Time: 1 hour 30 minutes

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1 Paper Reference (complete below) Centre No. Surname Initial(s) Candidate No. Signature Paper Reference(s) 6663 Edexcel GCE Pure Mathematics C Advanced Subsidiary Specimen Paper Time: hour 30 minutes Examiner s use only Team Leader s use only Materials required for examination Items included with question papers 7 Answer Book (AB6) Nil 8 Mathematical Formulae (Lilac) Graph Paper (ASG) Calculators may NOT be used in this examination. Instructions to Candidates Tour candidate details are printed next to the bar code above. Check that these are correct and sign your name in the signature box above. If your candidate details are incorrect, or missing, then complete ALL the boxes above. When a calculator is used, the answer should be given to an appropriate degree of accuracy. You must write your answer for each question in the space following the question. If you need more space to complete your answer to any question, use additional answer sheets. Information for Candidates A booklet mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. This paper has 0 questions. Question Number Blank 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. Total Turn over

2 0. Calculate (5 r ). r (3)

3 . Find (5x + 3x) dx. (4) 3

4 3. (a) Express 80 in the form a5, where a is an integer. (b) Express (4 5) in the form b + c5, where b and c are integers. () (3) 4

5 4. The points A and B have coordinates (3, 4) and (7, 6) respectively. The straight line l passes through A and is perpendicular to AB. Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5) 5

6 5. Figure y (, ) (0, ) O (3, 0) x Figure shows a sketch of the curve with equation y = f(x). The curve crosses the coordinate axes at the points (0, ) and (3, 0). The maximum point on the curve is (, ). On separate diagrams in the space opposite, sketch the curve with equation (a) y = f(x + ), (b) y = f(x). (3) (3) On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes. 6

7 6. (a) Solve the simultaneous equations y + x = 5, x 3x y = 6. (b) Hence, or otherwise, find the set of values of x for which x 3x 6 > 5 x. (6) (3).. 7

8 8

9 7. Ahmed plans to save 50 in the year 00, 300 in 00, 350 in 003, and so on until the year 00. His planned savings form an arithmetic sequence with common difference 50. (a) Find the amount he plans to save in the year 0. (b) Calculate his total planned savings over the 0 year period from 00 to 00. () (3) Ben also plans to save money over the same 0 year period. He saves A in the year 00 and his planned yearly savings form an arithmetic sequence with common difference 60. Given that Ben s total planned savings over the 0 year period are equal to Ahmed s total planned savings over the same period, (c) calculate the value of A. (4) 9

10 0

11 8. Given that x + 0x + 36 (x + a) + b, where a and b are constants, (a) find the value of a and the value of b. (b) Hence show that the equation x + 0x + 36 = 0 has no real roots. (3) () The equation x + 0x + k = 0 has equal roots. (c) Find the value of k. (d) For this value of k, sketch the graph of y = x + 0x + k, showing the coordinates of any points at which the graph meets the coordinate axes. (4) ()

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13 9. The curve C has equation y = f(x) and the point P(3, 5) lies on C. Given that f (x) = 3x 8x + 6, (a) find f(x). (b) Verify that the point (, 0) lies on C. (4) () The point Q also lies on C, and the tangent to C at Q is parallel to the tangent to C at P. (c) Find the x-coordinate of Q. (5) 3

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15 0. The curve C has equation y = x 3 5x + x, x 0. The points A and B both lie on C and have coordinates (, ) and (, ) respectively. (a) Show that the gradient of C at A is equal to the gradient of C at B. (b) Show that an equation for the normal to C at A is 4y = x 9. (5) (4) The normal to C at A meets the y-axis at the point P. The normal to C at B meets the y-axis at the point Q. (c) Find the length of PQ. (4) 5

16 END 6

17 Paper Reference (complete below) Centre No. Surname Initial(s) Candidate No. Signature Paper Reference(s) 6664 Edexcel GCE Pure Mathematics C Advanced Subsidiary Specimen Paper Time: hour 30 minutes Examiner s use only Team Leader s use only Materials required for examination Items included with question papers 7 Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 9, Casio CFX 9970G, Hewlett Packard HP 48G.. Question Number Blank Instructions to Candidates Tour candidate details are printed next to the bar code above. Check that these are correct and sign your name in the signature box above. If your candidate details are incorrect, or missing, then complete ALL the boxes above. When a calculator is used, the answer should be given to an appropriate degree of accuracy. You must write your answer for each question in the space following the question. If you need more space to complete your answer to any question, use additional answer sheets. Information for Candidates A booklet mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. This paper has 9 questions. 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 Total Turn over 7

