MT-03 December - Examination 06 B.A. / B.Sc. Pt. I Examination Co-ordinate Geometry & Linear Programming Paper - MT-03 Time : 3 Hours ] [ Max. Marks :- 66 Note: {ZX}e : Note: {ZX}e : The question paper is divided into three sections A, B and C. Write answers as per the given instructions. Use of non-programmable scientific calculator is allowed in this paper. àíz nì VrZ IÊS>m "A', "~'Am a "g' {d^m{ov h & àë oh$ IÊS> Ho$ {ZX}emZwgma àízm Ho$ CÎma Xr{OE& Bg àíznì Zm Z-àmoJ«m o~b gmb Q>r{ $H$ Ho$ëHw$boQ>a Ho$ Cn moj H$s AZw {V h & Section - A 6 = 6 (Very Short Answer Questions) Section A contain six (06) Very Short Answer Type Questions. Examinees have to attempt all questions. Each question is of 0 mark and maximum word limit may be thirty words. IÊS> - "A' (A{V bkw CÎmar àíz) IÊS> "E' N> 06 A{V bkwcîmamë H$ àíz h, narjm{w m H$mo g^r àízm H$mo hb H$aZm h & àë oh$ àíz Ho$ 0 A H$ h Am a A{YH$V eãx gr m Vrg eãx h & MT-03 / 900 / 6 () (P.T.O.)
) (i) Write the condition that the conic section ax by hxy gx fy c = 0 represent two straight lines. em H$d n[aàn>ox ax by hxy gx fy c = 0 Ho$ Xmo gab aoimam Ho$ {Zê${nV H$aZo H$m à{v~ Y {bimo& (ii) Define Sphere. Jmobo H$mo n[a^m{fv H$s{OE& (iii) Write the equation of enveloping cone of a sphere x y = a Jmobo x y = a Ho$ EÝdbmonr e Hw$ H$m g rh$au {b{ie& (iv) Define cylinder. ~obz H$mo n[a^m{fv H$s{OE& (v) Write tangent plane to the sphere at the point (a, b, γ) x y ux vy w d = 0 {ZåZ Jmobo Ho$ {~ÝXþ (a, b, γ) na ñne g Vb H$m g rh$au {b{ie& x y ux vy w d = 0 (vi) Define convex set. Ad wi g wàm H$mo n[a^m{fv H$s{OE& MT-03 / 900 / 6 () (Contd.)
Note: {ZX}e : 678 Section - B 4 8 = 3 (Short Answer Questions) Section B contain Eight (08) Short Answer Type Questions. Examinees will have to answer any four (04) questions. Each question is of 08 marks. Examinees have to delimit each answer in maximum 00 words. (IÊS> - ~) (bkw CÎmar àíz) IÊS> "~r" 08 bkw CÎma àh$ma Ho$ àíz h, narjm{w m H$mo {H$Ýht ^r Mma (04) gdmbm Ho$ Odm~ XoZm h & àë oh$ àíz 08 A H$ H$m h & narjm{w m H$mo A{YH$V 00 eãxm àë oh$ Odm~ n[agr{ V H$aZo h & ) Obtain the centre of the conic section ax by hxy gx fy c = 0 e Hw$ n[aàn>ox ax by hxy gx fy c = 0 Ho$ Ho$ÝÐ Ho$ {ZX}em H$ kmv H$s{OE& 3) Find the equation of a tangent plane to the sphere x y x - 4y 6-7 = 0 which intersect in the line 6x - 3y - 3= 0 = 3 Jmobo x y x y - 4 6-7 = 0 Ho$ Cg ñne g Vb H$m - - = = Ho$ g rh$au kmv H$s{OE Omo aoimam 6x 3y 3 0 3 à{vàn>ox go JwOao& 4) Find the equation of right circular cylinder whose radius is 4 and axis is x = y = - Cg bå~d Îmr dobz H$m g rh$au kmv H$s{OE {OgH$s {ÌÁ m 4 h VWm Aj x = y = - h & MT-03 / 900 / 6 (3) (P.T.O.)
x y 5) If the tangent plane to the ellipsoid a = cuts off the b c a b c intercepts a, b and g to the axes. Show that γ = α β {X XrK d VO a x I S> H$mQ>Vm h Vmo {gõ H$s{OE& y = H$m ñne Vb Ajm go a, b, g A V b c a b c γ = α β 6) Show that the section of the surface y x xy = a by the plane lx my n = p is a parabola, if l m n = 0 àx{e V H$s{OE {H$ n îr> y x xy = a H$m g Vb lx my n = p Ûmam n[aàn>ox EH$ nadb hmojm {X l m n = 0 7) Solve the following L.P.P. Graphically Max () = 5x 3x S.t. 3x x G x x G 4 x x G 8 x, x H 0 AmboIr {d{y go C³V ao{ih$ àmoj«m{ J g ñ m H$m hb kmv H$s{OE& 8) Solve the following L.P.P. Using simplex method Min () = 4x 3x S.t. 00x 00 x H 4000 x x H 400 x x H 35 x, x H 0 C³V L.P.P. g ñ m H$mo qgnbo³g {d{y go hb H$s{OE& MT-03 / 900 / 6 (4) (Contd.)
9) Solve the following transportation problem using Vogel s method in order to minimie total transportation cost. {ZåZ transportation ( mvm mv) g ñ m H$m mvm mv yë (cost) H$mo {ZåZV H$aZo Ho$ {be "dmojb' {d{y go hb H$s{OE& Destination Origin D D D 3 D 4 D 5 Availability O 3 5 8 9 0 O 5 4 0 7 0 40 O 3 5 8 7 5 30 Requirement 0 5 5 30 0 00/00 Note: Section - C 4 = 8 (Long Answer Questions) Section C contain 04 Long Answer Type Questions. Examinees will have to answer any two (0) questions. Each question is of 4 marks. Examinees have to delimit each answer in maximum 500 words. (IÊS> - g) (XrK CÎmar àíz) {ZX}e : IÊS> "gr" 04 {Z~ YmË H$ àíz h, narjm{w m H$mo {H$Ýht ^r Xmo (0) gdmbm Ho$ Odm~ XoZm h & àë oh$ àíz 4 A H$m H$m h & narjm{w m H$mo A{YH$V 500 eãxm àë oh$ Odm~ n[agr{ V H$aZo h & MT-03 / 900 / 6 (5) (P.T.O.)
0) Solve the L.P.P. 678 Max () = 4x 5x 3x 3 S.t. x x x 3 = 0 x x H x 3x x 3 G 30 x, x, x 3 H 0 C³V L.P.P. Problem H$mo hb H$s{OE& ) Find the condition that the lines x - α y β γ = - = - and l m n should be polar lines w.r.t. the sphere x y = a do à{v~ Y kmv H$s{OE {H$ aoime± = Ho$ gmnoj Yw«dr aoime± hmo & x y a x - α y β γ l m n = - = - Am a Jmobo ) Find the equation to the generators of the hyperboloid x y - = which passes through the point a b c (a cos a, b sin a, 0) A{Vdadb O a x y - = Ho$ {~ÝXþ (a cos a, b sin a, 0) go b c OmZodmbo OZH$mo Ho$ g rh$au kmv H$s{OE& 3) Find the principal planes and principal directions of the following conicoid 8x 7y 3-8y 4x - xy x - 8y = 0 {ZåZ em H$dO H$s w» {XemE± Ed w» g Vb kmv H$s{OE& 8x 7y 3-8y 4x - xy x - 8y = 0 MT-03 / 900 / 6 (6)