2021 NTSS Prelim 4E AM P2 QP v20210803a

This document consists of 16 printed pages. Setter: Mr Alan Cheng NEW TOWN SECONDARY SCHOOL Preliminary Examination Secondary 4 Express NAME CLASS INDEX NUMBER Additional Mathematics Paper 2 4049/02 2 August 2021 09:05 – 11:20 2 hours 15 min READ THESE INSTRUCTIONS FIRST Write your name, register number and class in the spaces provided above and 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 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. For Examiner’s Use

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , 2( 4 ) 2 b b acx a −  −= . Binomial Expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21)( , where n is a positive integer and ( ) ! ( 1) ( 1) ! ! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC A a sin = B b sin = C c sin a2 = b2 + c2 − 2bc cos A  = 2 1 bc sin A

3 1 Express ( )( ) 2 2 2 19 13 xx xx +− −+ as a sum of three partial fractions. [5]

4 2 The table below shows experimental values of two variables x and y. x 2 4 6 8 y 8.48 5.99 4.90 4.24 It is known that x and y are connected by the equation nyx k= , where k and n are constants. (a) Plot ln y against ln x , using a scale of 4 cm for 1 unit on both axes, for the given data and draw a straight line graph on the grid on page 5. [2] (b) Use your graph to estimate the value of n and of k. [3] (c) Use your graph to estimate the value of x when 2ye= . [2]

5 (d) On the same diagram, draw the straight line representing the equation 3yx= and hence find the value of x for which 3 nxk+ = [3]

6 3 The diagram shows part of the curve ( ) 2 9 1 3 y x =− − which intersects the x-axis at the origin and at the point P. The normal at the point Q on the curve cuts the x-axis at R. The gradient of the curve at Q is 18. (a) Find the coordinates of Q. [3] y x Q P R

7 (b) Find the coordinates of R. [3] (c) Find the area of the shaded region. [4]

8 4 (a) The equation of a curve is ( ) ( ) 22 2 4 8y b ac x a c x= − + + − . (i) Show that the roots of the equation are real if a, b and c are real. [3] (ii) State the conditions that the roots are equal. [2] (b) The equation of a curve is 2 43mx x m− + − , where m is a constant. Find the range of values of m for which 2 43mx x m− + − is not negative for all real values of x. [5]

9 5 (a) Show that 31 11 3 1 3 x xx=−−− and hence find 3 13 x dxx− . [4] (b) Given that ( )ln 1 3y x x=− , find an expression for dy dx . [2] (c) Using the results from (a) and (b), find ( ) 0 1 ln 1 3 x dx − − . [4]

10 6 In Figure 1, 3AE= cm and angle BAE x= . Triangle ABE is cut and rotated clockwise about E until AE rests on ED. The resulting shape is shown in Figure 2 below. (a) Show that the perimeter, P cm of the resulting shape BEBADCB is given by 14sin 8cos 2P x x= + + [4] Figure 1 Figure 2 A B C 3 cm 5 cm E D B E B C D A x x