MTE-04 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA

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1 No. of Printed Pages : 15 MTE-04 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA Time : 11/2 hours Maximum Marks : 25 Instructions : Weightage : 70% 1. Students registered for both MTE-4 & MTE-5 courses should answer both the question papers in two separate answer books entering their enrolment no. course code and course title clearly on both the answer books. 2. Students who have registered for MTE-4 or MTE-5 should answer the relevant question paper after entering their enrolment number, course code and course title on the answer book. Note : Answer any three questions from question Nos. 2 to 5. Question No. 1 is compulsory. Calculators are not allowed. 1. Which of the following statements are true and 10 which are false? Justify your answers. (a) {, IGNOU, algebra} is a set. (b) All roots of the equation 32x +81 =30 x ax are real. MTE-04 1 P.T.O.

2 (c) (d) (e) The arithmetic mean of 3 non - zero real numbers is greater than their harmonic mean. Moivre was the mathematician who first popularised the name, 'complex number'. Any pair of linear equations in two variables is consistent. 2. (a) Can we solve the following by Cramer's 21/2 rule? If yes, solve it. If no, then find solutions by any other method. x + 2y + 3z = 3 4x + y 4 2x + 4v + 6z = 6 (b) Show, by induction, that y n 1, 21/ (2n 1)2 = 3 (4n2 1) 3. (a) Let A and B be sets of natural numbers N 2 given by A = {3x xen}, B = (5x x N). Find AnB. Is AuB = N? Why? (b) Let x l, x2, xn be positive real numbers and s = x1 + x2 + + x n. Apply the Cauchy - Schwarz inequality to prove that xi s xi n 1 MTE-04 2

3 4. (a) If a is a negative real number and n = 2m, 2 with meni, then prove that there is no nth - root of a. (b) Give a direct proof, and a proof by contradiction, of the following statement : The equations 3x =y and 3y = x are consistent. 3 Find the polar form of 1+i. If the sum of two roots of the equation 3x4 28.x3 3x x 36 = 0 is zero, find all its roots. 1 4 MTE-04 3 P.T.O.

4 iricr) zmitt ch 11. if ) ti :I I h R1T qh4-ci, 2012 tr-ww : : VrtriTW 41,ii 71:R7 :1% -tv 3 -ri 3.7*w : 25 : 70% 7. W) E9/W7T.27'4`-4 ql.(//datsb41 k C/k1 71-4,F7 qi-d 30,-/colY dri< F7T,Jrf< yr.??73# airrt ardwrifw, RTIT r17-7:1" 3174" Rvoch< 2. *uīv7zie.e.-471t7t.e.1.-5chk1c-n ri-47f dq? sr1`01 dri, Sat-YrT,V7 3.7TRT 37--dwg-4, wcs4;5q 9h?5 721T El/creishH T qt(h-qiy) fcvoch( I TIF : sort?irw :1" 4 WI R.77. so,-/ f 7 WI 1 WRT.wtqt7 ti men,' ardrr ielt/ 1. fl qt.4 T t 10 4i-4i.4 am-ff? atcrk -afq --1F77 (a) {if, IGNOU, \Tvinim} ti (b).h4-nchtui 32x +81 =30 x 3x Tr%ft ar.ciract) kit41qti MTE-04 5 P.T.O.

5 (c) 3!1,, [ 141V4 4.1c-4-1T TīF-Ta:R4.q7s7 cie t (d),91f- 711 ti--4 1T- r7r ct;:l I MI (e) i fl EThF HUI1 51TU5ft TR Arm 1 2. (a) :z[t -44 oh'f obj, 21/2? Trf t '-.-t-i9-74t3 oh 24±,,37R `91. ' fttfr 31- J f--q-n -ot 711c-f x+ 2y+ 32 4x+y=4 2x + 4y + 6z 6 (b) 347[149 f9r414 A zit f-q-5-r7 -ft V fl?;1, 21/ ±...+ ( )2 = 11,1 (4112 1). 3. (a) c4mtr A B, A = t3x I xeni, B 15x I x N}. TT[IiTTb 4,&413 N AnB '\71 1 I AuB=N? MTE-04 6

6 (b) x i, x,?,-ft rinki 4,--1, , 3 S = X + X 2+!1%: ",r,={Thq-7174 ft:f.-1 ql"ir\31 ) E Xj 11 i=1 S n 1 4. (a) r, a -137 ak-ci cich n 2m, 2 rn N, a -wr 14.1 n t I (b) -Vc1 r 3- RT Li It air F47)--T TTI q11-,itj, I 3,k-14 11,4,01 3x = 3i)J. 3y = x, Tfl-TU 5. (a) 1+1 chf -A01,tr\ ;-1Id Thr717,1 1 (b) k71t-11-* x4 --28x3-3x2 +112x 36 = 0 4 cri77 I 1,1 t 7=1* -m-ft tc.1 TIN MTE-04 7 P.T.O.

