Gujarat University 2005 B.Sc Mathematics First Year - Question Paper

FYBSC Mathematics-II New

Seat No.

FB-02

Time : 3 Hours]

Mathematics Paper-Ii (New Course)

[Total Marks: 105

*Wl : (1) 4L UMULHL <H Slid HMl

(2)    LLL > VlMl-il <a"l.

(3)    &$. Vi-i-iL p %R*il .

1. (a) y.4cm> (Adjoint) iMis-fl. -LL"m 4LUl. >L A = [a_lJ]_n 4 n-i"iL-il =Mk%l iLli i*L ll y.lfclL h3.1 h A (adj A) = (adj A) A = |A| I_n.

4-1=11

(a) i.[L-LL UCrl (Transpose) L[Lh-{l c-.i"--LL 4LVlL.

>\ A 4 m x n UM-il ii[Lh 4-i B 4 n x p HiL-il i.[L l-L ll Li[LL hlL h (AB)^T = B^T a^t.

(b) L[ih-iL hi[-Ll -LL"m 4LVlL.

3 2 0 -1

1 -12 2

0 1 -3 -1

(c) Li L 4b LLi :

(1) i[irL 4-i [C&LPid L[ih-il cLL"LL 4LUl.

2 3 1

4 3 1

11 2 4

(2) iLli A =

2. (a) iky. i.[L HL LL"i[L i5-L 4-i C'LL"i[Lh *l[-$L-{l c-ll"-ll 4lv1

10 0 "

2    10 -il <rLL"i[L iS-Ll 4-l <ai"lllh %lfeL SlM.

3    0 0

rt tl"t[li .dlist stM yd

A⁸ - 5A⁷ + 7A⁶ - 3A⁵ + A⁴ - 5A³ + 8A² - 2A + I cQ [dUSl xtl*tctl it[Sli

tttt.

%ttli2Sl [rt 5x + 3y + 7z = 4

3x + 26y + 2z = 9

7x + 2y + 11z = 5 dt Wnl [dq.Hdt QudtH i& Qk-t d.ct\.

y-l=ll

Std-Plcar. ntl ct"t\ yd yl it.

%ttli2l lCrt

x + y + z = 3 x + 2y + 3z = 4

X + 4y + 9z = 6 dl lOldcll *lh lLt yd [".Vl lUHdl t QCt

feqat tdliSl

⁴ 4- /iKv³ -i- 6cx²

a, b, c, d, e e R, a * 0 d QisCtctldl 2l2ldl 2ld y.H'lctl.

y-l=ll

[t-tlrl ldliSl

ax³ + 3bx² + 3cx + d = 0.

a, b, c, d e R, a * 0 d Qk-tctl HlXdl iliddl 2ld %tmct\.

UllC-ld %L4dl R² Hi Ll4q.-l y>ll4L - = 1 + e cos 0 *L.cl.

MWl

a [lllcll.l M-l (5, a) 4-q.L.L    cfti.    >1 cl. HciHM

XRLR -LLL rll    L .cl!

LL rl *l LsH :

(1) >l m4cHl UWR C-lH dl[lciLM\ P'SP M-l Q'SQ li. rH LL[ll 41 4

11

+    = m=m. .

SPSP' SQSQ'

(2)    y>ll4L r = a cos 0 + b sin 0, a, b tR MLC-l, M q.-L [-lL 4. $ LH LrLlcll Aw A-. 4- mA Bmi SlM.

(3)    >l R³ Hi 416 [L--LL [[Cl-i. ll*l (4, , 4) \ii Ll Adi 4Lrlll mA Llctll ll*l *1.4.

Llci4 x² + y² + z² = a² XRdl [Sl- P(a, p, y) MLL.dl *UldCL-L y>ll4L A.cl.

_z2 = _a2

MWl

3

LlClrL 41 4 R Hi %LHdCL M-L LlCl4-H LLLLL Mi cl. $.

Ll A *l U3ll :

(1)    a -Ll 46 OtHd Hl %LHdCL x + y + z = a,

UCLA x² + y² + z² - 2x - 2y - 2z - 13 = 0 -L *U$T ? 4ill Pl-m! M-m! -Lli. ?

