The document contains a math test with multiple choice and free response questions. Some key points:
- It asks to find the domain and range of the function f(x) = 3x^2 - 6x + 1.
- It asks to solve the system of equations x^2 + y^2 - 4x + 2y - 5 = 0 and 2x - y = 0 to find the points A and B where the line is tangent to the circle.
- It asks to find the real root of the equation z^4 - 22z + 5 = 0.
- It provides two parts (A and B) for the free response section and asks to choose one. Part
1. The document provides information about a math exam, including the exam time of 180 minutes and 6 questions ranging from 1 to 2 points each. The questions cover topics such as solving equations, finding roots of equations, integrals, geometry problems, and systems of equations.
2. The responses provide solutions to each question, showing the steps and reasoning for obtaining the answers. Solutions include solving equations, finding integrals, using geometry relationships, and solving a system of inequalities.
3. Diagrams and calculations are shown to visually depict the solutions to the geometry problems involving shapes, angles, and areas.
This math test contains 6 questions about calculus, geometry, and equations. Question 1 involves finding the derivative of a function and finding critical points. Question 2 involves solving equations. Question 3 involves finding integrals and values that satisfy an equation. Question 4 is about finding properties of a square pyramid. Question 5 involves finding points on lines and writing equations of circles. Question 6 is about finding the minimum value of an expression involving integrals.
1. The document provides information about a math exam, including the duration of 180 minutes.
2. It lists 7 questions on the exam covering topics like functions, derivatives, integrals, geometry, and inequalities.
3. The questions involve solving equations, finding maximum/minimum values, determining tangent lines, proving relationships between angles and lines, and optimizing expressions.
This document contains equations related to fluid mechanics and wave theory. Some key equations include:
1) Equations for velocity profiles in boundary layers and free surface waves.
2) Equations describing linear wave theory including wave celerity, wave frequency, and potential functions for wave propagation.
3) Equations for hydrostatic pressure and continuity.
4) Equations describing forces on structures like drag and inertia.
This document appears to be a 10 question math exam in Vietnamese with the following key details:
1. The exam is 180 minutes long with no additional time for submission.
2. The questions cover topics like graphing functions, solving equations, integrals, geometry problems finding volumes of shapes, and coordinate geometry.
3. The final question asks students to find the minimum value of an expression given certain constraints on variables a, b, and c.
The document appears to contain mathematical equations and calculations involving trigonometric functions like cosine, sine and complex numbers. It includes equations for calculating the cosine and sine of various angles, and using those values to solve for complex numbers and their components. The equations are not clearly labeled or explained, so the overall meaning and context is difficult to discern from the summary.
1. The document is a math exam with 7 questions in Vietnamese.
2. Question 1 involves analyzing the behavior and limits of the function y = -x^2 - 4x + 1.
3. Question 2 involves solving two equations for the intersection points of lines and finding the minimum length between two points.
4. The remaining questions involve additional math problems like solving equations, finding limits, calculating areas, and expansions of functions.
1. Derive the expression for stress tensor in cylindrical coordinates (r, θ, z). Calculate the volume integral of r-1 over the sphere of radius R.
2. Find the solution u(r,t) of the heat equation with initial and boundary conditions given.
3. Find the temperature distribution inside a cylindrical rod at any time t > 0, given the initial temperature distribution and boundary conditions.
4. Find the function u(r,φ) satisfying the Laplace's equation and boundary condition on the surface of a sphere.
5. Find the solution u(r,y) of the given partial differential equation satisfying the boundary conditions.
1. The document provides information about a math exam, including the exam time of 180 minutes and 6 questions ranging from 1 to 2 points each. The questions cover topics such as solving equations, finding roots of equations, integrals, geometry problems, and systems of equations.
2. The responses provide solutions to each question, showing the steps and reasoning for obtaining the answers. Solutions include solving equations, finding integrals, using geometry relationships, and solving a system of inequalities.
3. Diagrams and calculations are shown to visually depict the solutions to the geometry problems involving shapes, angles, and areas.
