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Transcript of Sophia Girls' College, Ajmer
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2015- 16
End Semester Examination โ II
Class : B.Sc.[Math] โ SEM II
(Sub : Math)
Paper I : [MAT-201]: Vector Calculus and Geometry โ 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 = |
๐โ | , where
๐โ = xi + yj + jk. Prove that
๐๐๐๐ (1
๐) = โ
๐
๐
;fn r = | ๐โ | tgkยก
๐โ = xi + yj + jk fl) dhft,
๐๐๐๐ (1
๐) = โ
๐
๐
Define directional Derivative.
fnd vodyt dยจ ifjHkkfโkr dhft, |
2. Define curl of a vector point function.
lfnโk Qyu dยข dyZ dยจ ifjHkkfโkr dhft, |
3. State Stokeโs Theorem.
LVยจDl รes; dk dFku fyf[k, |
4. Define Line Integral.
js[kk lekdy dยจ ifjHkkfโkr dhft, |
5. Define conic section.
โkkado dยจ ifjHkkfโkr dhft, |
6. Write equation of the conic referred to centre as origin.
ewy fcUnq dยจ dยขUรฆ ekurs gq, โkkado dk lehdj.k fyf[k, |
7. Find the equation of sphere whose centre is ( ยฝ , - ยฝ , 1) and radius is 2.
xยจys dk dยขUรฆยผ ยฝ , - ยฝ , 1ยฝ rFkk f=T;k 2 bdkร ysrs gq, lehdj.k Kkr dhft, |
8. Define cone and right circular cone.
โkadq ,oa yEcdยจ.kh; โkadq dยจ ifjHkkfโkr dhft, |
9. Define reciprocal cone.
O;qRรe โkadq dยจ ifjHkkfโkr 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 ๐โ =xi+yj+zk and
๐โ is any constant vector then prove that
๐๐ข๐๐๐๐ฅ๐
๐3 = โ๐
๐3 +3(๐.๐)๐
๐5
;fn r =xi+yj+zk ,oa ๐โ o ,d vpj lfnโk gS
rยจ fl) dhft, ๐๐ข๐๐๐๐ฅ๐
๐3 = โ๐
๐3 +3(๐.๐)๐
๐5
11. Prove that div(axb) = b.curl a โ a.curl b
fl) fdft, div(axb) = b.curl a โ a.curl b
12. Evaluate โซ๐น.๐๐
๐, where
๐นโ= (x2 + y2)i โ 2xyj and curve c is the rectangle in the xy plane bounded by x=0,
x=a, y=0, y=b
Kkr dhft, โซ๐น.๐๐
๐, tgkยก
๐นโ= (x2 + y2)i โ 2xyj rFkk oร 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 โkkado dk dยขUรฆ Kkr dhft, ,oa โkkado dk dยขUรฆ ewy fcUnq ysrs gq, lehdj.k Kkr dhft,
36x2+24xy+29y2-72x+126y+81=0
14. Trace the parabola
fuEu oร 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 ๐
๐ฅ +
๐
๐ฆ +
๐
๐ง = 2
,d lery vpj fcUnq ยผ a,b,cยฝ ls xqtjrk gS rFkk v{kยจa dยจ ij dkVrk gS fl) dhft, fd xยจys dยข dยขUรฆ dk fcUnq iFk gS ๐
๐ฅ +
๐
๐ฆ +
๐
๐ง =
2
16. Find the equation of the cylinder whose generators are parallel to the line ๐ฅ
1 =
๐ฆ
โ2 =
๐ง
3 and whose guiding curve
is the ellipse x2 + 2y2 =1 , z = 0
ml csyu dk lehdj.k Kkr dhft, ftldk tud ๐ฅ
1 =
๐ฆ
โ2 =
๐ง
3 dยข lekUrj gS rFkk ftldk funZsโkd oร nh?kZo`r gS x2 + 2y2 =1 , z
= 0
Section C [45 Marks]
Section C โ 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 โ 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 oร x=2t2, y=t2 โ 4t, z=3t-5 osx o Roj.k dยข lfnโk i-3j+2k at t=1 dยข vuqfnโk ij ?kVd Kkr dhft, |
b) Prove that
fl) dhft, |
โ2 f (r) = f (r)+ 2
๐ f (r)
Where r = | ๐โ | and
๐โ = xi+yj+ zk
OR
Verify Stokeโs theorem for the function F=zi + xj + yk where curve C is the unit circle in xy plane bounding
the hemisphere.
Z = 1 โ โ1 โ ๐ฅ2 โ ๐ฆ2
LVยจDl รes; dk lR;kiu dhft, tgkยก F=zi + xj + yk rFkk C v)Zxยจys dยจ Z = 1 โ โ1 โ ๐ฅ2 โ ๐ฆ2 ifjc) djus okyk
v)Zo`r gS |
UNIT II
18. A) Discuss the tracing of conic ๐
๐ = 1 + e cos ๐
โkkado ๐
๐ = 1 + e cos ๐ dยข fp=.k dh foospuk dhft, |
OR
Find the asymptotes of the following hyperbolas and equation to their conjugate hyperbola fuEu vfrijoy; dh vUUr Liโkร Kkr dhft, ,oa bldยข 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 xยจyk ftldh f=T;k vpj r gS] ewy fcUnq ls xqtjrk gS rFkk A,B,C v{kยจa dยจ ij dkVrk gS- fl) dhft, dh ewy fcUnq ls lery
ABC ij Mkys x;s yEc dยข 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 dยขUรฆ dยข funZsโkkad Kkr dhft, |
x2 + y2 -2y โ 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 โkadq ax2+by2+cz2+2fyz+2gzx+2hxy=0dยข rhu ijLij yEcor LiโkZ 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 dยข lekarj rFkk lrg ax2+by2+cz2=1 rFkk lx + my + nz = p dยจ
dkVrk gS |
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2015- 16
End Semester Examination โ II
Class : B.Sc.[Math] โ SEM II
(Sub : Math)
Paper II: [MAT-202]: Calculus โ 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 Liโkร rFkk /kzoh v?k% Liโkร dh yEckร fyf[k;s |
2. Write the condition for the existence of the asymptote.
vuUr Liโkร dยข vfLrRo dยข fy, รfrcU/k fyf[k;s|
3. Write the coordinates of centre of curvature.
oรrk dยขUรฆ dยข funZsโkkad fyf[k;s | 4. Write the condition for a point to be a point of inflextion.
,d fcUnq dยข ufr ifjorZu fcUnq gยจus dh โkrZ fyf[k;s |
5. Define partial differential coefficient.
vakfโkd vody xq.kkad dยจ ifjHkkfโkr dhft, |
6. If u = ๐ฅ 2 and v=y2 then find the value of ๐(๐ข,๐ฃ)
๐(๐ฅ,๐ฆ)
;fn u = ๐ฅ 2 rFkk v=y2
rยจ ๐(๐ข,๐ฃ)
๐(๐ฅ,๐ฆ) dk eku Kkr dhft, |
7. Write the necessary conditions for f(a,b) to be an extreme value of f(๐ฅ,y)
Qyu f (๐ฅ,y) dยข pjeeku f (a,b) gยจus dk vkoโ;d รfrcU/k fyf[k;s |
8. Show that
รnรโkr dhft, |
โซ1
0 ๐ฅ n-1 (log
1
๐ฅ )m-1 d ๐ฅ =
โ(๐)
๐๐ , m , n > 0
9. Write double integral in polar coordinates.
/kqzoh funZsโkkadยจa esa f}lekdy dยจ fyf[k;s |
10. State Liouvilles extension of Dirichletโs 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 2๐(๐)
๐(๐)
รnรโkr dhft, fd oร p=f(r) dh /kzo ls xqtjus okyh oรrk thok dh yEckร 2๐(๐)
๐(๐) gS |
12. Find all the asymptotes of the following curve:
fuEu oร dh lHkh vuUr Liรโk;kยก Kkr dhft, |
(y-2๐ฅ)2(3๐ฅ +4y) + 3(y-2 ๐ฅ)(3 ๐ฅ +4y) โ 5 =0
UNIT II
13. If u = sin-1 ๐ฅ+2๐ฆ+3๐ง
๐ฅ 8+ ๐ฆ8 + ๐ง8 find the value of
๐ฅ ๐๐ข
๐๐ฅ + y
๐๐ข
๐๐ฆ + z
๐๐ข
๐๐ง
;fn u = sin-1 ๐ฅ+2๐ฆ+3๐ง
๐ฅ 8+ ๐ฆ8 + ๐ง8 rยจ ๐ฅ ๐๐ฅ
๐๐ฆ + y
๐๐ข
๐๐ฆ + z
๐๐ข
๐๐ง
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 oร dยข fdUgha nยจ fcUnqvยจa ij oรrk f=T;kvยจa dk vUrj rFkk mldยข dยขUรฆt dยข laxr fcUnqvยจa dยข chp dยข pki dh yEckร dk
vuqikr Kkr dhft, |
UNIT III
15. If โซ1
0 ๐ฅ m (1- ๐ฅ)n d ๐ฅ =โซ
1
0 ๐ฅ n (1- ๐ฅ)p, then find the value of p.