(2)    yiPld 41 4 LlCL4\

x² + y² + z² + 4x + 4y + 4z - 13 = 0 mA

x² + y² + z² - 20x - 36y - 14z + 73 = 0 HR*ll *M.1 $.

₂

(3)    cl.. - - 2- . (1, -2, 3) + 3 = 0, - . (1, 5, -7) = 45 -LL 4- M-l &W A.cjl.

R³ Hi Pi- (a, p, y) Hl-ll UR -Lrll M-l LlCL4 x² + y² + z² = a² -L wlrll 4 ".LM\ cii.L ufeusil y.*ll4L A.cl.

MWl

R³ Hi "U ^:x~r- = ^yp = ^zY >H\ M"i li. M-l (3mi r li. M=U    l    m    n

y.Hd.L4R-l %LtlJR'3L A.cH.

(b) olHl L *l U3lL :

(1)    RlpM- (a, p, y) H-L HLHLPq.' y¹ = 4ax, z = 0 H=LL Si-i PlMliL Hm.cH.

(2)    yiLd iPl i xy + yz + zx = 0 PLHi fcSlM. $, L-lL $LM, H*l (Miilm H-L H"L Hm\.

(3)    Hi PLH-WLiLP (4, -5, 3) Pl-hM UR -LLL L-tl H"L z-H"L-l PLHMP L-LL [5|- (5, -2, 6) h|L-LI UR -LLL . HL PLH-MLMP-i PLMliL Hm\.

7. (a) PLHtLCL lx + my + nz = p -Im. SlLi cc> ax² + by² + cz² = 1 -l L HL-ll ?LPL Hm\ Lhl> PU$1[Sl-.-U M.LH hLcH.

WWl

(a)    UPci-lW ax² + by² = 2z -ll L-Ll UP-ll [S{- (a, p, y) HI-lL [LCL-l PLMliL hlcCl.

(b)    Lh. L *l U3ll :

(1)    Hi Hlxmw 5x² - 4y² + 7z² = 139 -ll PLHtLCL 20x + 4y - 21z = 19 -l PLHldP ?M.lriCL\ -ll yMlill H-l Lhc-LL PUL-Hl HCcLC.

_x2 _y2 _z²

a b c

Mathematics Paper-Ii (New Course)

[Total Marks: 105

(1)    There are seven questions.

(2)    Attempt all questions.

(3)    All questions carry equal marks.

1. (a) Define adjoint of a matrix. If A = [a is a square matrix of order n, prove that A (adj A) = (adj A) A = |A| I_n.

OR

(a)    Define Transpose of a matrix.

If A is m x n matrix and B is n x p matrix then prove that

(AB)^T = B^T a^t.

(b)    Define rank of a matrix.

Find the rank of a matrix

(c) Attempt any one :

(1) Define symmetric and skew-symmetric matrices.

2 3 1

Express the matrix A =

as a sum of symmetric and skew-

symmetric matrices.

(2) For the matrix A =

In usual notations obtain the polar equation ~ = 1 + e cos 9 of a conic in R².

OR

Obtain the polar equation of circle having centre at (5, a) and radius a. If circle passes through pole, then what is its equation ?

Attempt any two :

(1)    If P'SP and Q'SQ are mutually perpendicular focal chords of conic, then

, 1 1 prove that _{cn cn}, +    = constant.

SP-SP SQ-SQ

(2)    Prove that the equation r = a cos 9 + b sin 9 (a, b non-zero constants) represents a circle. Also find its centre and radius.

(3)    Find Cartesian and spherical co-ordinates of a point whose cylindrical co

n

ordinates are (4, 4 , 4).

Obtain the equation of the tangent plane to the sphere x² + y² + z² = a² at the point P(a, P, y) in R³.

OR

Prove that the intersection of sphere and plane is a circle.

Attempt any two :

(1)    For what value of a the plane x + y + z = a touches the sphere x² + y² + z² - 2x - 2y - 2z - 13 = 0 ? Obtain the point of contact.

