Bangalore University 2008 B.Sc Mathematics III Maths V - - Question Paper
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AN - 1318
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III Year B.A./B.Sc. Examination, November/December 2008
(1999-2000 and Onwards)
MATHEMATICS (Paper j V)
Time: 3 Hours i Max. Marks: 100
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Instructions : 1) Answer all the questions. ;
2) Answers should be writen completely in Kannada or in English. !
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I. Answer any twenty questions : j (20x2=40)
1) Define a ring with unity and give an example of a commutative ring without unity.
2rProve that S = over integers.
b 0
3) Prove that the subring 2Z = j 2/ e z j isan Idel of (Z, +,.)
4) If f :R R' is a homomorphism, then prove that if (0) = 0'.
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5) Express (-2, 2, 3) as linear combination of (1, -1, j3) and (-1, 1, 0) of V3(R).
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Tind whether vectors (1, 0, 1); (1, 1, 0) and (-1, 0, j-1) are linearly dependent.
7) Prove that T (xY, x2, x3) = (0, x2, x3) is a linear transformation of V3 (R) to V3 (R).
8) Find the matrix of the linear transformation T: j/2 (R)V2 (R) defined by T (x,y) - (x, - y) w.r.t. the standard basis. |
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9) Draw the graphs of 2x + 5y < 180; 4x + 2y < 80; x > 0; y > 0.
10) Define i) basic solution ii) degenerate solution. j
1.1) Prove that intersection of two convex sets is a convex set.
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12) Show that S = j x 2y = 5 j ls a convex setJ
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13) If 0.777 is the approximate value of , find the absolute error and percentage errors.
14) Find a real root of x3 - x2 -1- 0 in (1,2) using bisection method in two stages.
| P.T.O.
15) Write the Newton-Raphson formula for the solution of f(x) = 0.
16) Evaluate A (tan-1 x).
17) With usual notations, prove that A = E -1.
18) Express f(x) = 3x2 + 2x -15 in factorial notation. l)HEvaluate j
i) A2 [2x(3)] ii) A2 01 j
26)Evaluate: |
21) Write the Lagranges interpolation formula for unequally spaced arguments. 22Construct the forward difference table from thJ table
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X |
1 |
2 |
3 |
4 |
5 |
|
f(x) |
4 |
13 |
4 |
73 |
136 |
dy |
23) Find the two solutions of = 1 + xy; given y(0) = 0 by Picards method.
24) Write the Runge-Kutta formula for solving I dy
= f(x,y)at (x,y0)
dx
II. Answer any two questions : !
1) Prove that every finite integral domain is a field. I
2) Prove that the intersection of two subrings is a subring.
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If P is an integer, then prove that pz ~ G Z | *s a maximal ideal of (Z, +,.)
if and only if p is a prime. j
AN - 1318
III. Answer any two questions :
(2x5=10)
} is linearly dependent iff
1) Prove that the ordered set of vectors {ap a2,......, an
some one of a-t (2 < i < n) is a linear combination of its preceeding vectors.
2)Find the dimension and basis of the subspace spanned by the vectors (2, 4, 2);
(1, -1,0), (1,2, 1) and (0,2, l)of V,(R). |
3) Find the matrix of the linear transformation
T: V2(R) V3(R) defined by T(x, y) = (x + y, x, 3x - y) w.r.t. standard basis.
4) Find the rank and nullity of a linear transformation defined by T(x, y, z) = (y - x, y - z) and verily rank-nullity theorem.
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IV. Answer any two questions : (2x5=10)
1) Prove that any convex combination of two points ofj a convex set S is also in S.
2) Maximise graphically Z - 3x + 5y subject to the constraints x + y < 30; x - y > 0; x<20, y >3 and y <12.
3) Using simplex method, maximise P =-5x + y + 4z x-Fy + z<5;y + z<3;x-fz<8, x, y, z > 0.
subject to the constraints
A company produces two types of pens A and B, where A is a superior quality and B is a lower quality. The profit on each of A and B are Rs. 5 and Rs. 3. The raw material required for each pen A is twice as that olf B. The supplyis sufficient only for 1000 pens of B. The pen A requires a special clip and only 400 such clips are available. For the pen B, 700 clips are available. What is the production plan to get maximum profit ? Solve by graphical method.
V. Answer any three questions :
1) Obtain the approximation of loge (1 + x) in the form of a second degree polynomial and hence evaluate loge (1.2).
2) Find a real root of x4 - x -10 = 0 by Regula-Falsi method which lies between 1.8 and 2. I
3) Using Newton-Raphson method, find the root of x3 - 2x - 5 = 0 which lies between 2 and 3.
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4) Find a real root of cos x ~ xex = 0 by iterative method which is near x = . "Solve 5x - y - 2z = 142, x - 3y - z = - 30 and 2x - y - 3z = 5 by using
Gauss-elimination method.
jStjsing Jacobis iterative method, solve 10x + y + z = 12; 2x+10y + z = 13 and 2x + 2y +10 z = 14 .
(3x5=15)
(3x5=15)
Answer any three questions :
1) Using separation of symbols, show that
U0 -U, + U2 - U3 +.....= ~- U0 - j A U0 + A2 U0A3 U +
2 4 o ; 16
Tind the polynomial f (x) from the data
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X |
0 |
1 |
2 |
3 |
4 |
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f(x) |
3 |
6 |
11 |
18 |
27 |
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3)Estimate the population of a town in the year 1955 from the data. |
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Year |
1921 |
1931 |
1941 |
1951 |
1961 |
1971 |
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Population (in lakhs) |
20 |
24 |
29 |
36 |
46 |
51 |
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Find f (8) and f (15) from the data. |
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X |
4 |
5 |
7 |
10 |
11 |
13 |
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f (x) |
48 |
100 |
294 |
900 |
1210 |
2028 |
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5) Evaluate J loge x dx by Simpsons - rule for six intervals.
dy
6) Using Picards method, solve = x + y for x = 0. II. Given that y =*1 when x = 0.
dx
a,beZ sbe02x2 doojbd ?roddoo>dodD
i/seoa.
