安徽大学 10-11(2)高数C(二)答案
再对两边求导得
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∑ (n + 1)nx
n =1
∞
n −1
=
2 , (1 − x)3
两边同乘以 x 得 ∑ (n + 1)nx n =
n =1
∞
2x , 且该级数的收敛域为(-1,1). (1 − x)3
令x=
∞ 1 1 得 ∑ n(n + 1)( ) n = 8 . 2 2 n =1
⎛E 其中 E 为 n 阶单位矩阵. 由于 ⎜ ⎝0 ⎛E ⎜ ⎝0 ⎛ AB 0 ⎞ ⎛ 0 即⎜ ⎟和⎜ ⎝ B 0⎠ ⎝ B
−1
A ⎞ ⎛ AB 0 ⎞⎛ E ⎟ ⎜ ⎟⎜ E ⎠ ⎝ B 0 ⎠⎝ 0
0 ⎞ ⎟ 具有相同的特征值. BA ⎠
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⎛0 征值和 n 个 0 构成, ⎜ ⎝B
⎛ 1 1 −a ⎞ ⎜ ⎟ A = ⎜ 1 a −1 ⎟ . ⎜ −a −1 1 ⎟ ⎝ ⎠
由于二次型的正、负惯性指数都是 1, 则矩阵 A 的含有一个特征值 0, 且秩为 2. 由此可得 | A |= − a 3 + 3a − 2 = 0. 于是, a = 1 或 a = −2 .
⎛ 1 1 −1⎞ ⎛1 1 2 ⎞ ⎜ ⎜ ⎟ ⎟ 由于 a = 1 时, A = ⎜ 1 1 −1⎟ 的秩为 1; a = −2 时, A = ⎜ 1 −2 −1⎟ 的秩 ⎜ −1 −1 1 ⎟ ⎜ 2 −1 1 ⎟ ⎝ ⎠ ⎝ ⎠
因此, α1 , α 2 , α 3 , α 4 线性相关, 且 r (α1 , α 2 , α 3 , α 4 ) = 3 . 由于 β1 , β 2 , β 4 为 β1 , β 2 , β 3 , β 4 的一个极大无关组, 且 β 3 = β1 − 5β 2 , 故
α1 , α 2 , α 4 为 α1 , α 2 , α 3 , α 4 的一个极大无关组, 且 α 3 = α1 − 5α 2 .
解 矩阵 A 的特征多项式
λ −1
det(λ E − A) = 0 0
0 λ −2 −1
0 −1 = (λ − 1) 2 (λ − 3),
λ −2
故 A 的特征值为 λ1 = 1 (二重), λ2 = 3. 对于 λ1 = 1 , 解齐次方程组 ( E − A) X = 0 , 得到一个基础解系为
α1 = (1, 0, 0)T , α 2 = (0,1, −1)T .
安徽大学 2010—2011 学年第二学期 《高等数学 C（二） 》 （A 卷）参考答案及评分标准
一、填空题（每小题 2 分，共 10 分） 1. 收敛. 3.
n(n − 1) −k . 2
2. C1 cos x + C2 sin x , 其中 C1 , C2 为任意常数. 4. −2 .
5. yn = C 2n − 1 , 其中 C 为任意常数. 二、选择题（每小题 2 分，共 10 分） 1. A. 2. C. 3. C. 三、计算题（每小题 10 分，共 60 分） x a " a a x " a 1. 计算行列式 . # # # a a " x x a " a a x " a 解 令 Dn = , 将各行加到第一行, 则第一行有公因子 x + ( n − 1) a , # # # a a " x 将其提到行列式之外得 1 1 " 1 a x " a . Dn = [ x + (n − 1)a ] # # # a a " x 将各行加上第一行的 −a 倍得 1 1 " 1 0 x−a " 0 Dn = [ x + (n − 1)a ] = ( x − a ) n −1[ x + (n − 1)a ]. # # # 0 0 " x−a 2. 设 AX = b 为三阶非齐次线性方程组， 已知 r ( A) = 1 ， η1 , η 2 , η3 是它的三个解， 并且
α 4 = (7,1, 0, −1,3)T .
（1）判定 α1 , α 2 , α 3 , α 4 的线性相关性，并求秩. （2）求 α1 , α 2 , α 3 , α 4 的一个极大无关组，并用这个极大无关组表示其余向量.
⎛6 1 ⎜ ⎜4 0 解 令 A = (α1 , α 2 , α 3 , α 4 ) = ⎜ 1 2 ⎜ ⎜ −1 3 ⎜ 2 −4 ⎝ 通过初等行变换将 A 化为阶梯形矩阵.
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1 7⎞ ⎟ 4 1⎟ −9 0 ⎟ . ⎟ −16 −1⎟ 22 3 ⎟ ⎠
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−9 0 ⎞ 1 7 ⎞ ⎛ 1 2 −9 0 ⎞ ⎛ 1 2 ⎛6 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 4 1⎟ ⎜4 0 4 1 ⎟ ⎜ 0 −8 40 1 ⎟ ⎜4 0 ⎜ 1 2 −9 0 ⎟ → ⎜ 6 1 1 7 ⎟ → ⎜ 0 −11 55 7 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −1 3 −16 −1⎟ ⎜ −1 3 −16 −1⎟ ⎜ 0 5 −25 −1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 −4 22 3 ⎠ ⎝ 2 −4 22 3 ⎠ ⎝ 0 −8 40 3 ⎠ ⎛ 1 2 −9 0 ⎞ ⎛ 1 2 −9 ⎜ ⎟ ⎜ ⎜ 0 −8 40 1 ⎟ ⎜ 0 0 0 → ⎜ 0 −1 5 5 ⎟ → ⎜ 0 −1 5 ⎜ ⎟ ⎜ ⎜ 0 5 −25 −1⎟ ⎜ 0 0 0 ⎜ ⎟ ⎜ ⎝ 0 −8 40 3 ⎠ ⎝ 0 0 0 0⎞ ⎛1 ⎟ ⎜ 1⎟ ⎜0 5⎟ → ⎜0 ⎟ ⎜ 1⎟ ⎜0 ⎜ 1⎟ ⎠ ⎝0 0 1 1 −5 0 0 0 0 0 0 0⎞ ⎟ 0⎟ 1 ⎟ =: ( β1 , β 2 , β3 , β 4 ) ⎟ 0⎟ 0⎟ ⎠
0 ⎞ ⎛ E A ⎞ ⎛ AB ABA ⎞ ⎟, ⎟⎜ ⎟=⎜ BA ⎠ 0⎠⎝ 0 E ⎠ ⎝ B A ⎞⎛ 0 0 ⎞ ⎛ AB ABA ⎞ ⎟, ⎟⎜ ⎟=⎜ E ⎠⎝ B BA ⎠ ⎝ B BA ⎠ A⎞ ⎟ 为可逆矩阵, 故 E⎠ A⎞ ⎛ 0 0 ⎞ ⎟=⎜ ⎟, E ⎠ ⎝ B BA ⎠ ⎛ AB 0 ⎞ 由于 ⎜ ⎟ 特征值是由 AB 的特 ⎝ B 0⎠
4. B.
5. D
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⎛1 ⎞ ⎟ η1 + η2 = ⎜ ⎜ 2⎟, ⎜3⎟ ⎝ ⎠
求该方程组的通解. 解
⎛0 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎟ η2 + η3 = ⎜ −1⎟ , η3 + ⎝ ⎠
因为 r ( A) = 1 , 故导出组 AX = 0 的基础解系含有两个解向量.
五、证明题（共 10 分） 设 A, B 均为 n 阶方阵， （1）若 A 或 B 可逆， 证明 AB 与 BA 具有相同的特征值. （2）若 A, B 均不可逆，上述结论是否正确？并说明理由. 证 (1) 不妨假设 A 可逆, 此时 A−1 ABA = BA , 即 AB 与 BA 相似, 故具有相 同的特征值. (2) 结论正确. (方法一) 考虑分块矩阵 ⎛ AB ⎜ ⎝ B ⎛E ⎜ ⎝0
确.
0 ⎞ ⎟ 的特征值是由 BA 的特征值和 n 个 0 构成, 故结论正 BA ⎠
(方法二) 设 λ 为 AB 的特征值, 对应特征向量为 X , 则 ABX = λ X . 两边同时左乘 B 得
BABX = BA( BX ) = λ ( BX ) .
若 BX ≠ 0 , λ 是 BA 的特征值. 若 BX = 0 , 由于 X ≠ 0 , 则此时 λ = 0 , 即 | B |= 0 . 因为 | BA |=| B || A |= 0 , 所以 λ = 0 也是 BA 的一个特征值. 同理可证若 λ 是 BA 的一个特征值, 则 λ 也是 AB 的一个特征值. 综上所述, 结论正确.
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= x( ∫
ln x dx + C ) x2
1 = x[ (ln x + 1) + C ] x = ln x + 1 + Cx,
其中 C 为任意常数.
将 x = 1, y = 1 带入得 C = 0 , 故原方程在初始条件 y (1) = 1 下的解为 y = ln x + 1.
