Sophia Girls' College, Ajmer

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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 = | โ†’ | , where โ†’ = xi + yj + jk. Prove that ( 1 )=โˆ’ ;fn r = | โ†’ | tgkยก โ†’ = xi + yj + jk fl) dhft, ( 1 )=โˆ’ Define directional Derivative. fnd vodyt dยจ ifjHkkfโ€œkr dhft, | 2. Define curl of a vector point function. lfnโ€˜k Qyu dยข dyZ dยจ ifjHkkfโ€œkr dhft, | 3. State Stokeโ€™s Theorem. LVยจDl รes; dk dFku fyf[k, | 4. Define Line Integral. js [kk lekdy dยจ ifjHkkfโ€œkr dhft, | 5. Define conic section. โ€˜kkado dยจ ifjHkkfโ€œkr dhft, | 6. Write equation of the conic referred to centre as origin. ewy fcUnq dยจ dยขUรฆ ekurs gq , โ€˜kkado 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 bdkรƒ ys rs gq, lehdj.k Kkr dhft, | 8. Define cone and right circular cone. โ€˜kadq ,oa yEcdยจ.kh; โ€˜kadq dยจ ifjHkkfโ€œkr dhft, | 9. Define reciprocal cone. O;q Rร˜e โ€˜kadq dยจ ifjHkkfโ€œkr 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 โ†’ =xi+yj+zk and โ†’ is any constant vector then prove that 3 = โˆ’ 3 + 3(.) 5 ;fn r =xi+yj+zk ,oa โ†’ o ,d vpj lfnโ€˜k gS rยจ fl) dhft, 3 = โˆ’ 3 + 3(.) 5 11. Prove that div(axb) = b.curl a โ€“ a.curl b

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 = |

๐‘Ÿโ†’ | , where

๐‘Ÿโ†’ = xi + yj + jk. Prove that

๐‘”๐‘Ÿ๐‘Ž๐‘‘ (1

๐‘Ÿ) = โˆ’

๐‘Ÿ

๐‘Ÿ

;fn r = | ๐‘Ÿโ†’ | tgkยก

๐‘Ÿโ†’ = xi + yj + jk fl) dhft,

๐‘”๐‘Ÿ๐‘Ž๐‘‘ (1

๐‘Ÿ) = โˆ’

๐‘Ÿ

๐‘Ÿ

Define directional Derivative.

fnd vodyt dยจ ifjHkkfโ€œkr dhft, |

2. Define curl of a vector point function.

lfnโ€˜k Qyu dยข dyZ dยจ ifjHkkfโ€œkr dhft, |

3. State Stokeโ€™s Theorem.

LVยจDl รes; dk dFku fyf[k, |

4. Define Line Integral.

js[kk lekdy dยจ ifjHkkfโ€œkr dhft, |

5. Define conic section.

โ€˜kkado dยจ ifjHkkfโ€œkr dhft, |

6. Write equation of the conic referred to centre as origin.

ewy fcUnq dยจ dยขUรฆ ekurs gq, โ€˜kkado 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 bdkรƒ ysrs gq, lehdj.k Kkr dhft, |

8. Define cone and right circular cone.

โ€˜kadq ,oa yEcdยจ.kh; โ€˜kadq dยจ ifjHkkfโ€œkr dhft, |

9. Define reciprocal cone.

O;qRร˜e โ€˜kadq dยจ ifjHkkfโ€œkr 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 ๐‘Ÿโ†’ =xi+yj+zk and

๐‘Žโ†’ is any constant vector then prove that

๐‘๐‘ข๐‘Ÿ๐‘™๐‘Ž๐‘ฅ๐‘Ÿ

๐‘Ÿ3 = โˆ’๐‘Ž

๐‘Ÿ3 +3(๐‘Ž.๐‘Ÿ)๐‘Ÿ

๐‘Ÿ5

;fn r =xi+yj+zk ,oa ๐‘Žโ†’ o ,d vpj lfnโ€˜k gS

rยจ fl) dhft, ๐‘๐‘ข๐‘Ÿ๐‘™๐‘Ž๐‘ฅ๐‘Ÿ

๐‘Ÿ3 = โˆ’๐‘Ž

๐‘Ÿ3 +3(๐‘Ž.๐‘Ÿ)๐‘Ÿ

๐‘Ÿ5

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 โˆซ๐น.๐‘‘๐‘Ÿ

๐‘, where

๐นโ†’= (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, โˆซ๐น.๐‘‘๐‘Ÿ

๐‘, tgkยก

๐นโ†’= (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 โ€˜kkado dk dยขUรฆ Kkr dhft, ,oa โ€˜kkado 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‘Ž

๐‘ฅ +

๐‘

๐‘ฆ +

๐‘

๐‘ง = 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‘Ž

๐‘ฅ +

๐‘

๐‘ฆ +

๐‘

๐‘ง =

2

16. Find the equation of the cylinder whose generators are parallel to the line ๐‘ฅ

1 =

๐‘ฆ

โˆ’2 =

๐‘ง

3 and whose guiding curve

is the ellipse x2 + 2y2 =1 , z = 0

ml csyu dk lehdj.k Kkr dhft, ftldk tud ๐‘ฅ

1 =

๐‘ฆ

โˆ’2 =

๐‘ง

3 dยข lekUrj gS rFkk ftldk funZsโ€˜kd 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 lfnโ€˜k i-3j+2k at t=1 dยข vuqfnโ€˜k ij ?kVd Kkr dhft, |

b) Prove that

fl) dhft, |

โˆ‡2 f (r) = f (r)+ 2

๐‘Ÿ f (r)

Where r = | ๐‘Ÿโ†’ | and

๐‘Ÿโ†’ = xi+yj+ zk

OR

Verify Stokeโ€™s theorem for the function F=zi + xj + yk where curve C is the unit circle in xy plane bounding

the hemisphere.

Z = 1 โˆ’ โˆš1 โˆ’ ๐‘ฅ2 โˆ’ ๐‘ฆ2

LVยจDl รes; dk lR;kiu dhft, tgkยก F=zi + xj + yk rFkk C v)Zxยจys dยจ Z = 1 โˆ’ โˆš1 โˆ’ ๐‘ฅ2 โˆ’ ๐‘ฆ2 ifjc) djus okyk

v)Zo`r gS |

UNIT II

18. A) Discuss the tracing of conic ๐‘™

๐‘Ÿ = 1 + e cos ๐œƒ

โ€˜kkado ๐‘™

๐‘Ÿ = 1 + e cos ๐œƒ dยข fp=.k dh foospuk dhft, |

OR

Find the asymptotes of the following hyperbolas and equation to their conjugate hyperbola fuEu vfrijoy; dh vUUr Liโ€˜kรŠ 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 funZsโ€˜kkad 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 โ€˜kadq ax2+by2+cz2+2fyz+2gzx+2hxy=0dยข rhu ijLij yEcor Liโ€˜kZ 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 Liโ€˜kรŠ rFkk /kzoh v?k% Liโ€˜kรŠ dh yEckรƒ fyf[k;s |

2. Write the condition for the existence of the asymptote.

vuUr Liโ€˜kรŠ dยข vfLrRo dยข fy, รfrcU/k fyf[k;s|

3. Write the coordinates of centre of curvature.

oร˜rk dยขUรฆ dยข funZsโ€˜kkad fyf[k;s | 4. Write the condition for a point to be a point of inflextion.

,d fcUnq dยข ufr ifjorZu fcUnq gยจus dh โ€˜krZ fyf[k;s |

5. Define partial differential coefficient.

vakfโ€˜kd vody xq.kkad dยจ ifjHkkfโ€œkr dhft, |

6. If u = ๐‘ฅ 2 and v=y2 then find the value of ๐œ•(๐‘ข,๐‘ฃ)

๐œ•(๐‘ฅ,๐‘ฆ)

;fn u = ๐‘ฅ 2 rFkk v=y2

rยจ ๐œ•(๐‘ข,๐‘ฃ)

๐œ•(๐‘ฅ,๐‘ฆ) dk eku Kkr dhft, |

7. Write the necessary conditions for f(a,b) to be an extreme value of f(๐‘ฅ,y)

Qyu f (๐‘ฅ,y) dยข pjeeku f (a,b) gยจus dk vkoโ€˜;d รfrcU/k fyf[k;s |

8. Show that

รnร†โ€˜kr dhft, |

โˆซ1

0 ๐‘ฅ n-1 (log

1

๐‘ฅ )m-1 d ๐‘ฅ =

โˆš(๐‘š)

๐‘›๐‘š , m , n > 0

9. Write double integral in polar coordinates.

/kqzoh funZsโ€˜kkadยจa esa f}lekdy dยจ fyf[k;s |

10. State Liouvilles extension of Dirichletโ€™s 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‘Ÿ)

๐‘“(๐‘Ÿ)

รnร†โ€˜kr dhft, fd oร˜ p=f(r) dh /kzo ls xqtjus okyh oร˜rk thok dh yEckรƒ 2๐‘“(๐‘Ÿ)

๐‘“(๐‘Ÿ) gS |

12. Find all the asymptotes of the following curve:

fuEu oร˜ dh lHkh vuUr Liร†โ€˜k;kยก Kkr dhft, |

(y-2๐‘ฅ)2(3๐‘ฅ +4y) + 3(y-2 ๐‘ฅ)(3 ๐‘ฅ +4y) โ€“ 5 =0

UNIT II

13. If u = sin-1 ๐‘ฅ+2๐‘ฆ+3๐‘ง

๐‘ฅ 8+ ๐‘ฆ8 + ๐‘ง8 find the value of

๐‘ฅ ๐œ•๐‘ข

๐œ•๐‘ฅ + y

๐œ•๐‘ข

๐œ•๐‘ฆ + z

๐œ•๐‘ข

๐œ•๐‘ง

;fn u = sin-1 ๐‘ฅ+2๐‘ฆ+3๐‘ง

๐‘ฅ 8+ ๐‘ฆ8 + ๐‘ง8 rยจ ๐‘ฅ ๐œ•๐‘ฅ

๐œ•๐‘ฆ + y

๐œ•๐‘ข

๐œ•๐‘ฆ + z

๐œ•๐‘ข

๐œ•๐‘ง

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 oร˜rk f=T;kvยจa dk vUrj rFkk mldยข dยขUรฆt dยข laxr fcUnqvยจa dยข chp dยข pki dh yEckรƒ dk

vuqikr Kkr dhft, |

UNIT III

15. If โˆซ1

0 ๐‘ฅ m (1- ๐‘ฅ)n d ๐‘ฅ =โˆซ

1

0 ๐‘ฅ n (1- ๐‘ฅ)p, then find the value of p.

