2022 TKSS AMath Prelims P2

2 This document consists of 19 printed pages and 1 blank page. CANDIDATE NAME ADDITIONAL MATHEMATICS 4049/02 Paper 2 Wednesday 24 Aug 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 4049/02/Sec4Prelim2022 [Turn over 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 4049/02/Sec4Prelim2022 [Turn over Answer all questions. 1 Solve the simultaneous equations [4] 3 4 54x y  , 4 3 1 0x y   .

4 4049/02/Sec4Prelim2022 [Turn over 2(a) Express 3 sin 3cos   in the form cos( )R   , where 0 and 0 2R    . [3] (b) Find the minimum value of 1 3 sin 3cos 1 y       and the corresponding value of  when it occurs. [3]

5 4049/02/Sec4Prelim2022 [Turn over 3 (a) Solve the equation 2 2( 3) 28x xe e   . [3] (b) Solve the equation 4 16 4 16log log (log )(log )x x x x  . [5]

6 4049/02/Sec4Prelim2022 [Turn over 4 The diagram shows a circle passing through the points A, B and C. The straight lines SAT and RBT are tangents to the circle at A and B respectively. AB and BC are chords of the circle. ADCS is a parallelogram. (a) Prove that triangle ADC is similar to triangle ACB. [2] S A T D B C R

7 4049/02/Sec4Prelim2022 [Turn over (b) Prove that 2AC CS AB  . [2] (c) Explain why 180CBA CAD ABT     . [2]

8 4049/02/Sec4Prelim2022 [Turn over 5 (a) The equation of a curve is 2 6 2 4y x x   . Find the range of values of x for 0y  . [3] (b) Given that 2 ( 8) 6m x x m   is never negative for all values of x, find the smallest integer value of m. [5]

9 4049/02/Sec4Prelim2022 [Turn over 6 (a) Write down, in descending powers of x, the first three terms in the expansion of 9 2 ax x      , where a is an integer. [2] (b) The coefficient of 3x in the expansion of 9 2 ax x      is 126 , find the value of a. [3] (c) Hence, find the term independent of x in the expansion of 9 2 3 21 axx x         . [3]

10 4049/02/Sec4Prelim2022 [Turn over 7 [The area of a sector is 21 2 r  , where  is the angle, in radians, subtended by the arc at the centre of the circle of radius r.] In the diagram, O is the centre of the circle with diameter BC and radius 10 cm. A is the shaded area enclosed by the chord CD, the diameter BC and the minor arc BD. Angle radiansBCD x . (a) Show that 100 50sin 2A x x  . [3] B D O C x 10 cm 2x 10 cm