Further core maths cheat sheet

Further core maths cheat sheet. Further Algebra and Functions IV Cheat Sheet AQA A Level Further Maths: Core Graphs of Rational Functions with Linear Terms A rational function with linear terms has Example 3:the form: = + > + @ where ≠− . In general, you can remember that ∑ 2=(∑ )2− t(∑ ) Further Algebra and Functions II Cheat Sheet AQA A Level Further Maths: Core Maclaurin Series of a Function and the General Term (A-Level Only) The Maclaurin series is a method of approximating a function by expressing the function as an infinite series polynomial. This series in particular is an arithmetic series which is introduced in Pure Year 2. In zero matrices, all elements are zero. 1 Roots of Polynomials for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. − Notes and questions on A-Level Further maths. KWWSV ELW O\ SPW HGX FF KWWSV ELW O\ SPW FF Edexcel Further Maths A-level . Another set which could be used are called polar coordinates where each point is described by its 2radial distance from the origin and an angle. In springs, when they Title: Ch. Further Vectors I Cheat Sheet AQA A Level Further Maths: Core Equation of a plane (A Level Only) A plane can also be described using a vector equation: !=#+%&+’( As before, # is a vector to the plane, & and ( are two non-parallel vectors along the plane, % and ’ parameters which have a unique value for each point on the plane. Complex Numbers IV Cheat Sheet (A Level Only) AQA A Level Further Maths: Core In this topic, we will make use of the results sin (− )=−sin( )and cos(− =cos ). 2 Correlation Cheat Sheet - Edexcel Further Maths A-level - FS2 Author: PMT Education Created Date: 9/22/2021 10:30:42 AM . # × Matrices For a matrix #=[ ], | #|= − Differential Equations V Cheat Sheet (A Level Only) AQA A Level Further Maths: Core Using Hooke’s Law to Formulate Differential Equations for Simple Harmonic Motion The process of observing a system and modelling it using a differential equation requires some knowledge of the forces acting in the system. % passes through both 1 and 2 so we can use !. Not provided in formula book Further Algebra and Functions III Cheat Sheet AQA A Level Further Maths: Core Inequalities Involving Polynomial Equations This section will be largely familiar and looks at sketching graphs and solving inequalities for polynomial equations. In other words, how the change of an independent variable affects a dependent variable. Two cases A Level Further Formula Sheet Summation Results Vectors Properties "(3) (These work similar to integrals)) • Can take constants out: ∑ "#!=" Example 1: Prove that arsinh(𝑥𝑥) = ln 𝑥𝑥+ √𝑥𝑥2+ 1 . On this page you can find formula sheets, cheat sheets and questions by topic for Further Additional Pure for Edexcel, AQA and OCR (A). AQA A Level Further Maths: Core. Sometimes one, or both, of the limits of integration will extend to infinity, with the upper limit approaching +∞ and the lower −∞. Find a vector equation of the straight line which passes through the points 1 and 2, with position vectors 44 +56−8 and 64+36+28 respectively. By integrating, all the values that the function takes within the given interval are Matrices Cheat Sheet II AQA A Level Further Maths: Core Determinant of a Matrix The determinant of a matrix is a scalar value which can be calculated from a square matrix. Differential equations are incredibly useful and can be used to model various phenomena since they describe the rate of change of a quantity with respect to another quantity. 7 %âãÏÓ 176 0 obj > endobj 207 0 obj >/Encrypt 177 0 R/Filter/FlateDecode/ID[09C8CDCBDD7F7A4FA8304B894A8050C8>]/Index[176 67]/Info 175 0 R/Length 133/Prev Further Calculus II Cheat Sheet AQA A Level Further Maths: Core Volumes of Revolution A solid of revolution is obtained by rotating a curve =𝑓( ) about some axis, with a full turn corresponding to rotation by an angle of 2𝜋 radians. Improper Integrals (A Level Only) Integrals With an Infinite Range . Further Vectors II Cheat Sheet AQA A Level Further Maths: Core ; Begin by calculating the vectors of two sides of this triangle. our last term will be y −( t4) = −1 ∑ This is the value of r where our series starts, i. Further Calculus VI Cheat Sheet AQA A Level Further Maths: Core Mean Value of a Function Integration can be used to find the mean value of a function between two points. Identities and equations You will need to be able to use and provehyperbolic identities, which are very similar to their trigonometric Polar Coordinates Cheat Sheet AQA A Level Further Maths: Core The familiar and axes of the 2D plane are just one set of coordinates which can be used to describe each point in the plane. 