# MEI STRUCTURED MATHEMATICS 4753/1

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6 Section A (36 marks) Given that 3, show that d d. [4] A population is P million at time t ears. P is modelled b the equation where a and b are constants. P 5 ae bt, The population is initiall 8 million, and declines to 6 million after ear. (i) Use this information to calculate the values of a and b, giving b correct to 3 significant figures. [5] (ii) What is the long-term population predicted b the model? [] 3 (i) Epress ln ln 3 as a single logarithm. [] (ii) Hence, given that satisfies the equation ln ln 3 ln (5 ), show that is a root of the quadratic equation [] (iii) Solve this quadratic equation, eplaining wh onl one root is a valid solution of ln ln 3 ln (5 ). [3] 4753/ Januar 006

7 3 4 Fig. 4 shows a cone. The angle between the ais and the slant edge is 30. Water is poured into the cone at a constant rate of cm 3 per second. At time t seconds, the radius of the water surface is r cm and the volume of water in the cone is V cm 3. rcm 30 Fig. 4 dv (i) Write down the value of [] dt. (ii) Show that V = 3 dv p r 3, and find [3] 3 dr. [You ma assume that the volume of a cone of height h and radius r is ] dr (iii) Use the results of parts (i) and (ii) to find the value of when r. [3] dt 5 A curve is defined implicitl b the equation 3. d (i) Show that [4] d ( ) 3. d (ii) Hence write down in terms of and. [] d 3p r h. 4753/ Januar 006 [Turn over

8 4 6 The function f () is defined b f () sin for p p. (i) Show that f - = Ê - ( ) arcsin ˆ and state the domain of this function. [4] Ë Fig. 6 shows a sketch of the graphs of f () and f (). A = f() C O B = f - () Fig. 6 (ii) Write down the coordinates of the points A, B and C. [3] 4753/ Januar 006

9 5 Section B (36 marks) 7 Fig. 7 shows the curve ln, where 0. The curve crosses the -ais at A, and has a turning point at B. The point C on the curve has -coordinate. Lines CD and BE are drawn parallel to the -ais. B C Not to scale O D E A Fig. 7 (i) Find the -coordinate of A, giving our answer in terms of e. [] (ii) Find the eact coordinates of B. [6] (iii) Show that the tangents at A and C are perpendicular to each other. [3] (iv) Using integration b parts, show that ln d ln 4 c. Hence find the eact area of the region enclosed b the curve, the -ais and the lines CD and BE. [7] [Question 8 is printed overleaf.] 4753/ Januar 006

10 8 The function f () sin has domain cos 6 p p. Fig. 8 shows the graph of f () for 0 P p. O p Fig. 8 (i) Find f ( ) in terms of f (). Hence sketch the graph of f () for the complete domain. [3] p p cos (ii) Show that f () Hence find the eact coordinates of the turning point P. ( cos ). State the range of the function f (), giving our answer eactl. [8] (iii) Using the substitution u cos or otherwise, find the eact value of Û Ù ı p sin d. cos [4] 0 - (iv) Sketch the graph of f (). [] (v) Using our answers to parts (iii) and (iv), write down the eact value of Û Ù ı p 0 sin d. - cos [] 4753/ Januar 006

12 Section A (36 marks) Solve the equation 3. [3] 6 Û p Show that Ù sin d =. [6] ı 4 0 p 3 Fig. 3 shows the curve defined b the equation arcsin ( ), for 0. 0 Fig. 3 d (i) Find in terms of, and show that cos. [3] d (ii) Hence find the eact gradient of the curve at the point where.5. [4] 4753/ June 006

13 4 Fig. 4 is a diagram of a garden pond. 3 hm Fig. 4 The volume V m 3 of water in the pond when the depth is h metres is given b V 3 p h (3 h). dv (i) Find [] dh. Water is poured into the pond at the rate of 0.0 m 3 per minute. dh (ii) Find the value of when h 0.4. [4] dt 5 Positive integers a, b and c are said to form a Pthagorean triple if a b c. (i) Given that t is an integer greater than, show that t, t and t form a Pthagorean triple. [3] (ii) The two smallest integers of a Pthagorean triple are 0 and. Find the third integer. Use this triple to show that not all Pthagorean triples can be epressed in the form t, t and t. [3] 6 The mass M kg of a radioactive material is modelled b the equation M M 0 e kt, where M 0 is the initial mass, t is the time in ears, and k is a constant which measures the rate of radioactive deca. (i) Sketch the graph of M against t. [] (ii) For Carbon 4, k Verif that after 5730 ears the mass M has reduced to approimatel half the initial mass. [] The half-life of a radioactive material is the time taken for its mass to reduce to eactl half the initial mass. (iii) Show that, in general, the half-life T is given b T ln [3] k. (iv) Hence find the half-life of Plutonium 39, given that for this material k [] 4753/ June 006 [Turn over

