Methods for Advanced Mathematics (C3) FRIDAY 11 JANUARY 2008

Transcription

1 ADVANCED GCE 4753/0 MATHEMATICS (MEI) Methods for Advanced Mathematics (C3) FRIDAY JANUARY 008 Additional materials: Answer Booklet (8 pages) Graph paper MEI Eamination Formulae and Tables (MF) Morning Time: hour 30 minutes INSTRUCTIONS TO CANDIDATES Write your name in capital letters, your Centre Number and Candidate Number in the spaces provided on the Answer Booklet. Read each question carefully and make sure you know what you have to do before starting your answer. Answer all the questions. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the contet. INFORMATION FOR CANDIDATES 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 7. You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used. This document consists of 4 printed pages. OCR 008 [M/0/65] OCR is an eempt Charity [Turn over

2 Section A (36 marks) Differentiate The functions f() and g() are defined for all real numbers by f() =, g() =. (i) Find thecompositefunctionsfg() and gf(). [3] (ii) Sketch the curves y = f(), y = fg() and y = gf(), indicating clearly which is which. [] 3 The profit P made by a company in its nth year is modelled by the eponential function P = Ae bn. In the first year (when n = ), the profit was In the second year, the profit was (i) Show that e b =.6, and find b and A. [6] (ii) What does this model predict the profit to be in the 0th year? [] 4 When the gas in a balloon is kept at a constant temperature, the pressure P in atmospheres and the volume V m 3 are related by the equation P = k V, where k is a constant. [This is known as Boyle s Law.] When the volume is 00 m 3, the pressure is 5 atmospheres, and the volume is increasing at a rate of 0 m 3 per second. (i) Show that k = 500. [] (ii) Find dp dv in terms of V. [] (iii) Find the rate at which the pressure is decreasing when V = (i) Verify the following statement: p is a prime number for all prime numbers p less than. [] (ii) Calculate 3 89, and hence disprove this statement: p is a prime number for all prime numbers p. [] OCR /0 Jan08

3 3 6 Fig. 6 shows the curve e y = + y. y O P Fig. 6 (i) Show that dy d = e y. (ii) Hence find to 3 significant figures the coordinates of the point P, shown in Fig. 6, where the curve has infinite gradient. Section B (36 marks) 7 A curve is defined by the equation y = ln( + ). (i) Find dy and hence verify that the origin is a stationary point of the curve. d (ii) Find d y, and use this to verify that the origin is a minimum point. [5] d (iii) Using the substitution u = +, showthat + d = (u + u ) du. Hence evaluate 0 d,givingyouranswerinaneactform. [6] + (iv) Usingintegrationbypartsandyouranswertopart(iii), evaluate ln( + ) d. 0 OCR /0 Jan08 [Turn over

4 8 Fig. 8 shows the curve y = f(), wheref() = + sin for 4 π 4 π. 4 y 4 O 4 Fig. 8 (i) State a sequence of two transformations that would map part of the curve y = sin onto the curve y = f(). (ii) Find the area of the region enclosed by the curve y = f(),the-ais and the line = π. 4 (iii) Find the gradient of the curve y = f() at the point (0, ). Hence write down the gradient of the curve y = f () at the point (, 0). (iv) State the domain of f (). Addasketchofy = f () to a copy of Fig. 8. [3] (v) Find an epression for f (). [] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Eaminations Syndicate (UCLES), which is itself a department of the University of Cambridge. OCR /0 Jan08

