[Jia Liwei university mathematics magazine] 322nd sets of simulation papers for Mathematics Competition Training of Gannan Normal University

 

 

Mathematical Analysis

 

 

1. known functions $ f (x) = \ ln x-Ax $, where $ A $ is a constant. if $ f (x) $ has two zeros $ x_1, x_2 $. test Certificate: $ x_1x_2> E ^ 2 $.

 

2. set the equation $ \ SiN x-x \ Cos x = 0 $ in $ (0, + \ infty) $ to resolve the $ N $ to $ x_n $. proof: $ \ Bex n \ PI + \ cfrac {\ PI} {2}-\ cfrac {1} {n \ PI} <x_n <n \ PI + \ cfrac {\ pi} {2 }. \ EEx $

 

3. discuss the consistency of $ f (x) = x \ SiN x $, $ g (x) = x \ ln x $ on $ [1, \ infty) $.

 

4. set $ f (x) $ to have a second-order continuous derivative on $ [a, B] $. The test evidence shows $ \ Xi \ In (a, B) $ make $ \ Bex F (a) + F (B) + 2f \ sex {\ frac {a + B} {2 }}+ \ frac {1} {4} (B-a) ^ 2f ''(\ XI ). \ EEx $

 

5. curved points $ \ Bex \ iint_s Z \ RD x \ RD y, \ EEx $ where $ S $ is the surface $ \ DPS {\ frac {x ^ 2} {A ^ 2} + \ frac {y ^ 2} {B ^ 2} + \ frac {z ^ 2} {C ^ 2} = 1} $, the direction is external.

 

6. set $ F $ to continuous on $ [a, B] $, $ G $ to micro on $ [a, B] $ and $ G' \ Leq 0 $. proof: $ \ Xi \ In (a, B) $ exists, making $ \ Bex \ int_a ^ B f (x) g (x) \ RD x = g () \ int_a ^ \ Xi f (x) \ RD x + g (B) \ int _ \ Xi ^ B f (x) \ RD X. \ EEx $

 

7. set $ F $ to be consistent and continuous on $ (0, \ infty) $, and to $ \ forall \ h> 0 $, $ \ DPS {\ vlm {n} f (NH)} $ exists. test Certificate: $ \ DPS {\ vlm {x} f (x)} $ exists.

 

8. set the non-negative functions on $ [0, T] $ F, G, h $ satisfies the differential inequality $ \ Bex \ cfrac {\ RD f} {\ RD t} + H \ Leq GF, \ quad 0 \ Leq t \ Leq T. \ EEx $ trial certificate: $ \ Bex F (t) + \ int_0 ^ t h (s) \ RD s \ Leq F (0) \ SEZ {1 + \ int_0 ^ T g (s) \ RD s \ cdot \ exp \ sex {\ int_0 ^ T g (s) \ RD s }}, \ quad 0 \ Leq t \ Leq T. \ EEx $

 

9. set $ f (x) $ to any order on $ \ BBR $, and $ \ Bex \ forall \ n \ In \ bbz ^ +, \ f \ sex {\ frac {1} {n} = 0. \ EEx $ test certificate: $ f ^ {(n)} (0) = 0 $.

 

10. set $ f (x) $ to micro on $ [a, B] $, $ F (a) = F (B) = 0 $, for $ \ forall \ x \ in [a, B] $, $ \ Xi \ In (a, B) $ exists, making $ \ Bex f (x) = \ frac {f'' (\ xi)} {2} (X-A) (X-B ). \ EEx $

 

11. set $ f (x) $ to be micro on $ [a, B] $ Level 2. Test Certificate: For any $ C \ In (a, B) $, $ \ Xi \ In (a, B) $ causes $ \ Bex \ frac {f'' (\ xi)} {2 }=\ frac {f ()} {(a-B) (a-c)} + \ frac {F (B)} {(B-A) (B-c )} + \ frac {f (c)} {(c-A) (C-B )}. \ EEx $

 

12. set $ F $ on $ \ BBR $ (n + 1) $ to export and test the certificate: For $ \ forall \ A \ In \ BBR $, $ \ Bex \ frac {\ RD ^ n} {\ RD x ^ n} \ SEZ {\ frac {f (x)-f ()} {X-A }}_{ x = A }=\ frac {f ^ {(n + 1)} (a)} {n + 1 }. \ EEx $

 

13. set $ f (x) $ to be exported on $ [a, B] $, and $ f' (A) = f' (B) $. test certificate: $ \ Bex \ exists \ Xi \ In (a, B), \ st f' (\ xi) (\ Xi-a) = f (\ XI) -F (). \ EEx $

