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Transcript of Sophia Girls' College, Ajmer
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2015- 16
End Semester Examination β II
Class : B.Sc.[Math] β SEM II
(Sub : Math)
Paper I : [MAT-201]: Vector Calculus and Geometry β I
Time : 2 Hrs. M.M: 70 Marks
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions, at
least 3 questions from each unit. Each question is of one mark.
I. Answer the following questions 1. If r = |
πβ | , where
πβ = xi + yj + jk. Prove that
ππππ (1
π) = β
π
π
;fn r = | πβ | tgkΒ‘
πβ = xi + yj + jk fl) dhft,
ππππ (1
π) = β
π
π
Define directional Derivative.
fnd vodyt dΒ¨ ifjHkkfβkr dhft, |
2. Define curl of a vector point function.
lfnβk Qyu dΒ’ dyZ dΒ¨ ifjHkkfβkr dhft, |
3. State Stokeβs Theorem.
LVΒ¨Dl Γes; dk dFku fyf[k, |
4. Define Line Integral.
js[kk lekdy dΒ¨ ifjHkkfβkr dhft, |
5. Define conic section.
βkkado dΒ¨ ifjHkkfβkr dhft, |
6. Write equation of the conic referred to centre as origin.
ewy fcUnq dΒ¨ dΒ’UΓ¦ ekurs gq, βkkado dk lehdj.k fyf[k, |
7. Find the equation of sphere whose centre is ( Β½ , - Β½ , 1) and radius is 2.
x¨ys dk d’Uæ¼ ½ , - ½ , 1½ rFkk f=T;k 2 bdkà ysrs gq, lehdj.k Kkr dhft, |
8. Define cone and right circular cone.
βkadq ,oa yEcdΒ¨.kh; βkadq dΒ¨ ifjHkkfβkr dhft, |
9. Define reciprocal cone.
O;qRΓe βkadq dΒ¨ ifjHkkfβkr dhft, |
Section B [15 Marks]
Section B contains 7 questions (50 words each) and a candidate is required to attempt 5 questions, at least 1 from
each unit. Each question is of 3 marks.
II. Answer the following questions
UNIT I
10. If πβ =xi+yj+zk and
πβ is any constant vector then prove that
ππ’ππππ₯π
π3 = βπ
π3 +3(π.π)π
π5
;fn r =xi+yj+zk ,oa πβ o ,d vpj lfnβk gS
rΒ¨ fl) dhft, ππ’ππππ₯π
π3 = βπ
π3 +3(π.π)π
π5
11. Prove that div(axb) = b.curl a β a.curl b
fl) fdft, div(axb) = b.curl a β a.curl b
12. Evaluate β«πΉ.ππ
π, where
πΉβ= (x2 + y2)i β 2xyj and curve c is the rectangle in the xy plane bounded by x=0,
x=a, y=0, y=b
Kkr dhft, β«πΉ.ππ
π, tgkΒ‘
πΉβ= (x2 + y2)i β 2xyj rFkk oΓ C] xy lery esa x=0, x=a, y=0, y=b }kjk fn;k x;k vk;r gS
|
UNIT II
13. Find the co ordinates of the centre of the conic. Also find the equation of the conic referred to the centre as
origin in the standard form. fuEu βkkado dk dΒ’UΓ¦ Kkr dhft, ,oa βkkado dk dΒ’UΓ¦ ewy fcUnq ysrs gq, lehdj.k Kkr dhft,
36x2+24xy+29y2-72x+126y+81=0
14. Trace the parabola
fuEu oΓ dk fp=.k dhft, |
9x2+24xy+16y2-2x+14y+1=0
UNIT III
15. A plane passes through a fixed point (a,b,c) and cut the axes in A,B,C. Show that the locus of the centre of
the sphere oABC is π
π₯ +
π
π¦ +
π
π§ = 2
,d lery vpj fcUnq ΒΌ a,b,cΒ½ ls xqtjrk gS rFkk v{kΒ¨a dΒ¨ ij dkVrk gS fl) dhft, fd xΒ¨ys dΒ’ dΒ’UΓ¦ dk fcUnq iFk gS π
π₯ +
π
π¦ +
π
π§ =
2
16. Find the equation of the cylinder whose generators are parallel to the line π₯
1 =
π¦
β2 =
π§
3 and whose guiding curve
is the ellipse x2 + 2y2 =1 , z = 0
ml csyu dk lehdj.k Kkr dhft, ftldk tud π₯
1 =
π¦
β2 =
π§
3 dΒ’ lekUrj gS rFkk ftldk funZsβkd oΓ nh?kZo`r gS x2 + 2y2 =1 , z
= 0
Section C [45 Marks]
Section C β Answer any three questions (400 words each), selecting one from each unit. Each question is of 15
marks.
III. Answer the following questions
UNIT I
17. a) A particle moves along the curve x=2t2, y=t2 β 4t, z=3t-5. Find the components of velocity and
acceleration of the particle in the direction of the vector i-3j+2k at t=1.
,d d.k oΓ x=2t2, y=t2 β 4t, z=3t-5 osx o Roj.k dΒ’ lfnβk i-3j+2k at t=1 dΒ’ vuqfnβk ij ?kVd Kkr dhft, |
b) Prove that
fl) dhft, |
β2 f (r) = f (r)+ 2
π f (r)
Where r = | πβ | and
πβ = xi+yj+ zk
OR
Verify Stokeβs theorem for the function F=zi + xj + yk where curve C is the unit circle in xy plane bounding
the hemisphere.
Z = 1 β β1 β π₯2 β π¦2
LVΒ¨Dl Γes; dk lR;kiu dhft, tgkΒ‘ F=zi + xj + yk rFkk C v)ZxΒ¨ys dΒ¨ Z = 1 β β1 β π₯2 β π¦2 ifjc) djus okyk
v)Zo`r gS |
UNIT II
18. A) Discuss the tracing of conic π
π = 1 + e cos π
βkkado π
π = 1 + e cos π dΒ’ fp=.k dh foospuk dhft, |
OR
Find the asymptotes of the following hyperbolas and equation to their conjugate hyperbola fuEu vfrijoy; dh vUUr LiβkΓ Kkr dhft, ,oa bldΒ’ la;qXeh vfrijoy; dh lehdj.k Kkr dhft,
y2-xy-2x2-5y+x-6=0
UNIT III
20. a) A sphere of constant radius r passes through the origin O and cuts the axis in A,B,C. Prove that the locus of
the foot of the perpendiculars drawn from O to the plane ABC is given by
(x2+y2+z2)2 (x-2+y-2+z-2) = 4r2
,d x¨yk ftldh f=T;k vpj r gS] ewy fcUnq ls xqtjrk gS rFkk A,B,C v{k¨a d¨ ij dkVrk gS- fl) dhft, dh ewy fcUnq ls lery
ABC ij Mkys x;s yEc dΒ’ ikn dk fcUnqiFk gS |
(x2+y2+z2)2 (x-2+y-2+z-2) = 4r2
b) Find the coordinates of the centre and radius of the circle
fuEu or dh f=T;k o dΒ’UΓ¦ dΒ’ funZsβkkad Kkr dhft, |
x2 + y2 -2y β 4z = 20 , x+27+2z=21
OR
Prove that the cone ax2+by2+cz2+2fyz+2gzx+2hxy=0 may have three mutually perpendicular tangent planes if
bc+ca+ab = f2+g2+h2
fl) dhft, fd βkadq ax2+by2+cz2+2fyz+2gzx+2hxy=0dΒ’ rhu ijLij yEcor LiβkZ ry gS ;fn bc+ca+ab = f2+g2+h2
Find the equation of Right circular cylinder whose generators are parallel to z axis and intersect the surfaces
ax2+by2+cz2=1 and lx + my + nz = p
yEc o`rh; csyu dk lehdj.k Kkr dhft, ftldk tud z v{k d’ lekarj rFkk lrg ax2+by2+cz2=1 rFkk lx + my + nz = p d¨
dkVrk gS |
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2015- 16
End Semester Examination β II
Class : B.Sc.[Math] β SEM II
(Sub : Math)
Paper II: [MAT-202]: Calculus β I
Time : 2 Hrs. M.M: 70 Marks
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following questions
1. Write the lengths of polar tangent and polar sub tangent.
?kqzoh LiβkΓ rFkk /kzoh v?k% LiβkΓ dh yEckΓ fyf[k;s |
2. Write the condition for the existence of the asymptote.
vuUr LiβkΓ dΒ’ vfLrRo dΒ’ fy, ΓfrcU/k fyf[k;s|
3. Write the coordinates of centre of curvature.
oΓrk dΒ’UΓ¦ dΒ’ funZsβkkad fyf[k;s | 4. Write the condition for a point to be a point of inflextion.