5 4753 Mark Scheme January (C3) Methods for Advanced Mathematics Section A /3 y= (+ 6 ) dy ( 6 ) = +. d 3 /3 = 4 (+ 6 ) /3 (i) fg() = f( ) = ( ) gf() = g( ) =. [3] chain rule used /3 3 u cao (must resolve /3 ) Mark final answer forming a composite function mark final answer If fg and gf the wrong way round, A0A0 (ii) - gf() fg() ft ft [] fg must have (, 0)labelled (or inferable from scale). Condone no y-intercept, unless wrong gf must have (0, ) labelled (or inferable from scale) Condone no -intercepts, unless wrong Allow ft only if fg and gf are correct but wrong way round. 3 (i) When n =, = A e b when n =, = A e b b 6000 Ae b = = e b 0000 Ae e b =.6 b = ln.6 = A = 0000/.6 = 650. (ii) When n = 0, P = 650 e = 75,550,000 E [6] [] soi soi eliminating A (do not allow verification) SCB if initial B s are missing, and ratio of years =.6 = e b ln.6 or 0.47 or better (mark final answer) cao allow recovery from ineact b s substituting n = 0 into their equation with their A and b Allow answers from to (i) 5 = k/00 k = 500* E [] (ii) dp 500 = 500V = dv V [] NB answer given ( )V o.e. allow k/v (iii) dp dp dv =. dt dv dt When V = 00, dp/dv = 500/0000 = 0.05 dv/dt = 0 dp/dt = = 0.5 So P is decreasing at 0.5 Atm/s ft chain rule (any correct version) (soi) (soi) 0.5 cao 7

6 4753 Mark Scheme January 008 5(i) p =, p = 3, prime p = 3, p = 7, prime p = 5, p = 3, prime p = 7, p = 7, prime E [] Testing at least one prime testing all 4 primes (correctly) Must comment on answers being prime (allow ticks) Testing p = is E0 (ii) 3 89 = 047 = is prime, 047 is not So statement is false. E [] must state or imply that is prime (p = is sufficient) 6 (i) e y = + y y dy dy e = + d d y dy (e ) = d dy = * y d e E Implicit differentiation allow one slip (but with dy/d both sides) collecting terms (ii) Gradient is infinite when e y = 0 e y = ½ y = ln ½ y = ½ ln ½ = (3 s.f.) = e y y = ½ ( 0.347) = = 0.90 must be to 3 s.f. substituting their y and solving for cao must be to 3 s.f., but penalise accuracy once only. 8

7 4753 Mark Scheme January 008 Section B 7(i) y = ln( + ) dy = + ln( + ) d + When = 0, dy/d = 0 + ln = 0 origin is a stationary point. E product rule d/d(ln(+)) = /(+) soi www (i.e. from correct derivative) (ii) d y ( + ).. = + d ( + ) + = + ( + ) + When = 0, d y/d = + = 4 > 0 (0, 0) is a min point ft E [5] Quotient or product rule on their /( + ) correctly applied to their /(+) o.e., e.g. 4+ cao ( + ) substituting = 0 into their d y/d www dep previous (iii) Let u = + du = d ( u ) d = du + u = ( u u+ ) du u = ( u + ) du * u d = ( u + ) du 0+ u ln u u u = + E ( u ) u www (but condone du omitted ecept in final answer) changing limits (or substituting back for and using 0 and ) + ln u u u = 4 + ln ( ½ + ln ) = ln ½ [6] substituting limits (consistent with u or ) cao (iv) A= ln( + ) d 0 Parts: u = ln( + ), du/d = /( + ) dv/d = v = = ln( + ) d = ln ln + ½ = ½ soi substituting their ln ½ for cao d 0 + 9

8 4753 Mark Scheme January (i) Stretch in -direction s.f. ½ translation in y-direction unit up (in either order) allow contraction dep stretch allow move, shift, etc direction can be inferred from 0 or 0 dep translation. 0 alone is A0 (ii) π /4 A = ( + sin d ) π /4 π /4 = cos π /4 = π/4 ½ cos π/ + π/4 + ½ cos ( π/) = π/ correct integral and limits. Condone d missing; limits may be implied from subsequent working. substituting their limits (if zero lower limit used, must show evidence of substitution) or.57 or better cao (www) (iii) y = + sin dy/d = cos When = 0, dy/d = So gradient at (0, ) on f() is gradient at (, 0) on f () = ½ ft ft differentiating allow error (but not + cos ) If, then must show evidence of using reciprocal, e.g. / (iv) Domain is 0. y Allow 0 to, but not 0 < < or y instead of clear attempt to reflect in y = correct domain indicated (0 to ), and reasonable shape π/4 O π/4 [3] (v) y = + sin y = + sin y sin y = y = arcsin( ) y = ½ arcsin( ) [] or sin = y cao 0