;fn โˆซ1

0 ๐‘ฅ m (1- ๐‘ฅ)n d ๐‘ฅ =โˆซ

1

0 ๐‘ฅ n (1- ๐‘ฅ)p, rยจ p dk eku Kkr dhft, |

16. Evaluate the following integral by changing the order of integration.

fuEu lekdy dk ร˜e cnydj eku Kkr dhft, |

โˆซ1

0

โˆซ๐‘’

๐‘’๐‘ฅ

๐‘‘๐‘ฆ๐‘‘๐‘ฅ

log ๐‘ฆ

17. Evaluate

eku Kkr dhft, |

โˆซ1

โˆ’1 โˆซ๐‘ง

0 โˆซ๐‘ฅ+๐‘ง

๐‘ฅโˆ’๐‘ง (๐‘ฅ +y+z) d ๐‘ฅ 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 Liร†โ€˜k;kยก Kkr dhft, |

(2 ๐‘ฅ -3y+1)2 (๐‘ฅ +y) = 8 ๐‘ฅ โ€“ 2y + 9

b. Find the angle of intersection of the following cardioids : fuEu รขn;ke dk รfrPNsnu dยจ.k Kkr dhft,

r = a(1+ Cos ๐œƒ) and r = b(1-Cos ๐œƒ)

OR

a. Show that the circle of curvature at the origin of the parabola y=mx+( ๐‘ฅ2

๐‘Ž ) is ๐‘ฅ

2+y2=a(1+m)2(y-m ๐‘ฅ).

รnร†โ€˜kr dhft, fd ewy fcUnq ij i[kyo y=m ๐‘ฅ +( ๐‘ฅ2

๐‘Ž ) dk oร˜rk o`r ๐‘ฅ2+y2=a(1+m)2(y-m ๐‘ฅ) gS |

b. Examine the nature of the origin on the following curve: fuEu oร˜ ij ewy fcUnq dh รร‘fr dh tkยกp dhft,

๐‘ฅ 7+2 ๐‘ฅ 4+2 ๐‘ฅ 3y+ ๐‘ฅ 2+2 ๐‘ฅ y+y2=0

UNIT II

19. a. If ๐‘ฅ +y = 2eฦŸ Cos โˆ… and ๐‘ฅ -y = 2ieฦŸ Sin โˆ…

show that ๐œ•2๐‘ข

๐œ•๐œƒ2 + ๐œ•2๐‘ข

๐œ•โˆ…2 = 4xy ๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ

;fn ๐‘ฅ+y = 2eฦŸ Cos โˆ… and ๐‘ฅ-y = 2ieฦŸ Sin โˆ…

rยจ รnร†โ€˜kr dhft,

๐œ•2๐‘ข

๐œ•๐œƒ2 + ๐œ•2๐‘ข

๐œ•โˆ…2 = 4xy ๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ

b. if u = ๐‘ฅ โˆš(1 โˆ’ ๐‘ฆ)2 + y โˆš(1 โˆ’ ๐‘ฅ)2 ; v = sin-1 ๐‘ฅ sin-1y

Show that u, v are functionally related and find the relationship.

;fn u = ๐‘ฅ โˆš(1 โˆ’ ๐‘ฆ)2 + y โˆš(1 โˆ’ ๐‘ฅ)2 ; v = sin-1 ๐‘ฅ sin-1y

รnร†โ€˜kr 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; โˆ being the parameter:

fuEu ljy js[kk dยข dqy dk vUokyยจi Kkr dhft, tgkยก โˆ รkpy gS |

a xsec โˆ- bycosec โˆ = a2-b2

b. Find the points where the function x3+y3-3axy has maximum or minimum value.

mu fcUnqvยจa dยจ Kkr dhft, tgka Qyu ๐‘ฅ 3+y3-3axy dk eku mPpre ;k U;wure gS |

UNIT III

20. Prove that fl) dhft,

โˆซโˆž

0

๐‘ฅ

1+๐‘ฅ6 dx = ๐‘ฅ

3โˆš3

a. โˆš(โˆ’1

2 )

b. โˆซ โˆš(1 โˆ’ ๐‘ฅ)1

0 dx

OR

a. Change the order of integration in the following integral: fuEu lekdy esa lekdyu dk ร˜e ifjorZu dhft,

โˆซ4๐‘Ž

0 โˆซ2โˆš(๐‘Ž๐‘ฅ)

๐‘ฅ2

4๐‘Ž

๐‘“ (๐‘ฅ,y) d๐‘ฅ dy

b. Evaluate eku Kkr dhft,

โˆญ ๐‘ฅ๐‘ฆ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง ,

Where region of integration is the complete ellipsoid :

tgkยก lekdy dk {ks= lEiw.kZ fn/kZo`Rrt gS

๐‘ฅ2

๐‘Ž2 + ๐‘ฆ2

๐‘2 + ๐‘ง2

๐‘2 โ‰ค 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 ifjHkkโ€œkk nhft,

2. Define double point. f}d fcUnq dh ifjHkkโ€œkk nhft,

3. Write the formula to find the radius of curvature for Cartesian curves. dkrรŠ; oร˜ dยข fy, oร˜rk f=T;k Kkr djus dk lw= fyf[k,

4. Write the degree of homogenous function

๐‘ข = ๐‘ฅ๐‘ฆ๐‘“(๐‘ฆ

๐‘ฅ).

lEk/kkr Qyu ๐‘ข = ๐‘ฅ๐‘ฆ๐‘“(๐‘ฆ

๐‘ฅ) dh /kkr fyf[k,

5. Define Envelope. vUokyยจi dh ifjHkkโ€œkk 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 รfrcU/k fyf[k,

7. Define Gamma Function. xkek Qyu dh ifjHkkโ€œkk nhft,

8. Evaluate. eku Kkr dhft, โˆซ โˆš๐‘ฅ. ๐‘’โˆ’๐‘ฅ3๐‘‘๐‘ฅ

โˆž

0

9. State Dirichletโ€™s 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.

๐‘ฆ2 = 4๐‘Ž(๐‘ฅ + ๐‘Ž)

ijoy; ๐‘ฆ2 = 4๐‘Ž(๐‘ฅ + ๐‘Ž) dk ifnd lehdj.k Kkr dhft,

OR

Find the asymptotes of the following curve: fuEu oร˜ dยข vUuRkLiร†โ€˜k;kยก Kkr dhft,

x3 + yx2 - xy2 - y3 - 3x โ€“ y โ€“ 1 = 0

UNIT II

12. If ;fn ๐‘ข = ๐‘ ๐‘–๐‘›โˆ’1 (โˆš๐‘ฅโˆ’โˆš๐‘ฆ

โˆš๐‘ฅ+โˆš๐‘ฆ) then show that. rc รnร†โ€˜kr dhft, fd

๐‘ฅ๐œ•๐‘ข

๐œ•๐‘ฅ+ ๐‘ฆ

๐œ•๐‘ข

๐œ•๐‘ฆ0

OR

Find the envelope of the circles drawn upon the central radii of the ellipse ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1as diameter.

mu o`Rrยจa dk vUokyยจi Kkr dhft, tยจ nh/kZoRr ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1 dh /kqzokUrr js[kkvยจa dยจ O;kl ekudj [khps x, gSa

UNIT III

13. To show that รnร†โ€˜kr dhft,

โˆš๐‘š โˆš๐‘š + 12โ„ =

โˆš๐œ‹

22๐‘šโˆ’1 โˆš2๐‘š ; ๐‘š โˆˆ 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 ร˜e ifjoร†rr dhft, โˆซ โˆซ ๐‘“(๐‘ฅ, ๐‘ฆ)โˆš๐‘Ž๐‘ฅ

2

๐‘ฅ24๐‘Žโ„

๐‘‘๐‘ฅ ๐‘‘๐‘ฆ4๐‘Ž

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

๐‘ฆ2(๐‘Ž + ๐‘ฅ) = ๐‘ฅ2(3๐‘Ž๐‘ฅ)

OR

(a) Show that for the radius of curvature at a point (๐‘Ž ๐‘๐‘œ๐‘ 3๐œƒ, ๐‘Ž ๐‘ ๐‘–๐‘›3๐œƒ) on the curve

๐‘ฅ2

3โ„ + ๐‘ฆ2

3โ„ = ๐‘Ž2

3โ„ is 3๐‘Ž

2๐‘ ๐‘–๐‘›2๐œƒ.

fl) dhft, fd oร˜ ๐‘ฅ2

3โ„ + ๐‘ฆ2

3โ„ = ๐‘Ž2

3โ„ dยขa fcUnq (๐‘Ž ๐‘๐‘œ๐‘ 3๐œƒ, ๐‘Ž ๐‘ ๐‘–๐‘›3๐œƒ) ij oร˜rk f=T;k

3๐‘Ž

2๐‘ ๐‘–๐‘›2๐œƒ gยจrh gS

(b) Find the point of inflection of the curve ๐‘ฆ(๐‘Ž2 + ๐‘ฅ2) = ๐‘ฅ3.

oร˜ ๐‘ฆ(๐‘Ž2 + ๐‘ฅ2) = ๐‘ฅ3 dยข ufr ifjorZu fcUnq Kkr dhft,

UNIT II

15. (a) If ;fn ๐‘ข = ๐‘ฅ ๐‘ ๐‘–๐‘›โˆ’1 ๐‘ฆ

๐‘ฅโ„ then prove that (rc fl) dhft, ) ๐‘ฅ2๐œ•2๐‘ข

๐œ•๐‘ฅ2 +2๐‘ฅ๐‘ฆ๐œ•2๐‘ฅ

๐œ•๐‘ฅ ๐œ•๐‘ฆ+ ๐‘ฆ2 ๐œ•2๐‘ข

๐œ•๐‘ฆ2 = 0

(b) If ;fn ๐‘‰ = ๐‘“(๐‘ฅ โˆ’ ๐‘ฆ, ๐‘ฆ โˆ’ ๐‘ง, ๐‘ง โˆ’ ๐‘ฅ)then prove that rc fl) dhft, ๐œ•๐‘‰

๐œ•๐‘ฅ+

๐œ•๐‘‰

๐‘ฆ+

๐œ•๐‘‰

๐œ•๐‘ง= 0

OR

(a) Find the envelope of the family of ellipse

๐‘ฅ2

๐‘Ž2+

๐‘ฆ2

๐‘2= 1

When a + b = c, c being a constant.

nh/kZorยจ ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 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.

