No. of Printed Pages : 11 BACHELOR'S DEGREE PROGRAMME

MTE-04 : ELEMENTARY ALGEBRA

MTE-05 : ANALYTICAL GEOMETRY

Instructions : 1. Students registered for both MTE-04 & MTE-05

courses should answer both the question papers in two separate answer books entering their enrolment number, course code and course title clearly on both the answer books.

2. Students who have registered for MTE-04 or MTE-05 should answer the relevant question paper after entering their enrolment number, course code and course title on the answer book.

tiilflet) 441 14)1 5114.1

71.t.t.-04 : 5n t1 eh a1Ailpti

711.t.t.-05 : ch ,reu 41:

1. * 77i.e.1.-04 ark7-Air.e.f.-05 cil-0-41WITT ralTu- ff, 047 517-4T-47 37( arov-arev 5P1W.WY 3FRT 373-A-7:17*.; 1/ wow Ida err

zugebty 717:T iv'-ffich Ik4W I 2. oa 7.27.e.-614 TIT F.e.e.-05 /# kkt

3F/4 Yggrff J-dr dri(--SPIWT ART 315-11/W, zifietow m1,5 ffen- westif 7PT 1155-61Y, ikgq5<

MTE-04/05 1 P.T.O.

MTE-04

BACHELOR'S DEGREE PROGRAMME Term-End Examination

June, 2014

MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA

Time : 1 1 hours 2

Maximum Marks : 25

(Weightage : 70%)

Note : Question no. 5 is compulsory. Do any 3 questions from Questions no. 1 to 4. Calculators are not allowed.

1. (a) Find a, b so that the system x + y + z = 6 x + 2y + 3z = 10 x + 2y + az = b has a unique solution. 3

(b) Plot in the Argand diagram 1 + 3i, 3 + i. 2

2. Solve the equation

3x4 25x3 + 50x2 50x + 12 = 0

given that the product of two of its roots is 2. 5

MTE-04 2

3. (a) Check if the following system of equations can be solved by Cramer's rule. If so, apply it for solving the system. Otherwise, solve it by the method of Gaussian elimination. 3

x + y + z = 4 x y + 2z = 3 2x + 3y z = 6

(b) Show that for any n E N, 2n > n. 2

4. (a) Prove by the principle of mathematical induction that 32n + 7 is divisible by 8 for all n E N. 3

(b) A shop sells two kinds of packs, Pack A containing 2 pencils and 3 pens and Pack B containing 4 pencils and 2 pens. Formulate as a set of linear equations the problem of finding the number of packs of each type you have to buy to get 14 pencils and 9 pens. 2

5. Which of the following statements are true and which are false ? Justify your answer with a short proof or a counter example. 10 (i) If A, B, C are three sets such that A B,

B C, then A C. (ii) All the roots of the equation x3 + px + q = 0

can be positive. MTE-04 3 P.T.O.

(iii) lxyl < I xj I yl for all x,ye R. (iv) The geometric representation of an

inconsistent set of equations is a point.

(v) If all the coefficients in a linear system of equations are positive, the solution set will also consist of positive numbers.

MTE-04 4

711.t.t.-04

trl 1 CM 4411 1 chi Shil

Trliff

2014

711.t.t.-04 : mtiNch in

/ uu2 arrw-dv aiw: 25 2 (pci WT: 70%) 4iZ : 37P. F. 5 ch

3. (*) z1111 Arr-4R -ft1R. 4-11,:nku wr -Wrzt )1-R CV-14.1 71- zfr rie I zit ti l t, ct)k * i Mr

104 11 ANR I 3T- f2TF 111\idiel ritIchtul f4R-T

C-A-R I 3 x+y+z=4 x y + 2z = 3 2x + 3y z = 6

n E N* R'R tUFIR .k2n > n 2

4. (*) Trrurtzr aTrrri:R * 11 clIk;. 7 fp-4

n E N 32n + 7, 8 A t I

-S*T4 SichR t, th-di-

A 142 44ra atIT 3 tai Ilh#M3 4 ift0 afIT 2 4q t I ZIR aTrcr 14 4W * 9 4q wft-4-41 aTrq skT mew

1*--dt ifteezrt Tgft-4, r Trm:Err

tru- cnkui Awr-zr*q Tkr-i-d A=rr-4R I

5 cl gci *24-4 fir4 2.rr ci.)1-14 ? aTET4 drR Tiff 31:FAerr - IcgWuI gRI aR.