;fn โซ1
0 ๐ฅ m (1- ๐ฅ)n d ๐ฅ =โซ
1
0 ๐ฅ n (1- ๐ฅ)p, rยจ p dk eku Kkr dhft, |
16. Evaluate the following integral by changing the order of integration.
fuEu lekdy dk รe cnydj eku Kkr dhft, |
โซ1
0
โซ๐
๐๐ฅ
๐๐ฆ๐๐ฅ
log ๐ฆ
17. Evaluate
eku Kkr dhft, |
โซ1
โ1 โซ๐ง
0 โซ๐ฅ+๐ง
๐ฅโ๐ง (๐ฅ +y+z) d ๐ฅ dydz.
Section C [30 Marks]
Section C โ 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 oร dh lHkh vuUr Liรโk;kยก Kkr dhft, |
(2 ๐ฅ -3y+1)2 (๐ฅ +y) = 8 ๐ฅ โ 2y + 9
b. Find the angle of intersection of the following cardioids : fuEu รขn;ke dk รfrPNsnu dยจ.k Kkr dhft,
r = a(1+ Cos ๐) and r = b(1-Cos ๐)
OR
a. Show that the circle of curvature at the origin of the parabola y=mx+( ๐ฅ2
๐ ) is ๐ฅ
2+y2=a(1+m)2(y-m ๐ฅ).
รnรโkr dhft, fd ewy fcUnq ij i[kyo y=m ๐ฅ +( ๐ฅ2
๐ ) dk oรrk o`r ๐ฅ2+y2=a(1+m)2(y-m ๐ฅ) gS |
b. Examine the nature of the origin on the following curve: fuEu oร ij ewy fcUnq dh รรfr dh tkยกp dhft,
๐ฅ 7+2 ๐ฅ 4+2 ๐ฅ 3y+ ๐ฅ 2+2 ๐ฅ y+y2=0
UNIT II
19. a. If ๐ฅ +y = 2eฦ Cos โ and ๐ฅ -y = 2ieฦ Sin โ
show that ๐2๐ข
๐๐2 + ๐2๐ข
๐โ 2 = 4xy ๐2๐ข
๐๐ฅ๐๐ฆ
;fn ๐ฅ+y = 2eฦ Cos โ and ๐ฅ-y = 2ieฦ Sin โ
rยจ รnรโkr dhft,
๐2๐ข
๐๐2 + ๐2๐ข
๐โ 2 = 4xy ๐2๐ข
๐๐ฅ๐๐ฆ
b. if u = ๐ฅ โ(1 โ ๐ฆ)2 + y โ(1 โ ๐ฅ)2 ; v = sin-1 ๐ฅ sin-1y
Show that u, v are functionally related and find the relationship.
;fn u = ๐ฅ โ(1 โ ๐ฆ)2 + y โ(1 โ ๐ฅ)2 ; v = sin-1 ๐ฅ sin-1y
รnรโkr 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; โ being the parameter:
fuEu ljy js[kk dยข dqy dk vUokyยจi Kkr dhft, tgkยก โ รkpy gS |
a xsec โ- bycosec โ = a2-b2
b. Find the points where the function x3+y3-3axy has maximum or minimum value.
mu fcUnqvยจa dยจ Kkr dhft, tgka Qyu ๐ฅ 3+y3-3axy dk eku mPpre ;k U;wure gS |
UNIT III
20. Prove that fl) dhft,
โซโ
0
๐ฅ
1+๐ฅ6 dx = ๐ฅ
3โ3
a. โ(โ1
2 )
b. โซ โ(1 โ ๐ฅ)1
0 dx
OR
a. Change the order of integration in the following integral: fuEu lekdy esa lekdyu dk รe ifjorZu dhft,
โซ4๐
0 โซ2โ(๐๐ฅ)
๐ฅ2
4๐
๐ (๐ฅ,y) d๐ฅ dy
b. Evaluate eku Kkr dhft,
โญ ๐ฅ๐ฆ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง ,
Where region of integration is the complete ellipsoid :
tgkยก lekdy dk {ks= lEiw.kZ fn/kZo`Rrt gS
๐ฅ2
๐2 + ๐ฆ2
๐2 + ๐ง2
๐2 โค 1
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2016- 17
End Semester Examination
Class : B.Sc [Maths]
Sub : Mathematics
Paper II : [MAT-202]: Calculus
Time : 2 ยฝ 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 ifjHkkโkk nhft,
2. Define double point. f}d fcUnq dh ifjHkkโkk nhft,
3. Write the formula to find the radius of curvature for Cartesian curves. dkrร; oร dยข fy, oรrk f=T;k Kkr djus dk lw= fyf[k,
4. Write the degree of homogenous function
๐ข = ๐ฅ๐ฆ๐(๐ฆ
๐ฅ).
lEk/kkr Qyu ๐ข = ๐ฅ๐ฆ๐(๐ฆ
๐ฅ) dh /kkr fyf[k,
5. Define Envelope. vUokyยจi dh ifjHkkโkk 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) dยข vf/kdre eku f(a,b) gยจus dh vkoโ;d o i;kZIr รfrcU/k fyf[k,
7. Define Gamma Function. xkek Qyu dh ifjHkkโkk nhft,
8. Evaluate. eku Kkr dhft, โซ โ๐ฅ. ๐โ๐ฅ3๐๐ฅ
โ
0
9. State Dirichletโs Integral for triple integral. f=lekdyu dยข 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.
๐ฆ2 = 4๐(๐ฅ + ๐)
ijoy; ๐ฆ2 = 4๐(๐ฅ + ๐) dk ifnd lehdj.k Kkr dhft,
OR
Find the asymptotes of the following curve: fuEu oร dยข vUuRkLiรโk;kยก Kkr dhft,
x3 + yx2 - xy2 - y3 - 3x โ y โ 1 = 0
UNIT II
12. If ;fn ๐ข = ๐ ๐๐โ1 (โ๐ฅโโ๐ฆ
โ๐ฅ+โ๐ฆ) then show that. rc รnรโkr dhft, fd
๐ฅ๐๐ข
๐๐ฅ+ ๐ฆ
๐๐ข
๐๐ฆ0
OR
Find the envelope of the circles drawn upon the central radii of the ellipse ๐ฅ2
๐2 +๐ฆ2
๐2 = 1as diameter.
mu o`Rrยจa dk vUokyยจi Kkr dhft, tยจ nh/kZoRr ๐ฅ2
๐2 +๐ฆ2
๐2 = 1 dh /kqzokUrr js[kkvยจa dยจ O;kl ekudj [khps x, gSa
UNIT III
13. To show that รnรโkr dhft,
โ๐ โ๐ + 12โ =
โ๐
22๐โ1 โ2๐ ; ๐ โ ๐
where m is positive. tgkยก m /kukRed gSa
OR
Change the order of integration in the following double integral.
fuEu f}lekdy es lekdyu dk รe ifjoรrr dhft, โซ โซ ๐(๐ฅ, ๐ฆ)โ๐๐ฅ
2
๐ฅ24๐โ
๐๐ฅ ๐๐ฆ4๐
0
Section C [45 Marks]
Section C โ 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 oร dk vUkjs[k.k dhft, LVยชยจQkยก;M
๐ฆ2(๐ + ๐ฅ) = ๐ฅ2(3๐๐ฅ)
OR
(a) Show that for the radius of curvature at a point (๐ ๐๐๐ 3๐, ๐ ๐ ๐๐3๐) on the curve
๐ฅ2
3โ + ๐ฆ2
3โ = ๐2
3โ is 3๐
2๐ ๐๐2๐.
fl) dhft, fd oร ๐ฅ2
3โ + ๐ฆ2
3โ = ๐2
3โ dยขa fcUnq (๐ ๐๐๐ 3๐, ๐ ๐ ๐๐3๐) ij oรrk f=T;k
3๐
2๐ ๐๐2๐ gยจrh gS
(b) Find the point of inflection of the curve ๐ฆ(๐2 + ๐ฅ2) = ๐ฅ3.
oร ๐ฆ(๐2 + ๐ฅ2) = ๐ฅ3 dยข ufr ifjorZu fcUnq Kkr dhft,
UNIT II
15. (a) If ;fn ๐ข = ๐ฅ ๐ ๐๐โ1 ๐ฆ
๐ฅโ then prove that (rc fl) dhft, ) ๐ฅ2๐2๐ข
๐๐ฅ2 +2๐ฅ๐ฆ๐2๐ฅ
๐๐ฅ ๐๐ฆ+ ๐ฆ2 ๐2๐ข
๐๐ฆ2 = 0
(b) If ;fn ๐ = ๐(๐ฅ โ ๐ฆ, ๐ฆ โ ๐ง, ๐ง โ ๐ฅ)then prove that rc fl) dhft, ๐๐
๐๐ฅ+
๐๐
๐ฆ+
๐๐
๐๐ง= 0
OR
(a) Find the envelope of the family of ellipse
๐ฅ2
๐2+
๐ฆ2
๐2= 1
When a + b = c, c being a constant.
nh/kZorยจ ๐ฅ2
๐2 +๐ฆ2
๐2 = 1 dยข dqy dk vUokyยจi Kkr dhft,] tcfd a + b = c tgkยก c vpj gSa
(b)Show the minimum value of the following function
is 3a2.
