64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

26
88
m kj izns’k jktf"kZ V.Mu eqDr fo’ofo|ky;] bykgkckn
fo’k; % xf.kr
Subject : Mathematics
dkslZ “kh’kZd % Advance Calculus
Courset Title : Advance Calculus
vf/kU;kl (Assignment) 2012-2013
Lukrd dyk dk;ZØe
Under Graduate Art Programme
Section - A
[k.M & d
fo’k; dksM% ;w0th0,e0,e0
Subject Code : UGMM
dkslZ dksM % ;w0th0,e0,e0-07
Course Code : UGMM-07
vf/kdre vad % 30
Maximum Marks:30
vf/kdre vad % 18
Max. Marks: 18
uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh
iz”u vfuok;Z gSA
Note : Long Answer Questions. Answer should be given in 800 to 1000 words.
Answer all questions.
1(a)
(b)
(
x,
y,
z)
Calculate the Jacobian
for 3
(
r,
q,
z)
tSdksfc;u Kkr djsaA
x = r cosq, y = r sinq , z = z
2 2
x
x
Prove that F = TR ® TR defined by f(x,y) = ( e cos y,
e sin y)
is
not invertible on the whole of TR 2 , but it is locally invertible at each point of
TR 2 . 3
2(a) Find
ò ò
D
Kkr djsaA
f ( x,
y)
dxdy where
ìSiny
f ( x,
y)
= í
î y
, y ¹ 0, y = 0
3
(b) Find the volume of the region lying in the first octant which is common to
the two cylinders x 2 + y 2 = a 2 and x 2 + z 2 = a 2
3
ml ifj{ks= dk vk;ru Kkr djsa tks x 2 + y 2 = a 2 rFkk x 2 + z 2 = a 2 ds chp esa
fLFkr gSA
3(a) Evaluate
lim x + cos x
(i)
x ® -¥ x + sin x
3
Kkr djsa
lim 1
(ii)
x ® ¥ 1+ log( x - 2)
(b) If A, B, C are the angles of a triangle such that 3
2
2
2
sin A + sin B + sin c = K
dA tanC
- tan B
=
dB tan A - tanC
2
2
2
;fn A, B, C ,d f=Hkqt ds dks.k gS rFkk sin A + sin B + sin c = K gks
dA tanC
- tan B
rks fn[kk;sa =
dB tan A - tanC
Section - B
[k.M & [k
vf/kdre vad % 12
Max. Marks: 12
uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh
iz”u vfuok;Z gSA
Note : Short Answer Questions. Answer should be given in 200 to 300 words.
Answer all questions.