Time: 1 hour 30 minutes

Transcription

1 Paper Reference(s) 6665/0 Edexcel GCE Core Mathematics C3 Gold Level (Hard) G Time: hour 30 minutes Materials required for examination Mathematical Formulae (Green) 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 formulas stored in them. Instructions to Candidates Write the name of the examining bo (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. There are 8 questions in this question paper. The total mark for this paper is 75. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. Suggested grade boundaries for this paper: A* A B C D E Gold This publication may only be reproduced in accordance with Edexcel Limited copyright policy Edexcel Limited.

2 . The point P lies on the curve with equation The y-coordinate of P is 8. y = 4e x +. (a) Find, in terms of ln, the x-coordinate of P. (b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found. () June 008. A curve C has equation 3 y = ( 5 3x), x 3 5. The point P on C has x-coordinate. Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers. (7) June A curve C has equation y = x e x. d y (a) Find, using the product rule for differentiation. (b) Hence find the coordinates of the turning points of C. d y (c) Find. (d) Determine the nature of each turning point of the curve C. () () June 007 Gold : 9/

3 4. (i) Given that y = ln( x +), find x d y. (ii) Given that x = tan y, show that d y = + x. (5) January Figure Figure shows a sketch of the curve C with the equation y = (x 5x + )e x. (a) Find the coordinates of the point where C crosses the y-axis. (b) Show that C crosses the x-axis at x = and find the x-coordinate of the other point where C crosses the x-axis. (c) Find d y. (d) Hence find the exact coordinates of the turning points of C. () (5) June 00 Gold :9/ 3

4 6. (a) Express 3 sin x + cos x in the form R sin (x + α) where R > 0 and 0 < α < π. (b) Hence find the greatest value of (3 sin x + cos x) 4. (c) Solve, for 0 < x < π, the equation 3 sin x + cos x =, giving your answers to 3 decimal places. (5) June 007 () 7. (a) Express 4 cosec θ cosec θ in terms of sin θ and cos θ. (b) Hence show that 4 cosec θ cosec θ = sec θ. (c) Hence or otherwise solve, for 0 < θ < π, () 4 cosec θ cosec θ = 4 giving your answers in terms of π. June 0 8. (a) Starting from the formulae for sin (A + B) and cos (A + B), prove that tan A + tan B tan (A + B) =. tan Atan B (b) Deduce that π + 3tanθ tan θ + =. 6 3 tanθ (c) Hence, or otherwise, solve, for 0 θ π, + 3 tan θ = ( 3 tan θ) tan (π θ). Give your answers as multiples of π. (6) January 0 END TOTAL FOR PAPER: 75 MARKS Gold : 9/ 4

5 . (a) e x+ = x + = ln (b) x = ( ln ) A () 8e x+ = B x = ( ln ) = 6 B y 8 = 6 x ( ln ) y = 6x+ 6 8ln A ( 4) (6 marks) Question. At P, y = 3 B d y 3( ) ( 5 3 ) 3 8 = x ( 3) or 3 (5 3 x) 8 3 { 8} d x = (5 3()) = m(n) = or 8 8 N: y 3 = ( x ) 8 N: x 8 y+ 5 = 0 A A [7] Gold :9/ 5

6 3. (a) x x = x e+ e x,a,a d y (b) If = 0, e x (x + x) = 0 setting (a) = 0 [e x 0] x(x + ) = 0 ( x = 0 ) x = A x = 0, y = 0 and x =, y = 4e ( = 0.54 ) A (c) d y d x x x x x e xe xe e x = x = ( + 4x+ )e x, A () (d) d y d y x = 0, > 0 (=) x =, < 0 [ = e ( = 0.70 )] : Evaluate, or state sign of, candidate s (c) for at least one of candidate s x value(s) from (b) minimum maximum A (cso) () (0 marks) Gold : 9/ 6

7 Q4 (i) y = ln( x + ) x u du x = ln( x + ) = x x + d A Apply quotient rule: u = ln( x + ) v = x du x dv = = x + x x + = ( x) x x ln( + ) A d x ( x + ) x = ln( x + ) (ii) x = tan y d d x sec y y = * A = { = cos y} d* sec y = + tan y d* Hence, = + x, (as required) A AG (5) [9] Gold :9/ 7

8 5. (a) y ( 0, 5 ) A Question (b) x = 0 B x 5 = (5 + x) ; x = ;A oe. 0 3 (c) fg() = f ( 3) = ( 3) 5 ; = = ;A (d) g( x) = x 4x+ = ( x ) 4 + = ( x ) 3. Hence gmin = 3 Either gmin = 3 or g( x) 3 B or g(5) = = 6 3 g( x) 6 or 3 y 6 A 6. (a) Complete method for R: e.g. R cosα = 3, Rsinα =, R = (3 + ) R = 3 or 3.6 (or more accurate) A 3 Complete method for tanα = [Allow tan α = ] 3 α = (Allow 33.7 ) A (b) Greatest value = ( ) 4 3 = 69, A () (c) O 5 (,0 ) x sin( x ) = ( = ) 3 sin(x + their α) = their R ( x ) = 0.8( 03 ) or 6. A (x ) = π Must be π their 0.8 or 80 their 6. or (x ) = π Must be π + their 0.8 or their 6. x =.73 or x = (awrt) Both (radians only) A (5) If 0.8 or 6. not seen, correct answers imply this A mark ( marks) () () [0] Gold : 9/ 8

9 7. (a) 4 4cosec θ cosec θ = sin θ sin θ = 4 (sinθcos θ) sin θ B B 4 4 = (sinθcos θ) sin θ 4sin θcos θ sin θ (b) (c) Using cos θ = sin θ cos θ = sin θcos θ sin θcos θ sin θ sin θcos θ = = sec θ cos θ A* sec θ = 4 secθ =± cosθ =± π π θ =, 3 3 A,A (9 marks) () Gold :9/ 9

10 number 8. (a) A ( A * (b) A * (c) θ= A θ= A (6) (3 marks) Gold : 9/ 0

11 Statistics for C3 Practice Paper G Mean score for students achieving grade: Qu Max Modal Mean score score % ALL A* A B C D E U Gold :9/