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

Ex/ML/12B/143/07(OLD)
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FIRST ENGG. (Mech.) EXAMINATION, 2007
(2nd Semester, Old Syllabus)
MATHEMATICS—II J
Time : 3 hours Full Marks : 100
ans any 5 ques. .
1. a) From the definition of definite integral obtain the value of
(i) ?
a
b
exdx (ii) ?
a
b
Sinx dx 3+3
b) If f(t) is bounded and integrable on [ a, b ] and if
F(x) = ?
a
x
f(t) dt, oeoeoeoeoe x ?( a, b ]
then prove that
(i) F is continuous on [ a, b ], 7
(ii) if f is continuous on (a, b) then F/(x) exists in (a, b)
and F/(x) = f(x) on (a, b). 7
c) Evaluate ?
0
p
x log sin x dx. 7
2. a) Prove that lt = (m > –1).
4
b) Show that ?
0
p
x Sin2 x dx = . 5
c) Test the convergence of the integrals
(i) ?
0
p/2
xm cosecnx dx (ii) ?
0
8
1m+2m +3m +....+ nm
nm+1
1
n?8 m+1
p2
4
dx
(1+x2)4
6. a) Evaluate
(i) Lim ( )1/x (ii) Lt (1 + 1—
x )x 7+4
b) State and prove Lagrange's Mean Value Theorem. 6
c) Use Mean Value Theorem to prove that
| sin x – sin y | = | x – y | oeoeoeoeoe x, y ? R. 3
7. a) Examine the subsequent functions for maximum and
minimum values
(i) f(x) = x5 – 5x4 + 5x3 – 1
(ii) f(x) = x4 – 8x3 + 22x2 – 24x + one 6+6
b) obtain the infinite Maclaurin's series of the function
f(x) = log(1+x). 8
8. a) Show that for the function
f(x, y) = xy . , x2 + y2 ? 0
= 0 x2 + y2 = 0
fxy(0, 0) ? fyx(0,0). 8
b) State and prove Euler's theorem for homogeneous
functions of 3 variables. 8
c) When a function of 2 variables is called
differentiable ? State a sufficient condition for a function
f(x, y) to be differentiable. 4
––––––––×–––––––
( two ) ( three )
(iii) ?
–8
8
4+4+3
3. a) Show that G (n + 1—
2 ) = 7
b) Show that ?
0
p/2
Sin4 ?Cos6 d? = 3
c) Show that ?
0
8
e–x4x2dx × ?
0
8
e–x4dx = 6
d) Show that ?
0
p/2
d? = 1—2
4
4. a) Show that
?
0
1 dx ?
0
1 dy ? ?
0
1dy ?
0
1 dx 8
b) Evaluate ??
R
(1 – – )dxdy where R is the part of
the ellipse + = one in the positive quadrant. 8
c) Evaluate ?
0
p/2
?
0
p
cos (x + y) dxdy. 4
5. a) State and prove Leibnite's Theorem. 9
b) If y = cos (sin–1x) then prove that (1 – x2)yn+2 –
(2n–1)xyn+1 + (1 – n2)yn = 0 and hence obtain (yn)0, where
yn denotes the nth derivative of y. 9
c) State Rolle's Theorem. 2
dx
x3
G(2n + 1) p
22n G(n + 1)
3p
512
p
8 2
sin2m–1?cos2n–1?
(asin2? +bcos2?)m+n
G(m) G(n)
ambn G(m + n)
x–y
(x+y)3
x–y
(x+y)3
x2
a2
y2
b2
x2
a2
y2
b2
tan x
x?0+ x x?8
x2– y2
x2 +y2

Earning:   Approval pending.