PAPER 1: Core Pure Mathematics 1

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 (a) Determine (i) the value of k, (ii) the smallest value of θ M =  4 3  4  (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) Question 1 continued (Total for Question 1 is 6 marks) 2. (a) Use the Maclaurin series expansion for cos x to determine the series expansion of cos2  x  in ascending powers of x, up to and including the term in x4  3 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  2  1 cos2  x   dx  x π 6  3  (3) (c) Use the integration function on your calculator to evaluate π  2  1 cos2  x   dx  x π 6  3  Give your answer to 5 decimal places. (d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the appro