Cambridge A Level Mathematics 9709 — 2002 May/June Paper 3 · Variant 1

CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level General Certificate of Education Advanced Level HIGHER MATHEMATICS 8719/3 MATHEMATICS 9709/3 PAPER 3 Pure Mathematics 3 (P3) MAY/JUNE SESSION 2002 1 hour 45 minutes Additional materials: Answer paper Graph paper List of Formulae (MF9) TIME = 1: hour 45 minutes INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper/answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. 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 75. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This question paper consists of 4 printed pages. UNIVERSITY of CAMBRIDGE © CIE 2002 Local Examinations Syndicate [Turn over

Prove the identity cot @ — tan @ = 2cot20. [3] = 3 Expand (1 — 3x) 3 in ascending powers of x, up to and including the term in x3, simplifying the coefficients. [4] The polynomial x* + 4x” + x + a is denoted by p(x). It is given that (x? + x + 2) is a factor of p(x). Find the value of a and the other quadratic factor of p(x). [4] The sequence of values given by the iterative formula 2 Xn = 3(s, +3 with initial value x,= 1, converges to a. @_ Use this formula to find a correct to 2 decimal places, showing the result of each iteration. [3] (ii) State an equation satisfied by a, and hence find the exact value of o. [2] The equation of a curve is y = 2cosx + sin 2x. Find the x-coordinates of the stationary points on the curve for which 0 < x < 2, and determine the nature of each of these stationary points. {7] 4x Tel = Ga pes De (i) Express f(x) in partial fractions. (5] 1 (ii) Hence show that i f(x) dx = 1 — In2. [5] ‘0 9709/S/M1S02

7 3 In a certain chemical process a substance is being formed, and t minutes after the start of the process there are m grams of the substance present. In the process the rate of increase of m is proportional to (50 — my’. When ¢ = 0, m= 0 and = 5, (i) Show that m satisfies the differential equation dm _ pn? a7 0.002(50 — m). {2] (ii) Solve the differential equation, and show that the solution can be expressed in the form mas 20, a (iii) Calculate the mass of the substance when t = 10, and find the time taken for the mass to increase from 0 to 45 grams. [2] (iv) State what happens to the mass of the substance as ¢ becomes very large. inal The straight line / passes through the points A and B whose position vectors are i+ k and 4i — j + 3k respectively. The plane p has equation x + 3y — 2z = 3. @ Given that / intersects p, find the position vector of the point of intersection. [4] (ii) Find the equation of the plane which contains / and is perpendicular to p, giving your answer in the form ax + by + cz =1. [6] The complex number 1 + i \/3 is denoted by u. @ Express u in the form r(cos @ + isin @), where r > 0 and —m < @ < 2. Hence, or otherwise, find the modulus and argument of w and wu’. [5] (ai) Show that u is a root of the equation z? — 2z-+ 4 = 0, and state the other root of this equation. (2] (iii) Sketch an Argand diagram showing the points representing the complex numbers i and u. Shade the region whose points represent every complex number z satisfying both the inequalities |z-i]<1 and argz> argu. [4] S70e/MAO2 [Turn over

10 The function f is defined by f(x) = (nx)? for x > 0. The diagram shows a sketch of the graph of y = f(x). The minimum point of the graph is A. The point B has x-coordinate e. (i) State the x-coordinate of A. 1) (id) Show that f’(x) = 0 at B. [4] iii) Use the substitution x = e“ to show that the area of the region bounded by the x-axis, the line x =e, and the part of the curve between A and B is given by 1 { we" du. 3] 0) (iv) Hence, or otherwise, find the exact value of this area. [3] S709/3/M/.V02

CAMBRIDGE INTERNATIONAL EXAMINATIONS JUNE 2002 GCE Advanced Level GCE Advanced Subsidiary Level MARK SCHEME MAXIMUM: MARK : 75 SYLLABUS/COMPONENT :9709 /3, 8719 /3 MATHEMATICS (Pure 3) EA University of CAMBRIDGE &F Local Examinations Syndicate

