ENGINEERING MATHEMATICS-II
FE 2015-16
Theory Question Bank
UNIT I
………………………………………………………………………………………………
A)
Formation of Differential Equations
y  ae9t cos(3t  b)
1. Form the differential equation whose general solution is
where a and b are arbitrary constants.
2. Find the differential equation of all circles touching Y-axis at the origin and
centers on the X-axis.
3. Form the differential equation whose general solution is
y  log cos( x  a)  b
where a and b are arbitrary constants.
4. Form the differential equation whose general solution is
y  ae2 x  be3 x
where a and b are arbitrary constants.
5. Form
the
differential
equation
whose
general
solution
is
y  A cos(log x)  B sin(log x) where A and B are arbitrary constants.
B
Solve following Differential equations
6. y  x
dy
dy 

 a  y2  .
dx
dx 

7. x cos x cos y  sin y

dy
0
dx

8. x tan ydx  x 2  1 sec 2 ydy  0
dy
2
 4 x  y 
dx
dy y 2
10.
x 
 y.
dx x
9.
11.
12.
13.
14.
15.
dy y
 y
  tan 
dx x
x
2
y  2 xy dx  2 x 2  3xy dy  0
dy
dy
y2  x2
 xy
dx
dx
 2x  y  1 dy   x  2 y  3 dx  0.




6x  9 y  6dy  2x  3 y  1dx
1
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
dy
x  y 1

.
dx 2 x  2 y  1
y e
2
x
y
2
3
xy 2


2
 xy  y 2  dx  xydy  0.
 2 x 2 y  dx   2 xy 2  x3  dy  0.
y 1  xy  dx  x 1  xy  x 2 y 2  dy  0.
 x y  5xy  2 ydx   x y  4xy  2 xdy  0.
x  y  6xdx  x  y  6ydy  0
 2 y  6xy  dx  3x  8x y  dy  0.
2
2
2
2
2
2
2
2
2
2
dy 1  y 2  3x 2 y

dx 1  2 xy  x 3
dy
 tan 2  x  y  .
dx
 x2 y  y4  dx   2x3  4xy3  dy  0.
2 y  6xy dx  3x  8x y dy  0
2
2
2
dy
 y 3e  x .
dx
 2x  ex log y  ydx  ex dy  0.
xy 
y
x
4
 2 x3 y  dx   x 4  2 xy 3  dy  0.
2
1

3


2
dy
 4x x 2  1 y  1
dx
dy
 e x y  e x  e y  .
dx
dy
x cos x   x sin x  cos x  y  1.
dx
 y  2x3  dx  x 1  xy  dy  0.
35.
y  2 x 2 y  e x  dx   e x  y 3  dy.
36.
cos y  x sin y
37.
dy
2
  x  y  1   x  y  .
dx
dy
 y cot x  sin 2 x
dx
dy
 x sin 2 y  x3 cos 2 y.
dx
x 3 y 2  xy dx  dy
38.
39.
40.

 4 x3 dx  2 xye xy  3 y 2 dy  0.

dy
 sec2 x.
dx

2
Answers
dy 

2. x 2  y 2  2 x  x  y 
dx 

4. y 2  5 y1  6 y  0
1. y2  18 y1  90 y  0.
3. y2  y12  1  0.
d2y
dy
x y0
2
dx
dx
7. ( x sin x  cos x)  log sec y  c
5.
6.  a  x 1  ay   yc.
x2

9. tan 1 (4 x  1)  x  c
 y 
  cx 4
12. 
 y x
 y
11. x sin   c
x
y
13. log y   c
x
14. 2 tan
17. e xy  x 4  y 3  c.
2
x2 y3  2 x2 y4  c.
y  x  sin  x  y  cos  x  y   c.
27.
x2 y3  2 x2 y4  c.
11
28. e x  y 2  2 x  c  .
2
x2  e x log y  c.
30. x3  y3  cxy.

x
2
32. e y  e y  1  cee .
y
y2
2
 c.
34.  x 
x
2
xy sec x  tan x  c.
2 3 ex y2
x  
 c.
3
y 2
37. x  y  x3  x  y  3 c.
39. tan y 
2
1 2
x  1  ce  x .

2
7
26. 7 x 2 y11  11x 2 y14  c.
31. x 2  1 y  tan  x  c
35.

  c.

22. x 4  2 x 2 y 2  6 x 2  6 y 2  y 4  c
24. x  xy 2  x 3 y  y  c
xy  5log x  4log y  c.
25.
33.
2
18. y  x log( x  y)  cx
1
1
20.
   log y  c.
2 2
2y x
xy
19. xy y 2  x 2  c.

