Jadavpur University 2007 B.E Mechanical Engineering MATHEMATICS-IVM - Question Paper

FIRST ENGG. (Mech.) EXAMINATION, 2007
(2nd Semester)
MATHEMATICS–IV M
Time : 3 hours Full Marks : 100
(50 marks for every part)
Use a separate Answer-Script for every part.
PART–I
ans any 6 ques..
All ques. carry equal marks.
Two marks for general proficiency
(Symbols : p = ; q = , D = ; D/ = )
1. Form the partial differential formula by eliminating f, where
f (x + y , e2 y (( x + y)2 + z2)) = 0.
2. Solve : (x – y)p + (x +y)q = 2xz
3. obtain the integral surface of
x(y2 + z) p – y (x2 + z) = z(x2 – y2) which contains the line
x + y = 0, z = 1.
4. Using Charpits method solve : 2(z + xp + yq) = yp2.
5. Solve : (4D2 – 4DD/ + D/2)z = 16 log (x + 2y).
Ex\ME\MATH\T\122\116\07
[ Turn Over ]
? z
? x
? z
? y
?
? x
?
? y
6. Solve : (D2 – DD/ + D/ – 1)z = cos (x + 2y) + ey.
7. Solve, the 1 dimensional heat flow formula :
= c2
8. decrease cannocial form and solve :
– y4 – 2y3 = 0.
PART–II
ans any 5 ques..
All ques. carry equal marks.
9. obtain the Fourier series to represent ( x + x2) in (– p , p)
and hence deduce that
= one – + – + ............. .
10. a) State the Dirichlet’s conditions for a function f(x) described
in the interval (a, b).
b) obtain the Fourier series for
f(x) = – one , – p < x < 0
0 , x = 0
1 , 0 < x < p
and hence deduce that = one – + – + .........
? u
? t
? 2z
? x2
? 2z
? x2
? 2u
? y2
? z
? y
p2
12
1
32
1
42
1
22
p
4
13
15
17
11. find the Fourier series for f(x) = e–x in the interval
( 0, 2p).
12. obtain the Fourier series for f(x) where
f(x) =
0 , – p < x < 0
, 0 < x < p
and hence deduce that = one + + + .........
13. obtain the Fourier series of f(x) where
f(x) =
x , 0 = x < p
2p – x , p = x = 2p
and hence deduce that = one + + + .........
14. Express f(x) = x as a half-range sine and cosine series in
0 < x < 2.
15. Determine the Fourier series expression for y as a function
of x upto the 2nd harmonics from the subsequent table :
x 0 30 60 90 120 150 180 210 240 270 300 330
y 1.80 1.10 0.30 0.16 1.50 1.30 2.16 1.25 1.30 1.52 1.76 2.00
px
4
p2
8
1
32
1
52
p2
8
1
32
1
52
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Earning:   Approval pending.