Download PTU B-Tech AR-Automation-And-Robotics 2020 Dec 3rd Sem 76502 Mathematics Iii Question Paper

Download PTU (I.K.Gujral Punjab Technical University (IKGPTU)) B-Tech (Bachelor of Technology) (AR)- Automation-And-Robotics 2020 December 3rd Sem 76502 Mathematics Iii Previous Question Paper

Roll No.
Total No. of Pages : 02
Total No. of Questions : 18
B.Tech. (Automation & Robotics) (2018 Batch) (Sem.?3)
MATHEMATICS-III
Subject Code : BTAR-303-18
M.Code : 76502
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 q ues tions.
3 .
SECT ION-C contains THREE questions carrying T EN marks e ach and s tudents
have to atte mpt any T WO questio ns.
SECTION-A
Write briefly :
1.
Find the Fourier series of the function f (x) = | x | over the interval [?2, 2].
2.
Find Laplace transform of e?t sin2 t.
3.
State and prove Second Shifting Property for Laplace transform.
2s 3
4.
Find inverse Laplace transform of
.
2
s 4s 13
5.
Express sum of Legendre polynomials 8P4(x) + 2P3(x) + P0(x) in terms of powers of x.
6.
For Legendre polynomial Pn(x), show that Pn (?x) = (?1)n Pn(x)
7.
Form a partial differential equation by eliminating arbitrary function f from the relation
2
1
z y 2 f
log y
.
x
8.
Solve z (xp ? yq) = y2 ? x2.
9.
Show that the function u (x, y) = 2x + y3 ? 3x2y is harmonic.
10. State Cauchy Integral Theorem.
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SECTION-B
11. Find the Fourier series expansion of the function
0,
for x 0

1
1
1
f (x)
. Deduce that
1
.....
1,
for 0 x

4
3
5
7
12. State and prove Convolution Theorem for Laplace transform.
13. For Legendre polynomial Pn(x), show that :
1
2
P (x) P (x) dx
,
m
n
for m = n.
2n 1
1
14. Solve by Charpit's method z = p2x + q2y.
3z 5
15. Evaluate
dz,
C : | z | = 1.
2
z 2z
C
SECTION-C
16. a) Using Laplace transform, solve y ? 6y + 9y = e3t t2, y (0) = 2, y (0) = 6.
2s 1
b) Find inverse Laplace transform of
.
2
2
(s 2) (s 1)
17. a) Solve Legendre differential equation (1 ? x2) y ? 2xy + n (n + 1) y = 0.
b) Using the method of separation of variables, solve
2
u
u
k
, u (x, 0) = x2, u (0, t) = u (2, t) = 0.
2
t
x
1
18. a) Find all Taylor and Laurent series expansions of f (z)
about the point z = i.
2
z 1
b) Compute the residues at the singular points z = 1, ? 2 of
2
1 z z
f (z)
2
(z 1) (z 2)
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.
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