RI C1 Basics Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _______________ Tutorial 1: Basics Page 1 of 4 Tutorial 1: Basics Section A (Basic Questions) 1 Find the range of values of k for the line 2yx k to intersect the curve 2228yx at two distinct points. [  ] 2 Sketch the graph of ln 2yx for 2x , showing clearly the equations of any asymptotes and the coordinates of any intersec tions with the axes. By drawing a suitable straight line graph on the same diagram, dete rmine the number of solutions to the equation 21e2x x  . [2] 3 Express 22 s i n 22 c o s in the form sin( )R  , where 0R and 0. 2  [4sin ] 4   4 Given that cos Ap and 270 360A , express each of the following in terms of p. (a) sin A , (b) sin 2A , (c) cos .2 A [(a) 21 p (b) 221pp (c) 1 2 p ] 5 ABCD is a parallelogram with 3 4AB   . Coordinates of points A and C are (1,1) and 8,8 respectively. (i) Find the coordinates of point D. (ii) AC and BD intersect at the point E. Find the coordinates of E. (iii) Find the length of the two sides of the parallelogram AB and BC. Deduce the geometrical relationship between A, B, C and D, and state the relationship between the two diagonals. [(i) (5,4) (ii) (4.5,4.5) (iii) 5] 6 Solve the equation 23 5x using 2 different methods, namely: (i) numerically, (ii) graphically. [ 4 or 1x ]

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ ______________ Tutorial 1: Basics Page 2 of 4 7 Given that 4x  , find the possible values of 13 x . [11, 13] 8 Draw the graphs of 1yx and 12 2yx on the same diagram. Hence, solve the equation 112 2x x . [ 2 or 63x ] Section B (Discussion Questions) 1 By expressing 2cos cosx xa  in the form  2 cosbx c  , where ,,abc   , find the range of values of a such that 2cos cosx xa  is always negative for all real values of x. 1, 4   2 The line with equation ym x is a tangent to the curve with equation    22 8 14 52.xy   (a) Show that m satisfies the equation 235 6 3 6 0mm  . A and B are points on the curve. The tangent at A and the tangent at B intersect at the origin. (b) Find the coordinates of A and B. [(b)  12,8 and 0.8,14.4 ] 3 (i) Determine constants A, B, C and D such that     32 22 461 12 112 1 1 xx B C D A xxx xx      . (ii) Hence find the exact value of    324 22 461 d 12 1 xx x xx   . [(i) 422, , , 1 33AB C D    (ii) 25 5 8ln37 1 5 ] 4 On a certain date, 80 cases of smallpox was recorded in a small city. This number increased with time and after t days the number of recorded cases was N. It is believed that N can be modelled by the formula e 19 e kt kt aN   , where a and k are real constants. (i) Given that the number of cases of smallp ox was recorded as 1296 after 4 days, find the value of k in the form ln b, where b is a real number. (ii) Find an expression for t in terms of N and hence estimate the number of days it takes for the number of cases of smallpox in the small city to reach 1482, giving your answer to the nearest integer. [(i) k=ln3 (ii) 11 9lnln 3 1600 Nt N  ; 5]

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ ______________ Tutorial 1: Basics Page 3 of 4 c a b A C B 5 Do not use a calculator in answering this question. In triangle ABC, the ratio of the sides are such that a : b : c = 2 : 3 : 4. (i) Find the ratio of sin A : sin B : sin C. (ii) Calculate the value of cos C and hence find the value of sin C. (iii) Given that the area of the triangle is 540 cm2, find the length of the shortest side and deduce that the perpendicular distance of A to BC is 135 2 cm. [(i) 2:3:4 (ii) 11 5,44 (iii) 42 cm] 6 Find an expression for 5tan 12  in the form ab c , where , a n d ab c   . [ 23 ] 7 Find all possible exact values of cos cos (2 1)36 2sin 4 n         , where n is a positive integer. 26 2 26[, ,] 444 

Raffles Institution H2 Mathematics 2025 Year 5 _______________________________________________________________________________________________ ______________ Tutorial 1: Basics Page 4 of 4 8 In triangle PQR, the point N on PR is such that PN = 2 3 PR . M is the mid-point of PQ, L is the mid-point of MN and PL produced meets RQ at K. 43RK KQ     , 7 12PL PK    , 2PN  a   and 2PM  b  . (a) Express, as simply as possible, in terms of a and /or b (i) NM  , (ii) NL  , (iii) PK   , (iv) PR  , (v) PQ   . (b) Express RQ   as simply as possible, in terms of a and b. (c) Calculate the value of KR QR . (d) Show that 3 347KR  ab  . (e) Calculate the value of (i) the area of the area of PKR PQR   , (ii) the area of the area of PKN PQR   . [(a)(i) 22ab (ii) ab (iii) 12 ()7 ab (iv) 3a (v) 4b (b) 34ab (c) 3 7 (e)(i) 3 7 (ii) 2 7 ] 9 The diagram shows part of the graph of 72 .yx The line 2,ym x where m is a constant, cuts 72yx at two distinct points. Explain why 4 2.7 m 2b L K N P Q R 2a M