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This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.

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segundo parcial de analisis del cbc exactas e ingenieria

Este documento contiene 4 ejercicios de análisis matemático con sus respectivas respuestas: 1) Hallar el polinomio de Taylor de orden 3 de una función, 2) Calcular una integral definida, 3) Hallar el área de una región delimitada por curvas, 4) Estudiar la convergencia de una serie de potencias.

Methods of successive over relaxation

This presentation is about the successive over relaxation method. (SOR Method). It is the proof of the Sor Method.

Trigonometry 10th edition larson solutions manual

Trigonometry 10th edition larson solutions manual.
Full download: https://goo.gl/gFVG5A
Peope also search:
algebra and trigonometry ron larson 9th edition pdf
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algebra and trigonometry ron larson pdf
ron larson trigonometry 9th edition pdf
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algebra and trigonometry 10th edition pdf
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Trigonometric function

This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Volume and surface area formulae

This document provides formulas for calculating geometric properties of various 2D and 3D shapes. It includes formulas for calculating the perimeter and area of rectangles, squares, triangles, and circles. It also provides formulas for calculating the surface area and volume of 3D shapes like rectangular prisms, triangular prisms, cylinders, cones, spheres, and square-based pyramids.

Aptitude Training - TIME AND DISTANCE 1

I have taken coaching from NARESH INSTITUTE for CRT (Campus Recruitment Training). In these videos, I have explained all the questions with answer and how to approach for the question etc, in the same manner how they have taught to me at the time of training. Hope u like it.
Aptitude training playlist link :
https://www.youtube.com/playlist?list=PL3v9ipJOEEPfumKHa02HWjCfPvGQiPZiG
For full playlist of Interview puzzles videos :
https://www.youtube.com/playlist?list=PL3v9ipJOEEPfI4zt4ExamGJwndkvg0SFc
24 standard interview puzzles:
https://www.youtube.com/playlist?list=PL3v9ipJOEEPefIF4nscYOobim1iRBJTjw
for C and C++ questions, that are asked in the interviews, go through the posts in the link : http://comsciguide.blogspot.com/
for more videos, my youtube channel :
https://www.youtube.com/channel/UCvMy2V7gYW7VR2WgyvLj3-A

Coordinate geometry

This document provides information on various concepts in elementary and additional mathematics including:
- The distance, midpoint, and gradient formulas for lines
- Equations of lines
- Parallel and perpendicular lines
- Intersecting lines and finding intersection points
- Perpendicular bisectors
- Finding the area of polygons
It includes examples of applying these concepts to solve problems involving lines, midpoints, gradients, intersections, perpendiculars, and calculating areas.

segundo parcial de analisis del cbc exactas e ingenieria

Este documento contiene 4 ejercicios de análisis matemático con sus respectivas respuestas: 1) Hallar el polinomio de Taylor de orden 3 de una función, 2) Calcular una integral definida, 3) Hallar el área de una región delimitada por curvas, 4) Estudiar la convergencia de una serie de potencias.

Methods of successive over relaxation

This presentation is about the successive over relaxation method. (SOR Method). It is the proof of the Sor Method.

Trigonometry 10th edition larson solutions manual

Trigonometry 10th edition larson solutions manual.
Full download: https://goo.gl/gFVG5A
Peope also search:
algebra and trigonometry ron larson 9th edition pdf
trigonometry larson 9th edition pdf
algebra and trigonometry ron larson pdf
ron larson trigonometry 9th edition pdf
trigonometry 10th edition solutions pdf
algebra & trigonometry
algebra and trigonometry 10th edition pdf
algebra and trigonometry 10th edition free

Trigonometric function

This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.

Sec 3 A Maths Notes Indices

1) The document provides examples of solving exponential equations with various methods depending on whether the equation has two terms, three or more terms, or involves indices as products or quotients.
2) Key steps include splitting equations, letting one term equal a variable, raising both sides to the same power, and changing all terms to have the same base before equating exponents.
3) Examples range from simple equations like 82=x to more complex ones involving subtraction, addition, and multiplication of terms with different bases and exponents like (23)3=x+2-x.

Volume and surface area formulae

This document provides formulas for calculating geometric properties of various 2D and 3D shapes. It includes formulas for calculating the perimeter and area of rectangles, squares, triangles, and circles. It also provides formulas for calculating the surface area and volume of 3D shapes like rectangular prisms, triangular prisms, cylinders, cones, spheres, and square-based pyramids.