四、分析计算题（共 10 分） 给定向量组 α1 = (6, 4,1, −1, 2)T , α 2 = (1, 0, 2,3, −4)T , α 3 = (1, 4, −9, −16, 22)T ,
y ln x + = 0 在初始条件 y (1) = 1 下的解. x x 1 ln x 解 原方程可化为 y '− y = − , 为一阶非齐次线性方程. x x 故原方程的通解为
6． 求微分方程 y '−
y = e ∫ x [ ∫ (−
dx
1
)dx ln x ∫ ( − 1 )e x dx + C ] x
因此, 原方程组的通解为
⎛ −1⎞ ⎛1 ⎞ ⎛1 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 / 2 ⎟ + C1 ⎜ 3 ⎟ + C2 ⎜ −1⎟ , ⎜1/ 2 ⎟ ⎜ 2⎟ ⎜2 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
其中 C1 , C2 为任意常数. (注: 本题的答案不唯一, 只要结论正确均给分)
⎛1 0 0⎞ ⎜ ⎟ 3. 设矩阵 A = ⎜ 0 2 1 ⎟ ， 求正交矩阵 Q ，使 Q −1 AQ 为对角矩阵. ⎜0 1 2⎟ ⎝ ⎠
⎛1 ⎞ ⎟ ξ1 = η1 − η3 = (η1 + η2 ) − (η2 + η3 ) = ⎜ ⎜3⎟, ⎜ 2⎟ ⎝ ⎠ ⎛ −1⎞ ⎟ ξ 2 = η2 − η1 = (η2 + η3 ) − (η1 + η3 ) = ⎜ ⎜ −1⎟ , ⎜2 ⎟ ⎝ ⎠
是导出组的两个线性无关的解向量.
⎛1 ⎞ 1 ⎟ η1 = [(η1 − η3 ) + (η1 + η3 )] = ⎜ 3/ 2⎟ . ⎜ 2 ⎜1/ 2 ⎟ ⎝ ⎠
安徽大学 10-11(2)高数A(二)、B(二)答案
+
2x
∂z
⎤ ⎥
∂y ⎥⎦
=
x2z2
sin
z + 2x2z cos z (cos z − xy)3
−
2x3 yz
．
四、应用题（每小题 8 分，共 16 分）
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1． 解． 构造 Lagrange 函数 L(x, y, z, λ, μ) = x + 2 y + 3z + λ(x2 + y2 − 2) + μ( y + z −1) ． 求偏导得 Lx = 1+ 2λ x, Ly = 2 + 2λ y + μ, Lz = 3 + μ , Lλ = x2 + y2 − 2, Lμ = y + z −1, 联立解得 x = −1, y = 1, z = 0 或 x = 1, y = −1, z = 2 ． 代入原函数得 f (−1,1, 0) = 1, f (1, −1, 2) = 5 ． 故所求最大值为 5, 最小值为1.
∫ 2． 解． 所求金属丝的质量为 m = ρds ． L
弧微分 ds = [x '(t)]2 + [ y '(t)]2 + [z '(t)]2 dt = 3etdt ．
∫ ∫ 故 m =
11 0 2e2t
3etdt = 3 1e−tdt = 3 (1− e−1) ．
20
2
五、证明题（每小题 5 分，共 10 分）
1 ． 证 明 ． 设 f (x) = x ， 则 f '(x) = 2011− x ， 显 然 x ≥ 2011 时 ，
x + 2011
2 x (x + 2011)
2012-2013学年安徽大学《高等数学 C(二)》(A卷)考试试题及参考答案
安徽大学2012—2013学年第二学期 《高等数学C (二)》考试试卷（A 卷）（闭卷 时间120分钟）院/系 年级 专业 姓名 学号答 题 勿 超 装 订 线 ------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------考场登记表序号_______题 号 一 二 三 四 五 总分 得 分阅卷人得分一、填空题（每小题3分，共15分）1、设，1224311A t−⎛⎞⎜⎟=⎜3⎜⎟−⎝⎠⎟B 为三阶非零矩阵，若0AB =，则__________． t =2、若A 为三阶矩阵，行列式 2A =，2B A =，B ∗为B 的伴随矩阵，则 B ∗=__________．3、若行列式 33 ij D a ×=满足111a =，122a =，130a =，且余子式，31 8M =−32M x =，，则3319M =x =__________．4、已知向量组，，，，若1(1, 0, 2)T α=2(1, 1, 3)T α=3(1,1, 2)T k α=−+(1, 2, 5)T β=β不．能．由12,,3ααα线性表示，则k =__________．5、若阶矩阵n A 的秩为，且1n −A 的各行元素之和均为，则齐次线性方程组00AX =的通解是__________．二、选择题（每小题3分，共15分）得分6、已知A ，B ，C 均为阶矩阵，则下列结论正确的是 （ ）n A ． 22()2A B A AB B +=++2m B ．，其中为正整数 ()m m AB A B =m C ．若AB AC =且，则0A ≠B C =D ．若，则ABCE =BCA E =，其中E 为n 阶单位矩阵7、设1α，2α均为维向量，向量n 1β，2β，3β均可以由1α，2α线性表示，则下列结论正确的是 （ ） A ．1β，2β，3β必线性无关 B ．1β，2β，3β必线性相关C ．仅当1α，2α线性无关时，1β，2β，3β线性无关D ．仅当1α，2α线性相关时，1β，2β，3β线性相关8、设A 为矩阵，则下列结论正确的是 （ ） m n × A ．若，则方程组m n <AX b =必有无穷多解B ．若，则方程组m n <0AX =必有非零解，且基础解系含有个线性无关解向量 n m −C ．若A 有阶子式不为零，则方程组n 0AX =仅有零解D ．若A 有n 阶子式不为零，则方程组AX b =有唯一解9、下列选项中，哪个不是．．“()ij n n A a ×=为正交矩阵”的充分条件 （ ） A ．A 的行向量组与列向量组均为正交向量组 B ．1A =，且对任意i j ,1,2,,n "，有ij ij a A = =C ．为正交矩阵 T A D ．1T A A −=10、若三阶矩阵A 有特征值122λλ==，E 为三阶单位矩阵，且|，则||A E −=0|A 为 （ ）A ．−B ．C ．224−D ．4三、计算题（每小题9分，共54分）得分11、计算n 阶行列式1211111111n n a a D a ++=+"""""""1，其中．120n a a a ≠"答 题 勿 超 装 订 线 ------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------12、若三维向量123(,,)a a a α=，123(,,)b b b β=，且211211211T A αβ⎛⎞⎜⎟==−−−⎜⎟⎜⎟⎝⎠，求:（1）T βα；（2）． 2A13、已知矩阵，判断021332121A ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠A 是否可逆．如果可逆，求；如果不可逆，请说明理由． 1A −14、求向量组，，，的秩和一个极大线性无关组，并把其余向量用该极大无关组线性表示． 1(1,0,2,0)T α=2(0,1,1,2)T α=−3(1,2,4,4)T α=−4(2,1,4,2)T α=−15、已知，，若20000101A a ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠20003402B b ⎛⎞⎜=⎜⎜⎟−⎝⎠⎟⎟A 与B 相似，求a ，b 的值．答 题 勿 超 装 订 线 ------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------16、已知方程组有无穷多个解，求123123123112x x x x x x x x x λλλ++=⎧⎪++=⎨⎪++=−⎩λ的值及方程组的通解．四、分析计算题（每小题10分，共10分）得分17、设二次型222123123121323(,,)4484f x x x x x x x x x x x x =++−−−,（1）判断二次型是否正定；（2）利用正交变换X QY =化二次型为标准形，并求出相应的正交矩阵． Q得分五、证明题（每小题6分，共6分）18、已知n 阶矩阵A 满足 32A E =，其中E 为阶单位矩阵，若n 2B A A =+，证明B 可逆，并求B 的逆矩阵．安徽大学2012—2013学年第二学期 《高等数学C （二）》考试试卷（A 卷）参考答案与评分标准一、填空题（每小题3分，共15分）1、；2、256；3、；4、3−4−1−；5、，其中为任意常数(1,1,,1)T k "k二、选择题（每小题3分，共15分）6、D ；7、B ；8、C ；9、A ； 10、D三、计算题（每小题9分，共54分）11、解：从第二行起，每行减去第一行，再从第二列起，第i 列的1ia a 倍加到第一列上，得(2,3,,i n =")111221311111110011111100n nna a a a a D a a a a ++−+==−+−""""""""""""""""10a ．．．．．．（4分） 112212131111001(1)000000ni in n i ina a a a a a a a a a ==++==∑+∑""""""""""．．．．．．．（9分） 12、解：（1）因为，()111121321232122233313233211211211T a a b a b a b a b b b a b a b a b a a b a ba b αβ⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟===−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠−−+=所以． ()1123211223332(1)12T a b b b a a b a b a b a βα⎛⎞⎜⎟==++=+−⎜⎟⎜⎟⎝⎠．．．．．．（5分）（2）2422()22422422T T T A A αβαβαβ⎛⎞⎜⎟====−−−⎜⎜⎟⎝⎠⎟． ．．．．．．（9分）13、解：利用初等变换法可以直接判断A 是否可逆，并求出1A −：()021100,332010121001A E ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠121001021100332010⎛⎞⎜⎟→⎜⎟⎜⎟⎝⎠10010102022613001322⎛⎞⎜⎟−⎜⎟→−−⎜⎟⎜⎟−⎜⎟⎝⎠100101010113001326−⎛⎞⎜⎟→−−⎜⎟⎜⎟−⎝⎠，．．．．．．（7分）故A 可逆，且1101113326A −−⎛⎞⎜=−−⎜⎜⎟⎟⎟−⎝⎠． ．．．．．．（9分）（注：若先由02133210121A ==≠判断出A 可逆，则给3分；之后正确求出1A −，则给9分．）14、解：依题意，将向量组按列排成矩阵并作初等行变换()123410120121,, , 21440242αααα⎛⎞⎜⎟−−−⎜⎟=⎜⎟⎜⎟⎝⎠1012012101200242⎛⎞⎜⎟−−−⎜⎟→⎜⎟⎜⎟⎝⎠1012012100010000⎛⎞⎜⎟−−−⎜⎟→⎜⎟−⎜⎟⎝⎠1010012000010000⎛⎞⎜⎟⎜⎟→⎜⎟⎜⎟⎝⎠， ．．．．．．（5分）故，()1234, , , 3r αααα=124,,ααα为向量组的一个极大无关组，且3122ααα=+． ．．．．．．（9分）15、解：由相似矩阵的性质，一方面A B =，即381b +=−，得．3b =− ．．．．．．（5分）另一方面，相似矩阵有相同的特征值，故()()tr A tr B =， 即2，得．5a +=+b 0a =．．．．．．（9分）16、解：依题意，对方程组的增广矩阵作初等行变换111112111111112111A λλλλλλ−⎛⎞⎛⎞⎜⎟⎜⎟=→⎜⎟⎜⎟⎜⎟⎜⎟−⎝⎠⎝⎠2112011301112λλλλλλ−⎛⎞⎜⎟→−−⎜⎟⎜⎟−−+⎝⎠ 112011300(1)(2)2(2)λλλλλλ−⎛⎞⎜⎟→−−⎜⎟⎜⎟−++⎝⎠， 故当2λ=−时，()()2r A r A ==，方程组有无穷多个解． ．．．．．．（4分）此时对应的同解方程组为1232322333x x x x x +−=−⎧⎨−+=⎩，令自由未知量，得该方程组的一个特解．30x =(1,1,0)T η=−−其对应齐次方程组1232320330x x x x x +−=⎧⎨−+=⎩的基础解系为，(1,1,1)T ξ=因此原方程组的通解为，其中为任意常数． ．．．．．．（9分）(1,1,1)(1,1,0)T x k k ξη=+=+−−T k四、分析计算题（每小题10分，共10分）17、解：（1）因为二次型的矩阵为124242421A −−⎛⎞⎜⎟=−−⎜⎟⎜⎟−−⎝⎠，2124242(5)(4)421E A λλλλλλ−−=−=−+=−0，所以A 的特征值为125λλ==，34λ=−．由于A 有一个特征值为负数，故A 不正定，该二次型不正定．．．．．．．（4分）（2）对于方程组(5，)0E A x −=解得基础解系为11(,1,0)2T ξ=−，．2(1,0,1)T ξ=−先正交化，得111(,1,0)2T ηξ==−，2122111(,)42(,,1)(,)55T ξηηξηηη=−=−−，再单位化，得111(T )ηγη==，222(Tηγη==． 对于方程组(4，解得基础解系， )0E A x −−=3(2,1,2)T ξ=单位化得333212(,,333T ξγξ==． ．．．．．．（6分） 故所求正交矩阵()123,,0Q γγγ⎛⎜⎜⎜==⎜⎜⎜⎜⎝， f 的标准形为221255423f y y y =+−． ．．．．．．（10分）五、证明题（每小题6分，共6分）18、证明：一方面，由32A E =知，A 可逆且1212A A −=． 另一方面，由32A E =得，33A E E +=，即2()()3A E A A E E +−+=，所以A E +可逆，且121()(3)A E A A −E +=−+． ．．．．．．（4分）由A ，A E +均可逆知，2()B A A A A E =+=+也可逆，且11(())()11B A A E A E A −−=+=+−−2243111()(3262)A A E A A A A =−+=−+． ．．．．．．（6分）。
2008-2009学年安徽大学《高等数学 C(二)》(A卷)考试试题及参考答案