18 . Find the first 3 terms, in ascending powers of x, of the binomial expansion of ( + 3x) 6. (4) 8

19 . The circle C has centre (3, 4) and passes through the point (8, 8). Find an equation for C. (4) 9

20 3. The trapezium rule, with the table below, was used to estimate the area between the curve y = (x 3 + ), the lines x =, x = 3 and the x-axis. x y (a) Calculate, to 3 decimal places, the values of y for x =.5 and x = 3. (b) Use the values from the table and your answers to part (a) to find an estimate, to decimal places, for this area. (4) () 0

21 4. Solve, for 0 x < 360, the equation 3 sin x = + cos x, giving your answers to the nearest degree. (7)

22 5. Figure B 8 mm 8 mm O A 8 mm 8 mm C The shaded area in Fig. shows a badge ABC, where AB and AC are straight lines, with AB = AC = 8 mm. The curve BC is an arc of a circle, centre O, where OB = OC = 8 mm and O is in the same plane as ABC. The angle BAC is 0.9 radians. (a) Find the perimeter of the badge. (b) Find the area of the badge. () (5)

23 3

24 6. At the beginning of the year 000 a company bought a new machine for Each year the value of the machine decreases by 0% of its value at the start of the year. (a) Show that at the start of the year 00, the value of the machine was () When the value of the machine falls below 500, the company will replace it. (b) Find the year in which the machine will be replaced. (4) To plan for a replacement machine, the company pays 000 at the start of each year into a savings account. The account pays interest at a fixed rate of 5% per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced. (c) Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced. (4) 4

25 5

26 7. (a) Use the factor theorem to show that (x + ) is a factor of x 3 x 0x 8. (b) Find all the solutions of the equation x 3 x 0x 8 = 0. () (4) (c) Prove that the value of x that satisfies log x + log (x ) = + log (5x + 4) (I) is a solution of the equation x 3 x 0x 8 = 0. (d) State, with a reason, the value of x that satisfies equation (I). (4) () 6

27 7

28 8. Figure y 8 B 5 A O x The line with equation y = x + 5 cuts the curve with equation y = x 3x + 8 at the points A and B, as shown in Fig.. (a) Find the coordinates of the points A and B. (b) Find the area of the shaded region between the curve and the line, as shown in Fig.. (7) (5) 8

29 9

30 9. Figure 3 Q R (x + ) P 30 (4 x) Figure 3 shows a triangle PQR. The size of angle QPR is 30, the length of PQ is (x + ) and the length of PR is (4 x), where x R. (a) Show that the area A of the triangle is given by A = 4 (x 3 7x + 8x + 6). (b) Use calculus to prove that the area of PQR is a maximum when x. Explain 3 clearly how you know that this value of x gives the maximum area. (6) (c) Find the maximum area of PQR. (d) Find the length of QR when the area of PQR is a maximum. (3) () (3) 30

31 END 3

32 6665 Edexcel GCE Pure Mathematics C3 Advanced Level Specimen Paper Time: hour 30 minutes Materials required for examination Answer Book (AB6) Mathematical Formulae (Lilac) Graph Paper (ASG) Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI-89, TI-9, Casio cfx 9970G, Hewlett Packard HP 48G. Instructions to Candidates In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Pure Mathematics C3), the paper reference (6665), your surname, other name and signature. 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. This paper has seven questions. 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. 003 Edexcel This publication may only be reproduced in accordance with Edexcel copyright policy. 3

33 . The function f is defined by f: x x 3, x R. (a) Solve the equation f(x) =. (3) The function g is defined by g: x x 4x +, x 0. (b) Find the range of g. (c) Find gf( ). (3) (). f(x) = x 3 x 5. (a) Show that there is a root of f(x) = 0 for x in the interval [, 3]. () The root is to be estimated using the iterative formula 5 x n + =, x n x 0 =. (b) Calculate the values of x, x, x 3 and x 4, giving your answers to 4 significant figures. (c) Prove that, to 5 significant figures, is (3) (3) 3. (a) Using the identity for cos (A + B), prove that cos sin ( ). (b) Prove that + sin cos sin ( )[cos ( ) + sin ( )]. (c) Hence, or otherwise, solve the equation sin cos 0, 0. (3) (3) (4) 33

