AJC H2 MATHS P1

© AJC Prelim 9758/ 01/ 2017 Page 1 of 5 ANDERSON JUNIOR COLLEGE 2017 Preliminary Examination H2 Mathematics Paper 1 (9758/01) Duration: 3 hours 1 Mr Tan invested a total of $25,000 in a st ructured deposit account, bonds and an estate fund. He invested $7,000 more in bond s than in estate fund. The projected annual interest rates for structured de posit account, bonds and estate fund are 2%, 3% and 4.5% respectivel y. Money that is not drawn out at the end of the year will be re-invested for the following year. Mr Tan plans to draw out hi s money from all investments at the end of the second year and estimates that he will receive a total of $26,300. Find the amount of money Mr Tan invested in each investment, givi ng your answer to the nearest dollar. [5] 2 Show that the differential equation 2 d3 10d1 3 yx y xxx  may be reduced by means of the substitution 213yu x to 2 d1 d 13 ux x x   . Hence find the general solution for y in terms of x. [5] 3 The diagram above shows a quadrilateral ABCD, where AB = 2, 2BC , angle 4ABC   radians and angle CAD  radians. Show that 64 c o s 4 s i nAC     . [2] Given that  is small enough for  3 and higher powers of  to be neglected, show that 2 ,AD a b c    where a, b and c are constants to be determined. [5] C A B D  2

© AJC Prelim 9758/ 01/ 2017 Page 2 of 5 4 (a) Given that  2 1 11 1 41 2 2 21 N n nN  , find  2 2 1 1 41 1 N n n   . Deduce that  2 2 1 1 23 N n n  is less than 1 6 . [5] (b) The sum to n terms of a series is given by 2 1ln 2 e n nSn  . Find an expression for the nth term of the series, in terms of n. Show that the terms of the series follow an arithmetic progression. [4] 5 A curve C has equation y = f( x). The equation of the tangent to the curve C at the point where x = 0 is given by 2x  ay = 3 where a is a positive constant. It is also given that y = f( x) satisfies the equation  2 2 dd12 0 dd yyxy xx  and that the third term in the Maclaurin’s expansion of f(x) is 21 .3 x Find the value of a. Hence, find the Maclaurin’s series for f( x) in ascending powers of x, up to and including the term in x3. [7] 6 The diagram below shows the line l that passes through the origin and makes an angle  with the positive real axis, where 0 2  . Point P represents the complex number 1z where 10a r g z  and length of OP is r units. Point P is reflected in line l to produce point Q, which represents the complex number 2z . Prove that arg 1z + arg 2z = 2. [2] Deduce that  2 12 cos 2 sin 2zz r i   . [1] Let R be the point that represents the complex number 12zz . Given that 4  , write down the cartesian equation of the locus of R as 1z varies. [2]   l P Q  O y x

© AJC Prelim 9758/ 01/ 2017 Page 3 of 5 7 Figure 1 shows a solid metal hexagonal prism of height h cm. Figure 2 shows the hexagonal cross-section ABCDEF of the prism where AD = 3x cm, BC = FE = x cm and the remaining 4 sides are of length kx cm each, where k is a constant. Show that  2281 2 1 2Sx k x h k   , [3] where S is the surface area of this solid hexagonal prism. (a) If the volume of the prism is fixed at 400 cm 3, use differentiation to find, in terms of k, the exact value of x that gives a stationary value of S. [3] Let k = 2. (b) The prism is heated and it expands in such a way that, at time t seconds, the rate of increase of x is the same as the rate of increase of its height h. At the instant when x = 3, the prism’s height is 8 cm and its surface area is increasing at a constant rate of 0.5 cm 2/s. Find the rate of change of the volume of the prism at this instant. [6] 8 The curve C has equation 242 2 xk xy x    , where k is a constant. (i) Show that curve C has stationary points when k < 9. [3] (ii) Sketch the graph of C for the case where 69 k  , clearly indicating any asymptotes and points of intersection with the axes. [4] (iii) Describe a sequence of transformati ons which transforms the graph of 12yx x to the graph of 248 2 2 xxy x    . [3] (iv) By drawing a suitable graph on the same diagram as the graph of C, solve the inequality 2 2 48 2 1 2 xx x x   . [3] x  kxkx  h kx  x  kx Fig 1  3x  x  kx kx kx kx  x  A F E D B C Fig 2

© AJC Prelim 9758/ 01/ 2017 Page 4 of 5 9 The position vectors of A, B and C with respect to the origin O are a, b and c respectively. It is given that 4ACC B   and 222 ab a b . (i) By considering   ab . ab , show that a and b are perpendicular. [2] (ii) Find the length of the projection of c on a in terms of a . [3] (iii) Given that F is the foot of the perpendicular from C to OA and f denotes the position vector OF  , state the geometrical meaning of cf . [1] (iv) Two points X and Y move along line segments OA and AB respectively such that 1(cos3 ) (sin 3 ) , 2 (sin ) (cos ) 2 , OX t t OY t t     ij k ij k where t is a real parameter, 02 πt  . By expressing the scalar product of OX  and OY  in the form of  sinpq t r  where p, q and r are real values to be determined, find the greatest value of the angle XOY. [5] 10 There are 25 toll stations, represented by T 1, T 2, T 3,……, T 25 along a 2000 km stretch of highway. T 1 is located at the start of the highway and T 2 is located x km from T1. Subsequently, the distance between tw o consecutive toll stations is 2 km more than the previous distance. Find the range of values x can take. [3] Use 60x for the rest of this question. Each toll station charges a fee based on the distance travelled from the previous toll station. The fee structure at each toll station is as follows: For the first 60 km, the fee per km will be 5 cents. For every additional 2 km, the fee per km will be 2% less than the previous fee per km. (i) Find, in terms of n, the amount of fees a driver will need to pay at Tn . [3] (ii) Find the total amount of fees a driver w ill need to pay, if he drives from T 1 to Tn . Leave your answer in terms of n. [4] More toll stations are built along the highway in the same manner, represented by T 26, T27, T28,…….. beyond the 2000 km stretch. (iii) If a driver starts driving from T1 and only has $200, at which toll station will he not have sufficient money for the fees? [2] 11 (i) Show by integration that

© AJC Prelim 9758/ 01/ 2017 Page 5 of 5 22 2 21es i n d es i n ec o s55 xx x x xx x A   where A is an arbitrary constant. [3] The diagram below shows a sketch of curve C, with parametric equations e tx  , es i ntyt  , t   . Point P lies on C where 2t  . (ii) Find the equation of the normal at P. [3] (iii) Find the exact area bounded by the curve C for 0 t   , the line 1x and the normal at P. [5] (iv) The normal at P cuts the curve C again at two points where tq and tr . Find the values of q and r. [3] End of paper y x  P 1