安徽大学2008—2009学年第二学期院/系 专业 姓名 学号答 题 勿 超 装 订 线------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------《高等数学C （二）》考试试卷（A 卷）（闭卷 时间120分钟）题 号 一 二 三 四 五 总 分得 分阅卷人得分一、填空题（每小题2分，共10分）1．已知两个4维向量与21(1,,1,0)t α=2(2,1,3,2)t α=−正交，则＝ t . 2．幂级数221212n nn n x ∞−=−∑的收敛半径为 . 3．设，100220345A ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠A ∗是A 的伴随矩阵，则1()A ∗−= .4．设平面区域：0,D 01x y y ≤≤≤≤(,)，f x y 在上连续，则利用极坐标变换可将二重积分D (,)Df x y d σ∫∫ 化为 .5．二次型22212312224243x x x x x x ++++x 的秩为 .得分 二、单项选择题（每小题2分，共10分）6. 二元函数222222,0(,)0,0xyx y x y f x y x y ⎧+≠⎪+=⎨⎪+=⎩在点处( ).(0,0)A. 连续，偏导数也存在 B. 连续，偏导数不存在C. 不连续，偏导数存在D. 不连续，偏导数也不存在7．若,A B 均为同阶可逆矩阵，则必有( ) . A. A 可经行初等变换变到B B. A B =C. 存在可逆矩阵，使得P 1P AP B −=D. A B +为可逆矩阵8．若阶矩阵n A 的一个特征值为2，则23A A E ++必有一个特征值为( ) .A. 0B. 1C. 11D. 不能确定9．若级数收敛，则( ) .1(n n n a b ∞=+∑)A. 、中至少有一个收敛 B. 1n n a ∞=∑1n n b ∞=∑1n n a ∞=∑、1n n b ∞=∑均收敛C. 1n n n a b ∞=+∑收敛 D. 1n n a ∞=∑、1n n b ∞=∑敛散性相同10. 差分方程的通解为 ( ) (其中为任意常数) .2132t t t y y y ++−+=02222C 1,C C A. B. C. 1C t C +12t C C +1(2)t C −+ D.12(1)t C C −+三、计算题得分(第11小题至第14小题每题8分，第15小题至第17小题每题10分，共62分)11. 已知sin y z x =，求(1) zx ∂∂、z y ∂∂； (2) ； (3) d z 2z x y ∂∂∂.12. 求二重积分cos Dxdxdy x∫∫，其中为直线D y x =与抛物线2y x =所围成的区域.院/系 专业 姓名 学号答 题 勿 超 装 订 线------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------13. 求微分方程32x y y y e −′′′−+=的通解.14. 将1()f x x=展开成的幂级数，并求该幂级数的收敛半径、收敛域. (3x −)⎟⎟15. 已知，. 若201030202A ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠100010000B ⎛⎞⎜=−⎜⎜⎟⎝⎠X 满足22AX B BA X +=+，求X .16．求矩阵的特征值和特征向量；判断它是否可以对角化，并说明理由.110430102A −⎛⎞⎜⎟=−⎜⎜⎟⎝⎠⎟0,院/系 专业 姓名 学号答 题 勿 超 装 订 线------------------------------装---------------------------------------------订----------------------------------------线----------------------------------------17．对于非齐次线性方程组1231231231,220.x x x x x x x x ax ++=⎧⎪++=⎨⎪−+=⎩(1) a 为何值时，方程组无解；(2) a 为何值时，方程组有解，并求其解.得分 四、应用题（本题10分）18．在平面上求一点，使它到三条直线0x =、0y =、2160x y +−=距离的平方和最小.五、证明题（本题8分） 得分19．设A 为矩阵，其秩为，m n ×AX b =r β是非齐次线性方程组的一个解，0AX =12,,,n r ααα−"是对应的齐次线性方程组的一个基础解系.证明：向量组12,,,,n r ααα−"β 线性无关.安徽大学2008－2009学年第二学期《高等数学 C（二）》考试试卷（A 卷）参考答案及评分细则一、填空题（每小题2分，共10分）1．1或； 3. 110A ； 4.csc 204(cos ,sin )d f r r r πθπθθ∫∫dr θ； 5. .2二、单项选择题（每小题2分，共10分）6. C；7. A；8. C；9. D； 10. B.三、计算题(第11小题至第14小题每题8分， 第15小题至第17小题每题10分，共62分)11. 已知sin yz x =，求(1) z x ∂∂、z y ∂∂； (2) ； (3) d z 2z x y ∂∂∂.解：2cos z y y x x x ∂=−∂,1cos z y y x x∂=∂ 21cos cos y y ydz dx dy x x x x=−+22(cos )z y y x y y x ∂∂=−∂∂∂x 231cos sin y y y x x x x =−+ 12. 求二重积分cos Dxdxdy x∫∫，其中为直线D y x =与抛物线2y x =所围成的区域. 解：cos Dxdxdy x ∫∫210cos x x x dx dy x=∫∫120cos ()xx x dx x=−∫1(cos cos )x x x d =−∫x=1cos1−13. 求微分方程32x y y y e −′′′−+=的通解.解：方程对应的齐次微分方程为：32y y y 0′′′−+= 0 其特征方程为，解得232λλ−+=121, 2λλ==.故齐次方程的通解为：212x x C e C e +. 设非齐次方程的一个特解为x y Ae ∗−=代入原方程得到32x x x x Ae Ae Ae e −−−++=−，故16A =这样原方程的通解为：21216x x x C e C e e −++.14. 将1()f x x =展开成的幂级数，并求该幂级数的收敛半径、收敛域.解：(3x −)1111()33331()3f x x x x ===⋅−+−+ 而01(1)1n n n x x ∞==−+∑，，(1,1)x ∈− 故11331()3x ⋅−+013(1)()33n n n x ∞=−=−∑＝1(3)(1)3n n n n x ∞+=−−∑ 且313x −<，于是33x −<，收敛半径为3r =， 收敛区域为.(0,6)15．已知，.若201030202A ⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠100010000B ⎛⎞⎜⎟⎟=−⎜⎜⎟⎝⎠X 满足22AX B BA X +=+，求X . 解：由 22AX B BA X +=+得到：(2)(2)A E X B A E −=−，从而1(2)(2)X A E B A E −=−−又，001(2)010200A E ⎛⎞⎜⎟−=⎜⎟⎜⎟⎝⎠11002(2)010100A E −⎛⎞⎜⎟⎜⎟−=⎜⎟⎜⎟⎜⎟⎝⎠这样，1200100001010010010100000200X ⎛⎞⎜⎟⎛⎞⎛⎞⎟⎟⎟⎠000010001⎛⎞⎜⎟=−⎜⎟⎜⎟⎝⎠⎜⎟⎜⎟⎜=−⎜⎟⎜⎟⎜⎜⎟⎜⎜⎟⎝⎠⎝⎜⎟⎝⎠⎟⎟16．求矩阵的特征值和特征向量；判断它是否可以对角化，并说明理由.110430102A −⎛⎞⎜=−⎜⎜⎟⎝⎠解：1104301022(1)(2λλE A λλλλ+−−=−−−)=−− 令0E A λ−=解得特征值为12λ=，231λλ== 对于12λ=，解方程组，得基础解系为：123(2)0x E A x x ⎛⎞⎜⎟−⎜⎟⎜⎟⎝⎠=1(0,0,1)T η=故属于12λ=的全部特征向量为1(0,0,1)T k 1(0k )≠ 对于231λλ==，解方程组，得基础解系为：123()x E A x x ⎛⎞⎜⎟−⎜⎟⎜⎟⎝⎠0=2(1,2,1)T η=−故属于231λλ==的全部特征向量为2(1,2,1)T k −2(0k )≠ 因A 只有两个线性无关的特征向量，故A 不能对角化.17．对于非齐次线性方程组1231231231,220.x x x x x x x x ax 0,++=⎧⎪++=⎨⎪−+=⎩(1) 为何值时，方程组无解；a (2) 为何值时，方程组有解，并求其解. a 解：方程组对应系数的增广矩阵为：11 1 112 2 011 0A a ⎛⎞⎜⎟=⎜⎟⎜⎟−⎝⎠111 1011 102 1 1 a ⎛⎞⎜⎟→−⎜⎟⎜⎟−−−⎝⎠11 1 1011 100 13 a ⎛⎞⎜⎟→−⎜⎟⎜⎟+−⎝⎠(1) 当时方程组无解；10a +=(2) 当即时，方程组有唯一解，其解为：10a +≠1a ≠− 123 23 113 1x x a x a ⎧⎪=⎪⎪=−⎨+⎪⎪=−⎪+⎩. 四、应用题（本题10分）18．在平面上求一点，使它到直线0x =，0y =及2160x y +−=的距离的平方和最小.解：设所求的点为(,)x y ，则它到0x =，0y =及2160x y +−=的距离分别为x ，y，于是由题意，距离的平方和为：221(216)5s x y x y =+++−2令22(216)0542(216)05s x x y x s y x y y∂⎧=++−=⎪∂⎪⎨∂⎪=++−=∂⎪⎩，解得唯一驻点816(,)55根据实际意义所求的点一点存在，即为816(,55.五、证明题（本题8分）设β是非齐次线性方程组AX b =的一个解，12,,,n r ααα−"是对应的齐次方程组的一个基础解系，证明：12,,,,n r ααα−"β线性无关.证明：设11220n r n r k k k k ααα−−++++="βr ，因为0,(1,2,,)i A i n α=="−，于是A 左乘上式两端得到0kA β=，而0A b β=≠，故0k =于是11220n r n rk k k ααα−−+++="，而12,,,n r ααα−"是0AX =的一个基础解系，从而线性无关，故，这样120n r k k k k −====="12,,,,n r ααα−"β线性无关.。
安徽大学期末试卷MK_10-11(1)高数A(三)答案.pdf
安徽大学2010—2011学年第一学期《高等数学A （三）》（B 卷）考试试题参考答案及评分标准一、选择题（本大题共5小题，每小题2分，共10分） 1.A 2. C 3. D 4.C 5. B二、填空题（本大题共5小题，每小题2分，共10分） 6.22E A + 7. 8.24649 9.43 10.]15.20,87.19[三、计算题（本大题共10分）11.解：将的第2列直到第列依次加到第1列得,n D n ma a a m a a a m a ma a maa m a m a a a maD n nn ni i n ni in ni i n ni in −−−=−−−−−=∑∑∑∑====""""""""""""""2221212121111)(,再将第1行乘上1−分别加到第2行直到第行得,n ).()(00001)(1121m a m mm a a m a D n i i n nn i i n −−=−−−=∑∑=−="""""""四、分析题（本大题共6小题，共62分） 12.（本小题12分）解：方程组的增广矩阵为:⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+−−−−−→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=a a a a a a a a a a a a A 2111031102111111112112111111112, ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛++−−−−→a a a a a a 24)2)(1(003110211于是,(1)当时,,1=a 1)(=A r 3)(=A r ,方程组无解;(2)当21−≠≠a a 且时,3)()(==A r A r ,方程组有唯一解;(3)当时,2−=a 2)()(==A r A r ,方程组有无穷多解. 当方程组有无穷多组解时,此时⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−→⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−=000033302211221111211112A ,对应的线性方程组为:⎩⎨⎧=+−−=−+3332232321x x x x x 令，得到原非齐次线性方程组的一个特解：. 03=x T )0,1,1(0−−=γ原非齐次线性方程组对应的导出组为:⎩⎨⎧=+−=−+0330232321x x x x x , 令，得到基础解系为； 13=x T )1,1,1(=η故原非齐次线性方程组的结构解为:ηγk X +=0，k 为任意常数。
2年高考系列——安徽省2010-2011年理科高考数学题试题及答案解析