รnรโkr dhft, fd fuEu Qyu dk fufEuโV eku 3a2 gS%
๐ข = ๐ฅ๐ฆ + ๐3(1
๐ฅ+
1
๐ฆ)
UNIT III
15. (a) Prove that :- fl) dhft, dh
โซ(๐ฅ โ ๐)๐โ1(๐ โ ๐ฅ)๐โ1 ๐๐ฅ
๐
๐
= (๐ โ ๐)๐+๐โ1 ๐ฝ(๐, ๐)
(b) Show that :- รnรโkr dhft, dh
โ๐ โ1 โ ๐ =๐
sin ๐ ๐
OR
(a) Evaluate :- eku Kkr dhft,
โซ โซ ๐ฅ๐ฆ(๐ฅ + ๐ฆ)๐๐ฅ ๐๐ฆ over the area between the parabola y = x2 and line y = x.
ijoy; y = x2 rFkk y = x dยข e/; {ks= ij
(b) Evaluate :- eku Kkr dhft,
โซ โซ โซ ๐๐ฅ+๐ฆ+๐ง ๐๐ฅ ๐๐ฆ ๐๐ง๐ฅ+log ๐ฆ
0
๐ฅ
0
๐๐๐2
0
The End
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2016- 17
End Semester Examination โ I
Class : B.Sc. Maths
Sub : Mathematic
Paper I : [MAT-201]:Vector Calculus and Geometry โ I
Time : 2 ยฝ 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 vfnโk fcUnq Qyu vpj gS ;fn grad f=0 |
b. If ๐ = |๐โ| where
๐โ = ๐ฅ๐ + ๐ฆ๐ + 2๐ prove that (fl) dhft, ) grad ๐ = ๐
^ |
c. If ๐โ = ๐ฅ๐ + ๐ฆ๐ + 2๐, prove that div
๐โ = 3
;fn ๐โ = ๐ฅ๐ + ๐ฆ๐ + 2๐, rยจ fl) dhft, fd div
๐โ = 3
d. State Gaussโ theorem.
xkWl รes; dk dFku fyf[k, |
e. Define conic section.
โkadq ifjPNsn dยจ ifjHkkfโkr dhft, |
f. Write polar equation of a straight line.
ljy rs[kk dยข /kqzoh; lehdj.k dยจ fyf[k, |
g. Write equation of the directrix.
fujkrk dยข lehdj.k dยจ fyf[k, |
h. Write equation of sphere in diameter form.
O;kl #i es xยจys dk lehdj.k fyf[k, |
i. Define enveloping cone.
vUokyยจih โkadq dยจ ifjHkkfโkr dhft, |
j. Define Right circular cylinder.
yEcoRrh; csyu dยจ ifjHkkfโkr 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 ๐โ = ๐ฅ๐ + ๐ฆ๐ + 2๐ and ๐ = |
๐โ| , prove (fl) dhft,) that curl ๐๐
๐โ = 0.
OR
Prove that fl) dhft, |
๐๐๐ฃ(๐ข๐) = ๐ข. ๐๐๐ฃ ๐ + ๐. ๐๐๐๐ ๐ข
UNIT II
12. (a) Find the co-ordinates of the centre of the following conic :
fuEu โkado dยข dยขUรฆ dยข funZsโkkad Kkr dhft, |
(b) Find the equation of the following conic referred to centre as origin in the standard form:-
dยขUรฆ dยจ ewy fcUnq ysdj fuEu โkkado 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 2๐๐
โ๐2+๐2.
fl) djยจ fd ijoy; (a2+b2) (x2+y2) = (bx+ay-ab)2 dh ukfHkyEc
2๐๐
โ๐2+๐2 gยจrh gS |
UNIT III
13. Two spheres of radii r1 and r2 cut orthogonally: Prove the radius of their common circle is. ๐1๐2
โ๐12 + ๐2
2
r1 rFkk r2 f=T;k dยข nยจ xยจys ykfEod #i ls dkVrs gS fl) dhft, fd mHk;fuโB o`Rr dh f=T;k gS | ๐1๐2
โ๐12 + ๐2
2
OR
If ๐ฅ
1=
๐ฆ
2=
๐ง
3 represent one of the three mutually perpendicular generators of then cone 5yz- 8xz - 3xy =
0 find the equation of the other two.
;fn ๐ฅ
1=
๐ฆ
2=
๐ง
3 โkadq 5yz - 8xz - 3xy = 0 dยข rhu ijLij lejhf.kd tudยจa esa ,d gS rยจ vU; nยจ tudยจa dยข 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 nยจ vfnโk fcUnq Qyu gยจ rยจ fl) dhft,
๐๐๐๐ (๐
๐) =
๐(๐๐๐๐ ๐)โ๐(๐๐๐๐ ๐ )
๐2, ๐ โ 0
(b) Find the equation of the tangent plane and
normal to the surface x2+y2+z2=25 at point
(4,0,3).
i`โB x2+y2+z2=25 dยข fcUnq (4,0,3) ij LiโkZ
Rky rFkk vfHkyEc dยข Lkehdj.k Kkr dhft, |
OR
(a) If ๐โ = ๐ฅ๐ + ๐ฆ๐ + 2๐ and ๐ = |
๐โ| , prove (fl) dhft,) that
div ๐๐
๐โ = (๐ + 3)
๐โ ๐
(b) Evaluate (eku Kkr dhft,)
โซ (๐ฆ๐ง๐๐ฅ + (๐ฅ๐ฆ + 1)๐๐ฆ + ๐ฅ๐ฆ๐๐ง) Where c is any path form (1,0,0) to (2,1,4)
tgka c fcUnq (1,0,0) ls (2,1,4) rd dยจร iFk gS |
UNIT II
15. Find the nature of the conic represented by the equation x2 + 2xy + y2 - 2x โ 1 = 0 and also trace
it.
Lkehdj.k x2 + 2xy + y2 - 2x โ 1 = 0 }kjk รnรโkr โkkado dh รรfr Kkr dhft, rFkk bldk vuqjs[k.k Hkh dhft, |
OR
Trace the curve. oร 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) vยฉj (0,0,1) ls xqtjus
okys ml xยจys dk lehdj.k Kkr dhft,
ftldh f=T;k U;wUkre gยจ |
(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 ๐โ2, ๐โ2, ๐โ2 respectively. Find its equation.
,d xยจys dk dยขUรฆ /ku vโVkaโkd esa gS vยฉj og ewy fcUnq ls xqtjrk gS funZsโkh lery x=y=z=0 ls bldk รfrPNsn
รeโk% ๐โ2, ๐โ2, ๐โ2 f=T;k okys o`Rr gS bl xยจys 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).
xยจys x2+y2xz2+2x-2y-2=0 dยข ml vUokyยจih โkadq dk lehdj.k Kkr dhft, ftldk โkhโkZ fcUnq (1,1,1) gS |
(b) Find the equation of right circular cylinder whose guiding curve is the circle
๐ฅ2 + ๐ฆ2 + ๐ง2 = 9, x - 2y + 2z = 3
ml yEcoRrh; csyu dk lehdj.k Kkr dhft, ftLkdk funZsโkd oRr ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9, x - 2y + 2z = 3 gS |
The End
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2017- 18
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper I : [MAT-201]: Vector Calculus & Geometry
Time : 2 ยฝ 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 โ๐, ๐๐ ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ2๐ฆ + ๐ฆ2๐ฅ + ๐ง2
;fn ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ2๐ฆ + ๐ฆ2๐ฅ + ๐ง2 , โ๐ eku Kkr dhft,A
2. Define Divergence of a vectors and Solenoidal vector.
Lkfnโk dk vilj.k o ifjufydk lfnโk dks ifjHkkf"kr dhft,A
3. Find the divergence of ๏ฟฝ๏ฟฝ. where ๏ฟฝ๏ฟฝ = ๐ฅ๐ฆ๐ง๐ + 3๐ฅ2๐ฆ๐ + (๐ฅ๐ง2 โ ๐ฆ2๐ง)๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ dk vilj.k Kkr dhft,] tgkยก ๏ฟฝ๏ฟฝ = ๐ฅ๐ฆ๐ง๐ + 3๐ฅ2๐ฆ๐ + (๐ฅ๐ง2 โ ๐ฆ2๐ง)๏ฟฝ๏ฟฝ
4. State Greenโs theorem.
xzhu izes; dk dFku fyf[k,A
5. Write the necessary conditions for conic ๐๐ฅ2 + ๐๐ฆ2 + 2โ๐ฅ๐ฆ + 2๐๐ฅ + 2๐๐ฆ + ๐ = 0 to represents an
equation of ellipse.
