题目讨论2

毛小小猫 发表于 2013-10-14 18:25
第36题我也说不清楚，感觉就是选E。。。坐等别的大牛来解释

第66题做变量替换，令x=pi/2-t带入 ...

变量替换太复杂了

dennis_szsy 发表于 2013-10-15 16:23
变量替换太复杂了

╮(╯▽╰)╭ 你总是能想到好简单的办法来做 厉害厉害~\(≧▽≦)/~
怪不得我们会做不完题，ets估计都是按你这种速度来出题的

毛小小猫 发表于 2013-10-15 14:58
还是没有找到有像lz有66道题这样完整的。。。
能不能给我发一份你11年的那个回忆题呢？

[数学] 2011 math sub 回忆题

说明：因时间所限没有抄完，不用给选项的题就都没抄选项。int(f(x),x=a..b)表示函数f(x)从a到b的定积分或广义积分或瑕积分，Pi表示圆周率，sqrt表示平方根，ln表示自然对数，exp(a)表示e的a次方，sigma表示求和，lim表示极限。很多题的措辞和试卷不同，但意思相同。
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1 f(x) is function, for x>0 lnf(x)=ln2-ln(x), find f'(x)
2 T1 is a equilateral triangle with side length 1. T2 is a triangle whose vertices are the midpoints of T1's sides. Find the area of T2. inter
3 A B C are three sets. |A|=17, |B|=15, |C|=20, |A intersect B|=10, |A intersect C|=4, |C intersect B|=2, |A intersect B intersect C|=1, find |A union B union C|5 Z: z3 b' V$ e6 @0 @2 J
4 Given that int(exp(-x^2), x=0..infinity)=sqrt(Pi)/2, a>0, find int(exp(-x^2/a^2), x=0..infinity)" Q! k( ]- ]+ m9 k2 H6 ]
5 A bag contains 4 green balls and 5 black balls. One picks two balls randomly from it without putting back. Find the probability of picking two green balls.
6 f(x,y) is continuous, int((int(f(x,y),y=0..x^2))x=0..2)= what (exchange the order of the two integrals)
7 f(x)=int((t^2-1)/(t^2+1),t=0..x), what is the set of the critical points of f(x)?6 U/ @" E& K! w
8 In a vector space with inner product <a,b> and norm ||a||, a,b,c are vectors, <a,b>=2, <b,c>=5, <a,c>=-3, ||a||=1, ||b||=2, ||c||=7. Find <a+b,b+c>
9 What the graph of y=3x^4-4x^3+1 like?( M  C0 E9 ^+ {: E3 z
10 Which one function is not even?
(a) f(x)=sin(xsin(x)) (b) f(x)=sin(cos(x)) (c) f(x)=cos(sin(x)) (d) f(x)=exp(cos(x))-exp(cos(-x)) (e) f(x)=cos(exp(2x))5 L3 A% R/ k. @
11 M2 is the vector space of all 2*2 matrices with real entries. For what real number r the set {(a b)(c d)|a+b+c+d=r} is a subspace of M2 ?
12 Find int(sqrt(exp(x)+exp(-x)+2),x=0..1)
13 Find the congruence of 4^578 (mod 7)
14 ABCD is a quadralateral, AB=BC=3, AD=DC=4, BD=5. find AC. X* F( t1 n& _7 S: r0 n$ E9 f
15 f and g are bijections from interval [0,1] to itself. which map must be an one-to-one map?; {' }7 f/ b! t( o7 ?
(a) f+g (b)f-g (c) f*g (d) (f^2+g^2)/2 (e) f(g)) ?& e2 x. z5 F% z
16 一张边长24的正方形纸，四个角各剪掉一个边长为h的正方形，剩下的部分可以折成一个无盖的长方体，求h使长方体体积最大。- e8 O  \; @1 E) s' p' d9 H
17 A and B are two invertible 5*5 matrices, B^T is the transposition of B. X is a 5*1 vector. Which statement must be true?
I. (B^T)A is invertible
II. the equation BX=AX has only the trivial solution X=0.5 A; D  h8 f1 C, }) D
III. the dimension of the column space of A equals to the dimension of the column space of B
18 g is a twice-differentiable function, g(0)=g'(0)=g''(0)=1, f(x)=g(x^2)+g(x)+1, find f''(0)