รnร†โ€˜kr dhft, fd fuEu Qyu dk fufEuโ€œV eku 3a2 gS%

๐‘ข = ๐‘ฅ๐‘ฆ + ๐‘Ž3(1

๐‘ฅ+

1

๐‘ฆ)

UNIT III

15. (a) Prove that :- fl) dhft, dh

โˆซ(๐‘ฅ โˆ’ ๐‘Ž)๐‘šโˆ’1(๐‘ โˆ’ ๐‘ฅ)๐‘›โˆ’1 ๐‘‘๐‘ฅ

๐‘

๐‘Ž

= (๐‘ โˆ’ ๐‘Ž)๐‘š+๐‘›โˆ’1 ๐›ฝ(๐‘š, ๐‘›)

(b) Show that :- รnร†โ€˜kr dhft, dh

โˆš๐‘› โˆš1 โˆ’ ๐‘› =๐œ‹

sin ๐‘› ๐œ‹

OR

(a) Evaluate :- eku Kkr dhft,

โˆซ โˆซ ๐‘ฅ๐‘ฆ(๐‘ฅ + ๐‘ฆ)๐‘‘๐‘ฅ ๐‘‘๐‘ฆ 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,

โˆซ โˆซ โˆซ ๐‘’๐‘ฅ+๐‘ฆ+๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง๐‘ฅ+log ๐‘ฆ

0

๐‘ฅ

0

๐‘™๐‘œ๐‘”2

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 vfnโ€˜k fcUnq Qyu vpj gS ;fn grad f=0 |

b. If ๐‘Ÿ = |๐‘Ÿโ†’| where

๐‘Ÿโ†’ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + 2๐‘˜ prove that (fl) dhft, ) grad ๐‘Ÿ = ๐‘Ÿ

^ |

c. If ๐‘Ÿโ†’ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + 2๐‘˜, prove that div

๐‘Ÿโ†’ = 3

;fn ๐‘Ÿโ†’ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + 2๐‘˜, rยจ fl) dhft, fd div

๐‘Ÿโ†’ = 3

d. State Gaussโ€™ theorem.

xkWl รes; dk dFku fyf[k, |

e. Define conic section.

โ€˜kadq ifjPNsn dยจ ifjHkkfโ€œkr 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 โ€˜kadq dยจ ifjHkkfโ€œkr dhft, |

j. Define Right circular cylinder.

yEcoRrh; csyu dยจ ifjHkkfโ€œkr 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 ๐‘Ÿโ†’ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + 2๐‘˜ and ๐‘Ÿ = |

๐‘Ÿโ†’| , prove (fl) dhft,) that curl ๐‘Ÿ๐‘›

๐‘Ÿโ†’ = 0.

OR

Prove that fl) dhft, |

๐‘‘๐‘–๐‘ฃ(๐‘ข๐‘Ž) = ๐‘ข. ๐‘‘๐‘–๐‘ฃ ๐‘Ž + ๐‘Ž. ๐‘”๐‘Ÿ๐‘Ž๐‘‘ ๐‘ข

UNIT II

12. (a) Find the co-ordinates of the centre of the following conic :

fuEu โ€˜kado dยข dยขUรฆ dยข funZsโ€˜kkad 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 โ€˜kkado 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 2๐‘Ž๐‘

โˆš๐‘Ž2+๐‘2.

fl) djยจ fd ijoy; (a2+b2) (x2+y2) = (bx+ay-ab)2 dh ukfHkyEc

2๐‘Ž๐‘

โˆš๐‘Ž2+๐‘2 gยจrh gS |

UNIT III

13. Two spheres of radii r1 and r2 cut orthogonally: Prove the radius of their common circle is. ๐‘Ÿ1๐‘Ÿ2

โˆš๐‘Ÿ12 + ๐‘Ÿ2

2

r1 rFkk r2 f=T;k dยข nยจ xยจys ykfEod #i ls dkVrs gS fl) dhft, fd mHk;fuโ€œB o`Rr dh f=T;k gS | ๐‘Ÿ1๐‘Ÿ2

โˆš๐‘Ÿ12 + ๐‘Ÿ2

2

OR

If ๐‘ฅ

1=

๐‘ฆ

2=

๐‘ง

3 represent one of the three mutually perpendicular generators of then cone 5yz- 8xz - 3xy =

0 find the equation of the other two.

;fn ๐‘ฅ

1=

๐‘ฆ

2=

๐‘ง

3 โ€˜kadq 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 vfnโ€˜k fcUnq Qyu gยจ rยจ fl) dhft,

๐‘”๐‘Ÿ๐‘Ž๐‘‘ (๐‘“

๐‘”) =

๐‘”(๐‘”๐‘Ÿ๐‘Ž๐‘‘ ๐‘“)โˆ’๐‘“(๐‘”๐‘Ÿ๐‘Ž๐‘‘ ๐‘” )

๐‘”2, 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`โ€œB x2+y2+z2=25 dยข fcUnq (4,0,3) ij Liโ€˜kZ

Rky rFkk vfHkyEc dยข Lkehdj.k Kkr dhft, |

OR

(a) If ๐‘Ÿโ†’ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + 2๐‘˜ and ๐‘Ÿ = |

๐‘Ÿโ†’| , prove (fl) dhft,) that

div ๐‘Ÿ๐‘›

๐‘Ÿโ†’ = (๐‘› + 3)

๐‘Ÿโ†’ ๐‘›

(b) Evaluate (eku Kkr dhft,)

โˆซ (๐‘ฆ๐‘ง๐‘‘๐‘ฅ + (๐‘ฅ๐‘ฆ + 1)๐‘‘๐‘ฆ + ๐‘ฅ๐‘ฆ๐‘‘๐‘ง) Where c is any path form (1,0,0) to (2,1,4)

tgka c fcUnq (1,0,0) ls (2,1,4) rd dยจรƒ 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 รnร†โ€˜kr โ€˜kkado dh รร‘fr 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 ๐‘Žโˆš2, ๐‘โˆš2, ๐‘โˆš2 respectively. Find its equation.

,d xยจys dk dยขUรฆ /ku vโ€œVkaโ€˜kd esa gS vยฉj og ewy fcUnq ls xqtjrk gS funZsโ€˜kh lery x=y=z=0 ls bldk รfrPNsn

ร˜eโ€˜k% ๐‘Žโˆš2, ๐‘โˆš2, ๐‘โˆš2 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 โ€˜kadq dk lehdj.k Kkr dhft, ftldk โ€˜khโ€œkZ fcUnq (1,1,1) gS |

(b) Find the equation of right circular cylinder whose guiding curve is the circle

๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 9, x - 2y + 2z = 3

ml yEcoRrh; csyu dk lehdj.k Kkr dhft, ftLkdk funZsโ€˜kd oRr ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 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 โˆ‡๐‘“, ๐‘–๐‘“ ๐‘“(๐‘ฅ, ๐‘ฆ, ๐‘ง) = ๐‘ฅ2๐‘ฆ + ๐‘ฆ2๐‘ฅ + ๐‘ง2

;fn ๐‘“(๐‘ฅ, ๐‘ฆ, ๐‘ง) = ๐‘ฅ2๐‘ฆ + ๐‘ฆ2๐‘ฅ + ๐‘ง2 , โˆ‡๐‘“ eku Kkr dhft,A

2. Define Divergence of a vectors and Solenoidal vector.

Lkfnโ€™k dk vilj.k o ifjufydk lfnโ€™k dks ifjHkkf"kr dhft,A

3. Find the divergence of ๏ฟฝ๏ฟฝ. where ๏ฟฝ๏ฟฝ = ๐‘ฅ๐‘ฆ๐‘ง๐‘– + 3๐‘ฅ2๐‘ฆ๐‘— + (๐‘ฅ๐‘ง2 โˆ’ ๐‘ฆ2๐‘ง)๏ฟฝ๏ฟฝ

๏ฟฝ๏ฟฝ dk vilj.k Kkr dhft,] tgkยก ๏ฟฝ๏ฟฝ = ๐‘ฅ๐‘ฆ๐‘ง๐‘– + 3๐‘ฅ2๐‘ฆ๐‘— + (๐‘ฅ๐‘ง2 โˆ’ ๐‘ฆ2๐‘ง)๏ฟฝ๏ฟฝ

4. State Greenโ€™s theorem.

xzhu izes; dk dFku fyf[k,A

5. Write the necessary conditions for conic ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + 2โ„Ž๐‘ฅ๐‘ฆ + 2๐‘”๐‘ฅ + 2๐‘“๐‘ฆ + ๐‘ = 0 to represents an

equation of ellipse.

'kkod ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + 2โ„Ž๐‘ฅ๐‘ฆ + 2๐‘”๐‘ฅ + 2๐‘“๐‘ฆ + 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 ๐‘Ž๐‘ฅ2 + 2โ„Ž๐‘ฅ๐‘ฆ + ๐‘๐‘ฆ2 = 1 in terms of the length ๐‘Ÿ of its

semi axis.

