BACHELOR'S DEGREE PROGRAMME MTE-05 : ANALYTICAL GEOMETRY

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1 CY3413S No. of Printed Pages : 11 ATTE-04/MTE-05 BACHELOR'S DEGREE PROGRAMME Instructions : MTE-04 : ELEMENTARY ALGEBRA 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. met) Itrliti chi chi.' :,1141 ict 1. oa 31 km71e m7r7 rivff ff, 047 3w-vgi. are7t-37r7t 3fff SPI*7311 7:7 NWT a; rte,.115?.5 UTT Vicvaw 377:1F15--R7F ftwf I. min 74-7.j.-04 ari f(n g, 37:It N77-17V 3W-3P/Wf ' 3131P7*, Itiesosi qds U7T VAVIShif WTI 67W-NFF fkiw / MTE-04/05 1 P.T.O.

2 I MTE-04 I BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 018 ELECTIVE COURSE : MATHEMATICS MTE-04 : ELEMENTARY ALGEBRA Time :- hours Maximum Marks : 5 (Weightage : 70%) Note : Attempt any three questions from questions no. 1 to 4. Question no. 5 is compulsory. Use of calculators is not allowed. 1. (a) (i) Apply Gaussian elimination method to solve the following system of equations : 5x + 7y + 3z = 4 6x + 3y = 8 3x y = z (ii) On the basis of the solution you have obtained in (i) above, could you have used the Cramer's rule to obtain the solution? Give reasons for your answer. 3 (b) For sets A, B and C in a universal set U, show that A x (B C) = (A x B) N (A x C). MTE-04

3 . (a) Solve the equation 54x3 3x 6x + 16 = 0, given that the foots are all real and are in G.P. (b) The annual bonus given to the'employees of a company is 5% of,their taxable incomes, after the State and Central taxes are deducted. The State tax is 10% of taxable income. The Central tax is 0% of taxable income after deducting the State tax. Formulate this situation for determining the bonus, as a linear system. 3. (a) Use the principle of mathematical induction to prove that (b) For a, b, c E R, check whether or not ab bc ca > abc (a + b + c) (a) Solve for x, a m x where a, b, m E R and m 0. (b) Find all the roots of Z5 + 3 = and show them in an Argand diagram. MT E-04 P.T.O.

4 5. Which of the following statements are true and which are false? Give a short proof or a counter-example to justify your answers. 5x=10 (a) {Indira Gandhi National Open University, Vi, (1)} is a set. (b) fax )+( x 3)+...+n(n+1)] n+1 n(n + 3) - 4 for n?_ 1. (c) The contrapositive of the statement, 'If it is raining in Delhi, then all the telephone instruments in Mumbai are black' is 'If it is not raining in Delhi, then all the telephone instruments in Mumbai are black'. (d) The equation x7-53x5 + 3x + 7x - 3 = 0 has at least one real root. (e) Any system of three linear equations over R is consistent. MTE-04 4

5 ituet.t. -04 truico Witt chitistoi ) 1:ffiff x,018 *-14.col-rickcichei : 7r.t.t.-04 : 51IkI ch al ifiWAr 3/ : 5 (pri : 70%) NO wv - #.144#4*---eRI7JR-4*- 3ffTeaq /MR Ft. 5 31/4"drif g l 4e,e.le(1 5M7T 04 * aryl* qeil I 1. () (i) ki&nehtul Pehlel cvl el Aticbtui -PA w -ti r4r : 5x+7y+3z=4 6x + 3y = 8 3x y = z (ii) (i) 4 -srrqr fkr 1 * eicm %WI 3111T 1' e R )sit mei)ii tic0 4? 3ltr4 zuk chikui tfa ligttzr u kt-ccitil A, B c* ttstr4r Ax(BNC).(AxB)N(AxC). MTE-04 5 P.T.O.

6 . ctrkul 54x3 3x 6x + 16 = 0 NO ii.drich att G.P. t 741 t *Pt vi ct, 4,0+11 * 341 *TA RI atrzt 4 14 ti,74 t*-4tzr eme.4 * 1 -alga atrzr W. 5%, aigeh 6.1)-R1 rcqi S i d i t I ti ro-1 Wit, *7-e1) 144 3Tt it 10%' I *4T threi chld * 1:frffs air W 0%' I a1-r-iatiftur TPIRT Irtsich fiwzi Vtf 4 L-P-Aw -tfa n = (n - 1) 1t v ct)t fa7 feria mei)ii *ti* I a, b, c e R* trnr ab bc ca > abc (a + b +.71T.0 1 a a x 4. () m m m = 0 * tom, b x b 7-4 a,b,meitailtm*o. NO z5 + 3 = lit 4,71 Ta 311T atrtit 3 4 TzkR I 3 MTE-04 6