34 4. f(x) = x + 3 x x, x R, x >. x 3 (a) Show that f(x) = x 3x 3. x 3 (5) (b) Solve the equation f (x) = 5. (5) 5. Figure y (0, q) (p, 0) O x Figure shows part of the curve with equation y = f(x), x R. The curve meets the x-axis at P (p, 0) and meets the y-axis at Q (0, q). (a) On separate diagrams, sketch the curve with equation (i) y = f(x), (ii) y = 3f( x). In each case show, in terms of p or q, the coordinates of points at which the curve meets the axes. (5) Given that f(x) = 3 ln(x + 3), (b) state the exact value of q, (c) find the value of p, (d) find an equation for the tangent to the curve at P. () () (4) 34

35 6. As a substance cools its temperature, T C, is related to the time (t minutes) for which it has been cooling. The relationship is given by the equation T = e 0.t, t 0. (a) Find the value of T when the substance started to cool. (b) Explain why the temperature of the substance is always above 0C. (c) Sketch the graph of T against t. (d) Find the value, to significant figures, of t at the instant T = 60. (e) Find (f) dt dt. Hence find the value of T at which the temperature is decreasing at a rate of.8 C per minute. (3) () () () (4) () 7. (i) dy Given that y = tan x + cos x, find the exact value of at x =. dx 4 (3) (ii) Given that x = tan d y y, prove that = dx x. (4) (iii) Given that y = e x dy sin x, show that dx can be expressed in the form R e x cos (x + ). Find, to 3 significant figures, the values of R and α, where 0 < α <. (7) END 35 Turn over

36 Paper Reference(s) 6666 Edexcel GCE Pure Mathematics C4 Advanced Level Specimen Paper Time: hour 30 minutes Materials required for examination Answer Book (AB6) Mathematical Formulae (Lilac) Graph Paper (ASG) Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI-89, TI-9, Casio CFX-9970G, Hewlett Packard HP 48G. Instructions to Candidates In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Pure Mathematics C4), the paper reference (6666), your surname, other name and signature. 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. This paper has eight questions. 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. This publication may only be reproduced in accordance with London Qualifications Limited copyright policy. Edexcel Foundation is a registered charity. 003 London Qualifications Limited

37 . Use the binomial theorem to expand (4 3x ), in ascending powers of x, up to and including the term in x 3. Give each coefficient as a simplified fraction. (5). The curve C has equation 3x + 3y 0xy = 5. Find an expression for dy dx as a function of x and y, simplifying your answer. (6) 3. Use the substitution x = tan to show that ( x 0 ) dx 8 4. (8) 37

38 4. Figure y O x Figure shows part of the curve with parametric equations x = tan t, y = sin t, t. (a) Find the gradient of the curve at the point P where t = 3. (4) (b) Find an equation of the normal to the curve at P. (3) (c) Find an equation of the normal to the curve at the point Q where t = 4. () 38

39 5. The vector equations of two straight lines are r = 5i + 3j k + (i j + k) and r = i j + ak + (3i 4j + 5k). Given that the two lines intersect, find (a) the coordinates of the point of intersection, (b) the value of the constant a, (c) the acute angle between the two lines. (5) () (4) 6. Given that ( x) x ( 3x) A ( x) + B ( x) + C, ( 3x) (a) find the values of A, B and C. (4) x (b) Find the exact value of dx, giving your answer in the form k + ln a, where 0 ( x) ( 3x) k is an integer and a is a simplified fraction. (7) 39 Turn over

40 x du 7. (a) Given that u = sin 4x, show that 8 dx = sin x. (4) Figure y O x x 4 Figure shows the finite region bounded by the curve y = x sin x, the line x = 4 and the x-axis. This region is rotated through radians about the x-axis. (b) Using the result in part (a), or otherwise, find the exact value of the volume generated. (8) 40

41 8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time t seconds is A cm, (a) show that da dt. A (6) Another stain, which is growing more quickly, has area S cm² at time t seconds. It is given that ds dt = t e S. Given that, for this second stain, S = 9 at time t = 0, (b) solve the differential equation to find the time at which S = 6. Give your answer to significant figures. (7) END 4 Turn over

