RI C1 Basics Add Prac (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________________________________ Additional Practice Questions for Chapter 1: Basics Page 1 of 9 Additional Practice Questions for Chapter 1: Basics 1 Find the set of values of p for which the equation 2 23 1xx p p x   has no real roots. [ 160, 9   ] Solution      2 2 2 2 2 23 1 23 10 For no real roots, 23 4 1 0 91 2 4 4 4 0 91 6 0 91 60 160 9 xx p p x xp x p pp pp p pp pp p                Set of values of p is 160, 9   . 2 Find the range of values of t for which the equation 2xt x has real roots, leaving your answer in surd form. 22 o r 22tt  Solution    2 22 2 20 For 2 0 to have real roots, 4 2 0 2 2 2 2 0 2 2 or 2 xt x xt x xt x t tt tt          2

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 2 of 9 3 Given that q is a constant, show that 2(2 ) (2 )qx x x q is positive for all real values of x. Solution          22 2 2 2 2 2 2 2 22 22 (2 ) (2 ) 4 4 2 2 3 4 Discriminant 3 4 2 4 73 2 32 (Since 0, so 7 0) 0 In addition, coefficient of = 2 0, qx x x q q x qx x qx q x q x qq q qq xq                   Hence 2(2 ) (2 )qx x x q is positive for all real values of x. Alternatively, complete the square to get                 2 22 22 22 2 22 2 22 2 2 2 22 2 2 2 2 2 2 3322 4 22 22 91 6 232 22 42 37 3 22 224 ( 2 73 2 3 0 for all since 2 0 and 2)( 4 2) 23 4 22 2 qqqx q qq qqqqx qq qqqx qq qq qq x q qx x x q qx q x q                                     2 0    4 Express   2 2 12 5 12 1 x x x x   as partial fractions. [ 2 12 12 1 x x x ] Solution             2 22 22 2 2 22 12 5Let 12 112 1 12 5 1 12 15 5When , 1. 24 4 When 0, 1 1 1 0 By comparing coefficients of , 5 1 2 2 12 5 1 2 12 112 1 xx A B x C xxxx xxA x B x C x xA A xC C xB B xx x xxxx                      

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 3 of 9 5 J2021/Cambridge O level Additi onal Mathematics/4037/11/Q8 A curve has equation  223 1yx x   . The x-coordinate of a point A on the curve is 31 23   . (a) Show that the coordinates of A can be written in the form  3, 3pq rs , where p, q, r and s are int