RI H2 Math Promo Paper 2023 (Qn)

 RI2023   [Turn over                 CANDIDATE NAME   CLASS 24     MATHEMATICS 9758 3 hours Candidates answer on the Question Paper. Note that the list of formulae can be found on page 2. READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. You may use the blank page on page 24 if necessary and you are reminded to indicate the question number(s) clearly. 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 are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. For examiner’s use only Q1 Q2 Q3 Q4 Q5 Q6 / 4 / 8 / 8 / 7 / 8 / 8 Q7 Q8 Q9 Q10 Q11 TOTAL / 12 / 9 / 12 / 12 / 12 / 100     This document consists of 23 printed pages and 1 blank page.   RAFFLES INSTITUTION Mathematics Department  RAFFLES INSTITUTION 2023 YEAR 5 PROMOTION EXAMINATION

 RI2023   [Turn over    PURE MATHEMATICS Algebraic series Binomial expansion: nnnnnn bbanbanbanaba         33221 3 2 1)( where n is a positive integer and )!(! ! rnr n r n    Maclaurin expansion:   )0(f!)0(f!2)0(f)0f()f( )( 2 n n n xxxx   rn xr rnnnxnnnxx ! )1()1( !2 )1(1)1( 2  1 x   !!3!21e 32 r xxxx r x (all x)     )!12( )1( !5!3sin 1253 r xxxxx rr (all x)   )!2( )1( !4!21cos 242 r xxxx rr (all x)    r xxxxx rr 132 )1( 32)1ln( ( 11  x ) Partial fractions decomposition Non-repeated linear factors: () () () () px q A B ax b cx d ax b cx d     Repeated linear factors: 2 22() () () () () px qx r A B C ax b cx d ax b cx d cx d      Non-repeated quadratic factor: 2 22 22() ( ) () ( ) pxq x r A B x C ax b x c ax b x c        Trigonometry BABABA sincoscossin)sin(  BABABA sinsincoscos)cos(  BA BABA tantan1 tantan)tan(   sin 2 2sin cosAA A 22 2 2cos 2 cos sin 2 cos 1 1 2 sinA AA A A    2 2t a ntan 2 1t a n AA A  )(cos)(sin2sinsin 2 1 2 1 QPQPQP  )(sin)(cos2sinsin 2 1 2 1 QPQPQP  )(cos)(cos2coscos 2 1 2 1 QPQPQP  )(sin)(sin2coscos 2 1 2 1 QPQPQP  Principal values:   2 11 2 1 sin   x  1 x   x1cos0  1 x   2 11 2 1 tan   x Derivatives: f( )x f( )x 1sin x 2 1 1 x 1cos x 2 1 1 x   1tan x 2 1 1 x cosec x cosec cotx x secx sec tanx x Integrals (Arbitrary constants are omitted; a denotes a positive constant.) )f(x xx d )f( 22 1 ax     a x a 1tan1 22 1 xa     a x1sin  ax  22 1 ax       ax ax a ln2 1 )( ax 22 1 xa       xa xa a ln2 1  ax  tan x ln(sec )x  2 1 x cot x ln(sin )x 0 x  cosec x ln(cosec cot )x x 0 x  secx ln(sec tan )x x  2 1 x Vectors The point dividing AB in the ratio  : has position vector       ba Vector product: 11 2 3 3 2 22 3 1 1 3 33 1 2 2 1 ab a b a b ab a b a b ab a b a b                 ab       2

3     H2 MA 9758/2023 RI Year 5 Promotion Examination [Turn over    1 The first 3 terms of a sequence are given by 1 1823u  , 2 200u  and 3 2023.u  Given that nu is a quadratic polynomial in n, find nu in terms of n. [4]

4     H2 MA 9758/2023 RI Year 5 Promotion Examination  2 Do not use a calculator in answering this question. Solve the inequality 2 565 31 xxx x   . [4]

5     H2 MA 9758/2023 RI Year 5 Promotion Examination [Turn over    Hence solve (a) 2 42 2 565 31 xxx x   , [2]                     (b) 2 5l n(ln ) 6 ln 5 13 l n xxx x   . [2]

6     H2 MA 9758/2023 RI Year 5 Promotion Examination  3 Vectors a and b are such that the magnitude of  a is 2 and b is a unit vector perpendicular to a. (a) Find the area of the parallelogram with adjacent sides formed by the vectors 3ab  and 54ab . [3]                 A vector c is such that 21 bc ab . (b) Show that 21cb a , where  is a constant. [2]

7     H2 MA 9758/2023 RI Year 5 Promotion Examination [Turn over    (c) Give the geometrical meaning of b.c and find the possible values of  if 5b.c . [3]

8     H2 MA 9758/2023 RI Year 5 Promotion Examination  4 (a) Write 1 (1 )rr  in partial fractions. [1]           (b) Using your answer to part (a), find 2 1 1n r rr  . [2]

9     H2 MA 9758/2023 RI Year 5 Promotion Examination [Turn over    Hence find (i) 2 2 1 1n rn rr  , [2]                     (ii) 111 23 34 45  [2]

10     H2 MA 9758/2023 RI Year 5 Promotion Examination  5 In this question you may use expansions from the List of Formulae (MF26). (a) Find the first four non-zero terms of the Maclaurin series of e1 c o s 3x x  , in ascending powers of x. [4]