'kkod ๐๐ฅ2 + ๐๐ฆ2 + 2โ๐ฅ๐ฆ + 2๐๐ฅ + 2๐๐ฆ + ๐ = 0 ,d nh?kZor ds lehdj.k dks fu:fir djrk gS
rks vkoโ;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 ๐๐ฅ2 + 2โ๐ฅ๐ฆ + ๐๐ฆ2 = 1 in terms of the length ๐ of its
semi axis.
'kkdo ๐๐ฅ2 + 2โ๐ฅ๐ฆ + ๐๐ฆ2 = 1 ds v/kZ&v{k dh yEckbZ ๐ ds :Ik mlds v{k dh lehdj.k fyf[k,A
8. Define Reciprocal cone.
O;qRรe 'kdq dks ifjHkkf"kr dhft,A
9. Define Cylinder.
csyu dks ifjHkkf"kr dhft,A
10. Find the centre and radius of the sphere 7๐ฅ2 + 7๐ฆ2 + 7๐ง2 โ 6๐ฅ โ 3๐ฆ โ 2๐ง = 0
xksy 7๐ฅ2 + 7๐ฆ2 + 7๐ง2 โ 6๐ฅ โ 3๐ฆ โ 2๐ง = 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 ๏ฟฝ๏ฟฝ and ๏ฟฝ๏ฟฝ are constant vectors and ๐ = ๐ฅ๐ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ then show that : ๐๐ข๐๐ [(๐ ร ๏ฟฝ๏ฟฝ) ร ๏ฟฝ๏ฟฝ] = ๏ฟฝ๏ฟฝ ร ๏ฟฝ๏ฟฝ
;fn ๏ฟฝ๏ฟฝ o ๏ฟฝ๏ฟฝ vpj lfnโk gS o ๐ = ๐ฅ๐ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ rks fl) dhft, fd ๐๐ข๐๐ [(๐ ร ๏ฟฝ๏ฟฝ) ร ๏ฟฝ๏ฟฝ] = ๏ฟฝ๏ฟฝ ร ๏ฟฝ๏ฟฝ
OR
Prove that :
fl) dhft, fd&
๐๐๐ฃ (๐
๐3) = 0
Where ๐ = ๐ฅ๐ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ
UNIT II
12. Find the condition so that the straight line ๐
๐= ๐ด ๐๐๐ ๐ + ๐ต ๐ ๐๐๐ may touch the conic
๐
๐= 1 +
๐ cos(๐ โ ๐ผ).
izfrcU/k Kkr dhft, tc fd ljy js[kk ๐
๐= ๐ด ๐๐๐ ๐ + ๐ต ๐ ๐๐๐ 'kkado
๐
๐= 1 + ๐ cos(๐ โ ๐ผ) dks
LiโkZ djrh gSaA
OR
What conic does equation 13๐ฅ2 โ 18๐ฅ๐ฆ + 37๐ฆ2 + 2๐ฅ + 14๐ฆ โ 2 = 0 represent? Find its centre
and the equation to the conic referred to the centre as origin.
lehdj.k 13๐ฅ2 โ 18๐ฅ๐ฆ + 37๐ฆ2 + 2๐ฅ + 14๐ฆ โ 2 = 0 dkSuls 'kkado dks fu:frr djrk gSa\ blds
dsUnz ds funsZโkkad 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 ๐ฅ2 + ๐ฆ2 + ๐ง2 + 5๐ฅ โ 7๐ฆ + 2๐ง โ 8 = 0,3๐ฅ โ
2๐ฆ + 4๐ง + 3 = 0 at the point (โ3,5,4) .
o`Rr ๐ฅ2 + ๐ฆ2 + ๐ง2 + 5๐ฅ โ 7๐ฆ + 2๐ง โ 8 = 0, 3๐ฅ โ 2๐ฆ + 4๐ง + 3 = 0 ds fcUnq (โ3,5,4) ij LiโkZ
js[kk ds lehdj.k Kkr dhft,A
OR
Find the equation of the sphere having the circle ๐ฅ2 + ๐ฆ2 + ๐ง2 + 10๐ฆ โ 4๐ง = 8, ๐ฅ + ๐ฆ + ๐ง = 3 as
a great circle.
xksys dk lehdj.k Kkr dhft, ftldk or ๐ฅ2 + ๐ฆ2 + ๐ง2 + 10๐ฆ โ 4๐ง = 8, ๐ฅ + ๐ฆ + ๐ง = 3 ,d
o`gr or gksA
Section C [45 Marks]
Section C โ 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 ๏ฟฝ๏ฟฝ = ๐ฅ๐ฆ๐ + ๐ง2๐ + 2๐ฆ๐ง๏ฟฝ๏ฟฝ on the tetrahedron ๐ฅ = ๐ฆ = ๐ง = 0,
๐ฅ + ๐ฆ + ๐ง = 1.
prq"Qyd ๐ฅ = ๐ฆ = ๐ง = 0, ๐ฅ + ๐ฆ + ๐ง = 1 ij ๏ฟฝ๏ฟฝ = ๐ฅ๐ฆ๐ + ๐ง2๐ + 2๐ฆ๐ง๏ฟฝ๏ฟฝ ds fy, vilj.k izes; dk
lR;kiu dhft,A
OR
Evaluateโซ ๐๐ข๐๐ ๏ฟฝ๏ฟฝ๐
. ๏ฟฝ๏ฟฝ๐๐ , where ๏ฟฝ๏ฟฝ = (๐ฆ2 + ๐ง2 โ ๐ฅ2)๐ + (๐ง2 + ๐ฅ2 โ ๐ฆ2)๐ + (๐ฅ2 + ๐ฆ2 โ
๐ง2)๏ฟฝ๏ฟฝ ๐๐๐ ๐ is the portion of the surface ๐ฅ2 + ๐ฆ2 โ 29๐ฅ + 9๐ง = 0 above the plane ๐ง = 0, and verify
the Stokeโs theorem.
โซ ๐๐ข๐๐ ๏ฟฝ๏ฟฝ๐
. ๏ฟฝ๏ฟฝ๐๐ dk eku Kkr dhft,A tgkยก ๏ฟฝ๏ฟฝ = (๐ฆ2 + ๐ง2 โ ๐ฅ2)๐ + (๐ง2 + ๐ฅ2 โ ๐ฆ2)๐ +
(๐ฅ2 + ๐ฆ2 โ ๐ง2)๏ฟฝ๏ฟฝ gS rFkk ๐, ๐ง = 0 ry ds mij Qyd ๐ฅ2 + ๐ฆ2 โ 29๐ฅ + 9๐ง = 0 dk Hkkx gS rFkk
LVkd izes; dk lR;kiu dhft,A
UNIT II 15. Trace the conic 14๐ฅ2 โ 4๐ฅ๐ฆ + 11๐ฆ2 โ 44๐ฅ โ 58๐ฆ + 71 = 0 . Also find the focus, eccentricity and
length of latus rectum.
'kkdo 14๐ฅ2 โ 4๐ฅ๐ฆ + 11๐ฆ2 โ 44๐ฅ โ 58๐ฆ + 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 ๐ ๐โ = 1 + ๐ ๐๐๐ ๐ is ๐2(1 โ ๐2) +2๐๐๐ ๐๐๐ ๐ โ 2๐2 = 0 .
fl) djks fd 'kkado ๐๐โ = 1 + ๐ ๐๐๐ ๐ ds fu;ked o`r dk lehdj.k ๐2(1 โ ๐2) + 2๐๐๐ ๐๐๐ ๐ โ
2๐2 = 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 ๐ก๐๐โ1โ2
;fn ,d yEc o`rh; 'kadq ds rhu tud ijLij yEc gS] rks fl) dhft, fd v)Zโkh"kZ dks.k dk eku
๐ก๐๐โ1โ2 gSA
(b) Show that the equation 4๐ฅ2 โ ๐ฆ2 + 2๐ง2 + 2๐ฅ๐ฆ โ 3๐ฆ๐ง + 12๐ฅ โ 11๐ฆ + 6๐ง + 4 = 0 represents a
cone with vertex (โ1, โ2, โ3).
fl) dhft, fd lehdj.k 4๐ฅ2 โ ๐ฆ2 + 2๐ง2 + 2๐ฅ๐ฆ โ 3๐ฆ๐ง + 12๐ฅ โ 11๐ฆ + 6๐ง + 4 = 0 ,d 'kadq
dks fu:fir djrk gS ftldk 'kh"kZ (โ1, โ2, โ3) gSaA
OR
(a) Find the equation of the enveloping cylinder of the surface ๐๐ฅ2 + ๐๐ฆ2 + ๐๐ง2 = 1 and whose
generators are parallel to the line ๐ฅ
๐=
๐ฆ
๐=
๐ง
๐ .
