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