3E NV AM EOY 2023

NORTH VISTA SECONDARY SCHOOL End-of-Year Examination 2023 Secondary 3 Express CANDIDATE NAME CLASS ADDITIONAL MATHEMATICS 4049 4 OCTOBER 2023 2 hour 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your centre number, index number and name 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. [Turn over INDEX NUMBER For Examiner’s Use Category Question No. Accuracy Brackets Fractions Units Others Marks Deducted 90

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

3 [Turn Over Answer all questions. 1 Do not use a calculator in this question. The curved surface area of a cylinder is 10 5 18 − cm2. The radius of the base of the cylinder is 3 2 5+ cm. Find an expression for the height of the cylinder in the form 5ab+ , where a and b are integers. [4]

4 2 (a) The angles A and B are such that 5sin 13A= , where oo90 270A and 4cos 5B= , where B is acute. Calculate (i) tan( )B− , [1] (ii) ( )tan 90 A− , [1] (iii) cos sinAB . [2] (b) Without using a calculator, find the exact value of sin cos tan6 4 3    + . Leave your answer in a single fraction. [2]

5 [Turn Over 3 (a) Express 22 8 14y x x=− + + in the form of 2()y a x b c= + + , where a, b and c are constants. [2] (b) Sketch the graph of 22 8 14y x x=− + + indicating the coordinates of the turning point and of the 𝑦-intercept. [2]

6 4 Express ( ) 32 2 7 5 12 ( 1) 2 x x x xx + − + −+ in partial fractions. [6]

7 [Turn Over 5 The expression 324 12x ax bx+ + − , where a and b are constants, has a factor of 2x− and leaves a remainder of 18− when divided by 21x− . (a) Show that 2a= and find the value of b. [4] (b) Using these values of a and b, solve the equation 324 12 0x ax bx+ + − = . [4]

8 6 (a) Find the coordinates of the points of intersection between the straight line 3 2 5xy+= and the curve 32 10xy+= . [4]

9 [Turn Over 6 (b) Show that the line 21y ax a= + − cuts the curve 282y x x= + − at two distinct points for all values of a. [3] (c) Find the range of values of p for which the graph 2 4y px x p= + + lies entirely below the line 3y=− . [4]

10 7 (a) Find the first 3 terms in the expansion, in descending powers of x, of 513 2x − . Give the terms in their simplest form. [2] 7 (b) (i) By considering the general term in the the binomial expansion of 7 2 3x x  − , explain why every term is dependent on x. [3] (ii) Hence find the term independent of x in the expansion of ( ) 7 2 3 43xx x  −− . [3]