1
16. 6 y  3x  log 3x  3 y  2   c.
2
29.
 y  1  1 log   x  1   y  1

2
 x  1 2 
 x  1
2
15. e xy  x 4  y 3  c.
21.
23.

8. x 2  1 tan 2 y  c
x
10.  log x  c
y
36. x cos y  tan x  c.
38. y 
40.
2 2
sin x  C cos ecx
3


 x2
1
 2  x 2  Ce 2
y
UNIT II
……………………………………………………………………………………
3
1) Find the orthogonal trajectories of straight lines 2x 2  y 2  cx . Ans: x2   y 2 log(cy) .
2) Find the orthogonal trajectories of r  a(1  cos ) .
Ans: r  c(1  cos )
3) If the temperature of the body drops from 100 c to 60 c in one minute when the
temperature of

surrounding is 20 c , what will be the temperature of the at the end of the second minutes.
4) A body originally at 80 c cools down to 60 c in 20 minutes, the temperature of the air being 40 c .
What will be the temperature of the body after 40 minutes from original?
5) If a thermometer is taken outdoors where the temperature is 0 c , from a room in which the temp.is
21 c and the reading drops to 10 c in one minute, how long after its removal will the reading be 5 c .
6) A body at temperature 100 c is placed in a room whose temp. is 20 c and cools to 65 c in 5
minutes. Find its temperature after a further interval of 3 minutes.
8) The resistance of the motion of a car of mass m varies as the square of its speed () and the effective
horse-power exerted at the road wheel is constant and equal to p. Show that the distance in which
 p  kv03 
m
the car can accelerate from speed to is given by log 
.
3 
3k
 p  kv1 
9) A body start moving from rest is opposed by a force per unit mass of value cx and resistance per unit
mass of value bv 2 , where x and v are displacement and velocity of the particle at that instant. Show
c
cx
that velocity of particle is given by v 2  2 1  e2bx   .
2b
b

a4 
10) A particle is moving in a straight line with an acceleration k  x  3  directed to words origin. If it
x 

starts from rest at a distance a from the origin, prove that it will arrive at origin at the end of time

.
4 k
11) Assuming that the resistance to movement of a ship through water in the form of a 2  b2v2 , where v
is velocity ,a & b are constants .Write down the differential equation for retardation of the ship
moving with engine stopped. Also prove that the time in which the speed falls to one half to its
w
abu


original value u is given by
, where w is weight of ship.
tan 1 
2
2 2 
abg
 2a  b u 

dv
v2 
 g 1  2  , where v is the
12) The distance x descended by a parachuter satisfies D.E. v
dx
k 

k2
 gt 
log cosh 
.
g
 k 
13) The equation of emf in terms if current i for an electric circuit having resistance R and condenser
i
of capacity C, in series E  Ri   dt . Find the current i at any time t, when E  E0 sin wt .
C
14) Find the current i in the circuit having resistance R and condenser of capacity C in series with
emf , E sin wt .
15) A voltage Ee at is applied at t = 0 to a circuit containing inductance L and resistance R . Show that
 Rt
  at

E
e L .
the current at any time t is
e
R  aL 

velocity k, g are constants .If v = 0 and x = 0 at time t = 0 ,show that x 
4
16) In a circuit containing inductance L, resistance R and voltage E, the current I is given by
dI
.Given L = 640H, R  250 , E = 500 volts .I being zero when t = 0 . Find the time
E  RI  L
dt
that elapses, before reaches 90% of its maximum value.
17) A circuit consists of resistance R ohms and a condenser of C farads connected to a constant emf .If
q
is the voltage of the condenser at a time t after closing the circuit, show that the voltage at a time t
c
t



RC
is E 1  e  .


18) Show that the D.E. for current i in an electric circuit containing an inductance L and resistance R in
di
series and acted on by an emf E sin wt satisfying the equation L  Ri  E sin wt .Find the value
dt
of current at an time t, initially there is no current in a circuit .
dI
19) The equation of L-R circuit given by L  RI  10sin t . If I = 0, at t = 0, express I as a function
dt
of t.
20) A constant emf E volts is applied to a circuit containing a constant resistance R ohm in series and
constant inductance L henries. If the initial current is zero .show that the current builds up to half its
L log 2
theoretical maximum is
R
21) The distance x descended by a person falling by means of parachute satisfies the equation
2
 dx 
2
2 gx / k 2 
where k & g are constants and x = 0 when t = 0.Then show that
   k 1  e