(2)    Prove that the spheres

x² + y² + z² + 4x + 4y + 4z - 13 = 0 and

x² + y² + z² - 20x - 36y - 14z + 73 = 0 touch each other externally.

(3)    Find centre and radius of the circle

-² - 2- . (1, -2, 3) + 3 = 0, - . (1, 5, -7) = 45.

Obtain the equation of an enveloping cone having generator touching a sphere

ugh

OR

x - a y - P z - y Obtain the equation of right circular cylinder having axis = _m = _n

and radius r in R³.

(1)    Obtain the equation of a cone having vertex at (a, P, y) and guiding curves y² = 4ax, z = 0.

(2)    Prove that the equation xy + yz + zx = 0 represents a right circular cone and also find its axis, the vertex and the semi-vertical angle.

(3)    If the axis of the right circular cylinder passing through (4, -5, 3) is parallel to z-axis and passes through point (5, -2, 6) then find the equation of right circular cylinder.

7. (a) Obtain the condition that the plane lx + my + nz = p touches the central conicoid ax² + by² + cz² = 1 and also find point of their contact.

OR

(a)    Obtain the equation of the tangent plane at point (a, P, y) to the paraboloid

ax² + by² = 2z.

(b)    Attempt any two :

(1) Find the equation of the tangent plane and the point of contact to the hyperboloid 5x² - 4y² + 7z² = 139 of one sheet parallel to the plane 20x + 4y - 21z = 19.

_x² _y² _z² id + y2 + = 1

(2) If the tangent plane to the ellipsoid _a2 + t2 + _c2

a b c

ordinate axes in A, B, C then prove that the locus of centroid of A ABC is

a² b² c²

x² + y² + z² = ⁹.

(3) Prove that for all values of X, the plane

- + _b + ²¹ + X    ^z - 21 = 1

a b c    Va b c

FB-02

Mathematics Paper-II (Old Course)

Time : 3 Hours]    [Total Marks : 105

*l*Hl : (1) 4L UMULHL 4.<H Slid VLMi $.

(2)    HHL > UMidl d"iL.

(3)    -2.4 VLMdl OL %i2*U .

1.    (a) &[-?L 4ci4L$L'{l ciL"ii 4Lui.

>l - = (xi, X2), y = (yi, y2) g R² li. L-LL - + y = X + X2, yi + y) 4-1

a - = (ax₂, 0) a g r ci cm"m[iiL ii R² qmPUs &fe?L 4c.4in ?

LHL2L WH'. LL-L'L 42.1.

WWl

(a)    y.[-n cLLL'LL GVLLCILL'-Il cii"m 4iiii. >\ A 4-1 B 4 2l[-$L 4C.4L$L V dl L GvllclLLl Ll ll[il 42i 4 A + B VSl V -j GULC.4L?L .

(b)    Ll L *l LLl :

(1)    y.lPiL 42.1 4 A = {(x, y, z) | x + y + z = 0} 4 R³ GULci4L$L .

(2)    R³ -LL Guici4L?L SP {(1, 2, 1), (-1, 3, 2), (4, 5, -3)} Hi 2l[M (2, 1, 1) 41

(1, 1, 2) 4lCIC'LL 4 dfe L d' 42l.

(3)    >\ A 4 y.[-$l 4C.4L$L V dl 4[24L GviOLL li. ll 2Ll[&LL 42l 4

(i)    [A] = A o A 4 V -j Gvllci4ll .

(ii)    [[A]] = [A].

2.    (a) Rⁿ dl y.M'l $,2"l ciLiiiLL 4-. "i 4ci<rLHd'{l CMA". 411.il.

Li[lL 42l 4 L"l 2411x1 ill'll -2.4 Gvioil 2.2.". 2cUi.Tl .

WWl

(a)    LL-rl UHlLl-l l[-SL 4c.hLSL V -il hl&M "i *41-1x1 ll-L V -ii 4lHl *Ll [CilL hl Slhl-L lH -LrLlcil.

