RI C1 Basics Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________________ Additional Practice Questions for Chapter 1: Basics Page 1 of 3 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   ] 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  3 Given that q is a constant, show that 2(2 ) (2 )qx x x q is positive for all real values of x. 4 Express   2 2 12 5 12 1 xx xx   as partial fractions. [ 2 12 12 1 x x x ] 5 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.  153 3 , 1 81 1 3 1 3 2     (a) (b) 6 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  (a)(ii) (b)

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 2 of 3 7 Let f t  1 1 2 e2t . (a) By solving f0t  , 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. 8 Solve the equation 4cos 2 cosec 4 0xx  for 02 x  . 5 or 66    9 Find all the angles between 0 and 2 π exclusive which satisfy the equation 32sin 3cos sinxx x . [ 5,, 33   ] 10 Relative to an origin O, the position vectors of the points A, B, C and D are 61 0 1 2, , and .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) 11 In the above diagram, 6OA a  and 6.OB b  The points C and D lie on OA and BA respectively such that :: 1 : 3OC OA BD BA . (a) Express as simply as possible, in terms of a and/or b, (i) ,BD   (ii) .CD   (b) What type of quadrilateral is OBDC? (c) State the ratio of the area of triangle AOB to area of triangle ACD. [(a)(i) 2 ab (ii) 4b (c) 9:4] O B C D A

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ __________________________________ Additional Practice Questions for Chapter 1: Basics Page 3 of 3 12 Solve the following equations. (a) 2 21x  (b) 23 5xx (c) 21 4 46xx  [(a) 3 or 1x  (b) 2 or 83x (c) 2 or 4x ] 13 2020/Nan Hua High School/4047/02/Q5(i)-(iv) The diagram below shows the graph of 2y ax b  , where a and b are constants. It is given that the y-intercept of the graph is 2 and the maximum value of y is 4. (i) State the value of a. [1] (ii) Find the value of b. [2] (iii) State the value of k for which the equation 2ax b k  has only one solution. [1] (iv) State the range of values of m for which the equation 21ax b m x  has exactly two distinct solutions. [2] [(i) 4 (ii) 3 (iii) 4 (iv) 22 m  ]