3) 2Z = j Zrj/ e 2 | doccbs), (Z, +,.) doo>d >d>eF <aodb &wo*>.
4) f : R-R' a&aed/dedrdf Add, f (0) = O' <od}(AjDQ&.
5) V3(R) (-2,2,3) <ao&oodsk(1, -1,3) doo(-1,1,0) dbdtf edraatorbdoudotoO.
6) (1,0,1); (1,1,0);(-1,0,-l)7oarddoor(od3db.
7) T: V3 (R)-> V3 (R)?3bT(xl, x2, x3) = (0, x2, x3) <aoci> tfyStop&azri, &
dQ5330d >OC>
8) T: V2 (R) -> V2 (R); T (x,y) = (x, - y) zSdtf d/ssraodd $ccb ejpad&dbrtbrssrod
u
&sedd?db tfodb&cOoO.
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9) 2x + 5y < 180; 4x + 2y < 80; x > 0; y > 0 cDoeo, .essfefrdz&rttf de<sro aasfcdb >s&.
10) S33 30V.
.1) doodOarod 2) &orfc3 dossd :
eJ j
11) <addb ercte rtorte* eed?5oo db3 Adbd <aodb
12) S = j x y+ 2y 5 r <aodb &ae.
13) -wororaddcto 0.777 ycrt'dbddtoed arto
9 & . <
14) x3 - x2 -1 = 0,dd s3s?od dbjs<yd?3b, (1,2) d<) db-Dras dbOod<addb dod<) odb 6oScCoO.
15) f(x) = 0 riaoetfdra dosrodd?fodA*bd 35?- ody ?fodcfc> wdotoo.
16) A (tarT1 x) fk tfodb&ScCo.
17) Aisdta e&rteod A = E -1 aorib.a3QAi. ;
18) f (x) = 3x2 + 2x -15 'sdsb speOcdbcf dddOwdcOoO.
19) 1) A2 [2x(3)] 2) A2 03 c) tfodb&josoO.
A2
x odbfioo2oo.
21) dnozSodb sfed sdjdb e?db oodacrort todcooo.
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22) dd dbcg eiswo dyo&>sb wdocoo.
-c t so eJ |
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X |
1 |
2 |
3 |
4 |
5 |
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f(x) |
4 |
13 |
4 |
73 |
136 |
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dy
23) dodsfo wdoiGeftA = 1 + xy; y(0) = 0 <sidd eadsk dosudnVq
dx
odb2oo!oQ.
24) = f(x,y)oi>?b (x 0, y ) aid 5ico dort- udowa.
dx
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AN - 1318 -6-
(2x5=10)
II. oiro)cradcta *>d:& wdOA):
1) Kc&raorib doaod sod Aribddodb
2) <adc> eroddoo&rte* qJed eraddoccbAdbdd <aodb ;Lpa.
3) Pz = |%Gz J dSdbe/ soQo&o* t?r|ddd, p sbdb do s>;!b
&dedraft;p*>0. . i
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4) f: R R' stoedrsedjro wcrort, f(R); R' eraddocabdod)
III. (c53dd<dci)r(av)0 (2x5=10)
1) n dotod *Osfrte>d {a,,a2,......, an} z*odo A>dtf odoo&cdtarfdedd
oatoscjDddta SoodbAiOd at (2 < i < n) odbo wdd koQcbLOdrfe?a>c& eiQedraoiraAddeb dsb ?3D5b cDod) stoeO&.
-D
2) (2,4,2); (1, -1,0); (1, 2, 1) db&(0,2, 1) Sod eruo&srtod V3 (R) erod Qd db&p> wpsd dssb dodjrarttfe&tfodb fioQcCoo.
3) T: V2(R) -> V3(R); T(x, y) = (x + y, x, 3x - y) dd djasraoddd >o3bdbd?& ocsbfcsouoo.
4) T (x, y, z) = (y - x, y - z) <odb 4i dtf djssjDoddd cos* dbdb
) 2) [ 0 C
tfOkS ri&QodD&Qocoo dodo * oos*- ctf* sdDecdbdjdb 33s?5e@.
IV. oifcs$c&dd/D <sdc> enOA): (2x5=10)
1) zodo eroc&rtrad ddo &ot)rte* erod e&set&lcdbo e? em?& rradOabc dbddodb ;sdQ*>.
2) $8 dbdjdb m)doif3eA : !
7 u\ t< I
rta&pfo&: Z = 3x + 5y
9>2Jo$rteb: x + y < 30; x ~ y > 0; x < 20,y>3db& y< 12
3) ftosdDddafoeAS) rfOC&: P = 5x + y + 4z ; w>zjo$rteb: x + y + z < 5; y + z < 3; x + z < 8, x, y, z > 0.
4) toodb odDcdbo <add) dddd s&forteftb A dbdb B dojteosbdd. A eruddo dbudo , B
7 <=< -O I _o -e eJ Q
dcsdburib. cdrsodo A dbsb B dskrteod &7t)d Rs. 5 dbdj Rs. 3. A sftart derbd &3dJb B dddddx 1000 B dbrtert sarbdddj <sd. 400 A dart, 700 dcrtebB dart o<Dd, Oa3ex>osoiod A dbB do!jsQde?b <doiood?i) rfsa tfdbaod
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