9 Report on the Units taken in January : Methods for Advanced Mathematics (C3) (Written Eamination) General Comments The paper proved to be a fair test of students ability. There were plenty of accessible marks, and even weak candidates managed to score around 0 marks. There were also many ecellent scripts over 60 marks. The scripts suggested signs of improvement on some topics which have caused problems in the past, such as implicit differentiation and inverse trigonometric functions. Although virtually all candidates attempted all the questions, there were signs in question 8 of a few candidates running out of time. The standard of presentation was variable. Algebraic immaturity is a common source of problems. For eample, the inability to eliminate variables from pairs of simultaneous equations in question 3 and errors in simplification of epressions in questions and 7(ii), were common sources of weakness. Another general point is candidates sensitivity to command words such as show and verify, for eample using verification in question 3(i) and showing in 7(i). Comments on Individual Questions Section A ) This was a straightforward test of the chain rule, in which most competent candidates scored full marks. Sources of error were using the wrong inde for the cube root, derivative formula errors, and faulty simplification of the final result. ) This question was well done, with nearly all candidates applying f and g in the correct order for each composite function, and sketching the resulting quadratic functions. In the sketch, only the -intercept for fg and y-intercept for gf were required, though some burned their boats by getting the other intercept wrong. 3) Weaker candidates made heavy weather of eliminating A and establishing e b =.6 many substituted the given result prematurely. However, nearly all candidates scored for A = 650 and b = 0.470, and the easy marks for part (ii). The answers to part (ii) were often written to improbable degrees of accuracy, though this was not penalised. 4) This question was less successfully answered. Part (i) was a gift of a mark for everyone, but the easy derivative in part (ii) was disappointingly done, the main errors being (v-k)/v and k lnv. The use of the chain rule in part (iii) was less secure, with many candidates thinking that the rate of change of P was dp/dv. 5) This little 4-mark question rarely achieved full marks. In part (i), we penalised those who tested p = clearly many candidates classify as a prime. In part (ii), although most candidates showed that = 047 for, many failed to complete the proof convincingly by pointing out that is prime, and 047 is not. 0

10 Report on the Units taken in January 008 6) The implicit differentiation in part (i) was usually either 0 marks or 4 marks, with many candidates scoring full marks. Part (ii) was less well done, with numerator = 0, or numerator = denominator, common errors. A fair number of candidates lost an accuracy mark by writing = 0.9 instead of Section B 7) Most candidates scored over half marks for this question, but a lot of marks were lost if they failed to find the correct derivative of ln( + ). (i) (ii) (iii) (iv) The product rule was well known, though the derivative of ln( + ) was occasionally wrong / a common error. It is important that candidates understand the meaning of command words such as verify : having found dy/d, a significant number of students tried to solve dy/d = 0 instead of verifying that =0 is a solution. Some candidates lost their way with the second derivative by combining the fractions instead of differentiating them separately, then using the quotient rule on the more complicated epression. The quotient rule was generally well done. The second derivative rule was also well known, though the final E mark was reserved for better candidates who had managed the algebra securely. This was a fairly easy integration by substitution, with the result given. However, candidates needed to show du = d, and to epand (u ), to gain the E mark. Integration errors were usually from the /u term, and some candidates got the limits for u and mied up. There were plenty of correct applications of integration by parts, but occasionally flawed notation led to the substitution of the answer from part (iii) incorrectly for eample, it was included inside the brackets with the limits resulting in it being added and subtracted. 8) This question tested a variety of topics, and rarely gained full marks. Some attempts showed evidence of rushing through lack of time. (i) (ii) (iii) To gain full marks, we wanted the transformations described correctly as translation and stretch, and the directions clearly indicated. Getting the directions mied up was quite a common error. A surprising number of candidates, even competent ones, thought that the limits of this integral were from 0 (rather than π/4) to π/4, and integration errors were quite common. The first three marks were very straightforward, though miing derivative and integral results for trigonometric functions is quite common. The derivative of the inverse function as the reciprocal was often not known some candidates tried to derive the inverse function, and others confused this with the perpendicular gradient result, giving the answer as ½ instead of ½.

11 Report on the Units taken in January 008 (iv) (v) Domains and ranges often cause problems, and correct domains here were relatively rare. Most candidates attempted a reflection in y = and gained, but few showed the correct domain for the inverse function for the. Many candidates got these two marks, and even weaker ones usually got for attempts to invert. The most common error was confusing the inverse with the reciprocal of the function.