,d fcUnq dΒ’ ufr ifjorZu fcUnq gΒ¨us dh βkrZ fyf[k;s |
5. Define partial differential coefficient.
vakfβkd vody xq.kkad dΒ¨ ifjHkkfβkr dhft, |
6. If u = π₯ 2 and v=y2 then find the value of π(π’,π£)
π(π₯,π¦)
;fn u = π₯ 2 rFkk v=y2
rΒ¨ π(π’,π£)
π(π₯,π¦) dk eku Kkr dhft, |
7. Write the necessary conditions for f(a,b) to be an extreme value of f(π₯,y)
Qyu f (π₯,y) dΒ’ pjeeku f (a,b) gΒ¨us dk vkoβ;d ΓfrcU/k fyf[k;s |
8. Show that
ΓnΓβkr dhft, |
β«1
0 π₯ n-1 (log
1
π₯ )m-1 d π₯ =
β(π)
ππ , m , n > 0
9. Write double integral in polar coordinates.
/kqzoh funZsβkkadΒ¨a esa f}lekdy dΒ¨ fyf[k;s |
10. State Liouvilles extension of Dirichletβs Integral.
fMfjpfyV lekdy dk fyosyh O;kidhdj.k dk dFku fyf[k;s |
Section B [10 Marks]
Section B contains 7 questions (50 words each) and a candidate is required to attempt 5 questions, at least 1 from
each unit. Each question is of 2 marks.
II. Answer the following questions.
UNIT I
11. Show that the chord of curvature through the pole for the curve p=f(r) is 2π(π)
π(π)
ΓnΓβkr dhft, fd oΓ p=f(r) dh /kzo ls xqtjus okyh oΓrk thok dh yEckΓ 2π(π)
π(π) gS |
12. Find all the asymptotes of the following curve:
fuEu oΓ dh lHkh vuUr LiΓβk;kΒ‘ Kkr dhft, |
(y-2π₯)2(3π₯ +4y) + 3(y-2 π₯)(3 π₯ +4y) β 5 =0
UNIT II
13. If u = sin-1 π₯+2π¦+3π§
π₯ 8+ π¦8 + π§8 find the value of
π₯ ππ’
ππ₯ + y
ππ’
ππ¦ + z
ππ’
ππ§
;fn u = sin-1 π₯+2π¦+3π§
π₯ 8+ π¦8 + π§8 rΒ¨ π₯ ππ₯
ππ¦ + y
ππ’
ππ¦ + z
ππ’
ππ§
dk eku Kkr fdft, |
14. Find the ratio of the difference between the radii of curvature at any two points of a curve and length of the
arc of the evolute between the two corresponding points. fdlh oΓ dΒ’ fdUgha nΒ¨ fcUnqvΒ¨a ij oΓrk f=T;kvΒ¨a dk vUrj rFkk mldΒ’ dΒ’UΓ¦t dΒ’ laxr fcUnqvΒ¨a dΒ’ chp dΒ’ pki dh yEckΓ dk
vuqikr Kkr dhft, |
UNIT III
15. If β«1
0 π₯ m (1- π₯)n d π₯ =β«
1
0 π₯ n (1- π₯)p, then find the value of p.
;fn β«1
0 π₯ m (1- π₯)n d π₯ =β«
1
0 π₯ n (1- π₯)p, rΒ¨ p dk eku Kkr dhft, |
16. Evaluate the following integral by changing the order of integration.
fuEu lekdy dk Γe cnydj eku Kkr dhft, |
β«1
0
β«π
ππ₯
ππ¦ππ₯
log π¦
17. Evaluate
eku Kkr dhft, |
β«1
β1 β«π§
0 β«π₯+π§
π₯βπ§ (π₯ +y+z) d π₯ dydz.
Section C [30 Marks]
Section C β Answer any three questions (400 words each), selecting one from each unit. Each question is of 10
marks.
III. Answer the following questions.
UNIT I
18. Find all the asymptotes of the following curve:
fuEu oΓ dh lHkh vuUr LiΓβk;kΒ‘ Kkr dhft, |
(2 π₯ -3y+1)2 (π₯ +y) = 8 π₯ β 2y + 9
b. Find the angle of intersection of the following cardioids : fuEu Γ’n;ke dk ΓfrPNsnu dΒ¨.k Kkr dhft,
r = a(1+ Cos π) and r = b(1-Cos π)
OR
a. Show that the circle of curvature at the origin of the parabola y=mx+( π₯2
π ) is π₯
2+y2=a(1+m)2(y-m π₯).
ΓnΓβkr dhft, fd ewy fcUnq ij i[kyo y=m π₯ +( π₯2
π ) dk oΓrk o`r π₯2+y2=a(1+m)2(y-m π₯) gS |
b. Examine the nature of the origin on the following curve: fuEu oΓ ij ewy fcUnq dh ΓΓfr dh tkΒ‘p dhft,
π₯ 7+2 π₯ 4+2 π₯ 3y+ π₯ 2+2 π₯ y+y2=0
UNIT II
19. a. If π₯ +y = 2eΖ Cos β and π₯ -y = 2ieΖ Sin β
show that π2π’
ππ2 + π2π’
πβ 2 = 4xy π2π’
ππ₯ππ¦
;fn π₯+y = 2eΖ Cos β and π₯-y = 2ieΖ Sin β
rΒ¨ ΓnΓβkr dhft,
π2π’
ππ2 + π2π’
πβ 2 = 4xy π2π’
ππ₯ππ¦
b. if u = π₯ β(1 β π¦)2 + y β(1 β π₯)2 ; v = sin-1 π₯ sin-1y
Show that u, v are functionally related and find the relationship.
;fn u = π₯ β(1 β π¦)2 + y β(1 β π₯)2 ; v = sin-1 π₯ sin-1y
ΓnΓβkr dhft, fd u, v Qyuh; lcaf/kr gS rFkk mlesa lEcU/k Kkr dhft, |
OR
a. Find the envelope of the family of the following straight lines; β being the parameter:
fuEu ljy js[kk dΒ’ dqy dk vUokyΒ¨i Kkr dhft, tgkΒ‘ β Γkpy gS |
a xsec β- bycosec β = a2-b2
b. Find the points where the function x3+y3-3axy has maximum or minimum value.
mu fcUnqvΒ¨a dΒ¨ Kkr dhft, tgka Qyu π₯ 3+y3-3axy dk eku mPpre ;k U;wure gS |
UNIT III
20. Prove that fl) dhft,
β«β
0
π₯
1+π₯6 dx = π₯
3β3
a. β(β1
2 )
b. β« β(1 β π₯)1
0 dx
OR
a. Change the order of integration in the following integral: fuEu lekdy esa lekdyu dk Γe ifjorZu dhft,
β«4π
0 β«2β(ππ₯)
π₯2
4π
π (π₯,y) dπ₯ dy
b. Evaluate eku Kkr dhft,
β π₯π¦π§ ππ₯ ππ¦ ππ§ ,
Where region of integration is the complete ellipsoid :
tgkΒ‘ lekdy dk {ks= lEiw.kZ fn/kZo`Rrt gS
π₯2
π2 + π¦2
π2 + π§2
π2 β€ 1
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2016- 17
End Semester Examination
Class : B.Sc [Maths]
Sub : Mathematics
Paper II : [MAT-202]: Calculus
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following questions 1. Define Pedal equation.
ifnd Lkehd.kZ dh ifjHkkβkk nhft,
2. Define double point. f}d fcUnq dh ifjHkkβkk nhft,
3. Write the formula to find the radius of curvature for Cartesian curves. dkrΓ; oΓ dΒ’ fy, oΓrk f=T;k Kkr djus dk lw= fyf[k,
4. Write the degree of homogenous function
π’ = π₯π¦π(π¦
π₯).
lEk/kkr Qyu π’ = π₯π¦π(π¦
π₯) dh /kkr fyf[k,
5. Define Envelope. vUokyΒ¨i dh ifjHkkβkk fyf[k,
6. Give the necessary and sufficient condition for f(a,b) to be a maximum value of f(x,y).
Qyu f(x,y) dΒ’ vf/kdre eku f(a,b) gΒ¨us dh vkoβ;d o i;kZIr ΓfrcU/k fyf[k,
7. Define Gamma Function. xkek Qyu dh ifjHkkβkk nhft,
8. Evaluate. eku Kkr dhft, β« βπ₯. πβπ₯3ππ₯
β
0
9. State Dirichletβs Integral for triple integral. f=lekdyu dΒ’ fy, Qyu esa lEcU/k fyf[k,
10. Write down the relation between Beta and Gamma function. chVk rFkk xkek Qyu esa lEcU/k fyf[k,
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following questions
UNIT I
11. Find the Pedal equation of the parabola.
π¦2 = 4π(π₯ + π)
ijoy; π¦2 = 4π(π₯ + π) dk ifnd lehdj.k Kkr dhft,
OR
Find the asymptotes of the following curve: fuEu oΓ dΒ’ vUuRkLiΓβk;kΒ‘ Kkr dhft,
x3 + yx2 - xy2 - y3 - 3x β y β 1 = 0
UNIT II
12. If ;fn π’ = π ππβ1 (βπ₯ββπ¦
βπ₯+βπ¦) then show that. rc ΓnΓβkr dhft, fd
π₯ππ’
ππ₯+ π¦
ππ’
ππ¦0
OR
Find the envelope of the circles drawn upon the central radii of the ellipse π₯2
π2 +π¦2
π2 = 1as diameter.
mu o`RrΒ¨a dk vUokyΒ¨i Kkr dhft, tΒ¨ nh/kZoRr π₯2
π2 +π¦2
π2 = 1 dh /kqzokUrr js[kkvΒ¨a dΒ¨ O;kl ekudj [khps x, gSa
UNIT III
13. To show that ΓnΓβkr dhft,
βπ βπ + 12β =
βπ
22πβ1 β2π ; π β π
where m is positive. tgkΒ‘ m /kukRed gSa
OR
Change the order of integration in the following double integral.
fuEu f}lekdy es lekdyu dk Γe ifjoΓrr dhft, β« β« π(π₯, π¦)βππ₯
2
π₯24πβ
ππ₯ ππ¦4π
0
Section C [45 Marks]
Section C β contains 6 questions. Answer any three questions (400 words each), selecting one from each
unit. Each question is of 15 marks.
