Cambridge A Level Mathematics 9709 — 2002 Oct/Nov Paper 2 · Variant 1

CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level MATHEMATICS 9709/2 PAPER 2 Pure Mathematics 2 (P2) OCTOBER/NOVEMBER SESSION 2002 1 hour 15 minutes Additional materials: Answer paper Graph paper List of Formulae (MF9) TIME 1 hour 15 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 50. 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 3 printed pages and 1 blank page. CIE 2002 [Turn over

2 1 Solve the inequality

2x
1



3x

. [4] 2 The cubic polynomial 2x3  ax2  b is denoted by f(x). It is given that (x  1) is a factor of f(x), and that when f(x) is divided by (x  2) the remainder is
5. Find the values of a and b. [5] 3 (i) Express 9x in terms of y, where y  3x. [1] (ii) Hence solve the equation 2(9x)
7(3x)  3  0, expressing your answers for x in terms of logarithms where appropriate. [5] 4 (i) By sketching a suitable pair of graphs, show that there is only one value of x in the interval 0  x  1 2π that is a root of the equation sin x  1 x2 . [2] (ii) Verify by calculation that this root lies between 1 and 1.5. [2] (iii) Show that this value of x is also a root of the equation x  (cosec x). [1] (iv) Use the iterative formula xn
1  (cosecxn) to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate. [3] 5 The angle x, measured in degrees, satisfies the equation cos(x
30 Æ)  3 sin(x
60 Æ). (i) By expanding each side, show that the equation may be simplified to (2 3) cos x  sin x. [3] (ii) Find the two possible values of x lying between 0 Æ and 360 Æ. [3] (iii) Find the exact value of cos 2x, giving your answer as a fraction. [3] 9709/2/O/N/02

3 6 (a) Find the value of
1 2π 0 (sin 2x
cos x) dx. [4] (b) The diagram shows part of the curve y  1 x
1. The shaded region R is bounded by the curve and by the lines x  1, y  0 and x  p. (i) Find, in terms of p, the area of R. [3] (ii) Hence find, correct to 1 decimal place, the value of p for which the area of R is equal to 2. [2] 7 The equation of a curve is 2x2
3y2  2xy  10. (i) Show that dy dx  y  2x 3y  x. [4] (ii) Find the coordinates of the points on the curve where the tangent is parallel to the x-axis. [5] 9709/2/O/N/02

CAMBRIDGE INTERNATIONAL EXAMINATIONS NOVEMBER 2002 GCE Advanced Subsidiary Level UNiversirTy of CAMBRIDGE Local Examinations Syndicate

Page 1 Mark Scheme Sia] Paper AS Level Examinations — November 2002 9709 1 EITHER: State or imply non-modular inequality (2x — 1)? < (3x)*, or corresponding equation BL Expand and make reasonable solution attempt at Z/pt 3-term quadratic, or equivalent Mil Obtain critical values —1.and ~ 1 Al State correct answer x<~—l, x>1 FA Al OR: State dae correct equation)for a critical value e.g. 2x ~ 1 = 3x MI [o} State two relevant equations separately e.g. 2x - 1 =3x and2x-1 Al Obtain critical values —1 and 1 Al State correct answer x<—1, x>1 . Al OR: State one critical value (probably, x =-1), from a graphical method or by inspection or by solving a linear inequality Bl State the other critical value correctly B2 State correct answer x <1, x> + BL 4 [The answer 1 <x<-I1 scores BO.] ——_— eee 2° State or obtain -2 + a+ b=0, or equivalent Bl Substitute x =—2 and equate to -5 Mi Obtain 3-term equation, or equivalent Al Solve a relevant pair of equations, obtaining a or b MI Obtain both answers a=3 and b=—1 Al 5 3 (i) State or imply that 97 =" Bl 1 (ii) Carry out recognisable solution method for quadratic in y * M1 Obtain y=4 and y=3 from 2yY-Ty+3=0 Al Use log method to solve an equation of the form 3*=k Mi In z . Obtain answer x= 8 , or exact equivalent t Ye ANY base 5 Al © State exact answer x= 1 (no penalty if logs used) Bl 5 4 (i) Make recognisable sketches over the given range of a suitable pair of graphs e.g. y = sin x andy = 4. Bl x State or imply connection between intersections and roots and justify given statement Bl 2 (ii) Calculate values (or signs) of sin x -4 at x= land x=1.5 Mi Derive given result correctly * Al 2 (iii) Rearrange sin x =i and obtain given answer Bl 1 (iv) Use the iterative formula correctly with 1< x, $1.5 M1 Obtain final answer 1.07 Al Show sufficient iterations to justify its accuracy to 3d.p., or show there is a sign change in the interval (1.065, 1.075) __ _ Al 3

Page 2 Mark Scheme Syllabus Paper AS Level Examinations - November 2002 9709 2 5 (i) Use relevant formulae for cos (x — 30°) and sin (x — 60°) { ckbow cue 5 “0 avr ] MI* Use sin 30° = cos 60° =4 and sin 60° = cos 30° -£ Mi (dep*) Collect terms and obtain given answer correctly Al 3 (ii) Carry out correct processes to evaluate a single trig ratio M1 Obtain answer 73.9° Al Obtain second answer 253.9° and no others Al/ 3 (iii) State or imply that cos*x=5 or sin? x=12 BI Use a relevant trig formula to evaluate cos 2x MI Obtain exact ariswer ~ 1 correctly Al 3 (Use of only say cos x= ae , probably from a right triangle, can eam B1MI1A0.] NM: 6 — (a) Obtain indefinite integral -t cos 2x + sin x BI+BI Use limits with attempted integral MI Obtain answer 2 correctly with no errors Al 4 (b) (i) Identify R with correct definite integral and attempt to integrate Mi Obtain indefinite integral In (x + 1) BL Obtain answer R= In (p+ 1)—In2 Al 3 (ii) Use exponential method to solve an equation of the form Inx =k MI Obtain answer p = 13.8 Al 2 7 (i State 62 as the derivative of 3° BI State + ax + 2y as the derivative of -2xy (allow any combination of signs here) Bl Equate attempted derivative of LHS to 0 (or 10) and solve for 2 Mi Obtain the given answer correctly Al 4 [The M1 is dependent on at least one of the B marks being earned.]. . (ii) State or imply the points lieon y-2r=O0 9 ¥/ (4-24 3y J =O BI ® Carry out complete method for finding one coordinate of a point of intersection of y = kx with the given curve . MI Obtain 10x? = 10 or 2 ¥ = 10 or 2-term equivalent Al Obtain one correct point e.g. (1,2) Ny J vakaes 4 me (w 4) Obtain a second correct point e.g. (-1, -2 ) ~ Al ( Al, 50