 

14. try $ \ Bex \ vlm {n} n ^ 2 \ sex {x ^ \ frac {1} {n}-x ^ \ frac {1} {n + 1} }, \ quad x> 0. \ EEx $

 

15. try $ \ Bex \ vlm {n} \ sex {\ int_0 ^ \ pi x ^ {2013} \ sin ^ N x \ RD x} ^ \ frac {1 }{ n }. \ EEx $

 

16. evaluate the range of the parameter $ A, B, and C $ so that $ \ sum _ {n = 3} ^ \ infty a ^ n ^ B \ ln ^ C N $

(1) absolute convergence;

(2) Conditional convergence

(3) divergence.

 

17. the center of the ball with a radius of $ r$ is on the surface of the unit ball $ x ^ 2 + y ^ 2 + Z ^ 2 = 1 $, ask $ r$ what is the maximum surface area of the ball in the unit ball?

 

18. Set $ a> 0 $ and find the area enclosed by $ \ DPS {Y = \ frac {x ^ 3} {2a-x }}$ and $ x = 2a $.

 

19. discuss $ f (x) = x \ SiN x, x ^ \ Al \ ln x (0 <\ Al <1) $ in $ [1, \ infty) $ indicates whether it is consistent and continuous, and explains the reason.

 

20. set $ \ Bex f (x) = \ int_x ^ {x ^ 2} \ sex {1 + \ frac {1} {2 t }}^ t \ SEZ {e ^ {\ frac {1} {\ SQRT {T }}- 1} \ RD t, \ quad T> 0. \ EEx $ evaluate $ \ Bex \ vlm {n} f (n) \ sin \ frac {1} {n }. \ EEx $

 

 

Advanced Algebra

 

 

1. set $ \ mathbb {p} $ to a number field. If $ P_1 (x), \ cdots, P_r (X) $ is an irrevocable polynomial of $ r$ on the number field $ \ mathbb {p} $ with different first coefficients of $1 $. Proof: $ f (x) = P_1 (X) \ cdots P_r (x) $ there is no duplicate root in the field $ \ mathbb {p} $.

 

2. set $ V $ to the vector space composed of all real-system unary Polynomials with a number of no more than $4 $. for any polynomial on $ V $ f (x) $, divide $ f (x) with $ x ^ 2-1 $) $ the formula and remainder are $ Q (x) $ and $ R (x) $ respectively. Note $ \ Bex f (x) = Q (x) (x ^ 2-1) + R (X ). \ EEx $ set $ \ SCRA $ to map $ V $ to $ V $, so that $ \ Bex \ SCRA (f (x) = R (X ). \ EEx $ test certificate: $ \ SCRA $ is a linear transformation, and calculates it about the base $ \ sed {1, x, x ^ 2, x ^ 3, x ^ 4} $ matrix.

 

3. calculation criterion: $ \ Bex D_n =\sev {\ BA {cccccc} x_1y_1 & x_1y_2 & x_1y_3 & \ cdots & x_1y _ {n-1} & {\ x_1y_2 & x_2y_2 & x_2y_3 & \ cdots & x_2y _ {n-1} & x_2y_n \ x_1y_3 & x_2y_3 & x_3y_3 & \ cdots & x_3y _ {n-1} & \\\ vdots & \ ddots & \ vdots & \ vdots \ x_1y _ {n-1} & x_2y _ {n-1} & x_3y _ {n-1} & \ cdots & X _ {n-1} y _ {n-1} & X _ {n-1} Y_n \ x_1y_n & x_2y_n & x_3y_n & \ cdots & X _ {n-1} Y_n & x_ny_n \ EA }. \ EEx $

 

4. set $ \ BBP $ to a number field, $ f (x), g (x) \ In \ BBP [x] $. proof: $ \ BEX (f (x), g (x) = 1 \ LRA (f (x ^ N), g (x ^ n) = 1, \ EEx $ here, $ N $ is a given natural number.

 

5.

(1) If $ \ lm \ NEQ 0 $ is an feature value of matrix $ A $, then $ \ DPS {\ frac {1} {\ lm} | A |}$ is an feature value of $ A ^ * $;

(2) If $ \ Al $ is a feature vector of $ A $, $ \ Al $ is also a feature vector of $ A ^ * $;

(3) $ (AB) ^ * = B ^ * a ^ * $.

