The document provides practice questions on surds and indices, differentiation, integration, and quadratics. For surds and indices, questions involve simplifying expressions with surds and rationalizing denominators. Differentiation questions involve taking derivatives of functions and finding derivatives from given information. Integration questions involve taking integrals and finding functions from their derivatives. Quadratic questions involve solving quadratic equations, finding the discriminant, and determining conditions on constants for the equation to have certain root properties.
This document provides a summary of core mathematics concepts including:
1) Linear graphs and equations such as y=mx+c and finding the equation of a line.
2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry.
3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs.
4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
1. Indices involve rules for exponents like xa+b = xaxb and (xa)b = xab. Solving exponential equations uses these rules.
2. Graph transformations include translations, stretches, reflections, and asymptotes. Translations replace x with (x-a) and y with (y-b).
3. Sequences are functions with successive terms defined by a rule. Geometric sequences multiply successive terms by a constant ratio while arithmetic sequences add a constant.
This document contains information about the format and topics covered in papers 1 and 2 of an exam. Paper 1 has 25 questions to be answered in 2 hours, with 10 questions of low difficulty, 6 of moderate difficulty, and 1 of high difficulty. Paper 2 has 3 sections, with the first section containing 6 questions to answer, the second 5 questions where the test taker must choose 4, and the third 4 questions where they must choose 2. The total time for Paper 2 is 2.5 hours.
The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.
This document contains questions related to relations and functions, inverse trigonometric functions, and matrices and determinants. It includes 1 mark, 2 mark, 4 mark, and 6 mark questions on these topics that would be expected for an exam in 2018. The questions cover key concepts like equivalence relations, one-to-one and onto functions, evaluating inverse trigonometric functions, solving inverse trigonometric equations, properties of matrices including determinants, and solving matrix equations.
35182797 additional-mathematics-form-4-and-5-notesWendy Pindah
1) The function f(x) = 2x^2 + 8x + 6 can be written as f(x) = 2(x+2)^2 - 2. The maximum point is (-2, -2) and the equation of the tangent at this point is y = -2.
2) The function f(x) = -(x-4)^2 + h has a maximum point at (k, 9) so k = 4 and h = 9.
3) The function y = (x+m)^2 + n has an axis of symmetry at x = -m. Given the axis is x = 1, m = -1 and the minimum point is (1
This document provides a summary of Chapter 5 on Indices and Logarithms from an Additional Mathematics textbook. It includes examples and explanations of:
1. Laws of indices such as addition, subtraction, multiplication and division of indices.
2. Converting expressions between index form and logarithmic form using common logarithms and other bases.
3. Applying the laws of logarithms including addition, subtraction, and change of base.
4. Solving equations involving indices and logarithms through appropriate applications of index laws and logarithmic properties.
This document contains a mathematics sample paper with 29 questions across 4 sections - A, B, C and D. The questions cover a wide range of mathematics topics including calculus, algebra, trigonometry, matrices, probability, linear programming, and geometry. The paper is designed to test students' mathematical skills and knowledge over 3 hours with the maximum marks being 100.
This document provides a summary of core mathematics concepts including:
1) Linear graphs and equations such as y=mx+c and finding the equation of a line.
2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry.
3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs.
4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
1. Indices involve rules for exponents like xa+b = xaxb and (xa)b = xab. Solving exponential equations uses these rules.
2. Graph transformations include translations, stretches, reflections, and asymptotes. Translations replace x with (x-a) and y with (y-b).
3. Sequences are functions with successive terms defined by a rule. Geometric sequences multiply successive terms by a constant ratio while arithmetic sequences add a constant.
This document contains information about the format and topics covered in papers 1 and 2 of an exam. Paper 1 has 25 questions to be answered in 2 hours, with 10 questions of low difficulty, 6 of moderate difficulty, and 1 of high difficulty. Paper 2 has 3 sections, with the first section containing 6 questions to answer, the second 5 questions where the test taker must choose 4, and the third 4 questions where they must choose 2. The total time for Paper 2 is 2.5 hours.
The document then lists topics that will be covered in the exam, grouped under the categories of Algebra, Geometry, Calculus, Trigonometry, Statistics, Science and Technology. Specific topics include functions, quadratic equations
This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.
This document contains questions related to relations and functions, inverse trigonometric functions, and matrices and determinants. It includes 1 mark, 2 mark, 4 mark, and 6 mark questions on these topics that would be expected for an exam in 2018. The questions cover key concepts like equivalence relations, one-to-one and onto functions, evaluating inverse trigonometric functions, solving inverse trigonometric equations, properties of matrices including determinants, and solving matrix equations.
35182797 additional-mathematics-form-4-and-5-notesWendy Pindah
1) The function f(x) = 2x^2 + 8x + 6 can be written as f(x) = 2(x+2)^2 - 2. The maximum point is (-2, -2) and the equation of the tangent at this point is y = -2.
2) The function f(x) = -(x-4)^2 + h has a maximum point at (k, 9) so k = 4 and h = 9.
3) The function y = (x+m)^2 + n has an axis of symmetry at x = -m. Given the axis is x = 1, m = -1 and the minimum point is (1
This document provides a summary of Chapter 5 on Indices and Logarithms from an Additional Mathematics textbook. It includes examples and explanations of:
1. Laws of indices such as addition, subtraction, multiplication and division of indices.
2. Converting expressions between index form and logarithmic form using common logarithms and other bases.
3. Applying the laws of logarithms including addition, subtraction, and change of base.
4. Solving equations involving indices and logarithms through appropriate applications of index laws and logarithmic properties.
This document contains a mathematics sample paper with 29 questions across 4 sections - A, B, C and D. The questions cover a wide range of mathematics topics including calculus, algebra, trigonometry, matrices, probability, linear programming, and geometry. The paper is designed to test students' mathematical skills and knowledge over 3 hours with the maximum marks being 100.
The document provides information about quadratic functions including:
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- A quadratic function has a minimum or maximum point which can be used to find the axis of symmetry.
- The relationship between the discriminant (b2 - 4ac) and the position of the graph is explained. If it is greater than 0, the graph cuts the x-axis at two points. If it is equal to 0, the graph touches the x-axis at one point. If it is less than 0, the graph does not cut or touch the x-axis.
- Quadratic inequalities can be solved by sketching
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document contains sample problems related to quadratic equations and quadratic functions for Form 4 Additional Mathematics. It is divided into three sections - the first two sections contain sample problems testing concepts related to solving quadratic equations and inequalities. The third section contains sample problems related to identifying properties of quadratic functions such as finding the minimum or maximum value, range of a quadratic function, expressing a quadratic in standard form and sketching its graph.
Chapter 9- Differentiation Add Maths Form 4 SPMyw t
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
The document discusses complex roots of the characteristic equation arising from assuming exponential solutions to a differential equation. It shows that complex roots lead to complex-valued solutions, but linear combinations of solutions can give real-valued solutions in the form of sine and cosine functions. Several examples are worked out to find the general solution of differential equations and determine the time for the solution to drop below a given value based on its graph.
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
The document contains a worksheet with multiple choice questions related to relations, functions, and matrices. Some questions test concepts like equivalence relations, inverse functions, and properties of matrices. The questions cover topics such as determining if a relation is reflexive, transitive, or symmetric; evaluating composite and inverse functions; finding the order and properties of matrices; and evaluating determinants.
Additional Mathematics form 4 (formula)Fatini Adnan
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.
The document discusses simplifying expressions involving radicals. It provides examples of simplifying expressions using properties of radicals, such as √x∙y = √x∙√y and √x∙√x = x. One example simplifies the expression 3√3√2∙2√2√3√2 to 36√2. Another example simplifies the expression √12(√3 + 3√2) to 12√6.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
1. This document contains a math exam with 31 questions testing various math skills like algebra, geometry, statistics, and problem solving.
2. The exam is broken into questions with points allocated for each part. An assessment sheet is provided to track points earned for each question.
3. The questions range in difficulty from basic operations to multi-step word problems. Various math concepts are covered, including fractions, ratios, graphs, equations, probability, and more.
1) Edexcel is an examining and awarding body that provides qualifications worldwide. It supports centers that offer education programs to learners through a network of UK and international offices.
2) Candidates' work will be marked according to principles such as marking positively and awarding all marks deserved according to the mark scheme. Subject specialists are available to help with specific content questions.
3) The document provides notes on marking principles for a GCSE mathematics exam, including how to apply the mark scheme and address various student responses.
The document provides information about quadratic functions including:
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- A quadratic function has a minimum or maximum point which can be used to find the axis of symmetry.
- The relationship between the discriminant (b2 - 4ac) and the position of the graph is explained. If it is greater than 0, the graph cuts the x-axis at two points. If it is equal to 0, the graph touches the x-axis at one point. If it is less than 0, the graph does not cut or touch the x-axis.
- Quadratic inequalities can be solved by sketching
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document contains sample problems related to quadratic equations and quadratic functions for Form 4 Additional Mathematics. It is divided into three sections - the first two sections contain sample problems testing concepts related to solving quadratic equations and inequalities. The third section contains sample problems related to identifying properties of quadratic functions such as finding the minimum or maximum value, range of a quadratic function, expressing a quadratic in standard form and sketching its graph.
Chapter 9- Differentiation Add Maths Form 4 SPMyw t
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
The document discusses complex roots of the characteristic equation arising from assuming exponential solutions to a differential equation. It shows that complex roots lead to complex-valued solutions, but linear combinations of solutions can give real-valued solutions in the form of sine and cosine functions. Several examples are worked out to find the general solution of differential equations and determine the time for the solution to drop below a given value based on its graph.
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
1. The document provides an overview of important topics covered in Form 4 and Form 5 mathematics. These include functions, quadratic equations, trigonometry, statistics, calculus, and coordinate geometry.
2. Examples of how to solve different types of problems are given for each topic, such as finding the sum and product of roots for quadratic equations or using rules of logarithms to simplify logarithmic expressions.
3. Strategies for solving problems involving concepts like differentiation, integration, progressions, and linear laws are outlined. Methods for finding volumes or areas under curves are also summarized briefly.
