B.A/B.Sc. (PART I) PRE. UNIVERSITY EXAMINATIONMATHEMATICSTHIRD PAPER

(vector calculus and Geometry)

uksV & izR;sd iz'u ds Hkkx (a),(b) 1½ vad ] Hkkx (c), (d) 3 vad o

Hkkx (e) 6 vad dk gSA

UNIT - I

1. (a)Define vector function of a single scalar variable.

,d vfn'k pj ds lfn'k Qyu dks ifjHkkf"kr dhft;sA

(b) If r⃗=acost .i+atsin t . j+bt k find|d r⃗dt | ;fn r⃗=acost .i+atsin t . j+bt k rks|d r⃗dt | Kkr dhft;s

(c) If r⃗=(cosnt )a+(sinnt)b where a and b are constant vectors. prove that.

d2 r⃗d t 2 +n2 r⃗=0

;fn r⃗=(cosnt )a+(sinnt)b tgka a vkSj b vpj lfn'k gS] rks fl) dhft;s &

d2 r⃗d t 2 +n2 r⃗=0

(d) Find the directional derivative of f = xy + yz +zx in the direction of the vector i + 2j+2k at

the point (1,2,0).

fcUnq (1,2,0) ij f = xy + yz +zx dk i + 2j+2k dh fn'kk esa fnd vodyu Kkr

dhft;sA

(e) Find the angle between surfaces x2 +y2 + y3 = 9 and z = x2 + y2 - 3 at the point (2,-1,2).

lrg x2 +y2 + y3 = 9 rFkk z = x2 + y2 - 3 ds chp dk dks.k fcUnq (2,-1,2) ij Kkr

dhft;sA

2. (a) Define the divergence of a vector point function.

lfn'k fcUnq Qyu dk vilj.k dks ifjHkkf"kr dhft;sA

(b) Find divergence of the following vector point function :

f = x2z2i - 2y3z2j +xy2zk

fuEu lfn'k Qyu dk vilj.k Kkr dhft;s %

f = x2z2i - 2y3z2j +xy2zk

(c) Prove that the curl of the gradient of a scalar function u is zeor.

fl) dhft;s fd fdlh vfn'k Qyu u dh izo.krk dk dqUry 'kwU; gksrk gSA

(d) If f = x2y + 2xy + z2, then find :

curl (grad f).

;fn f = x2y + 2xy + z2 ] rc Kkr dhft;s%

curl (grad f).

(e) If r = xi + yj +zk and r = |r|, then prove that.

V 2 (n2 r )=n(n+3)rn−2 r

;fn r = xi + yj +zk vkSj r = |r|, fl) dhft;s %

V 2 (n2 r )=n(n+3)rn−2 r

Unit - II

3. (a) Write the cartesian form of the line integral.

js[kk lekdy dk dkrhZ; :i fyf[k;sA

(b) Write the statement of Gauss Divergence theorem.

xkWl dk vilj.k izes; dk izdFku fyf[k;sA

(c) If F = x2yi + (x - z)j + xyz k and c is the curve y = x2, z = 2 from (0,0,2) to (1,1,2) then find :

∫c

❑

F .dr

;fn x2yi + (x - z)j + xyz k vkSj c oØ y = x2, z = 2 ij fcUnq (0,0,2) ls (1,1,2) rd dk

pki gks rks Kkr dhft;s

∫c

❑

F .dr

(d) If V is th volume enclosed by any closed surfaces show that :

∫s

❑

n̂. ds = 0

;fn V fdlh cUn i`"B S }kjk ifjc) vk;ru gks rks fl) dhft;sA

∫s

❑

n̂. ds = 0

(e) Evaluate : ∫s

❑

F .n̂ . ds where F = xi - yj+(z2 - 1)k and S is the surface of the cylinder

bounded by z =0, z = 1, x2+y2=4

eku Kkr dhft;s ∫s

❑

F .n̂ . ds

tgk¡ F = xi - yj+(z2 - 1)k vkSj S ml csyu dk i`"B gS tks z =0, z = 1 vkSj x2+y2=4

ls ifjc) gSA

4. (a) Write the statement of Stoke's theorem.

LVkWd izes; dk izdFku fyf[k;sA

(b) Write the cartesian form of Green's theorem.

xzhu izes; dks dkrhZ; :i esa ifjHkkf"kr dhft;sA

(c) Using stoke's theorem, evaluate :

∫c

❑

(xydx+x y2dy )

Where C is the square in the xy-plane with vertices respectively : (1,0);(-1,0);(0,1);(0,-1).

