Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation; numerical methods. C3.2 Assessment information Prerequisites and preamble Prerequisites A knowledge of the specifications for C and C2, their preambles, prerequisites and associated formulae, is assumed and may be tested. Preamble Methods of proof, including proof by contradiction and disproof by counter-eample, are required. At least one question on the paper will require the use of proof. Eamination Calculators The eamination will consist of one ½ hour paper. It will contain about seven questions of varying length. The mark allocations per question which will be stated on the paper. All questions should be attempted. Students are epected to have available a calculator with at least the following keys: +,,,,, 2,,, y, ln,e,!, sine, cosine and tangent and their inverses in degrees and decimals of a degree, and in radians; memory. Calculators with a facility for symbolic algebra, differentiation and/or integration are not permitted. Edecel GCE in Mathematics Edecel Limited 2007 Section C 3
Unit C3 Core Mathematics C3 (A2) Formulae Formulae which students are epected to know are given below and these will not appear in the booklet, Mathematical Formulae including Statistical Formulae and Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage. This section lists formulae that students are epected to remember and that may not be included in formulae booklets. Trigonometry cos 2 A + sin 2 A sec 2 A + tan 2 A cosec 2 A + cot 2 A sin 2A 2 sina cos A cos2a cos 2 A sin 2 A 2 tan A tan 2A tan 2 A function sin k cos k e k ln f ( ) + g ( f ( ) g ( ) f (g () derivative k cos k kk sin ke k k f ( ) + g ( f ( g ( +f ( ) g ( f (g ( ) g ( 32 Section C Edecel Limited 2007 Edecel GCE in Mathematics
Core Mathematics C3 (A2) Unit C3 Algebra and functions Simplification of rational epressions including factorising and cancelling, and algebraic division. Denominators of rational epressions will be linear or quadratic, eg a + b, a + b 2 p + q + r 3 2 +. Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs. The concept of a function as a one-one or many-one mapping from (or a subset of ) to. The notation f : and f( will be used. Students should know that fg will mean do g first, then f. f Students should know that if f eists, then f f( = ff ( =. The modulus function. Combinations of the transformations y = f( as represented by y = af(( ), y = f( + a, y = f( + a), y = f(a ). Students should be able to sketch the graphs of y = a + b and the graphs of y = f( and y = f( ), given the graph of y = f( ). Students should be able to sketch the graph of, for eample, y = 2f(3, y = f( ) +, given the graph of y = f( or the graph of, for eample, y = 3 + sin 2, π y = cos + 4. The graph of y = f(a + b) will not be required. Edecel GCE in Mathematics Edecel Limited 2007 Section C 33
Unit C3 Core Mathematics C3 (A2) 2 Trigonometry Knowledge of secant, cosecant and cotangent and of arcsin, arccos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains. Angles measured in both degrees and radians. Knowledge and use of sec 2 = + tan 2 and cosec 2 = + cot 2. Knowledge and use of double angle formulae; use of formulae for sin(a B), cos (A B) and tan (A B) and of epressions for a cos + b sin in the equivalent forms of r cos ( a) or r sin ( a). To include application to half angles. Knowledge of the t (tan θ ) 2 formulae will not be required. Students should be able to solve equations such as a cos + b sin = c in a given interval, and to prove simple identities such as cos cos 2 + sin sin 2 cos. 3 Eponentials and logarithms The function e and its graph. The function ln and its graph; ln as the inverse function of e. To include the graph of y = e a + b +c. Solution of equations of the form e a + b = p and ln (a + b) = q is epected. 34 Section C Edecel Limited 2007 Edecel GCE in Mathematics
Core Mathematics C3 (A2) Unit C3 4 Differentiation Differentiation of e, ln, sin, cos, tan and their sums and differences. Differentiation using the product rule, the quotient rule and the chain rule. The use of d y =. d d dy Differentiation of cosec, cot and sec are required. Skill will be epected in the differentiation of functions generated from standard forms using products, quotients and composition, such as 2 4 sin, e 3, cos 2 and tan 2 2. Eg finding d y d for = sin 3y. 5 Numerical methods Location of roots of f( = 0 by considering changes of sign of f( in an interval of in which f( ) is continuous. Approimate solution of equations using simple iterative methods, including recurrence relations of the form n+ = f( n ). Solution of equations by use of iterative procedures for which leads will be given. Edecel GCE in Mathematics Edecel Limited 2007 Section C 35