This math test contains 6 questions about calculus, geometry, and equations. Question 1 involves finding the derivative of a function and finding critical points. Question 2 involves solving equations. Question 3 involves finding integrals and values that satisfy an equation. Question 4 is about finding properties of a square pyramid. Question 5 involves finding points on lines and writing equations of circles. Question 6 is about finding the minimum value of an expression involving integrals.
1. The document provides information about a math exam, including the duration of 180 minutes.
2. It lists 7 questions on the exam covering topics like functions, derivatives, integrals, geometry, and inequalities.
3. The questions involve solving equations, finding maximum/minimum values, determining tangent lines, proving relationships between angles and lines, and optimizing expressions.
This document contains equations related to fluid mechanics and wave theory. Some key equations include:
1) Equations for velocity profiles in boundary layers and free surface waves.
2) Equations describing linear wave theory including wave celerity, wave frequency, and potential functions for wave propagation.
3) Equations for hydrostatic pressure and continuity.
4) Equations describing forces on structures like drag and inertia.
This document appears to be a 10 question math exam in Vietnamese with the following key details:
1. The exam is 180 minutes long with no additional time for submission.
2. The questions cover topics like graphing functions, solving equations, integrals, geometry problems finding volumes of shapes, and coordinate geometry.
3. The final question asks students to find the minimum value of an expression given certain constraints on variables a, b, and c.
The document appears to contain mathematical equations and calculations involving trigonometric functions like cosine, sine and complex numbers. It includes equations for calculating the cosine and sine of various angles, and using those values to solve for complex numbers and their components. The equations are not clearly labeled or explained, so the overall meaning and context is difficult to discern from the summary.
1. The document is a math exam with 7 questions in Vietnamese.
2. Question 1 involves analyzing the behavior and limits of the function y = -x^2 - 4x + 1.
3. Question 2 involves solving two equations for the intersection points of lines and finding the minimum length between two points.
4. The remaining questions involve additional math problems like solving equations, finding limits, calculating areas, and expansions of functions.
1. Derive the expression for stress tensor in cylindrical coordinates (r, θ, z). Calculate the volume integral of r-1 over the sphere of radius R.
2. Find the solution u(r,t) of the heat equation with initial and boundary conditions given.
3. Find the temperature distribution inside a cylindrical rod at any time t > 0, given the initial temperature distribution and boundary conditions.
4. Find the function u(r,φ) satisfying the Laplace's equation and boundary condition on the surface of a sphere.
5. Find the solution u(r,y) of the given partial differential equation satisfying the boundary conditions.
This document is unintelligible as it contains random characters and symbols with no discernible words or meaning. No important information can be summarized from this document as it does not communicate any ideas or facts in an understandable way.
This document is unintelligible as it contains random characters and symbols with no discernible words or meaning. No important information can be summarized from this document.
This document is entirely nonsensical, containing random symbols and characters with no coherent words, sentences, or meaning. It appears to be gibberish with no real information or essential details that could be summarized.
1. The document presents solutions to several quadratic equations involving variables x and y.
2. Methods shown include factoring, using the quadratic formula, and completing the square.
3. Solutions expressed in terms of radicals include x = 3 ±√3, x = -3 ±√15, and x = 2 ±√6.
This document contains information about derivatives, integrals, and trigonometric identities. It lists rules for derivation and integration along with the derivatives of common functions like exponentials, logarithms, trigonometric functions, and their inverses. It also provides formulas for integrals of rational, logarithmic, irrational and trigonometric functions. Finally, it lists 15 trigonometric identities.
The document discusses the relationship between various mathematical formulas and concepts. It explores equations for energy, frequency, and wavelength. Several Greek letters and scientific symbols are used in multi-variable formulas representing physical properties.
The document provides information about the constellation Orion. It lists the main stars in Orion including Betelgeuse, Rigel, and Mintaka. It mentions the Orionid meteors and the Great Orion Nebula. Orion is named after the hunter from Greek mythology and is one of the most visible and recognizable constellations in the sky.
The document provides instructions for candidates taking an exam. It includes details like writing the hall ticket number on the answer sheet, checking the question paper for defects, not using electronic devices during the exam, marking answers on the answer sheet, not leaving the exam hall early, and signing documents before and after the exam. Candidates are instructed to return the answer sheet before leaving and keep the question paper afterwards.