0= 2+2 +1=( +1)2 ⇒ = −1 Substitute into the general solution for Proof Cheat Sheet AQA A Level Further Maths: Core Proof by Induction Proof by induction is a robust and diverse method of mathematical proof used when the result or final expression is already known. Further Calculus V Cheat Sheet (A Level Only) AQA A Level Further Maths: Core Arc Length of a Curve Arc Length of a Curve Expressed in Cartesian Coordinates Integration is used to find the arc length of a curve between two end points. Instead of using the endpoints of each vertical strip, the -value of the curve at each midpoint is used to fit a rectangle. They have a range of applications in mathematics and physics such as Further Vectors I Cheat Sheet AQA A Level Further Maths: Core Equation of a plane (A Level Only) A plane can also be described using a vector equation: !=#+%&+’( As before, # is a vector to the plane, & and ( are two non-parallel vectors along the plane, % and ’ parameters which have a unique value for each point on the plane. Core Pure 1. On this page you can find formula sheets, cheat sheets and questions by topic for Further Core Pure for Edexcel, AQA and OCR (A). We want to find the velocity after 2𝑠𝑠 which means we first need to solve it. Consider a differentiable curve = ( ), given in cartesian coordinates. These decompositions Hyperbolic Functions II Cheat Sheet AQA A Level Further Maths: Core Hyperbolic Identities Just as there are identities linking the trigonometric functions together, there are similar identities linking hyperbolic functions together. Since being introduced to the modern number system, they have proved useful in many fields - including quantum mechanics and electronics. A trick is to look at the term involving 𝑑𝑑𝑑𝑑 Complex Numbers II Cheat Sheet AQA A Level Further Maths: Core Geometric Representation of Complex Numbers Argand Diagrams All real numbers exist on a straight, infinite number line; all complex numbers exist beyond the real number line in the complex plane. In certain cases, the integral will still be a finite number. =’+>" ’=$ 4 −1 & vector ", subtract the position vectors. our first term is y − = w. 7 Linear Transformations Cheat Sheet - Edexcel Further Maths A-level - CP1 Author: PMT Education Created Date: 20190916222029Z Matrices I Cheat Sheet AQA A Level Further Maths: Core 2 Matrix Arithmetic and Multiplying a Matrix by a Scalar A matrix is an array of numbers, or elements, arranged in rows and columns. 𝑑𝑑𝑣𝑣 𝑑𝑑𝑡𝑡 −2𝑣𝑣𝑡𝑡= 𝑡𝑡 Use the integrating factor method since this DE is non - separable. Roots of polynomials Cheat Sheet Edexcel Core Pure 1 You may decide not to remember these results since they are quite easy to prove; in each case you simply need to combine the fractions to achieve the . Make sure you’re comfortable interpreting the question is giving you, no matter the form. Revision notes on 3. ෍ ( v 𝑟3− u2 + )=( 𝑘 s)3 t 𝑘+1 𝑟=1 We start with the basis step; we show the 𝐿𝐻 = 𝐻 : 3 Complex Numbers I Cheat Sheet AQA A Level Further Maths Core Complex Numbers and Complex Algebra Complex numbers are a superset of the real numbers. Complex Numbers. 0=𝜆2−3𝜆+2=Ὠ𝜆−1ὩὨ𝜆−2Ὡ⇒𝜆 1=1, 𝜆2=2 Write complementary function 𝐶𝐹. != # 1 2 6 ’, (=# 3 4 5 ’ The cross product can now be calculated, and the modulus can be taken. De Moivre’s Theorem When raising complex numbers to a power, we can use De Moivre’s Theorem on their modulus-argument form, Proof by induction Cheat Sheet Edexcel Core Pure 1 Example 1: +Prove by induction that, for ∈ℤ ෍( v𝑟3− u𝑟2+𝑟)= 3( + s) 𝑛 𝑟=1 Start by making a note of what you want to prove in the inductive step. If you run out of questions by topic for your own exam board you should move onto another exam board’s tab. Polar Coordinates Cheat Sheet 𝑟𝑟2= 𝑥𝑥2 𝑦𝑦2 tan𝜃𝜃= 𝑑𝑑 𝑑𝑑 ⇒ Edexcel Pure Year 2 𝐴𝐴𝑟𝑟𝑎𝑎= 1 2 𝑟𝑟2 𝑏𝑏 𝑎𝑎 Edexcel Pure Year 2 KWWSV ELW O\ SPW HGX KWWSV ELW O\ SPW FF KWWSV ELW O\ SPW FF Further Calculus III Cheat Sheet (A-Level Only) AQA A-Level Further Maths: Core Integration Using Partial Fractions Integration using Partial Fractions with Linear Factors Partial fraction decomposition is used to simplify integrals involving the product of two (or more) binomials in the denominator of the integrand. 