SPS AM Prelim Papers

NAME CLASS INDEX NO. ST. PATRICK'S SCHOOL PRELIMINARY EXAMINATION 2021 SUBJECT : Additional Mathematics Paper 1 (4049/01) DATE : 17 August 2021 LEVEL : Secondary 4 Express DURATION : 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your Name, Class and Index No. in the spaces at the top of this page. Write in dark blue or back 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. For Examiner’s Use Score /90 This question paper consists of 23 printed pages, including the cover page. Candidates answer on the Question Paper.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  Binomial expansion nrrnnnnn bbar nbanbanaba                  ......21)( 221 , where n is a positive integer and .! )1(...)1( !)!( ! r rnnn rrn n r n       2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin    A AA AAAAA AAA BA BABA BABABA BABABA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(         Formulae for ABC C c B b A a sinsinsin  Abccba cos2222  Area of  = Abcsin2 1

3 1. Given that x2 + 2x – 3 is a factor of f(x), where f(x) = x4 + 4x3 + 2x2 + ax – b. (a) Find the values of a and b. [4] (b) Hence, find the other quadratic factor of f(x). [2]

4 2. The diagram shows a triangle PQR with an area of 18 cm2. The base QR has a length of ( 4 2 8 ) cm. The line RQ is extended to the point S where the angle PSR = 45. (a) Find the perpendicular distance from P to QR, leaving your answer in the form 2ab , where a and b are rational numbers. [3] P Q S R 45

5 (b) Find the exact length of PS, leaving your answer in the form 2mn , where m and n are integers. [2]

6 3. Given the curve 231 1 1 2 3 3 xxy e e    and the normal to the curve at point 55ln 2, 24P  crosses the y – axis at Q. (a) Show that 5 3 1x x dy e dx e  . [2]

7 (b) Find the equation of the normal at 55ln 2, 24P  , leaving your answer in exact form. [3] (c) Hence, find the coordinates of Q. [1]

8 4. (a) Find the coordinates of the stationary points on the graph of 322 9 13y x x   . [4] (b) Hence, determine the nature of the stationary point(s). [3]

9 5. (a) Differentiate 2 cos 2xx with respect to x. [2] (b) Hence, evaluate  2 0 1 4 sin 2 x x dx   . [5]

10 6. It is given that    3 2 2 5y x x   . (a) Obtain an expression for dy dx in the form    2 25ax b x , where a and b are integers. [2] (b) Determine the values of x for which y is a decreasing function. [2]