SAJC P1

2 [Turn Over 1 In triangle ABC, angle BAC is 6  radians, angle ABC is 3 x  radians. Given that x is sufficiently small, show using the sine rule that 2 2 1 31 2 BC a bx cxAC x x     , where a, b and c are constants to be determined. [4] 2 A sequence of real numbers 1u , 2u , 3u ,… satisfies the recurrence relation 1 2 nn nuu n   , 1n . (i) Show that 1 ( 1) 2 k kkuu  . (ii) Given that 1 2u  , use the method of mathematical induction to show that   123  n nS n n for all positive integers n, where nS denotes the sum of the first n terms of the sequence {}nu . [1] [5] 3 (i) Solve the inequality 2 23 , , 02x x x xx    . (ii) Without the use of graphic calculator, find the exact value of a such that 32 2 1 2 3 3 1 d , 22 3 4 4 a aax x x where a ax       [2] [4] 4 The complex number z satisfies the relations | 3| 3iz and 3arg 3 4z i      . (i) Illustrate both of these relations on a single Argand diagram. (ii) Hence find in exact values, the range of possible values of (a) 33zi-- (b) arg( 3 3 )zi-- [3] [3]

3 [Turn Over 5 Let (i) Sketch the graph of for . (ii) Find the series expansion of in ascending powers of , up to and including the term in . Denote the answer to (ii) by . (iii) By substituting into , show that . (iv) By substituting into , find another approximation for . (v) Explain why the value of x used for approximation in (iii) is better than that in (iv). [1] [3] [1] [1] [1] 6 A curve has parametric equations 2x at , 3y at where t and a > 0. Find, in terms of a, (i) the equation of the tangent to the curve at the point 125 8 2 ,4 5 aa  . (ii) the coordinates of the point where this tangent meets the curve again. (iii) the exact coordinates of the point(s) on the curve at which the normal to the curve passes through the point 21 2 ,0a  . [3] [2] [3] 7 The functions f, g and f–1h are given by 2 2 1 23f : , , 1 where is a constant, ,1 g : ( 1) 0.25, ,0 3.5, f h : , 1 , 0.x x axx x x a ax x x x x x e x x            Find the range of values of a for which f has stationary points. Hence, find the set of values of a for which 1f exists. Given that 3a , (i) find the exact range of fg. (ii) find h in a similar form. [4] [2] [2]

4 [Turn Over 8 (a)