1 1... L..4 No. of Printed Pages : 11 I MTE-04/MTE-05 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. tq11c1ch 441141 cbl $.)41 : 71.t.t.-05 : a )1.444) Iti I 1. Epic F.&$`.-04 37 x 1. e. 1.-05 c')47 VW AY g, 3-dz arev-aov drff 3PWfaii 4: 3797T ardwx* I/wow ads SIT ltigeotoi are flly,-flly, 1k4w.. TT.e.e.-04 TIT T4T.,:e.-05 k7w 1f4ifff WO"3 E7W drff J-d?--37Plw 71 37rrqr 3.13-A-Tr* litgersiw ads u17 litgetow WTI oty -RTW A7 W I MTE-04/05 1 P.T.O.
BACHELOR'S DEGREE PROGRAMME Term-End Examination December, 014 MTE-04 MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA Time : 11 hours Maximum Marks : 5 (VVeightage : 70%) Note : Question no. 1 is compulsory. Attempt any three questions from Q. No. to 5. Use of calculators is not allowed. 1. Which of the following statements are true and which are false? Justify your answer giving valid proof or counterexample. 10 (a) A\(B fl C) = (AB) fl (A\C) for subsets A, B, C of a set U. (b) A fourth degree equation with real coefficients has a real root. (c) 1 = x + iy for any real numbers x and y x - iy (at least one of them non-zero) and i = (d) Nrd is a rational number. (e) Any system of two linear equations in two unknowns has finite number of solutions. MTE-04
. (a) Apply De Moivre's theorem to express cos 40 in terms of powers of cos 0. (b) Prove that a real non-zero cube root of a positive real number is unique. 3. (a) Check if the following system of linear equations can be solved by Cramer's rule : 3x + y z = 7 5x 3y + z = 5 9x lly + 10z = 9 Obtain the solution. 3 (b) Show that nn > 1.3.5... (n 1). 4. (a) If a, 6, y are the roots of a cubic equation, then find the sum of the cubes of the roots, for the equation x3 4x + 6x 8 = 0. (b) Prove by induction that the product of any three consecutive natural numbers is divisible by 3. 3 5. (a) Show that 1 a a3 1 b b3 = (a b) (b c) (c a) (a + b + c). 3 1 c c3 (b) For any subsets A, B, C of a set U, prove the following : (An B)xC=(AxC)n(BxC) MTE-04 3 P.T.O.
t-ime) 3QT chi soii trft4tt Rtil-Git, 014 71:1:e1.-04 : siitflc, 4141 in 0414 : 1-1 EP7-317TIW-d-47 34-: 5 (poi W: 70%) TiF Mq-1 W. 1 c#w41 / 3041 4. 4 5 TT 4 ch #T W 6c/ 4,r7q / 47(-35(-?ej meng 31-jx/ft Wig / 1. P4-.1 kice4 t A 4i-14? 3ErcA. zrr atcr4 3TR ttrtr () U 311-19WA A, B, C A\(B n C) = (A\B) n (A\C). (w) Trr* airy..410 4 wilctkui 'WC - ar.ctidal alai t I (ii) 1*"4t aitclidch tit-041 x A y ("WTA- 1:1-141 t) A = r(s-r (Er) -Jd 1 x-iy x + iy..) tritglai titer i t atta- 4:0-r-41 f*-#t 19-*-P4*(-11 - =1- tritrrra- t-dt t MTE-04 4 10
., cbt cos 40 t cos 0 'Err-d*Tr41-11- *'1 NR (w) Th *li*1 aik.tifdal 3. Tt9' ' ti*r Pcblei Peiii 4 3x+y z= 7 5x 3y + z = 5 9x lly + 10z = 9 alfaftalai t PHRIRsti ~tiali art 3rrt:r tria 3 (31) turf fi nn > 1.3.5... (n 1). 4. () zlit a, (3, y *, w-11.kui v-41 zortm oict*'rr--4r : x 3-4x +6X-8=0 (ig) al-p-mq 4 kg -Q-P4R1 i#r9. 3ired titsenall Itiii14,0 3 4 "i4vt1 t I 3 5. tut-47 fi 1 a a3 1 b b3 = (a b) (b c) (c a) (a + b + c). 3 1 c ti -cati U f -41 411k1T:c1a1 A, B, C (AnB)xc=(Axc)n(Bxc) MTE-04 5 P.T.O.