๐๐ฅ2 + ๐๐ฆ2 + ๐๐ง2 = 1 ds vUokyksih csyu ds lehdj.k Kkr dhft, ftldh tud js[kk, ๐ฅ
๐=
๐ฆ
๐=
๐ง
๐ ds lekUrj gSaA
(b) Find the equation of the right circular cylinder whose guiding circle is ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 , ๐ฅ โ๐ฆ + ๐ง = 3 .
ml yEco`Rrh; csyu dk lehdj.k Kkr dhft, ftldk funsZโkkd o`r ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 , ๐ฅ โ ๐ฆ +
๐ง = 3 gSaA
The End
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2017- 18
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]:Calculus
Time : 2 ยฝ 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:
oรrk f=T;k Kkr dhft, ;fn nh/kZor dk ifnd lehdj.k gS %
1
๐2=
1
๐2+
๐
๐2โ
๐2
๐2๐2
2. Define derivative of the length of arc.
pki dh yEckร dยข vodyt dh ifjHkkโkk fnft,
3. Write the Eulerโs theorem for homogenous functions.
le?kkr Qyu dยข fy;s vkW;yj รes; fyf[k,
4. Define evaluate.
dยขUรฆt dh ifjHkkโkk fyf[k,
5. Give the sufficient condition for ๐(๐, ๐) to be a maximum value of ๐(๐ฅ, ๐ฆ).
Qyu ๐(๐ฅ, ๐ฆ dยข vf/kdre eku ๐(๐, ๐) gยจus dh i;kZIr รfrca/k fyf[k,
6. Evaluate the following. โซ ๐ ๐๐5๐ฅ๐
2โ
0 ๐๐๐ 6๐ฅ ๐๐ฅ.
fuEu dk eku Kk dhft,
7. Define Beta function.
chVk Qyu dh ifjHkkโkk nhft,
8. Change the order of integration in the following double integral.
fuEu f}&lekdy eSa lekdyu dk รe ifjoรrr dhft,%
โซ โซ๐โ๐ฆ
๐ฆ ๐๐ฅ๐๐ฆ
โ
๐ฅ
โ
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:
oร dยข ufr ifjorZu fcUnq Kkt dhft,%
๐ฆ = 3๐ฅ4 โ 4๐ฅ3 + 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. ๐ฅ = ๐ ๐๐๐ 3๐ก, ๐ฆ = ๐ ๐ ๐๐3๐ก
,LVยชk;M ๐ฅ = ๐ ๐๐๐ 3๐ก, ๐ฆ = ๐ ๐ ๐๐3๐ก dk ifnd lehdj.k Kkr dhft,
OR
Find the asymptotes of the following curve:
fuEu oร dh vUkUrLiรโk;kยก Kkr dhft,
๐ฆ3 โ ๐ฅ๐ฆ2 โ ๐ฅ2๐ฆ + ๐ฅ3 + ๐ฅ2 โ ๐ฆ2 โ 1 = 0
UNIT II
12. If ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง = ๐ then show that ๐2๐ง
๐๐ฅ๐๐ฆ= โ(๐ฅ log ๐๐ฅ)โ1.
;fn ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง = ๐ rยจ fl) dhft, fd ๐2๐ง
๐๐ฅ๐๐ฆ= โ(๐ฅ log ๐๐ฅ)โ1
OR
Find the envelope of the circles which passes through the vertex of the vertex of the parabola ๐ฆ2 =
4๐๐ฅ and whose centre lies on it.
,d o`r dk dยขUรฆ lnSo ijoy; ๐ฆ2 = 4๐๐ฅ ij jgrk gS rFkk ยธg ij oy; dยข โkhโkZ ls xqtjrk gS o`r
dqy dk vUokyยจi Kkr dhft,
UNIT III
13. To show that: รnรโr dhft,
โ๐ โ1 โ ๐ =๐
sin ๐๐, 0 < ๐ < 1.
OR
Change the order of integration in the following double integral:
fuEu f}&lekdy esa lekdyu dk รe ifjoรrr dhft,
โซ โซ ๐ (๐ฅ, ๐ฆ)๐๐ฅ ๐๐ฆ.2๐โ๐ฅ
๐ฅ2๐โ
๐
0
Section C [45 Marks]
Section C โ 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,
๐ฆ2(๐ + ๐ฅ) = ๐ฅ2(3๐ โ ๐ฅ)
OR
a. Prove that the radius of curvature of the centenary
๐ฆ = ๐ cosh (๐ฅ
๐) ๐๐ก (๐ฅ, ๐ฆ)๐๐
๐ฆ2
๐.
fl) djยจ fd dSVsujh ๐ฆ = ๐ cosh (๐ฅ
๐) dยข fcUnq (๐ฅ, ๐ฆ) ij oรrk f=T;k
๐ฆ2
๐ gS
b. Find the point inflection of the curve ๐ฆ(๐2 + ๐ฅ2) = ๐ฅ3
oร ๐ฆ(๐2 + ๐ฅ2) = ๐ฅ3 dยข ufr ifjorZu fcUnq Kkr dhft,
UNIT II
15. (a) If ๐ข = ๐ก๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅ+๐ฆ), then prove that.
;fn ๐ข = ๐ก๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅ+๐ฆ) rยจ fl) dhft,
๐ฅ2๐ฟ2๐ข
๐ฟ๐ฅ2+ 2๐ฅ๐ฆ
๐ฟ2๐ข
๐ฟ๐ฅ๐ฟ๐ฆ+ ๐ฆ2
๐ฟ2๐ข
๐ฟ๐ฆ2= sin 2๐ข (1 โ 4 ๐ ๐๐2๐ข)
(b) If ๐ฅ3 + ๐ฆ3 = 3๐๐ฅ๐ฆ, then find ๐2๐ฆ
๐๐ฅ2
;fn ๐ฅ3 + ๐ฆ3 = 3๐๐ฅ๐ฆ Kkr dhft, ๐2๐ฆ
๐๐ฅ2
OR
(a) Find the envelope of the straight lines ๐ฅ cos ๐ผ + ๐ฆ sin ๐ผ = ๐ sin ๐ผ cos ๐ผ; ๐ผ is a parameter.
๐ผ dยจ รkpyd ysrs gq, ljy js[kkvยจ ๐ฅ cos ๐ผ + ๐ฆ sin ๐ผ = ๐ sin ๐ผ cos ๐ผ ; dk vUoyยจi Kkr
dhft;s
(b) Find the maximum value of ๐ฅ3๐ฆ2(1 โ ๐ฅ โ ๐ฆ).
mfPpโB eku Kkr dhft,
UNIT III
16. (a) Show that :- รnรโkr dhft, fd
โซ ๐ก๐๐๐๐ฅ๐๐ฅ =๐
2sec (
๐๐
2)
๐2โ
0
(b) Prove that:- fl) dhft, fd
22๐โ1๐ต (๐, ๐) =โ๐โ๐
โ(๐ +12)
OR
(a) Evaluate:- eku Kr dhft, โซ โซ (๐ฅ + ๐ฆ)2๐๐ฅ ๐๐ฆ๐
. Where the area R is bounded by ellipse tgkยก
{ks= nh/kZo`r ls ifjc) gS
๐ฅ2
๐2 +๐ฆ2
๐2 = 1
(b) Evaluate: โซ โซ โซ๐๐ฅ ๐๐ฅ ๐๐ง
(๐ฅ+๐ฆ+๐ง+1)3
1โ๐ฅโ๐ฆ
0
1โ๐ฅ
0
1
0
The End
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2018- 19
End Semester Examination
Class : B.Sc. Math
Sub : Mathematics
Paper I : [MAT 201]: Vector Calculus & Geometry
Time : 2 ยฝ 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 ๐๐๐๐ ๐ = ๏ฟฝ๏ฟฝ , when ๐ = ๐ฅ๐ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ
2. Find the greatest rate of increase of โ = 2๐ฅ๐ฆ๐ง2 at (1,0, โ3).
โ = 2๐ฅ๐ฆ๐ง2 dk (1,0, โ3) 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 ๐๐ฅ2 + ๐๐ฆ2 + 2โ๐ฅ๐ฆ + 2๐๐ฅ + 2๐๐ฆ + ๐ = 0 to represent an equation of
hyperbola.
'kkado ๐๐ฅ2 + ๐๐ฆ2 + 2โ๐ฅ๐ฆ + 2๐๐ฅ + 2๐๐ฆ + ๐ = 0 ds ,d vfrijoy; dks iznfโkZr djus dh 'krZ
fyf[k,A
5. Write the equation of directive of conic ๐
๐= 1 + ๐ ๐๐๐ ๐, ๐คโ๐๐ ๐ < 1.
'kkado ๐
๐= 1 + ๐ ๐๐๐ ๐] tcfd ๐ < 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 LifโkZrk 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`รkh; 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 โ2๐๐ and show that โ2 ( 1
๐ ) = 0 where ๐ = (๐ฅ2 + ๐ฆ2 + ๐ง2)1/2
โ2๐๐ dk eku Kkr djks rFkk fl) djksa โ2 ( 1
๐ ) = 0 ] tgkยก ๐ = (๐ฅ2 + ๐ฆ2 + ๐ง2)1/2
OR
Evaluate โซ ๐ ๐
. ๐๐ , where ๐ = ๐ฅ๐ฆ๐ + (๐ฅ2 + ๐ฆ2)๐ and C is the rectangle in the ๐ฅ๐ฆ plane bounded
by lines ๐ฅ = 1, ๐ฅ = 4 , ๐ฆ = 2, ๐ฆ = 10.
