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 of e. [2]

16 7 (a) (i) Find 2 2 d 21 x x x³ , using the substitution cosșx where ʌ0 ș 2 . [5] (ii) Find 1sin dxx x ³ . [2] (b ) Find cos sin dax bx x³ , where a and b are real numbers and abz . [3]

18 8 A curve C has p arametric equations 32 s e c ,x T 3tan 1 ,y T .22 SS T (a) Find the range of values of x. [2] (b ) Find a cartesian equation of C. [2] ( c ) Hence or otherwise, sketch C, stating the coordinates of any vertices and equations of any asymptotes. [3] (d) (i) F ind the equation of the tangent to C at the point P where pT , .22 pSS [You do not need to simplify your answer.]. [3] (ii) Show that the gradient of the tangent at P cannot lie in the interval 33,22 ªº«»¬¼ as p vari es. [2]

22 9 (a) The diagram shows part of the graph of 2 2yx from 1x to 2x . The area under the curve in this interval may be approximated by the total area of n rectangles, A , as shown. The width of each rectangle is 1 n . Given that 2 1 12 16 n r nrnn ¦ , show that 2 13 3 1 32 6A nn . Deduce the exact value of 2 2 1 2d xx³ , justifying your answer. [6] ( b ) Region R is bounded by the curve y = ln x, x = 1 and y = ln 5. (i ) F ind the area of R. [2] (ii) F ind the exact volume of the solid generated when R is rotated through 4 right angles about the y-axis. [4] 1 2 x y … 2 O

26 10 Glucose is a simple carbohydrate th at can be easily absorbed by the body and provides instant energy. When a patient is dehydrated or unable to take food orally, glucose is given intravenously to the patient via a glucose drip. Glucose given intravenously enters the bloodstream at a constant rate of p units per hour. It is absorbed by the body, leaving the bloodstream, at a rate propor tional to the amount of glucose present in the bloodstream. G denotes the number of units of glucose in the bloodstream at time t hours after the glucose drip is administered. (a) Write down a differential equation relating G and t. [1] ( b ) Suppose there are 3 units of gluc ose in the bloodstream when t = 0 a n d t h a t t h e amount of glucose in the bloodstream remains constant when 8G . Show that the particular solutio n of the differe ntial equation in part (a) i s 885 e pt G

. [6] (c) Sketch the graph of G against t. [ 2] Instead of the glucose drip, a patient who is well enough will get his glucose from the food he consumes. At the end of each meal, the rate of change in glucose in the bloodstream can be modelled as 1 sin2 t§· ¨¸©¹ units per hour where 30 2t Sdd . (d) Find the time at which the amount of glucose in the bloodstream (i ) incre ases most rapidly after the meal, [2] ( ii) s t a r t s t o d e c r e a s e . [2]