This document provides information about polynomial operations including:
1) Defining polynomials as algebraic expressions involving integer powers of a variable and real number coefficients.
2) Examples of adding, subtracting, and multiplying polynomials by using vertical or FOIL methods.
3) Important formulas for polynomial operations such as (a + b)(a - b) = a2 - b2 and (a + b)2 = a2 + 2ab + b2.
4) Worked examples of applying these formulas and methods to polynomials involving single and multiple variables.
The document contains a worksheet with multiple choice questions related to relations, functions, and matrices. Some questions test concepts like equivalence relations, inverse functions, and properties of matrices. The questions cover topics such as determining if a relation is reflexive, transitive, or symmetric; evaluating composite and inverse functions; finding the order and properties of matrices; and evaluating determinants.
Additional Mathematics form 4 (formula)Fatini Adnan
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
This document provides an algebra cheat sheet that summarizes many common algebraic properties, formulas, and concepts. It covers topics such as arithmetic operations, properties of inequalities and absolute value, exponent properties, factoring formulas, solving equations, graphing functions, and common algebraic errors. The cheat sheet is a concise 3-page reference for the basics of algebra.
The document is a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. All questions are compulsory. The paper tests concepts related to matrices, trigonometry, calculus, differential equations, and vectors. Internal choices are provided in some questions. Calculators are not permitted.
The document discusses simplifying expressions involving radicals. It provides examples of simplifying expressions using properties of radicals, such as √x∙y = √x∙√y and √x∙√x = x. One example simplifies the expression 3√3√2∙2√2√3√2 to 36√2. Another example simplifies the expression √12(√3 + 3√2) to 12√6.
The document describes three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It can only be used if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
This document provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
1. This document contains a math exam with 31 questions testing various math skills like algebra, geometry, statistics, and problem solving.
2. The exam is broken into questions with points allocated for each part. An assessment sheet is provided to track points earned for each question.
3. The questions range in difficulty from basic operations to multi-step word problems. Various math concepts are covered, including fractions, ratios, graphs, equations, probability, and more.
1) Edexcel is an examining and awarding body that provides qualifications worldwide. It supports centers that offer education programs to learners through a network of UK and international offices.
2) Candidates' work will be marked according to principles such as marking positively and awarding all marks deserved according to the mark scheme. Subject specialists are available to help with specific content questions.
3) The document provides notes on marking principles for a GCSE mathematics exam, including how to apply the mark scheme and address various student responses.
The document is a mark scheme for GCSE Mathematics (2MB01) Foundation 5MB2F (Non-Calculator) Paper 01 exam from March 2012. It provides notes on marking principles and guidance for how to apply marks for specific types of questions and responses. It also includes worked examples showing the breakdown of method and accuracy marks for sample multi-step questions.
This document provides instructions and information for a practice GCSE mathematics exam. It outlines what materials are allowed, how to answer questions, how marks are allocated, and advice for taking the test. The exam contains 20 multiple-choice and written-response questions testing a range of math skills, including algebra, geometry, statistics, and transformations. It is 80 marks total and lasts 1 hour and 30 minutes.
This document is a mark scheme for the January 2013 A-level Mathematics exam. It provides guidance for examiners on how to mark students' responses consistently. The mark scheme was developed by the Principal Examiner and a panel of teachers, and was refined through a standardization process where examiners analyzed sample scripts. The mark scheme is a working document that may be expanded based on students' actual responses. Details of the mark scheme can change between exam sittings depending on the specific questions asked.
This document provides the final mark scheme for Edexcel's Core Mathematics C1 exam from January 2012. It lists the questions, schemes for awarding marks, and total marks for each question. The six mark questions cover topics like algebra, inequalities, coordinate geometry, and calculus. The longer questions involve multi-step problems applying these concepts, including sketching curves, finding equations of tangents and normals, and solving word problems involving formulas.
3 revision session for core 1 translations of graphs, simultaneous equation...claire meadows-smith
The Community Maths School has structured a revision programme to prepare students for the Core 1 exam. The programme is based on the AQA AS exam but is suitable for most boards. Over six revision sessions in March and April, the school will provide hints, exam solutions, and practice questions on topics like translations of graphs, simultaneous equations, and inequalities. Additional exam practice sessions will be held in May to help students for the Core 1 exam on May 19th.
The document is a mark scheme for a GCSE mathematics exam. It provides guidance to examiners on how to mark students' responses, including what constitutes correct working and answers for different parts of questions. The document also provides background information on the exam board and qualifications.
This document contains a mark scheme for a GCSE mathematics exam. It provides guidance for examiners on how to apply marks for different parts of student responses. Some key points include:
1) Examiners must mark all students equally and reward students for what they show they can do rather than penalize for omissions.
2) Full marks should be awarded if the answer matches the mark scheme.
3) Working should be considered, even if the final answer is incorrect, to award method marks where appropriate.
4) Follow through marks can be awarded if subsequent working is based on a previous correct response.
5) Marks cannot be awarded for one part of a question in another part
The first session in a structured revision programme for AS Core 1 Maths - includes Key points, Hints, links to exam solutions and practice questions. Created for the AQA AS Level,
1) The document is a mark scheme for GCSE Mathematics (Linear) 1MA0 Higher (Calculator) Paper 2H exam from Summer 2012.
2) It provides notes on marking principles for examiners, such as marking all candidates equally, awarding marks for correct working shown, and following standard procedures around parts of questions and probability answers.
3) The mark scheme then provides detailed guidance on marking for each question, including expected methods, intermediate working, and final answers for full marks.
5th sessions of a structured revision course for core 1 maths exam - diffe...claire meadows-smith
The document outlines a structured revision programme for a Core 1 math exam. It provides the dates for 6 revision sessions covering topics like differentiation, equations of tangents and normals, stationary points, and increasing and decreasing functions. It also lists exam practice dates and resources like a revision website and mobile app to support students' preparation for the Core 1 exam.
This document provides a mark scheme for GCSE Mathematics (Linear) 1MA0 Higher (Calculator) Paper 2H from March 2013. It outlines the general principles that examiners should follow when marking, such as awarding all marks that are deserved and following through correct working. It also provides specific guidance on marking certain types of questions involving areas like probability, linear equations, and multi-step calculations. The document aims to ensure examiners apply marks consistently across all candidates.
This document contains the mark scheme for a mathematics exam involving several multi-part questions.
In question 1, students could earn up to 3 marks for correctly factorizing a quadratic expression in one or two steps.
Question 2 was worth up to 2 marks for correctly writing the equation of a straight line in y=mx+c form.
Question 3 involved solving equations and inequalities across three parts, with a total of 6 available marks through setting up and solving the relevant expressions.
The remaining questions addressed topics including arithmetic and geometric sequences, calculus, coordinate geometry, and quadratic functions. Students could earn marks for setting up correct expressions and equations and obtaining the right numerical or algebraic solutions at each stage.
Mark Scheme (Results) June 2012 GCSE Mathematics (2MB01) Higher Paper 5MB3H_01 (Calculator) provides guidance for examiners on marking the GCSE Mathematics exam from June 2012. It includes notes on general marking principles, how to award marks for various parts of questions, and specific guidance for marking some sample questions from the exam. The document is published by Pearson Education and provides information to ensure accurate and consistent marking of the GCSE exam.
1. This document appears to be an exam paper for the Edexcel GCSE Methods in Mathematics exam. It provides instructions for students on how to complete the exam.
2. The exam consists of multiple choice and free response questions covering topics like operations with fractions, probability, geometry, and algebra. It is 1 hour and 45 minutes long.
3. Students are provided a formula sheet but are instructed not to write on it. Calculators are not permitted. Questions are worth varying point values adding up to a total of 100 points.
This document provides instructions and information for a practice GCSE Mathematics exam. It specifies that the exam is 1 hour and 45 minutes long and covers various topics in mathematics. It provides the materials allowed, instructions on completing the exam, information about marking and time allocation, and advice to students. The exam contains 18 questions testing skills in algebra, graphs, geometry, statistics, and problem solving. It is out of a total of 80 marks.
The document is a mark scheme for GCSE Mathematics (2MB01) Higher 5MB2H (Non-Calculator) Paper 01 exam from March 2012. It provides notes on marking principles and guidance for examiners on how to apply the mark scheme and award marks for questions 1 through 9 on the exam. The summary includes key details about the document type and content while being concise in 3 sentences or less.
The document provides instructions for a mathematics exam. It instructs students to fill out their personal information, use black or blue ink, answer all questions in the spaces provided, and show their working. It notes the total marks for the paper is 60 and which questions require clear written communication. The document advises students to read questions carefully, keep track of time, try to answer every question, and check their work. It also includes a blank formulae page.
The document appears to be a blueprint for a mathematics exam for class 12. It lists various topics that could be covered in the exam such as functions, derivatives, integrals, differential equations, 3-dimensional geometry, and matrices. For each topic it indicates the number and type of questions that may be asked, such as very short answer (1 mark), short answer (4 marks), and long answer (6 marks). The total number of questions is 29 with 10 short answer questions worth 1 mark each, 12 questions worth 4 marks each, and 7 questions worth 6 marks each. The document also includes sample questions that cover the listed topics as examples of what may be asked on the exam.
The document provided is a blue print for a mathematics exam for class 12. It lists various topics that could be included in the exam such as functions, derivatives, integrals, differential equations, 3 dimensional geometry etc. It specifies the number and type of questions (VSA, SA, LA) that may be asked from each topic along with the marks allocated. A total of 100 marks have been allocated with 29 questions. Section A will have 10 one mark questions, Section B will have 12 four mark questions and Section C will have 7 six mark questions. An example question paper format in line with this blue print is also provided.