14 4 Section B (36 marks) 7 Fig. 7 shows the curve 3 It has a minimum at the point P. The line l is an asmptote to. the curve. P O l Fig. 7 (i) Write down the equation of the asmptote l. [] (ii) Find the coordinates of P. [6] (iii) Using the substitution u, show that the area of the region enclosed b the -ais, the curve and the lines and 3 is given b Û Ê 4ˆ Ù u + + d. u ı Ë u Evaluate this area eactl. [7] d (iv) Another curve is defined b the equation e 3 Find in terms of and b. d differentiating implicitl. Hence find the gradient of this curve at the point where. [4] 4753/ June 006

15 8 Fig. 8 shows part of the curve where f e f (), ( ) 5 sin, for all. 5 = - P p p 0 p p Fig. 8 (i) Sketch the graphs of (A) f (), (B) f ( p ). [4] (ii) Show that the -coordinate of the turning point P satisfies the equation tan 5. Hence find the coordinates of P. [6] (iii) Show that f( + ) = -e - p p 5 f( ). Hence, using the substitution u p, show that p Û - p Ù f( ) d =-e 5 Û Ù f( u) du. ı ı p Interpret this result graphicall. [You should not attempt to integrate f ().] [8] p / June 006

17 Section A (36 marks) Fig. shows the graphs of and. The point P is the minimum point of, and Q is the point of intersection of the two graphs. = Q P = - + O Fig. (i) Write down the coordinates of P. [] (ii) Verif that the -coordinate of Q is. [4] Û Evaluate Ù ln d, giving our answer in an eact form. [5] ı 3 The value V of a car is modelled b the equation V Ae kt, where t is the age of the car in ears and A and k are constants. Its value when new is 0 000, and after 3 ears its value is (i) Find the values of A and k. [5] (ii) Find the age of the car when its value is 000. [] 4 Use the method of ehaustion to prove the following result. No - or -digit perfect square ends in, 3, 7 or 8 State a generalisation of this result. [3] 5 The equation of a curve is. d (i) Show that [4] d ( ) ( ). (ii) Find the coordinates of the stationar points of the curve. You need not determine their nature. [4] OCR /0 Jan 07

18 3 6 Fig. 6 shows the triangle OAP, where O is the origin and A is the point (0, 3). The point P(, 0) moves on the positive -ais. The point Q(0, ) moves between O and A in such a wa that AQ AP 6. A(0, 3) Q(0, ) O P(, 0) Fig. 6 (i) Write down the length AQ in terms of. Hence find AP in terms of, and show that ( 3) 9. [3] d (ii) Use this result to show that d 3. [] d d (iii) When and,. Calculate at this time. dt dt [3] OCR /0 Jan 07 [Turn over

19 4 Section B (36 marks) 7 Fig. 7 shows part of the curve f(), where f ( ) = +. The curve meets the -ais at the origin and at the point P. P O Fig. 7 (i) Verif that the point P has coordinates (, 0). Hence state the domain of the function f(). [] d + 3 (ii) Show that =. [4] d + (iii) Find the eact coordinates of the turning point of the curve. Hence write down the range of the function. [4] (iv) Use the substitution u to show that 0 Û Ù + d = Û Ù u -u d u. ı ı ( ) Hence find the area of the region enclosed b the curve and the -ais. [8] OCR /0 Jan 07

20 5 8 Fig. 8 shows part of the curve f(), where f() (e ) for 0. O Fig. 8 (i) Find f (), and hence calculate the gradient of the curve f() at the origin and at the point (ln, ). [5] The function g() is defined b g( ) = ln + for 0. (ii) Show that f() and g() are inverse functions. Hence sketch the graph of g (). Write down the gradient of the curve g () at the point (, ln ). [5] Û ( ) (iii) Show that Ù e - d = e - e + + c. ı ln ( ) Hence evaluate Û Ù e - d, giving our answer in an eact form. [5] ı 0 ( ) (iv) Using our answer to part (iii), calculate the area of the region enclosed b the curve g (), the -ais and the line. [3] OCR /0 Jan 07