III. Answer the following questions.
UNIT I
14. Trace the following curve: fuEu oΓ dk vUkjs[k.k dhft, LVΒͺΒ¨QkΒ‘;M
π¦2(π + π₯) = π₯2(3ππ₯)
OR
(a) Show that for the radius of curvature at a point (π πππ 3π, π π ππ3π) on the curve
π₯2
3β + π¦2
3β = π2
3β is 3π
2π ππ2π.
fl) dhft, fd oΓ π₯2
3β + π¦2
3β = π2
3β dΒ’a fcUnq (π πππ 3π, π π ππ3π) ij oΓrk f=T;k
3π
2π ππ2π gΒ¨rh gS
(b) Find the point of inflection of the curve π¦(π2 + π₯2) = π₯3.
oΓ π¦(π2 + π₯2) = π₯3 dΒ’ ufr ifjorZu fcUnq Kkr dhft,
UNIT II
15. (a) If ;fn π’ = π₯ π ππβ1 π¦
π₯β then prove that (rc fl) dhft, ) π₯2π2π’
ππ₯2 +2π₯π¦π2π₯
ππ₯ ππ¦+ π¦2 π2π’
ππ¦2 = 0
(b) If ;fn π = π(π₯ β π¦, π¦ β π§, π§ β π₯)then prove that rc fl) dhft, ππ
ππ₯+
ππ
π¦+
ππ
ππ§= 0
OR
(a) Find the envelope of the family of ellipse
π₯2
π2+
π¦2
π2= 1
When a + b = c, c being a constant.
nh/kZorΒ¨ π₯2
π2 +π¦2
π2 = 1 dΒ’ dqy dk vUokyΒ¨i Kkr dhft,] tcfd a + b = c tgkΒ‘ c vpj gSa
(b)Show the minimum value of the following function
is 3a2.
ΓnΓβkr dhft, fd fuEu Qyu dk fufEuβV eku 3a2 gS%
π’ = π₯π¦ + π3(1
π₯+
1
π¦)
UNIT III
15. (a) Prove that :- fl) dhft, dh
β«(π₯ β π)πβ1(π β π₯)πβ1 ππ₯
π
π
= (π β π)π+πβ1 π½(π, π)
(b) Show that :- ΓnΓβkr dhft, dh
βπ β1 β π =π
sin π π
OR
(a) Evaluate :- eku Kkr dhft,
β« β« π₯π¦(π₯ + π¦)ππ₯ ππ¦ over the area between the parabola y = x2 and line y = x.
ijoy; y = x2 rFkk y = x dΒ’ e/; {ks= ij
(b) Evaluate :- eku Kkr dhft,
β« β« β« ππ₯+π¦+π§ ππ₯ ππ¦ ππ§π₯+log π¦
0
π₯
0
πππ2
0
The End
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2016- 17
End Semester Examination β I
Class : B.Sc. Maths
Sub : Mathematic
Paper I : [MAT-201]:Vector Calculus and Geometry β I
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following questions. a. Prove that A scalar point function f is constant iff grad f=0.
fl) dhft, fd ,d vfnβk fcUnq Qyu vpj gS ;fn grad f=0 |
b. If π = |πβ| where
πβ = π₯π + π¦π + 2π prove that (fl) dhft, ) grad π = π
^ |
c. If πβ = π₯π + π¦π + 2π, prove that div
πβ = 3
;fn πβ = π₯π + π¦π + 2π, rΒ¨ fl) dhft, fd div
πβ = 3
d. State Gaussβ theorem.
xkWl Γes; dk dFku fyf[k, |
e. Define conic section.
βkadq ifjPNsn dΒ¨ ifjHkkfβkr dhft, |
f. Write polar equation of a straight line.
ljy rs[kk d’ /kqzoh; lehdj.k d¨ fyf[k, |
g. Write equation of the directrix.
fujkrk d’ lehdj.k d¨ fyf[k, |
h. Write equation of sphere in diameter form.
O;kl #i es x¨ys dk lehdj.k fyf[k, |
i. Define enveloping cone.
vUokyΒ¨ih βkadq dΒ¨ ifjHkkfβkr dhft, |
j. Define Right circular cylinder.
yEcoRrh; csyu dΒ¨ ifjHkkfβkr dhft, |
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following questions
UNIT I
11. If πβ = π₯π + π¦π + 2π and π = |
πβ| , prove (fl) dhft,) that curl ππ
πβ = 0.
OR
Prove that fl) dhft, |
πππ£(π’π) = π’. πππ£ π + π. ππππ π’
UNIT II
12. (a) Find the co-ordinates of the centre of the following conic :
fuEu βkado dΒ’ dΒ’UΓ¦ dΒ’ funZsβkkad Kkr dhft, |
(b) Find the equation of the following conic referred to centre as origin in the standard form:-
dΒ’UΓ¦ dΒ¨ ewy fcUnq ysdj fuEu βkkado dk ekud #i Kkr dhft, |
36x2+24xy+29y2-72x+126y+81=0
OR
Show that the latus rectun of the parabola (a2+b2)(x2+y2)=(bx+ay-ab)2 is 2ππ
βπ2+π2.
fl) dj¨ fd ijoy; (a2+b2) (x2+y2) = (bx+ay-ab)2 dh ukfHkyEc
2ππ
βπ2+π2 gΒ¨rh gS |
UNIT III
13. Two spheres of radii r1 and r2 cut orthogonally: Prove the radius of their common circle is. π1π2
βπ12 + π2
2
r1 rFkk r2 f=T;k dΒ’ nΒ¨ xΒ¨ys ykfEod #i ls dkVrs gS fl) dhft, fd mHk;fuβB o`Rr dh f=T;k gS | π1π2
βπ12 + π2
2
OR
If π₯
1=
π¦
2=
π§
3 represent one of the three mutually perpendicular generators of then cone 5yz- 8xz - 3xy =
0 find the equation of the other two.
;fn π₯
1=
π¦
2=
π§
3 βkadq 5yz - 8xz - 3xy = 0 dΒ’ rhu ijLij lejhf.kd tudΒ¨a esa ,d gS rΒ¨ vU; nΒ¨ tudΒ¨a dΒ’ Lkehdj.k
Kkr dhft, |
Section C [45 Marks]
Section C contains 6 questions (400 words each) and a candidate is required to attempt 3 question,
at least 1 from each unit. Each question is of 15 marks.
III. Answer the following questions.
UNIT I
14. (a) If f and g are scalar point function, then
Prove that
;fn f rFkk g nΒ¨ vfnβk fcUnq Qyu gΒ¨ rΒ¨ fl) dhft,
ππππ (π
π) =
π(ππππ π)βπ(ππππ π )
π2, π β 0
(b) Find the equation of the tangent plane and
normal to the surface x2+y2+z2=25 at point
(4,0,3).
i`βB x2+y2+z2=25 dΒ’ fcUnq (4,0,3) ij LiβkZ
Rky rFkk vfHkyEc dΒ’ Lkehdj.k Kkr dhft, |
OR
(a) If πβ = π₯π + π¦π + 2π and π = |
πβ| , prove (fl) dhft,) that
div ππ
πβ = (π + 3)
πβ π
(b) Evaluate (eku Kkr dhft,)
β« (π¦π§ππ₯ + (π₯π¦ + 1)ππ¦ + π₯π¦ππ§) Where c is any path form (1,0,0) to (2,1,4)
tgka c fcUnq (1,0,0) ls (2,1,4) rd d¨à iFk gS |
UNIT II
15. Find the nature of the conic represented by the equation x2 + 2xy + y2 - 2x β 1 = 0 and also trace
it.
Lkehdj.k x2 + 2xy + y2 - 2x β 1 = 0 }kjk ΓnΓβkr βkkado dh ΓΓfr Kkr dhft, rFkk bldk vuqjs[k.k Hkh dhft, |
OR
Trace the curve. oΓ vuqjs[k.k dhft, |
x2 +y2 + xy + x + y + 1 = 0
UNIT III
16. (a) Find the equation of the sphere which passes
through the points (1,0,0,)(0,1,0) and (0,0,1)
and has its radius as small as possible.
fcUnq (1,0,0,)(0,1,0) vΒ©j (0,0,1) ls xqtjus
okys ml x¨ys dk lehdj.k Kkr dhft,
ftldh f=T;k U;wUkre g¨ |
(b) A sphere whose centre lies in the positive octant, passes through the origin and cuts the plane x
= y = z = 0 in circles of radii πβ2, πβ2, πβ2 respectively. Find its equation.