legalization of archive s and information. Investee 4. i nsist on t he logisti cs, will be around strong str ong reforms, cadres of service management. Office w orkers is to be for thi s, de epe n reform, strengthen management, a nd conti nuously meet the ca dres of the organs function properly a nd the nee d for material and cult ural life. One is to enhance the l ogistica l support capabilit y. Ada pt to the new situation of the market economy, maki ng full use of social pow er, flexibility in the use of market mecha nisms a nd a ccel erate the pr oce ss of marketization and sociali zation of logisti c servi ces. A dhe re to the institutionalizati on of the logisti cs management, esta blish and impleme nt a financial manageme nt system, trolley ma nagement systems, healt h manageme nt systems, se curity sy stems, such a s the l ogistics manageme nt system, strengt hen t he manageme nt of human, fina ncial , material, and l og istical service s standardized a nd i nstitutionalize d track. Se condly, we shoul d strengt hen t he manageme nt ofconstructi on. Increasi ng investment, im proving t he worki ng conditions of organs; overall civilizati on create an a ctivity, a nd constantly improv e the level of civilizati on; t o prom ote greeni ng, lighti ng, purification, flower s, landscapi ng, Office w orkers create a g ood environment for work a nd life. T hir d, to improve staff welfare. To in poli cy allow s of range wit hin, tryi ng to for Office ca dres w orkers sol ution housing, a nd couple s separati on, a nd chil dren employme nt, aspects of actual difficult pr ovide s hel p, ha s conditi ons of pla ce and units, also should a ppropriate to i ncrea sed ca dres w orkers of welfare, real do "care er left people, feeling s left peo ple, a ppr opriate treatment left pe opl e", let General Office staff in pay har d efforts of while , feel organizati on of warm, get due of affordable. Investee 5. i nsist on the team, will focus on stimul ating vitality who care a nd re spe ct for pe ople. o Office ca dres for the cut ... Lear ning style , strengthen the theor etical knowle dge, optimize the struct ure of knowledgeand conti nuously im prove the quality of cadres a nd w orkers. Attention to the ir physi cal health, a nd a ctively carry out regular w hole some re creationaleople-oriente d in the name of reduci ng our enthusia sm. Premier We n Jiaba o's Governme nt work report this yea r, ide ntified as 7% gr owth, 2.1% low er than la st year's 9.1%, but i n 8 work remai ns to a ccelerate development i n the first pla ce ... Work standards. Second is to use the scale of developm ent to understand. Society is ev olving, human nee ds are also changi ng. Pra ctice of humani sm is a long-term pr ocess and ca nnot be achieve d overnig ht. To treat people if you stand still and rigid dogma, it is possible t o return to t his, see thing s but not people's old ways. T herefore, w e have to foll ow the development of the times, a nd continue de buggi ng the humani st perspective , innovative pe ople -oriented wa y. Under the new histori cal conditions, we are re que sted by the times, set up a bra nd new concept, a high starting poi nt, hig h standard of armed people, shapi ng, exerci se, people, consta ntly taking the new connotation of the times. Terms of Office, i s to lead t he vast num bers of ca dres to a ctivelyadapt to t he knowledge faster a nd a ctively ada pt to the i ncrea singly high level of leader shi p situati on, a ctively ada pt to th e dema nd of the masses m ore and more sit uations, eager t o learn, enhance accum ulation, efforts, hone d the a bility to overcome a g ood servi ce, pani c. Thir d is to use reali stic understanding. dher e to pe opl e-oriente d must be base d on reality, not divorced from reality. Desk no water cups here i n Hong Kong, staff need to drink some water to t he dri nking fountai n on a floor, the gla ss used i s a one-time non-cone type. Drink t his cup can only hol d, ca nnot be put dow n. This is not only to save on glass materials, more important is t o for ce you to get dri nking wate r, go back to y our seat busine ss. Our empha sis on people-oriente d, does not imply y ou ca n indulg e the desires, t he comfortable, t he pursuit of enjoyment. Office of the party committees at all level s of our comrade s, must conti nue t o maintain t he style of plain livi ng and har d struggl e, to sta nd up to t he tests oftemptations, ex citing "energy" conservation "disposition" to t he spirit of selfle ss de dicati on a nd provi de servi ces. Investe e 4. clearly defi ned responsi bilitie s. Putting people first is a common vi sion of dev elopment, every one i s a liability. People -orie nted, a bove a nd below with t he responsi bility for coordi nating system must be e stablished and hold every one a ccountable. First, we m ust strengt hen t he lea dershi p. "In its pla ce, its g overna nce, its level, bear the responsibility of the Division." Thi s is the minimum re quireme nt for leader s, is a measure of the ba sic criteria for wh ether a lea ding cadre is com petent. Leaders ne ed to "it is a n official term for the be nefit of" mission, consci ously assume responsibility for implementi ng pe opl e-oriente d. Dire ctor of the Office of the party committees at all level s is the responsibility of the units t o impleme nt pe ople -ori ented, to manage t heir people, "pl ug in own re sponsibility", effectively e nha nce t he e ducation of leadi ng ca dres, cadres a nd cadre s work2010年高考安徽卷理科数学试题第Ⅰ卷（选择题，共50分）一、选择题：本大题共10小题，每小题5分，共50分，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，=+i i33（A ）12341- （B ）i 12341- （C ）i 6321+ （D ）i 6321- （2）若集合}21log |{21≥=x x A ，则=A C R（A ）⎪⎪⎭⎫ ⎝⎛+∞⋃-∞,22]0,( （B ）⎪⎪⎭⎫ ⎝⎛+∞,22 （C ）⎪⎪⎭⎫⎢⎣⎡+∞⋃-∞,22]0,( （D ）⎪⎪⎭⎫⎢⎣⎡+∞,22 （3）设向量)21,21(),0,1(==b a ，则下列结论中正确的是（A ）||||b a = （B ）22=⋅b a （C ）b b a 与-垂直 （D ）b a // （4）若)(x f 是R 上周期为5的奇函数，且满足,2)2(,1)1(==f f 则)4()3(f f -=（A ）-1（B ）1 （C ）-2 （D ）2 （5）双曲线方程为1222=-y x ，则它的右焦点坐标为（A ）)0,22( （B ）)0,25( （C ）)0,26( （D ）)0,3(（6）设0>abc ，二次函数c bx ax x f ++=2)(的图象可能是legalization of archive s and information. Investee 4. i nsist on t he logisti cs, will be around strong str ong reforms, cadres of service management. Office w orkers is to be for thi s, de epe n reform, strengthen management, a nd conti nuously meet the ca dres of the organs function properly a nd the nee d for material and cult ural life. One is to enhance the l ogistica l support capabilit y. Ada pt to the new situation of the market economy, maki ng full use of social pow er, flexibility in the use of market mecha nisms a nd a ccel erate the pr oce ss of marketization and sociali zation of logisti c servi ces. A dhe re to the institutionalizati on of the logisti cs management, esta blish and impleme nt a financial manageme nt system, trolley ma nagement systems, healt h manageme nt systems, se curity sy stems, such a s the l ogistics manageme nt system, strengt hen t he manageme nt of human, fina ncial , material, and l ogistical service s standardized a nd i nstitutionalize d track. Se condly, we shoul d strengt hen t he manageme nt ofconstructi on. Increasi ng investment, im proving t he worki ng conditions of organs; overall civilizati on create an a ctivity, a nd constantly improv e the level of civilizati on; t o prom ote greeni ng, lighti ng, purification, flower s, landscapi