22 Section A (36 marks) (i) Differentiate +. [3] (ii) Show that the derivative of ln ( e ) is e. [4] Given that f() and g(), write down the composite function gf(). On separate diagrams, sketch the graphs of f() and gf(). [3] 3 A curve has equation 9. d (i) Find in terms of and. Hence find the gradient of the curve at the point A (, ). [4] d d (ii) Find the coordinates of the points on the curve at which 0. [4] d 4 A cup of water is cooling. Its initial temperature is 00 C. After 3 minutes, its temperature is 80 C. (i) Given that T 5 ae kt, where T is the temperature in C, t is the time in minutes and a and k are constants, find the values of a and k. [5] (ii) What is the temperature of the water (A) after 5 minutes, (B) in the long term? [3] 5 Prove that the following statement is false. For all integers n greater than or equal to, n 3n is a prime number. [] OCR /0 June 07

23 6 Fig. 6 shows the curve f (), where f() arctan. 3 O Fig. 6 (i) Find the range of the function f(), giving our answer in terms of p. [] (ii) Find the inverse function f (). Find the gradient of the curve f () at the origin. [5] (iii) Hence write down the gradient of arctan at the origin. [] Section B (36 marks) 7 Fig. 7 shows the curve It is undefined at a; the line a is a vertical asmptote. 3. = a O Fig. 7 (i) Calculate the value of a, giving our answer correct to 3 significant figures. [3] d 4 (ii) Show that Hence determine the coordinates of the turning points of the d ( 3 ). curve. [8] (iii) Show that the area of the region between the curve and the -ais from 0 to is 6 ln 3. [5] OCR /0 June 07 [Turn over

24 4 8 Fig. 8 shows part of the curve cos, together with a point P at which the curve crosses the -ais. O P Fig. 8 (i) Find the eact coordinates of P. [3] (ii) Show algebraicall that cos is an odd function, and interpret this result graphicall. [3] (iii) Find d d. [] (iv) Show that turning points occur on the curve for values of which satisf the equation tan. [] (v) Find the gradient of the curve at the origin. Show that the second derivative of cos is zero when 0. [4] (vi) Evaluate Û Ù ı 4 0 p cos d, giving our answer in terms of p. Interpret this result graphicall. [6] Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (OCR) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 June 07

26 Section A (36 marks) Solve the inequalit 3. [4] Find e 3 d. [4] 3 (i) State the algebraic condition for the function f() to be an even function. What geometrical propert does the graph of an even function have? [] (ii) State whether the following functions are odd, even or neither. (A) f() = 3 (B) g() =sin + cos (C) h() = + 3 [3] 4 Show that 4 + d = ln 6. [4] 5 Show that the curve = ln has a stationar point when = e. [6] 6 In a chemical reaction, the mass m grams of a chemical after t minutes is modelled b the equation m = e 0.t. (i) Find the initial mass of the chemical. What is the mass of chemical in the long term? [3] (ii) Find the time when the mass is 30 grams. [3] (iii) Sketch the graph of m against t. [] 7 Given that + + =, find d in terms of and. [5] d OCR /0 Jun08

27 8 Fig. 8 shows the curve = f(), wheref() = 3 Section B (36 marks) P is the point on the curve with -coordinate 3 π. + cos,for0 π. P =f( ) O 3 Fig. 8 (i) Find the -coordinate of P. [] (ii) Find f (). Hence find the gradient of the curve at the point P. [5] sin (iii) Show that the derivative of + cos is. Hence find the eact area of the region enclosed + cos b the curve = f(),the-ais, the -ais and the line = π. [7] 3 (iv) Show that f () =arccos( ). State the domain of this inverse function, and add a sketch of = f () to a cop of Fig. 8. [5] [Question 9 is printed overleaf.] OCR /0 Jun08

28 9 The function f() is defined b f() = 4 for. 4 (i) Show that the curve = 4 is a semicircle of radius, and eplain wh it is not the whole of this circle. [3] Fig. 9 shows a point P (a, b) on the semicircle. The tangent at P is shown. P ( ab, ) O Fig. 9 (ii) (A) Use the gradient of OP to find the gradient of the tangent at P in terms of a and b. (B) Differentiate 4 anddeducethevalueoff (a). (C) Show that our answers to parts (A) and(b) are equivalent. [6] The function g() is defined b g() =3f( ), for0 4. (iii) Describe a sequence of two transformations that would map the curve = f() onto the curve = g(). Hence sketch the curve = g(). [6] (iv) Show that if = g() then 9 + = 36. [3] Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (OCR) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jun08