LVkWd izes; dk mi;ksx djds fuEu dk eku Kkr dhft;s %

∫c

❑

(xydx+x y2dy )

tgk¡ C, xy lery esa ,d oxZ gS ftlds 'kh"kZ Øe'k% (1,0);(-1,0);(0,1) vkSj (0,-

1) gSA

(d) Evaluate by Green's theorem :

∫c

❑

[ (cos x sin y−xy )dx+sin x cos y dy ]

Where C is the circle x2 +y2 = 1

eku Kkr dhft;sA

∫c

❑

[ (cos x sin y−xy )dx+sin x cos y dy ]

tgk¡ C] o`r x2 +y2 = 1 gSA

(e) Apply stoke's theorem to prove that

div(curl F) = 0

LVkWd izes; dk iz;ksx djds fuEu fl) dhft;s &

div(curl F) = 0

Unit - III

5. (a) Which curve is represented by the equation :

x2 + 2xy + y2 - 2x - 1 = 0

fuEu lehdj.k }kjk fu:fir oØ Kkr dhft;sA

x2 + 2xy + y2 - 2x - 1 = 0

(b) Write down the equation of axis of the conic ax2 + 2hxy + by2 = 1 in terms of the length R

of its semi axis.

%'kkado ax2 + 2hxy + by2 = 1 ds v)Z v{k dh yEckbZ R ds :i esa mlds v{k

dh lehdj.k fyf[k;sA

(c) Find the length of the latus rectum of the parabola

(a2 +b2) (x2 + y2) = (bx +ay - ab)2

ijoy; (a2 +b2) (x2 + y2) = (bx +ay - ab)2 ds ukfHkyEc dh yEckbZ Kkr dhft;sA

(d) Find the centre of the conic :

17x2 - 12xy + 8y2 + 46x - 28y + 17 = 0

'kkado dk dsUnz Kkr dhft;sA

17x2 - 12xy + 8y2 + 46x - 28y + 17 = 0

(e) Trace the conic x2 - 3xy +y2 + 10x - 10y + 21 =0 .

'kkado x2 - 3xy +y2 + 10x - 10y + 21 =0 dk vuqjs[k.k dhft;sA

6. (a) Write the polar equation of a conic, the focus being the pole.

'kkado dk /kzqoh lehdj.k fyf[k;s tc /kzqo 'kkado dh ukfHk ij fLFkr

gksA

(b) Write the polar equation of the directrix (near the pole) of the conic 1r=1+ecosθ .

'kkado 1r=1+ecosθ dh fu;rk ¼tks /kzqo ds ikl gS½ dk /kzqoh lehdj.k

fyf[k;sA

(c) Show that the equations 1r=1+ecosθ and

1r=−1+ecosθ represent the same conic.

iznf'kZr dhft;s fd lehdj.k 1r=1+ecosθ rFkk

1r=−1+ecosθ ,d gh 'kkado dks

fu:fir djrs gSaA

(d) Find the condition that the staight line 1r=A cosθ+B sin θ . may touch the circle

r = 2a cos θ .

og izfrca/k Kkr dhft;s tcfd ljy js[kk 1r=A cosθ+B sin θ o`r r = 2a cos θ dks

Li'kZ djsaaA

(e) Find the locus of the pole of a chord of the conic 1r=1+ecosθ . which subtends a constant

angle 2α at the focus .

'kkado 1r=1+ecosθ dh thok ds /kzqo dk fcUnqifk Kkr dhft;s tks

'kkado dh ukfHk ij vpj dks.k 2α vUrfjr djrh gSA

Unit - IV

7. (a) Find the radius of the sphere.

x2 +y2+z2 - 2x +4y - 6z = 11

fuEu xksys fd f=T;k Kkr dhft;sA

x2 +y2+z2 - 2x +4y - 6z = 11

(b) Define power of a point.

fcUnq dh 'kfDr dks ifjHkkf"kr dhft;sA

(c) Find the equation of the sphere circumscribing the tetrahedron whose faces are :

yb+ zc=0 ;

zc+ xz=¿ 0 ;

xa+ yb=0 ;

xa+ yb+ zc=1

bl prq"Qyd dks ifjxr djus okys xksys dk lehdj.k Kkr dhft;s ftldh

Qydksa ds lehdj.k fuEu gS %

yb+ zc=0 ;

zc+ xz=¿ 0 ;

xa+ yb=0 ;

xa+ yb+ zc=1

(d) Find the equation of sphere which touches the plane 3x +2y - z + 2 = 0 at the piont (1,-2,1)

and also cuts orthogonally the sphere

x2+y2+z2 - 4x+6y+4 = 0

ml xksys dk lehdj.k Kkr dhft;s tks lery 3x +2y - z + 2 = 0 dks fcUnq (1,-

2,1) ij Li'kZ djrk gS rFkk xksys x2+y2+z2 - 4x+6y+4 = 0 dks ykfEcd :i ls dkVrk

gSA

(e) A sphere of constant radius (r) passes through the origin (O) andcuts the axes in A,B,C.

