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

18
83
mRrj iznsk jktfkZ V.Mu eqDr fo'ofo|ky;] bykgkckn
vf/kU;kl (Assignment) l= 2012&13
Lukrd dyk dk;ZØe
Under Graduate Art Programme
fo’k; % xf.kr
Subject : Mathematics
dkslZ “kh’kZd % Calculus
Course Title : Calculus-01
Section - A
[k.M & d
fo’k; dksM% ;w0th0,e0,e0
Subject Code : UGMM
dkslZ dksM % ;w0th0,e0,e0-01
Course Code : UGMM-01
vf/kdre vad % 30
Maximum Marks:30
vf/kdre vad % 18
Max. Marks:18
uksV % All qestions ar compulsory
Note : Section A is long Answer and Section B is Short Answer Questions.
1(a)
Evaluate
Kkr djsaA
lim
x ® 0
a
æ tan
x ÷ ö
ç
è ø
2
1/ x
(b) Evaluate
ò ( a - x)(
x - b ) dx
2
b
Kkr djsaA
d
tan x x
(c) Find ((cos x)
+ x )
2
dx
Kkr djsaA
2(a) State and prove Rolle's theorem. 3
2
2
-1æ1-
x
(b) Differentiate
÷ ö
-1
ç
æ 2x
ö
cos with respect to tan
2
ç
2
÷
è1+
x ø
è1-
x ø
2(a) jkWy izes; dks fy[kdj fl) djsaA
(b)
2
-1æ
1-
x
÷ ö
-1
ç
æ 2x
ö
cos dks tan
2 ç
2
÷
è1+
x ø è1-
x ø
ds lkis{k vodfyr djsaA
3(a)
2
ìx
ü
ï - a x < aï
ï
a
ï
If F( x)
= í0
x = 0ýthen discuss the Continuity and
ï 2
ï
ï
a
a - x > aï
î x
þ
differentiability at x = a
3
(b)
2 2 4 4 6 6 8 8
2 x 2 x 2 x 2 x
Show that cos 2x = 1-
+ - + - - - - -
2! 4! 6! 8!
3
2
ìx
ü
ï - a x < aï
ï
a
ï
3(a) ;fn F( x)
= í0
x = 0ý
rks x = a ij Fex)
dh lR;rk
ï 2
ï
ï
a
a - x > aï
î x
þ
,oa vodyuh;rk Kkr djsaA
2 2 4 4 6 6 8 8
2 x 2 x 2 x 2 x
(b) fn[kk;sa fd cos 2x
= 1-
+ - + - - - - - -
2! 4! 6! 8!
Section - B
[k.M & [k
vf/kdre vad % 12
Max. Marks:12
-
4. If (sin 1 2
2
y = x)
then show that (1 x ) y -(2n
+ 1) xy -n
y = 0 3
-
n+ 2
n+
1 n
3