Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 3rd Sem Mathematics Advanced Calculus Previous Question Paper
B.Sc. (Part?II) Semester?III Examination
MATHEMATICS
(Advanced Calculus)
Paper-V
Time : Three Hours] [Maximum Marks : 60
Note :?(1) Question No. l is compulsory. Attempt once.
1.
(2) Attempt ONE question from each unit.
Choose the correct alternative :?
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Every Cauchy sequence of real number is _ _
(a) unbounded (b) bounded
(c) bounded as well as unbounded (d) None of these
11
The sequence
" n n+1 _
(a) monotonically increasing (b) monotonically decreasing
(c) constant sequence _ (d) None of these
. . 1 .
The harmomc senes 2; lS _ _
(a) Convergent (b) Oscillatory
(c) Divergent (d) None of these
Let 2a" be a series with positive terms and lim 3:.? = I, then the series Ea? is convergent
' n--)co
if
(a) I=l (b) l>l
(c) l < l (d) None of these
If 11:3? f(P) = f(Po); where P, Po 6 R2 then __
(a) f is discontinuous at P0 (b) f is continuous at P0
(c) f is continuous at P (d) Nonevof these
If lim f(x, y) = I exist then repeated limits are
(X. )?)~*( x9 .m
(a) equal (b) not equal
(c) not exist ? (d) None of these
The function f(P) has absolute minima at P0 in D if .
(a) f(P) S f(Po) ; V P e D (b) f(P) 2 f(Po) ; ? P e D
(c) f(P) = f(Po) ; V P e D (d) None of these
YBC?15253 l (Contd.)
(8) If u : 2x ? y and V : x + 4y thzn J? ?v _
(a) 1 (h) 9
1 .
(c) 5 (d) None of these
(9) The value of I I Xzy dydx is W '7
l l
_ 3
(3) ~1 (b) ?g
a d 1
(c) 3 < >
1 I l
(10) The value of J- I Idx dy :12 is __ _--
O 0 f)
(a) 0 (b) 2
(c) -1 (d) 1 IO
UNlT?I
2 (a) Prove that a convergent sequence of a real numbers is bounded. 5
(b) Show that the sequence
. . n.
3. (p) Prove that every convergent sequence of real numbers is a Cauchy sequence. 5
(q) Show that the sequence Sn, where S" = (I + l] , is convergent and that Hm (I + 1) lies
\ [1,, n-+z~ n
between 2 and 3. 5
UNIT?ll
4 (a) Prove that the series an converges if and only if for every 6 > 0, 3 a M(e) e N such
thathnZMslxw+xM+ ...... +xm e. 5
~ ?I + 1 + 1 +
(b) Test the convergence of the seams x(x +2) (x + 2) (x + 4) (x + 4) (x + 6) -------- ,
x e R, x at 0. 5
5. (P) Prove that p-series 2?1?- is convergent for p > 1 and divergent for p S l. 6
n
. n1 + a
(q) Test the convergence of the sencs Z n v ne N. 4
YBC??15253
+8.
I?)
(Contd.)
6 (a)
(b)
(C)
7. (p)
(q)
(r)
8 (a)
(b)
9. (p)
(q)
10. (a)
(b)
11. (p)
(q)
UNlT?III
Prove that < x.y )L'B.-n(3" ?2y) = 14 by using e - 8 de?nition of a limit of a function.
4
Expand x3 + y3 ? 3xy in powers of x - 2 and y ? 3. 3
Let real valued functions f and g be continuous in an open set D g R2 then prove that
f + g is continuous in D. 3
Prove that the function f(x, y) = x + y is continuous v (x, y) e R2. 4
Expand e"y at the point (2, l) upto ?rst three terms. 3
Let f(x, y) = ?2?-y?y2-, show that the Stmultaneous ltmtt does not etht at the origin in
spite of the fact that the repeated limits exist at the origin and each equals to zero.
3
UNlT?IV
Find the maximum and minimum values of x3 + y3 - 3axy. 5
Find the least distance of the origin from the plane x ? 2y + 22 = 9 by using Lagrange?s
method of multipliers. 5
If x, y are differentiable functions of u, v and u, v are di?'erentiable ?mctions of r, s
then prove that
00w) 6(M) = 6(w)
6(u,v) 6(r,s) 6(r,s) '
6(x,y,z)
lf xu = yz, yv = xz and zw = xy ?nd the value of .
6(u,v,w)
UNlT?V
Evaluate by changing the order of integration :
j T ;dy dX. 5
Evaluate I(2x+y)dv, where v is the closed region bounded by the cylinder
z=4?x2vandtheplanesx=O,x=2,y=0,y=2,z=0. 5
Evaluate by Stoke?s theorem I(e?dx+2ydy?dz) , where c is the curve x2 + y2 = 4,
z = 2. ? 5
Evaluate by Gauss Divergence theorem H ?.?ds; where
f = (x2 ? yz)i +(y2 --zx)j+(z2 - xy)k and s is the surface of rectangular parallelepiped
OSxSa;OSySb;OSzSc. 5
YBC?l5253 3 625
This post was last modified on 10 February 2020