Prove that the locus of the foot of perpendicular drawn from O to the plane ABC is given by

(x2+y2+z2)2 (x-2 +y-2 +z-2) = 4r2

vpj f=T;k (r) dk ,d xksyk ewy fcUnq (O) ls xqtjrk gS vkSj funsZ'kh

v{kksa dks Øe'k% A,B,C ij dkVrk gSAfl) dhft;s fd (O) ls lery ABC ij

[khaps x;s yEc ds ikn dk fcUnqiFk gS %

(x2+y2+z2)2 (x-2 +y-2 +z-2) = 4r2

8. (a) Define :

(i) Enveloping cone

(ii) Right circular cylinder

ifjHkkf"kr dhft;sA

(i) vUokyksih 'kadq

(ii) yEco`rh; csyu

(b) Find the equation of a right circular cylinder whose axis is x−2

2= y−1

1= z

3 and which

passes through (0,0,1).

ml yEco`rh; csyu dk lehdj.k Kkr dhft;s ftldh v{k x−2

2= y−1

1= z

3 gS

rFkk tks fcUnq (0,0,1) ls xqtjrk gSA

(c) Find the enveloping cone of the sphere x2+y2+z2 +2x - 2y - 2 = 0 with its vertex at (1,1,1) .

xkssys x2+y2+z2 +2x - 2y - 2 = 0 ds ml vUokyksih 'kadq dk lehdj.k Kkr

dhft;s ftldk 'kh"kZ fcUnq

(1,1,1) gSA

(d) If x1= y

2= z

3 represents one of the three mutually perpendicular generators of the cone

5yz - 8xz - 3xy = 0 . Find the equations of the other two.

;fn x1= y

2= z

3 'kadq 5yz - 8xz - 3xy = 0 ds rhu ijLij ledksf.kd tudksa esa

ls ,d gS rks vU; nks tudksa ds lehdj.k Kkr dhft;sA

Unit - V

9. (a) Define :

(i) Pole

(ii) Polar plane

ifjHkkf"kr dhft;s %

(i) /kzqo

(ii) /kzqoh; rRo

(b) A tangent plane to the ellipsoid x2

a2 + y2

b2 + z2

c2=1 meets the coordinate axes in points A,B

and C. Prove that the centroid of the triangle ABC lies on the locus a2

x2 + b2

y2 + c2

z2 =9 .

nh?kZo`rt x2

a2 + y2

b2 + z2

c2=1 ds Li'kZ ry funsZ'kkad v{kksa dks A,B rFkk C

fcUnqvksa ij feyrk gSA fl) dhft;s fd f=Hkqt ABC dk dsUnzd dk

fcUnqiFk a2

x2 + b2

y2 + c2

z2 =9 ij fLFkr gksxkA

(c) Find the equation of the polar line of x+1

2= y−2

3= z+3

1 with respect to the sphere

x2 +y2 + z2 = 1 .

xksys x2 +y2 + z2 = 1 ds lkis{k js[kk x+1

2= y−2

3= z+3

1 dh /kzqoh; js[kk dk

lehdj.k Kkr dhft;sA

(d) Prove that the sum of the squares of the reciprocals of any three mutually perpendicular

semidiameters of an elipsoid x2

a2 + y2

b2 + z2

c2=1 is constant .

fl) dhft;s fd nh?kZo`rt x2

a2 + y2

b2 + z2

c2=1 ds fdUgha rhu ijLij yEcor~

v)ZO;kl ds O;qRØeksa ds oxksZa dk ;ksxQy vpj gksrk gSA

10. (a) Define paraboloid.

ijoy;t dks ifjHkkf"kr dhft;sA

(b) Write the equation of enveloping cone of the paraboloid ax2 + by2 = 2cz having vertex at

(x1,y1,z1).

ijoy;t ds vUokyksih 'kadq dk 'kh"kZ fcUnq (x1,y1,z1) ij lehdj.k fyf[k;sA

tgk¡ ijoy;t

ax2 + by2 = 2cz gSA

(c) The plane 3x + 4y = 1 is a diametral plane of the paraboloid 5x2 + 6y2 = 2z. Find the

equation of the chord through (3,4,5) which it bisects.

lery 3x + 4y = 1 ijoy;t 5x2 + 6y2 = 2z dk O;klh; lery gSA fcUnq (3,4,5) ls

xqtjus okyh rFkk bl ij lef}Hkkftr gksus okyh thok dk lehdj.k Kkr

dhft;sA

(d) Find the equation of the normal at the point (x1,y1,z1) of the paraboloid ax2 + by2 = 2cz

ijoy;t ax2 + by2 = 2cz ds fcUnq (x1,y1,z1) ij vfHkyEc dk lehdj.k Kkr dhft;sA

(e) Show that the section of the surface yz +zx +xy = a2 by the plane lx +my+nz = p is parabola

if √ l+√m+√n=0 .

iznf'kZr dhft;s fd i`"B yz +zx +xy = a2 dk lery lx +my+nz = p }kjk ifjPNsn

ijoy; gS ;fn

√ l+√m+√n=0 .