โซ ๐ ๐
. ๐๐ dk eku Kkr dhft,] tcfd ๐ = ๐ฅ๐ฆ๐ + (๐ฅ2 + ๐ฆ2)๐ o C, xy iVy es ๐ฅ = 1, ๐ฅ =
4 , ๐ฆ = 2, ๐ฆ = 10 js[kkvksa ls f?kjk vk;r gSaA
UNIT II
12. Find the lengths and the equation of the axes of the conic 2๐ฅ2 + 5๐ฅ๐ฆ + 2๐ฆ2 = 1 .
'kkado 2๐ฅ2 + 5๐ฅ๐ฆ + 2๐ฆ2 = 1 ds v{kksa dh yEckb;kยก o lehdj.k Kkr dhft,A
OR
A chord ๐๐ 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 ๐๐ ukfHk ๐ ij ledks.k vUrfjr djrh gSaA ;fn 'kkado dh mRdsUnzrk ๐ v)ZukfHkyEc ๐ gks rks fl) djksa&
(1
๐ ๐โ
1
๐)
2
+ (1
๐๐โ
1
๐)
2
=๐2
๐2
UNIT III
13. Find the equation of the sphere having the circle ๐ฅ2 + ๐ฆ2 + ๐ง2 + 10๐ฆ โ 4๐ง = 8 , ๐ฅ + ๐ฆ + ๐ง = 3 as
a great circle.
ml xksys dk lehdj.k Kkr djks ftldk or ๐ฅ2 + ๐ฆ2 + ๐ง2 + 10๐ฆ โ 4๐ง = 8 , ๐ฅ + ๐ฆ + ๐ง = 3 rd
o`gr or gksaA
OR
Find the equation of the cylinder whose generators are parallel to the line ๐ฅ
1=
๐ฆ
โ2=
๐ง
3 and passing
through the curve ๐ฅ2 + 2๐ฆ2 = 1 , ๐ง = 3.
ml csyu dk lehdj.k Kkr djksa ftldh tud js[kk,ยก ๐ฅ
1=
๐ฆ
โ2=
๐ง
3 ds lekUkkUrj gS rFkk ftldk
funsZโkd oร ๐ฅ2 + 2๐ฆ2 = 1 , ๐ง = 3 gSaA
Section C [45 Marks]
Section C โ 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&
๐๐ข๐๐ ๐ ร ๐
๐3= โ
๐
๐3+
3๐
๐5 ( ๐ . ๐ )
Where, ๐ = ๐ฅ ๏ฟฝ๏ฟฝ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ , ๐ = ๐1๐ + ๐2๐ + ๐3๏ฟฝ๏ฟฝ is a constant vector and |๐ | = ๐.
tgkยก ๐ = ๐ฅ ๏ฟฝ๏ฟฝ + ๐ฆ๐ + ๐ง๏ฟฝ๏ฟฝ , ๐ = ๐1๐ + ๐2๐ + ๐3๏ฟฝ๏ฟฝ ,d vpj lafnโk gS rFkk |๐ | = ๐
OR
Verify Gaussโ Divergence theorem for ๐ = 4๐ฅ๐ โ 2๐ฆ2๐ + ๐ง2๏ฟฝ๏ฟฝ , taken over the region bounded by
๐ฅ2 + ๐ฆ2 = 4 , ๐ง = 0, ๐ง = 3.
๐ฅ2 + ๐ฆ2 = 4 , ๐ง = 0, ๐ง = 3 }kjk ifjc) {ks= ij Qyu ๐ = 4๐ฅ๐ โ 2๐ฆ2๐ + ๐ง2๏ฟฝ๏ฟฝ ds fy, xkWl
vilj.k izes; lR;kfir dhft,A
UNIT II
15. Trace the curve.
oร vuqjs[k.k dhft,A
๐ฅ2 + ๐ฅ๐ฆ + ๐ฆ2 + ๐ฅ + ๐ฆ = 1
OR
Find the condition that the straight line ๐
๐= ๐ด ๐๐๐ ๐ + ๐ต๐ ๐๐๐ touches the circle ๐ = 2๐ ๐๐๐ ๐.
og izfrcU/k Kkr djks tcfd ljy js[kk ๐
๐= ๐ด ๐๐๐ ๐ + ๐ต๐ ๐๐๐ ] o`r ๐ = 2๐ ๐๐๐ ๐ dks LiโkZ djsaA
UNIT III
16. Find the equation of cone whose vertex is (5,4,3) and the guiding curve is 3๐ฅ2 + 2๐ฆ2 = 6 , ๐ฆ +
๐ง = 0.
ml 'kadq dk lehdj.k Kkr dhft, ftldk 'kh"kZ (5,4,3) rFkk funsZโkd oร 3๐ฅ2 + 2๐ฆ2 = 6 , ๐ฆ +
๐ง = 0 gSaA
OR
Find the equation of the right circular cylinder whose guiding circle is ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 , ๐ฅ โ๐ฆ + ๐ง = 3 .
ml yEco`rh; csyu dk lehdj.k Kkr djks ftldk funsZโkd o`r ๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 , ๐ฅ โ ๐ฆ +
๐ง = 3 gSaA
The End
Sophia Girlsโ College, Ajmer (Autonomous)
Semester II โ 2018- 19
End Semester Examination
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]:Calculus
Time : 2 ยฝ 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 Liโkร dยจ ifjHkkfโkr dhft, |
2. Write the necessary condition for the existence of double point for the curve ๐(๐ฅ, ๐ฆ) = 0 and also
the condition for the double point to be a node, cusp or conjugate point.
oร ๐(๐ฅ, ๐ฆ) = 0 dยข f}d fcUnqvยจa dยข vfLrRo dยข fy, vkoโ;d รfrcU/k fyf[k, ,oa f}d fcUnq dยข
uยจM] mHk;kXkz ;k la;qXeh fcUnq gยจus gsrq รfrca/k Hkh fyf[k, |
3. Find the radius of curvature at the point (s,๐) for the curve ๐ = ๐ ๐๐๐(sec ๐ + tan ๐).
oร ๐ = ๐ ๐๐๐(sec ๐ + tan ๐) dยข fcUnq (s, ๐) ij oรrk fยซkT;k Kku dhft, |
4. State Eulerโs theorem for homogeneous functions.
le?kku Qyuยจa dยข fy, vk;dj รes; dk dFku fyf[k, |
5. If ๐ข =๐ฆ2
2๐ฅ, ๐ฃ =
๐ฅ2+๐ฆ2
2๐ฅ, find
๐(๐ข,๐ฃ)
๐ (๐ฅ,๐).
;fn ๐ข =๐ฆ2
2๐ฅ, ๐ฃ =
๐ฅ2+๐ฆ2
2๐ฅ rยจ
๐(๐ข,๐ฃ)
๐ (๐ฅ,๐) Kkr dhft, |
6. Define envelope of a one parameter family of plane curves.
,d รkpy okys lery oร dqy dยข vUokyยจi dยจ ifjHkkfโkr dhft,
7. Evaluate โซ ๐ฅ5๐โ๐ฅโ
0 ๐๐ฅ
eku Kkr dhft, โซ ๐ฅ5๐โ๐ฅโ
0 ๐๐ฅ
8. Change the order of integration in the following integral
fuEu lekdy esa lekdyu dk รe ifjoรrr dhft, |
โซ โซ ๐ (๐ฅ, ๐ฆ)๐๐ฅ ๐๐ฆโ๐2โ๐ฅ2
0
๐
0
9. State Liouvilleโs extension of Dirichletโs integral.
fMjpfyV lekdy dยข fyosyh O;kidhdj.k dk dFku fyf[k, |
10. Check if the function ๐ฅ3 โ 4๐ฅ๐ฆ + 2๐ฆ2 attains its extreme value at the point (0, 0).
tkยกp dhft, fd D;k Qyu ๐ฅ3 โ 4๐ฅ๐ฆ + 2๐ฆ2 vius pje eku dยจ fcUnq (0, 0) ij รkIr 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 ๐๐
๐๐ for the curve
2๐
๐= 1 + cos ๐.
oร 2๐
๐= 1 + cos ๐ dยข fy,
๐๐
๐๐ Kkr dhft, |
OR
Find the radius of curvature at the point (r,๐) on the polar curve ๐ (1 + cos ๐) = ๐
/kqzoh oร ๐ (1 + cos ๐) = ๐ dยข fcUnq (r,๐) ij oรrk f=T;k Kkr dhft, |
UNIT II
12. If ๐ข = ๐ ๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅ+๐ฆ), then prove that ๐ฅ
๐๐ข
๐๐ฅ+ ๐ฆ
๐๐ข
๐๐ฆ= 2 cot ๐ข.
;fn ๐ข = ๐ ๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅ+๐ฆ) rยจ fl) dhft, ๐ฅ
๐๐ข
๐๐ฅ+ ๐ฆ
๐๐ข
๐๐ฆ= 2 cot ๐ข
OR
If ๐ข3 + ๐ฃ3 = ๐ฅ + ๐ฆ and ๐ข2 + ๐ฃ2 = ๐ฅ3 + ๐ฆ3, then find the value of ๐(๐ข,๐ฃ)
๐(๐ฅ,๐ฆ).