,d xΒ¨ys dk dΒ’UΓ¦ /ku vβVkaβkd esa gS vΒ©j og ewy fcUnq ls xqtjrk gS funZsβkh lery x=y=z=0 ls bldk ΓfrPNsn
Γeβk% πβ2, πβ2, πβ2 f=T;k okys o`Rr gS bl xΒ¨ys dk lehdj.k Kkr dhft, |
OR
(a) Find the enveloping cone of the sphere x2+y2xz2+2x-2y-2=0 with its vertex at (1,1,1).
xΒ¨ys x2+y2xz2+2x-2y-2=0 dΒ’ ml vUokyΒ¨ih βkadq dk lehdj.k Kkr dhft, ftldk βkhβkZ fcUnq (1,1,1) gS |
(b) Find the equation of right circular cylinder whose guiding curve is the circle
π₯2 + π¦2 + π§2 = 9, x - 2y + 2z = 3
ml yEcoRrh; csyu dk lehdj.k Kkr dhft, ftLkdk funZsβkd oRr π₯2 + π¦2 + π§2 = 9, x - 2y + 2z = 3 gS |
The End
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2017- 18
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper I : [MAT-201]: Vector Calculus & Geometry
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following questions
1. Find the value of βπ, ππ π(π₯, π¦, π§) = π₯2π¦ + π¦2π₯ + π§2
;fn π(π₯, π¦, π§) = π₯2π¦ + π¦2π₯ + π§2 , βπ eku Kkr dhft,A
2. Define Divergence of a vectors and Solenoidal vector.
Lkfnβk dk vilj.k o ifjufydk lfnβk dks ifjHkkf"kr dhft,A
3. Find the divergence of οΏ½οΏ½. where οΏ½οΏ½ = π₯π¦π§π + 3π₯2π¦π + (π₯π§2 β π¦2π§)οΏ½οΏ½
οΏ½οΏ½ dk vilj.k Kkr dhft,] tgkΒ‘ οΏ½οΏ½ = π₯π¦π§π + 3π₯2π¦π + (π₯π§2 β π¦2π§)οΏ½οΏ½
4. State Greenβs theorem.
xzhu izes; dk dFku fyf[k,A
5. Write the necessary conditions for conic ππ₯2 + ππ¦2 + 2βπ₯π¦ + 2ππ₯ + 2ππ¦ + π = 0 to represents an
equation of ellipse.
'kkod ππ₯2 + ππ¦2 + 2βπ₯π¦ + 2ππ₯ + 2ππ¦ + π = 0 ,d nh?kZor ds lehdj.k dks fu:fir djrk gS
rks vkoβ;d 'krZs fyf[k,A
6. Write polar equation of a circle.
o`r dh /kqzoh; lehdj.k dks fyf[k,A
7. Write down the equation of axis of the conic ππ₯2 + 2βπ₯π¦ + ππ¦2 = 1 in terms of the length π of its
semi axis.
'kkdo ππ₯2 + 2βπ₯π¦ + ππ¦2 = 1 ds v/kZ&v{k dh yEckbZ π ds :Ik mlds v{k dh lehdj.k fyf[k,A
8. Define Reciprocal cone.
O;qRΓe 'kdq dks ifjHkkf"kr dhft,A
9. Define Cylinder.
csyu dks ifjHkkf"kr dhft,A
10. Find the centre and radius of the sphere 7π₯2 + 7π¦2 + 7π§2 β 6π₯ β 3π¦ β 2π§ = 0
xksy 7π₯2 + 7π¦2 + 7π§2 β 6π₯ β 3π¦ β 2π§ = 0 dk dsUnz ,oa f=T;k Kkr dhft,A
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following questions
UNIT I
11. If οΏ½οΏ½ and οΏ½οΏ½ are constant vectors and π = π₯π + π¦π + π§οΏ½οΏ½ then show that : ππ’ππ [(π Γ οΏ½οΏ½) Γ οΏ½οΏ½] = οΏ½οΏ½ Γ οΏ½οΏ½
;fn οΏ½οΏ½ o οΏ½οΏ½ vpj lfnβk gS o π = π₯π + π¦π + π§οΏ½οΏ½ rks fl) dhft, fd ππ’ππ [(π Γ οΏ½οΏ½) Γ οΏ½οΏ½] = οΏ½οΏ½ Γ οΏ½οΏ½
OR
Prove that :
fl) dhft, fd&
πππ£ (π
π3) = 0
Where π = π₯π + π¦π + π§οΏ½οΏ½
UNIT II
12. Find the condition so that the straight line π
π= π΄ πππ π + π΅ π πππ may touch the conic
π
π= 1 +
π cos(π β πΌ).
izfrcU/k Kkr dhft, tc fd ljy js[kk π
π= π΄ πππ π + π΅ π πππ 'kkado
π
π= 1 + π cos(π β πΌ) dks
LiβkZ djrh gSaA
OR
What conic does equation 13π₯2 β 18π₯π¦ + 37π¦2 + 2π₯ + 14π¦ β 2 = 0 represent? Find its centre
and the equation to the conic referred to the centre as origin.
lehdj.k 13π₯2 β 18π₯π¦ + 37π¦2 + 2π₯ + 14π¦ β 2 = 0 dkSuls 'kkado dks fu:frr djrk gSa\ blds
dsUnz ds funsZβkkad rFkk blds dsUnz dks ewy fcUnw eku dj 'kkdo dk lehdj.k Kkr dhft,A
UNIT III
13. Find the equation of the tangent line to the circle π₯2 + π¦2 + π§2 + 5π₯ β 7π¦ + 2π§ β 8 = 0,3π₯ β
2π¦ + 4π§ + 3 = 0 at the point (β3,5,4) .
o`Rr π₯2 + π¦2 + π§2 + 5π₯ β 7π¦ + 2π§ β 8 = 0, 3π₯ β 2π¦ + 4π§ + 3 = 0 ds fcUnq (β3,5,4) ij LiβkZ
js[kk ds lehdj.k Kkr dhft,A
OR
Find the equation of the sphere having the circle π₯2 + π¦2 + π§2 + 10π¦ β 4π§ = 8, π₯ + π¦ + π§ = 3 as
a great circle.
xksys dk lehdj.k Kkr dhft, ftldk or π₯2 + π¦2 + π§2 + 10π¦ β 4π§ = 8, π₯ + π¦ + π§ = 3 ,d
o`gr or gksA
Section C [45 Marks]
Section C β contains 6 questions. Answer any three questions (400 words each), selecting one from each
unit. Each question is of 15 marks.
III. Answer the following questions.
UNIT I
14. Verify Gauss divergence theorem for οΏ½οΏ½ = π₯π¦π + π§2π + 2π¦π§οΏ½οΏ½ on the tetrahedron π₯ = π¦ = π§ = 0,
π₯ + π¦ + π§ = 1.
prq"Qyd π₯ = π¦ = π§ = 0, π₯ + π¦ + π§ = 1 ij οΏ½οΏ½ = π₯π¦π + π§2π + 2π¦π§οΏ½οΏ½ ds fy, vilj.k izes; dk
lR;kiu dhft,A
OR
Evaluateβ« ππ’ππ οΏ½οΏ½π
. οΏ½οΏ½ππ , where οΏ½οΏ½ = (π¦2 + π§2 β π₯2)π + (π§2 + π₯2 β π¦2)π + (π₯2 + π¦2 β
π§2)οΏ½οΏ½ πππ π is the portion of the surface π₯2 + π¦2 β 29π₯ + 9π§ = 0 above the plane π§ = 0, and verify
the Stokeβs theorem.
β« ππ’ππ οΏ½οΏ½π
. οΏ½οΏ½ππ dk eku Kkr dhft,A tgkΒ‘ οΏ½οΏ½ = (π¦2 + π§2 β π₯2)π + (π§2 + π₯2 β π¦2)π +
(π₯2 + π¦2 β π§2)οΏ½οΏ½ gS rFkk π, π§ = 0 ry ds mij Qyd π₯2 + π¦2 β 29π₯ + 9π§ = 0 dk Hkkx gS rFkk
LVkd izes; dk lR;kiu dhft,A
UNIT II 15. Trace the conic 14π₯2 β 4π₯π¦ + 11π¦2 β 44π₯ β 58π¦ + 71 = 0 . Also find the focus, eccentricity and
length of latus rectum.
'kkdo 14π₯2 β 4π₯π¦ + 11π¦2 β 44π₯ β 58π¦ + 71 = 0 dk vuqjs[k.k dhft,A blds ukfHk] mRdsUnzrk
rFkk ukf;yEc dh yEckbZ Hkh Kkr dhft,A
OR
Prove that the equation of the director circle of the conic π πβ = 1 + π πππ π is π2(1 β π2) +2πππ πππ π β 2π2 = 0 .
fl) djks fd 'kkado ππβ = 1 + π πππ π ds fu;ked o`r dk lehdj.k π2(1 β π2) + 2πππ πππ π β
2π2 = 0 gksxkA
UNIT III
16. (a) Prove that the semi-vertical angle of a right circular cone admitting sets of three mutually
perpendicular generators is π‘ππβ1β2
;fn ,d yEc o`rh; 'kadq ds rhu tud ijLij yEc gS] rks fl) dhft, fd v)Zβkh"kZ dks.k dk eku
π‘ππβ1β2 gSA
(b) Show that the equation 4π₯2 β π¦2 + 2π§2 + 2π₯π¦ β 3π¦π§ + 12π₯ β 11π¦ + 6π§ + 4 = 0 represents a
cone with vertex (β1, β2, β3).
fl) dhft, fd lehdj.k 4π₯2 β π¦2 + 2π§2 + 2π₯π¦ β 3π¦π§ + 12π₯ β 11π¦ + 6π§ + 4 = 0 ,d 'kadq
dks fu:fir djrk gS ftldk 'kh"kZ (β1, β2, β3) gSaA
OR
(a) Find the equation of the enveloping cylinder of the surface ππ₯2 + ππ¦2 + ππ§2 = 1 and whose
generators are parallel to the line π₯
π=
π¦
π=
π§
π .