This document contains a 14 question mathematics exam with questions covering a range of topics including:
- Matrix algebra and inverses
- Integration using substitution
- Graphing functions
- Probability
- Arithmetic and geometric progressions
- Complex numbers
- Linear transformations
- Logical statements
The exam consists of two sections - Section A contains 8 short answer questions, and Section B contains 6 multi-part questions where students must show working for partial or full marks. Overall the exam covers a wide breadth of mathematical concepts and techniques.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
1. This document contains mathematics questions on matrices and determinants ranging from 1 to 6 marks. Questions involve finding determinants, inverses, solving systems of equations using matrices, properties of determinants, and continuity and differentiability.
2. The document is divided into sections on 1, 2, 4 and 6 mark questions related to matrices and determinants as well as questions on continuity and differentiability ranging from 2 to 4 marks.
3. The questions cover a wide range of matrix and calculus concepts and techniques including finding determinants, inverses, solving systems of equations, properties of determinants, continuity, differentiation, mean value theorem, Rolle's theorem and related calculus topics.
1. The radius of curvature at a point on a curve is defined as the reciprocal of the curvature at that point. It represents the radius of the circle that best approximates the curve near that point.
2. For the circle x^2 + y^2 = 25, the radius of curvature at any point is equal to the radius of the circle, which is 25.
3. For the curve xy = c^2, the radius of curvature at the point (c, c) is c.
This document discusses identifying conic sections from their equations. It explains how to write equations of conic sections in standard and general form and use the method of completing the square to transform between the forms. Examples show how to identify equations as representing parabolas, circles, ellipses or hyperbolas based on their coefficients and whether they satisfy certain conditions. The objectives are to identify conic sections from their equations in standard or general form and graph the conic sections.
This document contains questions related to relations and functions, inverse trigonometric functions, and matrices and determinants. It includes 1 mark, 2 mark, 4 mark, and 6 mark questions. The questions cover topics such as equivalence relations, one-to-one and onto functions, composition of functions, inverse functions, inverse trigonometric functions, properties of matrices including symmetric, skew-symmetric and singular matrices, and evaluating determinants.
This document contains a practice exam for Math 220 with 16 multi-part questions covering topics such as functions, inequalities, trigonometry, logarithms, and graphing. The exam instructs students to show their work to receive partial credit and enough work to receive full credit. It tests skills like finding domains of functions, solving inequalities, determining equations of lines and circles, properties of functions, evaluating trigonometric and logarithmic expressions, solving equations, and graphing functions.
The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.
Successful people replace words like "wish", "try", and "should" with "I will" in their speech and thinking. Ineffective people do not make this replacement and continue using weaker language. The document provides mathematical formulas and properties involving complex numbers, including formulas for roots of unity, sums of trigonometric series, representations of lines and circles using complex numbers, and other identities.
The document contains 7 multi-part math problems involving polynomials, quadratics, differentiation, integration, trigonometry, and calculus concepts like finding derivatives, integrals, stationary/critical points, and sketching graphs. The problems cover factorizing polynomials, solving equations, finding rates of change, evaluating integrals, solving trigonometric equations, and applying calculus techniques like differentiation and integration to functions.
This document contains a mathematics exam with multiple choice and multi-part questions testing concepts like factors, rational numbers, geometry proofs, and polynomials. The exam has four sections with questions of varying point values testing skills such as simplifying expressions, factoring polynomials, solving equations, proving geometric relationships, and finding zeros of polynomials.
1
Week 2 Homework for MTH 125
Name___________________________________ Date: ___July 20, 2021______________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the equation by determining the missing values needed to plot the ordered pairs.
1) y + x = 3; ( 1, ), ( 3, ), ( 2, )
1) _______
A)
B)
C)
D)
2
Find the x- and y-intercepts. Then graph the equation.
2) 10y - 2x = -4
2) _______
A) ( -2, 0);
B) ; (0, -2)
3
C) ( 2, 0);
D) ; (0, 2)
Find the midpoint of the segment with the given endpoints.
4
3) ( 8, 4) and ( 7, 9) 3) _______
A) B) C) D)
Suppose that segment PQ has the given coordinates for one endpoint P and for its midpoint M. Find the coordinates
of the other endpoint Q.
4) P( 5, 5) and M 4) _______
A) Q( 4, 4) B) Q C) Q D) Q
Solve the problem.
5) The graphing calculator screen shows the graph of one of the equations below. Which equation is it?
5) _______
A) y + 3x = 15 B) y - 3x = 15 C) y = 3x + 3 D) y + 3x = 3
Find the slope.
6) m = 6) _______
A) -5 B) 5 C) 12 D) 8
Find the slope of the line through the given pair of points, if possible. Based on the slope, indicate whether the line
through the points rises from left to right, falls from left to right, is horizontal, or is vertical.
7) ( -3, -5) and ( 4, -4) 7) _______
A) - ; falls B) - 7; falls C) 7; rises D) ; rises
Find the slope of the line.
5
8)
8) _______
A) B) C) - D) -
Find the slope of the line and sketch the graph.
9) 2x + 3y = 10
9) _______
A) Slope:
6
B) Slope: -
C) Slope: -
7
D) Slope:
Decide whether the pair of lines is parallel, perpendicular, or neither.
10) 3x - 4y = 12 and 8x + 6y = -9 10) ______
A) Parallel B) Perpendicular C) Neither
Choose the graph that matches the equation.
11) y = 2x + 4 11) ______
A)
B)
C)
8
D)
Find the equation in slope-intercept form of the line satisfying the conditions.
12) m = 2, passes through ( 6, -3) 12) ______
A) y = 2x - 15 B) y = 3x + 16 C) y = 2x - 13 D) y = 2x + 14
Write the equation in slope-intercept form.
13) 17x + 5y = 7 13) ______
A) y = x + B) y = 17x - 7 C) y = - x + D) y = x -
Find the slope and the y-intercept of the line.
14) 7x + 5y = 48 14) ______
A) Slope - ; y-intercept B) Slope ; y-intercept
C) Slope ; y-intercept D) Slope - ; y-intercept
Find an equation of the line that satisfies the conditions. Write the equation in standard form.
9
15) Through ( 5, 4); m = - 15) ______
A) 4x - 9y = 56 B) 4x + 9y = -56 C) 4x + 9y = 56 D) 9x + 4y = -56
...
1 Week 2 Homework for MTH 125 Name_______________AbbyWhyte974
1
Week 2 Homework for MTH 125
Name___________________________________ Date: ___July 20, 2021______________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Graph the equation by determining the missing values needed to plot the ordered pairs.
1) y + x = 3; ( 1, ), ( 3, ), ( 2, )
1) _______
A)
B)
C)
D)
2
Find the x- and y-intercepts. Then graph the equation.
2) 10y - 2x = -4
2) _______
A) ( -2, 0);
B) ; (0, -2)
3
C) ( 2, 0);
D) ; (0, 2)
Find the midpoint of the segment with the given endpoints.
4
3) ( 8, 4) and ( 7, 9) 3) _______
A) B) C) D)
Suppose that segment PQ has the given coordinates for one endpoint P and for its midpoint M. Find the coordinates
of the other endpoint Q.
4) P( 5, 5) and M 4) _______
A) Q( 4, 4) B) Q C) Q D) Q
Solve the problem.
5) The graphing calculator screen shows the graph of one of the equations below. Which equation is it?
5) _______
A) y + 3x = 15 B) y - 3x = 15 C) y = 3x + 3 D) y + 3x = 3
Find the slope.
6) m = 6) _______
A) -5 B) 5 C) 12 D) 8
Find the slope of the line through the given pair of points, if possible. Based on the slope, indicate whether the line
through the points rises from left to right, falls from left to right, is horizontal, or is vertical.
7) ( -3, -5) and ( 4, -4) 7) _______
A) - ; falls B) - 7; falls C) 7; rises D) ; rises
Find the slope of the line.
5
8)
8) _______
A) B) C) - D) -
Find the slope of the line and sketch the graph.
9) 2x + 3y = 10
9) _______
A) Slope:
6
B) Slope: -
C) Slope: -
7
D) Slope:
Decide whether the pair of lines is parallel, perpendicular, or neither.
10) 3x - 4y = 12 and 8x + 6y = -9 10) ______
A) Parallel B) Perpendicular C) Neither
Choose the graph that matches the equation.
11) y = 2x + 4 11) ______
A)
B)
C)
8
D)
Find the equation in slope-intercept form of the line satisfying the conditions.
12) m = 2, passes through ( 6, -3) 12) ______
A) y = 2x - 15 B) y = 3x + 16 C) y = 2x - 13 D) y = 2x + 14
Write the equation in slope-intercept form.
13) 17x + 5y = 7 13) ______
A) y = x + B) y = 17x - 7 C) y = - x + D) y = x -
Find the slope and the y-intercept of the line.
14) 7x + 5y = 48 14) ______
A) Slope - ; y-intercept B) Slope ; y-intercept
C) Slope ; y-intercept D) Slope - ; y-intercept
Find an equation of the line that satisfies the conditions. Write the equation in standard form.
9
15) Through ( 5, 4); m = - 15) ______
A) 4x - 9y = 56 B) 4x + 9y = -56 C) 4x + 9y = 56 D) 9x + 4y = -56
...
This module discusses coordinate proofs and properties of circles on the coordinate plane. It introduces coordinate proofs as an analytical method of proving geometric theorems by using the coordinates of points and algebraic relationships. Examples demonstrate proving properties of triangles and quadrilaterals analytically. The standard form of the equation of a circle is derived from the distance formula as (x - h)2 + (y - k)2 = r2, where (h, k) is the center and r is the radius. Finding the center, radius, and equation of circles in various forms are illustrated.
The document lists topics that could be assessed on the last of three papers, including: algebra, sequences, equations, graph transformations, functions, geometry concepts like area, volume, scale factors and shapes, trigonometry, vectors, and data/probability topics such as averages, graphs, and diagrams. Key mathematical areas covered are numbers, algebra, geometry, trigonometry, vectors, and statistics.
This document lists potential topics that could be assessed on the last foundation paper, including algebra, geometry, trigonometry, statistics, and probability concepts. Some examples are LCM and HCF, BIDMAS, exchange rates, coordinates and midpoints, volume and surface area, angles, arcs and sectors using trigonometry, speed-time graphs, averages, Venn diagrams, and two-way tables.
