeds 3.and 4.on 5.covered 6.another 7.more 8.Actually 9.help

10.volunteer

(三)[2014·福建莆田3月质检]

How time flies!Three years have passed since I 1.________(进入)the school.As a Senior Three student, it won't take long 2.________ I graduate.High school is regarded as the golden time in a person's life.Now, I have much to share 3.________ my schoolmates.Firstly，I'd like to show my appreciation to those 4.s________by me all the way，teachers，parents and 5.________(朋友)included.Without their help and advice，my life would be different.6．________，it's high time to say sorry to classmates whom I hurt or misunderstood.7.________(交际)and smiles act as bridges to friendship.More importantly，I've made up my mind to make every effort to study，for I believe hard work is the key to success.Just as the old saying 8.g________，“No pain, no gain.”

Finally，I hope that every one of us can 9.a________ his/her dream in the near future.I'll attach great significance to our friendship formed at school, and I'd like to keep in contact with you after graduation.Meanwhile，I suggest all the younger fellows make 10.f________use of time，because time waits for no one.1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．entered 2.before 3.with 4.standing 5.friends

6．Secondly 7.Communication 8.goes 9.achieve 10.full

(四)[2014·福建漳州3月质检]

Tony Mott was just an ordinary artist.But then, 1.________ the age of 36, he had an idea which made him famous.It started when he wanted to earn some money for Christmas one year.His product was simple, a 2.s________ message—five words on a T-shirt.He took the T­shirts to a clothes shop and they sold 40 in a week.3.________(马上), he decided to start his own business.He got the business plan right and 4.________ worked.In the last twelve months, he has sold 60，000 T­shirts worldwide.The phrases for the T­shirts come from the things he thinks 5.________ during the day and from conversations with friends at dinner.His 6.________(顾客)who include the rich and famous enjoy his imaginative phrases.Mott says，“I'm successful, but it hasn't 7.c________ my personal life.I still work at home on the same small desk 8.________ I produce all the designs.My friends, who I've 9.k________ for twenty years, are still my friends.In fact, they're as surprised about my 10.s________ as I am.”

1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．at 2.short 3.Immediately/Instantly 4.it 5.of

6．customers 7.changed 8.where 9.known 10.success

(五)[2014·福建厦门3月质检]

The old saying starts, “Give a man a fish and you feed him for a day„” Those words were taken to heart by Robert Egger, who used to be in the restaurant business.He knew from 1.________(经验)how much perfectly good food was 2.t________ away each day.So an organization 3.________ into being to collect leftovers, unserved food from restaurants in the neighbourhood.Volunteers put together more than 3，000 4.________(均衡的)meals a day and distribute them 5.________community centres and homeless shelters.But giving 6.a________ food was only step one.“I wanted to do more，” he says.As the rest of that old saying 7.g________，“Teach a man to fish and you feed him for a lifetime.” That's 8.________the organization also runs a training programme for people who are prepared

9.________ careers in the food service industry.They learn cooking methods.Many graduates find jobs and express their 10.s________ thanks to Egger's training programme.“Whether it is food, money or life，” Egger says, “we can't afford to waste any of them.”

1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．experience 2.thrown 3.came 4.balanced 5.to

6．away 7.goes 8.why 9.for 10.sincere

(六)[2014·福建福州质检]

We moved away from my granny when I was eight years old.I 1.________ her terribly.Two years later my mother and father 2.________(离婚).I felt as if my world was falling apart.My mum must have sensed my longing, so she took my little brother and I back to visit my granny once in a 3.w________．

Granny didn't live in a fancy house 4.________ have expensive things.But it was the little things she gave me that had always mattered.I always remember she saved her pennies in a glass jar.I am sure granny could have used those pennies 5.h________，but she saved them to give us when we came to visit.I don't remember how much we collected on our visits.Those 6.________(记忆)，of when I was a child，still give me 7.w________ feelings on days that I need them.A granny's love stays 8.________ a grandchild, down through the years, even when that child becomes a grandma.I often wonder, after all those years, when I am lucky

