Download SGBAU BSc 2019 Summer 5th Sem Mathematics Mathematical Analysis Question Paper

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 5th Sem Mathematics Mathematical Analysis Previous Question Paper

AW?l 736
B.Sc. (Part?III) Semester?V Examination
MATHEMATICS (New)
(Mathematical Analysis)
Paper?IX
Time : Three Hours] [Maximum Marks : 60
NB. :? (1) Question No. l is compulsory. Attempt once.
(2) Attempt ONE question from each Unit.
1. Choose the correct alternatives :
(i) Let f: [0, l] ?) R be Riemann integrable. Which of the following is always true :
(a) f is continuous
(b) f is monotone
(c) f has only ?nite number of discontinuities
(d) the set of discontinuties of f may be in?nite '? l
.. . . .3 dx . .
(n) An 1mproper mtegral I?p , a e R 15 convergent 1f :
x
(3) P<1 (b) p>l
(C) p21 (d) p=l 1
(iii) B(m,n)is:
(m + n)
(a) W l? (b) m m 7
IE I3 [E m
(C) (m + n) (?0 Wm 2?11) 1
(iv) In the real line R. which of the following is true '.?
(a) Every bounded sequence converges (b) Every sequence converges
(c) Every Cauchy sequence converges (d) None of the above 1
(v) Every neighbourhood is a/an :
(a) Closed set (b) Open set
(c) Open closed set (d) None of the above 1
(vi) A function u(x, y) is harmonic in region D if:
(a) u? ? uW = O (b) u"y + uyx = O
(c) uw-uyx=0 (d) un+uyy=0 1
YBC?l 5304 l (Conld.)

.. , . ? t . . .
(vu) fhe functmn t(z) 2 VI xyt IS at the orlgm.
(a) Harmonic function (b) Analytic function
(c) Conjugate ?mctior. (d) Not analytic fmction 1
(viii) Ii'f(7.) and {(?23 are both analytic functions then f(z) is :
(a) ldentically zero (b) (?onstant
(c) Unboundcd (d) None of the above 1
(ix) The points z where 'e? 10 form a :
(a) C mlc (b) Straight line
(c) Hmcrbola (d) Parabola 1
(x) A bilincar transformation vs ith two non-in?nite ?xed points a and [3 having Normal form
w ? a 7. ? a.
V;? = R[j] is Elliptic 1:?;
(a) |kl? Lkis real (b) k: l,kis not real
(c) lkl = 1 (d) None Ofthe above 1
UNlT?l
2. (a) Prove that every continuous function is integrable, 4
(b) Let the function fhe de?ned :15
f(x) = 1. when x ix rational
== ?1, when x i:- irrational
Show that f is not R-intcgrabie over [0, l] but It? 6 R [0, l]. 3
(c) Show that any constant function de?ned on a bounded Closed intc 'val is integrable. 3
b)
(p) Iffis a bounded and integrable function over [a, b] and M, m are bounds offovcr [a. b],
prove that :
b
m(b?a)s If(x)d.\ SM(h?a)_ 4
' Prove that ~2? < r ?k -/ ? "? ' 3
(q) 17 _Jl1+x? ' "'?'
b
(r) lffis continuous and non-negative on [a. b], then show that Jfo.) dx 2 0. 3
YBC?l 5304 ' 2 (Cnntd.)

(b)
(C)
5. (1))
(<0
(I)
6. (a)
(b)
(C)
7. (p)
(Q)
(r)
8. (a)
(b)
UNlT?ll
b
d . . . .
Prove that the integral I(x xa)? converges 1f p < 1 and dwerges 11 p 2 l. 4
a.
1 sin x
Show that I x2 dx converges absolutely. 3
|
Show that Ie'x dx converges. 3
0
m n
Prove that B(m, n) = FT- . 4
m + n
x/Z 1t
Prove that I sin2 6 cos? 6 d6 = ?. 3
0 32
Prove that (n + 1) = nl'(n?) . 3
UNlT?III
[f f(z) = u(x, y) + iv(x, y) be analytic in a region D, then prove that u(x, y) and V(X, y) satisfy
Cauchy-Riemann equations. 4
If f(z) and f (2) are analytic functions, prove that f(z) is constant. 3
Show that u = 2x ? x3 + 3xy2 is harmonic and ?nd its harmonic conjugate function. Hence
?nd f(z) = u T iv. 3
au av . au 6v .
If u and v are harmonic in region R, prove that 5;" ? 5x? + 1 a ? 5; IS analytic
in R. 4
If the function f(z) = u + iv be analytic in domain D then prove that, the family of curves
u(x, y) = c1 and V(x, y) = c2 form an orthogonal system, where el and c2 are arbitrary
constants. 3
Determine a, b, c, d so that the function f(z) = (x2 + axy + byz) + i (cx2 + dxy + yz)
is analytic. 3
UNlT?IV
Prove that, every bilincar transformation with two non in?nite ?xed points a, B is of the
form w _ o. = k z ? a , when k is constant. 5
W - B Z - B
11/4
Under the transformation w = x/Ee 2, ?nd the image of the rectangle bounded by
x=0,y=0,x=2andy=3. 5
? YBC?IS304 3 - (Contd.)

9. (p)
(q)
10. (a)
(b)
(c)
11. (p)
(q)
(0
UI
Prove that the croas ratio remains iman'unt under a bilinear transformation.
Prove that under the transformation w = i: M i the region Im(z) 2 0 is mapped into the
region |w| S 1. 5
lNlT?V
Show that d(x, y) = Ix )L Vx, y e R de?nes a metric on R. 5
De?ne :
(i) Limitpoint
(ii) Boundary point. 2
Prove that every neighbourhood is an open set. 3
De?ne :
(i) Complete metric space
(ii) Open set. 2
Prove that every convergent sequence in a metric space is a Cauchy sequence. 3
Let X be a metric space. If {x3} and {yd} are scqucnccs in X such that x? ?> x and
y" ?) y then, prove that d(xn, y") ?> d(x, y). 5
YBC?15304 4 525