This document provides an acronym "A ripe forest" to help with persuasive writing techniques. It lists persuasive writing elements such as anecdotes, repetition, imperatives, pronouns, exaggeration, facts, opinions, rhetorical questions, emotive language, statistics, and triples. It notes you wouldn't use all of these but should choose the most appropriate for the task and remember purpose, audience, language, and layout.
This document provides a list of structural elements that may be present in a writing sample, including changes in time, place, sentence structure, focus, setting, and order. It identifies patterns, dialogue, flashbacks, sentence length, introductions, climaxes, conclusions, contrasts, and other techniques that reveal how a text is organized and what occurs within it.
This document provides an outline for a GCSE revision session taking place in June 2017. The session includes 6 activities to help students understand exam topics and develop effective revision strategies. Students will analyze exam extracts, consolidate language skills, review persuasive writing techniques, choose individual writing activities, discuss exam strategies, and create a personal revision plan. Useful revision tips and websites are also provided to support students in their preparation for the upcoming GCSE exams.
1. The document provides revision notes and ideas for various science topics organized into different units including fitness and health, human health and diet, staying healthy, the nervous system, drugs, staying in balance, controlling plant growth, and variation and inheritance.
2. Each topic within the units outlines key information to revise and provides one or two revision ideas such as making flashcards, designing experiments or diagrams, producing posters or leaflets, or developing question and answer activities.
3. Some common themes across the topics include the human body systems, health and disease, genetics, plant science, chemicals and their reactions, and polymers. The information and revision suggestions are aimed at different grade levels from E to A.
The document advertises "GradeBooster" classes that aim to improve exam grades through one-day or two-day master classes costing £180 and £300 respectively. The classes will take place at Kesgrave Community Centre on May 30th and 31st and in Bury St. Edmunds on June 1st and 2nd. Additional "Maths drop-in" sessions costing £20 per session or £30 for all three will be held on various Wednesdays and Mondays in May and June to provide extra math help for the GradeBooster classes.
This document contains a series of 21 math questions with explanations and worked examples. The questions cover topics like time, distance, rate, money, graphs, conversions between units, straight line graphs, and coordinate geometry. For each question, the number of marks available is provided. This appears to be a practice exam or set of worksheet problems for a math course.
The document provides examiners' reports and mark schemes for 21 math exam questions:
1) Question 1 involved subtracting times on a travel graph. Most students successfully subtracted the times, though some struggled with converting minutes to hours.
2) Questions 2-7 covered topics like travel graphs, percentages, sponsorship amounts, and staged charging structures. Most students answered parts of these questions correctly, though some made errors in calculations or failed to show their work.
3) Questions 8-21 covered a range of math topics from currency conversions to graphing lines. Many students struggled with interpreting scales accurately and converting between units consistently. Common errors included incorrect values, plotting points inaccurately, and failing to show steps in solutions
This document contains 22 math questions with explanations and worked examples related to topics like pie charts, percentages, ratios, time, money, operations, geometry, and measurement. The questions range from 1 to 7 marks and cover skills such as interpreting data in tables and charts, calculating percentages, solving word problems involving rates and time, using scales on maps, and calculating bearings and distances on diagrams.
This document contains examiners' reports on 22 math exam questions:
- Many students had difficulty drawing accurate pie charts and calculating percentages, angles, and sectors. Use of protractors was inconsistent.
- Bearings, scale drawings, and conversions between units also posed challenges. Accuracy was an issue.
- Multi-step word problems involving rates, proportions, or staged charging structures caused errors, as students struggled with understanding the concepts.
- Familiar topics like addition, subtraction, multiplication were generally answered correctly, but negatives signs and order of operations led to mistakes.
- Pythagoras' theorem, trigonometry including bearings were attempted, but understanding was sometimes lacking, leading to inaccurate responses.
This document contains a 14 question math exam with questions covering various topics including trigonometry, algebra, geometry, and calculus. The exam has a total of 58 marks. Each question is broken down into parts and shows the working and/or final answers. Marking schemes are provided showing the number of marks allocated to each part.
This document summarizes examiners' reports on questions from a math exam. Key points include:
- For question 2, many students found the correct length using Pythagoras' theorem but some made mistakes in algebra. Others started correctly with trigonometry but could not continue.
- Question 5 caused issues as some students subtracted rather than added when using Pythagoras' theorem, losing accuracy.
- Question 6 stumped many students who did not recognize it as a trigonometry problem. Few managed the full correct solution.
- Question 8 was generally answered poorly with many not understanding how to factorize or change the subject of a formula.
- Question 10 saw the preferred method of finding side lengths
This document provides a list of useful websites for spelling, grammar, language devices, general writing practice, and revision techniques. Key resources include sites run by Aylsham High School, OCR, and Kent Schools that offer guides to spelling, punctuation, grammar, sentence starters, and vocabulary. YouTube channels like Mr. Bruff provide videos explaining AQA exam question structures. Other sites provide quizzes on ambitious vocabulary, as well as general writing packs and mind mapping tools to support creative revision practices.
Check the exam details and come prepared with the necessary equipment. Listen carefully to the instructions and time each question to move on if you exceed the allotted time. Read questions multiple times and highlight key words. Consider your reading approach and read the entire text. Plan for essay questions and stick to the outline while writing for the intended purpose and audience. Use techniques you've practiced and revision guides for advice.
The document provides various revision tips for students preparing for exams. It recommends creating a revision plan and sticking to a schedule that increases revision time as exams approach. Students should start revising early instead of cramming last minute. Taking regular breaks is also suggested to avoid burnout. The tips include organizing notes by subject, using memory techniques like mnemonics and flashcards, getting tested by others, and practicing past essays and short plans under timed conditions.
This document contains 18 math questions with varying levels of difficulty related to topics like Pythagoras' theorem, percentages, proportions, geometry, and financial calculations. The questions provide worked examples, diagrams, and multi-step word problems for students to practice solving. Scores are provided after each question indicating the total marks available for getting the problem correct.
The examiner's report discusses common mistakes students made on several math exam questions involving Pythagoras' theorem and trigonometry. For questions about right triangles, many students doubled instead of squaring lengths, added lengths instead of squaring and adding them, or subtracted squares. On questions involving finding perimeters or diameters of shapes, some students incorrectly found areas instead. The report provides insight into where additional instruction is needed, such as understanding differences between areas and perimeters, and properly applying trigonometric functions and formulas.
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdfMalak Abu Hammad
Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!SOFTTECHHUB
As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
1. Mr A. Slack (2012)
Core 1
Practice
Examination
Questions
2. Mr A. Slack (2012)
Assessment Criteria by examination paper: Core 1
AlgebraandFunctions
I can use the laws of indices.
I can use and manipulate surds.
I can describe a quadratic function and its graph.
I can find the discriminant of a quadratic function and explain it.
I can complete the square of a quadratic function and explain it.
I can solve a quadratic by various means.
I can solve a pair of simultaneous equations.
I can expand brackets and collect like terms.
I can sketch curves defined by equations.
I can transform graphs.
CoordinateGeometry
I can find the equation of a straight line given information.
I can find the equation of a perpendicular line.
I know the conditions for a line to be perpendicular to another.
I know the conditions for a line to be parallel to another.
I can find the length of a line segment.
I can write the equation of a straight line in different forms
ArithmeticSeries
I can find the nth
term of an arithmetic sequence.
I can prove the formula to find the sum of the first n terms of a series.
I can find the sum of the first n numbers in an arithmetic series.
I can generate sequences from a recurrence relation.
I can use the notation.
3. Differentiation
I can differentiate a function.
I know the differentiation is the gradient of the tangents of the function.
I can find and understand the second order derivative of a function.
I can manipulate functions in order to differentiate them.
I know the link between the order of a function and the order of the differential.
I can find the equation of a tangent at a point to a given curve.
I can find the equation of the normal at a point to a given curve.
Integration
I know that indefinite integration is the reverse of differentiation.
I can integrate functions.
I can use integration to find the equation of a curve, given f’(x).
I understand the term ‘constant of integration’ and can find it.
This appendix lists formulae that candidates are expected to remember and that
will not be included in formulae booklets.
Quadratic equations
ax bx c
b b ac
a
2
2
0
4
2
has roots
Differentiation
function derivative
xn
nxn 1
Integration
function integral
xn
1
1
1
n
x n
+ c, n 1
5. 1. (a) Write 45 in the form a5, where a is an integer.
(1)
(b) Express
)53(
)53(2
in the form b + c5, where b and c are integers.
(5)
2. Write
√(75) – √(27)
in the form k √x, where k and x are integers.
(2)
3. Simplify
32
35
,
giving your answer in the form a + b3, where a and b are integers.
(4)
4. (a) Express 108 in the form a3, where a is an integer.
(1)
(b) Express (2 – 3)2
in the form b + c3, where b and c are integers to be found.
(3)
5. Simplify (3 + 5)(3 – 5).
(2)
6. Expand and simplify (7 + 2)(7 – 2).
(2)
7. (a) Expand and simplify (7 + 5)(3 – 5).
(3)
(b) Express
53
57
in the form a + b5, where a and b are integers.
(3)
6. Mr A. Slack (2012)
8. Simplify
13
325
,
giving your answer in the form p + q3, where p and q are rational numbers.
(4)
9. Simplify
(a) (3√7)2
(1)
(b) (8 + √5)(2 − √5)
(3)
10. (a) Expand and simplify (4 + 3) (4 – 3).
(2)
(b) Express
34
26
in the form a + b3, where a and b are integers.
(2)
11. (a) Find the value of 3
4
8 .
(2)
(b) Simplify
x
x
3
15 3
4
.
(2)
12. (a) Write down the value of 2
1
16 .
(1)
(b) Find the value of 2
3
16
.
(2)
13. (a) Write down the value of 4
1
16 .
(1)
(b) Simplify 4
3
)16( 12
x .
(2)
7. 14. (a) Write down the value of 3
1
125 .
(1)
(b) Find the value of 3
2
125
.