ππ₯2 + ππ¦2 + ππ§2 = 1 ds vUokyksih csyu ds lehdj.k Kkr dhft, ftldh tud js[kk, π₯
π=
π¦
π=
π§
π ds lekUrj gSaA
(b) Find the equation of the right circular cylinder whose guiding circle is π₯2 + π¦2 + π§2 = 9 , π₯ βπ¦ + π§ = 3 .
ml yEco`Rrh; csyu dk lehdj.k Kkr dhft, ftldk funsZβkkd o`r π₯2 + π¦2 + π§2 = 9 , π₯ β π¦ +
π§ = 3 gSaA
The End
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2017- 18
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]:Calculus
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following questions
1. Find the radius of curvature if the pedal equation of the ellipsoid is:
oΓrk f=T;k Kkr dhft, ;fn nh/kZor dk ifnd lehdj.k gS %
1
π2=
1
π2+
π
π2β
π2
π2π2
2. Define derivative of the length of arc.
pki dh yEckΓ dΒ’ vodyt dh ifjHkkβkk fnft,
3. Write the Eulerβs theorem for homogenous functions.
le?kkr Qyu dΒ’ fy;s vkW;yj Γes; fyf[k,
4. Define evaluate.
dΒ’UΓ¦t dh ifjHkkβkk fyf[k,
5. Give the sufficient condition for π(π, π) to be a maximum value of π(π₯, π¦).
Qyu π(π₯, π¦ dΒ’ vf/kdre eku π(π, π) gΒ¨us dh i;kZIr Γfrca/k fyf[k,
6. Evaluate the following. β« π ππ5π₯π
2β
0 πππ 6π₯ ππ₯.
fuEu dk eku Kk dhft,
7. Define Beta function.
chVk Qyu dh ifjHkkβkk nhft,
8. Change the order of integration in the following double integral.
fuEu f}&lekdy eSa lekdyu dk Γe ifjoΓrr dhft,%
β« β«πβπ¦
π¦ ππ₯ππ¦
β
π₯
β
0
9. Write down the relation between Beta and Gamma function.
chVk rFkk xkek Qyu lsa laca/k fyf[k,
10. Find the points of inflection of the curve:
oΓ dΒ’ ufr ifjorZu fcUnq Kkt dhft,%
π¦ = 3π₯4 β 4π₯3 + 1
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following questions
UNIT I
11. Find the pedal equation of the asteroid. π₯ = π πππ 3π‘, π¦ = π π ππ3π‘
,LVΒͺk;M π₯ = π πππ 3π‘, π¦ = π π ππ3π‘ dk ifnd lehdj.k Kkr dhft,
OR
Find the asymptotes of the following curve:
fuEu oΓ dh vUkUrLiΓβk;kΒ‘ Kkr dhft,
π¦3 β π₯π¦2 β π₯2π¦ + π₯3 + π₯2 β π¦2 β 1 = 0
UNIT II
12. If π₯π₯π¦π¦π§π§ = π then show that π2π§
ππ₯ππ¦= β(π₯ log ππ₯)β1.
;fn π₯π₯π¦π¦π§π§ = π rΒ¨ fl) dhft, fd π2π§
ππ₯ππ¦= β(π₯ log ππ₯)β1
OR
Find the envelope of the circles which passes through the vertex of the vertex of the parabola π¦2 =
4ππ₯ and whose centre lies on it.
,d o`r dk dΒ’UΓ¦ lnSo ijoy; π¦2 = 4ππ₯ ij jgrk gS rFkk ΒΈg ij oy; dΒ’ βkhβkZ ls xqtjrk gS o`r
dqy dk vUoky¨i Kkr dhft,
UNIT III
13. To show that: ΓnΓβr dhft,
βπ β1 β π =π
sin ππ, 0 < π < 1.
OR
Change the order of integration in the following double integral:
fuEu f}&lekdy esa lekdyu dk Γe ifjoΓrr dhft,
β« β« π (π₯, π¦)ππ₯ ππ¦.2πβπ₯
π₯2πβ
π
0
Section C [45 Marks]
Section C β contains 6 questions. Answer any three questions (400 words each), selecting one from each
unit. Each question is of 15 marks.
III. Answer the following questions.
UNIT I
14. Trace the following curve:
fuEu odZ dk vuqj[k.k dhft,
π¦2(π + π₯) = π₯2(3π β π₯)
OR
a. Prove that the radius of curvature of the centenary
π¦ = π cosh (π₯
π) ππ‘ (π₯, π¦)ππ
π¦2
π.
fl) djΒ¨ fd dSVsujh π¦ = π cosh (π₯
π) dΒ’ fcUnq (π₯, π¦) ij oΓrk f=T;k
π¦2
π gS
b. Find the point inflection of the curve π¦(π2 + π₯2) = π₯3
oΓ π¦(π2 + π₯2) = π₯3 dΒ’ ufr ifjorZu fcUnq Kkr dhft,
UNIT II
15. (a) If π’ = π‘ππβ1 (π₯3+π¦3
π₯+π¦), then prove that.
;fn π’ = π‘ππβ1 (π₯3+π¦3
π₯+π¦) rΒ¨ fl) dhft,
π₯2πΏ2π’
πΏπ₯2+ 2π₯π¦
πΏ2π’
πΏπ₯πΏπ¦+ π¦2
πΏ2π’
πΏπ¦2= sin 2π’ (1 β 4 π ππ2π’)
(b) If π₯3 + π¦3 = 3ππ₯π¦, then find π2π¦
ππ₯2
;fn π₯3 + π¦3 = 3ππ₯π¦ Kkr dhft, π2π¦
ππ₯2
OR
(a) Find the envelope of the straight lines π₯ cos πΌ + π¦ sin πΌ = π sin πΌ cos πΌ; πΌ is a parameter.
πΌ dΒ¨ Γkpyd ysrs gq, ljy js[kkvΒ¨ π₯ cos πΌ + π¦ sin πΌ = π sin πΌ cos πΌ ; dk vUoyΒ¨i Kkr
dhft;s
(b) Find the maximum value of π₯3π¦2(1 β π₯ β π¦).
mfPpβB eku Kkr dhft,
UNIT III
16. (a) Show that :- ΓnΓβkr dhft, fd
β« π‘ππππ₯ππ₯ =π
2sec (
ππ
2)
π2β
0
(b) Prove that:- fl) dhft, fd
22πβ1π΅ (π, π) =βπβπ
β(π +12)
OR
(a) Evaluate:- eku Kr dhft, β« β« (π₯ + π¦)2ππ₯ ππ¦π
. Where the area R is bounded by ellipse tgkΒ‘
{ks= nh/kZo`r ls ifjc) gS
π₯2
π2 +π¦2
π2 = 1
(b) Evaluate: β« β« β«ππ₯ ππ₯ ππ§
(π₯+π¦+π§+1)3
1βπ₯βπ¦
0
1βπ₯
0
1
0
The End
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2018- 19
End Semester Examination
Class : B.Sc. Math
Sub : Mathematics
Paper I : [MAT 201]: Vector Calculus & Geometry
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following. 1. Prove that:
fl) djksaA ππππ π = οΏ½οΏ½ , when π = π₯π + π¦π + π§οΏ½οΏ½
2. Find the greatest rate of increase of β = 2π₯π¦π§2 at (1,0, β3).
β = 2π₯π¦π§2 dk (1,0, β3) ij o`f) dh vf/kdre nj Kkr djksA
3. State Gauss divergence theorem.
xkWl vilj.k izes; dk dFku fyf[k,A
4. Write the condition for conic ππ₯2 + ππ¦2 + 2βπ₯π¦ + 2ππ₯ + 2ππ¦ + π = 0 to represent an equation of
hyperbola.
'kkado ππ₯2 + ππ¦2 + 2βπ₯π¦ + 2ππ₯ + 2ππ¦ + π = 0 ds ,d vfrijoy; dks iznfβkZr djus dh 'krZ
fyf[k,A
5. Write the equation of directive of conic π
π= 1 + π πππ π, π€βππ π < 1.
'kkado π
π= 1 + π πππ π] tcfd π < 1 dh fu;rk dk lehdj.k fyf[k,A
6. Define Auxiliary circle of a conic.
'kkado ds lgk;d or dks ifjHkkf"kr dhft,A
7. Write the condition of tangency of a plane and sphere.
,d lery ,oa xksys dk LifβkZrk izfrcU/k fyf[k,A
8. Write the condition of orthogonality of two spheres.
nks xksyksa ds ijLij ykfEcd izfrPNsnu dk izfrcU/k fyf[k,A
9. Define enveloping cone.
vUokyksih 'kadq dks ifjHkkf"kr dhft,A
10. Define right circular cylinder.
yEco`Γkh; csyu dks ifjHkkf"kr dhft,A
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following.
