Hua Yi Sec Prelim 2024 A Math Paper 2_MS

[Turn Over HUA YI SECONDARY SCHOOL PRELIMINARY EXAM 2024 4-G3 / 5-G2 NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 PAPER 2 (XX) 26 August 2024 2 hour 15 minutes Candidates answer on the Question Paper No Additional Materials is required. ANSWER SCHEME Setter : Mdm Suzanne Lye

1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = -b ± b2 - 4ac 2a Binomial expansion a + b( ) n = an + n 1 æ èç ö ø÷ an-1b + n 2 æ èç ö ø÷ an-2b2 + ...+ n r æ èç ö ø÷ an-rbr + ...+ bn , where n is a positive integer and n r æ èç ö ø÷ = n! r!(n - r)! = n(n -1)...(n - r +1) r! 2. TRIGONOMETRY Identities sin2 A+ cos2 A = 1 sec2 A = 1+ tan2 A cosec2 A = 1+ cot2 A sin( A ± B) = sin Acos B ± cos Asin B cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin2A = 2sin Acos A cos2A = cos2 A- sin2 A = 2cos2 A-1= 1- 2sin2 A tan 2 A = 2 tan A 1- tan2 A Formulae for DABC a sin A = b sin B = c sinC a2 = b2 + c2 - 2bccos A Cabsin2 1=

3 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 2 1 Show that 𝑥 = 1 2 is a solution of the equation 2𝑥3 + 𝑥2 − 3𝑥 + 1 = 0 and hence solve the equation completely. [5] 𝑓(𝑥) = 2𝑥3 + 𝑥2 − 3𝑥 + 1 𝑓 ( 1 2) = 2 ( 1 2) 3 + ( 1 2) 2 − 3 ( 1 2) + 1 ---------------------M1 = 0 𝑓(𝑥) = (2𝑥 − 1)(𝑥2 + 𝑏𝑥 − 1) ------------------------M1 Compare coeff. of 𝑥2, 1 = 2𝑏 − 1 b= 1 ---------------------------M1 𝑓(𝑥) = (2𝑥 − 1)(𝑥2 + 𝑥 − 1)-------------------------M1 𝑥 = 1 2, −1+√5 2 , −1−√5 2 --------------------------------------A1 Alternative Method : Using long division to find (𝑥2 + 𝑥 − 1).

4 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 2 2 (a) By considering the general term in the binomial expansion of (𝑝𝑥 + 1 𝑥3) 9 , where p is a constant, explain why there are no even powers of x in this expansion. [3] General Term =(9 𝑟) (𝑝𝑥)9−𝑟 ( 1 𝑥3) 𝑟 -----------------M1 = (9 𝑟) 𝑝9−𝑟𝑥9−4𝑟 -------------------M1 Since 9 − 4𝑟 = 1 + 4(2 − 𝑟), one added to any even number will give an odd value, hence there are no even powers of x in this expansion. --------A1

5 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 2 (b) Given that the coefficient of 𝑥8 is equal to the coefficient of 𝑥 in the expansion of (2𝑥3 + 1) (𝑝𝑥 + 1 𝑥3) 9 , find the value of p. [4] (2𝑥3 + 1) (… … (9 1) (𝑝𝑥)8 ( 1 𝑥3) 1 + (9 2) (𝑝𝑥)7 ( 1 𝑥3) 2 ) 9 ---------------------M1 = (2𝑥3 + 1) (… + 9𝑝8𝑥5 + 36𝑝7𝑥 + ⋯ ) -----------------------------------M1 2(9𝑝8) = 36𝑝7 -------------------------------------------------M1 𝑝 = 2 ------------------------------------------------------------A1 (c) Using the value of p in (b), find the term independent of x in the expansion of (2𝑥3 + 1) (𝑝𝑥 + 1 𝑥