Atr7R I (i) eir4A,B,C4k#1(.1 -ccleltft AZB,BC,

(ii) (i41chtul x3 + px + q = 0 * Ito '4974** Tr*-4 t I

3

2

10

MTE-04 6

Rtx,yeR*R-R lxYI < lx1-13r1. (iv) 1Rsich (-14114)kul . 31141M eiltaq \Tell H

f4F:Tor RW 6)di t I

(v) ire 164, tii-114)kul Awrzt TrIft Tr*

t.FTT-gr t, err kt-coel 141 tRiT-Tm ki(s.e4

&ill

MT E-04 7

MTE-05

BACHELOR'S DEGREE PROGRAMME(BDP) Term-End Examination

June, 2014

ELECTIVE COURSE : MATHEMATICS MTE-05 : ANALYTICAL GEOMETRY

Time : 11 hours 2

Maximum Marks : 25

(Weightage : 70%)

Note : Question no. 5 is compulsory. Do any 3 questions from Questions no. 1 to 4. Calculators are not allowed.

1. (a) Identify the conic 9x2 6xy + y2 + 60x 20y + 75 = 0 and trace it. 3

(b) Find the radius and the centre of the circle x2 +y2 +z2 x+z-2=0, x+2yz=4. 2

2. (a) Show that the sum of focal distances of any point on an ellipse is equal to the length of its major axis. 2

(b) Find the angle between the lines of intersection of x + y z = 0 and yz + 6zx 12xy = O.

MTE-05 8

3. (a) Find the centre of the conicoid 3x2 + 5y2 + 3z2 + 2yz + 2zx + 2xy 4x 8z + 5 = 0.

What will be its new equation, if the origin is shifted to this point ? 3

(b) Find the equation of the cylinder whose base is the circle x2 + y2 = 9, z = 0 and the axis

x y z is 2 4 3 5

4. (a) Trace roughly the conicoid 3 (x 1)2 + 2 (y + 1)2 + 6 (z + 2)2 = 6. What is the section of this conicoid by the plane z = 2 ? 3

(b) Find the equations of tangents to the conic x2 + 4xy + 3y2 5x 6y + 3 = 0 which are parallel to the line x + 4y = 0. 2

5. Which of the following statements are true and which are false ? Give reasons for your answers. 10

(i) The line y = x is a tangent to the conic x2 y2 = 9 4

(ii) The angle between the planes x + 2y + 2z = 5 and 2x + 2y + 3 = 0 is 60.

(iii) Every section of a paraboloid by a plane is a central conic.

(iv) S = {(x, y, z) E R3 I y2 + z2 = 1} represents a circle.

(v) The projection of the line segment joining (1, 5, 0) and (-2, 4, 1) on the line with direction ratios 3, 1, 8 is 0.

MTE-05 9

k-imen Twit!. chi stoi trftur

2014

4 I t:ti Sh41-1 : 1J1 I CI

TT . -05 : cr) 4f-11

.twit : 1-1 wu2 2

31RW-d7:1 37 : 25

(25ei : 70%)

7iF .C/?..1 5 9W 1/ arf4-472f # 3777 T. 1 4 # .b?./ 9 UP' Ace417 I 1PT)77 7' if ;Aix #

1. () -4ticha 9x2 6xy + y2 + 60x 20y + 75 = 0 3tr( 31-ritra. *=07 I 3

No v. x2 +y2 4. z2 _ x z 2 =0, x+2yz=4

wrff A'tf77 I 2

2. () tUr4R i 1-*711tEi-19- 1-*711 A:1 R-riftzt 71-1-7t Fe 3141 34ki

loth -1(11 t I 2 (W) x + y z = 0 yz + 6zx 12xy = 0

cl Zt5ii34*tq w Qui vci I 3

MTE-05 10

3. () iii i 3x2 +5y2 +3z2 +2yz+2zx+2xy-4x-8z+5=0

TTcl *tr-A7 I eirq P.TTRT- ft-ff Ren \AR, c cITzrT choi f 611 ? 3

NO ZIT chtui quo *IftR 171:W aTrtw Tx2 +y2 =9,z=oatItalv:==it I

2

4. klichciA 3 (x 1)2 + 2 (y + 1)2 + 6 (z + 2)2 = 6 *1 3-111trd WA7 I wicid z = 2 -grtf TEF iict)ci,31 1:fit

? 3 -41icha x2 + 4xy + 3y2 5x 6y + 3 = 0 tr 34 ti-nchtuiTa. *'n1-A-R x + 4y = 0* ii1-11-clt t I

5 tsid 4 4 4-1-4 kicel 0-1-14 aRTF:f ? 31tr-4 \ilk* cr)Ruf %ft A--dr-47I

10 2 2

(1) y = x Ylighq 9 x Y 4 = 1 tr t-44f IN!

t I (ii) Tfird-01 x + 2y + 2z = 5 AT 2x + 2y + 3 = 0*

Alq. *T chiul 60 t I wacieizi 3rA- trfk-Qq *4t-4 @II t I

(iv) S = {(x, y, z) E R3 Iy2 + z2 = 1} cr -ftecr-d alkot 4

(v) Rch, 31-50 3, 1, 8 awn" 1131F TR (1, 5, 0) aTIT (- 2, 4, 1) *1 iirdi4 Itgitgus 3f44ti

t I MTE-05 11 5,000

Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11