UNIT I
11. Find β2ππ and show that β2 ( 1
π ) = 0 where π = (π₯2 + π¦2 + π§2)1/2
β2ππ dk eku Kkr djks rFkk fl) djksa β2 ( 1
π ) = 0 ] tgkΒ‘ π = (π₯2 + π¦2 + π§2)1/2
OR
Evaluate β« π π
. ππ , where π = π₯π¦π + (π₯2 + π¦2)π and C is the rectangle in the π₯π¦ plane bounded
by lines π₯ = 1, π₯ = 4 , π¦ = 2, π¦ = 10.
β« π π
. ππ dk eku Kkr dhft,] tcfd π = π₯π¦π + (π₯2 + π¦2)π o C, xy iVy es π₯ = 1, π₯ =
4 , π¦ = 2, π¦ = 10 js[kkvksa ls f?kjk vk;r gSaA
UNIT II
12. Find the lengths and the equation of the axes of the conic 2π₯2 + 5π₯π¦ + 2π¦2 = 1 .
'kkado 2π₯2 + 5π₯π¦ + 2π¦2 = 1 ds v{kksa dh yEckb;kΒ‘ o lehdj.k Kkr dhft,A
OR
A chord ππ of a conic subtends a right angle at focus S. If e be the eccentricity, l be the semi-laths
rectum, prove that
,d 'kkado dh thok ππ ukfHk π ij ledks.k vUrfjr djrh gSaA ;fn 'kkado dh mRdsUnzrk π v)ZukfHkyEc π gks rks fl) djksa&
(1
π πβ
1
π)
2
+ (1
ππβ
1
π)
2
=π2
π2
UNIT III
13. Find the equation of the sphere having the circle π₯2 + π¦2 + π§2 + 10π¦ β 4π§ = 8 , π₯ + π¦ + π§ = 3 as
a great circle.
ml xksys dk lehdj.k Kkr djks ftldk or π₯2 + π¦2 + π§2 + 10π¦ β 4π§ = 8 , π₯ + π¦ + π§ = 3 rd
o`gr or gksaA
OR
Find the equation of the cylinder whose generators are parallel to the line π₯
1=
π¦
β2=
π§
3 and passing
through the curve π₯2 + 2π¦2 = 1 , π§ = 3.
ml csyu dk lehdj.k Kkr djksa ftldh tud js[kk,Β‘ π₯
1=
π¦
β2=
π§
3 ds lekUkkUrj gS rFkk ftldk
funsZβkd oΓ π₯2 + 2π¦2 = 1 , π§ = 3 gSaA
Section C [45 Marks]
Section C β contains 6 questions. Answer any three questions (400 words each), selecting one from each
unit. Each question is of 15 marks.
III. Answer the following.
UNIT I
14. Prove that :
fl) djks&
ππ’ππ π Γ π
π3= β
π
π3+
3π
π5 ( π . π )
Where, π = π₯ οΏ½οΏ½ + π¦π + π§οΏ½οΏ½ , π = π1π + π2π + π3οΏ½οΏ½ is a constant vector and |π | = π.
tgkΒ‘ π = π₯ οΏ½οΏ½ + π¦π + π§οΏ½οΏ½ , π = π1π + π2π + π3οΏ½οΏ½ ,d vpj lafnβk gS rFkk |π | = π
OR
Verify Gaussβ Divergence theorem for π = 4π₯π β 2π¦2π + π§2οΏ½οΏ½ , taken over the region bounded by
π₯2 + π¦2 = 4 , π§ = 0, π§ = 3.
π₯2 + π¦2 = 4 , π§ = 0, π§ = 3 }kjk ifjc) {ks= ij Qyu π = 4π₯π β 2π¦2π + π§2οΏ½οΏ½ ds fy, xkWl
vilj.k izes; lR;kfir dhft,A
UNIT II
15. Trace the curve.
oΓ vuqjs[k.k dhft,A
π₯2 + π₯π¦ + π¦2 + π₯ + π¦ = 1
OR
Find the condition that the straight line π
π= π΄ πππ π + π΅π πππ touches the circle π = 2π πππ π.
og izfrcU/k Kkr djks tcfd ljy js[kk π
π= π΄ πππ π + π΅π πππ ] o`r π = 2π πππ π dks LiβkZ djsaA
UNIT III
16. Find the equation of cone whose vertex is (5,4,3) and the guiding curve is 3π₯2 + 2π¦2 = 6 , π¦ +
π§ = 0.
ml 'kadq dk lehdj.k Kkr dhft, ftldk 'kh"kZ (5,4,3) rFkk funsZβkd oΓ 3π₯2 + 2π¦2 = 6 , π¦ +
π§ = 0 gSaA
OR
Find the equation of the right circular cylinder whose guiding circle is π₯2 + π¦2 + π§2 = 9 , π₯ βπ¦ + π§ = 3 .
ml yEco`rh; csyu dk lehdj.k Kkr djks ftldk funsZβkd o`r π₯2 + π¦2 + π§2 = 9 , π₯ β π¦ +
π§ = 3 gSaA
The End
Sophia Girlsβ College, Ajmer (Autonomous)
Semester II β 2018- 19
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]:Calculus
Time : 2 Β½ Hrs. M.M: 70 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [10 Marks]
Section A contains 10 questions (20 words each) and a candidate is required to attempt all 10 questions.
Each question is of one mark.
I. Answer the following.
1. Define an asymptote.
vuUr LiβkΓ dΒ¨ ifjHkkfβkr dhft, |
2. Write the necessary condition for the existence of double point for the curve π(π₯, π¦) = 0 and also
the condition for the double point to be a node, cusp or conjugate point.
oΓ π(π₯, π¦) = 0 dΒ’ f}d fcUnqvΒ¨a dΒ’ vfLrRo dΒ’ fy, vkoβ;d ΓfrcU/k fyf[k, ,oa f}d fcUnq dΒ’
uΒ¨M] mHk;kXkz ;k la;qXeh fcUnq gΒ¨us gsrq Γfrca/k Hkh fyf[k, |
3. Find the radius of curvature at the point (s,π) for the curve π = π πππ(sec π + tan π).
oΓ π = π πππ(sec π + tan π) dΒ’ fcUnq (s, π) ij oΓrk fΒ«kT;k Kku dhft, |
4. State Eulerβs theorem for homogeneous functions.
le?kku QyuΒ¨a dΒ’ fy, vk;dj Γes; dk dFku fyf[k, |
5. If π’ =π¦2
2π₯, π£ =
π₯2+π¦2
2π₯, find
π(π’,π£)
π (π₯,π).
;fn π’ =π¦2
2π₯, π£ =
π₯2+π¦2
2π₯ rΒ¨
π(π’,π£)
π (π₯,π) Kkr dhft, |
6. Define envelope of a one parameter family of plane curves.
,d Γkpy okys lery oΓ dqy dΒ’ vUokyΒ¨i dΒ¨ ifjHkkfβkr dhft,
7. Evaluate β« π₯5πβπ₯β
0 ππ₯
eku Kkr dhft, β« π₯5πβπ₯β
0 ππ₯
8. Change the order of integration in the following integral
fuEu lekdy esa lekdyu dk Γe ifjoΓrr dhft, |
β« β« π (π₯, π¦)ππ₯ ππ¦βπ2βπ₯2
0
π
0
9. State Liouvilleβs extension of Dirichletβs integral.
fMjpfyV lekdy dΒ’ fyosyh O;kidhdj.k dk dFku fyf[k, |
10. Check if the function π₯3 β 4π₯π¦ + 2π¦2 attains its extreme value at the point (0, 0).
tkΒ‘p dhft, fd D;k Qyu π₯3 β 4π₯π¦ + 2π¦2 vius pje eku dΒ¨ fcUnq (0, 0) ij ΓkIr djrk gS|
Section B [15 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt 3 questions, at least 1
from each unit. Each question is of 5 marks.
II. Answer the following.
UNIT I
11. Find ππ
ππ for the curve
2π
π= 1 + cos π.
oΓ 2π
π= 1 + cos π dΒ’ fy,
ππ
ππ Kkr dhft, |
OR
Find the radius of curvature at the point (r,π) on the polar curve π (1 + cos π) = π
/kqzoh oΓ π (1 + cos π) = π dΒ’ fcUnq (r,π) ij oΓrk f=T;k Kkr dhft, |
UNIT II
12. If π’ = π ππβ1 (π₯3+π¦3
π₯+π¦), then prove that π₯
ππ’
ππ₯+ π¦
ππ’
ππ¦= 2 cot π’.
;fn π’ = π ππβ1 (π₯3+π¦3
π₯+π¦) rΒ¨ fl) dhft, π₯
ππ’
ππ₯+ π¦
ππ’
ππ¦= 2 cot π’
OR
If π’3 + π£3 = π₯ + π¦ and π’2 + π£2 = π₯3 + π¦3, then find the value of π(π’,π£)
π(π₯,π¦).