 

6. set $ W $ as the number field $ \ BBP $ on $ N $ Dimension Linear Space $ V $ of the sub-space, $ \ SCRA $ is a linear transformation of $ V $, $ \ scra w $ indicates the sub-space composed of the vectors in $ W $, so $ W_0 = W \ cap \ Ker \ SCRA $, which proves: $ \ Bex \ dim W = \ dim \ scra w + \ dim W_0. \ EEx $

 

7. set $ A $, $ E-A $, $ E-A ^ {-1} $ to reversible matrix, trial: $ \ BEX (E-A) ^ {-1} + (E-A ^ {-1}) ^ {-1} = E. \ EEx $

 

8. set $ \ SCRA $ to a linear transformation in the linear space $ V $. Note: $ \ Bex \ Ker \ SCRA =\sed {\ Al \ In V; \ SCRA \ Al = 0}, \ quad \ im \ SCRA = \ sed {\ SCRA \ al; \ Al \ In v }. \ EEx $ test certificate: $ \ Bex \ im \ SCRA \ cap \ Ker \ SCRA = \ sed {0} \ LRA \ Ker \ SCRA = \ Ker (\ SCRA ^ 2 ). \ EEx $

 

9. set $ U, W $ to two subspaces of $ N $ Dimension Linear Space $ V $, and $ \ Bex \ dim U + \ dim W = n. \ EEx $ proof: linear transformation on $ V $ \ SCRA $ makes $ \ Bex \ Ker \ SCRA = u, \ quad \ im \ SCRA = W. \ EEx $

 

10. set $ \ SCRA $ to a linear transformation on $ N $ Dimension Linear Space $ V $, and $ \ Bex \ exists \ r \ In \ bbz ^ +, \ ST \ Ker (\ SCRA ^ R) = \ Ker (\ SCRA ^ {R + 1 }). \ EEx $ test certificate: $ \ Bex \ forall \ s \ In \ bbz ^ +, \ quad \ Ker (\ SCRA ^ r) = \ Ker (\ SCRA ^ {R + S }). \ EEx $

 

11. set $ V $ to $ N $ Dimension Linear Space, $ \ SCRA $, $ \ scrb $ to two linear transformations on $ V $, and $ \ SCRA $ has $ N $ feature values that are different from each other. proof: $ \ SCRA \ scrb = \ scrb \ SCRA $ is required only when $ \ scrb $ is $ \ SCRA ^ 0 = \ scre $ (constant transformation ), $ \ SCRA $, $ \ SCRA ^ 2 $, $ \ cdots $, $ \ SCRA ^ {n-1} $ linear combination.

 

12. set $ \ SCRA $ to the linear space on the number field $ \ BBF $ V $ to the previous linear transformation. In $ \ BBF [x] $, $ \ Bex f (x) = p (x) Q (x ). \ EEx $ verification: $ \ Bex \ Ker (f (\ SCRA) = \ Ker (P (\ SCRA )) \ oplus \ Ker (Q (\ SCRA )). \ EEx $

 

13. set $ \ Bex a =\sex {\ BA {CCC} 0 & 10 & 30 \ 0 & 0 & 2010 \ 0 & 0 & 0 \ EA }. \ EEx $ test certificate: matrix equation $ x ^ 2 = B $ no solution.

 

14. set $ V $ to the finite dimension vector space on the number field $ \ BBF $, and $ \ SCRA $ to the linear transformation on $ V $. proof: $ V $ can be divided into two subspaces: Straight and $ v = U \ oplus W $, where $ U and W $ meet: for $ \ forall \ U \ In U $, $ k \ In \ bbz ^ + $ exists, making $ \ SCRA ^ K (u) = 0 $; for $ \ forall \ W \ in W, \ forall \ m \ In \ bbz ^ + $, $ w_m \ In V $ exists, make $ W = \ SCRA ^ m (w_m) $.

 

15. set $ V $ to a real number field $ \ BBR $ to $ N $ Dimension Linear Space. $ \ SCRA $ is a linear transformation on $ V $, yes $ \ SCRA ^ 2 =-\ scre $.

(1) proof: $ N $ is an even number;

(2) If $ \ scrb $ is a linear transformation on $ V $, and $ \ scrb \ SCRA = \ SCRA \ scrb $ is met, $ \ det (\ scrb) \ geq 0 $.

 

16. known Quadratic Form $ \ Bex f (x, y, z) = x ^ 2 + 3y ^ 2 + Z ^ 2 + 2bxy + 2xz + 2yz \ EEx $ the rank is $2 $. Evaluate the parameter $ B $, and pointed out the equation $ \ Bex f (x, y, z) = 4 \ EEx $ represents what kind of surface?