42 EDEXCEL PURE MATHEMATICS C (6663) SPECIMEN PAPER MARK SCHEME Question number Scheme. a = 7, d = B S 0 = 0 ( ) = 50 M A Marks (3 marks) 3 5x. ( 5x 3 x) dx x C M A A B (4 marks) 3. (a) 80 = 45 B () (b) (4 5) = = 85 M A A (3) (4 marks) 4 ( 6) 5 4. Gradient of AB = 3 7 Gradient of l = 5 M A M y 4 = 5 (x 3) x 5y + 4 = 0 M A (5) (5 marks) 5. (a) y Position, Shape B O x (0, ), (, 0) B B (3) (b) y Position, Shape B O x (0, ),,, 3, 0 B (, 0) (3) (6 marks) 4

43 EDEXCEL PURE MATHEMATICS C (6663) SPECIMEN PAPER MARK SCHEME Question number Scheme 6. (a) 5 x = x 3x 6 x x = 0 M A (x 7)(x + 3) = 0 x = 3, x = 7 M A Marks y =, y = M Aft (6) (b) Using critical values x = 3, x = 7 x < 3, x > 7 M M Aft (3) (9 marks) 7. (a) a + (n )d = 50 + (0 50) = 750 M A () (b) n a ( n ) d = 0 ( ), = 4500 M A, A (3) (c) B: 0 (A ) [= 0(A + 40)], = 4500 B, M Solve for A: A = 55 M A (4) (9 marks) 8. (a) a = 5, (x + 5) b = B, M A (3) (b) b 4ac = 00 44, roots < 0, therefore no real M A () (c) Equal roots if b 4ac = 0 4k = 00 k =5 M A () (d) y Shape, position B B O x (5, 0) (0, 5) B Bft (4) ( marks) 43

44 EDEXCEL PURE MATHEMATICS C (6663) SPECIMEN PAPER MARK SCHEME Question number Scheme 9. (a) f(x)= x 3 4x + 6x + C M A 0. (a) Marks 5 = C C = 4 M A (4) (b) x = : y = = 0 M A () (c) f (3) = = 9, Parallel therefore equal gradient B, M 3x 8x + 6 = 9 3x 8x 3 = 0 M (3x + )(x 3) = 0 Q: x = M A (5) 3 ( marks) d y = 3x 5 x M A(,0) dx At both A and B, (b) Gradient of normal = 4 d y = 3 5 dx (= 4) M A (5) M Aft y () = 4 (x ) 4y = x 9 M A (4) (c) Normal at A meets y-axis where x = 0: y = Similarly for normal at B: 4y = x + 9 y = 4 9 Length of PQ = 9 B 4 M A A (4) 4 4 (3 marks) 44

45 EDEXCEL PURE MATHEMATICS C (6664)SPECIMEN PAPER MARK SCHEME Question number Scheme. ( + 3x) 6 = x + 4 (3x) > term correct M Marks = 64, + 576x, + 60x B A A (4 marks). r = 3) ( 8 4), = 3 Method for r or ( 8 r M A Equation: (x 3) + (y 4) = 69 ft their r M Aft (4 marks) 3. (a) (x =.5) y = (x = 3) y = 5.9 B B () (b) A [ ( )] For B ft their y values M Aft = 6.6 = 6.6 ( d.p.) A (4) 4. 3( cos x) = + cos x Use of s c M 0 = 3 cos x + cos x 3TQ in cos x M 0 = (3cos x )(cos x +) Attempt to solve M cos x 3 or Both A (6 marks) cos x 3 gives x = 48, 3 B, Bft cosx = gives x = 80 B (7 marks) 5. (a) Arc length = r = M for use of r M Perimeter = 6+ r = 3. (mm) A () (b) Area of triangle = Area of sector =.8.sin(0.9) (0.9) 8.8 Area of segment = = 3.7(33..) M Area of badge = triangle segment, =.3 (mm ) M, A (5) M Aft (7 marks) 45