;fn π’3 + π£3 = π₯ + π¦ rFkk π’2 + π£2 = π₯3 + π¦3 rΒ¨ π(π’,π£)
π(π₯,π¦). dk eku Kkr dhft, |
UNIT III
13. Prove that. fl) dhft, β«ππ₯
βπ4βπ₯4
π
0=
{Ξ (1/4) }2
4π β(2π)
OR
Evaluate β¬ π₯π¦(π₯ + π¦)ππ₯ ππ¦π
where the area of integration R is the area between π¦ = π₯2 and π¦ =
π₯.
β¬ π₯π¦(π₯ + π¦)ππ₯ ππ¦π
dk eku Kkr dhft, tgkΒ‘ lekdyu dk {ks= R, π¦ = π₯2 rFkk π¦ = π₯ dΒ’ e/;
dk {ks= gS |
Section C [45 Marks]
Section C β contains 6 questions. Answer any three questions (400 words each), selecting one from each
unit. Each question is of 15 marks.
III. Answer the following.
UNIT I
14. Trace the following curves.
fuEu oΓΒ¨a dk vuqjs[k.k dhft, |
a. π¦2 (2π β π₯) = π₯3
b. π = π(1 + cos π)
OR
Find all the asymptotes of the curve (2π₯ β 3π¦ + 1)2 (π₯ + π¦) β 8π₯ + 2π¦ β 9 = 0 and show that
they intersect the curve again in three points which lie on a straight line. Find the equation of this
line.
oΓ (2π₯ β 3π¦ + 1)2 (π₯ + π¦) β 8π₯ + 2π¦ β 9 = 0 dh lHkh vuUr LiΓβk;Β¨a Kkr dhft, rFkk fl)
dhft, fd, ;s oà d¨ iqu% rhu fcUnqv¨a ij dkVrh gS t¨ ,d js[kk ij fLFkr gS | bl js[kk dk
lehdj.k Kkr dhft, |
UNIT II
15. (a) Prove Eulerβs theorem on homogeneous function.
le?kkr QyuΒ¨a dΒ’ fy, vk;yj Γes; dΒ¨ fl) djsa |
(b) Prove that π’ = ππ₯3π¦2 β π₯4π¦2 β π₯3π¦3 is maximum at (π
2,
π
3)
fl) dhft, fd π’ = ππ₯3π¦2 β π₯4π¦2 β π₯3π¦3 mPpre eku fcUnq (π
2,
π
3) ij gS |
OR
Find the envelope of the family of straight lines ππ₯ sec πΌ β π π¦ πππ ππ πΌ = π2 β π2, πΌ being the
parameter. Use the above result to show that the evolutes of the ellipse π₯2
π2+
π¦2
π2= 1 is (ππ₯)
23β +
(ππ¦)2
3β = (π2 β π2)2
3β .
ljy js[kk dqy ππ₯ sec πΌ β π π¦ πππ ππ πΌ = π2 β π2 dk vUokyΒ¨i Kkr dhft, tgkΒ‘ πΌ Γkpy gS
mDr ifj.kke dΒ’ Γ;Β¨x }kjk fl) dhft, fd nh/kZo`Rr π₯2
π2 +π¦2
π2 = 1 dk dΒ’UΓ¦ (ππ₯)2
3β + (ππ¦)2
3β =
(π2 β π2)2
3β gS |
UNIT III
16. (a) Prove that:
fl) dhft,
π(π, π) =β(π)β(π)
β(π + π) (π > 0, π > 0)
(b) Evaluate β π₯π¦π§ ππ₯ ππ¦ ππ§ where region of integration is the complete ellipsoid π₯2
π2 +π¦2
π2 +π§2
π2 β€
1.
β π₯π¦π§ ππ₯ ππ¦ ππ§ dk eku dhft, tgkΒ‘ lekdyu dk {ks= laiw.kZ nh/kZo`Rrt π₯2
π2 +π¦2
π2 +π§2
π2 β€ 1 gSa |
OR
a. Evaluate the following integral by changing the order of integration.
fuEu lekdy esa lekdyu dk Γe cnydj eku Kkr dhft, |
β« β« π₯2 cos(π₯2 β π₯π¦) ππ¦ ππ₯1
π¦
1
π¦=0
b. Evaluate β π§ ππ₯ ππ¦ ππ§π£
where region of integration is a cylinder v bounded by z=0, z=1, π₯2 +
π¦2 = 4.
β π§ ππ₯ ππ¦ ππ§π£
dk eku Kkr dhft, tgkΒ‘ lekdyu {ks= ,d csyu v gS tΒ¨ z=0, z=1, π₯2 +
π¦2 = 4 }kjk ifjc) gS |
The End
Sophia Girlsβ College (Autonomous)
Ajmer Semester II β 2021-22
End Semester Examination
Class : B.Sc. Math
Sub : Mathematics
Paper I : [MAT-201]: Vector Calculus and Geometry
Time : 1 Β½ Hrs. M.M: 40 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [12 Marks]
Section A contains 12 questions (20 words each) and a candidate is required to attempt any 6 questions.
Each question is of 2 marks.
I. Answer the following questions.
1. If οΏ½οΏ½ = 5π‘2π β cos π‘π πππ οΏ½οΏ½ = π‘π + πππ π‘π then find πππ‘
(οΏ½οΏ½ Β· οΏ½οΏ½)
;fn οΏ½οΏ½ = 5π‘2π β cos π‘π rFkk οΏ½οΏ½ = π‘π + πππ π‘π rc πππ‘
(οΏ½οΏ½ Β· οΏ½οΏ½)Kkr dhft,
2. Define Curl of a vector point function.
Lkfnβk fcU/kq Qyu dk dqUry ifjHkkfβkr dhft,A
3. State Gaussβs divergence thorem
xkWl dk vilj.k Γ§es; fyf[k,A
4. If F = x2yi + xz j + 2yzk then find div curl F
;nh F = x2yi + xz j + 2yzk rks div curl F Kkr djks
5. Write asymptotes of ax2 + 2bxy + by2 + 2gx + 2fy + c = 0
ax2 + 2bxy + by2 + 2gx + 2fy + c = 0 dh vuUrLiβkhZZ;Β‘k fyf[,
6. Write equation of circle in polar form when pole is on circle
o`r dk lehdj.k /kzqohi :i esa fy[kks tc /kzqo o`r ij gksA
7. Define auxiliary circle and write its equation in polar form.
Lgk;d o`r dks ifjHkkfβkr dhft, rFkk bldk lehdj.k /kzqoh; :i es fyf[k,A
8. Name the conic represented by x2 + 2xy + y2 β 2x β 1 = 0
Lehdj.k x2 + 2xy + y2 β 2x β 1 = 0 esa Γ§Lrqr βkkdao dk uke crkb,A
9. Find Radius of sphere x2 + y2 + z2 β 2x + 4y β 6z = 11
Xkksys x2 + y2 + z2 β 2x + 4y β 6z = 11 dh f+=T;k Kkr djksA
10. Write equation of tangent plane at (Ξ±,) of sphere x2 + y2 + z2 + 2ux + 2vy + 2wz +d = 0
Xkksys x2 + y2 + z2 + 2ux + 2vy + 2wz +d = 0 dk fcUnq (Ξ±,) ij LiβkZ lekUrk fyf[k,
11. Write equation of right circular cone.
yEco`rh; βkdqa dk lehdj.k fyf[k,A
12. Define enveloping cylinder.
vUokyksih csyu dks ifjHkkfβkr dhft,A
Section B [10 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt any 2 questions from
different units . Each question is of 5 marks.
II. Answer the following questions.
UNIT I
13. Find the directional derivative of f = xy + yz + zx in the direction of the vector π + 2π + 2οΏ½οΏ½ at the
point (1, 2, 0)
fcUnq (1, 2, 0) ij f = xy + yz + zx dk π + 2π + 2οΏ½οΏ½ dh fnβkk esa fnd~ vodyt Kkr dhft,A
OR
Find div V and curl V if V = (x3 + y3 + z3β 3xyz)
;fn V = (x3 + y3 + z3β 3xyz)gS rks div V rFkk curl V Kkr dhft,A
UNIT II 14. Find lengths and equations of axes of the following conic 2x2 + 5xy + 2y2 = 1
fuEu βkkado dh v{kks ds lehdj.k rFkk yEckj;kΒ‘ Kkr djk sA 2x2 + 5xy + 2y2 = 1
OR
Show that the following equations represent the same conic:
fl) dhft, dh fuEu lehdj.k ,d gh βkkado dks fu:fir djrsa gSA
π
π= 1 + π cos π πππ
π
π= β1 + π cos π
UNIT III 15. Find equations to the tangent planes to the sphere x2 + y2 + z2 = 16 which passes through the lines x +
y = 5; x β 2z = 7
xksys x2 + y2 + z2 = 16 dh mu nks LiβkZ ledks ds lehdj.k Kkr djks tks js[kkvks x + y = 5; x β 2z = 7
ls xqt+jsA
OR
Find equation of enveloping cone of the sphere x2 + y2 + z2 + 2x β 2 = 0 with its vertex at (1, 1, 1)
xksys x2 + y2 + z2 + 2x β 2 = 0 ds ml vUokyksi βkakdq dk lehdj.k djks ftldk βkhβkZ fcUnq (1, 1, 1)
gSA
Section C [18 Marks]
Section C contains 6 questions (400 words each) and a candidate is required to attempt any 2 questions
from different units. Each question is of 9 marks.