Aptitude Training - TIME AND DISTANCE 1

I have taken coaching from NARESH INSTITUTE for CRT (Campus Recruitment Training). In these videos, I have explained all the questions with answer and how to approach for the question etc, in the same manner how they have taught to me at the time of training. Hope u like it.
Aptitude training playlist link :
https://www.youtube.com/playlist?list=PL3v9ipJOEEPfumKHa02HWjCfPvGQiPZiG
For full playlist of Interview puzzles videos :
https://www.youtube.com/playlist?list=PL3v9ipJOEEPfI4zt4ExamGJwndkvg0SFc
24 standard interview puzzles:
https://www.youtube.com/playlist?list=PL3v9ipJOEEPefIF4nscYOobim1iRBJTjw
for C and C++ questions, that are asked in the interviews, go through the posts in the link : http://comsciguide.blogspot.com/
for more videos, my youtube channel :
https://www.youtube.com/channel/UCvMy2V7gYW7VR2WgyvLj3-A

Coordinate geometry

This document provides information on various concepts in elementary and additional mathematics including:
- The distance, midpoint, and gradient formulas for lines
- Equations of lines
- Parallel and perpendicular lines
- Intersecting lines and finding intersection points
- Perpendicular bisectors
- Finding the area of polygons
It includes examples of applying these concepts to solve problems involving lines, midpoints, gradients, intersections, perpendiculars, and calculating areas.

Partial differential equations

The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.

Systems of 3 Equations in 3 Variables

The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.

Integrals by Trigonometric Substitution

In this video we learn the basics of trigonometric substitution and why it works. We talk about all the basic cases of integrals you can solve using trigonometric substitution.
For more lessons and presentations:

05 perfect square, difference of two squares

This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that

Areas of bounded regions

The document discusses different types of bounded regions and calculating their areas using integrals. It defines three types of regions: 1) bounded by two curves and vertical lines, 2) bounded by two curves, and 3) bounded by modulus functions where one curve is greater than the other over some intervals. Examples are provided for each type, such as finding the area between a parabola and line, two parabolas, a parabola and circle, and two circles. The key idea is that the region's area can be expressed as a definite integral of the differences between the bounding curves.

Trigonometry cheat sheet

The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for domain, range, period, identities, inverses, and laws involving trig functions like the Law of Sines, Cosines, and Tangents. Key formulas include definitions of sine, cosine, tangent and their inverses, as well as the Pythagorean, double angle, and sum and difference identities.

Pythagoras' Theorem

This document explains the Pythagorean theorem and how to use it to solve problems involving right triangles. It begins by defining a right triangle as one with a 90 degree angle. It then explains that Pythagoras discovered that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Several examples are provided to demonstrate how to use the theorem to calculate missing side lengths of right triangles in various contexts like finding the diagonal of a rectangle or calculating distance traveled.

Curve tracing

Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying

maths jee formulas.pdf

This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.

Radian and Degree Measure ppt.pptx

This document discusses measuring angles in degrees and radians. It defines an angle, describes quadrant classification of angles, and conversions between degree and radian measure. Key concepts covered include: one radian is the measure of a central angle that intercepts an arc equal to the radius; to convert degrees to radians multiply by π/180; to convert radians to degrees multiply by 180/π.

Lesson 27: Integration by Substitution (slides)

Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200

Functions (Theory)

The document defines and provides properties of various mathematical functions including:
- Relations and sets including Cartesian products and relations.
- Functions including domain, co-domain, range, and the number of possible functions between sets.
- Types of functions such as polynomial, algebraic, transcendental, rational, exponential, logarithmic, and absolute value functions.
- Graphs of important functions are shown such as 1/x, sinx, logx, |x|, [x], and their key properties are described.

Application of definite integrals

Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.

Pythagoras theorem

This document presents information about Pythagoras and his famous theorem. It discusses that Pythagoras was a Greek philosopher from Samos in the 6th century BC who made important contributions to mathematics and philosophy. It provides two proofs of Pythagoras' theorem - that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document also includes animations demonstrating proofs of the theorem through manipulating squares built on the sides of right triangles.

Trigonometric graphs

This document provides a review of trigonometric graphs including how to draw and identify them. It discusses the maximum and minimum values, range, period and number of cycles for sin and cos graphs. It also covers shifting graphs horizontally or vertically and combining trig functions with constants. Examples are provided to illustrate identifying trig graphs from their equations and sketching shifted or combined trig graphs.