1 Complex Numbers Cheat Sheet - Edexcel Further Maths A-level - CP1 Author: PMT Education Polar Coordinates Cheat Sheet AQA A Level Further Maths: Core The familiar and axes of the 2D plane are just one set of coordinates which can be used to describe each point in the plane. They serve as Differential Equations II Cheat Sheet (A Level Only) AQA A Level Further Maths: Core a) Write auxiliary equation. Core Pure . The single value, known as a point estimate, is likely to be inaccurate whereas it is more probable that the true population statistic is held within Hyperbolic Functions IV Cheat Sheet AQA A Level Further Maths: Core The question asks for the derivative to be given in terms of 𝑢 √ Further Calculus with Inverse Hyperbolic functions (A Level Only) Revision notes on 7. To find the length Ref: CUP AQA A Level Further Mathematics Book 2 b) When two of the equations describe the same plane and the third plane is not parallel to this plane, the planes will intersect along a line. This is illustrated in the diagrams below. 1 Polar Coordinates for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. Relationship Between the Roots and Coefficients of a Polynomial. Let ⁻this function ⁺𝑥= 0+ 1𝑥+ 2𝑥2+ 3𝑥3+⋯ Proof Cheat Sheet AQA A Level Further Maths: Core Proof by Induction Proof by induction is a robust and diverse method of mathematical proof used when the result or final expression is already known. Further Calculus IV Cheat Sheet AQA A Level Further Maths: Core Differentiating Inverse Trigonometric Functions (A Level Only) To find the derivatives of inverse trigonometric functions, implicit differentiation is used. Formula Sheet . Remember that the same pieces of information are given in most – if not all – questions (i. Further Algebra and Functions I Cheat Sheet AQA A Level Further Maths: Core. Volumes of Revolution Cheat Sheet Edexcel Core Pure 1: The bounds of integration can be given to you in different ways . This is a first-order differential equation. Cubic and quartic inequalities are solved in precisely the same way as quadratic inequalities. Matrices Cheat Sheet Edexcel Core Pure 1 : det 𝑎𝑎𝑏𝑏 𝑐𝑐𝑑𝑑 = 𝑎𝑎𝑏𝑏 𝑐𝑐𝑑𝑑 = ad − bc 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅𝒆𝒆 𝒇𝒇 𝒈𝒈𝒉𝒉𝒊𝒊, 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅𝒆𝒆𝒇𝒇 𝒈𝒈𝒉𝒉𝒊𝒊 , 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅 𝒆𝒆𝒇𝒇 Differential Equations I Cheat Sheet (A Level Only) AQA A Level Further Maths: Core Classifying Differential Equations Differential equations are incredibly useful and can be used to model various phenomena since they describe the rate of change of a quantity with respect to another quantity. Below are all Below are some examples of matrices: #=[7 4 2 4 9 13] $=[6 4 8 Revision notes, summary sheets with key points, checklists, worksheets, topic questions and papers with model solutions for AQA, Edexcel, OCR, MEI Maths A-level Matrices IV Cheat Sheet AQA A Level Further Maths: Core 3 Eigenvalues show the scale factor Factorisation of Determinants Using Row and Column Operations Eigenvalues and Eigenvectors (A Level Only) 2 A triangular matrix is a matrix which has non-zero entries only on the leading diagonal (top left to bottom Matrices Cheat Sheet Edexcel Core Pure 1 : det 𝑎𝑎𝑏𝑏 𝑐𝑐𝑑𝑑 = 𝑎𝑎𝑏𝑏 𝑐𝑐𝑑𝑑 = ad − bc 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅𝒆𝒆 𝒇𝒇 𝒈𝒈𝒉𝒉𝒊𝒊, 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅𝒆𝒆𝒇𝒇 𝒈𝒈𝒉𝒉𝒊𝒊 , 𝒂𝒂𝒃𝒃𝒄𝒄 𝒅𝒅 𝒆𝒆𝒇𝒇 Further Algebra and Functions V Cheat Sheet AQA A Level Further Maths: Core Graphs of the Parabola, Ellipse and Hyperbola Graphs of the parabola, ellipse and hyperbola are called conic sections because they are all found to be the intersection of a plane and a cone. For a matrix #, the determinant can be denoted by t ( #), t #, | #| or ∆. If so, the integral is said to converge. , the equation of the curve, the Further Algebra and Functions IV Cheat Sheet AQA A Level Further Maths: Core Graphs of