2024 JC2 H2 Math prelim - CJC

1 2024 CJC JC2 H2 Maths Prelim Paper 1 1 Dif ferentiate 1 e2tn 1 2 a e x x §· ¨¸¨¸©¹

with respect to x, leaving your answer in the simplest form 2 e e x x a bc where ,a n dab c are integers to be determined. [4]

3 2 (a) Given that the real root 1z D , where 0D ! satisfies the equation 32 2 0,zzz DD D find the other roots, 2 z and 3z , in terms of D . [4] (b) Hence solve 32 2ii 0ww wDD D in terms of D . [2]

6 3 I n an Argand diagram, the points A and B represent the complex numbers i6eD and i6eE respectively, where 0 and 22 SSDE S . (a) Mark the points A and B on an Argand diagram. You are expected to clearly label the relevant moduli and arguments of the complex numbers represented by the points A and B, in terms of D and E . [2] HOT (b) Show that ii6e 6e sin p q DE ED§· ¨¸©¹ , where the integers p and q are to be determined. Hence write down, in terms of D and E , the perimeter of the triangle OAB. [4]

9 4 HOT (a) A sequence 123, , ,...uuu is defined by 1 1u and 3 1 0.5r r ruu r , where 2rt . By considering 1 2 n r r r uu ¦ , find an expression for nu in terms of n. [It is given that 22 3 1 1 4 n r nnr ¦ .] [5] The divergence test states that for a sequence 123, , , . . .aaa , if lim nn ac of , where c is a non- zero constant or lim nn a of does not exist, then the series 1 r r a f ¦ diverges. (b) Determine whether 1 n n u f ¦ diverges, justifying your answer. [2]

11 5 [It is given that the volume of a circular cone with base radius r and height h is 21ʌ3 rh .] A frustum of a cone is the porti on of the cone which remains af ter its upper part has been cut off by a plane parallel to its base. The diagram below shows a vessel of lava lamp. It can be modell ed by two open hollow frustums with a common base radius r cm. To form this vessel, two congruent open cones with base radius r cm are cut to create two frustums which are then joined along their common base. The bottom frustum h as a height of 5 cm. The radii of top and bottom surfaces of the vessel are (3 )r cm and 10 cm respectively. (a) Show that the height, in cm, of each of the original cones is 5 10 r r . [1] I t is given that 12r . (b ) Find the volume of the bottom frustum. [2] The vessel is now mounted on a flat base and mineral oil is pou red into the vessel at a rate of 310 cm per second. (c ) ( i ) F ind the depth of the mineral oil in the top frustrum when the volume of t he mine ral oil in the vessel is 32000 cm . [3] (ii) Hence, using differentiation, find the rate of increase of the depth of the mineral oil at the instant when the volume of the mineral oil in the vessel is 32000 cm . [2] r 10 r – 3 5

14 6 I t is given that ln 1 tan .yx (a) Show that 2 2 2 dd 2t a n s e c .dd yy xxxx [3] ( b ) By further differentiation of the result in part (a), find the Maclaurin series for y, up to and including the term in 3x . [3] (c) Hence find an estimate for the value o