 dt 
k2
 gt 
x  log cosh   .
g
k 
22) An elastic string without weight of natural length l and modulus of elasticity being weight of ngrams ,is suspended by one end and a mass is attached to the other , show that time of oscillation is
ml
.
2
ng
23) A particle executes SHM when it is 2cm from mid path, its velocity is 10 cm/sec. and when it is 6
cm from center of its path, its velocity is 2 cm/sec. find its period and greatest acceleration.
24) A spring of negligible weight hangs vertically. A mass m attached to the other end . If the mass is
moving with velocity v0 when the spring is unscratched, find the velocity v as function of the stretch
x. (Take  as Young’s modulus of the spring.)
25) A particle of mass m is attached to one end of a light elastic string of normal length a and modulus
mg
. The other end of the string is fixed to a point O and the particle is allowed to fall from rest at O.
k
Obtain velocity of a particle and show that the highest magnitude is ag (2  k ) .
26) A point executing SHM has velocity v1 and v2 acceleration a1 and a2 in two position respectively.
v12  v2 2
Show that the distance between the two positions is
.
a1  a2
27) Two identical loads are suspended from the end of the spring. Find the motion imparted to one load
if the other breaks loose in case the increase in length of spring under action of one load at rest is a.
5
1
28) A spring is loaded with a mass of 2kg. It produces a static deflection m . A mass of 2kg is suddenly
4
added to the original mass. Show that the maximum deflection produced is0.75m.
29) Amass hangs from a fixed point by means of tight elastic spring given a small vertical displacement
.If n be number of oscillation per second in SHM and L be a length of spring when the system is in
g
equilibrium. Show that the natural length of spring is L  2 2 .
4 n
30) A pipe 20cm in diameter contains steam at 150 c and is protected with a covering 5cm thick for
which k = 0.0025. If the temperature of outer surface of covering is 40 c .Find the temp half way through
the covering under steam state conditions.
Ans: 89.5 c
31) A steam pipe 20cm in diameter is protected with a covering 6cm thick for which the coefficient of
thermal conductivity is k = 0.0003 steady state. Find the heat lost per hour through a meter length of
pipe, if the surface of the pipe is at 200 c and the outer surface of the covering is at 30 c .
Ans. 245443.3861cal.
31) For steady state heat flow through the wall of a spherical shell of inner and outer radii r1 and r2
respectively, temp. at distance r from the center of the sphere is given by
d 2T
dT
2
 0 , if u1 and u2 are the temp. at inner and outer surfaces, find T in terms of r.
2
dt
dt
r1r2  u2  u1  
1 
31) T 
 u2 r2  u1r1  

r2  r1 
r

r
32)A pipe 10cm in diameter contains steam at 100 c , it covered with asbestos 5cm thick for which k =
0.0006 and outside surface is at 30 c . Find amount of heat lost per hour from a meter length of pipe.
33)A steam pipe 20cm in diameter contains a steam at 150 c and is covered by layer of insulation 5cm
thick. The outside temperature is kept at 60 c . By how much the thickness of the covering be increased
in order that the rate of heat loss is decreased by 25%.
Ans:
30) 2.16cm
…………………………………………………………………………………
UNIT III

1. If I n 
2. If I m,n
sin  2n  1 x 
 n 
dx , prove that n  I n1  I n   sin 
 . Hence find I 3 .
sin x
 2 
0
  cosm x sin  nx  dx , prove that  m  n  I m,n   cosm x cos  nx   mI m1,n1 .
4


Hence evaluate
2
 cos
5
x sin  3 x  dx .
0

3. If U n 
4
 tan
n
 d , prove that n U n1  U n1   1 . Hence find I 6 .
0
6
4. If f  m, n    x m 1  x  dx , then show that
n
x m1 1  x 
n
f  m, n  

f  m, n  1 .
 m  n  1  m  n  1
n

5. If I n 
 
x n sin x dx , prove that I n  n  
2
2

0

2n
0

2
 cos
7. If I m,n 
m
x cos  nx  dx , prove that I m,n 
0

Hence prove that
2
 cos
m
x cos  mx  dx 
0

8. If U n 
2
  cos
n
 d , prove that U n  
0

9. Evaluate
 n  n  1 I n2 .
1 
1

.
x dx , prove that U n  1 
 U n 1 
2n 
n 2n 1

4
 sin
6. If U n 
n 1
m
I m1,n1 .
mn

2m1
.
1
n 1

U n2 . Hence evaluate U 4 .
2
n
n
8
 sin 8x  co s  4 x  dx .
4
6
0

10.
Evaluate

x e x dx .
4
0
1
11.
Evaluate
x

1
log  
 x
0

12.
Evaluate
x e
4  x4
dx .
dx .
0

13.
Prove that
2


tan  d
0
Evaluate

cot  d 
0
1
14.
2
m 1
2
 x 1  x 
n 1
2
2
.
dx .
0
1
15.
Evaluate
x 2  x3
 1  x 
7
dx .
0
1
16.
Evaluate
  x log x 
4
dx .
0
7