This document provides information about physics concepts and calculus formulas taught by a physics lecturer at the Royal University of Phnom Penh. It contains 3 sections:
1. Derivatives - Defines various derivatives and their formulas up to order n.
2. Integrals - Covers integration techniques like integration by parts, trigonometric integrals, and improper integrals.
3. Differential Equations - Discusses first order differential equations, second order differential equations, and their applications to dynamics.
Kompendium DTP. Adobe Photoshop, Illustrator, InDesign i Acrobat w praktyceWydawnictwo Helion
Zdobądź niezbędną wiedzę, aby tworzyć profesjonalne publikacje!
* Jak skutecznie retuszować zdjęcia?
* Jak tworzyć ścieżki i obiekty wektorowe?
* Jak przygotować publikacje do druku?
Przygotowanie profesjonalnej i wyjątkowej ulotki reklamowej, informatora czy innej publikacji wcale nie jest łatwe. Może dlatego umiejętności te są obecnie bardzo pożądane na rynku pracy. Jednak współczesne oprogramowanie komputerowe oraz najnowsze narzędzia poligraficzne dają możliwość stworzenia niepowtarzalnej i doskonałej technicznie pracy nawet początkującym grafikom i operatorom DTP. Aby móc wykorzystać tę szansę, należy najpierw dokładnie poznać narzędzia przydatne przy takiej pracy. W tym na pewno pomoże Ci ta książka — prawdziwe kompendium wiedzy z zakresu programów graficznych i zagadnień poligrafii.
Książka "Kompendium DTP. Adobe Photoshop, Illustrator, InDesign i Acrobat w praktyce" to wyjątkowy podręcznik, opisujący nie tylko funkcje i narzędzia dostępne w przedstawionych programach, ale także zagadnienia związane z profesjonalnym przygotowaniem publikacji do druku. W związku z tym stanowi niezastąpiony poradnik dla wszystkich, którzy chcieliby szybko i bez problemu poznać zasady edycji i tworzenia grafiki oraz odpowiedniego jej opracowania na potrzeby drukarni — tak aby gotowy produkt był zgodny z oczekiwaniami odbiorcy. Korzystając z tej książki, można dowiedzieć się, jak stworzyć ulotkę reklamową lub wielostronicowy katalog, a także poznać podstawowe narzędzia i tajniki poligrafii.
* DTP i grafika komputerowa
* Adobe Photoshop
* Narzędzia retuszu i selekcji
* Warstwy, przekształcenia i montaże
* Wykorzystanie obiektów inteligentnych
* Praca z tekstem
* Korekcja barw i efekty specjalne
* Adobe Illustrator
* Tworzenie i edycja obiektów wektorowych
* Zaawansowana edycja ścieżek
* Maski, zniekształcenia i transformacje
* Adobe InDesign
* Style akapitowe, znakowe, obiektowe i tabel
* Przygotowanie publikacji do druku
Teraz także Ty możesz zostać specjalistą DTP i tworzyć profesjonalne publikacje!
This document is unintelligible as it contains random characters and symbols with no discernible words, sentences, or meaning. It does not provide any information that can be summarized.
This document is an edit decision list for a film or video project. It contains 16 entries with shot numbers, timecodes for shot starts and ends, and brief comments describing each shot or take. The comments indicate things like camera issues, actor performances, or desired edits. The list provides a high-level overview of the shots and edits needed to assemble the final video project.
This document contains 6 math equations:
1) yj = 0
2) x' - 14 = 0
3) r<1+l?-> l3r = c
4) xz-13>'Y[-lL
5) f7t6x-2,4 =to
6) \z-3>-sE:a
Each equation is set equal to a variable or constant with restrictions on x and r provided for some equations.
This document contains worked solutions to mathematics questions:
(1) It solves equations such as 3x + 4 = 12 and 4x - 8 + 3x - 3 = 1x - 5.
(2) It evaluates expressions like (a + b)2 - 2ab and solves equations such as 3x4 = (a + b)2 - 2ab.
(3) It solves equations involving indices, surds, and algebraic fractions.
This document is unintelligible as it contains random characters and symbols with no discernible words or meaning. No important information can be summarized from this document as it does not communicate any ideas or facts in an understandable way.