30 Section A (36 marks) Solve the inequalit 3. [4] Find e 3 d. [4] 3 (i) State the algebraic condition for the function f() to be an even function. What geometrical propert does the graph of an even function have? [] (ii) State whether the following functions are odd, even or neither. (A) f() = 3 (B) g() =sin + cos (C) h() = + 3 [3] 4 Show that 4 + d = ln 6. [4] 5 Show that the curve = ln has a stationar point when = e. [6] 6 In a chemical reaction, the mass m grams of a chemical after t minutes is modelled b the equation m = e 0.t. (i) Find the initial mass of the chemical. What is the mass of chemical in the long term? [3] (ii) Find the time when the mass is 30 grams. [3] (iii) Sketch the graph of m against t. [] 7 Given that + + =, find d in terms of and. [5] d OCR /0 Jun08

31 8 Fig. 8 shows the curve = f(), wheref() = 3 Section B (36 marks) P is the point on the curve with -coordinate 3 π. + cos,for0 π. P =f( ) O 3 Fig. 8 (i) Find the -coordinate of P. [] (ii) Find f (). Hence find the gradient of the curve at the point P. [5] sin (iii) Show that the derivative of + cos is. Hence find the eact area of the region enclosed + cos b the curve = f(),the-ais, the -ais and the line = π. [7] 3 (iv) Show that f () =arccos( ). State the domain of this inverse function, and add a sketch of = f () to a cop of Fig. 8. [5] [Question 9 is printed overleaf.] OCR /0 Jun08

32 9 The function f() is defined b f() = 4 for. 4 (i) Show that the curve = 4 is a semicircle of radius, and eplain wh it is not the whole of this circle. [3] Fig. 9 shows a point P (a, b) on the semicircle. The tangent at P is shown. P ( ab, ) O Fig. 9 (ii) (A) Use the gradient of OP to find the gradient of the tangent at P in terms of a and b. (B) Differentiate 4 anddeducethevalueoff (a). (C) Show that our answers to parts (A) and(b) are equivalent. [6] The function g() is defined b g() =3f( ), for0 4. (iii) Describe a sequence of two transformations that would map the curve = f() onto the curve = g(). Hence sketch the curve = g(). [6] (iv) Show that if = g() then 9 + = 36. [3] Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (OCR) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jun08

34 Section A (36 marks) Solve the inequalit < 3. [3] (i) Differentiate cos with respect to. [3] (ii) Integrate cos with respect to. [4] 3 Given that f() = ln( ) and g() = + e, show that g() is the inverse of f(). [3] 4 Find the eact value of d, showing our working. [5] 5 (i) State the period of the function f() = + cos, where is in degrees. [] (ii) State a sequence of two geometrical transformations which maps the curve = cos onto the curve = f(). [4] (iii) Sketch the graph of = f() for 80 < < 80. [3] 6 (i) Disprove the following statement. If p > q, then p <. [] q (ii) State a condition on p and q so that the statement is true. [] 7 The variables and satisf the equation = 5. (i) Show that d d = ( ) 3. [4] Both and are functions of t. (ii) Find the value of d dt when =, = 8 and d dt = 6. [3] OCR /0 Jan09

35 3 Section B (36 marks) 8 Fig. 8 shows the curve = ln. P is the point on this curve with -coordinate, and R is the 8 point (0, 7 8 ). P O Q R 7 (0, 8 ( Fig. 8 (i) Find the gradient of PR. [3] (ii) Find d. Hence show that PR is a tangent to the curve. [3] d (iii) Find the eact coordinates of the turning point Q. [5] (iv) Differentiate ln. Hence, or otherwise, show that the area of the region enclosed b the curve = ln, the 8 -ais and the lines = and = is 59 4 ln. [7] 4 [Question 9 is printed overleaf.] OCR /0 Jan09 Turn over

36 9 Fig. 9 shows the curve = f(), where f() = The curve has asmptotes = 0 and = a. 4. O Fig. 9 a (i) Find a. Hence write down the domain of the function. [3] (ii) Show that d d = ( ) 3. Hence find the coordinates of the turning point of the curve, and write down the range of the function. [8] The function g() is defined b g() =. (iii) (A) Show algebraicall that g() is an even function. (B) Show that g( ) = f(). (C) Hence prove that the curve = f() is smmetrical, and state its line of smmetr. [7] Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (OCR) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jan09