17.
4
 sin
If un 
2n
0
1 
1

.
x dx , prove that un  1 
 un1 
2n 
n 2n1

……………………………………………………………………………………
UNIT IV
a2
1. Verify the rule of DUIS for
 log  ax  dx .
a
a2
2. Verify the rule of DUIS for
 tan
0

3. Evaluate
1
x
2
1
 x
  dx .
a
log 1  ax 2  dx .
0

4. Prove that
5. Prove that
 a2  1 
e x  e ax
1
dx

log

, a  0 .
0 x sec x
2
 2 
t
t
0
0
 erf  ax  dx   erfc  ax  dx  t .
2
6. If   x  
7. Prove that

e
e

t2
2
dt , show that erf
 x    x

2 .
0
d
erf
dt

8. Prove that
x
 st
 
erf
0
t 
et
t

. Hence evaluate
 t  dt  s
e
t
erf
 t  dt .
0
1
s 1
.
Trace the following curves.
9. y 2   x  1 x  2  x  3 .
10. y 2  x 2  y 2   a 2  x 2  y 2   0 .
10.
y 2  x  a   x 2  2a  x  .
12. x 4  y 4  a 2  x 2  y 2  .
11.
y 2  x5  2a  x  .
13. x3  y 3  3axy .
12.
r 1  sin    2a .
16. r  a sin 2 .
13.
r  a cos 3 .
17. x  a  t  sin t  , y  a 1  cos t  .
14.
Find the arc length of the cycloid x  a   sin   , y  a 1  cos  
from one cusp to another cusp.
15.
Find the total arc length of the curve x
2
3
y
2
3
a
2
3
.
8
16.
Evaluate
x2
y2
xyds
along
the
arc
of
the
ellipse

 1 in the first

2
2
a
b
quadrant.
17. Find the length of arc of the curve r  ae
vectors r1 and r2 .
m
intercepted between radii
18. Find the whole length of loop of the curve 3 y 2  x  x  1 .
2
Find the length of the upper arc of one loop of the curve.
…………………………………………………………………………........
UNIT V
1. Find the equation of the sphere passes through the points (1,0,0),(0,2,0),
(0,0,3) and has radius as small as possible.
2. A sphere with positive octant passes through the origin and cuts the coordinates plane of radii 10, 2, 10 respectively. Find equation of the sphere.
3. A sphere of a constant radius r passes through the origin and meets the coordinates axis at A, B & C. Show that the locus of the centroid of a triangle
ABC is sphere 9  x 2  y 2  z 2   4r 2 .
4. The plane passes through the fixed point (a,b,c) and meet co-ordinate axis at
the points A,B,C. Show that the locus of the center of the sphere OABC is
a b c
   2.
x y z
5. Show that the plane 2 x  2 y  z  12  0 touches to the sphere
x2  y 2  z 2  2 x  4 y  2 z  3  0 .
6. Find the equation of the sphere which touches to the plane 4 x  3 y  47 at the
point (8,5,4) and the sphere internally x2  y 2  z 2  1 .
7. A point moves so that the sum of squares of its distance from the six faces of
a cube is constant, then show that its locus is a sphere.
8. Find the equation of the sphere passes through the points (1,0,0),(0,1,0),
(0,0,1) and has radius as small as possible.
9. Find the equation of the sphere passes through the points (3,1,2) and meets
XOY plane in the circle of the radius 3 units with the center at (1,-2,0).
10.
If the circles 2( x2  y 2  z 2 )  8x  13 y  17 z  17  0  2x  y  3z  1 and
x2  y 2  z 2  3x  4 y  3z  0  x  y  2 z  4 lie on same sphere. Find the equation of the
sphere.
11.
Find the equation of the sphere passes through the circle
2
x  y 2  z 2  1& 2 x  3 y  z  5 and which intersect the sphere
x2  y 2  z 2  3( x  y  z )  56  0 orthogonally.
12.
Find the equation of the sphere passes through the points (1,0,0), (0,1,2),
(1,1,1) and touches the line x = 0 & y = z.
CONE:
9
13.
Find the equation of the cone whose vertex is (1,2,3) and guiding curve is
circle x2  y 2  z 2  4, x  y  z  1.
14.
Find the equation of the cone whose vertex is (1,1,1) and base of the
circle x2  y 2  4, z  2.
15.
Find the equation of the cone whose vertex is (1,1,3) and passes through
ellipse 4 x2  z 2  1, y  4.
16.
Find the equation of the right circular cone generated with straight lines
2 x  3 y  6, z  0 revolves about Y-axis.
17.
Find the equation of the cone whose vertex is (1,2,-3), semivertical angle
 1  and line
cos 1 

 3
x 1
y2
z 1 .