'kkdo ๐‘Ž๐‘ฅ2 + 2โ„Ž๐‘ฅ๐‘ฆ + ๐‘๐‘ฆ2 = 1 ds v/kZ&v{k dh yEckbZ ๐‘Ÿ ds :Ik mlds v{k dh lehdj.k fyf[k,A

8. Define Reciprocal cone.

O;qRร˜e 'kdq dks ifjHkkf"kr dhft,A

9. Define Cylinder.

csyu dks ifjHkkf"kr dhft,A

10. Find the centre and radius of the sphere 7๐‘ฅ2 + 7๐‘ฆ2 + 7๐‘ง2 โˆ’ 6๐‘ฅ โˆ’ 3๐‘ฆ โˆ’ 2๐‘ง = 0

xksy 7๐‘ฅ2 + 7๐‘ฆ2 + 7๐‘ง2 โˆ’ 6๐‘ฅ โˆ’ 3๐‘ฆ โˆ’ 2๐‘ง = 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‘Ÿ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ then show that : ๐‘๐‘ข๐‘Ÿ๐‘™ [(๐‘Ÿ ร— ๏ฟฝ๏ฟฝ) ร— ๏ฟฝ๏ฟฝ] = ๏ฟฝ๏ฟฝ ร— ๏ฟฝ๏ฟฝ

;fn ๏ฟฝ๏ฟฝ o ๏ฟฝ๏ฟฝ vpj lfnโ€™k gS o ๐‘Ÿ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ rks fl) dhft, fd ๐‘๐‘ข๐‘Ÿ๐‘™ [(๐‘Ÿ ร— ๏ฟฝ๏ฟฝ) ร— ๏ฟฝ๏ฟฝ] = ๏ฟฝ๏ฟฝ ร— ๏ฟฝ๏ฟฝ

OR

Prove that :

fl) dhft, fd&

๐‘‘๐‘–๐‘ฃ (๐‘Ÿ

๐‘Ÿ3) = 0

Where ๐‘Ÿ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ

UNIT II

12. Find the condition so that the straight line ๐‘™

๐‘Ÿ= ๐ด ๐‘๐‘œ๐‘ ๐œƒ + ๐ต ๐‘ ๐‘–๐‘›๐œƒ may touch the conic

๐‘™

๐‘Ÿ= 1 +

๐‘’ cos(๐œƒ โˆ’ ๐›ผ).

izfrcU/k Kkr dhft, tc fd ljy js[kk ๐‘™

๐‘Ÿ= ๐ด ๐‘๐‘œ๐‘ ๐œƒ + ๐ต ๐‘ ๐‘–๐‘›๐œƒ 'kkado

๐‘™

๐‘Ÿ= 1 + ๐‘’ cos(๐œƒ โˆ’ ๐›ผ) dks

Liโ€™kZ djrh gSaA

OR

What conic does equation 13๐‘ฅ2 โˆ’ 18๐‘ฅ๐‘ฆ + 37๐‘ฆ2 + 2๐‘ฅ + 14๐‘ฆ โˆ’ 2 = 0 represent? Find its centre

and the equation to the conic referred to the centre as origin.

lehdj.k 13๐‘ฅ2 โˆ’ 18๐‘ฅ๐‘ฆ + 37๐‘ฆ2 + 2๐‘ฅ + 14๐‘ฆ โˆ’ 2 = 0 dkSuls 'kkado dks fu:frr djrk gSa\ blds

dsUnz ds funsZโ€™kkad 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 ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 5๐‘ฅ โˆ’ 7๐‘ฆ + 2๐‘ง โˆ’ 8 = 0,3๐‘ฅ โˆ’

2๐‘ฆ + 4๐‘ง + 3 = 0 at the point (โˆ’3,5,4) .

o`Rr ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 5๐‘ฅ โˆ’ 7๐‘ฆ + 2๐‘ง โˆ’ 8 = 0, 3๐‘ฅ โˆ’ 2๐‘ฆ + 4๐‘ง + 3 = 0 ds fcUnq (โˆ’3,5,4) ij Liโ€™kZ

js[kk ds lehdj.k Kkr dhft,A

OR

Find the equation of the sphere having the circle ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 10๐‘ฆ โˆ’ 4๐‘ง = 8, ๐‘ฅ + ๐‘ฆ + ๐‘ง = 3 as

a great circle.

xksys dk lehdj.k Kkr dhft, ftldk or ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 10๐‘ฆ โˆ’ 4๐‘ง = 8, ๐‘ฅ + ๐‘ฆ + ๐‘ง = 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 = ๐‘ฅ๐‘ฆ๐‘– + ๐‘ง2๐‘— + 2๐‘ฆ๐‘ง๏ฟฝ๏ฟฝ on the tetrahedron ๐‘ฅ = ๐‘ฆ = ๐‘ง = 0,

๐‘ฅ + ๐‘ฆ + ๐‘ง = 1.

prq"Qyd ๐‘ฅ = ๐‘ฆ = ๐‘ง = 0, ๐‘ฅ + ๐‘ฆ + ๐‘ง = 1 ij ๏ฟฝ๏ฟฝ = ๐‘ฅ๐‘ฆ๐‘– + ๐‘ง2๐‘— + 2๐‘ฆ๐‘ง๏ฟฝ๏ฟฝ ds fy, vilj.k izes; dk

lR;kiu dhft,A

OR

Evaluateโˆซ ๐‘๐‘ข๐‘Ÿ๐‘™ ๏ฟฝ๏ฟฝ๐‘ 

. ๏ฟฝ๏ฟฝ๐‘‘๐‘ , where ๏ฟฝ๏ฟฝ = (๐‘ฆ2 + ๐‘ง2 โˆ’ ๐‘ฅ2)๐‘– + (๐‘ง2 + ๐‘ฅ2 โˆ’ ๐‘ฆ2)๐‘— + (๐‘ฅ2 + ๐‘ฆ2 โˆ’

๐‘ง2)๏ฟฝ๏ฟฝ ๐‘Ž๐‘›๐‘‘ ๐‘† is the portion of the surface ๐‘ฅ2 + ๐‘ฆ2 โˆ’ 29๐‘ฅ + 9๐‘ง = 0 above the plane ๐‘ง = 0, and verify

the Stokeโ€™s theorem.

โˆซ ๐‘๐‘ข๐‘Ÿ๐‘™ ๏ฟฝ๏ฟฝ๐‘ 

. ๏ฟฝ๏ฟฝ๐‘‘๐‘  dk eku Kkr dhft,A tgkยก ๏ฟฝ๏ฟฝ = (๐‘ฆ2 + ๐‘ง2 โˆ’ ๐‘ฅ2)๐‘– + (๐‘ง2 + ๐‘ฅ2 โˆ’ ๐‘ฆ2)๐‘— +

(๐‘ฅ2 + ๐‘ฆ2 โˆ’ ๐‘ง2)๏ฟฝ๏ฟฝ gS rFkk ๐‘†, ๐‘ง = 0 ry ds mij Qyd ๐‘ฅ2 + ๐‘ฆ2 โˆ’ 29๐‘ฅ + 9๐‘ง = 0 dk Hkkx gS rFkk

LVkd izes; dk lR;kiu dhft,A

UNIT II 15. Trace the conic 14๐‘ฅ2 โˆ’ 4๐‘ฅ๐‘ฆ + 11๐‘ฆ2 โˆ’ 44๐‘ฅ โˆ’ 58๐‘ฆ + 71 = 0 . Also find the focus, eccentricity and

length of latus rectum.

'kkdo 14๐‘ฅ2 โˆ’ 4๐‘ฅ๐‘ฆ + 11๐‘ฆ2 โˆ’ 44๐‘ฅ โˆ’ 58๐‘ฆ + 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‘™ ๐‘Ÿโ„ = 1 + ๐‘’ ๐‘๐‘œ๐‘ ๐œƒ is ๐‘Ÿ2(1 โˆ’ ๐‘’2) +2๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘ ๐œƒ โˆ’ 2๐‘™2 = 0 .

fl) djks fd 'kkado ๐‘™๐‘Ÿโ„ = 1 + ๐‘’ ๐‘๐‘œ๐‘ ๐œƒ ds fu;ked o`r dk lehdj.k ๐‘Ÿ2(1 โˆ’ ๐‘’2) + 2๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘ ๐œƒ โˆ’

2๐‘™2 = 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 ๐‘ก๐‘Ž๐‘›โˆ’1โˆš2

;fn ,d yEc o`rh; 'kadq ds rhu tud ijLij yEc gS] rks fl) dhft, fd v)Zโ€™kh"kZ dks.k dk eku

๐‘ก๐‘Ž๐‘›โˆ’1โˆš2 gSA

(b) Show that the equation 4๐‘ฅ2 โˆ’ ๐‘ฆ2 + 2๐‘ง2 + 2๐‘ฅ๐‘ฆ โˆ’ 3๐‘ฆ๐‘ง + 12๐‘ฅ โˆ’ 11๐‘ฆ + 6๐‘ง + 4 = 0 represents a

cone with vertex (โˆ’1, โˆ’2, โˆ’3).

fl) dhft, fd lehdj.k 4๐‘ฅ2 โˆ’ ๐‘ฆ2 + 2๐‘ง2 + 2๐‘ฅ๐‘ฆ โˆ’ 3๐‘ฆ๐‘ง + 12๐‘ฅ โˆ’ 11๐‘ฆ + 6๐‘ง + 4 = 0 ,d 'kadq

dks fu:fir djrk gS ftldk 'kh"kZ (โˆ’1, โˆ’2, โˆ’3) gSaA

OR

(a) Find the equation of the enveloping cylinder of the surface ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + ๐‘๐‘ง2 = 1 and whose

generators are parallel to the line ๐‘ฅ

๐‘™=

๐‘ฆ

๐‘š=

๐‘ง

๐‘› .

๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + ๐‘๐‘ง2 = 1 ds vUokyksih csyu ds lehdj.k Kkr dhft, ftldh tud js[kk, ๐‘ฅ

๐‘™=

๐‘ฆ

๐‘š=

๐‘ง

๐‘› ds lekUrj gSaA

(b) Find the equation of the right circular cylinder whose guiding circle is ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 9 , ๐‘ฅ โˆ’๐‘ฆ + ๐‘ง = 3 .

ml yEco`Rrh; csyu dk lehdj.k Kkr dhft, ftldk funsZโ€™kkd o`r ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 9 , ๐‘ฅ โˆ’ ๐‘ฆ +

๐‘ง = 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:

oร˜rk f=T;k Kkr dhft, ;fn nh/kZor dk ifnd lehdj.k gS %

1

๐‘2=

1

๐‘Ž2+

๐‘Ž

๐‘2โˆ’

๐‘Ÿ2

๐‘Ž2๐‘2

2. Define derivative of the length of arc.

pki dh yEckรƒ dยข vodyt dh ifjHkkโ€œkk fnft,

3. Write the Eulerโ€™s theorem for homogenous functions.

le?kkr Qyu dยข fy;s vkW;yj รes; fyf[k,

4. Define evaluate.

dยขUรฆt dh ifjHkkโ€œkk fyf[k,

5. Give the sufficient condition for ๐‘“(๐‘Ž, ๐‘) to be a maximum value of ๐‘“(๐‘ฅ, ๐‘ฆ).

Qyu ๐‘“(๐‘ฅ, ๐‘ฆ dยข vf/kdre eku ๐‘“(๐‘Ž, ๐‘) gยจus dh i;kZIr รfrca/k fyf[k,

6. Evaluate the following. โˆซ ๐‘ ๐‘–๐‘›5๐‘ฅ๐œ‹

2โ„

0 ๐‘๐‘œ๐‘ 6๐‘ฅ ๐‘‘๐‘ฅ.

fuEu dk eku Kk dhft,

7. Define Beta function.

chVk Qyu dh ifjHkkโ€œkk nhft,

8. Change the order of integration in the following double integral.

fuEu f}&lekdy eSa lekdyu dk ร˜e ifjoร†rr dhft,%

โˆซ โˆซ๐‘’โˆ’๐‘ฆ

๐‘ฆ ๐‘‘๐‘ฅ๐‘‘๐‘ฆ

โˆž

๐‘ฅ

โˆž

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,%

๐‘ฆ = 3๐‘ฅ4 โˆ’ 4๐‘ฅ3 + 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. ๐‘ฅ = ๐‘Ž ๐‘๐‘œ๐‘ 3๐‘ก, ๐‘ฆ = ๐‘Ž ๐‘ ๐‘–๐‘›3๐‘ก

,LVยชk;M ๐‘ฅ = ๐‘Ž ๐‘๐‘œ๐‘ 3๐‘ก, ๐‘ฆ = ๐‘Ž ๐‘ ๐‘–๐‘›3๐‘ก dk ifnd lehdj.k Kkr dhft,

OR

Find the asymptotes of the following curve:

fuEu oร˜ dh vUkUrLiร†โ€˜k;kยก Kkr dhft,

๐‘ฆ3 โˆ’ ๐‘ฅ๐‘ฆ2 โˆ’ ๐‘ฅ2๐‘ฆ + ๐‘ฅ3 + ๐‘ฅ2 โˆ’ ๐‘ฆ2 โˆ’ 1 = 0

UNIT II

12. If ๐‘ฅ๐‘ฅ๐‘ฆ๐‘ฆ๐‘ง๐‘ง = ๐‘ then show that ๐œ•2๐‘ง

๐œ•๐‘ฅ๐œ•๐‘ฆ= โˆ’(๐‘ฅ log ๐‘’๐‘ฅ)โˆ’1.

;fn ๐‘ฅ๐‘ฅ๐‘ฆ๐‘ฆ๐‘ง๐‘ง = ๐‘ rยจ fl) dhft, fd ๐œ•2๐‘ง

๐œ•๐‘ฅ๐œ•๐‘ฆ= โˆ’(๐‘ฅ log ๐‘’๐‘ฅ)โˆ’1

OR

Find the envelope of the circles which passes through the vertex of the vertex of the parabola ๐‘ฆ2 =

4๐‘Ž๐‘ฅ and whose centre lies on it.

,d o`r dk dยขUรฆ lnSo ijoy; ๐‘ฆ2 = 4๐‘Ž๐‘ฅ ij jgrk gS rFkk ยธg ij oy; dยข โ€˜khโ€œkZ ls xqtjrk gS o`r

dqy dk vUokyยจi Kkr dhft,

UNIT III

13. To show that: รnร†โ€˜r dhft,

โˆš๐‘› โˆš1 โˆ’ ๐‘› =๐œ‹

sin ๐‘›๐œ‹, 0 < ๐‘› < 1.

OR

Change the order of integration in the following double integral:

fuEu f}&lekdy esa lekdyu dk ร˜e ifjoร†rr dhft,

โˆซ โˆซ ๐‘“ (๐‘ฅ, ๐‘ฆ)๐‘‘๐‘ฅ ๐‘‘๐‘ฆ.2๐‘Žโˆ’๐‘ฅ

๐‘ฅ2๐‘Žโ„

๐‘Ž

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,

๐‘ฆ2(๐‘Ž + ๐‘ฅ) = ๐‘ฅ2(3๐‘Ž โˆ’ ๐‘ฅ)

OR

a. Prove that the radius of curvature of the centenary

๐‘ฆ = ๐‘ cosh (๐‘ฅ

๐‘) ๐‘Ž๐‘ก (๐‘ฅ, ๐‘ฆ)๐‘–๐‘ 

๐‘ฆ2

๐‘.

fl) djยจ fd dSVsujh ๐‘ฆ = ๐‘ cosh (๐‘ฅ

๐‘) dยข fcUnq (๐‘ฅ, ๐‘ฆ) ij oร˜rk f=T;k

๐‘ฆ2

๐‘ gS

b. Find the point inflection of the curve ๐‘ฆ(๐‘Ž2 + ๐‘ฅ2) = ๐‘ฅ3

oร˜ ๐‘ฆ(๐‘Ž2 + ๐‘ฅ2) = ๐‘ฅ3 dยข ufr ifjorZu fcUnq Kkr dhft,

UNIT II

15. (a) If ๐‘ข = ๐‘ก๐‘Ž๐‘›โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅ+๐‘ฆ), then prove that.

;fn ๐‘ข = ๐‘ก๐‘Ž๐‘›โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅ+๐‘ฆ) rยจ fl) dhft,

๐‘ฅ2๐›ฟ2๐‘ข

๐›ฟ๐‘ฅ2+ 2๐‘ฅ๐‘ฆ

๐›ฟ2๐‘ข

๐›ฟ๐‘ฅ๐›ฟ๐‘ฆ+ ๐‘ฆ2

๐›ฟ2๐‘ข

๐›ฟ๐‘ฆ2= sin 2๐‘ข (1 โˆ’ 4 ๐‘ ๐‘–๐‘›2๐‘ข)

(b) If ๐‘ฅ3 + ๐‘ฆ3 = 3๐‘Ž๐‘ฅ๐‘ฆ, then find ๐‘‘2๐‘ฆ

๐‘‘๐‘ฅ2

;fn ๐‘ฅ3 + ๐‘ฆ3 = 3๐‘Ž๐‘ฅ๐‘ฆ Kkr dhft, ๐‘‘2๐‘ฆ

๐‘‘๐‘ฅ2

OR

(a) Find the envelope of the straight lines ๐‘ฅ cos ๐›ผ + ๐‘ฆ sin ๐›ผ = ๐‘™ sin ๐›ผ cos ๐›ผ; ๐›ผ is a parameter.

๐›ผ dยจ รkpyd ysrs gq, ljy js[kkvยจ ๐‘ฅ cos ๐›ผ + ๐‘ฆ sin ๐›ผ = ๐‘™ sin ๐›ผ cos ๐›ผ ; dk vUoyยจi Kkr

dhft;s

(b) Find the maximum value of ๐‘ฅ3๐‘ฆ2(1 โˆ’ ๐‘ฅ โˆ’ ๐‘ฆ).

mfPpโ€œB eku Kkr dhft,

UNIT III

16. (a) Show that :- รnร†โ€˜kr dhft, fd

โˆซ ๐‘ก๐‘Ž๐‘›๐‘š๐‘ฅ๐‘‘๐‘ฅ =๐œ‹

2sec (

๐‘š๐œ‹

2)

๐œ‹2โ„

0

(b) Prove that:- fl) dhft, fd

22๐‘›โˆ’1๐ต (๐‘›, ๐‘š) =โˆš๐œ‹โˆš๐‘›

โˆš(๐‘› +12)

OR

(a) Evaluate:- eku Kr dhft, โˆซ โˆซ (๐‘ฅ + ๐‘ฆ)2๐‘‘๐‘ฅ ๐‘‘๐‘ฆ๐‘…

. Where the area R is bounded by ellipse tgkยก

{ks= nh/kZo`r ls ifjc) gS

๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1

(b) Evaluate: โˆซ โˆซ โˆซ๐‘‘๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ง

(๐‘ฅ+๐‘ฆ+๐‘ง+1)3

1โˆ’๐‘ฅโˆ’๐‘ฆ

0

1โˆ’๐‘ฅ

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 ๐‘”๐‘Ÿ๐‘Ž๐‘‘ ๐‘Ÿ = ๏ฟฝ๏ฟฝ , when ๐‘Ÿ = ๐‘ฅ๐‘– + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ

2. Find the greatest rate of increase of โˆ… = 2๐‘ฅ๐‘ฆ๐‘ง2 at (1,0, โˆ’3).

โˆ… = 2๐‘ฅ๐‘ฆ๐‘ง2 dk (1,0, โˆ’3) 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 ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + 2โ„Ž๐‘ฅ๐‘ฆ + 2๐‘”๐‘ฅ + 2๐‘“๐‘ฆ + ๐‘ = 0 to represent an equation of

hyperbola.

'kkado ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + 2โ„Ž๐‘ฅ๐‘ฆ + 2๐‘”๐‘ฅ + 2๐‘“๐‘ฆ + ๐‘ = 0 ds ,d vfrijoy; dks iznfโ€™kZr djus dh 'krZ

fyf[k,A

5. Write the equation of directive of conic ๐‘™

๐‘Ÿ= 1 + ๐‘’ ๐‘๐‘œ๐‘ ๐œƒ, ๐‘คโ„Ž๐‘’๐‘› ๐‘’ < 1.