 

17. set $ \ SCRA $, $ \ scrb $ to two linear transformations in the $ N $ dimension linear space in a certain number field, $ \ Bex \ left \ {\ BA {ll} \ SCRA \ CIRC \ scrb = \ scrb \ CIRC \ SCRA, \ exists \ n \ In \ bbz ^ +, \ s. t. \ SCRA ^ n = \ scro. \ EA \ right. \ EEx $ proof: $ \ Bex \ SCRA + \ scrb \ mbox {reversible linear transformation} \ LRA \ scrb \ mbox {reversible linear transformation }. \ EEx $

 

18. set $ \ SCRA $ to the orthogonal transformation on the Euclidean Space $ V $, and $ \ SCRA ^ m = \ scre \ (M> 1) $. note that $ W _ \ SCRA =\sed {\ Al \ In V; \ SCRA \ Al =\al }$, $ W _ \ SCRA ^ \ perp $ is its orthogonal complement, and $ \ Bex \ forall \ Al \ In V, \ exists \ | \ beta \ in W _ \ SCRA, \ gamma \ in W _ \ SCRA ^ \ perp, \ ST \ Al = \ Beta + \ Gamma. \ EEx $ test certificate: $ \ Bex \ Beta = \ frac {1} {m} \ sum _ {I = 1} ^ m \ SCRA ^ {I-1} \ Al. \ EEx $

 

19. set $ v = \ BBC ^ {n \ times n} $ to represent $ N on the complex field $ \ BBC $. Linear Combination of the addition of the matrix and the multiplication of the number of Matrices space, $ A \ In V $, define the transform $ \ SCRA $ on $ V $ as follows: $ \ Bex \ SCRA (x) = AX-XA, \ quad \ forall \ x \ In v. \ EEx $ test certificate:

(1) $ \ SCRA $ is a linear transformation;

(2) $ \ SCRA (xy) = x \ SCRA (y) + \ SCRA (x) y $;

(3) $0 $ is an feature value of $ \ SCRA $;

(4) If $ A ^ K = 0 $, then $ \ SCRA ^ {2 k} = \ scro $.

 

20. set $ \ lm_1, \ cdots, \ lm_n $ to all feature values of $ N $ level matrix $ A $, but $-\ lm_ I \ (I = 1, 2, \ cdots, n) $ is not the feature value of $ A $. It defines a linear transformation of $ \ BBR ^ {n \ times n} $ \ Bex \ SCRA (x) = a ^ Tx + XA, \ quad \ forall \ x \ In \ BBR ^ {n \ times n }. \ EEx $

(1) test: $ \ SCRA $ is a reversible linear transformation;

(2) For any real symmetric matrix $ C $, a unique real symmetric matrix $ B $ must exist, making $ A ^ TB + BA = C $.

 

21. Set $ A and B $ to two N $ positive definite matrices. proof:

(1) If $ AB = BA $, $ AB $ is also a positive matrix;

(2) If $ A-B $ is positive, $ B ^ {-1}-a ^ {-1} $ is also positive.

 

22. Set $ N $ square matrix $ A $ positive. Reset

(1) $ B _1, \ cdots, B _n $ is any $ N $ non-zero real number. Test Certificate: matrix $ B = (A _ {IJ} B _ib_j) $ is also positive.

(2) $ B $ is $ n \ times M $ real matrix, and $ \ rank (B) = M $. Test Certificate: $ B ^ tab $ is also positive.

(3) $ B $ is $ N $ level positive definite matrix. Test Certificate: $ c = (A _ {IJ} B _ {IJ}) $ is also positive.

 

23. set $ A, B $ to $ N $, and $ A $ is a non-zero semi-Definite Matrix. $ B $ is a positive definite matrix: $ | a + B |> | B | $.

 

24. set $ A $ to $ N $ order reversible matrix. Test Certificate: orthogonal matrix exists $ p $, positive matrix $ u, v $ to make $ \ Bex a = Ru = VR. \ EEx $

 

25. set $ q $ to $ N $ positive sequence matrix, and $ x $ to $ N $ dimension column vector. Example: $ \ Bex 0 \ Leq x ^ t (q + XX ^ t) ^ {-1} x <1. \ EEx $

 

26. Set $ A $, $ B $ to a real symmetric matrix, $ A $ Positive Definite. Test Certificate: $ B $ positive definite when and only when the feature values of $ AB $ are all greater than zero.

