Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 5th Sem Mathematics Mathematical Methods Previous Question Paper
B.Sc. (Part?Ill) Semester?V Examination
SS 2 MATHEMATICS (New)
(Mathematical Methods)
Paper?X
Time : Three Hours] [Maximum Marks : 60
Note :? (1) Question No.1 is compulsory and attempt it once.
(2) Solve ONE question from each Unit.
1. Choose the correct alternative (1 mark each) 2
(1)
(ii)
(Iii)
(iv)
(V)
If pn(x) is the solution of Legendre?s D.E., then pn(?1) is :
(a) -1 (b) l
(C) (-1)? (d) 0
The value of integral IX P100 dX ,where p (x) is chendre? s polynomial of degree 1,
-l
equals:
, 3 b 1
i d
(c) 15 <) 0
The value of J 100?) equals :
) 1? i cos x b 3? sin x
(a mt ( ) mt
mt mt .
(c) T cos x (d) ~2? sm x
Eigen ?mctions corresponding to different Eigen values are :
(a) Linearly dependent (b) Linearly independent
(c) Real (d) None
The coef?cient in a halfrange sine series for the function f(x) = sin x de?ned on [0, i] is
given by :
i . mcx mrx
sm x cos ?? dx cos x cos ?? dx
2 ? . . ?
(c) E Ism x sm BZ?X dx (d)%jsin x sinn ?? de
o 0 .
YBC?15305 l (Contd.)
(w') The function f(xt = (7 sin x): is :
(a) Odd (b) liven
(c) Even and Odd (?d) None 0fthcse
(vii) If Lff(t)] = F(s). thcr L[f(at)] is :
1 t s
(a) F(s?a) (b) ""(?J
(viii) The value of L"[~Al J is :
_s ? a _;
(3) 1 (b) t
(c) c? (d) e?
(ix) The Fourier sinc transform af?x) = e ?x'. x 2 O is :
l X
(a) - + 2'? (b) 1 _ )3
27? 1
(C) 1 _ )3 (d) ET
(x) [fF[f(x)] = F(A), then the Fourier transform of f(ax) is :
K 1(7?
(a) F[?] (b) V-IF (?Ja?
a la! a
J?FO) a?O d ??F[)?] 3:0 10
(9) 12.1 H la! a,
UNlT?I
2. (a) Show that pn(x) is the coef?cient of h" in the ascending power series expansion of
(1 ? 2xh * h3)""'3. 5
(b) Prove that npn = xp'1 - pi, l 3?
l 2
(c) Prove that x =?p0(x) p(x). 2
YBC?l 5305 2 (Contd.)
(Q)
4. (a)
(b)
(C)
(q)
(r)
6. (a)
(b)
7. (p)
(q)
8. (a)
(b)
(C)
2
2n+ll
1
Prove that [[px(x)]zdx =
-l
d?
2?n! dx"
Prove that px(x) = (x2 -1)?.
UNIT?ll
Prove that 13,2(x) = "?2? (5m X ? cos x).
TIX x
Prove that XJL = pJp ? xde.
b
Evaluate [10(X) ?J.(x)dx,
Prove that Eigen values of the S-L problem are real.
Prove that (xP-Jp) = x"Jp_l
Prove that J_L,2(x) =41 cos x.
nx
UNlT?III
a0
L)
DJ
If the trigonometric series 3 + 2 (an cos nx + bn sin nx) converges unifome t0 f(x) in
n-I
c S x < c + 2n, then ?nd the Fourier coef?cient of f(x).
Obtain Fourier Series in [0, 2] for the function f(x) ?? x1.
Obtain Fourier Series in [?n, 1t] for the function :
f() ?1t,?1:
x ,0
UNIT?IV
?
Prove that th" -f(t)| = (?l)n :n
5
Find L[sin t - cos 2t - cos 3t].
F(s), n = 1,2,3 .....
1
Show that [.(t") = 1%? s > 0.
s
YBC?IS305 3
(J I
U!
(Contd. )
(q)
(r)
10. (a)
(b)
(c)
11. (p)
(q)
YBC?l 5305
Solve the 0.1:. y" ~ 4y' ~ -81, y(O) ? yw) = 0.
1
Fmd the Inverse Laplace transtorm of (S _ 23(5 + 2): by usmg Convoluuon theorem.
pmve chatL (mu) ?~ 53L(u(x. t): ? su(x, O) ? u?(x. O).
UNlT?V
Find the ?nite Fourier sine and cosine transform of f(x) = sin 2x in (0. 1:).
F ind the Fourier transfom1 0f the function :
f( )_ l, |x[
(
Prove 'that If?(x) sing?;? dx 2: ?n??Ttl?;(:n).
O i
Find the Fourier sine and cosine transfom of the function f(x) = x?? ', n > 0.
Find ?nite Fourier cosine transform of u? and u?; where u = u( <, t).
[J
'JI
525
This post was last modified on 10 February 2020