'kkado ๐‘™

๐‘Ÿ= 1 + ๐‘’ ๐‘๐‘œ๐‘ ๐œƒ] 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 Lifโ€™kZrk 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`ร™kh; 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 โˆ‡2๐‘Ÿ๐‘› and show that โˆ‡2 ( 1

๐‘Ÿ ) = 0 where ๐‘Ÿ = (๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2)1/2

โˆ‡2๐‘Ÿ๐‘› dk eku Kkr djks rFkk fl) djksa โˆ‡2 ( 1

๐‘Ÿ ) = 0 ] tgkยก ๐‘Ÿ = (๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2)1/2

OR

Evaluate โˆซ ๐‘“ ๐‘

. ๐‘‘๐‘Ÿ , where ๐‘“ = ๐‘ฅ๐‘ฆ๐‘– + (๐‘ฅ2 + ๐‘ฆ2)๐‘— and C is the rectangle in the ๐‘ฅ๐‘ฆ plane bounded

by lines ๐‘ฅ = 1, ๐‘ฅ = 4 , ๐‘ฆ = 2, ๐‘ฆ = 10.

โˆซ ๐‘“ ๐‘

. ๐‘‘๐‘Ÿ dk eku Kkr dhft,] tcfd ๐‘“ = ๐‘ฅ๐‘ฆ๐‘– + (๐‘ฅ2 + ๐‘ฆ2)๐‘— o C, xy iVy es ๐‘ฅ = 1, ๐‘ฅ =

4 , ๐‘ฆ = 2, ๐‘ฆ = 10 js[kkvksa ls f?kjk vk;r gSaA

UNIT II

12. Find the lengths and the equation of the axes of the conic 2๐‘ฅ2 + 5๐‘ฅ๐‘ฆ + 2๐‘ฆ2 = 1 .

'kkado 2๐‘ฅ2 + 5๐‘ฅ๐‘ฆ + 2๐‘ฆ2 = 1 ds v{kksa dh yEckb;kยก o lehdj.k Kkr dhft,A

OR

A chord ๐‘ƒ๐‘„ 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 ๐‘ƒ๐‘„ ukfHk ๐‘† ij ledks.k vUrfjr djrh gSaA ;fn 'kkado dh mRdsUnzrk ๐‘’ v)ZukfHkyEc ๐‘™ gks rks fl) djksa&

(1

๐‘ ๐‘โˆ’

1

๐‘™)

2

+ (1

๐‘†๐‘„โˆ’

1

๐‘™)

2

=๐‘’2

๐‘™2

UNIT III

13. Find the equation of the sphere having the circle ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 10๐‘ฆ โˆ’ 4๐‘ง = 8 , ๐‘ฅ + ๐‘ฆ + ๐‘ง = 3 as

a great circle.

ml xksys dk lehdj.k Kkr djks ftldk or ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 + 10๐‘ฆ โˆ’ 4๐‘ง = 8 , ๐‘ฅ + ๐‘ฆ + ๐‘ง = 3 rd

o`gr or gksaA

OR

Find the equation of the cylinder whose generators are parallel to the line ๐‘ฅ

1=

๐‘ฆ

โˆ’2=

๐‘ง

3 and passing

through the curve ๐‘ฅ2 + 2๐‘ฆ2 = 1 , ๐‘ง = 3.

ml csyu dk lehdj.k Kkr djksa ftldh tud js[kk,ยก ๐‘ฅ

1=

๐‘ฆ

โˆ’2=

๐‘ง

3 ds lekUkkUrj gS rFkk ftldk

funsZโ€™kd oร˜ ๐‘ฅ2 + 2๐‘ฆ2 = 1 , ๐‘ง = 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&

๐‘๐‘ข๐‘Ÿ๐‘™ ๐‘Ž ร— ๐‘Ÿ

๐‘Ÿ3= โˆ’

๐‘Ž

๐‘Ÿ3+

3๐‘Ÿ

๐‘Ÿ5 ( ๐‘Ž . ๐‘Ÿ )

Where, ๐‘Ÿ = ๐‘ฅ ๏ฟฝ๏ฟฝ + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ , ๐‘Ž = ๐‘Ž1๐‘– + ๐‘Ž2๐‘— + ๐‘Ž3๏ฟฝ๏ฟฝ is a constant vector and |๐‘Ÿ | = ๐‘Ÿ.

tgkยก ๐‘Ÿ = ๐‘ฅ ๏ฟฝ๏ฟฝ + ๐‘ฆ๐‘— + ๐‘ง๏ฟฝ๏ฟฝ , ๐‘Ž = ๐‘Ž1๐‘– + ๐‘Ž2๐‘— + ๐‘Ž3๏ฟฝ๏ฟฝ ,d vpj lafnโ€™k gS rFkk |๐‘Ÿ | = ๐‘Ÿ

OR

Verify Gaussโ€™ Divergence theorem for ๐‘“ = 4๐‘ฅ๐‘– โˆ’ 2๐‘ฆ2๐‘— + ๐‘ง2๏ฟฝ๏ฟฝ , taken over the region bounded by

๐‘ฅ2 + ๐‘ฆ2 = 4 , ๐‘ง = 0, ๐‘ง = 3.

๐‘ฅ2 + ๐‘ฆ2 = 4 , ๐‘ง = 0, ๐‘ง = 3 }kjk ifjc) {ks= ij Qyu ๐‘“ = 4๐‘ฅ๐‘– โˆ’ 2๐‘ฆ2๐‘— + ๐‘ง2๏ฟฝ๏ฟฝ ds fy, xkWl

vilj.k izes; lR;kfir dhft,A

UNIT II

15. Trace the curve.

oร˜ vuqjs[k.k dhft,A

๐‘ฅ2 + ๐‘ฅ๐‘ฆ + ๐‘ฆ2 + ๐‘ฅ + ๐‘ฆ = 1

OR

Find the condition that the straight line ๐‘™

๐‘Ÿ= ๐ด ๐‘๐‘œ๐‘ ๐œƒ + ๐ต๐‘ ๐‘–๐‘›๐œƒ touches the circle ๐‘Ÿ = 2๐‘Ž ๐‘๐‘œ๐‘ ๐œƒ.

og izfrcU/k Kkr djks tcfd ljy js[kk ๐‘™

๐‘Ÿ= ๐ด ๐‘๐‘œ๐‘ ๐œƒ + ๐ต๐‘ ๐‘–๐‘›๐œƒ ] o`r ๐‘Ÿ = 2๐‘Ž ๐‘๐‘œ๐‘ ๐œƒ dks Liโ€™kZ djsaA

UNIT III

16. Find the equation of cone whose vertex is (5,4,3) and the guiding curve is 3๐‘ฅ2 + 2๐‘ฆ2 = 6 , ๐‘ฆ +

๐‘ง = 0.

ml 'kadq dk lehdj.k Kkr dhft, ftldk 'kh"kZ (5,4,3) rFkk funsZโ€™kd oร˜ 3๐‘ฅ2 + 2๐‘ฆ2 = 6 , ๐‘ฆ +

๐‘ง = 0 gSaA

OR

Find the equation of the right circular cylinder whose guiding circle is ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 9 , ๐‘ฅ โˆ’๐‘ฆ + ๐‘ง = 3 .

ml yEco`rh; csyu dk lehdj.k Kkr djks ftldk funsZโ€™kd o`r ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 = 9 , ๐‘ฅ โˆ’ ๐‘ฆ +

๐‘ง = 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 Liโ€˜kรŠ dยจ ifjHkkfโ€œkr dhft, |

2. Write the necessary condition for the existence of double point for the curve ๐‘“(๐‘ฅ, ๐‘ฆ) = 0 and also

the condition for the double point to be a node, cusp or conjugate point.

oร˜ ๐‘“(๐‘ฅ, ๐‘ฆ) = 0 dยข f}d fcUnqvยจa dยข vfLrRo dยข fy, vkoโ€˜;d รfrcU/k fyf[k, ,oa f}d fcUnq dยข

uยจM] mHk;kXkz ;k la;qXeh fcUnq gยจus gsrq รfrca/k Hkh fyf[k, |

3. Find the radius of curvature at the point (s,๐œ‘) for the curve ๐‘  = ๐‘Ž ๐‘™๐‘œ๐‘”(sec ๐œ‘ + tan ๐œ‘).

oร˜ ๐‘  = ๐‘Ž ๐‘™๐‘œ๐‘”(sec ๐œ‘ + tan ๐œ‘) dยข fcUnq (s, ๐œ‘) ij oร˜rk fยซkT;k Kku dhft, |

4. State Eulerโ€™s theorem for homogeneous functions.

le?kku Qyuยจa dยข fy, vk;dj รes; dk dFku fyf[k, |

5. If ๐‘ข =๐‘ฆ2

2๐‘ฅ, ๐‘ฃ =

๐‘ฅ2+๐‘ฆ2

2๐‘ฅ, find

๐œ•(๐‘ข,๐‘ฃ)

๐œ• (๐‘ฅ,๐‘Œ).