This document is unintelligible as it contains random characters and symbols with no discernible words or meaning. No important information can be summarized from this document.
This document is entirely nonsensical, containing random symbols and characters with no coherent words, sentences, or meaning. It appears to be gibberish with no real information or essential details that could be summarized.
1. The document presents solutions to several quadratic equations involving variables x and y.
2. Methods shown include factoring, using the quadratic formula, and completing the square.
3. Solutions expressed in terms of radicals include x = 3 ±√3, x = -3 ±√15, and x = 2 ±√6.
This document contains information about derivatives, integrals, and trigonometric identities. It lists rules for derivation and integration along with the derivatives of common functions like exponentials, logarithms, trigonometric functions, and their inverses. It also provides formulas for integrals of rational, logarithmic, irrational and trigonometric functions. Finally, it lists 15 trigonometric identities.
The document discusses the relationship between various mathematical formulas and concepts. It explores equations for energy, frequency, and wavelength. Several Greek letters and scientific symbols are used in multi-variable formulas representing physical properties.
The document provides information about the constellation Orion. It lists the main stars in Orion including Betelgeuse, Rigel, and Mintaka. It mentions the Orionid meteors and the Great Orion Nebula. Orion is named after the hunter from Greek mythology and is one of the most visible and recognizable constellations in the sky.
The document provides instructions for candidates taking an exam. It includes details like writing the hall ticket number on the answer sheet, checking the question paper for defects, not using electronic devices during the exam, marking answers on the answer sheet, not leaving the exam hall early, and signing documents before and after the exam. Candidates are instructed to return the answer sheet before leaving and keep the question paper afterwards.
This document provides information about physics concepts and calculus formulas taught by a physics lecturer at the Royal University of Phnom Penh. It contains 3 sections:
1. Derivatives - Defines various derivatives and their formulas up to order n.
2. Integrals - Covers integration techniques like integration by parts, trigonometric integrals, and improper integrals.
3. Differential Equations - Discusses first order differential equations, second order differential equations, and their applications to dynamics.
Kompendium DTP. Adobe Photoshop, Illustrator, InDesign i Acrobat w praktyceWydawnictwo Helion
Zdobądź niezbędną wiedzę, aby tworzyć profesjonalne publikacje!
* Jak skutecznie retuszować zdjęcia?
* Jak tworzyć ścieżki i obiekty wektorowe?
* Jak przygotować publikacje do druku?
Przygotowanie profesjonalnej i wyjątkowej ulotki reklamowej, informatora czy innej publikacji wcale nie jest łatwe. Może dlatego umiejętności te są obecnie bardzo pożądane na rynku pracy. Jednak współczesne oprogramowanie komputerowe oraz najnowsze narzędzia poligraficzne dają możliwość stworzenia niepowtarzalnej i doskonałej technicznie pracy nawet początkującym grafikom i operatorom DTP. Aby móc wykorzystać tę szansę, należy najpierw dokładnie poznać narzędzia przydatne przy takiej pracy. W tym na pewno pomoże Ci ta książka &#8212; prawdziwe kompendium wiedzy z zakresu programów graficznych i zagadnień poligrafii.
Książka "Kompendium DTP. Adobe Photoshop, Illustrator, InDesign i Acrobat w praktyce" to wyjątkowy podręcznik, opisujący nie tylko funkcje i narzędzia dostępne w przedstawionych programach, ale także zagadnienia związane z profesjonalnym przygotowaniem publikacji do druku. W związku z tym stanowi niezastąpiony poradnik dla wszystkich, którzy chcieliby szybko i bez problemu poznać zasady edycji i tworzenia grafiki oraz odpowiedniego jej opracowania na potrzeby drukarni &#8212; tak aby gotowy produkt był zgodny z oczekiwaniami odbiorcy. Korzystając z tej książki, można dowiedzieć się, jak stworzyć ulotkę reklamową lub wielostronicowy katalog, a także poznać podstawowe narzędzia i tajniki poligrafii.