7 5. orid 4 14 A. WaR att zercrf4 zrr arem-? I'c4qwul it I 5x=10 () {-AU 'gw felidsimei,.5, 01 ti-cciti t n 1 1C.R [(1x)+(x3)±...+n(n-Fm > n + 1 n(n + 3) 4 (TO ceirq eilikki t Tel t, 4cii 4 Tal td=t0q siticbtui chi) w.srftwffw "zrft i'<`t 4 qiikki Tel 'Ott, 46ii 4 TO edtr:b dtiehtul ' (Er).o.11.htui x7 53x5 + 3x + 7x - 3 = 0 alt-ct 't I (Z) R f*-41 4t4 Ifispry chiti MTE-04 7 P.T.O.

8 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 018 ELECTIVE COURSE : MATHEMATICS MTE-05 : ANALYTICAL GEOMETRY I MTE-05 1 Time : 1 hours Maximum Marks : 5 (Weightage : 70%) Note : Question no. 1 is compulsory. Do any three questions from questions no. to 5. Use of calculators is not allowed. 1. Are the following statements true or false? Justify your answer with a short proof or a counter-example. 5x=10 (a) The line ax + by = 0 is tangent to the conic x + y + tax + by = 1, where a and b are non-zero constants. (b) The equation x + xy + A (x + y) = 0 represents a pair of straight lines for all E R. (c) (d) MTE-05 (e) The planes x + y z + 1 = 0 and 3x + 3y 3z = 0 are parallel. The xy-plane intersects the sphere x + y + z x y + 1 = 0 in a great circle. The section of, a paraboloid by a plane is a parabola. 8

9 . (a) Find the points of intersection of the conics ax by = 1 and bx ay = 1. (b) Find the centre and radius of the sphere passing through 11(1, 0, 0), (0, 1, 1), (0, 0, 1) and (.5 3. (a) Identify and trace the conic x -y +x+4y-3=0. (b) Find the equation of the plane passing through the line = Y 4 = z 1 and the x 3 point ( 1, 1, 1). 4. (a) Check whether the surface represented by x + y = (z + 1) is symmetric about the YZ-plane, ZX-plane and XY-plane. Do the coordinate axes intersect the surface? (b) Find the equation of the cylinder whose base is the circle x + y = 4, z = 0 and the axis is x= Y = z. 5. Find the, new equation of the conicoid 4x 4y 8z xy + 5yz + 5zx = 0 under the transformations given by the following table : x y z x' y' z' MTE-05 P.T.O.

10 71:Lt.*.-05 *411fIch 441Il chl ch41 (*Alt.*. ) Trgia tift4tt T4, 018 0;i.s ch gifrstoi : 71:t ct).rclil+i`f 1t / FP:Ri: *-677 : 5 (y)c-i : 70%) R"? TR7 WI 1 arftpig Nww eh ReiT f-4q. 4(--p-kj me/1,1 1)(4 4i` aryq& Re' I 1. w zrr TRT? zerea zrt sicgcwul 3irr4 3w{*1 - fsz *07 I 54=10 () 113T ax + by = 0, klittvi x + y + ax + by = 1 (n.) t \Tie a 31{ 1374T( 3TW t cbtul x + xy + X (x + y) = 0, Tr141 A E -R-R (Ito 1131T f fwd 4)01 t (ii) til-ric1 x+y z+1-=034 -(3x+3y-3z=0 t (1) x +y +z -x-y+1=0t 41 s Kol t I tikcirv4,= 1 4 trftqq TR4eaT di t MTE-05 10

11 . () 71i ax - by = 137 bx.srpdqq figs Tff *tp4r Rsaff (1, 0, 0), (0, 1, 1), (0, 0, 1) fiwrr *=rpar 4 3,71t 3. (*) 41Ichel x y + x + 4y *) TAR atrlfto trp4r x = y- = z-1 ' 4 3 ay = 1 1 1, 1) 1\yR gia (-MOM 'Wr ti 4-11ChtUI 41i Qr-AR I 4. (*), f i* x + y = (z + 1) 404a. 13 ZX-tvicm 3 XY-ti+400 * 311W tztt 'Ter I 14-4TiW 314T m cht;? *T w4ict)tul Tra. ttpar 'N/T+1 31TITE rx +y =4,z=4t 314Tx=- =zt I 5. fa+ -i1ir t tittun 4 to4latuil 3 ItTOFER 4x + 4y 8z xy + 5yz + 5zx = 0*T RIT ti ChtUi Ta- trrar : MTE-05 7,000