ng, Office w orkers create a g ood environment for work a nd life. T hir d, to improve staff welfare. To in poli cy allow s of range wit hin, tryi ng to for Office ca dres w orkers sol ution housing, a nd couple s separati on, a nd chil dren employme nt, aspects of actual difficult pr ovide s hel p, ha s conditi ons of pla ce and units, als o should a ppropriate to i ncrea sed ca dres w orkers of welfare, real do "care er left people, feeling s left people, a ppr opriate t reatment left pe opl e", let General Office staff in pay har d efforts of while , feel organizati on of warm, get due of affordable. Investee 5. i nsist on the team, will focus on stimul ating vitality who care a nd re spe ct for pe ople. o Office ca dres for the cut ... Lear ning style , strengthen the theor etical knowle dge, optimize the struct ure of knowledgeand conti nuously im prove the quality of cadres a nd w orkers. Attention to the ir physi cal health, a nd a ctively carry out regular w hole some re creationaleople-oriente d in the name of reduci ng our enthusia sm. Premier We n Jiaba o's Governme nt work report this yea r, ide ntified as 7% gr owth, 2.1% low er than la st year's 9.1%, but i n 8 work remai ns to a ccelerate development i n the first pla ce ... Work standards. Second is to use the scale of development to understand. Society is ev olving, human nee ds are also changi ng. Pra ctice of humani sm is a long-term pr ocess and ca nnot be achieve d overnig ht. To treat people if you stand still and rigid dogma, it is possible t o return to t his, see thing s but not people's old ways. T herefore, w e have to foll ow the development of the times, a nd continue de buggi ng the humani st perspective , innovative pe ople -oriented wa y. Under the new histori cal conditions, we are re que sted by the times, set up a bra nd new concept, a high starting poi nt, hig h standard of armed people, shapi ng, exerci se, people, consta ntly taking the new connotation of the times. erms of Office, i s to lead t he vast num bers of ca dres to a ctivelyadapt to t he knowledge faster a nd a ctively ada pt to the i ncrea singly high level of leader shi p situati on, a ctively ada pt to th e dema nd of the masses m ore and more sit uations, eager t o learn, enhance accum ulation, efforts, hone d the a bility to overcome a g ood servi ce, pani c. Thir d is to use reali stic understanding. A dher e to pe opl e-oriente d must be base d on reality, not divorced from reality. Desk no water cups here i n Hong Kong, staff need to drink some water to t he dri nking fountai n on a floor, the gla ss used i s a one-time non-cone type. Drink t his cup can only hol d, ca nnot be put dow n. This is not only to save on glass materials, more important is t o for ce you to get dri nking wate r, go back to y our seat busine ss. Our empha sis on people-oriente d, does not imply y ou ca n indulg e the desires, t he comfortable, t he pursuit of enjoyment. Office of the party committees at all level s of our comrade s, must conti nue t o maintain t he style of plain livi ng and har d struggl e, to sta nd up to t he tests oftemptations, ex citing "energy" conservation "disposition" to t he spirit of selfle ss de dicati on a nd provi de servi ces. Investe e 4. clearly defi ned responsi bilitie s. Putting people first is a common vi sion of dev elopment, every one i s a liability. People -orie nted, a bove a nd below with t he responsi bility for coordi nating system must be e stablished and hold every one a ccountable. First, we m ust strengt hen t he lea dershi p. "In its pla ce, its g overna nce, its level, bear the responsibility of the Division." Thi s is the minimum re quireme nt for leader s, is a measure of the ba sic criteria for whether a lea ding cadre is com petent. Leaders ne ed to "it is a n official term for the be nefit of" mission, consci ously assume responsibility for implementi ng pe opl e-oriente d. Dire ctor of the Office of the party committees at all level s is the responsibility of the units t o impleme nt pe ople -ori ented, to manage t heir people, "pl ug in own re sponsibility", effectively e nha nce t he e ducation of leadi ng ca dres, cadres a nd cadre s work（7）设曲线C 的参数方程为⎩⎨⎧+-=+=θθsin 31cos 32y x （θ为参数），直线l 的方程为023=+-y x ，则曲线C 到直线l 的距离为10107的点的个数为 （A ）1 （B ）2（C ）3 （D ）4（8）一个几何全体的三视图如图，该几何体的表面积为（A ）280 （B ）292（C ）360 （D ）372 （9）动点),(y x A 在圆122=+y x 上绕坐标原点沿逆时针方向匀速旋转，12秒旋转一周.已知定时t=0时，点A 的坐标是)23,21(，则当120≤≤t 时，动点A 的纵坐标y 关于t （单位：秒）的函数的单调递增区间是 （A ）[0，1] （B ）[1，7] （C ）[7，12] （D ）[0，1]和[7，12]、（10）设}{n a 是任意等比数列，它的前n 项和，前2n 项和与前3n 项和分别为X ，Y ，Z ，则下列等式中恒成立的是（A ）Y Z X 2=+ （B ）)()(X Z Z X Y Y -=- （C ）XZ Y =2（D ）)()(X Z X X Y Y -=-第Ⅱ卷（非选择题 共100分）考生注意事项：请用0.5毫米黑色墨水签字笔在答题卡上．．．．．作答，在试题卷上答题无效．．．．．．．．．． 二、填空题：本大题共5小题，每小题5分，共25分．把答案填在答题卡的相应位置．（11）命题“对任何3|4||2|,>-+-∈x x R x ”的否定是 ．（12）6⎪⎪⎭⎫ ⎝⎛-x y y x 的展开式中，3x 的系数等于 ．legalization of archive s and information. Investee 4. i nsist on t he logisti cs, will be around strong str ong reforms, cadres of service management. Office w orkers is to be for thi s, de epe n reform, strengthen management, a nd conti nuously m eet the ca dres of the organs function properly a nd the nee d for material and cult ural life. One is to enhance the l ogistica support capabilit y. Ada pt to the new situation of the market economy, maki ng full use of social pow er, flexibility in the use of market mecha nisms a nd a ccel erate the pr oce ss of marketization and sociali zation of logisti c servi ces. A dhe re to the institutionalizati on of the logisti cs management, esta blish and impleme nt a financial manageme nt system, trolley ma nagement systems, healt h manageme nt systems, se curity sy stems, such a s the l ogistics manageme nt system, strengt hen t he manageme nt of human, fina ncial , m aterial, and l ogistical service s standardized a nd i nstitutionalize d track. Se condly, we shoul d strengt hen t he manageme nt ofconstructi on. Increasi ng investment, im proving t he worki ng conditions of organs; overall civilizati on create an a ctivity, a nd constantly improv e the level of civilizati on; t o prom ote greeni ng, lighti ng, purification, flower s, landscapi ng, Office w orkers create a ood environment for work a nd life. T hir d, to improve staff welfare. To in poli cy allow s of range wit hin, tryi ng to for Office ca dres w orkers sol ution housing, a nd couple s separati on, a nd chil dren employme nt, aspects of actual difficult pr ovide s hel p, ha s conditi ons of pla ce and