Distance Formula - PPT Presentation.pptx

This document discusses coordinate geometry and the distance formula. It provides examples of using the distance formula to calculate the distance between points in the coordinate plane and to prove geometric properties, such as showing a triangle is isosceles or that four points form a square. It derives the distance formula and explains its components and how to apply it to find distances. Examples are provided to illustrate its use in solving geometry problems using algebraic methods in the coordinate plane.

03 open methods

This document outlines numerical methods for finding roots of nonlinear equations presented by Dr. Eng. Mohammad Tawfik. It introduces the fixed point, Newton-Raphson, and secant methods. The fixed point method rearranges the equation to an iterative form where the next estimate is a function of the previous. Newton-Raphson linearizes the function to get faster convergence. The secant method does not require derivatives by using the slope between previous points. Convergence conditions and algorithms are provided for each method. Students are assigned homework problems from the textbook.

Quadratic functions

The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the

Regresi Linear Kelompok 1 XI-10 revisi (1).pptx

regresi linear

Lesson 10: The Chain Rule

The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.

TABREZ KHAN.ppt

The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.

Ee gate'14-paper-01

This document contains a sample GATE paper with questions from various subjects like mathematics, physics, chemistry and general aptitude. The questions include multiple choice, numerical answer type and explanation type questions. Some questions test concepts like differential equations, complex numbers, Laplace transforms, electric circuits etc. The document also contains information about an online portal for GATE preparation that has trained over 1 lakh students across India.

Partial differential equations

The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.

Systems of 3 Equations in 3 Variables

The document discusses solving systems of 3 linear equations with 3 variables. It provides steps to set up the equations in standard form, eliminate one variable using two equations, eliminate the same variable from another pair of equations to get a system of 2 equations with 2 variables, solve this system to find the values of two variables, substitute these values into the original third equation to solve for the remaining variable, and check the solution. An example problem demonstrates applying these steps to solve for x, y, and z. The summary notes that the solution may not be unique depending on whether eliminating variables results in a true or false statement.

Integrals by Trigonometric Substitution

In this video we learn the basics of trigonometric substitution and why it works. We talk about all the basic cases of integrals you can solve using trigonometric substitution.
For more lessons and presentations:

05 perfect square, difference of two squares

This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that

Areas of bounded regions

The document discusses different types of bounded regions and calculating their areas using integrals. It defines three types of regions: 1) bounded by two curves and vertical lines, 2) bounded by two curves, and 3) bounded by modulus functions where one curve is greater than the other over some intervals. Examples are provided for each type, such as finding the area between a parabola and line, two parabolas, a parabola and circle, and two circles. The key idea is that the region's area can be expressed as a definite integral of the differences between the bounding curves.

Trigonometry cheat sheet

The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for domain, range, period, identities, inverses, and laws involving trig functions like the Law of Sines, Cosines, and Tangents. Key formulas include definitions of sine, cosine, tangent and their inverses, as well as the Pythagorean, double angle, and sum and difference identities.

Pythagoras' Theorem

This document explains the Pythagorean theorem and how to use it to solve problems involving right triangles. It begins by defining a right triangle as one with a 90 degree angle. It then explains that Pythagoras discovered that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Several examples are provided to demonstrate how to use the theorem to calculate missing side lengths of right triangles in various contexts like finding the diagonal of a rectangle or calculating distance traveled.

Curve tracing

Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying

maths jee formulas.pdf

This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.

Radian and Degree Measure ppt.pptx

This document discusses measuring angles in degrees and radians. It defines an angle, describes quadrant classification of angles, and conversions between degree and radian measure. Key concepts covered include: one radian is the measure of a central angle that intercepts an arc equal to the radius; to convert degrees to radians multiply by π/180; to convert radians to degrees multiply by 180/π.

Lesson 27: Integration by Substitution (slides)

Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200

Functions (Theory)

The document defines and provides properties of various mathematical functions including:
- Relations and sets including Cartesian products and relations.
- Functions including domain, co-domain, range, and the number of possible functions between sets.
- Types of functions such as polynomial, algebraic, transcendental, rational, exponential, logarithmic, and absolute value functions.
- Graphs of important functions are shown such as 1/x, sinx, logx, |x|, [x], and their key properties are described.

Application of definite integrals

Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.

Pythagoras theorem

This document presents information about Pythagoras and his famous theorem. It discusses that Pythagoras was a Greek philosopher from Samos in the 6th century BC who made important contributions to mathematics and philosophy. It provides two proofs of Pythagoras' theorem - that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document also includes animations demonstrating proofs of the theorem through manipulating squares built on the sides of right triangles.