* DTP i grafika komputerowa
* Adobe Photoshop
* Narzędzia retuszu i selekcji
* Warstwy, przekształcenia i montaże
* Wykorzystanie obiektów inteligentnych
* Praca z tekstem
* Korekcja barw i efekty specjalne
* Adobe Illustrator
* Tworzenie i edycja obiektów wektorowych
* Zaawansowana edycja ścieżek
* Maski, zniekształcenia i transformacje
* Adobe InDesign
* Style akapitowe, znakowe, obiektowe i tabel
* Przygotowanie publikacji do druku
Teraz także Ty możesz zostać specjalistą DTP i tworzyć profesjonalne publikacje!
This document is unintelligible as it contains random characters and symbols with no discernible words, sentences, or meaning. It does not provide any information that can be summarized.
This document is an edit decision list for a film or video project. It contains 16 entries with shot numbers, timecodes for shot starts and ends, and brief comments describing each shot or take. The comments indicate things like camera issues, actor performances, or desired edits. The list provides a high-level overview of the shots and edits needed to assemble the final video project.
This document contains 6 math equations:
1) yj = 0
2) x' - 14 = 0
3) r<1+l?-> l3r = c
4) xz-13>'Y[-lL
5) f7t6x-2,4 =to
6) \z-3>-sE:a
Each equation is set equal to a variable or constant with restrictions on x and r provided for some equations.
This document contains worked solutions to mathematics questions:
(1) It solves equations such as 3x + 4 = 12 and 4x - 8 + 3x - 3 = 1x - 5.
(2) It evaluates expressions like (a + b)2 - 2ab and solves equations such as 3x4 = (a + b)2 - 2ab.
(3) It solves equations involving indices, surds, and algebraic fractions.
This document discusses properties of fractional parts that are useful for solving olympiad math problems. It introduces 12 properties of fractional parts, such as the fractional part of a sum is less than or equal to the sum of the fractional parts. It then provides examples and explanations for each property. The document aims to help students better understand fractional parts and improve their math problem solving skills.
This document contains mathematical calculations and derivations involving parameters such as s, K, and z. It determines that certain expressions equal zero, finds limits, and identifies poles and zeros of transfer functions. The document also contains comments on closed-loop stability analysis and indicates that the poles are real when K is less than one.
Đề Kiểm Tra Chất Lượng Môn Toán Trường Chuyên Hà Nội AmsterdamMathX Thích Học Toán
1. The document provides 4 math problems involving solving equations and inequalities for unknown variables x and y.
2. Problem 1 involves solving a quadratic equation for x. Problem 2 involves solving a linear equation for x.
3. Problem 3 finds values of x and y that satisfy two simultaneous equations.
4. The final problem involves using given values to solve two equations for the unknowns x and y.
1. The document discusses reducing a given proposition to its minimal equivalent expression in sum of products form. It provides examples of reducing propositions such as A+BC to its minimal expression of A+B+C.
2. Methods for simplifying Boolean expressions using Boolean algebra rules are presented, including eliminating common factors, combining like terms, and removing redundant variables.
3. The process of obtaining the minimum equivalent expression for circuits and logic gates such as AND, OR, and NAND is explained step-by-step with examples.
The document contains solutions to calculus problems involving finding maximum and minimum values of functions, determining critical points, and investigating functions for maxima and minima. Several functions are given as examples, including f(x)=x^3-6x^2+12x-3, f(x)=e^x +2cosx+ e^-x, and f(x)=sin(x)(1+cos(x))^. The document examines these functions and discusses determining critical points and maxima/minima values.
The document contains solutions to calculus problems involving finding maximum and minimum values of functions, determining critical points, and investigating functions for maxima and minima. Several functions are given as examples, including f(x)=x^3-6x^2+12x-3, f(x)=e^x +2cosx+ e^-x, and f(x)=sin(x)(1+cos(x))^. The document examines these functions and discusses determining critical points and maxima/minima values.
Đại lý dây điện Cadivi 3.5 tại Tphcm chiết khấu caoVõ Thành Tiến
Đại lý dây điện Cadivi 3.5 tại Tphcm chiết khấu cao
Bảng giá dây điện cadivi mới nhất
Chiết khấu dâ điện Cadivi 2018 mới nhất tại Tphcm
https://thietbidandung.vn/day-cap-dien-cadivi
1. The document presents equations for several related differential equations involving functions of x (f(x), g(x), etc.) and their derivatives.
