Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 1st Sem Mathematics Differential Integral Calculus Previous Question Paper
B.Sc. Part-I (Scmcster?I) Examination
MATHEMATICS
(Differential & Integral Calculus)
Papcr?II
Time : Threg: IIours] [Maximum Marks : 60
Note :?(1) Question No. l is compulsory. Attempt oncc.
(2) Attempt ONE question from each unit.
1. Choose the correct alternatives (1 mark each) : l()
(i) The value of lim m is :
x?>0 X
(a) 0 (b) 1
(c) so (d) None of these
(ii) Ify ?? c 3?, thcn yll is :
(a) ?2? e z? (b) 2? 6?
(c) -?2? c?? (d) Nonc ot?thcsc
(iii) The series :
3 211?1
X-?x*~+ ........ +(?l)?"????+ .........
3! (211?1)!
is the expansion of function :
(a) sin x (b) sinh x
(c) cos x (d) cosh x
' l _ . i. .
(W) .X XUI<5 represents.
(a) x0?8
(a) Monotonic increasing in la, b] (b) Monotonic decreasing in [11, b]
(c) Constant in [a. b] (d) None of these
(vi) For f(x) = x3; in [1, 3] then the value of ?C? by Lagrange?s mean value theorem is :
6
(a) E (b) 2
(c) 0 (d) 1
YBC??15194 1 (Could)
(vii)'l?hc area boundcd by the curve )4 = g(y): yvaxis and y = a. y = b is :
b b
(a) lde (b) dey
'd a
b 2 b ,Zd.
(c) IV d" (d) J" -?
a 4
(viii) T he functions f and g be :
(N
(X)
(b)
(c)
3- (p)
(Q)
(i) continuous in [:1. h]
(ii) derivable in (a, b) and
(iii) g'(x) ,t 0 for all x e (a, bi.
These are the hypothesis of mean value theorem by :
(a) Rolle?s (b) Lttgrangc's
(c) Cauchy?s (d) Lcibnitz
The function f(x) has the removable discontinuity if :
(8) f0?) it f(X') (b) f(X?.) = f(x ) i f(X)
(c) f(x?), f(x') do not exist (d) None of these
d
a cosh x [S t
(a) sinh x (b) ?-sinh x
(c) h sinh x I (d) ??h sinh x
UNlT?I
ll? lim l?().)=? and lim g(xium, then prove that :
X?9Xo \-*.\i?
lim iif(x)+g(x)]: lim f(x)+ lim g(x)
X?PXO X??\U \-?)\'0
= 1? + tn. 4
Prove that the function (lCllllCd by ['(x') = x: is continuous l'Ur all x e R. 3
Using de?nition of limit. ?rms that :
2
. x ?2x ?x?6
hm ?????? : l4 3
x?a3 x ~ 3
De?ne limit of a function 3rd show that the limit of a function if it exist. is
unique. 1+3
Prove that lim x2 :4; by using e-d de?nition. 3
x??2
YBC?15194 2 (C0md.)
(r)
(b)
(C)
5- (p)
(q)
(r)
(b)
(e)
7- (p)
(q)
(r)
I A(X
lf f(x)=l?:~c?1?,x?0,
= 0, , x = 0
then show that f(x) has a simple discontinuity at x = 0.
UNlT?H
Prove that if f(x) is differentiable at x = x0. then it is continuous at x = x0. Is converse
of this statement true ? Justify.
Evaluate :
2
lim (cos x)cm "
x?>0
If y = A sin mx + B cos mx, then prove that y2 + m?y = 0.
[f y = sin(m sin 'x), then show that :
(i) (l ? x?)y2 - xy. + mzy = 0
(ii) (1 ? x2)ym2 ? (2n + 1)xyml ? (n2 ? m2))/" = 0.
[1 n
~ ?hcn Prove that y 42%
If = .
y ax+b " (alx+b)n+l
Evaluate :
x?sinx
hm 3
x?>0 x
UNIT?III
State and prove Lagrangc?s mean value theorem.
Verify Cauchy mean value theorem for the functions :
f(x) = e" and g(x) = c? in [a, bl.
Expand sin x in powers of x_?, upto ?rst four terms.
2
State and prove Cauchy?s mean value theorem.
Expand 3x3 + 4x2 + 5x ? 3 about the point x = 1 by Taylor?s theorem.
Verify the Rolle?s theorem for the function :
f(x) = i"; in [0, n],
C
UNIT?IV
If u = f(x, y, z) is a homogeneous function of degree n, then show that :
au Eu 6U
x?+y??+z-??=nu
8x ay 02 '
YBC?- 15194 3
5
(C ontd.)
(b) Verify liulcr's thcorcm for u 2:13;} + Z: 3
y z x
(c) Ifu = e? (x cos y -A y sin y), thcn ?nd the value u? + u?. 3
9. (p) If-u = f(x. y) be homogcncous function of degree n then prove that :
? Bu (?u ? _ ~ _
(1) 5g are homogcncous iuncltons of degree ?n ,, l 111 x, y and
-2 ?lu ?2?
.. u u c L .
(u) x2-r?7+2.\'\?;~?4?3?;?-n(nvl)u. 4
0x' ("My Ex"
(q) If u = 3(ax + by + Cl) 7 (3,3 + y; t 7}) and a: ? b" ?- c3 ; 1. then show that :
a2 2 -
0 u 0 u 6 u
6x By 67?
x ?v4
(r) [f u=log?-?'??, x :t y, thcn proxtc thut :
x ? y
(i) xu?+yu?=3
(ii) xzuH + 2xy u? v y" u? = ?3. 3
UNIT?\?
10. (a) Prove that:
? :n-l n?i
_ 5m xcos x 11?1 . n a
[Sirl'nx cos"x dx = ~ ?-??-?+ J?smhxcosn 2x dx. 4
m + n m + n
(b) Evaluate:
2
.\ +2x c 3
I?=:~-?dx t
A 3 J
V x? + x +1
(c) Show that ?8a? is the length of an arc of thc cycloid \' -, a(t a sin t). y ~? a(l ? cos I);
0 S t S 27:. 3
11. (p) Prove that :
1
14m?? x
J?tannxdxz-~7? ?l 7
1? _1 H ?...
Hence evaluate Itan3x d3: . 4
. . x .
(q) Fmd the area bounded b) the (-ales. the curve _\ = c cosh? and thc ordmatcs .\' = 0,
c
x = a. 3
(r) Show that length of the curu- _\ ?? log sec x bcmccn the points, where x = 0 and
n _ --
x = ?1510gc(2+\/3].
3
L.)
YBC?lSl94 4 52s
This post was last modified on 10 February 2020