9.________ to find a penny 10.l________ on the ground somewhere, if it could possibly be granny giving me pennies from heaven.1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．missed 2.divorced 3.while 4.or 5.herself

6．memories 7.warm/wonderful 8.with 9.enough 10.lying/left

(七)[2014·福建龙岩一检]

When you feel sad, you may think that the feeling will last forever.1.H________，feelings of sadness don't usually last very long—a few moments or maybe a day or two.But sometimes sad feelings can go 2.________ for a long time, hurt deeply, and make 3.________hard for you to enjoy the good things about your life.This 4.k________ of sadness that lasts a lot longer is called depression.People of all 5.________(年龄)can become depressed，6.i________ kids.Depression brings down a person's spirits and energy.It can affect 7.________ people think about themselves and their situations.If you think you have depression or you just have sadness that simply will not go away, sharing it 8.________someone who cares can help.There is always somebody to talk to when you are sad or depressed.You feel better when someone 9.________(知道)what you are going through.Plus，the other person can help you think of ways to make the situation better.But don't spend all your time 10.t________ about what is wrong.Be sure to share the good things, too.1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．However 2.on 3.it 4.kind 5.ages 6.including

7．how 8.with 9.knows 10.talking

(八)[2014·福建四校联考]

Dear future pen pal,My name is Tudor, which is a very Welsh name.Maybe you have never been to Wales, but perhaps you know where it is.It is a small country, but with a big heart, I think.My family live

1.________ a farm, where we mainly have sheep, but there are 2.a________ some cows, which I

like very much.In this part of the country it is quite mountainous.They're not big, 3.h________ mountains, but pretty impressive.Well, I'm just fifteen now, and my future work will be here in these 4.________(田野).My brother is an engineer, so no farming for him, and my sister is now 5.________ and has a small child.So, here I am, an uncle 6.a________，and with three years' school still ahead.I'm very 7.________ in things about China.In fact I'm learning some Chinese from a Chinese student.He 8.________(鼓励)me to find a pen pal in Beijing and that's 9.________ I'm writing this letter.I've put my 10.a________ on the top of the letter.Please write and tell me about yourself.Best wishes.Yours，Tudor

1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．on 2.also 3.high 4.fields 5.married 6.already

7．interested 8.encouraged 9.why 10.address

(九)[2014·福建三校联考]

For 25 years Terry Cemm was a policeman, but for the last 17 years he has been

1.________(行走)up and down five miles of beach every day, looking for things that might be

2.u________ to someone.Terry's a beachcomber(海滩拾荒者)．

Nearly everything in his cottage has come 3.________ the sea—chairs, tables, even tins of food.“I even found a box of beer just before Christmas.That was nice，” he remembers.He finds lots of bottles with messages in them, 4.m________ from children.They all get a

5.r________ if there's an address in the bottle.“Shoes? If you find one, you'll find the

6.________ the next week，” he says.But does he really 7.m________ a living? “Half a living，” he smiles.“Anyway I have my police pension.But I don't actually need money.My life is rich in 8.________(多样性)．” Terry is happy.“You have to find a way to live a simple life.”

“Some people say I'm mad，” says Terry.“9.________ there are many more who'd like to do 10.________ I do.Look at me.I've got everything I could possibly want.”

1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．walking 2.useful 3.from 4.mainly 5.reply

6．other 7.make 8.variety 9.But 10.what

(十)[2014·福建福州模拟]

A rich man was near death and was very upset.He had worked so hard for his money

1.________ he dreamt he could take it with him to heaven.So he 2.________(祈祷)his dream would come true.An angel appeared and said no.The man begged the angel to speak to God to see whether he might 3.________ the rules.The angel reappeared and said that God could permit him to take one

suitcase.4.________(激动地)，the man gathered his suitcase and filled it with pure gold bars.Afterwards, he died and showed up in heaven to greet St.Peter.5.S________ the suitcase，St.Peter said，“Hold 6.________； you can't bring that here！”The man explained that he had God's 7.p________.St.Peter checked it out and said，“You are right.You are allowed 8.o________ suitcase，but I'm supposed to check its contents 9.________ letting it through.”

