2025+Y5+H2+Math+Promo+ 28Qn 29

_________________________________________________________________________ This document consists of 8 printed pages. RAFFLES INSTITUTION  RI2025 Mathematics Department [Turn over RAFFLES INSTITUTION 2025 YEAR 5 PROMOTION EXAMINATION Higher 2 MATHEMATICS 9758 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question.

2 H2 MA 9758/2025 RI Year 5 Promotion Examination 1 Fig. 1 A tablet is dissolving in water and is modelled as a cylinder, as shown in Fig. 1. At t seconds after being dropped into the water, the radius of the tablet is x mm, and its thickness is 1 3 x mm. The circular cross-sectional area of the tablet is decreasing at a constant rate of 0.5 mm² per second. When x = 5, find the exact value of (a) d d x t , [2] (b) the rate of change of the volume of the tablet. [2] 2 Do not use a calculator in answering this question. Showing your working, find the complex numbers z and w which satisfy the simultaneous equations ( ) * 2i 5 7i, 4 i 3i. zw wz +=+ −= − [5]

3 H2 MA 9758/2025 RI Year 5 Promotion Examination [Turn over 3 The point A on the Argand diagram below represents the complex number 1w with modulus r and argument θ. It is given that the complex numbers 2w and 3w satisfy the equations 21ww=− and 32 iww= . Let B, C and D represent the complex numbers 2w , 3w and 23ww+ respectively. (a) On the copy of the Argand diagram in the Printed Answer Book, plot the points B, C and D, indicating clearly the modulus and argument of 2w and 3w . [3] (b) State, in radians, the angle BOC. [1] (c) By considering the quadrilateral OBDC , find 23ww+ in terms of r and 23arg ( )ww+ in terms of θ. [2] Im Re r

4 H2 MA 9758/2025 RI Year 5 Promotion Examination 4 (a) The diagram shows the curve f( )yx= , with turning points at (3, 2.5)A and (0, 1)B . The lines 1y= and 2x=− are asymptotes of the curve. On separate diagrams, sketch the following graphs, clearly stating the equations of the asymptotes and the coordinates of the points corresponding to A and B where appropriate. (i) f ( 2) 2yx= −+ [2] (ii) ( )fyx ′= [3] (b) The parametric equations of a curve are given by 23 2 , 3 ., for 0xt t yt t t= −= − > The curve is reflected in the y-axis. Find the exact coordinates of the point(s) where the reflected curve intersects the x-axis. [2] y O

5 H2 MA 9758/2025 RI Year 5 Promotion Examination [Turn over 5 In the triangle ABC, AB = 2, BC = 3 , AC x= and angle ABC = 6 π θ+ radians (see diagram). (a) Show that 2 7 6cos 2 3 sinx θθ= −+ . [2] (b) Show that 22 2 dd 3cos 3 sindd xxx θθθθ +=− . [2] (c) Use the result from part (b) to find the Maclaurin expansion for x, up to and including the term in 3θ . Give the coefficients as exact values in their simplest form. [3] (d) Using the result from part (c), deduce the approximate value of x when angle ABC is 35 , giving your answer correct to 6 decimal places. [2] 6 (a) The non -zero vectors a, b, and c are such that 32×= ×ab bc , where 32≠−ac . Find a linear relationship between a, b, and c. [3] (b) The variable position vector v satisfies the equation ( )2 a×−+ + = ++v i jk i jk , where a is a non-zero constant. (i) Show that 3a= . [1] (ii) Hence, by finding the set of position vectors v, describe geometrically the set of points represented by the position vectors v. [3] 3

6 H2 MA 9758/2025 RI Year 5 Promotion Examination 7 The curve 1C has equation 2 3 2 ax bxy x ++= + , where a and b are constants. It is given that 1C has an asymptote 3yx= − . (a) State the value of a and show that 1b=− . [3] (b) Using an algebraic method, find the set of values that y cannot take. [3] The locus of a point is defined as the path traced out by that point as it moves. (c) Let 1(, )Pxy be a point on the curve 1C and 2(, )Qxy be a point on the curve 2C with equation 23 2 xxy x −= + . Find the cartesian equation of the locus of the midpoint of P and Q as x varies. [2] 8 The function g is defined by ( )g : ln 3xx − , for x∈  , 3x> . (a) Explain how you know 1g− exists. [2] (b) Find 1g () x− and write down its domain. [3] (c) Solve g( ) 0x = exactly. [1] (d) Hence, without using a calculator, solve the inequality ( )( ) g( ) 0.31 x xx ≥−+ [3] 9 (a) It is given that ( )( ) ( )( )1 2 11 1 22 1 2 n r rr r n n= = −++ ++∑ . (i) State the sum to infinity of the series. [1] (ii) Find the sum of the first n terms of the series 222 17 18 19 18 19 20 19 20 21+++×× ×× ×× . [2] (iii) Find ( ) 2 2 1 1 n r rr= −∑ . [3] (b) The sequence 123,,,uuu  is defined by 1 3u = , 1 42 ,1n n un u + = −≥ . (i) Find the values of 2u , 3u , 4u and 100u . [2] (ii) Hence find 32 1 n r r u − = ∑ in terms of n , simplifying your answer. [2]

7 H2 MA 9758/2025 RI Year 5 Promotion Examination [Turn over 10 With reference to the origin O, the point A has position vector 3 3.−− +i jk The plane π and the line l have equations 4 12 13 3 12 4 λµ −         = ++         −     r and 23 1 12 tβ      = +     −   r , respectively, where , andλµ β are parameters and t is a real value. (a) It is given that l and π intersect at a point. (i) Find the range of values of t. [2] (ii) Find, in terms of t, the coordinates of the point of intersection, B, of l and π. [2] (b) It is given that A lies on l. (i) Show that 4t = . [1] (ii) Find the position vector of the foot of perpendicular, F, of A onto π. [3] (iii) Find the cartesian equation of another line m through A, that lies on the plane containing A, B and F, and m makes the same angle with π as l makes with π. [4] 11 An arithmetic series has positive first term a and common difference .d The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing. (a) Show that 7 16da=− . [3] (b) Give a reason why the geometric series converges. [2] It is given that the sum to infinity of the geometric series is 32 7 . (c) Find the sum of the first twenty terms of the arithmetic series. [2] (d) Given that the sum of all the terms after the nth term of the geometric series is greater