B.A/B.Sc. (PART I) PRE. UNIVERSITY EXAMINATIONMATHEMATICSSECOND PAPER

(Calculus )

uksV & izR;sd iz'u ds Hkkx (a),(b) 1½ vad ] Hkkx (c), (d) 3 vad o

Hkkx (e) 6 vad dk gSA

UNIT - I

1. (a)Define homegeneous function..

le?kkr Qyu dks ifjHkkf"kr dhft;sA

(b) Write geometrical meaning of partial derivative.

vkaf'kd vodyt dk T;kferh; vFkZ fyf[k;sA

(c) If u = (1- 2xy+y2¿−12 , then prove that :

∂∂ x {(1−x2) ∂u

∂ x }+ ∂∂ y {y2 ∂u

∂ y }=0

;fn u = (1- 2xy+y2¿−12 rks fl) dhft;s fd

∂∂ x {(1−x2) ∂u

∂ x }+ ∂∂ y {y2 ∂u

∂ y }=0

(d) If u = f ( xy , yz , zx ) ; then prove that :

x∂u∂ x

+ y ∂u∂ y

+z ∂u∂ z

=0

;fn u = f ( xy , yz , zx ) rks fl) dhft;s fd %

x∂u∂ x

+ y ∂u∂ y

+z ∂u∂ z

=0

(e) If u = xϕ(y/x)+ψ(y/x), then prove that :

x2 ∂2u∂ x2 +2 xy

∂2u∂ x∂ y

+ y2 ∂2u∂ y2 =0

;fn u = xϕ(y/x)+ψ(y/x) rks fl) dhft;sA

x2 ∂2u∂ x2 +2 xy

∂2u∂ x∂ y

+ y2 ∂2u∂ y2 =0

2. (a) If the relation f(x,y) = C. defines y as a differentiable function of x, then find dydx

.

;fn laca/k f(x,y) = C,x dk vodyuh; Qyu ifjHkkf"kr djrk gS rks dydx dk eku

Kkr dhft;sA

(b) If x = r cos θ, y = r sin θ, then find the value of J(x,y).

;fn x = r cos θ, y = r sin θ gks rks J(x,y) dk eku Kkr dhft;sA

(c) If xxyyzz = C, then prove that x = y = z ;

∂2 z∂ x∂ y

= −1x log ex

;fn xxyyzz = C rks x = y = z ij fl) dhft;s fd %

∂2 z∂ x∂ y

= −1x log ex

(d) If u = 1x

, v = x2

y and w = (x + y + zy2) then find :

∂(u , v ,w)∂(x , y , z)

;fn u = 1x , v =

x2

y rFkk w = (x + y + zy2) rks eku Kkr dhft;sA

∂(u , v ,w)∂(x , y , z)

(e) If u3 + v3 + w3 = x + y + z, u2 +v2 + w2 = x3 + y3 +z3 and u + v +w = x2 + y2 +z2, then show

that ∂(u , v ,w)∂(x , y , z)

=( y−a ) ( z−x )(x− y )(u−v ) ( v−w )(w−u)

;fn u3 + v3 + w3 = x + y + z, u2 +v2 + w2 = x3 + y3 +z3 rFkk u + v +w = x2 + y2 +z2 ] rks

iznf'kZr dhft;sA

∂(u , v ,w)∂(x , y , z)

=( y−a ) ( z−x )(x− y )(u−v ) ( v−w )(w−u)

Unit - II

3. (a) State the necessary and sufficient condition for extreme value of function f(x,y).

Qyu f(x,y) ds pje eku gksus dkd vko';d ,oa i;kZIr izfrca/k dk dFku

fyf[k;sA

(b) Find the minimum value of √ x2+ y2 , where x2 + y2 +xy = 1.

izfrcU/k x2 + y2 +xy = 1 ds vUrxZr √ x2+ y2 dk U;wure eku Kkr dhft;sA

(c) Find the maxima and minima of :

sin x + sin y + sin(x+y)

mfPp"B rFkk fufEu"B Kkr dhft;sA

sin x + sin y + sin(x+y)

(d) Explain Lagrange's method of undetermined multiplers.

vfu/kkZ;Z xq.kkadksa dh ykxzkat fof/k dks le>kb;sA

(e) Prove that a rectangular solid of maximum volxuame within a sphere is a cube.

fl) dhft;s fd ,d xksys ds vUrxZr vf/kdre vk;ru okyk vk;rkdkj

Bksl ,d ?ku gksrk gSA

4. (a) Define multiple points.

cgqy fcUnq dks ifjHkkf"kr dhft;sA

(b) Define point of inflection and write down test for point of inflexion.

ufr ifjorZu fcUnq dh ifjHkk"kk nsrs gq, bldk ijh{k.k Hkh fyf[k,A

(c) Prove that at the origin on the curve y2 = bx sin ( xa ), there is a node or a conjugate point

according as a and b have same or opposite signs.

fl) dhft;s oØ y2 = bx sin ( xa ) ij a rFkk b ds leku ;k foijhr fpUgksa ds

vuqlkj ewy fcUnq ij ,d uksM ;k la;qXeh fcUnq gSA

(d) Find the nature and position of double points of the curve y (y - 6) = x2(x-2)3 - 9.

fuEu oØ ds f}d fcUnqvksa dh fLFkfr ,oa iz—fr Kkr dhft;sA

y (y - 6) = x2(x-2)3 - 9

(e) Trace the following curve :

r2 = a2 cos 2θ

fuEu oØ dk vuqjs[k.k dhft;s

r2 = a2 cos 2θ

Unit - III

5. (a) Give the necessary and sufficient condition for f(a,b) to be a maximum value of f (x,y).