38 Section A (36 marks) Evaluate 6 π 0 sin 3 d. [3] A radioactive substance decas eponentiall, so that its mass M grams can be modelled b the equation M = Ae kt, where t is the time in ears, and A and k are positive constants. (i) An initial mass of 00 grams of the substance decas to 50 grams in 500 ears. Find A and k. [5] (ii) The substance becomes safe when 99% of its initial mass has decaed. Find how long it will take before the substance becomes safe. [3] 3 Sketch the curve = arccos for. [3] 4 Fig. 4 shows a sketch of the graph of =. It meets the - and -aes at (a, 0) and (0, b) respectivel. b O a Fig. 4 Find the values of a and b. [3] 5 The equation of a curve is given b e = + sin. (i) B differentiating implicitl, find d in terms of and. [3] d (ii) Find an epression for in terms of, and differentiate it to verif the result in part (i). [4] 6 Given that f() = +, show that ff() =. Hence write down the inverse function f (). What can ou deduce about the smmetr of the curve = f()? [5] OCR /0 Jun09

39 3 7 (i) Show that (A) ( )( + + ) = 3 3, (B) ( + ) = + +. [4] (ii) Hence prove that, for all real numbers and, if > then 3 > 3. [3] Section B (36 marks) 8 Fig. 8 shows the line = and parts of the curves = f() and = g(), where f() = e, g() = + ln. The curves intersect the aes at the points A and B, as shown. The curves and the line = meet at the point C. = C = f( ) B O A = g( ) Fig. 8 (i) Find the eact coordinates of A and B. Verif that the coordinates of C are (, ). [5] (ii) Prove algebraicall that g() is the inverse of f(). [] (iii) Evaluate f() d, giving our answer in terms of e. [3] 0 (iv) Use integration b parts to find ln d. Hence show that g() d = e e. [6] (v) Find the area of the region enclosed b the lines OA and OB, and the arcs AC and BC. [] OCR /0 Jun09 Turn over

40 4 9 Fig. 9 shows the curve = 3. P is a turning point, and the curve has a vertical asmptote = a. P = a O Fig. 9 (i) Write down the value of a. (ii) Show that d (3 ) = d (3 ). [] [3] (iii) Find the eact coordinates of the turning point P. Calculate the gradient of the curve when = 0.6 and = 0.8, and hence verif that P is a minimum point. [7] (iv) Using the substitution u = 3, show that 3 d = 7 (u + + u ) du. Hence find the eact area of the region enclosed b the curve, the -ais and the lines = 3 and =. [7] Copright Information OCR is committed to seeking permission to reproduce all third-part content that it uses in its assessment materials. OCR has attempted to identif and contact all copright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copright acknowledgements are reproduced in the OCR Copright Acknowledgements Booklet. This is produced for each series of eaminations, is given to all schools that receive assessment material and is freel available to download from our public website ( after the live eamination series. If OCR has unwittingl failed to correctl acknowledge or clear an third-part content in this assessment material, OCR will be happ to correct its mistake at the earliest possible opportunit. For queries or further information please contact the Copright Team, First Floor, 9 Hills Road, Cambridge CB PB. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jun09

42 Section A (36 marks) Solve the equation e 5e = 0. [4] The temperature T in degrees Celsius of water in a glass t minutes after boiling is modelled b the equation T = 0 + be kt, where b and k are constants. Initiall the temperature is 00 C, and after 5 minutes the temperature is 60 C. (i) Find b and k. [4] (ii) Find at what time the temperature reaches 50 C. [] 3 (i) Given that = 3 + 3, use the chain rule to find d in terms of. [3] d (ii) Given that 3 = + 3, use implicit differentiation to find d in terms of and. Show that this d result is equivalent to the result in part (i). [4] 4 Evaluate the following integrals, giving our answers in eact form. (i) (ii) d. + d. [3] [5] 5 The curves in parts (i) and (ii) have equations of the form = a + b sin c, where a, b and c are constants. For each curve, find the values of a, b and c. (i) 3 3 [] (ii) [] OCR /0 Jan0