Page 1 | Mark Scheme [_Syliabus | Paper | I A & AS Level Examinations — June 2002 [ 9709, 8719 [| 3 | 1 EITHER: Express LHS in terms of cos@ and sin@ or in terms of tand MI Make sufficient relevant use of double-angle formula(e) MI Complete proof of the result Al OR: Express RHS in terms of cos@and sin@ or in terms of 1an@ MI Express RHS as the difference (or sum) of two fractions MI Complete proof of the result Al 2 {SR: an attempt ending with Jit =cot@-tan@ earns M1 BI only,] 2 EITHER: Show correct (unsimplified) version of the x or the x* or the x? term Mi Obtain correct first two terms 1 + x AL Obtain correct quadratic term 2x* Al Obtain correct cubic term uy {allow 22, 4.67, 4.66 for the coefficient) Al [The M mark may be implied by correct simplified terms, if no working is shown. It is not earned by a1 unexpanded binomial coefficients involving -+,e.g. 4, or ( ) J [An attempt to divide | by the expansion of (1 - 3x)? earns M| if the expansion has a correct (unsimplified) x, x* , or x? term and if the partial quotient contains a term in x. The remaining A marks are awarded as above.] OR: Differentiate and calculate f(0), f(0), where f(x) = k(1 - ayer! M1 Obtain correct first two terms 1 + x Al Obtain correct quadratic term 2x? Al Obtain correct cubic term Bx (allow 2, 4.67, 4.66 for the coefficient) Al 3 Attempt to find @ and/or quadratic factor by division or by inspection M1 Obtain partial quotient or factor yex Al State answer a = 6 Bi State or imply the other factor is x - x + 3 Al (The M1 is earned if division has produced a partial quotient x? + bx, or if inspection has an unknown factor x? +6x +¢ and has reached an equation in 6 and/or c.] [SR: a correct division with unresolved constant remainder can earn MIAIBOAL,] [NB: successive division by a pair of incorrect linear factors, e.g. x - land x+2orx+landx+2, can earn M1A0 or MIA{if their product is of the form x? +x+k).]

Page 2 Mark Scheme Syllabus Paper A& AS Level Examinations — June 2002 9709, 8719 3 4 5 6 (i) Use the formuta correctly at least once ML State @ = 1.26 as final answer Al Show sufficient iterations to justify @ = 1.26 10 2d.p., or shaw there is a sign change in the interval (1.255, 1,265) Al (it) State any suitable equation in one unknown e.g. x = 3 (+ +4] BL x fy State exact value of a (or x) is #2 or2? Bl Obtain derivative + 2sin x + kcos 2x or + 2sin x +k(cos” x + sin? x} M1 Equate derivative to zero and use trig formula to obtain an equation involving only one trig function M1 Obtain a correct equation of this type e.g. 2sin? x + sin x - 1 =0 or cos 2x = cos G -x) Al Obtain value x= t # (allow 0.524 radians or 30°) Al Show by any method that the corresponding point is a maximum point Al Obtain second value x =n (altow 2.62 radians or 150°) , and no others in range Al’ Determine that it corresponds to a minimum point Al . ; -_A Bo, c {i) State or imply — f(x) Gr + me ep Bi State or obtain 4 = ~3 Bl State or obtain B= Bl Use any relevant method to find C Mi Obtain C=1 Al . . A Dxt+E [Special case: allow the form + and apply the above scheme (4 = -3, D = 1, E=3).] Gxt) +? {SR: if f(x) is given an incomplete form of partial fractions, give B) for 2 form equivalent to the omission of C, or E, or B in the above, and M1 for finding one coefficient.] 2 (ii) Integrate and obtain terms. -In (3x + 1) — arp tery Bl+Bl+BiY Use limits correctly MI Obtain the given answer correctly Al 5 5

