Centre No. Candidate No. Paper Reference(s) 6665/01 Edecel GCE Core Mathematics C3 Advanced Thursday 11 June 009 Morning Time: 1 hour 30 minutes Materials required for eamination Mathematical Formulae (Orange or Green) Paper Reference 6 6 6 5 0 1 Surname Signature Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Initial(s) Eaminer s use only Team Leader s use only Question Number Blank 1 3 4 5 6 7 8 Instructions to Candidates In the boes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. Answer ALL the questions. You must write your answer for each question in the space following the question. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g. (). There are 8 questions in this question paper. The total mark for this paper is 75. There are 8 pages in this question paper. Any pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the Eaminer. Answers without working may not gain full credit. This publication may be reproduced only in accordance with Edecel Limited copyright policy. 009 Edecel Limited. Printer s Log. No. H3464A W850/R6665/57570 4/5/5/3 *H3464A018* Total Turn over
1. y 15 10 5 A 1 O 1 3 5 Figure 1 Figure 1 shows part of the curve with equation -ais at the point A where = α. y 3 = + +, which intersects the To find an approimation to α, the iterative formula is used. + = + n 1 ( n ) (a) Taking 0 =.5, find the values of 1,, 3 and 4. Give your answers to 3 decimal places where appropriate. (b) Show that α =.359 correct to 3 decimal places. *H3464A08*
. (a) Use the identity cos θ + sin θ = 1 to prove that tan θ = sec θ 1. (b) Solve, for 0 θ < 360, the equation tan θ + 4 sec θ + sec θ = () (6) 4 *H3464A048*
3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation 5 P = 80e 1 t, t, t 0 (a) Write down the number of rabbits that were introduced to the island. (1) (b) Find the number of years it would take for the number of rabbits to first eceed 1000. () (c) Find d P. dt (d) Find P when d P = 50. dt () 6 *H3464A068*
4. (i) Differentiate with respect to (a) (b) cos3 ln( + 1) + 1 (4) (ii) A curve C has the equation y= ( 4+ 1 ), >, y > 0 The point P on the curve has -coordinate. Find an equation of the tangent to C at P in the form a + by + c = 0, where a, b and c are integers. (6) 1 4 10 *H3464A0108*
Question 4 continued *H3464A0118* 11 Turn over
5. y O B A Figure Figure shows a sketch of part of the curve with equation y = f(),. 1 The curve meets the coordinate aes at the points A(0,1 k) and B ( ln k,0 ), where k is a constant and k > 1, as shown in Figure. On separate diagrams, sketch the curve with equation (a) y = f( ), (b) y = 1 f ( ). () Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the aes. Given that f( ) = e k, (c) state the range of f, (1) (d) find 1 f ( ), (e) write down the domain of 1 f. (1) 14 *H3464A0148*
Question 5 continued *H3464A0158* 15 Turn over
6. (a) Use the identity cos( A+ B) = cos Acos B sin Asin B, to show that cos A = 1 sin A () The curves C 1 and C have equations C 1 : y = 3sin C : y = 4sin cos (b) Show that the -coordinates of the points where C 1 and C intersect satisfy the equation 4cos + 3sin = (c) Epress 4cos + 3sin in the form R cos ( α), where R > 0 and 0 < α < 90, giving the value of α to decimal places. (d) Hence find, for 0 < 180, all the solutions of 4cos + 3sin = giving your answers to 1 decimal place. (4) 18 *H3464A0188*
Question 6 continued *H3464A0198* 19 Turn over
7. The function f is defined by 8 f( ) = 1 +, ( + 4) ( )( + 4) 4, 3 (a) Show that f( ) = (5) The function g is defined by e 3 g( ) =, e, ln e (b) Differentiate g( ) to show that g( ) = (e ) (c) Find the eact values of for which g( ) = 1 (4) *H3464A08*
Question 7 continued *H3464A038* 3 Turn over
8. (a) Write down sin in terms of sin and cos. (b) Find, for 0 < < π, all the solutions of the equation cosec 8cos = 0 (1) giving your answers to decimal places. (5) 6 *H3464A068*