(2)
15. (a) Find the value of 4
1
16
.
(2)
(b) Simplify
4
4
1
2
xx .
(2)
16. (a) Write down the value of 3
1
8 .
(1)
(b) Find the value of 3
2
8
.
(2)
17. Find the value of
(a) 2
1
25 ,
(1)
(b) 2
3
25
.
(2)
18. Given that 32√2 = 2a
, find the value of a.
(3)
8. Mr A. Slack (2012)
Differentiation
and
Integration
9. 1. (a) Write
x
x 32
in the form 2xp
+ 3xq
, where p and q are constants.
(2)
Given that y = 5x – 7 +
x
x 32
, x > 0,
(b) find
x
y
d
d
, simplifying the coefficient of each term.
(4)
2. Given that
y = 8x3
– 4x +
x
x 23 2
, x > 0,
find
x
y
d
d
.
(6)
3. Given that
x
xx
2
3
2
2
can be written in the form 2xp
– xq
,
(a) write down the value of p and the value of q.
(2)
Given that y = 5x4
– 3 +
x
xx
2
3
2
2
,
(b) find
x
y
d
d
, simplifying the coefficient of each term.
(4)
4. Given that
x
xx
2
5
36
can be written in the form 6xp
+ 3xq
,
(a) write down the value of p and the value of q.
(2)
Given that
x
y
d
d
=
x
xx
2
5
36
and that y = 90 when x = 4,
(b) find y in terms of x, simplifying the coefficient of each term.
(5)
10. Mr A. Slack (2012)
5. f(x) =
x
x
2
)43(
, x > 0.
(a) Show that f(x) = BAxx
2
1
2
1
9 , where A and B are constants to be found.
(3)
(b) Find f'(x).
(3)
(c) Evaluate f'(9).
(2)
6. (a) Show that
x
x
2
)3(
can be written as 2
1
2
1
69 xx
.
(2)
Given that
x
y
d
d
=
x
x
2
)3(
, x > 0, and that y = 3
2
at x = 1,
(b) find y in terms of x.
(6)
7. f(x) = 3x + x3
, x > 0.
(a) Differentiate to find f (x).
(2)
Given that f (x) = 15,
(b) find the value of x.
8. Given that y = 3x2
+ 4x, x > 0, find
(a)
x
y
d
d
,
(2)
(b) 2
2
d
d
x
y
,
(2)
(c)
xy d .
(3)
11. 9. Given that y = 2x5
+ 7 + 3
1
x
, x ≠ 0, find, in their simplest form,
(a)
x
y
d
d
,
(3)
(b)
xy d .
(4)
10. (i) Given that y = 5x3
+ 7x + 3, find
(a)
x
y
d
d
,
(3)
(b) 2
2
d
d
x
y
.
(1)
(ii) Find
2
1
31
x
x dx.
(4)
11. Given that y = 2x3
+ 2
3
x
, x≠ 0, find
(a)
x
y
d
d
,
(3)
(b)
xy d , simplifying each term.
(3)
12. Given that y = 6x – 2
4
x
, x ≠ 0,
(a) find
x
y
d
d
,
(2)
(b) find y
dx.
(3)
12. Mr A. Slack (2012)
13. Given that y = 2x2
– 3
6
x
, x 0,
(a) find
x
y
d
d
,
(2)
(b) find
xy d .
(3)
14. (a) Show that (4 + 3x)2
can be written as 16 + kx + 9x, where k is a constant to be found.
(2)
(b) Find
xx d)34( 2
.
(3)
15. Find
xxx d)743( 52
.
(4)
16. Find
xxx d)568( 2
1
3
,
giving each term in its simplest form.
(4)
17. Find
xxx d)3812( 35
, giving each term in its simplest form.
(4)
18. Find
xxxx d)4312( 3
1
25
,
giving each term in its simplest form.
(5)
13. 19. Find
xx d)52( 2
.
(3)
20. Find xxx d)26( 2
1
2
, giving each term in its simplest form.
(4)
21. A curve has equation y = f(x) and passes through the point (4, 22).
Given that
f (x) = 3x2
– 2
1
3x – 7,
use integration to find f(x), giving each term in its simplest form.
(5)
22. Given that
y = 4x3
– 1 + 2 2
1
x , x > 0,
find
x
y
d
d
.
(4)
23. Differentiate with respect to x
(a) x4
+ 6x,
(3)
(b)
x
x 2
)4(
.
(4)
24. Given that y = x4
+ 3
1
x + 3, find
x
y
d
d
.
(3)
14. Mr A. Slack (2012)
25.
x
y
d
d
= 2
1
5
x + xx, x > 0.
Given that y = 35 at x = 4, find y in terms of x, giving each term in its simplest form.
(7)
26. The curve C has equation y = f(x), x > 0, where
x
y
d
d
= 3x –
x
5
– 2.
Given that the point P (4, 5) lies on C, find
(a) f(x),
(5)
(b) an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0,
where a, b and c are integers.
(4)
16. Mr A. Slack (2012)
1. The equation 2x2
– 3x – (k + 1) = 0, where k is a constant, has no real roots.
Find the set of possible values of k.
(4)
2. The equation x2
+ kx + (k + 3) = 0, where k is a constant, has different real roots.
(a) Show that 01242
kk .
(2)
(b) Find the set of possible values of k.
(4)
3. x2
– 8x – 29 (x + a)2
+ b,
where a and b are constants.
(a) Find the value of a and the value of b.
(3)
(b) Hence, or otherwise, show that the roots of
x2
– 8x – 29 = 0
are c d5, where c and d are integers to be found.
(3)
4. x2
+ 2x + 3 (x + a)2
+ b.
(a) Find the values of the constants a and b.
(2)
(b) Sketch the graph of y = x2
+ 2x + 3, indicating clearly the coordinates of any intersections
with the coordinate axes.
(3)
(c) Find the value of the discriminant of x2
+ 2x + 3. Explain how the sign of the discriminant
relates to your sketch in part (b).
(2)
The equation x2
+ kx + 3 = 0, where k is a constant, has no real roots.
(d) Find the set of possible values of k, giving your answer in surd form.
(4)
17. 5. The equation
x2
+ kx + 8 = k
has no real solutions for x.
(a) Show that k satisfies k2
+ 4k – 32 < 0.
(3)
(b) Hence find the set of possible values of k.
(4)
6. The equation kx2
+ 4x + (5 – k) = 0, where k is a constant, has 2 different real solutions for x.
(a) Show that k satisfies
k2
– 5k + 4 > 0.
(3)
(b) Hence find the set of possible values of k.
(4)
7. The equation x2
+ (k − 3)x + (3 − 2k) = 0, where k is a constant, has two distinct real roots.
(a) Show that k satisfies
k2
+ 2k − 3 > 0.
(3)
(b) Find the set of possible values of k.
(4)
8. Given that the equation 2qx2
+ qx – 1 = 0, where q is a constant, has no real roots,
(a) show that q2
+ 8q < 0.
(2)
(b) Hence find the set of possible values of q.
(3)
9. The equation x2
+ 3px + p = 0, where p is a non-zero constant, has equal roots.
Find the value of p.
(4)
18. Mr A. Slack (2012)
10. Given that the equation kx2
+ 12x + k = 0, where k is a positive constant, has equal roots, find the
value of k.
11. The equation x2
+ 2px + (3p + 4) = 0, where p is a positive constant, has equal roots.
(a) Find the value of p.
(4)
(b) For this value of p, solve the equation x2
+ 2px + (3p + 4) = 0.
(2)
12. (a) Show that x2
+ 6x + 11 can be written as
(x + p)2
+ q,
where p and q are integers to be found.
(2)
(b) Sketch the curve with equation y = x2
+ 6x + 11, showing clearly any intersections with the
coordinate axes.
(2)
(c) Find the value of the discriminant of x2
+ 6x + 11.
(2)
13. f(x) = x2
+ (k + 3)x + k,
where k is a real constant.
(a) Find the discriminant of f(x) in terms of k.
(2)
(b) Show that the discriminant of f(x) can be expressed in the form (k + a)2
+ b, where a and b
are integers to be found.
(2)
(c) Show that, for all values of k, the equation f(x) = 0 has real roots.
(2)
19. 3. On separate diagrams, sketch the graphs of
(a) y = (x + 3)2
,
(3)
(b) y = (x + 3)2
+ k, where k is a positive constant.
(2)
Show on each sketch the coordinates of each point at which the graph meets the axes.
10. Given that
f(x) = x2
– 6x + 18, x 0,
(a) express f(x) in the form (x – a)2
+ b, where a and b are integers.
(3)
The curve C with equation y = f(x), x 0, meets the y-axis at P and has a minimum point at Q.
(b) Sketch the graph of C, showing the coordinates of P and Q.
(4)
The line y = 41 meets C at the point R.
(c) Find the x-coordinate of R, giving your answer in the form p + q2, where p and q are
integers.
10. f(x) = x2
+ 4kx + (3 + 11k), where k is a constant.
(a) Express f(x) in the form (x + p)2
+ q, where p and q are constants to be found in terms of k.
(3)
Given that the equation f(x) = 0 has no real roots,
(b) find the set of possible values of k.
(4)
Given that k = 1,
(c) sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses
a coordinate axis.
(3)
21. 1. Solve the simultaneous equations
x + y = 2
x2
+ 2y = 12.
(6)
2. Solve the simultaneous equations
y = x – 2,
y2
+ x2
= 10.
(7)
3. Solve the simultaneous equations
y – 3x + 2 = 0
y2
– x – 6x2
= 0
(7)
4. Solve the simultaneous equations
x – 2y = 1,
x2
+ y2
= 29.
(6)
5. Solve the simultaneous equations
x + y = 2
4y2
– x2
= 11
(7)
22. Mr A. Slack (2012)
6. (a) By eliminating y from the equations
,4 xy
,82 2
xyx
show that
0842
xx .
(2)
(b) Hence, or otherwise, solve the simultaneous equations
,4 xy
,82 2
xyx
giving your answers in the form a ± b3, where a and b are integers.