units, also should a ppropriate to i ncrea sed ca dres w orkers of welfare, real do "care er left people, f eeling s left people, a ppr opriate treatment left pe opl e", let General Office staff in pay har d efforts of while , feel organizati on of warm, get due of affordable. Investee 5. i nsist on the team, will focus on stimul ating vitality who care a nd re spe ct for pe ople. o Office ca dres for the cut ... Lear ning style , strengthen the theor etical knowle dge, optimize the struct ure of knowledgeand conti nuously im prove the quality of cadres a nd w orkers. Attention to the ir physi cal health, a nd a ctively carry out regular w hole some re creationaleople-oriente d in the name of reduci ng our enthusia sm. Premier We n Jiaba o's Governme nt work report this yea r, ide ntified as 7% gr owth, 2.1% low er than la st year's 9.1%, but i n 8 work remai ns to a ccelerate development i n the first pla ce ... Work standards. Second is to use the scale of development to un derstand. Society is ev olving, human nee ds are also changi ng. Pra ctice of humani sm is a long-term pr ocess and ca nnot be achieve d overnig ht. To treat people if you stand still and rigid dogma, it is possible t o return to t his, see thing s but not people's old ways. T herefore, w e have to foll ow the development of the times, a nd continue de buggi ng the humani st perspective , innovative pe ople -oriented wa y. Under the new histori cal conditions, we are re que sted by the times, set up a bra nd new concept, a high starting poi nt, hig h standard of armed people, shapi ng, exerci se, people, consta ntly taking the new connotation of the times. erms of Office, i s to lead t he vast num bers of ca dres to a ctivelyadapt to t he knowledge faster a nd a ctively ada pt to the i ncrea singly high level of leader shi p situati on, a ctively ada pt to th e dema nd of the masses m ore and more sit uations, eager t o learn, enhance accum ulation, efforts, hone d the a bility to overcome a g ood servi ce, pani c. Thir d is to use reali stic understanding. A dher e to pe opl e-oriente d must be base d on reality, not divorced from reality. Desk no water cups here i n Hong Kong, staff need to drink som e water to t he dri nking fountai n on a floor, the gla ss used i s a one-time non-cone type. Drink t his cup can only hol d, ca nnot be put dow n. This is not only to save on glass materials, more important is t o for ce you to get dri nking wate r, go back to y our seat busine ss. Our empha sis on people-oriente d, does not imply y ou ca n indulg e the desires, t he comfortable, t he pursuit of enjoyment. Office of the party committees at all level s of our comrade s, must conti nue t o maintain t he style of plain livi ng and har d struggl e, to sta nd up to t he tests oftemptations, ex citing "energy" conservation "disposition" to t he spirit of selfle ss de dicati on a nd provi de servi ces. Investe e 4. clearly defi ned responsi bilitie s. Putting people first is a common vi sion of dev elopment, every one i s a liability. People -orie nted, a bove a nd below with t he responsi bility for coordi nating system must be e stablished and hold every one a ccountable. First, we m ust strengt hen t he lea dershi p. "In its pla ce, its g overna nce, its level, bear the responsibility of the Division." Thi s is the minimum re quireme nt for leader s, is a measure of the ba sic criteria for whether a lea ding cadre is com petent. Leaders ne ed to "it is a n official term for the be nefit of" mission, consci ously assume responsibility for implementi ng pe opl e-oriente d. Dire ctor of the Office of the party committees at all level s is the responsibility of the units t o impleme nt pe ople -ori ented, to manage t heir people, "pl ug in own re sponsibility", effectively e nha nce t he e ducation of leadi ng ca dres, cadres a nd cadre s work（13）设y x ,满足约束条件⎪⎩⎪⎨⎧≥≥≤--≥+-,0,0,048,022y x y x y x 若目标函数)0,0(>>+=b a y abx z 的最大值为8，则b a +的最小值为 ．（14）如图所示，程序框图（算法流程图）的输出值=x ．（15）甲罐中有5个红球，2个白球和3个黑球，乙罐中有4个红球，3个白球和3个黑球，先从甲罐中随机取出一球放入乙罐，分别以A 1，A 2和A 3表示由甲罐取出的球是红球，白球和黑球的事件；再从乙罐中随机取出一球，以B 表示由乙罐取出的球是红球的事件，则下列结论中正确的是 （写出所有正确结论的编号）．①52)(1=B P ； ②115)|(1=A B P ；③事件B 与事件A 1相互独立；④A 1，A 2，A 3是两两互斥的事件； ⑤)(B P 的值不能确定，因为它与A 1，A 2，A 3中究竟哪一个发生有关．三、解答题：本大题共6小题，共75分．解答应写出文字说明、证明过程或演算步骤，解答写在答题卡上的指定区域内．（16）（本小题满分12分）设ABC ∆是锐角三角形，c b a ,,分别是内角A ，B ，C 所对边长，并且.sin )3sin()3sin(sin 22B B B A +-+=ππ（Ⅰ）求角A 的值；（Ⅱ）若12,27AB AC a ⋅==，求c b ,（其中c b <）．（17）（本小题满分12分）设a 为实数，函数.,22)(R x a x e x f x∈+-=legalization of archive s and information. Investee 4. i nsist on t he logisti cs, will be around strong str ong reforms, cadres of service management. Office w orkers is to be for thi s, de epe n reform, strengthen management, a nd conti nuously meet the ca dres of the organs function properly a nd the nee d for material and cult ural life. One is to enhance the l ogistica l support capabilit y. Ada pt to the new situation of the market economy, maki ng full use of social pow er, flexibility in the use of market mecha nisms a nd a ccel erate the pr oce ss of marketization and sociali zation of logisti c servi ces. A dhe re to the institutionalizati on of the logisti cs management, esta blish and impleme nt a financial manageme nt system, trolley ma nagement systems, healt h manageme nt systems, se curity sy stems, such a s the l ogistics manageme nt system, strengt hen t he manageme nt of human, fina ncial , material, and l ogistical service s standardized a nd i nstitutionalize d track. Se condly, we shoul d strengt hen t he manageme nt of constructi on. Increasi ng investment, im proving t he worki ng conditions of organs; overall civilizati on create an a ctivity, a nd constantly improv e the level of civilizati on; t o prom ote greeni ng, lighti ng, purification, flower s, landscapi ng, Office w orkers create a g ood environment for work a nd life. T hir d, to improve staff welfare. To in poli cy allow s of range wit hin, tryi ng to for Office ca dres w orkers sol ution housing, a nd couple s separati on, a nd chil dren employme nt, aspects of actual difficult pr ovide s hel p, ha s conditi ons of pla ce and units, als o should a ppropriate to i ncrea sed ca dres w orkers of welfare, real do "care er left people, feeling s left people, a ppr opriate t reatment left pe opl e", let General Office staff in pay har d efforts of while , feel organizati on of warm, get due of affordable. Investee 5. i nsist on the team, will focus on stimul ating vitality who care a