(b)    Ll l *l Lll :

(1)    R² -LL 4LHL3. {(3, 4), (4, 3)} -i LlVl"l y.[-$L (25, 25) -il -LLH cl.

(2)    >\ R³ -LL ?U3L l[-SLl -, -, - "i *41-1x1 l-i ll &irLLciL h - + -, - + -,

- + - VLSI "{ cu-ixi .

(3)    U3L {(1, 0, 1)} -l R³ -LL L Oi-- 4lHl *{M1 cL&Ucd.

3.    (a) *"i Uft4h-{l c-LL"-ii 4ivl

..[.1 hl h >1 T : U V "i Uft4h l-i

o T(a - + P ^-) = a T(-) + PT(y)

V x, y e U, a, P e R.

421=11

(a)    HLl h T : U V "i Uft4h $. yiO*.! hL h T 4h-4h l-L ll 4-i ll > N(T) = {9u}.

(b)    Ll l *l Lll :

(1)    LL[lrL h3.1 h T : R³ R², T (x, y, z) = (x + y, y + z) "i uRLc..-!. $

lH> N(T) 4-i R(T) Sllil.

(2)    LL[lrL hl h "i vi[.cil--i t : R³ R³, T (x, y, z) = (x + y + z, y + z, z) c-LM %lVM. $ 4-i T ^-1 c{l.

(3)    "i Uft4h t : R² R³, T(1, 1) = (2, 0, 1) 4-i T(2, -1) = (1, -1, 1) l-l il T(x, y) cl Lh> T(2, 3) SlM.

4.    (a) *"i vi[cil-i yi-L Ll-lci iL[Lh-{l c-ii"-{. 4ivl

>1 T : R² R², T(x, y) = (x cos 9 - y sin 9, x sin 9 + y cos 9) 9 e R 4lV,Cl "i Ufccik l-L 4-i B₁ = B₂ = {e₁, e₂} 4-i R² -il 4lHl l-L ll il[Lh [T : B₁, B₂] Lcil.

.Hi. Sl'lici-. ML-liL r = 1 + e cos 0 *im\.

lL L LLl :

(1)    "L uR.clL-l T : R² R³    i A = [T : B₁, B₂] -LLU W

" 1 2 "

A = ⁰ 1 y-L B1 = {(1, 2), (-2, 1)} LW B2 = {(1, -1, -1),

_ -1 3 J

(1, 2, 3), (-1, 0, 2)} y y-L'*l R² y-L R³ Hi 'P-LL yiUlL .

(2)    ML-li3L 15 - 3r = r cos 0 iM.L UiL-il mic. -Slici ? L-LL -uMsi-Ll CLSaE Lh> L-L MrMq. ML-liL *Lqd.

(3)    LcflL ULH LL[tLL (2, n/6) y-L (3, n/3) (Sl-yinM UMLL -Led "LL-L

ML-liL $lM. l-LL cLMM -IAc-LL CLSi-Ll CLSaE $lM.

ik-li x² + y² + z² = a² U-LL [S- P(a, P, y) yiW-U MU$LlL-L ML-liL *Ld.

ywi

MLL[5LL iL i Hi R³ LLLCL y-L llLi-il MLLHL-L - yi clL .

l*L L U3lL :

(1)    a -Ll iE [iLL HlX MLHdCL x + y + z = a LlCLi

x² + y² + z² - 2x - 2y - 2z - 13 = 0 -L ? i*il [S-yl (Sl-yi -LLL ?

(2)    MLL[SLL i LLciii x² + y² + z² + 4x + 4y + 4z - 13 = 0 y-L x² + y² + z² - 20x - 36y - 14z + 73 = 0 S.-LVll MUl .

(3)    cilq - - 2 - (1, -2, 3) + 3 = 0, - (1, 5, -7) = 45 -LL is- y-L &W

?iM.

R³ Hi [Si- (a, P, y) *ilLl H-HL3. -Lll y-L liCLi x² + y² + z² = a² -L MUll MLi ."LiylcLLqi ufeULl ?L-L ML-liL *Ld.

ywi

r³ Hi ml    = -^z-Y >-ii y"L -k y-L [5lll r -k ycii

   l    m    n

MLH-LLil-L MLili 3J3L *Ld.