(5)
24. Mr A. Slack (2012)
1. The rth term of an arithmetic series is (2r – 5).
(a) Write down the first three terms of this series.
(2)
(b) State the value of the common difference.
(1)
(c) Show that
n
r
r
1
)52( = n(n – 4).
(3)
2. On Alice’s 11th birthday she started to receive an annual allowance. The first annual allowance
was £500 and on each following birthday the allowance was increased by £200.
(a) Show that, immediately after her 12th birthday, the total of the allowances that Alice had
received was £1200.
(1)
(b) Find the amount of Alice’s annual allowance on her 18th birthday.
(2)
(c) Find the total of the allowances that Alice had received up to and including her 18th
birthday.
(3)
When the total of the allowances that Alice had received reached £32 000 the allowance stopped.
(d) Find how old Alice was when she received her last allowance.
(7)
3. Ann has some sticks that are all of the same length. She arranges them in squares and has made
the following 3 rows of patterns:
Row 1
Row 2
Row 3
She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make
2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.
(a) Find an expression, in terms of n, for the number of sticks required to make a similar
arrangement of n squares in the nth row.
(3)
25. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row
and so on until she has completed 10 rows.
(b) Find the total number of sticks Ann uses in making these 10 rows.
(3)
Ann started with 1750 sticks. Given that Ann continues the pattern to complete k rows but does
not have sufficient sticks to complete the (k + 1)th row,
(c) show that k satisfies (3k – 100)(k + 35) < 0.
(4)
(d) Find the value of k.
(2)
4. The first term of an arithmetic sequence is 30 and the common difference is –1.5.
(a) Find the value of the 25th term.
(2)
The rth term of the sequence is 0.
(b) Find the value of r.
(2)
The sum of the first n terms of the sequence is Sn.
(c) Find the largest positive value of Sn.
(3)
5. The first term of an arithmetic series is a and the common difference is d.
The 18th term of the series is 25 and the 21st term of the series is 32 2
1
.
(a) Use this information to write down two equations for a and d.
(2)
(b) Show that a = –17.5 and find the value of d.
(2)
The sum of the first n terms of the series is 2750.
(c) Show that n is given by
n2
– 15n = 55 40.
(4)
26. Mr A. Slack (2012)
(d) Hence find the value of n.
(3)
6. Jill gave money to a charity over a 20-year period, from Year 1 to Year 20 inclusive. She gave
£150 in Year 1, £160 in Year 2, £170 in Year 3, and son on, so that the amounts of money she
gave each year formed an arithmetic sequence.
(a) Find the amount of money she gave in Year 10.
(2)
(b) Calculate the total amount of money she gave over the 20-year period.
(3)
Kevin also gave money to charity over the same 20-year period.
He gave £A in Year 1 and the amounts of money he gave each year increased, forming an
arithmetic sequence with common difference £30.
The total amount of money that Kevin gave over the 20-year period was twice the total amount
of money that Jill gave.
(c) Calculate the value of A.
(4)
7. An arithmetic sequence has first term a and common difference d. The sum of the first 10 terms
of the sequence is 162.
(a) Show that 10a + 45d = 162.
(2)
Given also that the sixth term of the sequence is 17,
(b) write down a second equation in a and d,
(1)
(c) find the value of a and the value of d.
(4)
8. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of
5 km on the first Saturday. Each Saturday she increases the length of her run from the previous
Saturday by 2 km.
(a) Show that on the 4th Saturday of training she runs 11 km.
(1)
(b) Find an expression, in terms of n, for the length of her training run on the nth Saturday.
27. (2)
(c) Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km.
(3)
On the nth Saturday Sue runs 43 km.
(d) Find the value of n.
(2)
(e) Find the total distance, in km, Sue runs on Saturdays in n weeks of training.
(2)
9. A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and
finished in 1990 (Year 40).
The numbers of houses built each year form an arithmetic sequence with first term a and
common difference d.
Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find
(a) the value of d,
(3)
(b) the value of a,
(2)
(c) the total number of houses built in Oldtown over the 40-year period.
(3)
10. An arithmetic series has first term a and common difference d.
(a) Prove that the sum of the first n terms of the series is
2
1
n[2a + (n – 1)d].
(4)
Sean repays a loan over a period of n months. His monthly repayments form an arithmetic
sequence.
He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on.
He makes his final repayment in the nth month, where n > 21.
(b) Find the amount Sean repays in the 21st month.
(2)
28. Mr A. Slack (2012)
Over the n months, he repays a total of £5000.
(c) Form an equation in n, and show that your equation may be written as
n2
– 150n + 5000 = 0.
(3)
(d) Solve the equation in part (c).
(3)
(e) State, with a reason, which of the solutions to the equation in part (c) is not a sensible
solution to the repayment problem.
(1)
11. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On
each day after the first day he runs further than he ran on the previous day. The lengths of his 11
practice runs form an arithmetic sequence with first term a km and common difference d km.
He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
Find the value of a and the value of d.
(7)
12. A girl saves money over a period of 200 weeks. She saves 5p in Week 1, 7p in Week 2,
9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.
(a) Find the amount she saves in Week 200.
(3)
(b) Calculate her total savings over the complete 200 week period.
(3)
13. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays
£a for their first day, £(a + d ) for their second day, £(a + 2d ) for their third day, and so on, thus
increasing the daily payment by £d for each extra day they work.
A picker who works for all 30 days will earn £40.75 on the final day.
(a) Use this information to form an equation in a and d.
(2)
A picker who works for all 30 days will earn a total of £1005.
(b) Show that 15(a + 40.75) = 1005.
(2)
29. (c) Hence find the value of a and the value of d.
(4)
14. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,
2 + 4 + 6 + ...... + 100.
(3)
(b) In the arithmetic series
k + 2k + 3k + ...... + 100,
k is a positive integer and k is a factor of 100.
(i) Find, in terms of k, an expression for the number of terms in this series.
(ii) Show that the sum of this series is
50 +
k
5000
.
(4)
(c) Find, in terms of k, the 50th term of the arithmetic sequence
(2k + 1), (4k + 4), (6k + 7), … ,
giving your answer in its simplest form.
(2)
30. Mr A. Slack (2012)
Transformations
of Functions
31. 1.
Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the
points (1, 0) and (4, 0). The maximum point on the curve is (2, 5).
In separate diagrams, sketch the curves with the following equations. On each diagram show
clearly the coordinates of the maximum point and of each point at which the curve crosses the
x-axis.
(a) y = 2f(x),
(3)
(b) y = f(–x).
(3)
The maximum point on the curve with equation y = f(x + a) is on the y-axis.
(c) Write down the value of the constant a.
(1)
2. Figure 1
y
O 2 4 x
P(3, –2)
1
(2, 5)
4 x
y
32. Mr A. Slack (2012)
Figure 1 shows a sketch of the curve with equation y = f(x). The curve crosses the x-axis at the
points (2, 0) and (4, 0). The minimum point on the curve is P(3, –2).
In separate diagrams sketch the curve with equation
(a) y = –f(x),
(3)
(b) y = f(2x).
(3)
On each diagram, give the coordinates of the points at which the curve crosses the x-axis, and the
coordinates of the image of P under the given transformation.
3. Figure 1
y
Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the points
(0, 3) and (4, 0) and touches the x-axis at the point (1, 0).
On separate diagrams, sketch the curve with equation
(a) y = f(x + 1),
(3)
(b) y = 2f(x),
(3)
(c) y = f
x
2
1
.
(3)
On each diagram show clearly the coordinates of all the points at which the curve meets the axes.
O (1, 0)
(4, 0)
(0, 3)
x
33. 4.
Figure 1
Figure 1 shows a sketch of the curve C with equation y = f(x). There is a maximum at (0, 0),
a minimum at (2, –1) and C passes through (3, 0).
On separate diagrams, sketch the curve with equation
(a) y = f(x + 3),
(3)
(b) y = f(–x).
(3)
34. Mr A. Slack (2012)
5.
Figure 1
Figure 1 shows a sketch of the curve C with equation y = f(x).
The curve C passes through the origin and through (6, 0).
The curve C has a minimum at the point (3, –1).
On separate diagrams, sketch the curve with equation
(a) y = f(2x),
(3)
(b) y = −f(x),
(3)
(c) y = f(x + p), where p is a constant and 0 < p < 3.
(4)
On each diagram show the coordinates of any points where the curve intersects the x-axis and of
any minimum or maximum points.
On each diagram show clearly the coordinates of the maximum point, the minimum point and
any points of intersection with the x-axis.
6.
x
y = f(x)
O
y = 1
(–2, 5)
y
35. Figure 1
Figure 1 shows a sketch of part of the curve with equation y = f(x).
The curve has a maximum point (–2, 5) and an asymptote y = 1, as shown in Figure 1.
On separate diagrams, sketch the curve with equation
(a) y = f(x) + 2,
(2)
(b) y = 4f(x),
(2)
(c) y = f(x + 1).
(3)
On each diagram, show clearly the coordinates of the maximum point and the equation of the
asymptote.
7.
Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the point
(0, 7) and has a minimum point at (7, 0).
On separate diagrams, sketch the curve with equation
(a) y = f(x) + 3,
(3)
(b) y = f(2x).
(2)
On each diagram, show clearly the coordinates of the minimum point and the coordinates of the
point at which the curve crosses the y-axis.
36. Mr A. Slack (2012)
8. Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x). The curve passes through the
origin O and through the point (6, 0). The maximum point on the curve is (3, 5).
On separate diagrams, sketch the curve with equation
(a) y = 3f(x),
(2)
(b) y = f(x + 2).
(3)
On each diagram, show clearly the coordinates of the maximum point and of each point at which
the curve crosses the x-axis.
y
x(6, 0)O
(3, 5)
37. 9.
Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x). The curve has a maximum point A at
(–2, 3) and a minimum point B at (3, – 5).
On separate diagrams sketch the curve with equation
(a) y = f (x + 3),
(3)
(b) y = 2f(x).
(3)
On each diagram show clearly the coordinates of the maximum and minimum points.
