BACHELOR'S DEGREE PROGRAMME MTE-04 : ELEMENTARY ALGEBRA MTE-05 : ANALYTICAL GEOMETRY

Transcription

1 No. of Printed Pages : 11 BACHELOR'S DEGREE PROGRAMME MTE-04 : ELEMENTARY ALGEBRA Instructions : MTE-05 : ANALYTICAL GEOMETRY 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.. 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) ) t.t.-04 : 5nt1 eh a1ailpti 711.t.t.-05 : ch,reu 41: 1. * 77i.e ark7-air.e.f.-05 cil-0-41witt raltu- ff, T-47 37( arov-arev 5P1W.WY 3FRT 373-A-7:17*.; 1/ wow Ida err zugebty 717:T iv'-ffich Ik4W I. oa 7.7.e.-614 TIT F.e.e.-05 /# kkt 3F/4 Yggrff Jḏr dri(--spiwt ART /W, zifietow m1,5 ffen- westif 7PT Y, ikgq5< MTE-04/05 1 P.T.O.

2 MTE-04 BACHELOR'S DEGREE PROGRAMME Term-End Examination June, 014 MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA Time : 1 1 hours Maximum Marks : 5 (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 + y + 3z = 10 x + y + az = b has a unique solution. 3 (b) Plot in the Argand diagram 1 + 3i, 3 + i.. Solve the equation 3x4 5x3 + 50x 50x + 1 = 0 given that the product of two of its roots is. 5 MTE-04

3 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 + z = 3 x + 3y z = 6 (b) Show that for any n E N, n > n. 4. (a) Prove by the principle of mathematical induction that 3n + 7 is divisible by 8 for all n E N. 3 (b) A shop sells two kinds of packs, Pack A containing pencils and 3 pens and Pack B containing 4 pencils and 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. 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.

4 (iii) lx yl < 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

5 711.t.t.-04 trl 1 CM chi Shil Trliff t.t.-04 : mtinch in / uu arrw-dv aiw: 5 (pci WT: 70%) 4iZ : 37P. F. 5 ch< "fri / WI Ft WV ; / 40/35e-?e< Teto.i, g 1. (ni) a, b* 79. t1d7 frt* c T i4*tzf x + y + z = 6 x + y + 3z = 10 x + y + az = b n4i 1/c t I 3 (13) 1 + 3i * 3 + i 3TT kilicm aiiltrd W-4I. ki4flchtu1 3x4 5x3 + 50x 50x + 1 = 0 rctil iv)!_tul-itho 5 MTE-04 5 P.T.O.

6 3. (*) z1111 Arṟ4R -ft1r. 4-11,:nku wr -Wrzt )1-R CV zfr rie I zit ti l t, ct)k * i Mr ANR I 3T- ftf 111\idiel ritichtul f4r-t C-A-R I 3 x+y+z=4 x y + z = 3 x + 3y z = 6 n E N* R'R tufir.kn > n 4. (*) Trrurtzr atrrri:r * 11 clik;. 7 fp-4 n E N 3n + 7, 8 A t I -S*T4 SichR t, th-di- A 14 44ra atit 3 tai Ilh#M3 4 ift0 afit 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 Tkṟi-d A=rr-4R I 5 cl gci *4-4 fir4.rr ci.)1-14? atet4 drr Tiff 31:FAerr - IcgWuI gri ar. Atr7R I (i) eir4a,b,c4k#1(.1 -ccleltft AZB,BC, 3 10 (ii) (i41chtul x3 + px + q = 0 * Ito '4974** Tr*-4 t I MTE-04 6

7 Rtx,yeR*R-R lx YI < lx1-13r1. (iv) 1Rsich (-14114)kul M 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

8 MTE-05 BACHELOR'S DEGREE PROGRAMME(BDP) Term-End Examination June, 014 ELECTIVE COURSE : MATHEMATICS MTE-05 : ANALYTICAL GEOMETRY Time : 11 hours Maximum Marks : 5 (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 9x 6xy + y + 60x 0y + 75 = 0 and trace it. 3 (b) Find the radius and the centre of the circle x +y +z x+z-=0, x+y z=4.. (a) Show that the sum of focal distances of any point on an ellipse is equal to the length of its major axis. (b) Find the angle between the lines of intersection of x + y z = 0 and yz + 6zx 1xy = O. MTE-05 8

9 3. (a) Find the centre of the conicoid 3x + 5y + 3z + yz + zx + xy 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 x + y = 9, z = 0 and the axis is x y z (a) Trace roughly the conicoid 3 (x 1) + (y + 1) + 6 (z + ) = 6. What is the section of this conicoid by the plane z =? 3 (b) Find the equations of tangents to the conic x + 4xy + 3y 5x 6y + 3 = 0 which are parallel to the line x + 4y = 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 x y = 9 4 (ii) The angle between the planes x + y + z = 5 and x + y + 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 y + z = 1} represents a circle. (v) The projection of the line segment joining (1, 5, 0) and (-, 4, 1) on the line with direction ratios 3, 1, 8 is 0. MTE-05 9

10 k-imen Twit!. chi stoi trftur I t:ti Sh41-1 : 1J1 I CI TT. -05 : cr) 4f-11.twit : 1-1 wu 31RW-d7:1 37 : 5 (5ei : 70%) 7iF.C/? W 1/ arf4-47f # 3777 T. 1 4 #.b?./ 9 UP' Ace417 I 1PT)77 7' if ;Aix # 1. () -4ticha 9x 6xy + y + 60x 0y + 75 = 0 3tr( 31-ritra. *=07 I 3 No v. x +y 4. z _ x z =0, x+y z=4 wrff A'tf77 I. () tur4r i 1-*711tEi-19-1-*711 A:1 R-riftzt t Fe ki loth -1(11 t I (W) x + y z = 0 yz + 6zx 1xy = 0 cl Zt5ii34*tq w Qui vci I 3 MTE-05 10

11 3. ( ) iii i 3x +5y +3z +yz+zx+xy-4x-8z+5=0 TTcl *tr-a7 I eirq P.TTRT- ft-ff Ren \AR, c citzrt choi f 611? 3 chtui quo *IftR 171:W atrtw NO ZIT Tx +y =9,z=oatItalv:==it I 4. klichcia 3 (x 1) + (y + 1) + 6 (z + ) = 6 * trd WA7 I wicid z = -grtf TEF iict)ci,31 1:fit? 3-41icha x + 4xy + 3y 5x 6y + 3 = 0 tr 34 ti-nchtuita. *' 1-A-R x + 4y = 0* ii1-11-clt t I 5 tsid kicel artf:f? 31tr-4 \ilk* cr)ruf %ft A--dr-47I 10 (1) y = x Ylighq x Y 4 = 1 tr t-44f IN! 9 t I (ii) Tfird-01 x + y + z = 5 AT x + y + 3 = 0* Alq. *T chiul 60 t I wacieizi 3rA- trfk-qq t I (iv) S = {(x, y, z) E R3 Iy + z = 1} cr -ftecr-d alkot 4 (v) Rch, , 1, 8 awn" 1131F TR (1, 5, 0) atit (-, 4, 1) *1 iirdi4 Itgitgus 3f44ti t I MTE ,000