43 6 Write down the conditions for f() to be an odd function and for g() to be an even function. 3 Hence prove that, if f() is odd and g() is even, then the composite function gf() is even. [4] 7 Given that arcsin = arccos, prove that + =. [Hint: let arcsin = θ.] [3] 8 Fig. 8 shows part of the curve = cos 3. The curve crosses the -ais at O, P and Q. Section B (36 marks) O P Q Fig. 8 (i) Find the eact coordinates of P and Q. [4] (ii) Find the eact gradient of the curve at the point P. Show also that the turning points of the curve occur when tan 3 = 3. [7] (iii) Find the area of the region enclosed b the curve and the -ais between O and P, giving our answer in eact form. [6] [Question 9 is printed overleaf.] OCR /0 Jan0 Turn over

44 4 9 Fig. 9 shows the curve = f(), where f() = + for the domain 0. O Fig. 9 (i) Show that f () = 6, and hence that f() is an increasing function for > 0. [5] ( + ) (ii) Find the range of f(). [] (iii) Given that f () = 6 8 ( + ) 3, find the maimum value of f (). [4] The function g() is the inverse function of f(). (iv) Write down the domain and range of g(). Add a sketch of the curve = g() to a cop of Fig. 9. [4] (v) Show that g() = +. [4] Copright Information OCR is committed to seeking permission to reproduce all third-part content that it uses in its assessment materials. OCR has attempted to identif and contact all copright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copright acknowledgements are reproduced in the OCR Copright Acknowledgements Booklet. This is produced for each series of eaminations, is given to all schools that receive assessment material and is freel available to download from our public website ( after the live eamination series. If OCR has unwittingl failed to correctl acknowledge or clear an third-part content in this assessment material, OCR will be happ to correct its mistake at the earliest possible opportunit. For queries or further information please contact the Copright Team, First Floor, 9 Hills Road, Cambridge CB GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jan0

46 Section A (36 marks) Evaluate 6 π 0 cos 3 d. [3] Given that f() = and g() = +, sketch the graphs of the composite functions = fg() and = gf(), indicating clearl which is which. [4] 3 (i) Differentiate + 3. [3] (ii) Hence show that the derivative of + 3 is [4] 4 A piston can slide inside a tube which is closed at one end and encloses a quantit of gas (see Fig. 4) cm Fig. 4 The pressure of the gas in atmospheric units is given b p = 00, where cm is the distance of the piston from the closed end. At a certain moment, = 50, and the piston is being pulled awa from the closed end at 0 cm per minute. At what rate is the pressure changing at that time? [6] 5 Given that 3 =, show that d d = 3. Hence show that the curve 3 = has a stationar point when = 8. [7] 6 The function f() is defined b f() = + sin 3, π 6 π 6. You are given that this function has an inverse, f (). Find f () and its domain. [6] 7 State whether the following statements are true or false; if false, provide a counter-eample. (i) If a is rational and b is rational, then a + b is rational. (ii) If a is rational and b is irrational, then a + b is irrational. (iii) If a is irrational and b is irrational, then a + b is irrational. [3] OCR /0 Jun0

47 8 Fig. 8 shows the curve = 3 ln +. 3 Section B (36 marks) The curve crosses the -ais at P and Q, and has a turning point at R. The -coordinate of Q is approimatel.05. R O P Q Fig. 8 (i) Verif that the coordinates of P are (, 0). [] (ii) Find the coordinates of R, giving the -coordinate correct to 3 significant figures. Find d, and use this to verif that R is a maimum point. [9] d (iii) Find ln d. Hence calculate the area of the region enclosed b the curve and the -ais between P and Q, giving our answer to significant figures. [7] [Question 9 is printed overleaf.] OCR /0 Jun0 Turn over

50 Section A (36 marks) Given that = 3 +, find d d. [4] Solve the inequalit + 4. [4] 3 The area of a circular stain is growing at a rate of mm per second. Find the rate of increase of its radius at an instant when its radius is mm. [5] 4 Use the triangle in Fig. 4 to prove that sin θ + cos θ =. For what values of θ is this proof valid? [3] C A Fig. 4 B 5 (i) On a single set of aes, sketch the curves = e and = e. [3] (ii) Find the eact coordinates of the point of intersection of these curves. [5] 6 A curve is defined b the equation ( + ) = 4. The point (, ) lies on this curve. B differentiating implicitl, show that d d = +. Hence verif that the curve has a stationar point at (, ). [4] OCR /0 Jan