Qyu f (x,y) ds vf/kdre eku f(a,b) gksus dk vko';d ,oa i;kZIr izfrca/k

fyf[k,A

(b) Find the points where the function x3 - 4xy + 2y2 has maximum or minimum value.

mu fcUnqvksa dks Kkr dhft, tgk¡ Qyu x3 - 4xy + 2y2 dk eku mPpre ;k

U;wure gSA

(c) Find the maximum value of

u = sin x sin y sin (x + y)

mPpre eku Kkr dhft, &

u = sin x sin y sin (x + y)

(d) Find the minimum value of u = x2 + y2+z2 when ax + by + cz = p .

U;wure eku Kkr dhft, &

u = x2 + y2+z2

tcfd ax + by + cz = p

(e) Prove that a rectangular solid of maximum volume. within a sphere is a cube.

fl) dhft, fd ,d xksy ds vUrxZr vf/kdre vk;ru okyk vk;rkdkj Bksl ,d ?

ku gksrk gSA

6. (a) Define a double point.

,d f}d fcUnq dks ifjHkkf"kr dhft,A

(b) Write down the equation of tangent at the origin to the curve a2y2 - b2x2 = x2y2 .

oØ a2y2 - b2x2 = x2y2 dh ewy fcUnq ij Li'kZ js[kk dk lehdj.k fyf[k,A

(c) Find the nature and position of the double points of the curve

x3 - y3 - 7x2 + 4y + 15x - 13 = 0 .

fuEu oØ ds f}d fcUnqvksa dh fLFkfr ,oa iz—fr Kkr dhft;s

x3 - y3 - 7x2 + 4y + 15x - 13 = 0

(d) Find the points of inflexion of the following curve y(a2 + x2) = x3 .

fuEu oØ ds ufr ifjorZu fcUnq Kkr dhft, &

y(a2 + x2) = x3

(e) Trace the curve x3 +y3 = 3axy.

fuEu oØ dk vuqjs[k.k dhft, &

x3 +y3 = 3axy

Unit - IV

7. (a) Show that -

B(m,n) = B(n,m) where , B(m,n) is Beta function.

iznf'kZr dhft, &

B(m,n) = B(n,m)

tgk¡ B(m,n) chVk Qyu gSA

(b) Show that -

∫0

∞

Xn−1e−ax dx=Γ (n)an

iznf'kZr dhft, &

∫0

∞

Xn−1e−ax dx=Γ (n)an

(c) prove that :

∫0

∞xdx

1+x6 =π

3√3

fl) dhft, &

∫0

∞xdx

1+x6 =π

3√3

(d) Show that -

∫0

1

xn−1( log1x )

m−1

dx=Γ (m)nm

%iznf'kZr dhft,&

∫0

1

xn−1( log1x )

m−1

dx=Γ (m)nm

(e) Prove that -

Γ (m )Γ (m+ 12 )= √π

22m−1Γ (2m)

where, m Z.

fl) dhft, &

Γ (m )Γ (m+ 12 )= √π

22m−1Γ (2m)

tgk¡ m Z.

8. (a) Evaluate .

∫0

1

∫0

2

( x+ y )dx dy

eku Kkr dhft, &

∫0

1

∫0

2

( x+ y )dx dy

(b) Define double integral.

f}lekdy dks ifjHkkf"kr dhft,A

(c) Change the order of integration in the following double integral.

∫0

a

∫y

√a2− x2

f ( x , y )dx dy

fuEu f}lekdy dk Øe ifjofrZr dhft, &

∫0

a

∫y

√a2− x2

f ( x , y )dx dy

(d) Change the following integral to polar co-ordinater and evaluate :

∫0

a

∫y

axdxdyx2+ y2

fuEu lekdy dks /kzqoh funsZ'kkadksa esa ifjofrZr dj eku Kkr dhft,

&

∫0

a

∫y

axdxdyx2+ y2

(e) Show that :

∫0

∞

e−x2

dx=√π2

iznf'kZr dhft,

∫0

∞

e−x2

dx=√π2

Unit - V

9. (a) Define Quadrature.

