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) R Q O P The diagram above shows a rectangle OPQR inscribed on the quadrant of a circle of fixed radius a with O as the center of the circle as shown. If OP = x, find the area A of the rectangle OPQR in terms of a and x. Hence, show that A is maximum when the perimeter of the rectangle OPQR is 4 times the length of OP. [5] (b) C 6 cm D P. The line CD is perpendicular to a horizontal plane through the point D and CD is 6 cm. A variable point P moves along a straight line through D on the horizontal plane. Given that P is moving away from D at a speed of 2 cms -1, find the rate of change of the angle CPD with respect to time when the distance of P from D is 36 cm. [5] 9 In the diagram below, OAB is the horizontal base, where OA = 6 units, OB = 4 units, and 90AOB   . Poles OP and BR are placed vertically, where OP = 5 units and = 2 unitsBR . The unit vectors i, j, k are parallel to OA, OB and OP respectively. The point O is taken as the origin. (i) Show that the equation of line PR is 00 04 53                    r , where  . [2] i O P B R j k A

5 [Turn Over (ii) A point Q divides AB such that AQ:QB = 1:3. Find OQ . (iii) A line l with equation 10; 8 3 yxz    intersects the line PR at a point X. Find the position vector of X. (iv) Let m be a unit vector along AB. Find | QX m |. Give a geometrical meaning of | QX m |. Hence, or otherwise, find the area of triangle AXB. [2] [3] [4] 10 A plane is given by the equation 25xy   . A sphere with centre represented by the position vector −4i  3j  k rests on the plane such that the plane touches the sphere only at one point A. (i) Find the position vector of A. Hence, or otherwise, find the radius of the sphere exactly. (ii) A line l1 that passes through the point ak ( where a > 0) and the centre of the sphere makes an angle of 300 with the plane  Find a in exact form. (iii) Another plane contains the line , t Î ¡ and is also parallel to vector −2i  j k. Find the vector equation of  in scalar product form. Hence, show that and are perpendicular. [4] [4] [3] 11 (a) (i) The region R is bounded by the curves 2 4 1y x  , ln( 1)yx and the line 1x = and the y-axis. Find the area of region R. [2] (ii) Find the exact volume of the solid formed when R is rotated 2p radians about the y-axis. [4] (b) The diagram shows part of the graph of 2 1 1y x  . 2 1 1y x  y x O 3 6 3 31 1 1 n n n n+ + +

6 [Turn Over (i) By considering 1n rectangles of equal width from 0x to 3x , show that for all non-negative integers n,   22 0 3( 1) 91 n r nA rn   , where A is the area bounded by the curve, the axes and the line x = 3. (ii) Deduce   22 0 3( 1)lim 91n n r n rn    exactly. [3] [2] 12 (a) By means of the substitution 1ux , find 2 d 1 x x x . [4] (b) Find 21 d (1 ) tan x xx  where x > 0. [2] (c) (i) Differentiate 21 xe  with respect to x for 1x £ . (ii) Hence, find the exact value of 21 1 0 dxx e x  . [2] [4] End of Paper