51 3 7 Fig. 7 shows the curve = f(), where f() = + arctan,. The scales on the - and -aes are the same. O Fig. 7 (i) Find the range of f, giving our answer in terms of π. [3] (ii) Find f (), and add a sketch of the curve = f () to the cop of Fig. 7. [5] OCR /0 Jan Turn over

52 8 (i) Use the substitution u = + to show that where a and b are to be found. Hence evaluate 0 4 Section B (36 Marks) 3 + d = Fig. 8 shows the curve = ln( + ). 0 a b (u 3u + 3 u ) du, 3 d, giving our answer in eact form. [7] + O Fig. 8 (ii) Find d d. Verif that the origin is a stationar point of the curve. [5] (iii) Using integration b parts, and the result of part (i), find the eact area enclosed b the curve = ln( + ), the -ais and the line =. [6] OCR /0 Jan

53 9 Fig. 9 shows the curve = f(), where f() = and = π. 5 cos, π < < π, together with its asmptotes = π O Fig. 9 (i) Use the quotient rule to show that the derivative of sin cos is cos. [3] (ii) Find the area bounded b the curve = f(), the -ais, the -ais and the line = π. [3] 4 The function g() is defined b g() = f( + 4 π). (iii) Verif that the curves = f() and = g() cross at (0, ). [3] (iv) State a sequence of two transformations such that the curve = f() is mapped to the curve = g(). On the cop of Fig. 9, sketch the curve = g(), indicating clearl the coordinates of the minimum point and the equations of the asmptotes to the curve. [8] (v) Use our result from part (ii) to write down the area bounded b the curve = g(), the -ais, the -ais and the line = π. [] 4 OCR /0 Jan

54 6 BLANK PAGE OCR /0 Jan

55 7 BLANK PAGE OCR /0 Jan

56 8 Copright Information OCR is committed to seeking permission to reproduce all third-part content that it uses in its assessment materials. OCR has attempted to identif and contact all copright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copright acknowledgements are reproduced in the OCR Copright Acknowledgements Booklet. This is produced for each series of eaminations and is freel available to download from our public website ( after the live eamination series. If OCR has unwittingl failed to correctl acknowledge or clear an third-part content in this assessment material, OCR will be happ to correct its mistake at the earliest possible opportunit. For queries or further information please contact the Copright Team, First Floor, 9 Hills Road, Cambridge CB GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jan

58 Section A (36 marks) Solve the equation =. [4] Given that f() = ln and g() = e, find the composite function gf(), epressing our answer as simpl as possible. [3] 3 (i) Differentiate ln, simplifing our answer. [4] (ii) Using integration b parts, show that ln d = ( + ln ) + c. [4] 4 The height h metres of a tree after t ears is modelled b the equation where a, b and k are positive constants. h = a be kt, (i) Given that the long-term height of the tree is 0.5 metres, and the initial height is 0.5 metres, find the values of a and b. [3] (ii) Given also that the tree grows to a height of 6 metres in 8 ears, find the value of k, giving our answer correct to decimal places. [3] 5 Given that = + 4, show that d d = (5 + ) + 4. [5] 6 A curve is defined b the equation sin + cos = 3. (i) Verif that the point P ( 6 π, π) lies on the curve. [] 6 (ii) Find d d in terms of and. Hence find the gradient of the curve at the point P. [5] 7 (i) Multipl out (3 n + )(3 n ). [] (ii) Hence prove that if n is a positive integer then 3 n is divisible b 8. [3] OCR /0 Jun

59 3 Section B (36 marks) 8 Fig. 8 shows the curve = f(), where f() = Fig. 8 e + e +. (i) Show algebraicall that f() is an even function, and state how this propert relates to the curve = f(). [3] (ii) Find f (). (iii) Show that f() = e (e + ). [3] [] (iv) Hence, using the substitution u = e +, or otherwise, find the eact area enclosed b the curve = f(), the -ais, and the lines = 0 and =. [5] (v) Show that there is onl one point of intersection of the curves = f() and = 4 e, and find its coordinates. [5] [Question 9 is printed overleaf.] OCR /0 Jun Turn over