;fn ๐‘ข =๐‘ฆ2

2๐‘ฅ, ๐‘ฃ =

๐‘ฅ2+๐‘ฆ2

2๐‘ฅ rยจ

๐œ•(๐‘ข,๐‘ฃ)

๐œ• (๐‘ฅ,๐‘Œ) Kkr dhft, |

6. Define envelope of a one parameter family of plane curves.

,d รkpy okys lery oร˜ dqy dยข vUokyยจi dยจ ifjHkkfโ€œkr dhft,

7. Evaluate โˆซ ๐‘ฅ5๐‘’โˆ’๐‘ฅโˆž

0 ๐‘‘๐‘ฅ

eku Kkr dhft, โˆซ ๐‘ฅ5๐‘’โˆ’๐‘ฅโˆž

0 ๐‘‘๐‘ฅ

8. Change the order of integration in the following integral

fuEu lekdy esa lekdyu dk ร˜e ifjoร†rr dhft, |

โˆซ โˆซ ๐‘“ (๐‘ฅ, ๐‘ฆ)๐‘‘๐‘ฅ ๐‘‘๐‘ฆโˆš๐‘Ž2โˆ’๐‘ฅ2

0

๐‘Ž

0

9. State Liouvilleโ€™s extension of Dirichletโ€™s integral.

fMjpfyV lekdy dยข fyosyh O;kidhdj.k dk dFku fyf[k, |

10. Check if the function ๐‘ฅ3 โˆ’ 4๐‘ฅ๐‘ฆ + 2๐‘ฆ2 attains its extreme value at the point (0, 0).

tkยกp dhft, fd D;k Qyu ๐‘ฅ3 โˆ’ 4๐‘ฅ๐‘ฆ + 2๐‘ฆ2 vius pje eku dยจ fcUnq (0, 0) ij รkIr 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 ๐‘‘๐‘ 

๐‘‘๐‘œ for the curve

2๐‘Ž

๐‘Ÿ= 1 + cos ๐œƒ.

oร˜ 2๐‘Ž

๐‘Ÿ= 1 + cos ๐œƒ dยข fy,

๐‘‘๐‘ 

๐‘‘๐‘œ Kkr dhft, |

OR

Find the radius of curvature at the point (r,๐œƒ) on the polar curve ๐‘Ÿ (1 + cos ๐œƒ) = ๐‘Ž

/kqzoh oร˜ ๐‘Ÿ (1 + cos ๐œƒ) = ๐‘Ž dยข fcUnq (r,๐œƒ) ij oร˜rk f=T;k Kkr dhft, |

UNIT II

12. If ๐‘ข = ๐‘ ๐‘’๐‘โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅ+๐‘ฆ), then prove that ๐‘ฅ

๐œ•๐‘ข

๐œ•๐‘ฅ+ ๐‘ฆ

๐œ•๐‘ข

๐œ•๐‘ฆ= 2 cot ๐‘ข.

;fn ๐‘ข = ๐‘ ๐‘’๐‘โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅ+๐‘ฆ) rยจ fl) dhft, ๐‘ฅ

๐œ•๐‘ข

๐œ•๐‘ฅ+ ๐‘ฆ

๐œ•๐‘ข

๐œ•๐‘ฆ= 2 cot ๐‘ข

OR

If ๐‘ข3 + ๐‘ฃ3 = ๐‘ฅ + ๐‘ฆ and ๐‘ข2 + ๐‘ฃ2 = ๐‘ฅ3 + ๐‘ฆ3, then find the value of ๐œ•(๐‘ข,๐‘ฃ)

๐œ•(๐‘ฅ,๐‘ฆ).

;fn ๐‘ข3 + ๐‘ฃ3 = ๐‘ฅ + ๐‘ฆ rFkk ๐‘ข2 + ๐‘ฃ2 = ๐‘ฅ3 + ๐‘ฆ3 rยจ ๐œ•(๐‘ข,๐‘ฃ)

๐œ•(๐‘ฅ,๐‘ฆ). dk eku Kkr dhft, |

UNIT III

13. Prove that. fl) dhft, โˆซ๐‘‘๐‘ฅ

โˆš๐‘Ž4โˆ’๐‘ฅ4

๐‘Ž

0=

{ฮ“ (1/4) }2

4๐‘Ž โˆš(2๐œ‹)

OR

Evaluate โˆฌ ๐‘ฅ๐‘ฆ(๐‘ฅ + ๐‘ฆ)๐‘‘๐‘ฅ ๐‘‘๐‘ฆ๐‘…

where the area of integration R is the area between ๐‘ฆ = ๐‘ฅ2 and ๐‘ฆ =

๐‘ฅ.

โˆฌ ๐‘ฅ๐‘ฆ(๐‘ฅ + ๐‘ฆ)๐‘‘๐‘ฅ ๐‘‘๐‘ฆ๐‘…

dk eku Kkr dhft, tgkยก lekdyu dk {ks= R, ๐‘ฆ = ๐‘ฅ2 rFkk ๐‘ฆ = ๐‘ฅ 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 oร˜ยจa dk vuqjs[k.k dhft, |

a. ๐‘ฆ2 (2๐‘Ž โˆ’ ๐‘ฅ) = ๐‘ฅ3

b. ๐‘Ÿ = ๐‘Ž(1 + cos ๐œƒ)

OR

Find all the asymptotes of the curve (2๐‘ฅ โˆ’ 3๐‘ฆ + 1)2 (๐‘ฅ + ๐‘ฆ) โˆ’ 8๐‘ฅ + 2๐‘ฆ โˆ’ 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˜ (2๐‘ฅ โˆ’ 3๐‘ฆ + 1)2 (๐‘ฅ + ๐‘ฆ) โˆ’ 8๐‘ฅ + 2๐‘ฆ โˆ’ 9 = 0 dh lHkh vuUr Liร†โ€˜k;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 Eulerโ€™s theorem on homogeneous function.

le?kkr Qyuยจa dยข fy, vk;yj รes; dยจ fl) djsa |

(b) Prove that ๐‘ข = ๐‘Ž๐‘ฅ3๐‘ฆ2 โˆ’ ๐‘ฅ4๐‘ฆ2 โˆ’ ๐‘ฅ3๐‘ฆ3 is maximum at (๐‘Ž

2,

๐‘Ž

3)

fl) dhft, fd ๐‘ข = ๐‘Ž๐‘ฅ3๐‘ฆ2 โˆ’ ๐‘ฅ4๐‘ฆ2 โˆ’ ๐‘ฅ3๐‘ฆ3 mPpre eku fcUnq (๐‘Ž

2,

๐‘Ž

3) ij gS |

OR

Find the envelope of the family of straight lines ๐‘Ž๐‘ฅ sec ๐›ผ โˆ’ ๐‘ ๐‘ฆ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐›ผ = ๐‘Ž2 โˆ’ ๐‘2, ๐›ผ being the

parameter. Use the above result to show that the evolutes of the ellipse ๐‘ฅ2

๐‘Ž2+

๐‘ฆ2

๐‘2= 1 is (๐‘Ž๐‘ฅ)

23โ„ +

(๐‘๐‘ฆ)2

3โ„ = (๐‘Ž2 โˆ’ ๐‘2)2

3โ„ .

ljy js[kk dqy ๐‘Ž๐‘ฅ sec ๐›ผ โˆ’ ๐‘ ๐‘ฆ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐›ผ = ๐‘Ž2 โˆ’ ๐‘2 dk vUokyยจi Kkr dhft, tgkยก ๐›ผ รkpy gS

mDr ifj.kke dยข ร;ยจx }kjk fl) dhft, fd nh/kZo`Rr ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1 dk dยขUรฆ (๐‘Ž๐‘ฅ)2

3โ„ + (๐‘๐‘ฆ)2

3โ„ =

(๐‘Ž2 โˆ’ ๐‘2)2

3โ„ gS |

UNIT III

16. (a) Prove that:

fl) dhft,

๐‘(๐‘š, ๐‘›) =โˆš(๐‘š)โˆš(๐‘›)

โˆš(๐‘š + ๐‘›) (๐‘š > 0, ๐‘› > 0)

(b) Evaluate โˆญ ๐‘ฅ๐‘ฆ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง where region of integration is the complete ellipsoid ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 +๐‘ง2

๐‘2 โ‰ค

1.

โˆญ ๐‘ฅ๐‘ฆ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง dk eku dhft, tgkยก lekdyu dk {ks= laiw.kZ nh/kZo`Rrt ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 +๐‘ง2

๐‘2 โ‰ค 1 gSa |

OR

a. Evaluate the following integral by changing the order of integration.

fuEu lekdy esa lekdyu dk ร˜e cnydj eku Kkr dhft, |

โˆซ โˆซ ๐‘ฅ2 cos(๐‘ฅ2 โˆ’ ๐‘ฅ๐‘ฆ) ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ1

๐‘ฆ

1

๐‘ฆ=0

b. Evaluate โˆญ ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง๐‘ฃ

where region of integration is a cylinder v bounded by z=0, z=1, ๐‘ฅ2 +

๐‘ฆ2 = 4.

โˆญ ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง๐‘ฃ

dk eku Kkr dhft, tgkยก lekdyu {ks= ,d csyu v gS tยจ z=0, z=1, ๐‘ฅ2 +

๐‘ฆ2 = 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 = 5๐‘ก2๐‘– โ€“ cos ๐‘ก๐‘— ๐‘Ž๐‘›๐‘‘ ๏ฟฝ๏ฟฝ = ๐‘ก๐‘– + ๐‘†๐‘–๐‘› ๐‘ก๐‘— then find ๐‘‘๐‘‘๐‘ก

(๏ฟฝ๏ฟฝ ยท ๏ฟฝ๏ฟฝ)

;fn ๏ฟฝ๏ฟฝ = 5๐‘ก2๐‘– โ€“ cos ๐‘ก๐‘— rFkk ๏ฟฝ๏ฟฝ = ๐‘ก๐‘– + ๐‘†๐‘–๐‘› ๐‘ก๐‘— rc ๐‘‘๐‘‘๐‘ก

(๏ฟฝ๏ฟฝ ยท ๏ฟฝ๏ฟฝ)Kkr dhft,

2. Define Curl of a vector point function.

Lkfnโ€k fcU/kq Qyu dk dqUry ifjHkkfโ€™kr dhft,A

3. State Gaussโ€™s 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 vuUrLiโ€khZZ;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 ifjHkkfโ€™kr 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 โ€œkkdao 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 Liโ€kZ lekUrk fyf[k,

11. Write equation of right circular cone.

yEco`rh; โ€œkdqa dk lehdj.k fyf[k,A

12. Define enveloping cylinder.

vUokyksih csyu dks ifjHkkfโ€™kr 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 fnโ€kk 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 โ€œkkado 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 โ€œkkado dks fu:fir djrsa gSA

๐‘™

๐‘Ÿ= 1 + ๐‘’ cos ๐œƒ ๐‘Ž๐‘›๐‘‘

๐‘™

๐‘Ÿ= โˆ’1 + ๐‘’ cos ๐œƒ

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 Liโ€kZ 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 โ€œkakdq dk lehdj.k djks ftldk โ€œkhโ€™kZ 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 = ๐‘ฅ2๐‘– + ๐‘ฅ๐‘ฆ๐‘— 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 = ๐‘ฅ2๐‘– + ๐‘ฅ๐‘ฆ๐‘— 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 vuqfnโ€k gSA