7 MTE-05 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2012 ELECTIVE COURSE : MATHEMATICS MTE-05 : ANALYTICAL GEOMETRY Time : 11/2 hours Maximum Marks : 25 Weightage : 70% Note : Question No. 5 is compulsory. Do any three questions from questions numbers.. 1 to 4. Calculators are not allowed. 1. (a) In a parabola show that the tangent at any 3 point makes equal angles with the focal radius of the point and the line parallel to the axis through the point. (b) Show that the plane x + 2y + 2z = 9 touches 2 the sphere x2 + y2 + z2 = 9. Find the point of contact. 2. (a) Find the length of the major axis of the ellipse 2e r 1 + e cos() MTE-05 9 P.T.O.

8 (b) Find the equation of right circular cone 3 whose axis is the y - axis, vertex is the origin and semi vertical angle is / (a) Prove that the equations of line of 2 intersection of planes 4x 4y 5z =12 and 8x +12y 13z =32 can be written as : x 1 1/ (b) Check whether the conicoid. x2 + y2 + z2 6yz 2ry 6x 2y-2z +1=- 0 has a centre or not. If so find the centre (a) Find the equation of tangent planes to 3 7x2 3y2 z = 0 which pass through the line 7x 6y + 9 = 0, z =3. (b) For the hyperbola 9x2 4y2 = 36, find the 2 vertices eccentricity, foci and equations of asymptotes. 5. Are the following statements true or false? Give reasons in support of your answers. (a) (b) All planar sections of an ellipsoid are ellipses. 5x 2y = 5 +2z represents a line. 10 MTE-05 10

9 (c) (d) (e) The projection of the line segment joining the points P(1, 2, 3) and Q( 2, 1, 4) on x- axis is 5. The length of the perpendicular from any point on the cylinder X2 + y2 =9 to its axis is 4. The equation. 2 Tr r =3 cos( 0 +2 sin( represents a straight line. MTE P.T.O.

10 ..-tictch Gil 141) 4-1 ( 't tft ) koici trftalt.kiti:eff, 2012 : fluid Tr1217 : V/2 : el)k)e.)1144) d 37"fiTTI 3fT 25 TOTh7 : 70% MT-q- (-1&3e.// 5 31-fg-d-Rit/ 777 i-ise-1/ dl-h 01-'4 j Tf 2t7T ch? 1. (a) (b) trciriq f Fch 1-ft TO tur f-- 1-r f-q f-qt m- t-{=n9-1 01'0(1k.47 Wdf % x2 + y2 + z2 = 9 ch) TcRi WTdl t I x + 2y +2z =9 ft1 T1*-1 t 3 2 f gild off A- I 2. (a) (b) 2e t7:* r 1 + e cos0 W ch1 (1-1 11C1 4FA7 I.11 (-14-4'01 71Id ṈTĪWE 31U 2 3 y-31 t f41 t 3-1- Tr/6 t I MTE P.T.O.

11 3. (a) ft, ":1-fq7, fc-t TTIT-171'1. 4x + 4y =12 8x +12y-132=32=t U-T3ft: :47 ch't Tq 4 Four : 2 x 1 y 2 z (b) AT,=I f 111-)ci 7 7 6yz 22:x 6x 2y - - 2z + 1=0 cfd Turief t 3 4. (a) 7x2 3y2 z = 0 1-,1t4 PTO cl-q 3 uc Lr 'f7 t -tur 7x 6y + 9 = 0, z 3 qz.r- (b) 31-R-TE17c x2 4 y2= 36 tfi ftlf, 2, T 3fR ci-v\317 I 5. rrr t` 3---rc74 a-)rur 10 (a)?th#ff (1c1 1 1:1-R--4q7.ff t I (b) 5x 2y =5 + 2z 'ft fthfcm chtdi t I MTE-05 14

12 _ 3IP,T -crt P(i, 2, 3) AT Q( 2, 1, 4) on') d, rft kj._11 et: TA 5 t- si c1-1 x2 + ti2 = 9 1-'ff 31-fiTrịq 4 t- I (e) tuf = 3 cos(u 4 1±2 sin 0 1- F-RcTLI,c-1 (4- c-11 t- I MTE-05 15