19 f is a function that is differentiable at 0, find lim(x->0, [f(x^5)-f(0)]/2x^3)
20 the graph of the derivative g'(x) of function y=g(x) has the graph as follows（题目给了图，我没法画，用解析式表示大概是）：当x不大于2，g'(x)=x-1；当2《x《3，g'(x)=3-x；当3《x《4，g'(x)=x-3；当x不小于4，g'(x)=5-x. Which one is not true; D) k/ g' I2 }1 {% r
(a) g has a local minimum at x=1  (b) g has a local maximum at x=2 (c) g has an inflection point at x=3 (d) g has an inflection point at x=4 (e) g has an absolute maximum at x=5 , [" l4 p  r. \6 z  p
21 Given that the 3*3 matrix: (three row vectors are (0,0,-2),(1,2,1),(1,0,3)
, respectively) has an eigenvalue of 2, find a base of the subspace of eigenvectors corresponding to 2.' |( B' o" c4 d3 B" I6 e
22 On a curve r=r(t)(r是向量)， a point is regular if dr/dt is not zero vector. Then how many points on the curve x(t)=[sin(t)]^3, y(t)=[cos(t)]^3 are not regular?' g, x% @3 q) I5 O9 K& k
ditons
23 Let z=x+iy, x and y are real numbers. f(z)=6x-4y+i(ax+by), a and b are real numbers. If f(z) is differentiable, find a and b.
24 a,b,c belongs to interval [0,1] and a<b<c. Which function's graph satisfy the following conditions: (1) f is nonnegative; (2) int(f(x),x=a..b)=int(f(x),x=a..c); (3) f(a)=f(c) 图像就不画了，反正很简单。
25 z=x+iy, x>0 and y>0. If z^2=-1/2+i*sqrt(3)/2, then z^3+2z+1=?7 @( m% m6 U- l2 g& j) d
26 {a_n} is a sequence, a_0=0,a_1=1, a_(n+1)=a_n+2*a_(n-1) for n>=1. Find lim(n->infinity, a_(n+1)/a_n)
27 Let A be the area of the triangle formed by x-axle, y-axle and the tangent line of y=x^p (p<0) at x=c. Find p such that A is independent of c.
28 Given that x-2y+3z=1, find the minimum value of x^2+y^2+z^2
29 P(x) is a quadratic polynomial of x that has the same value as sin(x) at x=0,x=Pi/4 and x=Pi/2. Find P(Pi)
30 For real x, let f(x)=lim(n->infinity, [cos(x)]^2n). Then what about the continuity of f(x). （选项不列了，就是问f在哪些点连续）* O( o( F& s, L5 ^! q
31 For real x, let f(x)=x-[x], i.e., f is the fractional part of x. If a and b are real numbers and b=a+1, find the image of f on interval [a,b].& q6 L/ e3 a+ j
32 G(x)=int(exp(-t^2),t=0..x^2), find G'(x)
33 老师给12个孩子分12个玩具，每人1个。只有4种不同的玩具，每种分别有4，5，2，1个，问有多少种分配的方法。
34 Find the intersection point of plane 3x+2y-z=5 and line x=1+2t,y=-1-t,z=-3t.
35 Find the mean value of function f(x)=xe^3x on interval [0,2]. R( N  e% j5 X& @
36 S and T are nonzero 3*3 matrices. Which of the following cannot be true?
(A) ST=0 (B) ST<>TS (C) ST=ST^2 (D) T=T^4 (E) S^4=0 but S^3<>0 ("<>" means "not equal to" )
37 {a_n} is a strictly increasing sequence of positive numbers, which of the following sum must converge?
I. sigma(1/(a_i)^2, i=1..infinity)0 }4 Q2 |* H, Q$ o) }
II. sigma((a_i)/(a_(i+1)), i=1..infinity)4 b% j4 K9 ^8 d5 a9 f# o
III. sigma(exp(-a_i), i=1..infinity)