nd re spe ct for pe ople. o Office ca dres for the cut ... Lear ning style , strengthen the theor etical knowle dge, optimize the struct ure of knowledge and conti nuously im prove the quality of cadres a nd w orkers. Attention to the ir physi cal health, a nd a ctively carry out regular w hole some re creationaleople-oriente d in the name of reduci ng our enthusia sm. Premier We n Jiaba o's Governme nt work report this yea r, ide ntified as 7% gr owth, 2.1% low er than la st year's 9.1%, but i n 8 work remai ns to a ccelerate development i n the first pla ce ... Work standards. Second is to use the scale of development to understand. Society is ev olving, human nee ds are also changi ng. Pra ctice of humani sm is a long-term pr ocess and ca nnot be achieve d overnig ht. To treat people if you stand still and rigid dogma, it is possible t o return to t his, see thing s but not people's old ways. T herefore, w e have to foll ow the development of the times, a nd continue de buggi ng the humani st perspective , innovative pe ople -oriented wa y. Under the new histori cal conditions, we are re que sted by the times, set up a bra nd new concept, a high starting poi nt, hig h standard of armed people, shapi ng, exerci se, people, consta ntly taking the new connotation of the times. erms of Office, i s to lead t he vast num bers of ca dres to a ctively adapt to t he knowledge faster a nd a ctively ada pt to the i ncrea singly high level of leader shi p situati on, a ctively ada pt to th e dema nd of the masses m ore and more sit uations, eager t o learn, enhance accum ulation, efforts, hone d the a bility to overcome a g ood servi ce, pani c. Thir d is to use reali stic understanding. A dher e to pe opl e-oriente d must be base d on reality, not divorced from reality. Desk no water cups here i n Hong Kong, staff need to drink some water to t he dri nking fountai n on a floor, the gla ss used i s a one-time non-cone type. Drink t his cup can only hol d, ca nnot be put dow n. This is not only to save on glass materials, more important is t o for ce you to get dri nking wate r, go back to y our seat busine ss. Our empha sis on people-oriente d, does not imply y ou ca n indulg e the desires, t he comfortable, t he pursuit of enjoyment. Office of the party committees at all level s of our comrade s, must conti nue t o maintain t he style of plain livi ng and har d struggl e, to sta nd up to t he tests of temptations, ex citing "energy" conservation "disposition" to t he spirit of selfle ss de dicati on a nd provi de servi ces. Investe e 4. clearly defi ned responsi bilitie s. Putting people first is a common vi sion of dev elopment, every one i s a liability. People -orie nted, a bove a nd below with t he responsi bility for coordi nating system must be e stablished and hold every one a ccountable. First, we m ust strengt hen t he lea dershi p. "In its pla ce, its g overna nce, its level, bear the responsibility of the Division." Thi s is the minimum re quireme nt for leader s, is a measure of the ba sic criteria for whether a lea ding cadre is com petent. Leaders ne ed to "it is a n official term for the be nefit of" mission, consci ously assume responsibility for implementi ng pe opl e-oriente d. Dire ctor of the Office of the party committees at all level s is the responsibility of the units t o impleme nt pe ople -ori ented, to manage t heir people, "pl ug in own re sponsibility", effectively e nha nce t he e ducation of leadi ng ca dres, cadres a nd cadre s work （I ）求)(x f 的单调区间与极值；（II ）求证：当012ln >->x a 且时，.122+->ax x e x（18）（本小题满分13分）如图，在多面体ABCDEF 中，四边形ABCD 是正方形，EF//AB ，EF ⊥FB ，AB=2EF ，,90︒=∠BFC BF=FC ，H 为BC 的中点.（I ）求证：FH//平面EDB ；（II ）求证：AC ⊥平面EDB ； （III ）求二面角B —DE —C 的大小.（19）（本小题满分13分）已知椭圆E 经过点A （2，3），对称轴为坐标轴，焦点F 1，F 2在x 轴上，离心率.21=e （I ）求椭圆E 的方程；（II ）求21AF F ∠的角平分线所在直线l 的方程；（III ）在椭圆E 上是否存在关于直线l 对称的相异两点？若存在，请找出；若不存在，说明理由.（20）（本小题满分12分）设数列,,,21 a a ,n a 中的每一项都不为0.证明，}{n a 为等差数列的充分必要条件是：对任何N n ∈，都有.1111113221++=+++n n n a a n a a a a a a （21）（本小题满分13分）品酒师需要定期接受酒味鉴别功能测试，一种通常采用的测试方法如下：拿出n 瓶外观相同但品质不同的酒让其品尝，要求其按品质优劣为它们排序，经过一段时间，等其记忆淡忘之后，再让其品尝这n 瓶酒，并重新按品质优劣为它们排序，这称为一轮测试.根据一轮测试中的两次排序的偏离程度的高低为其评分.现设n=4，分别以4321,,,a a a a 表示第一次排序时被排为1，2，3，4的四种酒在第二次排序时的序号，并令.|4||3||2||1|4321a a a a X -+-+-+-=则X 是对两次排序的偏离程度的一种描述.（I ）写出X 的可能值集合；AB C D E FHlegalization of archive s and information. Investee 4. i nsist on t he logisti cs, will be around strong str ong reforms, cadres of service management. Office w orkers is to be for thi s, de epe n reform, strengthen management, a nd conti nuously m eet the ca dres of the organs function properly a nd the nee d for material and cult ural life. One is to enhance the l ogistica support capabilit y. Ada pt to the new situation of the market economy, maki ng full use of social pow er, flexibility in the use of market mecha nisms a nd a ccel erate the pr oce ss of marketization and sociali zation of logisti c servi ces. A dhe re to the institutionalizati on of the logisti cs management, esta blish and impleme nt a financial manageme nt system, trolley ma nagement systems, healt h manageme nt systems, se curity sy stems, such a s the l ogistics manageme nt system, strengt hen t he manageme nt of human, fina ncial , m aterial, and l ogistical service s standardized a nd i nstitutionalize d track. Se condly, we shoul d strengt hen t he manageme nt of constructi on. Increasi ng investment, im proving t he worki ng conditions of organs; overall civilizati on create an a ctivity, a nd constantly improv e the level of civilizati on; t o prom ote greeni ng, lighti ng, purification, flower s, landscapi ng, Office w orkers create a ood environment for work a nd life. T hir d, to improve staff welfare. To in poli cy allow s of range wit hin, tryi ng to for Office ca dres w orkers sol ution housing, a nd couple s separati on, a nd chil dren employme nt, aspects of actual difficult pr ovide s hel p, ha s conditi ons of pla ce and units, also should a ppropriate to i ncrea sed ca dres w orkers of welfare, real do "care er left people, f eeling s left people, a ppr opriate treatment left pe opl e", let General Office staff in pay har d efforts of while , feel organizati on of warm, get due of affordable. Investee 5. i nsist on the team, will focus on stimul ating vitality who care a nd re spe ct for pe ople. o Office ca dres for the cut ... Lear ning style , strengthen the theor etical knowle dge, optimize the struct ure of knowledge and conti nuously im prove the quality of cadres