Download PTU B-Tech ME 2020 Dec 5th Sem 70601 Mathematics Iii Question Paper

Download PTU (I.K.Gujral Punjab Technical University (IKGPTU)) B-Tech (Bachelor of Technology) Mechanical Engineering 2020 December 5th Sem 70601 Mathematics Iii Previous Question Paper

Roll No.
Total No. of Pages : 02
Total No. of Questions : 18
B.Tech. (ME) (2012 Onwards) (Sem.?5)
MATHEMATICS-III
Subject Code : BTAM-500
M.Code : 70601
Time : 3 Hrs. Max. Marks : 60
INST RUCT IONS T O CANDIDAT ES :
1 .
SECT ION-A is COMPULSORY cons is ting of TEN questions carrying TWO marks
each.
2 .
SECT ION-B c ontains F IVE questions c arrying FIVE marks eac h and s tud ents
have to atte mpt ANY FOUR questio ns.
3 .
SECT ION-C contains THREE questions carrying T EN marks e ach and s tudents
have to atte mpt ANY TWO questions .
SECTION-A
Write briefly :
1.
Expand f (x) = | sin x | in Fourier series.
2.
Find Laplace transform of sin h t cos2 t.
at
bt
e
e
3.
Find Laplace transform of
.
t
7s
e
4.
Find inverse Laplace transform of
3
(s 3)
5.
Express x 4 + 2 x 3 ? 6 x 2 + 5x ? 3 in terms of Legendre polynomials.
n(n 1)
6.
For Legendre polynomial P
n(x), show that P (1)
n
2
7.
Form a partial differential equation by eliminating arbitrary functions from the relation z
= y f (x) + x g (y).
8.
Solve x p + yq = 3z.
9.
Show that the function f (z) = | z |4 satisfies the Cauchy-Riemann equations only at region.
10. State Cauchy Integral Theorem.
1 | M-70601
(S2)-907

---

SECTION-B
11. Find the Fourier series expansion of the function f (x) = x2, ? < x < . Deduce that
2

1
1
1
1
.....
2
2
2
2
6
1
2
3
4
12. State and prove Convolution theorem for Laplace transform.
13. For Bessel's function Jn(x), show that 2
2
2
2
J
0 2 ( 1
J
J2 J3 .....) 1
14. Solve by Charpit's method q + xp = p2.
dz
15. Evaluate
, C : | z i | 2

2
2
(z 4)
16
C
SECTION-C
16. a) Using Laplace transform, solve y + 2y = 1 ? H (t ? 1), y (0) = 2, where H (t) is
Heaviside's unit step function.
1
b) Find inverse Laplace transform of
.
2
s (s 1)
17. a) Using Frobenius method, find two linearly independent solutions of the equation
2x2y + xy ? (x2 + 1) y = 0.
b) A rod of length l with insulated side is initially at a uniform temperature u0. Its ends
are suddenly cooled at 0?C and kept at that temperature. Find the temperature
function u (x, t).
1
18. a) Find all Taylor and Laurent series expansions of f (z)
about the point z =0.
z (z 1)
2
z
b) Compute the residues at all the singular points of f (z)
.
2
2
(z 1)
NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.
2 | M-70601
(S2)-907

---