III. Answer the following questions.
UNIT I
16. Verify Stokes theorem for the function οΏ½οΏ½ = π₯2π + π₯π¦π integrated round the square in the plane z =
0, whose sides are along the lines x = y = 0 and x = y = a
Qyu οΏ½οΏ½ = π₯2π + π₯π¦π ds fy, LVkSd Γ§es; dk lR;kiu dhft,] tgk dk lekdyu rFkk z = 0 esa
fL;r oxZ ds pkjks vksj fd;k x;k gS% ftldh Hkqtk,a js[kk x = y = 0 rFkk x = y = a ds vuqfnβk gSA
OR
If π = π‘π β π‘2π + (π‘ β 1)οΏ½οΏ½ and π = 2π‘2π + 6π‘οΏ½οΏ½ then find value of
;fn π = π‘π β π‘2π + (π‘ β 1)οΏ½οΏ½ rFkk π = 2π‘2π + 6π‘οΏ½οΏ½ rks Kkr dhft,A
a. β« (π Β· π )ππ‘2
0
b. β« (π Γ π )ππ‘2
0
UNIT II
17. Trace the curve
oΓ dk vuqjs[k.k dhft,A
x2 + y2 + xy + x + y β 1 = 0
OR
A circle passing through the focus of a conic whose latus rectum is 2π meet the conic in four points
whose distance from the focus are π1 , π2 , π3 πππ π4 respectively , prove that
,d oRr fdlh βkkado ls xqtjrk gS ftldh ukfHkyEc 2π gS] rFkk βkkado dks pkj fcUnqvksa ij feyrk
gS ftudh ukfHk ls yEckb;kW dzeβk π1 , π2 , π3 vkSj π4 gS] rks fl) dhft, dh
1
π1+
1
π2+
1
π3+
1
π4=
2
π
UNIT III
18. Find the equation of right circular cylinder where guiding curve is the circle x2 + y2 + z2 = 9, x β 2y
+ 2z = 3
ml yEco`Rrh; csyu dk lehdj.k Kkr djks ftldk funsβkd oRr x2 + y2 + z2 = 9, x β 2y + 2z = 3
gSA
OR
Find coβordinates of centre and radius of the circle
fuEu o`r dk dsUnz vkSj f=T;k Kjr djksA
X2 + y2β2yβ4z = 20, x + 2y + 2z = 21
--The End--
Sophia Girlsβ College (Autonomous)
Ajmer Semester β 2021-22
End Semester Examination (May 2022)
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]: Advanced Calculus
Time : 1 Β½ Hrs. M.M: 40 Marks
Instruction : In case of any doubt, the English version of paper stands correct.
Section A [12 Marks]
Section A contains 12 questions (20 words each) and a candidate is required to attempt any 6 questions.
Each question is of 2 marks.
I. Answer the following questions.
1. Write the formula to find the derivative of an arc for the polar curve π = π(π)
pki dh yEckbZ dk vodyt Kkr djus dk lw= /kqoh; oΓ π = π(π) ds fy, fyf[k,A
2. Define parallel and oblique asymptotes.
lekUrj ,oa fr;Zd vuUrLifβkZ;ksa dh ifjHkkβkk nksA
3. Write the formula for radius of curvature of Cartesian curves.
dkrhZ; odzksa ds fy, odzrk f=T;k dk lw= fyf[k,A
4. Define single and double cusps.
,dy ,oa f)d mHk;k.r dh ifjHkkβkk nhft,A
5. Verify the theorem π2π’
ππ₯ππ¦=
π2π’
ππ¦ππ₯ , when π’ = π₯3 + π¦3 + 3ππ₯π¦
izes; π2π’
ππ₯ππ¦=
π2π’
ππ¦ππ₯ dh iqfβV dhft, tcfd π’ = π₯3 + π¦3 + 3ππ₯π¦
6. Write the formula for finding second differential coefficient of implicit functions.
vLiβV Qyuksa ds fn`rh; vody xq.kkad Kkr djus dk lw= fyf[k,A
7. Define envelope.
vUokyksi dh ifjHkkβkk nhft,A
8. Give criteria for extreme value of f(x, y).
Qyu f(x, y) ds pje eku dh dlkSVh nhft,A
9. What is Legendreβs duplication formula?
fytsUMj f)xq.ku lw= D;k gS\
10. State Dirichletβs theorem for triple integral.
f= lekdyu ds fy, fMfjpysVl~ izes; dk dFku fyf[k,A
11. Write polar form of the following:
fuEu dk /kzqoh; :Ik fyf[k,A
β¬ π(π₯, π¦)ππ₯ππ¦
12. Change the order of integration in the following integral:
fuEufyf[kr lekdy esa lekdyu dk dze cnfy,A
β« β« π(π₯, π¦)ππ₯ππ¦π₯
0
π
0
Section B [10 Marks]
Section B contains 6 questions (50 words each) and a candidate is required to attempt any 2 questions from
different units . Each question is of 5 marks.
II. Answer the following questions.
UNIT I
13. For the curve ππ = πππππ ππ , prove that ππ
ππ= π(π ππ ππ)
πβ1
π
odz ππ = πππππ ππ ds fy, fl) dhft, fd
ππ
ππ= π(π ππ ππ)
πβ1π
OR
Find the asymptotes of the following curve:
fuEu odz ds vuUrLifβkZ;kW Kkr dhft,A
π₯3 + π¦3 β 3ππ₯π¦ = 0
UNIT II
14. If π’ = π‘ππβ1 (π₯3+π¦3
π₯βπ¦), then prove that π₯
ππ’
ππ₯+ π¦
ππ’
ππ¦= π ππ2π’
;fn π’ = π‘ππβ1 (π₯3+π¦3
π₯βπ¦) rc π₯
ππ’
ππ₯+ π¦
ππ’
ππ¦= π ππ2π’ fl) dhft,A
OR
Find the envelope of the family of ellipse
π₯2
π2 +π¦2
π2 = 1 π€βππ π + π = π , c being a constant.
nh?kZo`Rrksa π₯2
π2 +π¦2
π2 = 1 ds dqy dk vUokyksi Kkr tcfd π + π = π tgkW π vpj gSaA
UNIT III 15. Evaluate Kkr dhft,
β« π₯πβ11
0(πππ
1
π₯)
πβ1
ππ₯ = Ξ(π)
ππ ; π, π > 0
OR
Evaluate eku Kkr dhft,
β« β« β« π₯π¦π§ ππ₯ ππ¦ ππ§
where the region of integration is the volume of the ellipsoid in the positive octant.
tgkW lekdyu dk {ks= /kukRed vCVkaβkd esa nh/kZo`=t gSA
π₯2
π2+
π¦2
π2+
π§2
π2β€ 1
Section C [18 Marks]
Section C contains 6 questions (400 words each) and a candidate is required to attempt any 2 questions
from different units. Each question is of 9 marks.
III. Answer the following questions.
UNIT I 16. Trace the following curve
fuEufyf[kr odz dk vuqjs[k.k dhft,A
π2 = π2 cos 2π
OR
Show that the radius of curvature at a point (π πππ 3π, ππ ππ3π) on the curve π₯2
3 + π¦2
3 = π2
3 is 3π
2sin 2π
fl) dhft, odzrk f=T;k fcUnq (π πππ 3π, ππ ππ3π) ] ij odz π₯2
3 + π¦2
3 = π2
3 dh 3π
2sin 2π gSA
UNIT II
17. If π’ = π₯π ππβ1 (π¦
π₯) then prove that
π₯2π2π’
ππ₯2+ 2π₯π¦
π2π’
ππ₯ππ¦+ π¦2
π2π’
ππ¦2= 0
;fn π’ = π₯π ππβ1 (π¦
π₯) rc fl) dhft,A
π₯2π2π’
ππ₯2+ 2π₯π¦
π2π’
ππ₯ππ¦+ π¦2
π2π’
ππ¦2= 0
OR
Find the maxima and minima of π’ = π₯2 + π¦2 + π§2 subject to the conditions ππ₯2 + ππ¦2 + ππ§2 =
1 πππ ππ₯ + ππ¦ + ππ§ = 0
izfrcU/kks ππ₯2 + ππ¦2 + ππ§2 = 1 rFkk ππ₯ + ππ¦ + ππ§ = 0 ds vUrZxr π’ = π₯2 + π¦2 + π§2 ds
mfPpβB vkSj fufEuβB Kkr dhft,A
UNIT III
18. Evaluate the following integral by changing the order:
fuEu lekdy dk dze cnydj eku Kkr dhft,A
β« β«πβπ¦
π¦
β
π₯
β
0
ππ₯ ππ¦
OR
Evaluate the following integral by changing to polar coordinates:
fuEu lekdy dks /kqzohZ; funsZβkkadksesa ifjofrZr dj eku Kkr dhft,A
β« β« βπ₯2 + π¦2β2π₯βπ₯2
π₯
1
0ππ₯ ππ¦
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