 

27. set $ A $ to $ N $ order positive definite matrix, $ x $, $ y $ to $ N $ dimension column vector and meet $ x ^ ty> 0 $. proof matrix $ \ Bex M = a + \ cfrac {XX ^ t} {x ^ ty}-\ cfrac {Ayy ^ Ta} {y ^ Tay} \ EEx $ Zhengding.

 

28. proof: $ A = \ sex {A _ {IJ} $ is a positive definite matrix, $ \ Bex a _ {IJ} =\frac {1} {I + J-1 }. \ EEx $

 

29. set $ W $ to the sub-space of the Euclidean Space $ V $, and define the distance from $ \ Al \ In V $ to $ W $ \ RD (\ Al, W) = | \ al-\ al' | $, where $ \ al' $ is an orthogonal projection of $ \ Al $ on $ W $. set $ \ al_1, \ cdots, \ al_m $ to a group of bases for $ W $. trial certificate: $ \ Bex \ RD (\ Al, W) =\ SQRT {\ cfrac {G (\ al_1, \ cdots, \ al_m, \ Al)} {G (\ al_1, \ cdots, \ al_m )}}, \ EEx $ where $ G (\ al_1, \ cdots, \ al_m) $ is the Gram matrix of $ \ al_1, \ cdots, \ al_m $.

 

30. Set $ N $ level-2 Symmetric Matrix $ A $ to zero for the sum of all level-1 primary keys and all level-2 primary keys. Prove that $ A $ is a zero matrix.

 

31. note $ \ lm_1 (a) \ geq \ lm_2 (a) \ geq \ cdots \ geq \ lm_n (a) $ is the feature value of $ N $ level real symmetric matrix $ A $. test certificate:

(1) If $ A and B $ are real symmetric matrices and the real number $0 \ Leq A \ Leq 1 $, then for $ I = 1, 2, \ cdots, N $, $ \ Bex a \ lm_ I (A) + (1-A) \ lm_n (B) \ Leq \ lm_ I (AA + (1-A) B) \ Leq A \ lm_ I () + (1-A) \ lm_1 (B); \ EEx $

(2) If $ B $ is semi-definite, then $ \ Bex \ lm_1 (a + B) \ geq \ lm_1 (A), \ quad \ lm_n (A + B) \ geq \ lm_n (). \ EEx $

 

32. set $ A and B $ to level $ M $ and level $ N $ matrices respectively. test evidence: $ A, B $ a matrix equation $ AX = XB $ only has zero solutions.

 

33. Test Certificate: $ N $ level matrix $ X, Y $ makes $ XY-YX = e $.

 

34. if $ A (t) = (a _ {IJ} (t) $ is set, each $ A _ {IJ} (t) $ is bootable. test certificate: $ \ Bex \ frac {\ RD} {\ RD t} | A (t) | = | A (t) | \ cdot \ tr \ SEZ {A ^ {-1} (t) \ cdot \ frac {\ RD a (t) }{\ rd T }}. \ EEx $

 

35. set $ N $ level antisymmetric matrix $ A = (A _ {IJ}) $ to the determinant of $1 $. for any $ x $, calculate the determinant of $ B = (A _ {IJ} + x) $.

 

36. Set $ A $ as an orthogonal array. The feature values of $ A $ are real numbers. test example: $ A $ is a symmetric matrix.

 

37.

(1) solutions to any matrix $ A $, matrix equation $ Axa = A $;

(2) If the matrix equations $ ay = C $ and $ ZB = C $ have solutions, then the equation $ AXB = C $ has solutions.

 

38. Set $ A $, $ B $ to a real symmetric matrix, $ A $ Positive Definite. Test Certificate: $ B $ positive definite when and only when the feature values of $ AB $ are all greater than zero.

 

39. set $ F $ to a linear ing from $ \ BBC ^ {n \ times n} $ to $ \ BBC $ to meet $ F (e) = N $, and for any matrix $ a, B \ In \ BBC ^ {n \ times N }$, $ F (AB) = f (BA) $. test Certificate: $ F = \ tr $.

 

40. set $ A $ as a symmetric matrix with Linearly Independent Vectors $ X_1 and X_2 $, so that $ X_1 ^ tax_1> 0 $, $ X_2 ^ tax_2 <0 $. proof: There is a linear unrelated vector $ X_3, x_4 $ to make $ x_1, x_2, X_3, x_4 $ linear correlation, and $ X_3 ^ tax_3 = x_4 ^ tax_4 = 0 $.

[Jia Liwei university mathematics magazine] 322nd sets of simulation papers for Mathematics Competition Training of Gannan Normal University