Trigonometric graphs

This document provides a review of trigonometric graphs including how to draw and identify them. It discusses the maximum and minimum values, range, period and number of cycles for sin and cos graphs. It also covers shifting graphs horizontally or vertically and combining trig functions with constants. Examples are provided to illustrate identifying trig graphs from their equations and sketching shifted or combined trig graphs.

Distance Formula - PPT Presentation.pptx

This document discusses coordinate geometry and the distance formula. It provides examples of using the distance formula to calculate the distance between points in the coordinate plane and to prove geometric properties, such as showing a triangle is isosceles or that four points form a square. It derives the distance formula and explains its components and how to apply it to find distances. Examples are provided to illustrate its use in solving geometry problems using algebraic methods in the coordinate plane.

03 open methods

This document outlines numerical methods for finding roots of nonlinear equations presented by Dr. Eng. Mohammad Tawfik. It introduces the fixed point, Newton-Raphson, and secant methods. The fixed point method rearranges the equation to an iterative form where the next estimate is a function of the previous. Newton-Raphson linearizes the function to get faster convergence. The secant method does not require derivatives by using the slope between previous points. Convergence conditions and algorithms are provided for each method. Students are assigned homework problems from the textbook.

Quadratic functions

The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the

Regresi Linear Kelompok 1 XI-10 revisi (1).pptx

regresi linear

Lesson 10: The Chain Rule

The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.

Partial differential equations

Partial differential equations

Systems of 3 Equations in 3 Variables

Systems of 3 Equations in 3 Variables

Integrals by Trigonometric Substitution

Integrals by Trigonometric Substitution

05 perfect square, difference of two squares

05 perfect square, difference of two squares

Areas of bounded regions

Areas of bounded regions

Trigonometry cheat sheet

Trigonometry cheat sheet

Pythagoras' Theorem

Pythagoras' Theorem

Curve tracing

Curve tracing

maths jee formulas.pdf

maths jee formulas.pdf

Radian and Degree Measure ppt.pptx

Radian and Degree Measure ppt.pptx

Lesson 27: Integration by Substitution (slides)

Lesson 27: Integration by Substitution (slides)

Functions (Theory)

Functions (Theory)

Application of definite integrals

Application of definite integrals

Pythagoras theorem

Pythagoras theorem

Trigonometric graphs

Trigonometric graphs

Distance Formula - PPT Presentation.pptx

Distance Formula - PPT Presentation.pptx

03 open methods

03 open methods

Quadratic functions

Quadratic functions

Regresi Linear Kelompok 1 XI-10 revisi (1).pptx

Regresi Linear Kelompok 1 XI-10 revisi (1).pptx

Lesson 10: The Chain Rule

Lesson 10: The Chain Rule

TABREZ KHAN.ppt

The document is a maths project report for class 12th student Tabrez Khan on the topic of determinants. It contains definitions and properties of determinants of order 1, 2 and 3 matrices. It discusses minors, cofactors and applications of determinants like solving systems of linear equations using Cramer's rule. It also contains examples of evaluating determinants and applying properties of determinants to simplify expressions.

Ee gate'14-paper-01

This document contains a sample GATE paper with questions from various subjects like mathematics, physics, chemistry and general aptitude. The questions include multiple choice, numerical answer type and explanation type questions. Some questions test concepts like differential equations, complex numbers, Laplace transforms, electric circuits etc. The document also contains information about an online portal for GATE preparation that has trained over 1 lakh students across India.

2.1 Rectangular Coordinates

This document defines and explains ordered pairs, rectangular coordinate systems, and formulas for distance and midpoint between points in a coordinate plane. It begins by defining ordered pairs and how they represent relationships between elements of two sets. It then introduces the rectangular coordinate system using perpendicular x and y axes intersecting at the origin, and how points are located using ordered pair coordinates. It provides the distance formula for finding the distance between any two points using their x and y coordinates. Examples are given to demonstrate using the formula and interpreting results. The midpoint formula is also defined for finding the midpoint of a line segment given the endpoints. Examples are worked through for both formulas.

maths 12th.pdf

This document contains a sample question paper for Class XII Mathematics. It has 5 sections (A-E). Section A contains 18 multiple choice questions and 2 assertion-reason questions worth 1 mark each. Section B has 5 very short answer questions worth 2 marks each. Section C contains 6 short answer questions worth 3 marks each. Section D has 4 long answer questions worth 5 marks each. Section E contains 3 case study/passage based questions worth 4 marks each with internal subparts. The document provides sample questions on topics including trigonometry, calculus, matrices, probability, linear programming and more.