2. The equations contain common functions like exponentials, logarithms, trigonometric functions, and their derivatives.
3. Boundary conditions or initial conditions are provided for solving some of the differential equations.
This document contains mathematical formulas and properties related to:
- Arithmetic and geometric sequences
- Trigonometric functions and their properties
- Logarithmic and exponential functions
- Derivatives of functions
- Integrals and properties of integrals
- Trigonometric identities
TMUA 2021 Paper 1 Solutions (Handwritten).pdfssuser625c41
(1) The document provides instructions for a test of mathematics paper with 20 questions and a time limit of 75 minutes. No calculators or additional materials are allowed.
(2) Candidates must fill out personal information on the answer sheet and choose one answer for each question, recording their choice on the answer sheet. There are no penalties for incorrect answers.
(3) The test consists of 20 multiple choice questions about mathematics, each worth one mark. Candidates should attempt all questions within the time limit.
1. This document is a marking scheme for Additional Mathematics paper 1 containing solutions and marks for various questions.
2. It provides the correct solutions, working and marks awarded for 16 questions ranging from simple calculations to complex problem solving.
3. The marking scheme acts as a guide for examiners to consistently and fairly award marks for student responses based on the level of accuracy and working shown.
1. The document describes several functions f(x) and their properties
2. It defines the functions f(x) = 6x-5, f(x) = bx-5+3, and f(x) = (x-3)2
3. It asks to find other functions that satisfy the given properties and conditions
1. The question asks to analyze the function y=x3+3x2-4 and find the point of inflection of the equation (x+2)'=4cos(x).
2. For the function y=x3+3x2-4, the point of inflection is (-1,-2) and the minimum point is (0,-4).
3. The point of inflection of the equation (x+2)'=4cos(x) is (-1,0).
1. The document is a practice test for a Japanese language exam containing 40 multiple choice questions testing reading comprehension and grammar.
2. The test has two sections - a multiple choice section worth 8 points and an essay section worth 2 points.
3. Test takers are not allowed to use any reference materials and must submit both the answer sheet and test paper after completion.
1) The document discusses components of forces and their resolution into perpendicular and parallel components using trigonometric identities.
2) It provides equations to calculate the horizontal and vertical components of an inclined force.
3) Conditions for equilibrium of rigid bodies under the action of coplanar forces are explained along with examples of stable, unstable and neutral equilibrium.
The document appears to be part of an exam for an engineering mathematics course. It contains instructions for answering questions, notes on objective type questions, and four practice problems:
1) Choose the correct answer for questions about electrochemical cells and redox reactions.
2) Solve the differential equation p' - 2p sinh x = -1.
3) Solve the differential equation y" + y = cos x.
4) Obtain the general and singular solutions of the Clairaut's equation (y - px)(p-1) = p.
- The code defines a class called PrintLoops that inherits from IRVisitor. It overrides the visit method to print the name of any For nodes visited.
- A print_loops function takes a statement and uses a PrintLoops visitor to print the name of all for loops in the statement.
Resolução do Livro Mecânica Vetorial para Engenheiros BEER 5ª Edição.pdfDeboraIshikawa
This document contains calculations for determining forces and reactions in structural analysis problems. It solves several systems of equations to find unknown forces and reactions given information about known forces, distances, and orientations. Key values calculated include forces of 200N, 750N, 1250N and reactions of 47lb and 117.7lb. Diagrams and step-by-step working are shown to arrive at the solutions.
1. The document discusses integration and properties of integrals. It shows that the integral of the derivative of a function equals the function evaluated from negative infinity to positive infinity.
2. Several integral properties are demonstrated, including properties related to adding or subtracting integrals and integrating with respect to different variables.
3. The document also explores integrals of functions over all real numbers and shows some integrals equal zero while others do not, depending on the properties of the functions.
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2. Chrmg minh rdng mdi ti6p tuy6n cira (C) chi ti6p xric vdi (C) tei dung mQt di6m.