{ks=Qyu dks ifjHkkf"kr dhft,A

(b) Write down the formula to find the common area of two Cartesian curves.

y = f1(x) and y = f2(x).

nks dkrhZ; oØks y = f1(x) rFkk y = f2(x) dk mHk;fu"B {ks=Qy Kkr djus

dk lw= fyf[k,A

(c) Find the length of the are of the semi-cubical parabola ay2=x3 from its vertex to the point

(a,a).

v)Z?ku ijoy; ay2=x3 ds 'kh"kZ ls fcUnq (a,a) rd ds pki dh yEckbZ Kkr

dhft,A

(d) Find the perimeter of the cardioid r = a(1 + cosθ). Also prove that the arc of the upper half of

the car-dioid is bisected by θ = π3

dkfMZ;ksbM r = a(1 + cosθ) dk ifjeki Kkr dhft,A ;g Hkh fl) dhft, fd

dkfMZ;ksbM dk Åijh v)ZHkkx pki θ = π3 ls lef}Hkkftr gksrk gSA

(e) Find the area included between the following curve and its asymptote xy2 = 4a2 (2a - x).

fuEu oØ rFkk mlds vuUrLi'khZ ds e/; dk {ks=Qy Kkr dhft,

xy2 = 4a2 (2a - x).

10. (a) Write a formula to find the volume of the solid generated by the revolution about the intial

line of the area bounded by the curve r = f(θ) and the radii vector θ = α, θ = β.

oØ r = f(θ) ij lfn'k dks.k θ = α rFkk θ = β ds ifjc) {ks= dk vkjfEHkd js[kk

ds ifjr ifjØe.k ls tfur ?kuk—fr dk vk;ru Kkr dhft;sA

(b) Write down the Liouville's extension of Dirichler's integral.

fMfjpysV lekdy dk fyosyh O;kihdj.k fyf[k,A

(c) Find the volume of the solid generated by revolving the ellipse x2

a2 + y2

b2 =1 about the axis

of x.

nh?kZo`r x2

a2 + y2

b2 =1 dk x-v{k ds ifjr ifjØe.k djus ij izkIr ?kukd`fr dk

vk;ru Kkr dhft;sA

(d) Find the volume of the solid generated by the revolution of the carioid r = a(1 + cos θ )

about the initial line.

dkfMZ;ksbM r = a(1 + cos θ ) ds vkjfEHkd js[kk ds ifjr ifjØe.k ls tfur ?

kuk—fr dk vk;ru Kkr dhft;sA

(e) Evaluate

∭V

❑

2 z dxdy dz

where, region of integration is a cone V bounded by the following curves . x2 + y2 = z2, z = 1

eku Kkr dhft;s &

∭V

❑

2 z dxdy dz

tgk¡] lekdyu dk {ks= ,d 'kadq gS tks fuEu oØksa ls ifjc) gS

x2 + y2 = z2, z = 1

B.A/B.Sc. (PART I) PRE. UNIVERSITY EXAMINATIONMATHEMATICS

FIRST PAPER(ALGEBRA )

uksV & izR;sd iz'u ds Hkkx (a),(b) 1½ vad ] Hkkx (c), (d) 3 vad o

Hkkx (e) 6 vad dk gSA

UNIT - I

1. (a) write the statement of factor theorem.

xq.ku[kaM+ izes; dk izdFku dhft;sA

(b) What is reciprocal equation of the given equation.

nh gqbZ lehdj.k dh O;qRØe lehdj.k D;k gksrh gS \

(c) Solve the equation

x4 + 4x3 + 6x2 + 4x + 5 = 0 if one root is √−1

lehdj.k x4 + 4x3 + 6x2 + 4x + 5 = 0 dks gy dhft;s ;fn bldk ,d ewy √−1 gksA

(d) If the roots of the equation

x3 + 3px2 + 3qx + r = 0 are in H.p., show that 2q3=r(3pq-r)

;fn lehdj.k x3 + 3px2 + 3qx + r = 0 ds ewy gjkRed Js.kh esa gksa rks fl)

dhft;s fd %

2q3=r(3pq-r)

(e) Slove the cubic equation

9x3+6x2-1=0 by Cardon's method.

f=?kkr lehdj.k 9x3+6x2-1=0 dks dkMZu fof/k ls gy dhft;sA

2. (a) Write the Descarte's Rule of Signs.

nsdkrsZ dk fpUg fu;e fyf[k;sA

(b) Find the nature of the roots of equation :

x4 - ax3 - bx - c = 0 where a, b and c are positive.

lehdj.k x4 - ax3 - bx - c = 0 ds ewyksa dh iz—fr dh tk¡p dhft;sA tgk¡ a, b

vkSj c /kukRed gSA

(c) Show that the following equation has atleast four imaginary roots :

2x7 - x4 + 4x3 - 5 = 0

fl) dhft;s fd lehdj.k ds de ls de pkj ewy dkYifud gksaxs &

2x7 - x4 + 4x3 - 5 = 0

(d)Find the roots of the equation :

x3 - 3x + 1 = 0

fuEu lehdj.k ds ewy Kkr dhft;sA

x3 - 3x + 1 = 0

(e) Solve the equation by Ferrari's method.