Stability criterion of periodic oscillations in a (9)

This document studies the fixed points and properties of the Duffing map, which is a 2-D discrete dynamical system. It finds that the Duffing map has three fixed points when a > b + 1. It divides the parameter space into six regions to determine whether the fixed points are attracting, repelling, or saddle points. It also determines the bifurcation points of the parameter space. Key properties of the Duffing map explored include that it is a diffeomorphism when b ≠ 0, its Jacobian is equal to b, and the eigenvalues of its derivative depend on a and b.

Quadratic Equation

This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.

2.1 Rectangular Coordinate Systems

* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.

Sec 3 A Maths Notes Indices

1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.

Banco de preguntas para el ap

This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.

Sec 4 A Maths Notes Maxima Minima

The document contains two examples of maximum and minimum problems involving differentiation.
Example 1 asks the reader to find the minimum volume of a cone given that a sphere must fit inside it. It is found that the minimum volume occurs when the radius of the cone is 28.577 cm.
Example 2 involves finding the maximum volume of a cylinder inscribed in a sphere. The maximum volume is calculated to be 104,000 cm3, occurring when the height of the cylinder is 28.5 cm.
The document provides guidance on solving maximum and minimum problems using differentiation, illustrated through these two examples involving geometric shapes.

Mid-Term ExamName___________________________________MU.docx

Mid-Term Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Fill in the blank with one of the words or phrases listed below.
distributive real reciprocals absolute value opposite associative
inequality commutative whole algebraic expression exponent variable
1) The of a number is the distance between the number and 0 on the number line.
A) opposite B) whole
C) absolute value D) exponent
1)
Find an equation of the line. Write the equation using function notation.
2) Through (1, -3); perpendicular to f(x) = -4x - 3
A) f(x) =
1
4
x -
13
4
B) f(x) = -
1
4
x -
13
4
C) f(x) = -4x -
13
4
D) f(x) = 4x -
13
4
2)
Multiply or divide as indicated.
3)
60
-5
A) -22 B) 12 C) - 1
12
D) -12
3)
Write the sentence using mathematical symbols.
4) Two subtracted from x is 55.
A) 2 + x = 55 B) 2 - x = 55 C) x - 2 = 55 D) 55 - 2 = x
4)
Name the property illustrated by the statement.
5) (-10) + 10 = 0
A) associative property of addition B) additive identity property
C) commutative property of addition D) additive inverse property
5)
Tell whether the statement is true or false.
6) Every rational number is an integer.
A) True B) False
6)
Add or subtract as indicated.
7) -5 - 12
A) 7 B) -17 C) 17 D) -7
7)
1
Name the property illustrated by the statement.
8) (1 + 8) + 6 = 1 + (8 + 6)
A) distributive property
B) associative property of addition
C) commutative property of multiplication
D) associative property of multiplication
8)
Simplify the expression.
9) -(10v - 6) + 10(2v + 10)
A) 30v + 16 B) -10v + 94 C) 10v + 106 D) 30v + 4
9)
Solve the equation.
10) 5(x + 3) = 3[14 - 2(3 - x) + 10]
A) -39 B) 3 C) -13 D) 39
10)
List the elements of the set.
11) If A = {x|x is an odd integer} and B = {35, 37, 38, 40}, list the elements of A ∩ B.
A) {35, 37}
B) {x|x is an odd integer}
C) {x|x is an odd integer or x = 38 or x = 40}
D) { }
11)
Solve the inequality. Graph the solution set.
12) |x| ≥ 4
A) (-∞, -4] ∪ [4, ∞)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
B) [-4, 4]
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
C) [4, ∞)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
D) (-∞, -4) ∪ (4, ∞)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
12)
Solve.
13) The sum of three consecutive even integers is 336. Find the integers.
A) 108, 110, 112 B) 110, 112, 114 C) 112, 114, 116 D) 111, 112, 113
13)
2
Solve the inequality. Write your solution in interval notation.
14) x ≥ 4 or x ≥ -2
A) (-∞, ∞) B) [4, ∞)
C) [-2, ∞) D) (-∞, -2] ∪ [4, ∞)
14)
Use the formula A = P 1 + r
n
nt
to find the amount requested.
15) A principal of $12,000 is invested in an account paying an annual interest rate of 4%. Find the
amount in the account after 3 years if the account is compounded quarterly.
A) $1521.9 B) $13,388.02 C) $13,498.37 D) $13,521.90
15)
Graph the solution set ...

