Annamalai University B.Sc. chemistry:MathematicsII question paper december 2014

Annamalai University question paper december 2014
B.Sc. DEGREE EXAMINATION, December 2014
(CHEMISTRY)
(SECOND YEAR)
(PART-III)
(GROUP-A: MAIN)
660/650. MATHEMATICS-II
(Common with B.Sc Applied Chemistry/Electronic Science/Physics)
Time: Three hours Maximum: 75 marks
 Answer any FIVE questions
 All questions carry equal marks (5×15=75)

 1. a)
If In= ∫
0
0
sin
π
x xdx n
and n is a positive integer, prove that
 In+n(n-1)In-2=n( 1
)
2
π n−
 b)
Evaluate ∫ ∫ +
2
0
2
0
sin( )
π π
θ φ dθ dφ
 2. a) The temperature at any point(x,y,z) at time t is
( , , , ) 2 cos( ). 2 2
φ x y z t = x y + yz t − xt find the rate of change of
temperature, with respect the point (3,-2,1) with velocity
v = 2i − j + k at time t=0.
 b) Show that A 6( xy z )i _ 3( x z) j 3( xz y)k
3 2 2
= + + − + − is irrotational.
 3. a)
Solve y =px+
p
a
 b) Solve (D2-3D+2)y=sin3x.
 4. a) Solve p4x2+2px-y=0
 b) Solve (D2-2D+4)y=x cos x.
 5. a) Using Lagrange’s method , solve (y+z)p+(z+x)q=x+y.
2
 5288
 b) Obtain the general solution of the equation
3p+2q2=4pq
 6. a) Solve p+q=sinx+siny
 b)
Solve y2z .
2 2
x y
y
z
x z
x
z
=
∂
∂
−
∂
∂
 7. Find a cosine series in the range 0 to π fir the function
f(x) = x 0<k<
2
π
 π − k π
π
< k <
2
Heme prove that f(x)= 





− + + +.....
5
cos10
3
cos6
1
2 cos2
4
2 2 2
k k x
π
π
 8. a)
Find L-1 





( +1)( + 2 + 2
1
2
s S s
 b) Find L[sin 2t cost cos 2t]
3
+
9. Solve the differential equation y''-3y'+2y=e3k given that y(0)=y'(0)=0, using
Laplace transform.
10. Find the Fourier transform of f(y) = 1-y2 for -1<y<1
0 otherwise
Hence, evaluate dy y
y
y y y













 −
∫
∞
2
cos
sin cos
0
3
********