Cfru II (2,0 tli6m)
1. Giai phuong trinh 9'io" * 4,gcosz
x= 13 +,*'z'+l-3'o'2'.
lx+ Y =g
2. Giai he Phuong trinh I r ^- r---:-
[r/xZ + 9 *^ly'+9 =10
(x'yeR)'
cflu IItr (1,0 tli6m) Tinh tich phdn 1 :pt 4*.
ix'
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Ggi A',8',C' vit D' lan luqt ld trung ,rliOm cira .!C,SD,SA vit SB. Chimg minh rdng AA',BB',CC' vh
DD' d6ng quy; Tfnh th6 tictr ctra hinh ch6p ^S'.1'B'
C' D' theo a, v&i ,S' h tam cfu hinh w0ng ABCD '
CiuV(1,0tli6m)Xicdinh m saocho xa -2x3 +8**l)*'-2mx+m'-4>A, Vxe [-f1]
II. PHAN RItNG (3,0 tli6m). Thf sinh chi iluqc chgn mQt trong hai phin (phin A ho{c phin B)
A. Phin A (theo chucrng trinh Chuin):
Cfiu VI.a (2,0 ili0m)
1. Trong mat phang tqa d0 Oxy , cho hinh r.u6ng c6 mQt dinh l(- 1;2) vA m$t duong ch6o nim trOn
ttucmg thing c6 phucmg trinh 2r - y -l = 0. Tim tqa d0 c6c dinh cdn lgi cira hinh vu6ng.
2. Vi6t phuong trinh m[t cAu (C) cO t6m thu$c dudng thlng (A) c6 phuong trinh
lx-vtz=oJ"
lZx+ Y+22-I=0
vdtitip xric voi hai m{t phang (a): 2x +2y - z + 6 =O va (B) ; 2x +2y - z-6 = 0.
Cffu VII.a (1,0 rli6m) Cho zr,z, ldhai nghiQm phrlc cria phuong frnh zz -22 +5 = 0. Tinh gi6 tri cua
bi€u thirc P =lr?l*l':l
B. Phin B (theo ehuong trinh Ning cao):
Cflu VI.b (2,0 tli6m)
1. Trongm{tphingtqad0 Oxychodudmgtdn(C): *2+y2-4x+2y-5=0.fhlrtA6euerngthing 4
d*! x - my=0 cat ducrng trdn (C) t?i hai di6m A, B pherr-biQt, sao cho dQ dii do4n AB nhtt nh6t.
z. Trong khdng gian tsa dQ Oxyz cho c6c tli6m l(l;O;t), f(tt;O), CQ;l;*l) vi m{t phdng (a) cO
phucrng trinhx + y + z-1 = 0. Tim to4 d0 diem M sao cho khoing c6ch tir M dln (a) Uang khoang
c6chtu M danm6iei6m A,B,C.
(r- 3
Ciu VII.b (1,0 tli6m) Tim sd phtrc z ,Ai6t Z =42-!
-------n6t-
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2. so crAo Duc vA o.A,o rAo HA NOI
rntldxc THPT CHU VAN AN
PAT AU - THANG DIEM
of rnr rrulDAr Hgc - DgT r nim zort
M6n Tofn - KhAi A
an- di6m 07 tran
Cfiu Dfrr 6n Di6m
I
(2,0 ili6m)
t. rt.O tli6m)
. T0P x6c dinh: B'
. Sg bi6n thi6n
- Gi6i han: lirn ! = -c; lim Y - +60.
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- Chi€ubii5nthi€n: !'=3x2 -6x; y'-0ex=0 hotrc x=2.
.y'>0e x<0 hoflc x>2; .y'<0<]0<x<2. HAm sO AOng bi6n tr€n cfc khoang
(-*,0), Q,**) vA nghich bi6n trOn ktroang (O,Z).
- Cuc tri: Hdm s5 dat cgc ct4i tq.i x = Q;yru = 1, d?t cgc tiAu tqi x:2i/cr =-3.
:-iia;s L-i6iitiliit;
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. EO thi
y"=6x-6; /"=0(i r=1
ximg cta AO tfri hnm s6.