Page 3 Mark Scheme Syllabus Paper A & AS Level Examinations -— June 2002 [_9709, 8719 3 7 @ State that = 4(50 ~ m)? Justify & = 0.002 (it) Separate variables and attempt to integrate —1_ (50 -m)? 1 (S0-m) Evaluate a constant or use limits f = 0, # = 0 Obtain + and 0,002¢, or equivalent Obtain any correct form of solution e.g. Bom = 0.0021 +t Obtain given answer correctly (iit) Obtain answer m = 25 when ¢= 10 Obtain answer f= 90 when m= 45 (iv) State that m approaches 50 (i) State or imply a simplified direction vector of / is 3i- j + 2k , or equivalent State equation of / is r=i+k + AGi-j+2k), or = Fy m or equivalent Substitute in equation of p and solve for A, or one of x, y, or z Obtain point of intersection -2i + j-k [Any notation is acceptable. ] (ii) State or imply a normal vector of p is i+ 3j~- 2k EITHER: Use scalar product to obtain a + 36 — 2c = 0 Use points on / to obtain two equations ina, b,c eg. atc=1, 4a-6+3c=1 Solve simultaneous equations, obtaining one unknown Obtain one correct unknown e.g. a=-2 Obtain the other unknowns ¢.g. 6= 4, ¢ =4 OR: Use scalar product to obtain a +36 -2c =0 Use scalar product to obtain 3a -6 + 2c = Solve simultaneous equations to obtain onc ratioe.g. a:b Obtain a: 6:¢=2:-4:-5, or equivalent Obtain a=-2, b= 4, ¢=4 (NB: candidates may transfer from the E/THER to OR scheme by subtracting the two “point” equations, or transfer from OR to EITHER by finding one of the “point” equations. } OR: Calculate the vector product (3i —j + 2k)x(i + 3j - 2k) Obtain answer ~ 4i + 8j + 10k, or equivalent Substitute in ~ 4x + 8y + 10z = ato find d, or equivalent Obtain d= 6 , or equivalent Obtain a=-4,b=4,c=4 OR: State or imply a correct equation of the plane ¢.g, r= AC3i — j+2k) + pi + 3) ~2k)+i+k State 3 equations in x,y,z, A.and 4,e.g.x=3At+prl,y=-At3y,2=2A-2yw4l Eliminate A and y . Obtain equation ~4x +8) +10z = 6, or equivalent ka Obtain az~t,b=9.c5 [SR: condone the use of xi + yj + zk for ai + 6 + ck in the E/THER scheme and the first OR scheme.] Bl Bl MI Al ML Al Al Bl Bl Bi Bl Bw Mi Al BI Ml Bid Al Al MI Bit Al Al MI Alt M1 Al Al M1 Alf Mi Al

Page 4 Mark Scheme [_ Syllabus Paper A&AS Level Examinations — June 2002 [ 9709, 8719 3 9 @) State or imply that ¢ = 2 BL State or imply that @ = 5 7 (allow 1.05 radians or 60°) Bl Obtain modulus 4, and argument 2 7 of u* (allow 27: 2.09 or 2.10 radians or 120°) Bl+Bl4 Obtain modulus 8 and argument zof u° (allow 2°; 3.14 or 3.15 radians or 180°) BIS [Fotlow through on wrong rand @] [SR: if u?.and w? are only given in polar form, give BI forw* and BLY’ foru?.) (ii) EITHER: Deduce that u? ~2u+4=0 from wu? +8=0 OR: Verify that u? - 2u + 4=0 by calculation Bl State that the other root is 1 ~ i 3, or equivalent Bl [NB: stating that the roots are it i¥3 is sufficient for both B marks.] Show both points correctly on an Argand diagram Bl Show the correct relevant circle BI Show line (segment) correctly Bl Shade the correct region Bl {SR: allow work on separate diagrams to be eligible for the first three B marks. ] 10 = {i) State at any stage that the x-coordinate of A is equal to 1, or that A is the point (1,0) Bl (ii) State f'{x) = 2tax , oF equivalent Bl ‘Use product or quotient rule for the next differentiation MI Obtain 2. 4 4 +2Inx. (3) , or any equivalent correct unsimplified form Al x Verify that F"(e}=0 Al ¢ State or imply area is fans? dx Bl 3 Use & =e", or equivalent, in substituting for x throughout Mi Obtain given answer cortectly (allow change of limits to be done mentally) Al (iv) Attempt the first integration by parts, going the correct way MI Obtain (a? - 2u + 2)e" , or equivalent, after two applications of the rule Al Obtain exact answer in terms of e, in any correct form, e.g. (e - 2¢ + 2e} - 2, ore - 2 Al (The substitution in (iii) may be done in reverse i.e. starting with the w integral and obtaining the x integral. The MIAL scheme applies, but only an explicit statement will earn the B1.} [The MIAIA1 in (iv) applies to those working in terms of x and obtaining x((In x) - 2 In x +2) , or equivalent,}