9231_w02_qp_1 - CAMBRIDGE INTERNATIONAL EXAMINATIONS...

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Unformatted text preview: CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level FU FITHEFI MATHEMATICS 9231 I 1 PAPER 1 DCTDBERJ'NOVEMBER SESSION 2002 3 hours Additional materials: Answer paper Graph paper List of Formulae {M F10) TIME 3 hours INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paperranswer booklet. Answer all the questions. Give non—exact numerical answers correct to 3 significant figures, or 1 decimal place in the case ct angles in degrees, unless a difierent level of accuracv is specified in the question. INFORMATION FOR CANDIDATES The number oi marks is given in brackets [ ] at the end of each question or part question. The use of a calculator is expected, where appropriate. Flesults obtained solely from a graphic caICulator, without supporting working or reasoning, will not receive credit. You are reminded ol the need tor clear presentation in your answers. This question paper consists of 5 printed pages and 3 blank pages. is n.- _' UNivuRsrrvqfCAMBRIDGE © CIE 2002 Local Examinations Syndicailc [Turn over Given that A. find 2 n” in terms MN and x. J'le H H _- + “I chi 1 ].-.' Hence determine the sel of values. 01‘): for which the infinite genes is, convergent and give lht: sum [0 infinity [301' cases where lhis exisls. The equation when: A is a constant. has mots; 0:. ,3. y, 35. Find a polynomial equation whose roots are Given that '3 ’3 ",I ’3 n'+;3'+y'+5'=—+ (I find the value (if/1, It is given that. for]; = If}. I. 2, 3. .. bn'nplil}.r am] It is given that, I’m' H E D, {i} Find {1 in terms; nfc. (ii) Show that {iii} Find Ii in [firms Ufa. HI 4 ill 0513'?“ +u.,+z£ + 1 l 5 j x + .13 +Ax“ + 4,: — 2 = fl. ci”:1?'2”+ 3(9)” + 20. 9231m0mi'02 — a“, and hence pmve by induction that a” is divisible by 24 for all n 1:: U. [3] [3] [6] [l] 5 3 The euwe C has polar equation HQ 2 I. for 0 c: IE3 Q 211'. sin 6 {i} Use the fact that a tends to I as 6 tends to 0 to show that the line with eartesian equation )2 = is an asymptote to C. [2] (iilI Sketch C. U] The paints P and Q on C cen‘espe-nd to 9 = 6 a' and t9 = £11 respectively. {iii} Find the area of the sector OPQ, where 0 is the origin. |3| (iv) Show that the length of the are PQ is J1 Mm; [2] 92 an" A curve has equation x3 + .1113 — y] = 3. . . . . . dv {I} Show that there Is Im point of the curve at which : 0. [4] C if d .r d2 .? (ii) Find the values (if '3 and at the point {1, —]}. [5] {it dr“ Given that z = cos 6 + isin El, shew that l {i} z——=Zisi119. [ll 2 (ii) 2" + z'” = 24:05:19. [2] Hence shew that sin‘:1 9 = %( lU — lS ens 26 + 6 ens 4B — ee-s 69). . . , . . tr t r a - ~ Find a similar expressmn for cos ' 9. and hence express cos ’6‘ — sin :9 In the 10m] nee-s 26 + I; cos (:8. [3] The value nl'the assets nl'a large commercial organisation at time :2 measured in years. is $t [03y + 109}. The variables y and r are related by the differential equation div t r _ _ g+5—' +6_i-'=15e053r— Bstnfi’n‘. dr- a: . . . . dr Find )3 In terms at t, given that y = 3 and = —2 when i" = 0. [9] t Show that, for large values of r. the value of the assets is less than $9.5 >< 10E for about a third of the time. [3] easimommz [Turn over 4 9 The planes H] and H,“ which meet in the line I, have vector equations :- = 2i+4j+6k+ Blt2i+3k)+¢l(—4j+5k), r = 2i+4j+6k+ 62(3j+k] +¢2[—i+j +2k], respectively. Find a veetor equation of the line I in the form r : a + rb. [5] Find a vector equation of the plane H: which contains .-' and which passes through the point with position vector 4i + 3j + 2k. Find also the equation of H3 in the form (LT + by + {'z = d. [4] Deduee, or prove otherwise, that the system of equations fix — 5}! — 42 = —32. 5o.- - y + 32 = 24. 9x — 2y + 52 : 4'0. has an infinite number of solutions. [3] 10 The linear translormatlion T : R4 —) lli’i'l is represented by the matrix H. where 2 —3 —5 —] 4 5 l H _ 2 3 0 —3 —3 5 7 2 {i} Find the dimension of the range space of T. [3] (ii) Find a basis for the null space ofT. [3] {iii} It is given that x satisfies the equation 2 Hit : _:ll:l —l 5 Using the fact that 1 2 —3 —lU H 1 _ —| ' —2 —15 find the least possible value of I'll _r| i h . _ J"? _ .n' .3 _2 .2 2 [Fort e vecto: x — I; , xl — (.11 +2.2 +13 +.t4).] .1; 9231HEOINJ'UE 5 11 Answer only one ofthe following two altemalives. EITHER The vector e is an eigenveelor of the square matrix C. Show that {i} e is an eigenveetor of G + kl. where I: is a sealar and I is an identity matrix, .. . . 1| (11) e :5 an eugenveetot' of G'. [51 Find the eigenvalues, and corresponding eigenvectors, of the matrices A and BE, where 3 -3 t] —5 -3 t} A: l 0 l and B: l —8 | . [9] —l 3 2 —l 3 —6 OR The curve C has equation __ (x — a) [,t‘ — b] "5 d .t.‘ — r. ' where a. b. c' are eonslants, and it is given that I} < u < b <: c". {i} Express y in the form .1’+ P+ Q . J." — (‘ giving the constants P and Q in terms of a. b and (n [3] (ii) Find the equations ofthe asymptotes of C. [2] {iii} Show that Chas [W0 stationary points. [5] (iv) Given also that a + b r:- c', sketch C, showing the flS}'ltlpl0l€.‘-§ and the coordinates oflhe points of intersection of C with the axes. [4] 9231momioz 6 BLANK PAGE 9231mommz 7 BLANK PAGE 9231mommz 8 BLANK PAGE 9231mommz ...
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9231_w02_qp_1 - CAMBRIDGE INTERNATIONAL EXAMINATIONS...

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