PAPER 1: Core Pure Mathematics 1

The transformation P is an enlargement, centre the origin, with scale factor k, where k > 0 The transformation Q is a rotation through angle θ degrees anticlockwise about the origin. The transformation P followed by the transformation Q is represented by the matrix  −4 −4 3 M =  4 3 −4  (a) Determine (i) the value of k, (ii) the smallest value of θ (4) A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant. The square S is transformed to the square S′ by the transformation represented by M. (b) Determine, in terms of a, the area of S′ (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks) *P66796A0536* 2. (a) Use the Maclaurin series expansion for cos x to determine the series expansion of cos2 3x in ascending powers of x, up to and including the term in x4 Give each term in simplest form. (2) (b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for π x 2 x dx cos   3  (3) (c) Use the integration function on your calculator to evaluate π x 2 x dx cos   3  Give your answer to 5 decimal places. (1) (d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found