PLMGS(S) 4049 AM Prelim P2 2021 (w final ans)

[Turn over Candidate Name: ___________________________ ( ) Class: _______ Centre Number S Index Number ADDITIONAL MATHEMATICS 4049/02 Paper 2 30 August 2021 Candidates answer on the Question Paper. Additional Materials: Writing Paper (1 sheet) 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, index number and class in the spaces at the top of this page. 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. 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. The use of an approved scientific calculator is expected, where appropriate. 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 90. This document consists of 19 printed pages and 1 blank page. Paya Lebar Methodist Girls’ School (Secondary) Preliminary Examination 2021 Secondary 4 Express For Examiner’s Use 90 Calculator Model: (if applicable)

2 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  Binomial Expansion ,21)( 221 nrrnnnnn bbar nbanbanaba                   where n is a positive integer and     ! 11 )!(! ! r rnnn rnr n r n        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 BA BABA tantan1 tantan)tan(   sin 2A = 2 sin A cos A cos 2A = cos2 A  sin2 A = 2cos2 A  1 = 1  2sin2 A A AA 2tan1 tan22tan  Formulae for ABC C c B b A a sinsinsin  a2 = b2 + c2 – 2bc cos A 1 sin2 bc A

3 [Turn over 1 (i) Differentiate 213 xxe  with respect to x. [2] (ii) Hence find 219 d .xxe x  [3]

4 [Turn over 2 (i) Prove that 2 2 1 tan cos sin sec 2 tan cos sin          . [4]

5 [Turn over (ii) Hence solve the equation 221 tan 2sec 4 tan     for 180 180 .    [4]

6 [Turn over 3 (i) Show that xa is a factor of 3 2 2 32 3 3 2x ax a x a   where a is a constant. [1] (ii) Express 3 2 2 3 3 2 3 3 2x ax a x a   as the sum of three partial fractions. [7]

7 [Turn over 4 (a) Express  332log 2 log 3xx   as a quadratic equation in x and explain why there are no real solutions. [4] (b) Without using a calculator, solve the equation   ln ln 2 2xx    using the substitution lnyx . [3]

8 [Turn over 5 (a) Find the values of x and y which satisfy the equations 4 12 16 xy  , 1 351 25 5 x y    . [4]

9 [Turn over (b) A trapezium of area  2 15 3 cm2 has a perpendicular height of  2 4 3 cm and the length of one of the parallel sides of  23 cm. Without using a calculator, obtain an expression for the length of the other parallel side in the form  3ab , where a and b are integers. [5]

10 [Turn over 6 The diagram shows two circles C1 and C2. The equation of circle C1, with centre B, is 22 8 84 0x y y    . The tangent to circle C1 at the point A has a gradient of 4 3 . Circle C2 has diameter AB. (i) Find the radius of circle C1 and the coordinates of its center. [2] x y O C1 C2 A B