Inspecting the things that the man found too 10.________(珍贵的)to leave behind, St.Peter exclaimed, “You brought pavement? As you can see, the street of heaven is made of gold！”

1．________ 2.________ 3.________ 4.________ 5.________ 6.________

7.________ 8.________ 9.________ 10.________

【答案】

1．that 2.prayed 3.break 4.Excitedly 5.Seeing/Shown

6．on 7.permission 8.one 9.before 10.precious

第五篇：2012年高考真题文科数学解析分类15：推理与证明1

2012高考文科试题解析分类汇编：推理和证明

1.【2012高考全国文12】正方形ABCD的边长为1，点E在边AB上，点F在边BC上，AEBF

13。动点P从E出发沿直线向F运动，每当碰到正方形的边时反弹，反弹时反

射角等于入射角，当点P第一次碰到E时，P与正方形的边碰撞的次数为

（A）8（B）6（C）4（D）

3【答案】B

【命题意图】本试题主要考查了反射原理与三角形相似知识的运用。通过相似三角形，来确定反射后的点的落的位置，结合图像分析反射的次数即可。

【解析】解：结合已知中的点E,F的位置，进行作图，推理可知，在反射的过程中，直线是平行的，那么利用平行关系，作图，可以得到回到EA点时，需要碰撞8次即可。

2n...sin2.【2012高考上海文18】若SnsinsinnN），则在S1,S2,...,S100777

中，正数的个数是（）

A、16B、72C、86D、100

【答案】C

【解析】依据正弦函数的周期性，可以找其中等于零或者小于零的项.【点评】本题主要考查正弦函数的图象和性质和间接法解题.解决此类问题需要找到规律，从题目出发可以看出来相邻的14项的和为0，这就是规律，考查综合分析问题和解决问题的能力.3.【2012高考江西文5】观察下列事实|x|+|y|=1的不同整数解（x,y）的个数为4，|x|+|y|=2的不同整数解（x,y）的个数为8，|x|+|y|=3的不同整数解（x,y）的个数为12 ….则|x|+|y|=20的不同整数解（x，y）的个数为

A.76B.80C.86D.92

【答案】B

【解析】本题主要为数列的应用题，观察可得不同整数解的个数可以构成一个首先为4，公差为4的等差数列，则所求为第20项，可计算得结果.4.【2012高考陕西文12】观察下列不等式

1

1121

22321

532，5

311

22132142

……

照此规律，第五个不等式为.．．．

【答案】1











116

.【解析】观察不等式的左边发现，第n个不等式的左边=111

2n11n1

n1，右边=5.【2012

k，所以第五个不等式为1





2







116

． 表示为

高考湖南文

k1

16】对于

nN，将n

nak2ak12a12a02,当ik时ai1，当0ik1时ai为0

或1，定义bn如下：在n的上述表示中，当a0,a1，a2，…，ak中等于1的个数为奇数时，bn=1；否则bn=0.（1）b2+b4+b6+b8=__；

（2）记cm为数列{bn}中第m个为0的项与第m+1个为0的项之间的项数，则cm的最大值是___.【答案】（1）3；（2）2.010【解析】（1）观察知1a02,a01,b11；21202,a11,a00,b21；

10210

一次类推31212,b30；4120202,b41；

5120212,b50；6121202，b60，b71,b81，210210

b2+b4+b6+b8=３；（2）由（1）知cm的最大值为２.【点评】本题考查在新环境下的创新意识，考查运算能力，考查创造性解决问题的能力.需要在学习中培养自己动脑的习惯，才可顺利解决此类问题.6.【2012高考湖北文17】传说古希腊毕达哥拉斯学派的数学家经常在沙滩上面画点或用小石子表示数。他们研究过如图所示的三角形数：