60 4 9 Fig. 9 shows the curve = f(). The endpoints of the curve are P ( π, ) and Q (π, 3), and f() = a + sin b, where a and b are constants. 3 = f( ) Q (, 3) P (, ) Fig. 9 (i) Using Fig. 9, show that a = and b =. [3] (ii) Find the gradient of the curve = f() at the point (0, ). Show that there is no point on the curve at which the gradient is greater than this. [5] (iii) Find f (), and state its domain and range. Write down the gradient of = f () at the point (, 0). [6] (iv) Find the area enclosed b the curve = f(), the -ais, the -ais and the line = π. [4] Copright Information OCR is committed to seeking permission to reproduce all third-part content that it uses in its assessment materials. OCR has attempted to identif and contact all copright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copright acknowledgements are reproduced in the OCR Copright Acknowledgements Booklet. This is produced for each series of eaminations and is freel available to download from our public website ( after the live eamination series. If OCR has unwittingl failed to correctl acknowledge or clear an third-part content in this assessment material, OCR will be happ to correct its mistake at the earliest possible opportunit. For queries or further information please contact the Copright Team, First Floor, 9 Hills Road, Cambridge CB GE. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge. OCR /0 Jun

### Tuesday 18 June 2013 Morning

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### * * MATHEMATICS (MEI) 4753/01 Methods for Advanced Mathematics (C3) ADVANCED GCE. Thursday 15 January 2009 Morning. Duration: 1 hour 30 minutes

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### Applications of Advanced Mathematics (C4) Paper A TUESDAY 22 JANUARY 2008

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### * * MATHEMATICS (MEI) 4753/01 Methods for Advanced Mathematics (C3) ADVANCED GCE. Friday 5 June 2009 Afternoon. Duration: 1 hour 30 minutes.

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### MATHEMATICS 4723 Core Mathematics 3

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### * * MATHEMATICS 4721 Core Mathematics 1 ADVANCED SUBSIDIARY GCE. Monday 11 January 2010 Morning QUESTION PAPER. Duration: 1 hour 30 minutes.

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### MATHEMATICS 4722 Core Mathematics 2

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### MATHEMATICS 4728 Mechanics 1

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### MATHEMATICS 4725 Further Pure Mathematics 1

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### Monday 6 June 2016 Afternoon

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OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 4752 Concepts for Advanced Mathematics (C2)

### 4754A * * A A. MATHEMATICS (MEI) Applications of Advanced Mathematics (C4) Paper A ADVANCED GCE. Friday 14 January 2011 Afternoon

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### Wednesday 25 May 2016 Morning

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### Thursday 12 June 2014 Afternoon

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### Wednesday 18 May 2016 Morning

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### Monday 14 January 2013 Morning

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### Friday 21 June 2013 Morning

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### Tuesday 10 June 2014 Morning

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### Wednesday 30 May 2012 Afternoon

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### Monday 10 June 2013 Morning

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### * * MATHEMATICS (MEI) 4751 Introduction to Advanced Mathematics (C1) ADVANCED SUBSIDIARY GCE. Monday 11 January 2010 Morning QUESTION PAPER

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### Two boats, the Rosemary and the Sage, are having a race between two points A and B. t, where 0 t (i) Find the distance AB.

5 8 In this question, positions are given relative to a fixed origin, O. The x-direction is east and the y-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are

### MEI STRUCTURED MATHEMATICS 4757

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### 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 72.

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### * * MATHEMATICS (MEI) 4757 Further Applications of Advanced Mathematics (FP3) ADVANCED GCE. Wednesday 9 June 2010 Afternoon PMT

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### B294A. MATHEMATICS B (MEI) Paper 4 Section A (Higher Tier) GENERAL CERTIFICATE OF SECONDARY EDUCATION. Friday 15 January 2010 Morning WARNING

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### Wednesday 3 June 2015 Morning

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### * * MATHEMATICS (MEI) 4764 Mechanics 4 ADVANCED GCE. Thursday 11 June 2009 Morning. Duration: 1 hour 30 minutes. Turn over

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### Thursday 26 May 2016 Morning

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### Wednesday 11 January 2012 Morning

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### Thursday 11 June 2015 Afternoon

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### * * MATHEMATICS 4732 Probability & Statistics 1 ADVANCED SUBSIDIARY GCE. Wednesday 27 January 2010 Afternoon. Duration: 1 hour 30 minutes.

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### Thursday 4 June 2015 Morning

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### * * MATHEMATICS (MEI) 4761 Mechanics 1 ADVANCED SUBSIDIARY GCE. Wednesday 21 January 2009 Afternoon. Duration: 1 hour 30 minutes.

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### Monday 16 January 2012 Morning

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### Thursday 9 June 2016 Morning

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