(1)    [MIS- (a, P, y) yrt ylRC y² = 4ax, z = 0 lt yctl sirt *tHlihSl SlM.

(2)    yifclcl ihl i xy + yz + zx = 0 fl.Hl &lM. A Vtl $tM, y*t RR:il.Sl yt

y"t Hml

(3)    yi %ttrllir (4, -5, 3) [S-hM xrlr u*t A 14 y"t z-y"t A lttl. t-tl (5, -2, 6) hM ur u*t A. yt %ttrtll2.rt .Hlist Hm\.

7. (a) y.HlCl Ix + my + nz = p iirUt Sl'liO ax² + by² + cz² = 1 A xtl t HlX-l $lhl Hm\ Lh> xt?[[5l-,rtt lt*t Hmt.

yl=ll

(a)    UVtCtW ax² + by² = 2z dl M xRdl [St- (a, p, y) ylMTtl *Utrt04 %l*ll.St

Hm\.

(b)    l*t t *l lSlt :

(1) yi y[tq.<aw 5x² - 4y² + 7z² = 139 dl y.Hlt 20x + 4y - 21z = 19 A y.HtR *UtlCtVtl y.HliRSll yl lHtrll xt?[[St-y\ Hlqctt.

_x2 _y2 _z2

(2) Qxlctctt>' + 72 + ~ = 1 rt lCt lt*tt"tl [St- A, B, C Hll A- A tl abc

_a² _b² _c²

ll[Stl iH i A ABC rtl    xt-t ~2 + T + T = 9 A.

xyz

(3) ll[Stl iht i X rtl -hi QtHrt HlX a.HrtCt

^2x + y + ²⁵ + X I^x - 2* - ^z - 2! = 1 , *liict> 4 + - 4 = 1 rt

+ r + ^_ + X I-- ~r- - - 21 = 1 , mio -2 + ,₉ - ₂

a b c    V.a b c j    a² b² c²

*HT A

Mathematics Paper-II (Old Course)

[Total Marks : 105

Instructions : (1) There are seven questions in this question paper.

(2)    Attempt all questions.

(3)    All questions carry equal marks.

1. (a) Define vector space.

If x = (x₁, x₂), y = (y₁, y₂) e R² and x + y = (x₁ + x₂, y₁ + y₂) and

a - = (ax₂, 0) a e R are defined in R², is R² a vector space ? Justify your answer.

OR

(a)    Define subspace of the vector space. If A and B are two subspaces of vector space V then prove that A + B is also subspace of V.

(b)    Attempt any two :

(1)    Prove that A = {(x, y, z) | x + y + z = 0} is a subspace of R³.

(2)    Determine whether the vectors (2, 1, 1) and (1, 1, 2) belongs to the subspace SP {(1, 2, 1), (-1, 3, 2), (4, 5, -3)} of R³ or not.

(3)    If A is a non-empty subset of a vector space V then prove that

(i)    [A] = A o A is a subspace of V

(ii)    [[A]] = [A]

(a)    Every linearly independent subset of a finite dimensional vector space V can be extended up to a basis of V.

(b)    Attempt any two :

(1)    Obtain co-ordinates of the vector (25, 25) of the vector space R² with respect to the basis {(3, 4), (4, 3)}.

(2)    If -, ^-, z are three linearly independent vectors of R³, prove that - + -, y + z, z + x are also linearly independent vectors.

(3)    Extend the set {(1, 0, 1)} to two different bases of vector space R³.

3.    (a) Define linear transformation. Prove that if T : U V is linear transformation

o T(a - + P ^-) = a T(-) + PT(y).

V x, y e U, a, P e R.

OR

(a)    T : U V is linear transformation. Prove that T is one-one if and only if N(T) = {0u}.

(b)    Attempt any two :

(1)    Prove that T : R³ R², T (x, y, z) = (x + y, y + z) is linear transformation. Also find N(T) and R(T).

