where is an integer. p (a) Express in the form + where and are real. Give your answer in its simplest form z a bi a b in terms of p. (4) (b) Given that arg(z) = where tan = 1 find the possible values of p. θ, θ (5) 5. (a) Use the standard results for and to show that (5) (b) Calculate the value of . (3) 6. and Given that M = (A + B)(2A B), – (a) calculate the matrix M, (6) (b) find the matrix C such that MC = A. (4) 7. The parabola has cartesian equation = 4ax, > 0. C y2 a The points 2ap) and lie on P(ap2, Pʹ(ap2, C. –2ap) (a) Show that an equation of the normal to at the point is C P + = 2ap + y px ap3 (5) (b) Write down an equation of the normal to at the point C Pʹ. (1) The normal to at meets the normal to at at the point C P C Pʹ Q. (c) Find, in terms of and the coordinates of a p, Q. (2) Given that is the focus of the parabola, S
where p is an integer. (a) Express z in the form a + bi where a and b are real. Give your answer in its simplest form in terms of p. (4) (b) Given that arg(z) = θ, where tan θ = 1 find the possible values of p. (5) 5. (a) Use the standard results for and to show that (5) (b) Calculate the value of . (3) 6. and Given that M = (A + B)(2A – B), (a) calculate the matrix M, (6) (b) find the matrix C such that MC = A. (4) 7. The parabola C has cartesian equation y2 = 4ax, a > 0. The points P(ap2, 2ap) and Pʹ(ap2, –2ap) lie on C. (a) Show that an equation of the normal to C at the point P is y + px = 2ap + ap3 (5) (b) Write down an equation of the normal to C at the point Pʹ. (1) The normal to C at P meets the normal to C at Pʹ at the point Q. (c) Find, in terms of a and p, the coordinates of Q. (2) Given that S is the focus of the parabola,
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