The graph of y = f(x) + a has a minimum at (3, 0), where a is a constant.
(c) Write down the value of a.
(1)
38. Mr A. Slack (2012)
10.
Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x) where
f(x) =
2x
x
, x 2.
The curve passes through the origin and has two asymptotes, with equations y = 1 and x = 2, as
shown in Figure 1.
(a) In the space below, sketch the curve with equation y = f(x − 1) and state the equations of the
asymptotes of this curve.
(3)
(b) Find the coordinates of the points where the curve with equation y = f(x − 1) crosses the
coordinate axes.
(4)
11. Given that f(x) =
x
1
, x 0,
(a) sketch the graph of y = f(x) + 3 and state the equations of the asymptotes.
(4)
(b) Find the coordinates of the point where y = f(x) + 3 crosses a coordinate axis.
(2)
40. Mr A. Slack (2012)
1. The line L has equation y = 5 – 2x.
(a) Show that the point P (3, –1) lies on L.
(1)
(b) Find an equation of the line perpendicular to L, which passes through P. Give your answer in
the form ax + by + c = 0, where a, b and c are integers.
(4)
2. The points P and Q have coordinates (–1, 6) and (9, 0) respectively.
The line l is perpendicular to PQ and passes through the mid-point of PQ.
Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are
integers.
(5)
3. The line l1 has equation 3x + 5y – 2 = 0.
(a) Find the gradient of l1.
(2)
The line l2 is perpendicular to l1 and passes through the point (3, 1).
(b) Find the equation of l2 in the form y = mx + c, where m and c are constants.
(3)
4. The point A(–6, 4) and the point B(8, –3) lie on the line L.
(a) Find an equation for L in the form ax + by + c = 0, where a, b and c are integers.
(4)
(b) Find the distance AB, giving your answer in the form k5, where k is an integer.
(3)
5. The line l1 passes through the point A(2, 5) and has gradient – 2
1
.
(a) Find an equation of l1, giving your answer in the form y = mx + c.
(3)
41. The point B has coordinates (–2, 7).
(b) Show that B lies on l1.
(1)
(c) Find the length of AB, giving your answer in the form k5, where k is an integer.
(3)
The point C lies on l1 and has x-coordinate equal to p.
The length of AC is 5 units.
(d) Show that p satisfies
p2
– 4p – 16 = 0.
(4)
6. The line 1L has equation 2y − 3x − k = 0, where k is a constant.
Given that the point A(1, 4) lies on 1L , find
(a) the value of k,
(1)
(b) the gradient of 1L .
(2)
The line 2L passes through A and is perpendicular to 1L .
(c) Find an equation of 2L giving your answer in the form ax + by + c = 0, where a, b and c are
integers.
(4)
The line 2L crosses the x-axis at the point B.
(d) Find the coordinates of B.
(2)
(e) Find the exact length of AB.
(2)
42. Mr A. Slack (2012)
7.
Figure 2
The points Q (1, 3) and R (7, 0) lie on the line 1l , as shown in Figure 2.
The length of QR is a√5.
(a) Find the value of a.
(3)
The line 2l is perpendicular to 1l , passes through Q and crosses the y-axis at the point P, as
shown in Figure 2. Find
(b) an equation for 2l ,
(5)
(c) the coordinates of P,
(1)
(d) the area of ΔPQR.
(4)
8. The line l1 passes through the point (9, –4) and has gradient 3
1
.
(a) Find an equation for l1 in the form ax + by + c = 0, where a, b and c are integers.
(3)
The line l2 passes through the origin O and has gradient –2. The lines l1 and l2 intersect at the
point P.
(b) Calculate the coordinates of P.
(4)
Given that l1 crosses the y-axis at the point C,
(c) calculate the exact area of OCP.
(3)
43. 9. (a) Find an equation of the line joining A(7, 4) and B(2, 0), giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(3)
(b) Find the length of AB, leaving your answer in surd form.
(2)
The point C has coordinates (2, t), where t > 0, and AC = AB.
(c) Find the value of t.
(1)
(d) Find the area of triangle ABC.
(2)
10. The curve C has equation y = f(x), x 0, and the point P(2, 1) lies on C. Given that
f (x) = 3x2
– 6 – 2
8
x
,
(a) find f(x).
(5)
(b) Find an equation for the tangent to C at the point P, giving your answer in the form
y = mx + c, where m and c are integers.
(4)
11. The curve with equation y = f(x) passes through the point (−1, 0).
Given that
f ′(x) = 12x2
− 8x + 1,
find f(x).
(5)
44. Mr A. Slack (2012)
12. Figure 2
y
A(1, 7) B(20, 7)
D(8, 2)
O
C(p, q)
The points A(1, 7), B(20, 7) and C( p, q) form the vertices of a triangle ABC, as shown in
Figure 2. The point D(8, 2) is the mid-point of AC.
(a) Find the value of p and the value of q.
(2)
The line l, which passes through D and is perpendicular to AC, intersects AB at E.
(b) Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers.
(5)
(c) Find the exact x-coordinate of E.
(2)
45. 13.
Figure 1
The points A and B have coordinates (6, 7) and (8, 2) respectively.
The line l passes through the point A and is perpendicular to the line AB, as shown in Figure 1.
(a) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.
(4)
Given that l intersects the y-axis at the point C, find
(b) the coordinates of C,
(2)
(c) the area of ΔOCB, where O is the origin.
(2)
14. The line l1 passes through the points P(–1, 2) and Q(11, 8).
(a) Find an equation for l1 in the form y = mx + c, where m and c are constants.
(4)
The line l2 passes through the point R(10, 0) and is perpendicular to l1. The lines l1 and l2 intersect
at the point S.
(b) Calculate the coordinates of S.
46. Mr A. Slack (2012)
(5)
(c) Show that the length of RS is 35.
(2)
(d) Hence, or otherwise, find the exact area of triangle PQR.
(4)
15. The line 1l has equation 23 xy and the line 2l has equation 0823 yx .
(a) Find the gradient of the line 2l .
(2)
The point of intersection of 1l and 2l is P.
(b) Find the coordinates of P.
(3)
The lines 1l and 2l cross the line 1y at the points A and B respectively.
(c) Find the area of triangle ABP.
(4)
48. Mr A. Slack (2012)
1. The curve C has equation y = 4x2
+
x
x5
, x 0. The point P on C has x-coordinate 1.
(a) Show that the value of
x
y
d
d
at P is 3.
(5)
(b) Find an equation of the tangent to C at P.
(3)
This tangent meets the x-axis at the point (k, 0).
(c) Find the value of k.
(2)
2. The curve C has equation
x
xx
y
)8)(3(
, x > 0.
(a) Find
x
y
d
d
in its simplest form.
(4)
(b) Find an equation of the tangent to C at the point where x = 2.
(4)
3. The curve C has equation
y = 3
2
1
x – 2
3
9x +
x
8
+ 30, x > 0.
(a) Find
x
y
d
d
.
(4)
(b) Show that the point P(4, –8) lies on C.
(2)
(c) Find an equation of the normal to C at the point P, giving your answer in the form
ax + by + c = 0 , where a, b and c are integers.
(6)
49. 4. The curve C has equation y = x2
(x – 6) +
x
4
, x > 0.
The points P and Q lie on C and have x-coordinates 1 and 2 respectively.
(a) Show that the length of PQ is 170.
(4)
(b) Show that the tangents to C at P and Q are parallel.
(5)
(c) Find an equation for the normal to C at P, giving your answer in the form ax + by + c = 0,
where a, b and c are integers.
(4)
5. The gradient of the curve C is given by
x
y
d
d
= (3x – 1)2
.
The point P(1, 4) lies on C.
(a) Find an equation of the normal to C at P.
(4)
(b) Find an equation for the curve C in the form y = f(x).
(5)
(c) Using
x
y
d
d
= (3x – 1)2
, show that there is no point on C at which the tangent is parallel to the
line y = 1 – 2x.
(2)
6. The curve C has equation y = f(x), x > 0, and f(x) = 4x – 6x + 2
8
x
.
Given that the point P(4, 1) lies on C,
(a) find f(x) and simplify your answer.
(6)
(b) Find an equation of the normal to C at the point P(4, 1).
(4)
50. Mr A. Slack (2012)
7. The curve C has equation y = 4x + 2
3
3x – 2x2
, x > 0.
(a) Find an expression for
x
y
d
d
.
(3)
(b) Show that the point P(4, 8) lies on C.
(1)
(c) Show that an equation of the normal to C at the point P is
3y = x + 20.
(4)
The normal to C at P cuts the x-axis at the point Q.
(d) Find the length PQ, giving your answer in a simplified surd form.
(3)
8. The curve C has equation
y = 9 – 4x –
x
8
, x > 0.
The point P on C has x-coordinate equal to 2.
(a) Show that the equation of the tangent to C at the point P is y = 1 – 2x.
(6)
(b) Find an equation of the normal to C at the point P.
(3)
The tangent at P meets the x-axis at A and the normal at P meets the x-axis at B.
(c) Find the area of the triangle APB.
(4)
51. 9. Figure 2
Figure 2 shows part of the curve C with equation
y = (x – 1)(x2
– 4).
The curve cuts the x-axis at the points P, (1, 0) and Q, as shown in Figure 2.
(a) Write down the x-coordinate of P and the x-coordinate of Q.
(2)
(b) Show that
x
y
d
d
= 3x2
– 2x – 4.
(3)
(c) Show that y = x + 7 is an equation of the tangent to C at the point (–1, 6).
(2)
The tangent to C at the point R is parallel to the tangent at the point (–1, 6).
(d) Find the exact coordinates of R.
(5)
10. The curve with equation y = f(x) passes through the point (1, 6). Given that
f (x) = 3 +
2
1
25 2
x
x
, x > 0,
find f(x) and simplify your answer.
(7)
y
C
xP O 1 Q
4
52. Mr A. Slack (2012)
11. The gradient of a curve C is given by
x
y
d
d
= 2
22
)3(
x
x
, x 0.