BACHELOR'S DEGREE PROGRAMME (BDP} Term-End Examination December, 014 MTE-05 ELECTIVE COURSE : MATHEMATICS MTE-05 : ANALYTICAL GEOMETRY Time : 1-1 hours Maximum Marks : 5 (Weightage : 70%) Note : Question no. 5 is compulsory. Answer any three questions from question no. 1 to 4. Use of calculators is not allowed. 1. (a) Find the equation of the conic section with eccentricity 1, (1, 0) as its focus and y = x as its directrix. x + y z = 1. (b) Trace the surface 9 5 4 Describe its sections by the planes z = ± 4. 3. (a) A circle cuts the parabola y = 4ax in the points ( ati, ati) for i = 1,, 3, 4. Prove that ti + t + t3 + t4 = O. (b) Show that if the sum of the squares of the distances of (a, b, c) from the planes x + y + z = 0, x = z and x + z = y is 3, then a + b + c = 3. 3 MTE-05 6
3. (a) Find the equation of the cylinder whose axis is x = y = - z and radius is. (b) Find the equation of the line parallel to y + x + 1 = 0 and passing through (1, 1). What is the angle between the line obtained and x = y? 3 4. (a) Show that the conicoid 3x + 7y + 3z + 10yz - zx + 10xy + 4x - 1y - 4z + 1 = 0 has a centre, and find the centre. 3 (b) Find the equation of a right circular cone with vertex at 0, axis at OX and semi-vertical angle -71 3. What will be its new equation when the origin is shifted to (0, 1, -1)? 5. Are the following statements true or false? Justify your answer. 10 (a) (b) The relation between the cartesian and the polar co-ordinates is x = r sin 0, y = r cos 0 and 0 = tan-1 y. x The conditions on m and c, so that y = mx + c will be a tangent to x = 4ay are m tan LC and c = - amt. (c) The line - x = y = ẕ is parallel to the x-axis. 3 MTE-05 7 P.T.O.
(d) x, s = 0 and s 9 + 4 I = xy 9 = 0, then the condition on k for which s + ksi = 0 will be an ellipse is k <. 9 (e) The condition for a line with direction ratios a, 3, y to be a tangent to the central conicoid ax + by = z at (x0, y0, z0) is ax, a by (3 = y. 0 0 MTE-05 8
till /et) 441 ti era 01.11.0 Trgiff IT Rtri-Git, 014 M : 'TRIM : cr, 7#41 OW/ : / / Etue 31AW747 : 5 (pci : 70%) *F: RI 5 0(4/ alftref / 37 Ft. 1 4 4 ri f:4-7-e 07 TRTi aff?" ef47 l ei,,,,cadj ym-ri37-17* 1. () 3-4--ḏt 1, -IA (1, 0) MIT.iticR Pelc11 y = x tfitq-4 ti41.*kui Ta- trf* I x NO 7 + + z 4 = 341Zifff WA I 5 1:11Td-M1 z = ± 4 'TU *(14) tlitqtr I crul tp.a.r I 3. 7" TR-474 y = 4ax -saff (aq,ati), = 1,, 3, 4 ', 7-51i cboi t I f..g*rnr t +t +t +t O. 1 3 4 - NO 'zit "Ril (a, b, c)*1141:1-d-01 x + y + z = 0, x = z x + z = y PA.* trfth-f 3 t, t "POW a + b + c = 3 I 3 MTE-05 9 P.T.O.
3. (*) 3T1 to-1 W ti4)chtui Vd tia -Nfrwr alat x=y.-z MIT i.,3-0-ii ' I NO y + x + 1= 0* tilii-cit 'ffit (1, 1).4 TIR al sii WF tii-ncntui vo te4r I TR. (.gi 3* x = y* ct).)ul WIT'? 3 4. (*) ittiii57-1* klichcl 71 3x + 7y + 3z + loyz - zx + 10xy + 4x - 1y - 4z + 1 = 0 w 4.- t, TAT d.).- Tff *'rn7 I NO trci o, am ox w ap-i-tti ch ul ' i girl 44-1-41-4 kit w tii414*11 gild -i=ri I zft iti -fk (0, 1, -1) 7fd1:flierd W( en,iir, t ichi 'WM W-fichkul ^EIT 411? 5. ;rit -14-1 0 tgi W-T4 Ore/ ' 3-1TT 3? 3itr4 31-lk41. tf-a7 1 (*) Wrefzi T4TF VRf fe4e4 *#1- Trgaim- x = r sin 0, y = r cos 0 3I 0 = tan-1 x (W) y = mx + c * X = 4ay ii '-'-iki Ztg i t I * fm m'ffttetranttm=tan a 3 c=-am t I 3 10 (TT) Itgi = y= -;, x-31.4 * tv11-cit t I MTE-05 10
0=0 Zifk s v v + 1 = 0 3.14R xy 9 = s + ksi = 0 R'W titiff +II k 1Tt -srpd-4.4 k < 9 Rct-31-dc1T61 a, p, y curd if *-41-4 ax + by = z *1 (x0, y0, z0) tft *4441 tgrr 3iNIT axoa by013 =y I MTE-05 11 6,000