3E NV AM EOY 2022

NORTH VISTA SECONDARY SCHOOL End-of-Year Examination 2022 Secondary 3 Express CANDIDATE NAME CLASS ADDITIONAL MATHEMATICS 4049 5 OCTOBER 2022 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 a 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 A function f(x) is given by the expression 322 5 4 3x x x− − + . (i) Find the remainder when f(x) is divided by x – 1. [1] (ii) Show that x – 3 is a factor of f(x). [1] (iii) Hence, solve f(x) = 0. [4]

4 2 Given that 3cos 8A=− , 9sin 10B= and both A and B are in the same quadrant, find the exact value of (i) sin A, [1] (ii) cos (−B), [1] (iii) tan tanAB . [2]

5 [Turn Over 3 By using a suitable substitution, solve the equation 2 2 42 2 2 4x x x++ + = + . [5]

6 4 (a) Sketch the graph of 3 1 x ye=+ . [2] (b) To solve the equation 3ln(3 4)xx=+ graphically, a suitable straight line can be [2] drawn on the same set of axes in (a). Find the equation of the straight line. (c) Hence, draw the line on the same diagram in (a) and state the number of solutions [2] of the equation 3ln(3 4)xx=+ . x y

7 [Turn Over 5 (a) Find the value of p for which the x-axis is the tangent to the curve 23 2 1y x x p= − + − . [2] (b) Find the range of values of k for which the line 3y x k=+ and the curve 22 10xy+= do not intersect. [5]

8 6 (i) Find the general term for the expansion of 7 2 3x x  − and write down, in terms of [2] r, the power of x in this general term. (ii) Hence, explain whether the term independent of x in the expansion of 7 2 3x x  − [2] exist.

9 [Turn Over 7 The right square pyramid has a square base ABCD with sides (√2 + √3) cm. Given that the volume of the pyramid is (22 + 9√6) cm3, find the vertical height, h cm, of the pyramid. Leave your answer in the form a + b √3 where a and b are integers. [4] C B D h A O

10 8 Find the exact value of 60x , given that 4 8 2 23 20xx+− = . [4]