Further core maths cheat sheet. %PDF-1. Further Algebra and Functions I Cheat Sheet AQA A Level Further Maths: Core. 𝑑𝑑𝑣𝑣 𝑑𝑑𝑡𝑡 −2𝑣𝑣𝑡𝑡= 𝑡𝑡 Use the integrating factor method since this DE is non - separable. 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. 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. In zero matrices, all elements are zero. 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. This series in particular is an arithmetic series which is introduced in Pure Year 2. In this cheat sheet we Put the loci equation into the form |#−#!|=&. If so, the integral is said to converge. 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. If you run out of questions by topic for your own exam board you should move onto another exam board’s tab. − Notes and questions on A-Level Further maths. They serve as Differential Equations II Cheat Sheet (A Level Only) AQA A Level Further Maths: Core a) Write auxiliary equation. Make sure you’re comfortable interpreting the question is giving you, no matter the form. Cubic and quartic inequalities are solved in precisely the same way as quadratic inequalities. 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. Since one of the points is the origin, the vectors for the two other points are the position vectors. Relationship Between the Roots and Coefficients of a Polynomial. e. 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. Imaginary and Complex Numbers. We can use loci to represent regions on an Argand diagram that correspond to given constraints such as inequalities. 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. In springs, when they Title: Ch. 1 Polar Coordinates for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. Volumes of Revolution Cheat Sheet Edexcel Core Pure 1: The bounds of integration can be given to you in different ways . 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. 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. Improper Integrals (A Level Only) Integrals With an Infinite Range . 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. We want to find the velocity after 2𝑠𝑠 which means we first need to solve it. KWWSV ELW O\ SPW HGX FF KWWSV ELW O\ SPW FF Edexcel Further Maths A-level . 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. our first term is y − = w. 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 ). These roots can be found by utilising De Moivre’s theorem. 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 . In AQA A-Level Further Mathematics, it is involved only in proving sums of series, divisibility, and powers of matrices. Ch. 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. Revision notes on 3. About Revision Course; Purchase Access To Recordings and Resources (£50) Login; AS Maths (Year 1) Consolidation Course 2021. Core Pure . 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. , 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 ≠− . 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. 1. The roots of a polynomial are the points at which the curve crosses the -axis. # × 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. Digestible notes on A-Level Further maths. 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. Back to Maths. Further Vectors II Cheat Sheet AQA A Level Further Maths: Core ; Begin by calculating the vectors of two sides of this triangle. Complex Numbers Cheat Sheet Edexcel Core 1 KWWSV ELW O\ SPW HGX Ch. Remember that the same pieces of information are given in most – if not all – questions (i. Since being introduced to the modern number system, they have proved useful in many fields - including quantum mechanics and electronics. One-To-One Maths/Further Maths Tuition; Maths Tuition Modelling with Differential Equations Cheat Sheet Edexcel Core Pure 2 . Provided in formula book . 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. 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. 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. % passes through both 1 and 2 so we can use !. Formula Sheet . Consider a differentiable curve = ( ), given in cartesian coordinates. 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. ෍ ( 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. 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. 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. Integration is used to find the volume of these solids, which are called volumes of revolution. 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. 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. 2 Correlation Cheat Sheet - Edexcel Further Maths A-level - FS2 Author: PMT Education Created Date: 9/22/2021 10:30:42 AM . Some classic examples include a pendulum moving back and forth or a spring bouncing up and down. 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. 1 Roots of Polynomials for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. For a matrix #, the determinant can be denoted by t ( #), t #, | #| or ∆. 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. In certain cases, the integral will still be a finite number. 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. AQA A Level Further Maths: Core. 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. |#−(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. Vectors Cheat Sheet Edexcel Core Pure 1 5 −’ %) % 6 & & = 7 ’ ’ in cartesian form. 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. Complex Numbers. 0=𝜆2−3𝜆+2=Ὠ𝜆−1ὩὨ𝜆−2Ὡ⇒𝜆 1=1, 𝜆2=2 Write complementary function 𝐶𝐹. 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. Classifying Differential Equations. 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. In other words, how the change of an independent variable affects a dependent variable. 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. 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. On this page you can find formula sheets, cheat sheets and questions by topic for Further Statistics for Edexcel, AQA and OCR (A). This is illustrated in the diagrams below. != # 1 2 6 ’, (=# 3 4 5 ’ The cross product can now be calculated, and the modulus can be taken. About AS (Year 1) Maths Revision Course; Purchase Access to Recordings and Resources (£50) Login; Private Tuition. They can come in any order. Sometimes one, or both, of the limits of integration will extend to infinity, with the upper limit approaching +∞ and the lower −∞. 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. Instead of using the endpoints of each vertical strip, the -value of the curve at each midpoint is used to fit a rectangle. On this page you can find formula sheets, cheat sheets and questions by topic for Further Core Pure for Edexcel, AQA and OCR (A). On this page you can find formula sheets, cheat sheets and questions by topic for Further Additional Pure for Edexcel, AQA and OCR (A). =’+>" ’=$ 4 −1 & vector ", subtract the position vectors. 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 . our last term will be y −( t4) = −1 ∑ This is the value of r where our series starts, i. This is a first-order differential equation. 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. 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 ≠− . 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. 𝑚 A Level Further Maths Easter Revision Course 2021. 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. Core Pure 1. !×(=# 10 − Revision notes for the Edexcel A Level Further Maths: Core Pure syllabus, written by the Further Maths experts at Save My Exams. ipq nlkrv xqu urgxj hovu gmmpi bcfatkxy bolnc azex ubc