38 a=(a1,a2,a3) is a nonzero vector. S is the linear transformation from R^3 to R^3 such that S(x)=a×x(差乘) for all vector x. Find the determinant of S.
39 P和Q是命题，则“非（P且Q）”等价于以下哪个？ 答案：P蕴含非Q4 z" V- I% k& e3 m: @# z
40 Which set is not the intersection of countably many open sets?* h5 o7 w& j/ e1 d2 P, k' L
(a) Z (b) Q (c) interval [0,infinity) (d) the complement of Q (e) the complement of Z) x; Y& b' t' x0 K' V3 D9 H
41
. Waral dи,42 h is a continuous function on interval [a,b] and h(x) belongs in Q for all x. Which statement is true?
(a) h is constant on the interval 后面的选项忘了( u3 @4 k7 ?/ m' Q1 Y
43 直线 y=1000x 与 y=999x 的夹角用弧度表示最接近哪个选项？  10^-2 10^-3 10^-4 10^-5 10^-6& V' B! o9 g. ?- i+ ^8 ]- ?
44 Find sigma([(-1)^n]*(n+1)/n!,n=0..infinity)
45 # y9 C& y1 Y% I* @. l
46 n>=3,k is real number, sigma((a_i)^3,i=1..n) <= k*sqrt(sigma((a_i)^6,i=1..n)) for all real numbers {a_i}. Find the minimum value of k.1 p! m% L) n; ]5 }2 D
47 C is the circle x^2+y^2=9, oriented counterclockwise, find 沿C的线积分（exp(Pi*z)/(z-4i)^2/(z-i) dz）.
48 For a commutative ring M and ideal A, let N(A)={x in M|there exists a nonnegative integer n such that x^n in A}. Which of following is true for N(A)=A?2 `" g+ g3 k& p! w2 w# j9 r: I' _
I. M=Z, A=(2)
II. M=Z[x], A=(x^2+2)
III. M=Z/27Z, A=(18+27Z)( `5 V* q" Z2 J4 T+ @3 F: a  _
49 X和Y两个正态分布，前者期望是52，标准差是6；后者期望是40，标准差是8。求Z=X+Y的标准差
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50 对不小于3的正整数n，求单位圆的外切正n边形与内接正n边形的周长之比。
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题号没记 How many distict real root does the equation x^4-x^3*sin(x)-x^2*cos(x)=0 have? 0 1 2 3 45 K4 r8 o1 i) h! T3 @
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题号没记 有多少个互不同构的3^2*11^4*17阶Abel群？
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题号没记 Ta是以点(0,0),(1,0),(0,a)为顶点的三角形，F（a）是函数f(x,y)=y^2在Ta上的积分，求F'(1)

55 The solution y(x) of y''+6y'+cy=0 satisfies that lim(x->positive infinity, y(x))=0, find all valid c
(a) c<=0 (b) c>0 (c) 0<c<=9 (d) c>=9 (e) R  % s: ?+ b, C& `9 q4 [) p, c
56 Line y=x is tangant to a circle at (3,3) and line y=2x passes the center of that circle. Which circle is possible?/ E& ?6 J& s8 p6 C8 W3 @6 q
Answer: (a) (x-2)^2+(y-4)^2=2
57 X is a compact metric space, f is a continuous function X->X. Which of the following must be true?1 h% ?2 m. L0 Y
I. f has fixed point
II. f is uniformly continuous
III. f is a closed map( R" G2 P5 }9 d! ]9 b
59 G is a group, a and b are non-unit elements of G, ab=bba. If the subgroup of G generated by a has order 3, what about the order of the subgroup of G generated by b?
(a) 3 (b) 5 (c) 7 (d) 9 (e) cannot determined by given information3 u+ X3 C' f2 o  N. H! z2 s. W& D
66 Find int([sin(t)]^3/{[sin(t)]^3+[cos(t)]^3},t=0..Pi/2)