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Structured Mathematics FURTHER PURE MATHEMATICS FP2 (September 2004 version, based on FPM) Assessment format Examination 1h 30 mins (72 marks) Section A: 3 compulsory questions, each worth about 18 marks, total: 54 marks. Section B: One question to be chosen from two, both worth 18 marks, total: 18 marks. Coursework None Assumed Knowledge Candidates are expected to know the content for C1, C2, C3, C4 and FP1. Topic Competence Book Reference Understand the meaning of polar co-ordinates (r, θ) and to be able to convert from polar to Cartesian coordinates and vice versa. Be able to sketch curves with simple polar equations. Be able to find the area enclosed by a polar curve. Ex 3B, q1-2, p44-45 The inverse functions of sine, cosine and tangent. Differentiation of arcsin x , arccos x and arctan x Understand the definitions of inverse trigonometric functions. Be able to differentiate inverse trigonometric functions. p31-32 Use of trigonometrical substitutions in integration. Recognise integrals of functions of the form 1 (a 2 x 2 ) 2 and (a 2 x 2 ) 1 and be able to integrate associated functions by using trigonometrical substitutions. Ex 2D, q2-6, p40 POLAR COORDINATES Polar co-ordinates in two dimensions. Ex 3C, p48 Ex 3D, p51 CALCULUS Ex 2D, q1, p40 Progres s SERIES Maclaurin series. Approximate evaluation of a function. Be able to find the Maclaurin series of a function, including the general term in simple cases. Ex 9D, p179 Appreciate that the series may converge only for a restricted set of values of x. Departmental notes Identify and be able to use the Ex 9D, p179 Maclaurin series of standard functions. COMPLEX NUMBERS Modulus-argument form Understand the polar (modulus-argument) form of a complex number, and the definition of modulus, argument. Definition, p6 Ex 1B, q2, 5,6,7, p9 Be able to multiply and divide complex numbers in polar form Departmental notes De Moivre’s Understand de Moivre’s theorem and simple theorem applications. Expression of complex numbers in the form Ex 15A, p334 Be able to apply de Moivre’s theorem to finding multiple angle formulae and to summing suitable series. Ex 15C, p352 Understand the definition e j cos j sin and hence the form z re j . Departmental notes The n nth roots of a complex number Know that every non-zero complex number has n nth roots, and that in the Argand diagram these are the vertices of a regular n-gon. Departmental notes Know that the distinct nth Ex 15B, q1ii, j 4, 5, 12, 14, roots of re are: 2k j sin 2k p342 r cos n n for k= 0, 1, …, n-1. 1 n Be able to explain why the sum of all the nth roots is zero. Departmental notes Appreciate the effect in the Argand diagram of multiplication by a complex number. Ex 15D, q1, 3, p359 Be able to represent complex roots of unity on an Argand diagram Departmental notes Be able to apply complex numbers to geometrical problems Ex 15D, p359 Determinant and inverse of a 3x3 matrix Be able to find the determinant of any 3x3 matrix and the inverse of a nonsingular 3x3 matrix. Ex 14A, q2c, d, e; q3, 6, 9, 10, 11, 12, p306 Eigenvalues and eigenvectors of 2x2 and 3x3 matrices Understand the meaning of eigenvalue and eigenvector, and be able to find these for 2x2 and 3x3 matrices whenever this is possible. Ex 14B, q514, p324 Applications of complex numbers in Geometry. MATRICES Diagonalisation and Be able to form the matrix of powers of 2x2 and eigenvectors and use this to 3x3 matrices reduce a matrix to diagonal form. Ex 14B, q7-9, p325 Solutions of equations The use of the Cayley-Hamiton Theorem Be able to find powers of a 2x2 or 3x3 matrix. Departmental notes Be able to solve a matrix equation or the equivalent simultaneous equations, and to interpret the solution geometrically. Understand the term characteristic equation of a 2x2 or 3x3 matrix Departmental notes Understand that every 2x2 or 3x3 matrix satisfies its own characteristic equation, and be able to use this. . Departmental notes Understand the definitions of hyperbolic functions and be able to sketch these graphs. P189-191 Be able to differentiate and integrate hyperbolic functions. Ex 10A, q3, 4, 7b, 10i, 12, 13, 16a Departmental notes Departmental notes SECTION B Option 1 HYPERBOLIC FUNCTIONS Hyperbolic functions: definitions, graphs, differentiation and integration. . Inverse hyperbolic functions, including the logarithmic forms. Use in integration. Understand and be able to use the definitions of the inverse hyperbolic functions. Be able to use the logarithmic forms of the inverse hyperbolic functions. Departmental notes Be able to integrate Ex 10B, q2-4, p206 x 2 a 1 2 2 and x 2 a related functions. Option 2 INVESTIGATION OF CURVES 1 2 2 and Curves Know vocabulary associated with curves Departmental notes Graphical Calculator Be able to use a suitable graphical calculator to draw curves. Departmental notes Properties of curves. Be able to find, describe and generalise properties of curves Departmental notes Be able to determine asymptotes Departmental notes Be able to identify cusps and loops Departmental notes Be able to find and work with equations of chords, tangents and normals Know the names and shapes of conics Departmental notes Know the standard Cartesian and parametric equations of conics. p218-229 Ex 11A-D Conics p218-229 Textbook: “Further Pure Mathematics” Brian and Mark Gaulter (FPM) [HR: 06/05, 08/08]