(a) Show that
x
y
d
d
= x2
+ 6 + 9x–2
.
(2)
The point (3, 20) lies on C.
(b) Find an equation for the curve C in the form y = f(x).
(6)
12. The curve C with equation y = f(x), x 0, passes through the point (3, 7 2
1
).
Given that f(x) = 2x + 2
3
x
,
(a) find f(x).
(5)
(b) Verify that f(–2) = 5.
(1)
(c) Find an equation for the tangent to C at the point (–2, 5), giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
13. The curve C has equation
y = x3
– 2x2
– x + 9, x > 0.
The point P has coordinates (2, 7).
(a) Show that P lies on C.
(1)
(b) Find the equation of the tangent to C at P, giving your answer in the form y = mx + c, where
m and c are constants.
(5)
The point Q also lies on C.
Given that the tangent to C at Q is perpendicular to the tangent to C at P,
(c) show that the x-coordinate of Q is
3
1
(2 + 6).
(5)
53. 14. The curve C has equation y = 3
1
x3
– 4x2
+ 8x + 3.
The point P has coordinates (3, 0).
(a) Show that P lies on C.
(1)
(b) Find the equation of the tangent to C at P, giving your answer in the form y = mx + c, where
m and c are constants.
(5)
Another point Q also lies on C. The tangent to C at Q is parallel to the tangent to C at P.
(c) Find the coordinates of Q.
(5)
55. 1. The sequence of positive numbers u1, u2, u3, ..., is given by
un + 1 = (un – 3)2
, u1 = 1.
(a) Find u2, u3 and u4.
(3)
(b) Write down the value of u20.
(1)
2. A sequence is given by
1x = 1,
xn + 1 = xn(p + xn),
where p is a constant (p ≠ 0).
(a) Find x2 in terms of p.
(1)
(b) Show that x3 = 1 + 3p + 2p2
.
(2)
Given that 3x = 1,
(c) find the value of p,
(3)
(d) write down the value of 2008x .
(2)
3. A sequence 1a , 2a , 3a , ... is defined by
1a = 2,
1na = na3 – c
where c is a constant.
(a) Find an expression for 2a in terms of c.
(1)
Given that
3
1i
ia = 0,
(b) find the value of c.
(4)
56. Mr A. Slack (2012)
4. A sequence 1x , 2x , 3x , … is defined by
1x = 1,
xn + 1 = axn – 3, n 1,
where a is a constant.
(a) Find an expression for 2x in terms of a.
(1)
(b) Show that 3x = a2
– 3a – 3.
(2)
Given that 3x = 7,
(c) find the possible values of a.
(3)
5. A sequence a1, a2, a3, ... is defined by
a1 = k,
an + 1 = 2an – 7, n 1,
where k is a constant.
(a) Write down an expression for a2 in terms of k.
(1)
(b) Show that a3 = 4k – 21.
(2)
Given that
4
1r
ra = 43,
(c) find the value of k.
(4)
6. A sequence a1, a2, a3, . . . is defined by
a1 = 3,
an + 1 = 3an – 5, n 1.
57. (a) Find the value a2 and the value of a3.
(2)
(b) Calculate the value of
5
1r
ra .
(3)
7. A sequence ...,,, 321 aaa is defined by
,1 ka
531 nn aa , 1n ,
where k is a positive integer.
(a) Write down an expression for 2a in terms of k.
(1)
(b) Show that 2093 ka .
(2)
(c) (i) Find
4
1r
ra in terms of k.
(ii) Show that
4
1r
ra is divisible by 10.
(4)
8. A sequence of positive numbers is defined by
1na = ( 2
na + 3), n 1,
1a = 2.
(a) Find 2a and 3a , leaving your answers in surd form.
(2)
(b) Show that 5a = 4.
(2)
58. Mr A. Slack (2012)
9. A sequence 1a , 2a , 3a , …, is defined by
1a = k,
1na = 5 na + 3, n 1,
where k is a positive integer.
(a) Write down an expression for 2a in terms of k.
(1)
(b) Show that 3a = 25k + 18.
(2)
(c) (i) Find
4
1r
ra in terms of k, in its simplest form.
(ii) Show that
4
1r
ra is divisible by 6.
(4)
(4)
60. Mr A. Slack (2012)
1. (a) On the same axes sketch the graphs of the curves with equations
(i) y = x2
(x – 2),
(3)
(ii) y = x(6 – x),
(3)
and indicate on your sketches the coordinates of all the points where the curves cross the x-axis.
(b) Use algebra to find the coordinates of the points where the graphs intersect.
(7)
2. (a) On the axes below sketch the graphs of
(i) y = x (4 – x),
(ii) y = x2
(7 – x),
showing clearly the coordinates of the points where the curves cross the coordinate axes.
(5)
(b) Show that the x-coordinates of the points of intersection of
y = x (4 – x) and y = x2
(7 – x)
are given by the solutions to the equation x(x2
– 8x + 4) = 0.
(3)
The point A lies on both of the curves and the x and y coordinates of A are both positive.
(c) Find the exact coordinates of A, leaving your answer in the form (p + q√3, r + s√3), where p,
q, r and s are integers.
(7)
3. The curve C has equation
y = (x + 3)(x – 1)2
.
(a) Sketch C, showing clearly the coordinates of the points where the curve meets the coordinate
axes.
(4)
(b) Show that the equation of C can be written in the form
y = x3
+ x2
– 5x + k,
61. where k is a positive integer, and state the value of k.
(2)
There are two points on C where the gradient of the tangent to C is equal to 3.
(c) Find the x-coordinates of these two points.
(6)
4. The point P(1, a) lies on the curve with equation y = (x + 1)2
(2 – x).
(a) Find the value of a.
(1)
(b) Sketch the curves with the following equations:
(i) y = (x + 1)2
(2 – x),
(ii) y =
x
2
.
On your diagram show clearly the coordinates of any points at which the curves meet the
axes.
(5)
(c) With reference to your diagram in part (b), state the number of real solutions to the equation
(x + 1)2
(2 – x) =
x
2
.
(1)
5. (a) Factorise completely x3
– 4x.
(3)
(b) Sketch the curve C with equation
y = x3
– 4x,
showing the coordinates of the points at which the curve meets the axis.
(3)
The point A with x-coordinate –1 and the point B with x-coordinate 3 lie on the curve C.
62. Mr A. Slack (2012)
(c) Find an equation of the line which passes through A and B, giving your answer in the form
y = mx + c, where m and c are constants.
(5)
(d) Show that the length of AB is k10, where k is a constant to be found.
(2)
6. (a) Sketch the graphs of
(i) y = x(x + 2)(3 − x),
(ii) y = –
x
2
.
showing clearly the coordinates of all the points where the curves cross the coordinate axes.
(6)
(b) Using your sketch state, giving a reason, the number of real solutions to the equation
x(x + 2)(3 – x) +
x
2
= 0.
(2)
7. The curve C with equation y = f(x) passes through the point (5, 65).
Given that f (x) = 6x2
– 10x – 12,
(a) use integration to find f(x).
(4)
(b) Hence show that f(x) = x(2x + 3)(x – 4).
(2)
(c) Sketch C, showing the coordinates of the points where C crosses the x-axis.
(3)
8. The curve C has equation y =
x
3
and the line l has equation y = 2x + 5.
(a) Sketch the graphs of C and l, indicating clearly the coordinates of any intersections with the
axes.
(3)
(b) Find the coordinates of the points of intersection of C and l.
(6)
63. 9. (a) Factorise completely x3
– 6x2
+ 9x
(3)
(b) Sketch the curve with equation
y = x3
– 6x2
+ 9x
showing the coordinates of the points at which the curve meets the x-axis.
(4)
Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation
y = (x – 2)3
– 6(x – 2)2
+ 9(x – 2)
showing the coordinates of the points at which the curve meets the x-axis.
(2)
10. The curve C has equation
y = (x + 1)(x + 3)2
.
(a) Sketch C, showing the coordinates of the points at which C meets the axes.
(4)
(b) Show that
x
y
d
d
= 3x2
+ 14x + 15.
(3)
The point A, with x-coordinate –5, lies on C.
(c) Find the equation of the tangent to C at A, giving your answer in the form y = mx + c, where
m and c are constants.
(4)
Another point B also lies on C. The tangents to C at A and B are parallel.
(d) Find the x-coordinate of B.
(3)
11. Factorise completely x3
– 9x.
(3)
64. Mr A. Slack (2012)
12. Factorise completely
x3
– 4x2
+ 3x.
(3)
13. Given that f(x) = (x2
– 6x)(x – 2) + 3x,
(a) express f(x) in the form x(ax2
+ bx + c), where a, b and c are constants.
(3)
(b) Hence factorise f(x) completely.
(2)
(c) Sketch the graph of y = f(x), showing the coordinates of each point at which the graph meets
the axes.
(3)
14.
Figure 1
Figure 1 shows a sketch of the curve with equation 0,
3
x
x
y .
(a) On a separate diagram, sketch the curve with equation ,2,
2
3
x
x
y showing the
coordinates of any point at which the curve crosses a coordinate axis.
(3)
(b) Write down the equations of the asymptotes of the curve in part (a).
(2)
x
y
O
65. 15. Find the set of values of x for which
(a) 3(2x + 1) > 5 – 2x,
(2)
(b) 2x2
– 7x + 3 > 0,
(4)
(c) both 3(2x + 1) > 5 – 2x and 2x2
– 7x + 3 > 0.
(2)
16. Find the set of values of x for which
x2
– 7x – 18 > 0.
(4)
17. Find the set of values of x for which
(a) 4x – 3 > 7 – x
(2)
(b) 2x2
– 5x – 12 < 0
(4)
(c) both 4x – 3 > 7 – x and 2x2
– 5x – 12 < 0
(1)
18. Find the set of values of x for which
(a) 3(x – 2) < 8 – 2x,
(2)
(b) (2x – 7)(1 + x) < 0,
(3)
(c) both 3(x – 2) < 8 – 2x and (2x – 7)(1 + x) < 0.
(1)