(2)    Prove that the linear transformation

T : R³ R³, T (x, y, z) = (x + y + z, y + z, z) is non-singular.

Also find T ^-1.

(3)    For linear transformation T : R² R³, T(1, 1) = (2, 0, 1) and T(2, -1) = (1, -1, 1), then find T(x, y). Also find T(2, 3).

4.    (a) Define matrix associated with a linear transformation.

If T : R² R², T(x, y) = (x cos 0 - y sin 0, x sin 0 + y cos 0) 0 e R is linear transformation and B₁ = B₂ = {e₁, e₂} are bases in R² then find [T : B₁, B₂].

OR

₂ l

(a) In usual notations obtain the polar equation of a conic in R as r = 1 + e cos 0. FB-02    14

(1)    Obtain the linear transformation T : R² R³ so that A = [T : B₁, B₂]

'1 2 "

where A = 0 1 and B₁ = {(1, 2), (-2, 1)} and

_ -1 3 J

B₂ = {(1, -1, -1), (1, 2, 3), (-1, 0, 2)} are ordered basis of R² and R³ respectively.

(2)    Which conic is represented by the equation 15 - 3r = r cos 9 ? Obtain length of latusrectum & its Cartesian equation.

(3)    Find the polar equation of the straight line passing through (2, n/6) and (/3, n/3). Find the length of perpendicular drawn from the pole upon it.

5. (a) Obtain the equation of the tangent plane to the sphere x² + y² + z² = a² at the point P(a, P, y) in R³.

OR

(a)    Prove that the intersection of sphere and plane is a circle.

(b)    Attempt any two :

(1) For what value of a the plane x + y + z = a touches the sphere

² + y² + z² - 2x - 2y - 2z - 13 = 0 ? Obtain the point of contact.

(2) Prove that the spheres x² + y² + z² + 4x + 4y + 4z - 13 = 0 and

² + y² + z² - 20x - 36y - 14z + 73 = 0 touch each other externally.

(3) Find centre and radius of the circle

-² - 2 ^- (1, -2, 3) + 3 = 0, ^- (1, 5, -7) = 45.

6. (a) Obtain the equation of an enveloping cone having generator touching a sphere x² + y² + z² = a² and passing through a point (a, P, y) in R³.

OR

(a) Obtain the equation of right circular cylinder having axis

x - a y - P z - y    . ₃

j =    = and radius r in R³.

l    m n

(1)    Obtain the equation of a cone having vertex at (a, P, y) and guiding curves y² = 4ax, z = 0.

(2)    Prove that the equation xy + yz + zx = 0 represents a right circular cone and also find its axis, the vertex and the semivertical angle.

(3)    If the axis of the right circular cylinder passing through (4, -5, 3) is parallel to z-axis and passes through point (5, -2, 6) then find the equation of right circular cylinder.

Obtain the condition that the plane Ix + my + nz = p touches the central conicoid ax² + by² + cz² = 1 and also find point of contact.

OR

Obtain the equation of the tangent plane at point (a, P, y) to the paraboloid

ax² + by² = 2z.

Attempt any two :

(1)    Find the equation of the tangent plane and the point of contact to the hyperboloid 5x² - 4y² + 7z² = 139 of one sheet parallel to the plane 20x + 4y - 21z = 139.

_x² _y² _z²

(2)    If the tangent plane to the ellipsoid ~ + , ₂ + ~ = 1 meets the co-

abc

ordinate axes in A, B, C then prove that the locus of centroid of A ABC is a² b² c² _

x² + y² + z² = ⁹.

(3)    Prove that for all values of X, the plane

2xy 2z . fx 2yz

+ , + + X I^--,- ^_ - 21 = 1 touches the conicoid a b c V.a b c j

x² y² z²

2 + u2 ^- 2 = ¹. _a² _b² _c²

FB-02    16

1

(a) Define eigen value and eigen vector of a square matrix. Find the eigen values and corresponding eigen vectors of the matrix

Prove that every subset of a linearly independent set is linearly independent.

OR

Attachment: gujarat-university-2005-b-sc-mathematics-33448-1.pdf

Earning:   Approval pending.