;fn ๐ข3 + ๐ฃ3 = ๐ฅ + ๐ฆ rFkk ๐ข2 + ๐ฃ2 = ๐ฅ3 + ๐ฆ3 rยจ ๐(๐ข,๐ฃ)
๐(๐ฅ,๐ฆ). dk eku Kkr dhft, |
UNIT III
13. Prove that. fl) dhft, โซ๐๐ฅ
โ๐4โ๐ฅ4
๐
0=
{ฮ (1/4) }2
4๐ โ(2๐)
OR
Evaluate โฌ ๐ฅ๐ฆ(๐ฅ + ๐ฆ)๐๐ฅ ๐๐ฆ๐
where the area of integration R is the area between ๐ฆ = ๐ฅ2 and ๐ฆ =
๐ฅ.
โฌ ๐ฅ๐ฆ(๐ฅ + ๐ฆ)๐๐ฅ ๐๐ฆ๐
dk eku Kkr dhft, tgkยก lekdyu dk {ks= R, ๐ฆ = ๐ฅ2 rFkk ๐ฆ = ๐ฅ dยข e/;
dk {ks= gS |
Section C [45 Marks]
Section C โ 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 oรยจa dk vuqjs[k.k dhft, |
a. ๐ฆ2 (2๐ โ ๐ฅ) = ๐ฅ3
b. ๐ = ๐(1 + cos ๐)
OR
Find all the asymptotes of the curve (2๐ฅ โ 3๐ฆ + 1)2 (๐ฅ + ๐ฆ) โ 8๐ฅ + 2๐ฆ โ 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.
oร (2๐ฅ โ 3๐ฆ + 1)2 (๐ฅ + ๐ฆ) โ 8๐ฅ + 2๐ฆ โ 9 = 0 dh lHkh vuUr Liรโk;ยจa Kkr dhft, rFkk fl)
dhft, fd, ;s oร dยจ iqu% rhu fcUnqvยจa ij dkVrh gS tยจ ,d js[kk ij fLFkr gS | bl js[kk dk
lehdj.k Kkr dhft, |
UNIT II
15. (a) Prove Eulerโs theorem on homogeneous function.
le?kkr Qyuยจa dยข fy, vk;yj รes; dยจ fl) djsa |
(b) Prove that ๐ข = ๐๐ฅ3๐ฆ2 โ ๐ฅ4๐ฆ2 โ ๐ฅ3๐ฆ3 is maximum at (๐
2,
๐
3)
fl) dhft, fd ๐ข = ๐๐ฅ3๐ฆ2 โ ๐ฅ4๐ฆ2 โ ๐ฅ3๐ฆ3 mPpre eku fcUnq (๐
2,
๐
3) ij gS |
OR
Find the envelope of the family of straight lines ๐๐ฅ sec ๐ผ โ ๐ ๐ฆ ๐๐๐ ๐๐ ๐ผ = ๐2 โ ๐2, ๐ผ being the
parameter. Use the above result to show that the evolutes of the ellipse ๐ฅ2
๐2+
๐ฆ2
๐2= 1 is (๐๐ฅ)
23โ +
(๐๐ฆ)2
3โ = (๐2 โ ๐2)2
3โ .
ljy js[kk dqy ๐๐ฅ sec ๐ผ โ ๐ ๐ฆ ๐๐๐ ๐๐ ๐ผ = ๐2 โ ๐2 dk vUokyยจi Kkr dhft, tgkยก ๐ผ รkpy gS
mDr ifj.kke dยข ร;ยจx }kjk fl) dhft, fd nh/kZo`Rr ๐ฅ2
๐2 +๐ฆ2
๐2 = 1 dk dยขUรฆ (๐๐ฅ)2
3โ + (๐๐ฆ)2
3โ =
(๐2 โ ๐2)2
3โ gS |
UNIT III
16. (a) Prove that:
fl) dhft,
๐(๐, ๐) =โ(๐)โ(๐)
โ(๐ + ๐) (๐ > 0, ๐ > 0)
(b) Evaluate โญ ๐ฅ๐ฆ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง where region of integration is the complete ellipsoid ๐ฅ2
๐2 +๐ฆ2
๐2 +๐ง2
๐2 โค
1.
โญ ๐ฅ๐ฆ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง dk eku dhft, tgkยก lekdyu dk {ks= laiw.kZ nh/kZo`Rrt ๐ฅ2
๐2 +๐ฆ2
๐2 +๐ง2
๐2 โค 1 gSa |
OR
a. Evaluate the following integral by changing the order of integration.
fuEu lekdy esa lekdyu dk รe cnydj eku Kkr dhft, |
โซ โซ ๐ฅ2 cos(๐ฅ2 โ ๐ฅ๐ฆ) ๐๐ฆ ๐๐ฅ1
๐ฆ
1
๐ฆ=0
b. Evaluate โญ ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง๐ฃ
where region of integration is a cylinder v bounded by z=0, z=1, ๐ฅ2 +
๐ฆ2 = 4.
โญ ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง๐ฃ
dk eku Kkr dhft, tgkยก lekdyu {ks= ,d csyu v gS tยจ z=0, z=1, ๐ฅ2 +
๐ฆ2 = 4 }kjk ifjc) gS |
The End
Sophia Girlsโ College (Autonomous)
Ajmer Semester II โ 2021-22
End Semester Examination
Class : B.Sc. Math
Sub : Mathematics
Paper I : [MAT-201]: Vector Calculus and Geometry
Time : 1 ยฝ 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 ๏ฟฝ๏ฟฝ = 5๐ก2๐ โ cos ๐ก๐ ๐๐๐ ๏ฟฝ๏ฟฝ = ๐ก๐ + ๐๐๐ ๐ก๐ then find ๐๐๐ก
(๏ฟฝ๏ฟฝ ยท ๏ฟฝ๏ฟฝ)
;fn ๏ฟฝ๏ฟฝ = 5๐ก2๐ โ cos ๐ก๐ rFkk ๏ฟฝ๏ฟฝ = ๐ก๐ + ๐๐๐ ๐ก๐ rc ๐๐๐ก
(๏ฟฝ๏ฟฝ ยท ๏ฟฝ๏ฟฝ)Kkr dhft,
2. Define Curl of a vector point function.
Lkfnโk fcU/kq Qyu dk dqUry ifjHkkfโkr dhft,A
3. State Gaussโs divergence thorem
xkWl dk vilj.k รงes; 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 vuUrLiโkhZZ;ยกk 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 ifjHkkfโkr dhft, rFkk bldk lehdj.k /kzqoh; :i es fyf[k,A
8. Name the conic represented by x2 + 2xy + y2 โ 2x โ 1 = 0
Lehdj.k x2 + 2xy + y2 โ 2x โ 1 = 0 esa รงLrqr โkkdao dk uke crkb,A
9. Find Radius of sphere x2 + y2 + z2 โ 2x + 4y โ 6z = 11
Xkksys x2 + y2 + z2 โ 2x + 4y โ 6z = 11 dh f+=T;k Kkr djksA
10. Write equation of tangent plane at (ฮฑ,) of sphere x2 + y2 + z2 + 2ux + 2vy + 2wz +d = 0
Xkksys x2 + y2 + z2 + 2ux + 2vy + 2wz +d = 0 dk fcUnq (ฮฑ,) ij LiโkZ lekUrk fyf[k,