Dd thi hdm sO c6 di6m uOn l(t,-l) vd n6 la tdm d6i
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2. fl.$ tli6m)
v,,{F*ALt - I t=-:
$ r.,t uJUcnn i
3. Ciu Eip 6n Di6m
ai di6m: Mo(to,Yo) vd M1(x1,Y1)
Khi d6 phucrng trinh cria ti6p tuy6n li
y=6*8-6xoh-'r-3'+3xfi+I vd ,:b*? -e'r! -z*l *?t:!
0'5
'-------'.
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#ii;fi;d'd #dil dcn ;ils i;-i,-6;ong liiirtr
"t
imqt tiiSp tuv6n n€n
3r& - 6xs =zxl - 6x1 ,
-2*3+lxfr +l=1xl +3xl +t .
Giai hQ trOn ta dugc xe = 11, do d6 ta c6 dpcm
II
(2,0 tli6m)
T (1"0 dtffi)
Phuong trinh dd cho tucrng duong vfi
9sin2.r * 4,gl-sn'zx =13+ 93/2-Zsintx -31-2sin2x
<) 9sin2r * 31
= 13 + 3- -:-gsin'x 92sn'r gstn- r
<+ 9sin" * jl - -4--- 13 = o'.
9sin2
x
(nrrr
r
),
Dflt r =9''" , 1 < r < 9, ta nhfln dugc phuong trinh , *+ -T -13 = 0
e (t *l)(t -3X/ - 9) = 0 <+ r =l;t = 3;t = 9.
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0025
Phrrcmstrinhddchotuonsduonsv6i sin2r=0 ho6c sin2x=t hoFc qr-41 f -=-!!-?
a
o
sin2x=0<+ x=kn.
sin2 r= 1 <> cosx = 0 e r = n l2+ kn 11:
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. sin2 x =ll2e cos2x= Q 49 x = t I 4+ ktr 12.
Vfly phucmg trinh dd cho c6 nghiQm * = k7 (k .4.
4
z. d.o tli6m)
-a c'6 y = 8 - x, thO vdo phuong trinh thrl hai cira hQ
"f7 .g *.{; -16r'173 =1g
r:l:
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€x2-8x*+t+@-59
e@=-x2+8x+9
f-*'*8x+920*
tft' * eh' -t6x +n)=l *' *sx+ef
f-ts x <v
<>{
[x'-8x+16:0
(3x=4.
. Ix=4
Suy ra he dA cho c6 mQt nghiQm duy nhdt j ., _ ,l"v--
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o:rs
III
(1,0 tli6m) Ta c6 t =2'pI4*
ix'
2
4. Cflu Edp 6n Ei6m
= -, Jf*)'' nxdx = ryli* r"l# 0,50
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22le 22 4I
- -!') -'t -------l
TL-L--
e xll e e e
4
Vqy I =)-*e
IV
(1,0 tli6m)
LSAC ta c6 AA',CC',,9,S' ld c6c dudng trung tuyOn n6n
G, cintam gi6c vi
,scr - 2cts',. (l)
AA' ftqngt4rx6t
tdm
'iic
'lt{--
-i".A
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Tucrng tp, trong LSBD ta cfing c6 BB' cdt DD' t4i trqng t6m G, vd
^sG, - 2c25',, (2)
TiI (1) viL Q)suy ra G, = Grhay AA',BB',CC' vd DD' d6ng quy.
Tt gia thii5t ta suy ra A'B'll =Jrcn, B'C'll =!ne,C'D'll =Iut2--' 2 2
D'A'll =!ac. Do d6 (A'B'c'D;)tt(aaco) vit A' B'c'D' h hinh vudng eqnh
2
Hcnrnir4 s'A'/l= ]s,a, mir Sl L('encn) nOn ,S'l'I {A'B'C'D').
z
va
a
2
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Ydy vs,.u,u,.,r, =
i
t' A' ft{A' B'C'r) =
i
23
doa_.-=+
24 24
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v
(1,0 tti6m)
Ta c6
xo -zxt +8*+l)x2 -Zmx+m'*4>-a, vxe[*1,1]<>
(*' - * * *f r-4vx e [*r,r]o Hfi(r' - * * *)' > 4. (3)
DAt t = xt -x, tac6 t'=2x-I.
xl I^ l-1 : 1
'r
t'l - 0 +
!.1 ?""....-.-.-- -a 9.
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