2022 TKSS AMath Prelims P1

2 This document consists of 21 printed pages and 3 blank pages. CANDIDATE NAME ADDITIONAL MATHEMATICS 4049/01 Paper 1 Thursday 11 August 2022 2 hours 15 minutes Candidates answer on the Question Paper. No additional materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number on 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, highlighters, glue or correction fluid / tape. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal in the case of angles in degree, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. 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 90. TANJONG KATONG SECONDARY SCHOOL Preliminary Examination 2022 Secondary 4 CLASS INDEX NUMBER

2 4049/1/Sec4Prelim2022 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x = a acbb 2 42  . Binomial Theorem (a + b)n = an +       1 n an  1 b +       2 n an  2 b2 + . . . +       r n an  r br + . . . + bn, where n is a positive integer and       r n = !)!( ! rrn n  = ! )1).......(1( r rnnn  2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ∓ sin A sin B tan (A ± B) = tan tan 1 tan tan A B A B   sin 2A = 2 sin A cos A cos 2A = cos2 A  sin2 A = 2 cos2 A  1 = 1  2 sin2 A tan 2A = A A 2tan1 tan2  Formulae for ABC C c B b A a sin sin sin  a2 = b2 + c2  2bc cos A  = 2 1 bc sin A

3 4049/1/Sec4Prelim2022 [Turn over [ Turn over for Question 1 ]

4 4049/1/Sec4Prelim2022 1 Find the set of values of m, for which the line 2y m x  cuts the curve 2 2 3 1 0x xy y    at two points and represent this set on a number line. [5]

5 4049/1/Sec4Prelim2022 2 A curve has equation cbxaxxy  23 . (i) Find d d y x . [1] (ii) It is given that y is a decreasing function for 31  x . Show that 3a and 9b . [3] [Turn over

6 4049/1/Sec4Prelim2022 3 (a) Given that 2 3100 5 5 2 3 56 2 a b c            , find the value of each of a, b and c. [3] (b) A man invested $50 000 of his savings at the beginning of 2010. The money he invested grew by 5% each year. Find the amount of money he will have in his savings, at the beginning of 2022. [2]

7 4049/1/Sec4Prelim2022 4 Express 2 2 2 1 2 x x x x     in partial fractions. [5] [Turn over

8 4049/1/Sec4Prelim2022 5 The graph of sin 3y p x q  , where p and q are positive, has a maximum value of 3 and passes through the point  ,1 . (i) Show that p = 2 and q = 1. [2] (ii) Sketch the graph of sin 3y p x q  for 0 x   . [2]

9 4049/1/Sec4Prelim2022 (iii) Sketch on the same axes, the graph of 2 1y x   . Explain why there are 5 solutions for 2sin 3 1p x q x    when 0 2 x   . [2] (iv) State the maximum value of the graph of 2( sin 3 )y p x q  . [1] [Turn over

10 4049/1/Sec4Prelim2022 6 Find the coordinates of the stationary points of the curve 2( 2)( 1)y x x   and determine the nature of each stationary point. [6]