46 EDEXCEL PURE MATHEMATICS C (6664) SPECIMEN PAPER MARK SCHEME Question Scheme Marks number 6. (a) 5000 (0.8) 9600 (*) M for by 0.8 M A cso () n (b) 5000 (0.8) 500 Suitable equation or inequality M n log(0.8) log( ) Take logs M 30 n > 5.(4 ) n = is OK A So machine is replaced in 05 A (4) (c) a = 000, r =.05, n = 6 ( correct) M S (.05 ).05 M A = = or or 3657 A (4) (0 marks) 7. (a) f() = f(+) or f( ) M = 0 so (x + ) is a factor = 0 and comment A () (b) 3 x x x (5 4) Out of logs M i.e. x 3 x 0x 8 = 0 (*) A cso (4) M x =,, 4 A(, 0) (4) (c) log x log ( x) log (5x4) Use of log x n M x ( x) log Use of log a log b M 5x 4 (d) x = 4, since x < 0 is not valid in logs B, B () ( marks) 46

47 EDEXCEL PURE MATHEMATICS C (6664)SPECIMEN PAPER MARK SCHEME Question number Scheme 8. (a) x 3x + 8 = x + 5 Line = curve M x 4x + 3 = 0 3TQ = 0 M 0 = (x 3)(x ) Solving M Marks A is (, 6); B is (3, 8) A; A (5) (b) 3 x 3x ( x 3x 8) dx 8x Integration M A(,0) Area below curve = (9 4) ( 8) Use of Limits M 3 3 Trapezium = (6 + 8) = 4 B Area = Trapezium Integral, = M, A (7) ( marks) ALT (b) x + 4x 3 Line curve M 3 x ( x 4x 3) dx = x 3x 3 Integration M A(,0) Area = 3 (...) dx = ( ) ( 3 + 3) Use of limits M = 3 A (7) 47

48 EDEXCEL PURE MATHEMATICS C (6664) SPECIMEN PAPER MARK SCHEME Question Scheme Marks number 9. (a) A = (x + ) (4 x) sin 30 Use of absin C M = 4 (x + )(6 8x + x ) Attempt to multiply out. M = 4 (x 3 7x + 8x + 6) (*) A cso (3) (b) da dx da dx = 4 (3x 4x + 8) Ignore the 4 M A = 0 (3x )(x 4) = 0 M So x 3 or 4 At least x = 3 or A d A e.g. = 4 (6x 4), when x 3 it is < 0, so maximum Any full method M dx So x 3 gives maximum area (*) Full accuracy A (6) (c) Maximum area = 5 0 4()( 3 3) 4.6 or 4.63 or B () (d) Cosine rule: QR ( ) ( ) ( ) cos30 M for QR or = QR M A QR = 9.7 or 9.70 or A (3) (3 marks) 48

49 EDEXCEL PURE MATHEMATICS C3 (6665) SPECIMEN PAPER MARK SCHEME Question number Scheme. (a) x 3 = x = 6 B Marks ( x ) 3 = x = M A (3) (b) g(x) = x 4x + = (x ) +7 or g(x) = x 4 M A g(x) = 0 x = Range: g(x) 7. A (3) (c) gf() = g(0) correct order; = M A (). (a) f() = = method shows change of sign M (8 marks) f(3) = = 6 root with accuracy A () (b) x =., x =.087, x 3 =.097, x 4 =.094 M A (, 0) (3) (c) Choosing suitable interval, e.g. [.09455,.09465] M f(.09455) = shows change of sign M f(.09465) = (099..) accuracy and conclusion A (3) 3. (a) cos (A + B) = cos A cos B sin A sin B (formula sheet) cos ( + ) = cos ( ) cos ( ) sin ( ) sin ( ) = cos ( ) sin ( ) M (8 marks) = { sin ( )} sin ( ) = sin ( ) M A (3) (b) sin + cos = sin ( ) cos ( ) + sin ( ) M M (c) sin ( ) [cos ( ) + sin ( )] = 0 = sin ( ) [cos ( ) + sin ( )] A (3) [M use of sin A = sin A cos A; M use of (a)] sin ( ) = 0 or cos ( ) + sin ( ) = 0 = 0 M B tan = ; = 3 M A (4) (0 marks) 49

50 EDEXCEL PURE MATHEMATICS C3 (6665) SPECIMEN PAPER MARK SCHEME Question Scheme number 4. (a) x + x 3 = (x + 3)(x ) B Marks f(x) = x( x x 3) 3( x 3) ( x 3)( x ) [= x 3 x 3 ] MA ( x 3)( x ) = = ( x )( x 3x 3) ( x )( x 3) ( x 3x 3) ( x 3) M A (5) (b) f (x) = ( x 3)(x 3) ( x ( x 3) 3x 3) [= x 6x 6 ] M A,, 0 ( x 3) Setting f (x) = 5 and attempting to solve quadratic M 3 ALT (b) ALT: f(x) =, x 3 x = (only this solution) A (5) 3 f ( x) ( x 3) x (0 marks) 50