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- 1. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 1 Be a lifelong student. The more you learn, the more you earn and more self confidence you will have. - Brian Tracy Notes: Coordinate Geometry [A] Distance Formula This formula is an application of Pythagoras' theorem for right triangles: Note that the distance is taken to be positive. Example 1: Given that is an isosceles triangle with vertices , and and , find the value of . Ans: Solution: ABC ( , 1)A p - (2, 5)B (3, 4)C ACAB = p 3- ACAB = Given two points and , the distance between these points is given by the formula: ( )1 1,x y ( )2 2,x y 2 21 2 21 )()( yyxxPQ -+-= Length same, apply distance formula
- 2. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 2 Example 2: In the diagram below, is a regular hexagon. , and has coordinates , and respectively. (i) State the coordinates of , in terms of . [1] (ii) Justify, showing all workings clearly, why the coordinate of will not be an integer. [2] Solution: (i) (ii) Let the midpoint of be . By Pythagoras theorem, Since 12 is not a perfect square, the coordinate of will not be an integer. Or can apply distance formula to and , whereby . ABCDEF A E F (0,6) ( , 0)s (0,2) B s x - D ( , 8)B s DF X 2 2 2 2 4s + = 12s = 2 12DF = x - D (2 , 2)D s ( , 0)E s 4DE =
- 3. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 3 Example 6: Given that the gradient of and is , find the possible coordinates of . Ans: Solution: Example 7: The diagram, which is not drawn to scale, shows the three lines , and . (a) Find the coordinates of , and . [4] (b) The point is the same distance from as it is from Find the value of . [1] Ans: (a) (b) 0.5 (3, )A p- 2 ( , )B p p- 2 1 - B ( 1,1), (1.5,2.25)B B- 2 ( ) 1 3 2 p p p - - = - - - 5=y xy -= 3 343 += xy A B C ( ,0)k A B k 6 1 ( 2,5), (3,5), ( ,2 ) 7 7 A B C- Casio mode 3, 3 y x O A B C horizontal Positive gradient Negative gradient
- 4. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 4 [C] Manipulate the Equation of a straight line Example 8: Determine the gradient and intercept for each of the straight lines in the table below. y - Equation Gradient intercept No need transform No need transform 12 0 No need transform 0 5 4 cmxy += -y xy 3 1 5 -= xy 12= 5=y 182 += xy 1 4 2 y x= + 1 2 105 += yx The equation of a straight line with gradient and intercept ism -y c y mx c= + cmxy += Gradient, must be subjecty
- 5. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 5 Example 9: The gradient of the line , is . Find the value of , where is a whole number. Ans: Ws 2 [D] Form equation of a straight line (1) Given gradient and pass through a point Example 10: Find the equation of the straight line whose gradient is 3 and passes through the point . Solution: l 2 2 6 0k x ky- - = 9 k k 18k = (4, 2)- y mx c= + 3y x c= + 2 3(4) c- = + 14c = - 3 14y x = - 1. Gradient Equation of Line 2. A point Sub grad 1st Sub point 2nd
- 6. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 6 (2) Given 2 points Example 11: Find the equation of the line passing through and . Solution: gradient Observe that the intercept is 11 Example 12: The point lies on the line . (a) Find the value of . [1] (b) Find the equation of the line parallel to , passing through the point . [2] Ans: (a) (b) (0,11)A )2,6(B 11 2 3 0 6 2 - = = - - y mx c= + 3 2 y x c= - + y - 3 11 2 y x = - + ( , 2)a 3 2 1 0x y+ - = a 3 2 1 0x y+ - = ( 1, 4)- 1a = - 3 5 2 2 y x= - + Find grad 1st 1. Gradient Equation of Line 2. A point Sub point into equation of line Grad same, manipulate to make the subjecty
- 7. 4048 3EM Coordinate Geometry (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. © www.MathAcademy.sg 7 Example 13: The diagram shows a regular hexagon, , where and . Given that the length of is units, find (a) the value of . [2] (b) the equation of , [2] Ans: (a) 18 (b) ABCDEF (0,10)B (4, )C p BC 80 p BC 2 10y x= +