将三角形数1,3，6,10，…记为数列{an}，将可被5整除的三角形数按从小到大的顺序组成一个新数列{bn}，可以推测：

（Ⅰ）b2012是数列{an}中的第______项；（Ⅱ）b2k-1=______。（用k表示）【答案】（Ⅰ）5030；（Ⅱ）

5k5k1

n(n1)2

【解析】由以上规律可知三角形数1,3,6,10,…,的一个通项公式为an，写出其若

干项有：1,3,6,10,15,21,28,36,45,55,66,78,91,105,110,发现其中能被5整除的为10,15,45,55,105,110,故b1a4,b2a5,b3a9,b4a10,b5a14,b6a15.从而由上述规律可猜想：b2ka5k

b2k1a5k1

(5k1)(5k11)



5k(5k1)

（k为正整数），5k(5k1)，故b2012a21006a51006a5030,即b2012是数列{an}中的第5030项.【点评】本题考查归纳推理，猜想的能力.归纳推理题型重在猜想，不一定要证明，但猜想需要有一定的经验与能力，不能凭空猜想.来年需注意类比推理以及创新性问题的考查.7.【2102高考北京文20】（本小题共13分）设A是如下形式的2行3列的数表，满足性质P：a，b，c，d，e，f∈[-1,1],且a+b+c+d+e+f=0.记ri（A）为A的第i行各数之和（i=1,2），Cj（A）为第j列各数之和（j=1，2，3）；记k（A）为|r1(A)|, |r2(A)|, |c1(A)|，|c2(A)|，|c3(A)|中的最小值。对如下数表A，求k（A）的值

设数表A形如

其中-1≤d≤0，求k（A）的最大值；

（Ⅲ）对所有满足性质P的2行3列的数表A，求k(A)的最大值。

【考点定位】此题作为压轴题难度较大，考查学生分析问题解决问题的能力，考查学生严谨的逻辑思维能力。

（1）因为r1(A)=1.2，r2(A)1.2，c1(A)1.1，c2(A)0.7，c3(A)1.8，所以

k(A)0.7

（2）r1(A)12d，r2(A)12d，c1(A)c2(A)1d，c3(A)22d.因为1d0，所以|r1(A)|=|r2(A)|d0，|c3(A)|d0.所以k(A)1d1.当d0时，k(A)取得最大值1.（3

任意改变A的行次序或列次序，或把A中的每个数换成它的相反数，所得数表A*仍满足性

*

质P，并且k(A)k(A)，因此，不妨设r1(A)0,c1(A)0,c2(A)0，由k(A)的定义

知

3k，1(A

k(

A)(A

r()c

A)(A,k，(A)

c

从)

(A

c而)

a

(A

()kb)r1

(abcdef)(abf)abf3

因此k(A)1，由（2）知，存在满足性质P的数表A，使k(A)1，故k(A)的最大值为1。

8.【2102高考福建文20】20.（本小题满分13分）

某同学在一次研究性学习中发现，以下五个式子的值都等于同一个常数。（1）sin213°+cos217°-sin13°cos17°（2）sin215°+cos215°-sin15°cos15°（3）sin218°+cos212°-sin18°cos12°

（4）sin2（-18°）+cos248°-sin2（-18°）cos248°（5）sin2（-25°）+cos255°-sin2（-25°）cos255° Ⅰ 试从上述五个式子中选择一个，求出这个常数

Ⅱ 根据（Ⅰ）的计算结果，将该同学的发现推广位三角恒等式，并证明你的结论。

考点：三角恒等变换。难度：中。

分析：本题考查的知识点恒等变换公式的转换及其应用。解答：

（I）选择（2）：sin15cos15sin15cos151

sin30

（II）三角恒等式为：sincos(30)sincos(30)

sincos(30)sincos(30)

sin234sin

234

cos

sin)sin2



sin)