This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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‘espend 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 — ees 69). . . , . . tr t r a  ~
Find a similar expressmn for cos ' 9. and hence express cos ’6‘ — sin :9 In the 10m] nees 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 ...
View
Full
Document
 Spring '11
 no

Click to edit the document details