ASRJC 2023 JC1 Further Math MYE

ASRJC 2023 1 It is given that    1f 12r rr  . (a) Show that      2f1 f 12rr rr r   and find  1 1 12 n r rr r  in terms of n. [4] (b) (i) Deduce the exact value of  1 1 12r rr r    . [1] ( i i ) For 3n , deduce an expression for   1 3 1 12 N r rr r    in terms of N. [2] 2 (a) A geometric progression G has positive first term a , common ratio r and sum to infinity S. ๠e sum to infinity of the even-numbered terms of G, i.e. the second, fourth, sixth, … terms, is 1 2 S . ( i ) Find the value of r . [3] (ii) In another geometric progression H, each term is the modulus of the corresponding term of G. Given that the third term of G is 2, show that the sum to infinity of H is 27. [3] (b) An arithmetic progression has first term 1000 and common difference 1.4 . Determine, with clear workings, the value of the first negative term of the sequence and the sum of all the positive terms. [4] 3 (a) Points A and B lie on a parabola such that the line segment AB passes through the focus. Points R and S are the feet of perpendiculars from A and B to the directrix respectively. It is given that RS is m units and the perimeter of ABSR is n units. Express the area of ABSR in terms of m and n. [3] (b) ๠e conic C has foci at (5, 2) and (5, 10) and it passes through the point (2, 6). ( i ) Find the cartesian equation of C in standard form. [3] (ii) When the variable y in the cartesian equation of C is replaced with 5y, the resultant equation represents another conic. Find the exact coordinates of the foci of this conic. [3] 4 Referring to the pole O, the curve C has polar equation cotr  , where 62   . (a) Sketch the curve C. [2] (b) Show that 2 d1 d( 2 ) y xr r  . Determine the exact range of values of the gradient of C. [5] (c) Obtain a cartesian equation of C in the form f( )y x . [3]

2 ASRJC 2023 5 Relative to an origin O, an object is placed at point P with coordinates (4 ,,)cc , where c is a positive real constant, and there is a mirror plane with equation 1xyz , as shown in the diagram (not drawn to scale). It is known that the shortest distance between P and the mirror is 33 . (a) Show that 7c . [3] A point A has coordinates ( 15,17,5) . (b) Find the coordinates of A, the point of reflection of A in the mirror. [4] A laser beam is directed from A towards a point on the mirror and is reflected to reach the object at P. (c) Find the acute angle that the laser beam makes with the mirror. [3] 6 Do not use a calculator in this question. (a) (i) It is given that 2iw  and 2 iiww z  . Find the complex number z in the form ixy , ,xy   , showing your workings. [3] (ii) Using the information in (i), determine two values of u that satisfy 23 i 0uui u z  , justifying your answer clearly. [3] (b) Given that 22sin i cos55p  , determine the three smallest positive integers n such that np is a negative real number. [4] A P mirror

3 ASRJC 2023 7 A straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. ๠e intensity of light at point P at ground level on the street is proportional to the angle  radians, where APB  as shown in the cross-sectional diagram below. (a) Given that the distance of P from the base of the wall of height 8 metres is x metres ( 02 0x ), show that 11 20tan tan81 3 xx   . [1] (b) Find an expression for d dx  . [3] (c) Hence, determine the value of x corresponding to the maximum light intensity at P. Give your answer to four signi ficant figures. You need not justify that the value of x obtained gives the maximum light intensity at P. [2] (d) Find the minimum value of  as x varies. [2] (e) ๠e point P moves across the street from the base of A to the base of B with speed 10.5 ms . Determine the rate of change of  with respect to time when P is at the midpoint of the street. [3]