51 EDEXCEL PURE MATHEMATICS C3 (6665) SPECIMEN PAPER MARK SCHEME Question number Scheme Marks 5. (a) (i) y Shape correct: B q p 0 x Intercepts B () (ii) y Shape correct B p 3q (p, 0) on x B (0, 3q) on y B (3) (b) q = 3 ln 3 B () (c) ln(p + 3) = 0 p + 3 = ; p = M A () (d) d y 6 = ; evaluated at x = p (6) M A dx x 3 Equation: y = 6(x + ) any form M Aft (4) ( marks) 5

52 EDEXCEL PURE MATHEMATICS C3 (6665) SPECIMEN PAPER MARK SCHEME Question Scheme Marks number 6. (a) T = 80 B () (b) e 0. t 0 or equivalent B () (c) T 80 Negative exponential shape t 0, 80 M clearly not x-axis A () (d) 60 = e 0. t 60 e 0. t = 40 M t (e) (f) 0. t = ln 3 MA t = 4. A (4) dt dt = 6 e 0. t MA () dt Using dt =.8 B Solving for t, or using value of e 0. t (0.3) M T = 38 A (3) (3 marks) 5

53 EDEXCEL PURE MATHEMATICS C3 (6665) SPECIMEN PAPER MARK SCHEME Question number 7. (i) (ii) (iii) Scheme d y = sec x sin x B B dx d x = sec dy d y = = dx y sec When x =, 4 Marks d y = B (3) dx y B = y x M M A (4) tan d y = e x cosx e x sin x = e x (cos x sin x) dx M A A Method for R: R =.4 (allow 5) M A Method for : = M A (7) (4 marks) 53

54 EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME Question number Scheme Marks. (4 3x ) = 3 x = 3 x x x = + x, x, B M 35 x. A, A, A 048 (5 marks). 6x + 6yy ; 0xy 0y = 0 MA; MA y (6y 0x) = 0y 6x y = 0y 6x 6y 0x = 5y 3x 3y 5x M A (6 marks) 3. x = tan Limits 4 and 0 dx = sec I = d sec sec 4 d M A B I = cos d = cos d M A sin = 4 = 4 0 M A (*) A cao 4 8 (8 marks) 54

55 EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME Question number Scheme Marks 4. (a) d x dt = sec t d y dt d y cos = cos t, = dx sec tt M A, M When t = 3 gradient is B (4) 4 (b) y = (x 3) P has coordinates (3, 3 m 3 ) B (c) y 3 = 4 (x 3) M 7 y = 4x 3 d y = 0 gradient of tan = 0, gradient of normal undefined M dx A (3) x = tan, i.e: x = 4 A () (9 marks) 5. (a) 5 + = 3; 3 = 4 B B = = = = 0 = = 3 M A point is (8, 3, 4) A (5) (b) a 0 = 4 a = 4 M A () (c) cos = M A = 5 35 = Angle = 45 M A (4) ( marks) 55

56 EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME Question number Scheme 6. (a) x A( + 3x) + B( x)( + 3x) + C( x) Putting x = A = Putting x = = C 3 9 C = 3 B B Marks cf x 0 = 3B + C B = MA (4) (b) 0 x ( x) 3 ( 3x) dx = ln x ln 3x M Aft Aft x Aft = [4 + ln ln 3 ( ln )] M = + ln 3 M = + ln 7 A (7) ( marks) 7. (a) du dx = cos 4x; = ( sin x) = sin x M A; M A (4) (b) V = x sin x dx M x = sin 4 x x x sin 4x dx M A A x = x x sin 4x cos 4x 4 0 M A = = 64 6 M A (8) ( marks) 56

57 EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME Question number 8. (a) d r k = dt r A = r da d r d A = r dt r da dt A k = k Scheme = r MA r ; = 3 k k = A A B M; M Marks (*) A (6) (b) S ds = et dt M S 3 = e t + C MA 3 t = 0, S = 9 C = 7 B S = e t + 7 and use S = 6 M = e t 77 t = ln 3 3 M =.6 A (7) (3 marks) 57