11. Write equation of right circular cone.
yEco`rh; โkdqa dk lehdj.k fyf[k,A
12. Define enveloping cylinder.
vUokyksih csyu dks ifjHkkfโkr 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 ๐ + 2๐ + 2๏ฟฝ๏ฟฝ at the
point (1, 2, 0)
fcUnq (1, 2, 0) ij f = xy + yz + zx dk ๐ + 2๐ + 2๏ฟฝ๏ฟฝ dh fnโkk esa fnd~ vodyt Kkr dhft,A
OR
Find div V and curl V if V = (x3 + y3 + z3โ 3xyz)
;fn V = (x3 + y3 + z3โ 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 โkkado dh v{kks ds lehdj.k rFkk yEckj;kยก 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 โkkado dks fu:fir djrsa gSA
๐
๐= 1 + ๐ cos ๐ ๐๐๐
๐
๐= โ1 + ๐ cos ๐
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 โ 2z = 7
xksys x2 + y2 + z2 = 16 dh mu nks LiโkZ ledks ds lehdj.k Kkr djks tks js[kkvks x + y = 5; x โ 2z = 7
ls xqt+jsA
OR
Find equation of enveloping cone of the sphere x2 + y2 + z2 + 2x โ 2 = 0 with its vertex at (1, 1, 1)
xksys x2 + y2 + z2 + 2x โ 2 = 0 ds ml vUokyksi โkakdq dk lehdj.k djks ftldk โkhโkZ 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 ๏ฟฝ๏ฟฝ = ๐ฅ2๐ + ๐ฅ๐ฆ๐ integrated round the square in the plane z =
0, whose sides are along the lines x = y = 0 and x = y = a
Qyu ๏ฟฝ๏ฟฝ = ๐ฅ2๐ + ๐ฅ๐ฆ๐ ds fy, LVkSd รงes; 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 vuqfnโk gSA
OR
If ๐ = ๐ก๐ โ ๐ก2๐ + (๐ก โ 1)๏ฟฝ๏ฟฝ and ๐ = 2๐ก2๐ + 6๐ก๏ฟฝ๏ฟฝ then find value of
;fn ๐ = ๐ก๐ โ ๐ก2๐ + (๐ก โ 1)๏ฟฝ๏ฟฝ rFkk ๐ = 2๐ก2๐ + 6๐ก๏ฟฝ๏ฟฝ rks Kkr dhft,A
a. โซ (๐ ยท ๐ )๐๐ก2
0
b. โซ (๐ ร ๐ )๐๐ก2
0
UNIT II
17. Trace the curve
oร dk vuqjs[k.k dhft,A
x2 + y2 + xy + x + y โ 1 = 0
OR
A circle passing through the focus of a conic whose latus rectum is 2๐ meet the conic in four points
whose distance from the focus are ๐1 , ๐2 , ๐3 ๐๐๐ ๐4 respectively , prove that
,d oRr fdlh โkkado ls xqtjrk gS ftldh ukfHkyEc 2๐ gS] rFkk โkkado dks pkj fcUnqvksa ij feyrk
gS ftudh ukfHk ls yEckb;kW dzeโk ๐1 , ๐2 , ๐3 vkSj ๐4 gS] rks fl) dhft, dh
1
๐1+
1
๐2+
1
๐3+
1
๐4=
2
๐
UNIT III
18. Find the equation of right circular cylinder where guiding curve is the circle x2 + y2 + z2 = 9, x โ 2y
+ 2z = 3
ml yEco`Rrh; csyu dk lehdj.k Kkr djks ftldk funsโkd oRr x2 + y2 + z2 = 9, x โ 2y + 2z = 3
gSA
OR
Find coโordinates of centre and radius of the circle
fuEu o`r dk dsUnz vkSj f=T;k Kjr djksA
X2 + y2โ2yโ4z = 20, x + 2y + 2z = 21
--The End--
Sophia Girlsโ College (Autonomous)
Ajmer Semester โ 2021-22
End Semester Examination (May 2022)
Class : B.Sc. [Maths]
Sub : Mathematics
Paper II : [MAT-202]: Advanced Calculus
Time : 1 ยฝ 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 ๐ = ๐(๐)
pki dh yEckbZ dk vodyt Kkr djus dk lw= /kqoh; oร ๐ = ๐(๐) ds fy, fyf[k,A
2. Define parallel and oblique asymptotes.
lekUrj ,oa fr;Zd vuUrLifโkZ;ksa dh ifjHkkโkk 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 ifjHkkโkk nhft,A
5. Verify the theorem ๐2๐ข
๐๐ฅ๐๐ฆ=
๐2๐ข
๐๐ฆ๐๐ฅ , when ๐ข = ๐ฅ3 + ๐ฆ3 + 3๐๐ฅ๐ฆ
izes; ๐2๐ข
๐๐ฅ๐๐ฆ=
๐2๐ข
๐๐ฆ๐๐ฅ dh iqfโV dhft, tcfd ๐ข = ๐ฅ3 + ๐ฆ3 + 3๐๐ฅ๐ฆ
6. Write the formula for finding second differential coefficient of implicit functions.
vLiโV Qyuksa ds fn`rh; vody xq.kkad Kkr djus dk lw= fyf[k,A
7. Define envelope.
vUokyksi dh ifjHkkโkk 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 Legendreโs duplication formula?
fytsUMj f)xq.ku lw= D;k gS\
10. State Dirichletโs 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
โฌ ๐(๐ฅ, ๐ฆ)๐๐ฅ๐๐ฆ
12. Change the order of integration in the following integral:
fuEufyf[kr lekdy esa lekdyu dk dze cnfy,A
โซ โซ ๐(๐ฅ, ๐ฆ)๐๐ฅ๐๐ฆ๐ฅ
0
๐
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 ๐๐ = ๐๐๐๐๐ ๐๐ , prove that ๐๐
๐๐= ๐(๐ ๐๐ ๐๐)
๐โ1
๐
odz ๐๐ = ๐๐๐๐๐ ๐๐ ds fy, fl) dhft, fd
๐๐
๐๐= ๐(๐ ๐๐ ๐๐)
๐โ1๐
OR
Find the asymptotes of the following curve:
fuEu odz ds vuUrLifโkZ;kW Kkr dhft,A
๐ฅ3 + ๐ฆ3 โ 3๐๐ฅ๐ฆ = 0
UNIT II
14. If ๐ข = ๐ก๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅโ๐ฆ), then prove that ๐ฅ
๐๐ข
๐๐ฅ+ ๐ฆ
๐๐ข
๐๐ฆ= ๐ ๐๐2๐ข
;fn ๐ข = ๐ก๐๐โ1 (๐ฅ3+๐ฆ3
๐ฅโ๐ฆ) rc ๐ฅ
๐๐ข
๐๐ฅ+ ๐ฆ
๐๐ข
๐๐ฆ= ๐ ๐๐2๐ข fl) dhft,A
OR
Find the envelope of the family of ellipse
๐ฅ2
๐2 +๐ฆ2
๐2 = 1 ๐คโ๐๐ ๐ + ๐ = ๐ , c being a constant.
nh?kZo`Rrksa ๐ฅ2
๐2 +๐ฆ2
๐2 = 1 ds dqy dk vUokyksi Kkr tcfd ๐ + ๐ = ๐ tgkW ๐ vpj gSaA
UNIT III 15. Evaluate Kkr dhft,
โซ ๐ฅ๐โ11
0(๐๐๐
1
๐ฅ)
๐โ1
๐๐ฅ = ฮ(๐)
๐๐ ; ๐, ๐ > 0
OR
Evaluate eku Kkr dhft,
โซ โซ โซ ๐ฅ๐ฆ๐ง ๐๐ฅ ๐๐ฆ ๐๐ง
where the region of integration is the volume of the ellipsoid in the positive octant.
tgkW lekdyu dk {ks= /kukRed vCVkaโkd esa nh/kZo`=t gSA
๐ฅ2
๐2+
๐ฆ2
๐2+
๐ง2
๐2โค 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
๐2 = ๐2 cos 2๐
OR
Show that the radius of curvature at a point (๐ ๐๐๐ 3๐, ๐๐ ๐๐3๐) on the curve ๐ฅ2
3 + ๐ฆ2
3 = ๐2
3 is 3๐
2sin 2๐
fl) dhft, odzrk f=T;k fcUnq (๐ ๐๐๐ 3๐, ๐๐ ๐๐3๐) ] ij odz ๐ฅ2
3 + ๐ฆ2
3 = ๐2
3 dh 3๐
2sin 2๐ gSA
UNIT II
17. If ๐ข = ๐ฅ๐ ๐๐โ1 (๐ฆ
๐ฅ) then prove that
๐ฅ2๐2๐ข
๐๐ฅ2+ 2๐ฅ๐ฆ
๐2๐ข
๐๐ฅ๐๐ฆ+ ๐ฆ2
๐2๐ข
๐๐ฆ2= 0
;fn ๐ข = ๐ฅ๐ ๐๐โ1 (๐ฆ
๐ฅ) rc fl) dhft,A
๐ฅ2๐2๐ข
๐๐ฅ2+ 2๐ฅ๐ฆ
๐2๐ข
๐๐ฅ๐๐ฆ+ ๐ฆ2
๐2๐ข
๐๐ฆ2= 0
OR
Find the maxima and minima of ๐ข = ๐ฅ2 + ๐ฆ2 + ๐ง2 subject to the conditions ๐๐ฅ2 + ๐๐ฆ2 + ๐๐ง2 =
1 ๐๐๐ ๐๐ฅ + ๐๐ฆ + ๐๐ง = 0
izfrcU/kks ๐๐ฅ2 + ๐๐ฆ2 + ๐๐ง2 = 1 rFkk ๐๐ฅ + ๐๐ฆ + ๐๐ง = 0 ds vUrZxr ๐ข = ๐ฅ2 + ๐ฆ2 + ๐ง2 ds
mfPpโB vkSj fufEuโB Kkr dhft,A
UNIT III
18. Evaluate the following integral by changing the order:
fuEu lekdy dk dze cnydj eku Kkr dhft,A
โซ โซ๐โ๐ฆ
๐ฆ
โ
๐ฅ
โ
0
๐๐ฅ ๐๐ฆ
OR
Evaluate the following integral by changing to polar coordinates:
fuEu lekdy dks /kqzohZ; funsZโkkadksesa ifjofrZr dj eku Kkr dhft,A
โซ โซ โ๐ฅ2 + ๐ฆ2โ2๐ฅโ๐ฅ2
๐ฅ
1
0๐๐ฅ ๐๐ฆ
--The End--