NBSS Prelims 2021 - 4E Add Maths Paper 1

Name ( ) Class ADDITIONAL MATHEMATICS Paper 1 4049/01 26 August 2021 Candidates answer on the Question Paper. 2 hour 15 minutes No Additional Materials are required READ THESE INSTRUCTIONS FIRST Write your name, class and index number 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, highlighters, glue or correction fluid. Answer all 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. 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. Item For examiner’s use Presentation Accuracy Units Total Parent’s Signature ________________________________________________________________________ This paper consists of 20 printed pages. NAVAL BASE SECONDARY SCHOOL PRELIMINARY EXAMINATION, 2021 O [Turn over

2 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 ! )1(...)1( )!(! ! r rnnn rnr n r n       2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin    BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(      A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin    Formulae for ABC 2 2 2 sin sin sin 2 cos 1 sin2 a b c A B C a b c bc A bc A      [Turn over

3 Answer all the questions. 1 The diagram shows an isosceles triangle DEF with vertices at D (8, 11), E (p, 12) and F (0, 2). (i) Find the value of p. [3] (ii) Find the equation of the perpendicular bisector of EF. [2] [Turn over y x O F (0, 2) E (p, 12) D (8, 11)

4 2 (i) On the same axes, sketch the graphs of 3cos2yx and 11 sin 2yx for 0  x  . [4] (ii) Hence, state the number of solutions for the equation 13cos2 sin 1 2xx for –2  x  2 . [1] [Turn over

5 3 The coefficient of x in the expansion of 4(1 ) ax is 12 . (i) Show thar a = 3. [3] (ii) Hence, find the coefficient of 2x in the expansion of 4(2 1)(1 )x ax . [3] [Turn over

6 4 (i) The function f is defined by 2 2 3f ( ) 5 xx x   , x > 0. Explain, with working, whether f is an increasing or a decreasing function. [4] (ii) A point K moves along the curve 2 2 3 5 xy x   in such a way that the y-coordinate of K is increasing at a rate of of 0.2 units per second. Find the rate of increase of the x-coordinate of K when x = 1. [2] [Turn over

7 5 The population of Singapore increased from 3.75 million at the beginning of 2000 to 5.79 million at the beginning of 2020. Suppose that the population, P million, is modelled by the formula, P = 3.75ekt, where k is a constant and t is the number of years starting from the beginning of 2000. (i) Find the estimated population of Singapore at the beginning of 2030, giving your answer correct to 3 significant figures. [4] (ii) Find the year in which the population of Singapore will exceed 10 million. [2] [Turn over

8 6 At any point (x, y) on a curve,   32 2 2 18 d d   xx y . The gradient of the curve at (5, 20) is 2. (i) Find the equation of the curve. [4] (ii) Find the equation of the normal to the curve at (5, 20). [2] [Turn over

9 7 (i) Express 23 6 5xx and 2 43xx   in the form   2 a x b c , where a, b and c are constants. [4] (ii) Using your answers from part (i), explain why the two curves 23 6 5y x x   and 2 43y x x   will not intersect. [3] [Turn over

10 8 Without using a calculator, (i) show that 26sin 75 4  . [2] (ii) Hence, express 2cosec 75 in the form 3ab , where a and b are integers. [5] [Turn over