Rational Functions with Linear Terms A rational function with linear terms has Example 3:the form: = + > + @ where ≠− . Provided in formula book . We can use loci to represent regions on an Argand diagram that correspond to given constraints such as inequalities. Ch. Example 2: Find the general solution to the differential equation You might at first find it difficult to spot the use of the product rule. |#−(6−3*)|=4 Thus, the loci of # is a circle of radius 4, centred at (6,−3) Argand diagrams look similar to Cartesian diagrams This chapter aims to build upon the complex numbers you learnt in Core Pure 1. Ref: CUP AQA A Level Further Mathematics Book 2 c) When all three equations describe different planes, but they all intersect along a line, they form a sheaf. Vectors Cheat Sheet Edexcel Core Pure 1 5 −’ %) % 6 & & = 7 ’ ’ in cartesian form. %PDF-1. Complex Numbers Cheat Sheet Edexcel Core 1 KWWSV ELW O\ SPW HGX Ch. 𝑚 A Level Further Maths Easter Revision Course 2021. 𝐶𝐹= Numerical Methods Cheat Sheet (A Level Only) AQA A Level Further Maths: Core The Mid-Ordinate Rule The mid-ordinate rule is a way of approximating the area under a curve by using vertical strips of equal width. Since one of the points is the origin, the vectors for the two other points are the position vectors. In other words, how the change Differential Equations IV Cheat Sheet (A Level Only) AQA A Level Further Maths: Core Introduction to Simple Harmonic Motion Simple harmonic motion describes systems involving regular oscillations. Imaginary and Complex Numbers. Title: Hyperbolic Functions III Cheat Sheet - AQA Further Maths A-level Author: PMT Education Created Date: 8/19/2022 9:03:48 AM Confidence Intervals I Cheat Sheet AQA A Level Further Maths: Statistics When estimating a population parameter, it is preferable to have a range of values rather than just one value. These roots can be found by utilising De Moivre’s theorem. 1. The roots of a polynomial are the points at which the curve crosses the -axis. Some classic examples include a pendulum moving back and forth or a spring bouncing up and down. About AS (Year 1) Maths Revision Course; Purchase Access to Recordings and Resources (£50) Login; Private Tuition. In this cheat sheet we Put the loci equation into the form |#−#!|=&. We will look at Euler’s formula and De Moivre’s theorem; two powerful ideas which will lay the foundation for most of the techniques you will encounter in this chapter. To geometrically represent complex numbers, we use an Argand diagram which Series Cheat Sheet Edexcel Core Pure 1 This tells us the last value of r for our sequence, i. e. Hyperbolic Functions II Cheat Sheet AQA A Level Further Maths: Core Hyperbolic Identities Just as there are identities linking the trigonometric functions together, there are similar identities linking hyperbolic functions together. Back to Maths. About Revision Course; Purchase Access To Recordings and Resources (£50) Login; AS Maths (Year 1) Consolidation Course 2021. Complex Numbers V Cheat Sheet AQA A Level Further Maths: Core Finding 𝐭𝐡𝐧 Roots of Complex Numbers Any complex number of the form 𝑧𝑛=𝑎 (where 𝑧 and 𝑎 are complex numbers and 𝑛 𝜖 ℤ+) has 𝑛 1distinct 𝑛𝑡ℎroots. One-To-One Maths/Further Maths Tuition; Maths Tuition Modelling with Differential Equations Cheat Sheet Edexcel Core Pure 2 . Complex Numbers III Cheat Sheet AQA A Level Further Maths: Core Loci A locus is defined as the set of all points that satisfy a given constraint. On this page you can find formula sheets, cheat sheets and questions by topic for Further Statistics for Edexcel, AQA and OCR (A). They can come in any order. Classifying Differential Equations. !×(=# 10 − Revision notes for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. In AQA A-Level Further Mathematics, it is involved only in proving sums of series, divisibility, and powers of matrices. Integration is used to find the volume of these solids, which are called volumes of revolution. Digestible notes on A-Level Further maths. Differential Equations III Cheat Sheet (A Level Only) AQA A Level Further Maths: Core Find the roots of the auxiliary equation to the associated homogeneous differential equation d 2 d 2 −3d d +2 =0. kjljge tsjfss ddw bijgyxf abir jkwz qgra fmrvl lmmnbvq zffppe