OR

If ๐‘Ÿ = ๐‘ก๐‘– โ€“ ๐‘ก2๐‘— + (๐‘ก โ€“ 1)๏ฟฝ๏ฟฝ and ๐‘  = 2๐‘ก2๐‘– + 6๐‘ก๏ฟฝ๏ฟฝ then find value of

;fn ๐‘Ÿ = ๐‘ก๐‘– โ€“ ๐‘ก2๐‘— + (๐‘ก โ€“ 1)๏ฟฝ๏ฟฝ rFkk ๐‘  = 2๐‘ก2๐‘– + 6๐‘ก๏ฟฝ๏ฟฝ rks Kkr dhft,A

a. โˆซ (๐‘Ÿ ยท ๐‘ )๐‘‘๐‘ก2

0

b. โˆซ (๐‘Ÿ ร— ๐‘ )๐‘‘๐‘ก2

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 ๐‘Ÿ1 , ๐‘Ÿ2 , ๐‘Ÿ3 ๐‘Ž๐‘›๐‘‘ ๐‘Ÿ4 respectively , prove that

,d oRr fdlh โ€œkkado ls xqtjrk gS ftldh ukfHkyEc 2๐‘™ gS] rFkk โ€œkkado dks pkj fcUnqvksa ij feyrk

gS ftudh ukfHk ls yEckb;kW dzeโ€k ๐‘Ÿ1 , ๐‘Ÿ2 , ๐‘Ÿ3 vkSj ๐‘Ÿ4 gS] rks fl) dhft, dh

1

๐‘Ÿ1+

1

๐‘Ÿ2+

1

๐‘Ÿ3+

1

๐‘Ÿ4=

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 funsโ€kd oRr x2 + y2 + z2 = 9, x โ€“ 2y + 2z = 3

gSA

OR

Find coโ€“ordinates of centre and radius of the circle

fuEu o`r dk dsUnz vkSj f=T;k Kjr djksA

X2 + y2โ€“2yโ€“4z = 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‘Ÿ = ๐‘“(๐œƒ)

pki dh yEckbZ dk vodyt Kkr djus dk lw= /kqoh; oร˜ ๐‘Ÿ = ๐‘“(๐œƒ) ds fy, fyf[k,A

2. Define parallel and oblique asymptotes.

lekUrj ,oa fr;Zd vuUrLifโ€kZ;ksa dh ifjHkkโ€™kk 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 ifjHkkโ€™kk nhft,A

5. Verify the theorem ๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ=

๐œ•2๐‘ข

๐œ•๐‘ฆ๐œ•๐‘ฅ , when ๐‘ข = ๐‘ฅ3 + ๐‘ฆ3 + 3๐‘Ž๐‘ฅ๐‘ฆ

izes; ๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ=

๐œ•2๐‘ข

๐œ•๐‘ฆ๐œ•๐‘ฅ dh iqfโ€™V dhft, tcfd ๐‘ข = ๐‘ฅ3 + ๐‘ฆ3 + 3๐‘Ž๐‘ฅ๐‘ฆ

6. Write the formula for finding second differential coefficient of implicit functions.

vLiโ€™V Qyuksa ds fn`rh; vody xq.kkad Kkr djus dk lw= fyf[k,A

7. Define envelope.

vUokyksi dh ifjHkkโ€™kk 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 Legendreโ€™s duplication formula?

fytsUMj f)xq.ku lw= D;k gS\

10. State Dirichletโ€™s 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

โˆฌ ๐‘“(๐‘ฅ, ๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ

12. Change the order of integration in the following integral:

fuEufyf[kr lekdy esa lekdyu dk dze cnfy,A

โˆซ โˆซ ๐‘“(๐‘ฅ, ๐‘ฆ)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘ฅ

0

๐‘Ž

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 ๐‘Ÿ๐‘š = ๐‘Ž๐‘š๐‘๐‘œ๐‘ ๐‘š๐œƒ , prove that ๐‘‘๐‘ 

๐‘‘๐œƒ= ๐‘Ž(๐‘ ๐‘’๐‘ ๐‘š๐œƒ)

๐‘šโˆ’1

๐‘š

odz ๐‘Ÿ๐‘š = ๐‘Ž๐‘š๐‘๐‘œ๐‘ ๐‘š๐œƒ ds fy, fl) dhft, fd

๐‘‘๐‘ 

๐‘‘๐œƒ= ๐‘Ž(๐‘ ๐‘’๐‘ ๐‘š๐œƒ)

๐‘šโˆ’1๐‘š

OR

Find the asymptotes of the following curve:

fuEu odz ds vuUrLifโ€kZ;kW Kkr dhft,A

๐‘ฅ3 + ๐‘ฆ3 โˆ’ 3๐‘Ž๐‘ฅ๐‘ฆ = 0

UNIT II

14. If ๐‘ข = ๐‘ก๐‘Ž๐‘›โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅโ€“๐‘ฆ), then prove that ๐‘ฅ

๐œ•๐‘ข

๐œ•๐‘ฅ+ ๐‘ฆ

๐œ•๐‘ข

๐œ•๐‘ฆ= ๐‘ ๐‘–๐‘›2๐‘ข

;fn ๐‘ข = ๐‘ก๐‘Ž๐‘›โˆ’1 (๐‘ฅ3+๐‘ฆ3

๐‘ฅโ€“๐‘ฆ) rc ๐‘ฅ

๐œ•๐‘ข

๐œ•๐‘ฅ+ ๐‘ฆ

๐œ•๐‘ข

๐œ•๐‘ฆ= ๐‘ ๐‘–๐‘›2๐‘ข fl) dhft,A

OR

Find the envelope of the family of ellipse

๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1 ๐‘คโ„Ž๐‘’๐‘› ๐‘Ž + ๐‘ = ๐‘ , c being a constant.

nh?kZo`Rrksa ๐‘ฅ2

๐‘Ž2 +๐‘ฆ2

๐‘2 = 1 ds dqy dk vUokyksi Kkr tcfd ๐‘Ž + ๐‘ = ๐‘ tgkW ๐‘ vpj gSaA

UNIT III 15. Evaluate Kkr dhft,

โˆซ ๐‘ฅ๐‘›โˆ’11

0(๐‘™๐‘œ๐‘”

1

๐‘ฅ)

๐‘šโˆ’1

๐‘‘๐‘ฅ = ฮ“(๐‘š)

๐‘›๐‘š ; ๐‘š, ๐‘› > 0

OR

Evaluate eku Kkr dhft,

โˆซ โˆซ โˆซ ๐‘ฅ๐‘ฆ๐‘ง ๐‘‘๐‘ฅ ๐‘‘๐‘ฆ ๐‘‘๐‘ง

where the region of integration is the volume of the ellipsoid in the positive octant.

tgkW lekdyu dk {ks= /kukRed vCVkaโ€kd esa nh/kZo`=t gSA

๐‘ฅ2

๐‘Ž2+

๐‘ฆ2

๐‘2+

๐‘ง2

๐‘2โ‰ค 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

๐‘Ÿ2 = ๐‘Ž2 cos 2๐œƒ

OR

Show that the radius of curvature at a point (๐‘Ž ๐‘๐‘œ๐‘ 3๐œƒ, ๐‘Ž๐‘ ๐‘–๐‘›3๐œƒ) on the curve ๐‘ฅ2

3 + ๐‘ฆ2

3 = ๐‘Ž2

3 is 3๐‘Ž

2sin 2๐œƒ

fl) dhft, odzrk f=T;k fcUnq (๐‘Ž ๐‘๐‘œ๐‘ 3๐œƒ, ๐‘Ž๐‘ ๐‘–๐‘›3๐œƒ) ] ij odz ๐‘ฅ2

3 + ๐‘ฆ2

3 = ๐‘Ž2

3 dh 3๐‘Ž

2sin 2๐œƒ gSA

UNIT II

17. If ๐‘ข = ๐‘ฅ๐‘ ๐‘–๐‘›โˆ’1 (๐‘ฆ

๐‘ฅ) then prove that

๐‘ฅ2๐œ•2๐‘ข

๐œ•๐‘ฅ2+ 2๐‘ฅ๐‘ฆ

๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ+ ๐‘ฆ2

๐œ•2๐‘ข

๐œ•๐‘ฆ2= 0

;fn ๐‘ข = ๐‘ฅ๐‘ ๐‘–๐‘›โˆ’1 (๐‘ฆ

๐‘ฅ) rc fl) dhft,A

๐‘ฅ2๐œ•2๐‘ข

๐œ•๐‘ฅ2+ 2๐‘ฅ๐‘ฆ

๐œ•2๐‘ข

๐œ•๐‘ฅ๐œ•๐‘ฆ+ ๐‘ฆ2

๐œ•2๐‘ข

๐œ•๐‘ฆ2= 0

OR

Find the maxima and minima of ๐‘ข = ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 subject to the conditions ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + ๐‘๐‘ง2 =

1 ๐‘Ž๐‘›๐‘‘ ๐‘™๐‘ฅ + ๐‘š๐‘ฆ + ๐‘›๐‘ง = 0

izfrcU/kks ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฆ2 + ๐‘๐‘ง2 = 1 rFkk ๐‘™๐‘ฅ + ๐‘š๐‘ฆ + ๐‘›๐‘ง = 0 ds vUrZxr ๐‘ข = ๐‘ฅ2 + ๐‘ฆ2 + ๐‘ง2 ds

mfPpโ€™B vkSj fufEuโ€™B Kkr dhft,A

UNIT III

18. Evaluate the following integral by changing the order:

fuEu lekdy dk dze cnydj eku Kkr dhft,A

โˆซ โˆซ๐‘’โˆ’๐‘ฆ

๐‘ฆ

โˆž

๐‘ฅ

โˆž

0

๐‘‘๐‘ฅ ๐‘‘๐‘ฆ

OR

Evaluate the following integral by changing to polar coordinates:

fuEu lekdy dks /kqzohZ; funsZโ€kkadksesa ifjofrZr dj eku Kkr dhft,A

โˆซ โˆซ โˆš๐‘ฅ2 + ๐‘ฆ2โˆš2๐‘ฅโˆ’๐‘ฅ2

๐‘ฅ

1

0๐‘‘๐‘ฅ ๐‘‘๐‘ฆ

--The End--