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Structured Mathematics PURE MATHEMATICS C2 (September 2004 version, based on IPM) Assessment format Examination 1h 30 mins (72 marks) Section A: 8-10 questions, each worth no more than 5 marks, total: 36 marks. Section B: 3 questions, each worth about 12 marks, total: 36 marks. Coursework None Topic Competence Book Reference Understand the meaning of the word logarithm. Understand the laws of logarithms and how to apply them. Know the value of log a a and log a 1 Know how to convert from index form to logarithmic form and vice versa. Know the function y a x and its graph. Be able to solve equations of the form ax b . Know how to reduce the equations y=axn and y=abx to linear form and, using experimental data, to draw a graph to find values of a, b and n. p410 definition p414 17C q1,2 p410 definition p414 17C q3 ALGEBRA Logarithms Structured Mathematics C2 (September 2004) page 1/5 Progress SEQUENCES AND SERIES Definitions of sequences Arithmetic series Geometric series Know what a sequence of numbers is and what are meant by finite and infinite sequences. Know that a sequence can be generated using a formula for the kth term, or a recurrence relation of the form ak 1 f (ak ) . Know what a series is, and the difference between convergent and divergent series. Be familiar with notation. Know and be able to recognise the periodicity. Know what is meant by an arithmetic series & sequences. Be able to use the standard formulae associated with arithmetic series. Know what is meant by a geometric series. Be able to use the standard formulae associated with geometric series. Know the condition for a geometric series to be convergent and be able to find the sum to infinity. Be able to solve problems involving arithmetic and geometric series. Structured Mathematics C2 (September 2004) p235 9A q1,4 p235 9A q1,4 p241 9B q1 p235 9A q4 p245 9C q1 p245 9C q2,3 p250 9D q1 p250 9D q2,3 p257 9E q1 p257 9E q10,14 page 2/5 TRIGONOMETRY Basic trigonometry Know how to solve rightangled triangles using trigonometry. The sine, cosine Be able to use the and tangent definitions of sin and cos for any angle. Know the graphs of sin , cos and tan for all values of , their symmetries and their periodicities Know the values of sin , cos and tan when is 0°, 30°, 45°, 60°, 90° and 180°. Identities Be able to use sin tan (for any cos angle). Be able to use the identity cos 2 sin 2 1 , and the equivalent forms. Be able to solve simple trigonometric equations in given intervals. Area of a triangle Be able to use the fact that the area of a triangle is given by 1 ab sin C . 2 The sine and cosine Know and be able to rules use the sine and cosine rules. Radians Understand the definition of a radian and be able to convert between radians and degrees. Be able to find the arc length and area of a sector of a circle, when the angle is given in radians. Structured Mathematics C2 (September 2004) P347 14A q3 P347 14A q1 P353 14B q2 P353 14B q3 P353 14B q5 P353 14B q2 P66 2C q3 P61 2B q1, 2 P70 2D q1, 2 P71 2D q4, 6 page 3/5 CALCULUS The basic process of differentiation. Applications of differentiation to the graphs of functions Integration as the inverse of differentiation Know that the gradient of a curve at a point is given by the gradient of the tangent at the point. Know that the gradient of the tangent is given by the limit of the gradient of a chord. Know that the gradient dy function gives the dx gradient of the curve and measures the rate of change of y with respect to x. Be able to differentiate y kx n where k is a constant, and the sums of such functions. Be able to find second derivatives. Be able to use differentiation to find stationary points on a curve: maxima, minima and points of inflection. Understand the terms increasing function and decreasing function. Be able to find the equation of a tangent and normal at any point on a curve. Know that integration is the inverse of differentiation. Be able to integrate functions of the form kx n where k is a constant and n ≠ -1 and the sum of such functions. Know what are meant by indefinite and definite integrals. Be able to evaluate definite integrals. Structured Mathematics C2 (September 2004) P171 6B q8a,b,c,d,g, h P163 6A q5 (except h, i, l) P163 6A, q11 P181 6C q3 P181 6C q5 P171 P193 6B q8a,b,c,d,g, h,j q9a,b,f,g 7A q10a,b,c,d,i, j 7A, q1 P205 7B, q1a-f P195 page 4/5 Integration to find the area under a curve. Be able to find a constant of integration given relevant information. Know that the area under a graph can be found as the limit of a sum of areas of rectangles. Know that an approximate value of a definite integral can be obtained using the trapezium rule, and comment sensibly on its accuracy. Be able to use integration to find the area between a graph and the x-axis. P195 7A, q10a,b,c,d,I, j P487 20B q2 P205 7B q24 CURVE SKETCHING Stationary points Be able to use stationary points when curve sketching. Stretches Know how to sketch curves of the form y af ( x) and y f (ax) , given the curve of y f ( x) . Textbook: “Introducing Pure Mathematics” R Smedley & G Wiseman (IPM) [WG: 09/95; PJM: 10/96; PJM: 10/97; DJR: 05/00; ET:04/02; HR: 08/04; TEK 04/05] Structured Mathematics C2 (September 2004) page 5/5