B.E-B.E 2nd Sem Engineering Mathematics -II(University of Pune, Pune-2013)

UNIVERSITY OF PUNE
[4361-7]
F.E. Examination 2013
Engineering Mathematics -II
(2008 pattern)

Time : 3 Hours                                                                                                  Max. Marks : 100

[Total No. of Question=12] [Total no. of printed pages= 5]

Instructions:

(1)In Section -I Solve Q.1 or Q.2,Q.3 or Q.4,Q.5 or Q.6. In Section-II Solve Q.7
or Q.8,Q.9 or Q.10,Q.11 or Q.12
(2)Neat diagrams must be drawn wherever necessary.
(3)Figures to the right indicate full marks.
(4)Use of electronic pocket calculator is allowed.
(5)Assume suitable data wherever necessary.
SECTION-I
Q.1 (a)Form the differential equation whose general solution is
X −A2Y−B2=16 where A and B are arbitrary constants . (6)
(b)Solve any two. (10)
(i) y  x2 ye x dx−e xdy=0
(ii) dy
dx= xy1
2x2y3
(iii) dy
dxy cot x=sin2x
OR
Q.2 (a)By eliminating arbitrary constants a & b find the differential equation whose
general solution is y=log cos (x−a)+b. (6)
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(b)Solve any two. (10)
(i) dy
dx=e x−y  ex−e y 
(ii) dy
dx= tan y−2xy−y
x2−x tan2 y+sec2 y
(iii)  y4−2x3 y dx x4−2xy3 dy=0
Q.3 Solve any three. (18)
(a)If the temperature of the body drops from 1000C to 600C in one minute,the
temperature of surrounding being 200C ,what will be the temperature of the
body after two minutes?
(b)In a circuit containing inductance L,resistance R and voltage E,the current I is
given by E=RIL dI
dt .Given L=320H , R=125Ω and E =250 volts, I being zero
when t=0. Find the time that elapses before current reaches half of its theoretical
maximum.
(c)A particles is moving in a straight line with an acceleration k [x a4
x 4 ] ,directed
towards origin. If it starts from rest at a distance 'a' from origin, prove that it will
arrive at origin at the end of time
π
4√k .
(d)Find orthogonal trajectories of the family of curves given by x2+2y2=c2
where c is arbitrary constant.
OR
Q.4 Solve any three. (18)
(a)A pipe 10 cm in diameter contains steam at 2000C .It is protected with a
covering 5cm thick, for which k=0.12 and outside surface is at 500C .Find the
temperature half way through the covering under steady state conditions.
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(b)A body of mass m falls from rest under the influence of gravity and a retarding
force, due to air resistance, proportional to instantaneous velocity of the body.
Find velocity and distance described as a function of time.
(c)The charge 'Q' on the plate of a condenser of capacity 'C' charged through a
resistance 'R' by a steady voltage 'V' satisfies the differential equation.
R dQ
dt Q
C =V .If Q=0 at t=0, show that Q=CV [1−e−t /RC ] .Find the current
flowing into the circuit.
(d)The amount x of a substance in a certain chemical reaction at time t is given by
dx
dt  x
10=2−1.5 e−t /10 .If at t=0, x=0.5, find x at t=10.
Q.5 (a)Find the Fourier series to represent the function f  x =x2 ,in the interval
−x and f  x2= f  x  for all x.
Deduce that 1
12 1
22 1
32....=2
6 . (9)
(b)If I n=∫0
4
cos2n x dx , prove that I n= 1
n 2n−1+2n−1
2n
I n−1
Hence evaluate ∫
0
4
cos6 x dx . (7)
OR
Q.6 (a)Find the Fourier series upto first harmonics to represent f(x) in the interval
0,2 from the tabulated values of x & f(x) given below. (8)
x 0 
3
2π
3
π 4
3
5
3
2
f(x) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
(b)Evaluate ; ∫
0
∞
 x e−x dx . (4)
3
(c) ∫
3
7
 x−31/ 47−x 1 / 4 dx . (4)
SECTION-II
Q.7 (a)Trace the following curves.(any two). (8)
(i) 3ay2=x  x−a 2
(ii) r=a 1cos
(iii) x=t2 , y=t−t3
3
(b)Using DUIS evaluate. (5)
∫0
∞ e−x
x (a−1
x+1
x
e−ax)dx
(c)Using proper rectification formula, find the circumference of the circle of
radius a. (4)
OR
Q.8 (a)Trace the following curves (any two). (8)
(i) a2 y2=x2 2a−x  x−a
(ii) x=a tsin t , y=a 1−cost 
(iii) r=a+b cosθ for a > b
(b)Show that ∫
0
∞
e−xa 2 dx=
2 [1−erf a ] (4)
(c)Find the whole length of the loop of the curve 9y2=(x+7)( x+4)2 (5)
Q.9 (a)A sphere of constant radius r passes through the origin and cuts
the axes in the points A,B,C. Find the locus of the foot of (6)
perpendicular from origin to the plane ABC.
(b)Find the equation of the right circular cone with vertex at (1,2,-3),
semivertical angle cos−1 1
3
 and axis is the line x−1
1 = y−2
2 = z3
−1 (6)
(c)Find the equation of the right circular cylinder of radius 2,whose axis
is the line x−1
2 = y−2
1 = z−3
2 . (5)
OR
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Q.10 (a)Find the equations of tangent planes to the sphere x2 y2z2=9 which pass
through the line x+y=6, x−2z=3 . (6)
(b)Find the equation of right circular cone which has its vertex at the
point (0,0,10)and whose intersection with the xoy plane is circle
of diameter 10. (6)
(c)Find the equation of right circular cylinder of radius 2 where axis
pass through (1,2,3) and has direction cosines proportional to 2,-3,6. (5)
Q.11 Solve ant two. (8)
(a)Evaluate ∫∫R
√ xy(1−x−y )dxdy where R is the area bounded by
x=0, y=0 and x+y=1. (8)
(b)Change the order of integration in double integral . (8)
∫0
a
∫
√a2−y2
y+a
f ( x , y)dx dy
(c)Find the area common to the circles. (8)
x2 y2=a2 And x2 y2=2ax
OR
Q.12 Solve any two.
(a)Express the following as a single term integral and evaluate: (8)
∫0
a
√2
∫0
x
cos k (x2+ y2)dx dy+∫a
√2
a
∫0
√a 2−x2
cos k (x2+y2)dx dy
(b)Evaluate ∫
0
∞
∫0
∞
∫0
∞ dxdydz
1x2y2z2 2 . (8)
(c)If the density at any point of the curve x=a sin  , y=a 1−cos varies as
its distance from the x-axis,find the distance of its C.G. of arc from the x-axis.(8)
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