6x4 - 7x3 + 8x2 - 7x + 2 = 0

fuEu lehdj.k dks QSjkjh fof/k ls gy dhft;sA

6x4 - 7x3 + 8x2 - 7x + 2 = 0

Unit - II

3. (a) Define Hermitian and Skew Hermitian matrix.

gehZf'k;u vkSj fo"ke gehZf'k;u eSfVªDl dks ifjHkkf"kr dhft;sA

(b) Define rank of a matrix.

eSfVªDl dh jSad ¼tkfr½ dks ifjHkkf"kr dhft;sA

(c) If A and B are unitary marrices of the same order then AB and BA are also unitary matrices.

;fn A vkSj B nks leku Øe dh ,sfdd eSfVªDl gksa rks AB rFkk BA

Hkh ,sfdd eSfVªDl gksrh gSaA

(d) Prove that the rank of a non-singular matrix is equal to the rank of its inverse.

fl) dhft;s fd fdlh O;qRØe.kh; eSfVªDl dh tkfr mlds O;qRØe dh tkfr

ds cjkcj gksrh gSA

(e) Find the eigen values and the corresponding eigen vectors of the following matrix A:

A = | 8 −6 2−6 7 −4

2 −4 3 | fuEu eSfVªDl A ds vfHkyk{kf.kd ewyksa ,oa muds laxr lfn'kksa dks

Kkr dhft;s %

A = | 8 −6 2−6 7 −4

2 −4 3 |4. (a) Write the statement of Cayley - Hamilton Theorem.

dSyh&gSfeyVu izes; dk izdFku fyf[k;sA

(b) write the condition of consistency of the system of equations AX = B .

lehdj.k fudk; AX = B dh laxrrk dh 'krZ fyf[k;sA

(c) Reduce the following matrix in normal form :

|0 1 24 0 22 1 3

−261 |

fuEu eSfVªDl dks vfHkyEc :i esa lekuhr dhft;s %

|0 1 24 0 22 1 3

−261 |

(d) For what value of k, the following system of equations has non-trivial solution :

2x + 3y + 4z = 0

x + y + z = 0

4x + 6y + kz = 0

k ds fdl eku ds fy;s fuEu lehdj.k fudk; ds gy lkFkZd gSa \

2x + 3y + 4z = 0

x + y + z = 0

4x + 6y + kz = 0

(e) Apply matrix theory to solve the following system of equations :

2x + 3y - z = 9

x + y + z = 9

3x - y - z = -1

eSfVªDl fl)kUr dk iz;ksx dj fuEu lehdj.k fudk; dks gy dhft;s %

2x + 3y - z = 9

x + y + z = 9

3x - y - z = -1

Unit - III

5. (a) Define binary operation on a set.

fdlh leqPp ij f}vk/kkjh lafØ;k dh ifjHkk"kk nhft;sA

(b) Define a group.

,d lewg dh ifjHkk"kk nhft;sA

(c) If G is a finite semi group such that for any a,b,c, G,

(i) ab = ac b = c

(ii) ba = ca b = c, then prove that G is a group.

;fn fdlh ifjfer lkfelewg esa fdUgha a,b,c, G, ds fy,

(i) ab = ac b = c

(ii) ba = ca b = c, rks fl) dhft;s fd G ,d lewg gSA

(d) If G is a group, then prove that for a, b G,

(ab)-1 = b-1c-1

;fn ,d lewg gks rks fl) dhft;s fdUgha a, b G, ds fy, %

(ab)-1 = b-1c-1

(e) If S is the set of real numbers other than -1, then show that (S,*) is a group where * is the

operation defined as :

a * b = a + b + ab ∀ a, b S

;fn &1 ds vfrfjDr lHkh okLrfod la[;kvksa dk lewPp; S gks rks fl)

dhft;s fd (S, *) ,d lewg gS tgk¡ lafØ;k * fuEu izdkj ifjHkkf"kr gS %

a * b = a + b + ab ∀ a, b S

6. (a) State Lagrange's theorem.

ykxzkUt izes; dk dFku nhft;sA

(b) Give an example to show that union of two subgroups of a group is not necessarily a

subgroup.

,d mnkgj.k nsdj fn[kkb;s fd fdlh lewg ds nks milewgksa dk la?k

vfuok;Zr% ,d milewg ugha gksrk gSA

(c) If H and K are two subgroup of a group G, then prove that HK is a subgroup of G iff

HK =KH.

;fn H vkSj K lewg G ds nks milewg gksa rks fl) dhft;s HK, G dk

milewg gksxk ;fn vkSj dsoy ;fn HK = KH.

