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 integers. (b) Find the x-coordinate of the stationary point on the curve, giving your answers in the form 3ab , where a and b are rational numbers.  1(a) 5 3 3,18 11 3 (b) 1 3 2     Solution 31 312 3 23 23 23 23 3 2 3 43 53 3       (a) Substitute 53 3x into  223 1yx x   :          2 23 5 3 3 5 3 3 1 2 3 25 27 30 3 4 3 3 2 3 52 30 3 4 3 3 104 90 3 60 52 4 3 3 18 11 3 y                   Coordinates of A are 5 3 3,18 11 3    223 1 d 2 2 3 1d yx x y xx       (b)   d 0d 12 3 2 3 2 3 1 1324 3 2 22322 3 y x x          The x-coordinate of the stationary point is 113 2

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 4 of 9 6 J2021/Cambridge O level Additional Mathematics/4037/21/Q8 In this question, a, b, c and d are positive constants. (a) (i) It is given that    log 3 log 2 1aayx x   . Explain why x must be greater than 1 .2 (ii) Find the exact solution of the equation  log 6 2.log 3 a a y  (b) Write the expression   log 9 log log 9aa bba in the form log 9 acd , where c and d are integers. 36 2 3 l o g 9 a  (ai) (b) Solution (ai)    log 3 log 2 1aayx x   For above logarithm functions to be defined, 30x and 21 0x 13 and 2xx   1 2x  (aii)  log 6 2log 3 a a y   log 6 2 log 3aa y   2 log 6 log 3 0aa y    2 6log 0 3 a y     0 2 2 6 1 3 36 36 a y y y         Since 30y for logarithm function to be defined, 36y       log 9log 9 log log 9 log 9 log log log 9 loglog 9 log 1 log2 log 9 2 log 9 1 23 l o g 9 a aa aa b a aa aa a aa a aba b b ab b          (b) logNote that log log , where , , 0log although log log log a aa a aa a c cb a b cb c cbb   

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 5 of 9 7 Let  21f1 e 2 tt  . (a) By solving f0 t  , show that 0.347t  , correct to 3 significant figures. (b) Sketch the graph of fyt for 1t , showing the equations of any asymptotes and the coordinates of any intersections with the axes. Solution (a) 2 2 11e 02 e2 2l n 2 ln 2 2 0.34657 0.347 (to 3 s.f) (shown) t t t t t         (b) 8 Solve the equation 4cos 2 cos 4 0xe c x  for 02 x  . 5 or 66   Solution 4c os2 co s 4 0x ecx  2 141 2 s i n 4 0 sinx x    38sin 1 0x  3 1sin 8 15sin or 26 6 x xx    y = 1 t (–1, –2.69) (–0.347 , 0 ) (0, 0.5)

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 6 of 9 9 Find all the angles between 0 and 2 π exclusive which satisfy the equation 32sin 3cos sinx xx . [ 5,, 33   ] Solution 3 3 2 2 2 2sin 3cos sin 2sin 3cos sin 0 sin (2sin 3cos ) 0 sin 0 or 2(1 cos ) 3cos 0 or 2cos 3cos 2 0 xx x xx x xx x xx x xx x          or (2cos 1)(cos 2) 0 1 cos or cos 2 (reject since cos 1) 2 5 , 33 5Thus , , 33 xx xxx x x          10 J2021/Cambridge O level Additi onal Mathematics/4037/21/Q10 Relative to an origin O, the position vectors of the points A, B, C and D are 61 0 1 2,, a n d .53 7 xOA OB OC OD y                     (a) Find the unit vector in the direction of .AB  (b) The point A is the mid-point of BC. Find the value of x and of y. (c) The point E lies on OD such that :1 : 1 .OE OD  Find the value of  such that BE  is parallel to the x-axis. 114 2, 13 2 35 xy     (a) (b) (c) Solution 41 482AB      (a) Unit vector in the direction of AB  is 11 25   (b) Point A is the mid-point of BC BAA C     164 25 x y       2 13 2, 13 x y xy        

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________