a nd w orkers. Attention to the ir physi cal health, a nd a ctively carry out regular w hole some re creationaleople-oriente d in the name of reduci ng our enthusia sm. Premier We n Jiaba o's Governme nt work report this yea r, ide ntified as 7% gr owth, 2.1% low er than la st year's 9.1%, but i n 8 work remai ns to a ccelerate development i n the first pla ce ... Work standards. Second is to use the scale of development to un derstand. Society is ev olving, human nee ds are also changi ng. Pra ctice of humani sm is a long-term pr ocess and ca nnot be achieve d overnig ht. To treat people if you stand still and rigid dogma, it is possible t o return to t his, see thing s but not people's old ways. T herefore, w e have to foll ow the development of the times, a nd continue de buggi ng the humani st perspective , innovative pe ople -oriented wa y. Under the new histori cal conditions, we are re que sted by the times, set up a bra nd new concept, a high starting poi nt, hig h standard of armed people, shapi ng, exerci se, people, consta ntly taking the new connotation of the times. erms of Office, i s to lead t he vast num bers of ca dres to a ctively adapt to t he knowledge faster a nd a ctively ada pt to the i ncrea singly high level of leader shi p situati on, a ctively ada pt to th e dema nd of the masses m ore and more sit uations, eager t o learn, enhance accum ulation, efforts, hone d the a bility to overcome a g ood servi ce, pani c. Thir d is to use reali stic understanding. A dher e to pe opl e-oriente d must be base d on reality, not divorced from reality. Desk no water cups here i n Hong Kong, staff need to drink som e water to t he dri nking fountai n on a floor, the gla ss used i s a one-time non-cone type. Drink t his cup can only hol d, ca nnot be put dow n. This is not only to save on glass materials, more important is t o for ce you to get dri nking wate r, go back to y our seat busine ss. Our empha sis on people-oriente d, does not imply y ou ca n indulg e the desires, t he comfortable, t he pursuit of enjoyment. Office of the party committees at all level s of our comrade s, must conti nue t o maintain t he style of plain livi ng and har d struggl e, to sta nd up to t he tests of temptations, ex citing "energy" conservation "disposition" to t he spirit of selfle ss de dicati on a nd provi de servi ces. Investe e 4. clearly defi ned responsi bilitie s. Putting people first is a common vi sion of dev elopment, every one i s a liability. People -orie nted, a bove a nd below with t he responsi bility for coordi nating system must be e stablished and hold every one a ccountable. First, we m ust strengt hen t he lea dershi p. "In its pla ce, its g overna nce, its level, bear the responsibility of the Division." Thi s is the minimum re quireme nt for leader s, is a measure of the ba sic criteria for whether a lea ding cadre is com petent. Leaders ne ed to "it is a n official term for the be nefit of" mission, consci ously assume responsibility for implementi ng pe opl e-oriente d. Dire ctor of the Office of the party committees at all level s is the responsibility of the units t o impleme nt pe ople -ori ented, to manage t heir people, "pl ug in own re sponsibility", effectively e nha nce t he e ducation of leadi ng ca dres, cadres a nd cadre s work （II ）假设4321,,,a a a a 等可能地为1，2，3，4的各种排列，求X 的分布列；（III ）某品酒师在相继进行的三轮测试中，都有2≤X ，（i ）试按（II ）中的结果，计算出现这种现象的概率（假定各轮测试相互独立）； （ii ）你认为该品酒师的酒味鉴别功能如何？说明理由.2011安徽数学（理科）试题第Ⅰ卷（选择题共50分）一、选择题：本大题共10小题，每小题5分，共50分。
2025届安徽江南十校高考数学二模试卷含解析
2025届安徽江南十校高考数学二模试卷注意事项：1．答卷前，考生务必将自己的姓名、准考证号、考场号和座位号填写在试题卷和答题卡上。
用2B 铅笔将试卷类型（B ）填涂在答题卡相应位置上。
将条形码粘贴在答题卡右上角"条形码粘贴处"。
2．作答选择题时，选出每小题答案后，用2B 铅笔把答题卡上对应题目选项的答案信息点涂黑；如需改动，用橡皮擦干净后，再选涂其他答案。
答案不能答在试题卷上。
3．非选择题必须用黑色字迹的钢笔或签字笔作答，答案必须写在答题卡各题目指定区域内相应位置上；如需改动，先划掉原来的答案，然后再写上新答案；不准使用铅笔和涂改液。
不按以上要求作答无效。
4．考生必须保证答题卡的整洁。
考试结束后，请将本试卷和答题卡一并交回。
一、选择题：本题共12小题，每小题5分，共60分。
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．设等比数列{}n a 的前项和为n S ，若2019201680a a +=，则63S S 的值为（ ） A ．32B ．12C ．78 D ．982．复数()()()211z a a i a R =-+-∈为纯虚数，则z =( )A ．iB ．﹣2iC ．2iD ．﹣i3．已知函数()sin(2)f x x ϕ=+，其中(0,)2πϕ∈，若,()6x R f x f π⎛⎫∀∈≤ ⎪⎝⎭恒成立，则函数()f x 的单调递增区间为（ ） A ．,()36k k k z ππππ⎡⎤-+∈⎢⎥⎣⎦B ．2,()33k k k z ππππ⎡⎤-+∈⎢⎥⎣⎦C ．2,()33k k k z ππππ⎡⎤++∈⎢⎥⎣⎦ D ．2,()3k k k Z πππ⎡⎤+⎢⎥⎣∈⎦ 4．已知复数()11z ai a R =+∈，212z i =+（i 为虚数单位），若12z z 为纯虚数，则a =（ ） A ．2-B ．2C ．12-D ．125．双曲线22221(0,0)x y a b a b -=>>的左右焦点为12,F F ，一条渐近线方程为:b l y x a=-，过点1F 且与l 垂直的直线分别交双曲线的左支及右支于,P Q ，满足11122OP OF OQ =+，则该双曲线的离心率为（ ） AB ．3CD ．26．执行如图所示的程序框图，若输出的值为8，则框图中①处可以填（ ）．A ．7?S ≥B ．21?S ≥C ．28?S ≥D ．36?S ≥7．若函数()f x 的图象如图所示，则()f x 的解析式可能是（ ）A ．()x e xf x x +=B ．()21x f x x-=C ．()x e xf x x-=D ．()21x f x x +=8．设0.380.3log 0.2,log 4,4a b c ===，则（ ）A ．c b a <<B ．a b c <<C ．a c b <<D ．b a c <<9．函数()sin 2sin 3f x x m x x =++在[,]63ππ上单调递减的充要条件是（ ）A ．3m ≤-B ．4m ≤-C ．33m ≤-D ．4m ≤10．已知七人排成一排拍照，其中甲、乙、丙三人两两不相邻，甲、丁两人必须相邻，则满足要求的排队方法数为（ ）. A ．432B ．576C ．696D ．96011．已知函数()[]f x x x =-，其中[]x 表示不超过x 的最大正整数，则下列结论正确的是（ ）A ．()f x 的值域是[]0,1B ．()f x 是奇函数C ．()f x 是周期函数D ．()f x 是增函数12．空间点到平面的距离定义如下：过空间一点作平面的垂线，这个点和垂足之间的距离叫做这个点到这个平面的距离．已知平面α，β，λ两两互相垂直，点A α∈，点A 到β，γ的距离都是3，点P 是α上的动点，满足P 到β的距离与P 到点A 的距离相等，则点P 的轨迹上的点到β的距离的最小值是（ ） A ．33-B ．3C ．332- D ．32二、填空题：本题共4小题，每小题5分，共20分。
安徽大学期末试卷MK08-09(1)高数C(三)答案.pdf
<
⎫ 1⎬
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= 2Φ(1) − 1
= 2 × 0.8413 −1 = 0.6826
（２）解法 1:
由正态分布的对称性知 c = 10.
解法 2:
因为 P ( X > c) = P ( X ≤ c)
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解：设 B = {取到的产品为次品}
10. (19.8728,20.1472)
A1 = {取到的产品来自于甲车间} A2 = {取到的产品来自于乙车间} A3 = {取到的产品来自于丙车间}
（１） 由全概率公式有
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安徽大学 10-11(1)高数C(三)试卷
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A． 事件 A 与事件 B 互不相容 C. 事件 A 与事件 B 互不独立
B. 事件 A 与事件 B 相容 D. 事件 A 与事件 B 对立
2．设随机变量 X 和Y 都服从正态分布 N (0,σ 2 ) ，且 P( X > 2,Y > −2) = 0.75 ，
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A. 必接受 H0
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MK_10-11(2)高数A(二)、B(二)试卷
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安徽大学 2010—2011 学年第二学期
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安徽大学高数A(二)期末试卷答案
安徽大学 2009—2010 学年第二学期 《高等数学 A（二）、B（二）》考试试卷（A 卷）
（闭卷 时间 120 分钟）
题号 一
二
三
四
五
总分
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学号
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四、应用题（本大题共 8 分）
19. 在椭圆 x2 + 4 y2 = 4 上求一点，使该点到直线 2x + 3y −12 = 0 的距离最短.
《高等数学 A（二) 、B（二）》（A 卷） 第 5 页 共 6 页
_________.
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