2024 NYJC Prelim Exam H2MATH Paper 1

2024 Nanyang Junior College Preliminary Examination H2 Mathematics Paper 1 1 Find the complex numbers v and w which satisfy the following simultaneous equations. 2 i 2 0vw− + = i3vw= + + Give your answers in the form iab+ , where a and b are real numbers. [4] 2 The path traced by a moving particle is a curve C, given by sinx a a=− and cosy a a =− , where a is a positive constant and 02 π . (a) Show that the gradient of C at a point with parameter  , is 1cot 2  . [2] (b) Find the equation of the tangent to C at the point where 3  = and show that this tangent passes through the point 1 , 23 aa  . [4] 3 (a) Given that ( ) 21sinyx −= , show that ( ) 2 2 2 dd1 2. dd yyxx xx − − = [2] (b) Hence obtain the Maclaurin expansion of y in terms of x, up to and including the term in 4x . [3] (c) By using 1 2x= , find an approximation for  in surd form. Find the percentage error of this approximation and comment on its accuracy. [3] 4 It is given that f ( ) 3cos sinx x x=+ . (a) Write f ( )x as ( )cosRx − , where R and α are constants to be found. [2] (b) Find the exact value of 26 0 1 df ( ) xx       . [2] (c) Find the exact value of 12 0 1 df (2 ) xx   . [3] 5 (a) Find 2 41 d 44 x x xx − ++  , giving your answer in simplest form. [4] (b) Find the value of 1 2 0 41 d 44 x x xx − ++  , expressing your answer in the form p ln q + r, where p, q and r are exact constants in simplest form to be found. [3]

2 NYJC 2024 JC2 Preliminary Examination 9758/01 6 (a) An arithmetic series has first term a and common difference d, where d  0. The 1st, 6th and 14th terms of this series are the 1st, 2nd and 3rd terms of a geometric series. Find d in terms of a. [3] (b) A geometric series has first term b, where b > 0, and common ratio 0.5. (i) Find the sum to infinity of this series in terms of b. [1] (ii) Find the smallest possible value of n for which the sum of the first n terms of the series differs from the sum of the first 2n terms of the series by less than 0.004b. [4] 7 An isosceles triangle ABC, with AB = AC, is inscribed in a fixed circle of radius 1 unit and centre O. It is given that angle BOC = 2, where  is acute. Using calculus, show that the area of the triangle ABC is a maximum when it is equilateral. State the maximum value of this area exactly. [8] 8 A curve C has equation 1 ,y ax b xa= + + − where a and b are positive real constants such that 1a and .xa (a) Sketch C on the axes below stating the equations of any asymptotes and the coordinates of the point where C crosses the y-axis. [4] (b) On the same axes, sketch the graph of ( ) ( ) 22 22 ,x a y a b r− + − − = where r is a positive constant, such that it intersects C at more than 2 points. ( ) 222y r x a a b= − − + + [2] x y

NYJC 2024 JC2 Preliminary Examination 9758/01 (c) By stating the values of