(d) Find all the Cosets of H = {0,4} in the group G = (Z8 +8).

lewg G = (Z8 +8). esa H = {0,4} ds lHkh lgleqPp; Kkr dhft;sA

(e) Prove that every subgroup of a cyclic group is also cyclic.

fl) dhft;s fd ,d pØh; lewg dk izR;sd milewg Hkh pØh; gksrk gSA

Unit - IV

7. (a) Define normal subgroup.

izlkekU; milewg dh ifjHkk"kk nhft;sA

(b) Prove that every quotient group of an abelian group is abelian.

fl) dhft;s fd ,d vkcsyh les dk izR;sd foHkkx lewg vkcsyh gksrk gSA

(c) Prove that the intersection of any two normal subgroups of a group is a normal subgroup.

fl) dhft;s fd fdUgha nks izlkekU; milewg dk loZfu"B ml lewg dk ,d

izlkekU; milewg gksrk gSA

(d) Let G be the group of all real matrices of types [a b0 d ] , ad 0 under matrix multiplication

and N = {|a b0 d|, b∈R} then prove that N G,

ekuk G okLrfod la[;kvksa ij [a b0 d ] , ad 0 vkdkj ds lHkh eSfVªDl dk

xq.kukRed lewg gS rFkk N = {|a b0 d|, b∈R} rks fl) dhft;s fd N G,

(e) Prove that the set of all cosets of a normal subgroup H of a group G, is a group with respect

to multiplicatioon of cosets defined as follows :

HaHb = Hab a,b G

fl) dhft;s fd fdlh lewg G esa mlds fdlh izlkekU; milewg H ds lHkh

lgleqPp;ksa dk leqPp; fuEu izdkj ifjHkkf"kr lgleqPp;ksa ds xq.ku ds

fy, ,d lewg gksrk gS %

HaHb = Hab a,b G

8. (a) Define homomorphism.

lekdkfjrk dh ifjHkk"kk nhft;sA

(b) Define kernel of a homomorphism.

lekdkfjrk dh vf"V dh ifjHkk"kk nhft;sA

(c) Prove that every group is homomorphic to its quotient group.

fl) dhft;s fd izR;sd lewg vius foHkkx lewg ds lekdkjh gksrk gSA

(d) If a is an element of a group G, then prove that the mapping fa : G → G defined by

fa (x) = axa-1 x G is an automorphism of G.

;fn a fdlh lewg G esa ,d vo;o gks rks fl) dhft;s fd izfrfp=.k %

fa : G → G defined by fa (x) = axa-1 x G

lewg G dh ,d Lokdkfjrk gSA

(e) If f is a homomorphism from a group G to G' with kernel K then prove that K is a normal

subgroup of G.

;fn f lewg G ls G' ij ,d lekdkfjrk gks rks fl) dhft;s fd f dh vf"V K,G

dk ,d izlkekU; milewg gksrh gSA

Unit - V

9. (a) Define cyclic permutation.

pØh; Øep; dh ifjHkk"kk nhft;sA

(b) Define order of a permutation.

Øep; dh dksfV dh ifjHkk"kk nhft;sA

(c) Find ρ-1 σρ when :

σ = (1 3 5 4) (2 6 8 ) ( 9 7)

ρ = ( 8 9 7 6) (5 4 1) (2 3)

find wherther ρ-1σρ is even or odd permutation. Also find ots order.

ρ-1 σρ Kkr dhft;s tcfd %

σ = (1 3 5 4) (2 6 8 ) ( 9 7)

ρ = ( 8 9 7 6) (5 4 1) (2 3)

crkb;s fd ρ-1 σρ le vFkok fo"ke Øep; gSA bldh dksfV Hkh Kkr dhft;sA

(d) Prove that the alternating group An of all even permutation of degree n is a normal subgroup

of the symetric group Sn.

fl) dhft;s fd n-va'kkad ds lHkh le Øep;ksa dk ,dkUrj lewg An ] Sn lefer

lewg dk izlkekU; milewg gksrk gSA

(e) State and prove Cayley theorem.

dsys izes; dks le>kb;s ,oa fl) dhft;sA

10. (a) Define order of an element of a group.

fdlh lewg ds vo;o dh dksfV ifjHkkf"kr dhft;sA

(b) If in a group G, a-1 b-1 ab = e a, b G, then prove that G is an abelian group.

;fn fdlh lewg esa a-1 b-1 ab = e a, b G rks fl) dhft;s fd G ,d vkcsyh lewg

gSA

(c) If in a group G, a5 = e and aba-1 = b2 a,b G then find the order of b.

;fn fdlh lewg G esa a5 = e rFkk aba-1 = b2 a,b G rks b dh dksfV Kkr

dhft;sA

(d) If a, b are two elements of a group G such that ambn = ba m, n Z, then prove that

ambn-2 , am-2bn, ab-1 are elements of the same order in G.

;fn lewg G esa a,b ,sls vo;o gksa fd ambn = ba m, n Z rks fl) dhft;s fd

ambn-2 , am-2bn rFkk ab-1 , G esa leku dksfV ds vo;o gSA

(e) If order of an element a of a group(G,*) is n, then prove that am = e iff m is a multiple of n.

;fn fdlh lewg ds ,d vo;o dh dksfV gks rks fl) dhft;s fd ;fn vkSj

dsoy ;fn dk xq.kt gksA