1
2
1
18.
Find the equation of the right circular cone passes through the point (2,x  2 y 1 z  2
2,1) with vertex at origin and axis parallel to the line


.
5
1
1
CYLINDER:
19.
Find the equation of the cylinder whose generators are parallel to X-axis
and whose guiding curve is with straight lines ax2  by 2  cz 2  1, lx  my  nz  p .
20.
Find the equation of the cylinder whose generators are parallel to the
line x  1  y  1  z  2 and whose guiding curve is x2  y 2  25, z  0 .
3
1
6
21.
Find the equation of the right circular cylinder of radius 2 whose axis
passes through the point (1,2,3) and has directional cosines proportional to
2,-3,6.
22.
Find the equation of the right circular cylinder of radius 2 whose axis is
the line x  1  y  2  z  3 .
2
23.
1
2
Find the equation of the right circular cylinder whose guiding curve is
2
x  y 2  z 2  9, x  y  z  3 .
24.
Find the equation of the right circular cylinder described on a circle
through the points (3,0,0), (0,3,0), (0,0,3).
ANSWERS:
a)  5 2,135 ,126.87  , 


14,143.3 ,116.56
 , b)
 3 3  9 3 3 3
,
,1 ,  ,
,  

2
 2 2  4 4
1) x2  y 2  z 2  4x  7 y  3z  15  0 ,2) 49( x2  y 2  z 2 ) 13x  80 y 135z  36  0 ,
3) x2  y 2  z 2  6 x  2 y  2z  0 ,
6) (-3,4,-2)
2
2
2
10) x  y  z  2x  4 y  4z  4  0
15) 2( x2  y 2  z 2 )  3x  3 y  7 z  1  0
16) 5x2  3 y2  z 2  2xy  6 yz  4zx  6x  8 y 10z  26  0
17) x2  y2  2z 2  2xz  6 yz  4x  4 y  4  0
18) 12x2  4 y2  3z 2  26 yz  8xy  32 x  34 y  69  0
19) 4x2  9 y2  4z 2  36 y  36  0 , 21) 8x2  4 y 2  4 z 2  5xy  yz  5xz  0
22) a(my  nz  p)2  bl 2 y 2  cl 2 z 2  l 2
10
23) 6x2  6 y 2  10 z 2  2 6 yz  6 6 xz 150  0
24) 45x2  40 y 2  13z 2  36zy  24xz  12xy  42x  280 y 126z  294  0 .
…………………………………………………………………………….
UNIT VI
Evaluate following integrals
1.
dxdy

Evaluate
1  x
R
2
y

2 3/ 2
where R
8.
Express as a single integral and
hence

Evaluate
V
2
2
   x  y  dxdy  
0 0
dxdydz
 x2  y 2  z 2

3
2
9.
spheres
x2  y 2  z 2  a 2 and x 2  y 2  z 2  b2 , ( a  b  0).
3.
4.
5.
a
a x
0
ax  x 2
 
Evaluate
2
 
Evaluate

xye
dxdy .
x2  y 2
2
2
a a a  y
Evaluate

 x2  y 2
0

4x 2  y 2 dxdy over the
region bounded by y=0, y=x and
6. Change the order of integration
ydxdy
  (1  xy) (1  y )
2
0 x
7. Express as a single integral and
hence
a/ 2 x

 xdxdy 
0
0
evaluate
a
a x
2
 
a/ 2
0
the
order
2 1 2 x  x 2
integration
 
of
f  x, y dxdy
1 1 2 x  x 2
10.
Transform to polar form
and
11.
 sin  x
evaluate
2
 y 2 dxdy
x2  y 2  a2 , x  0
Find the area outside circle
x2  y 2  a 2 and inside r  a 1  cos   .
12.
Find the area of the curve
a 2 x 2  y 3  2a  y  .
13.
x=1.
2
 y 2 dxdy
1 0
where R is
0
and evaluate
2
R
xy log( x  a)dxdy
( x  a)2
1 1/ x
 x
Change
where V is annulus between the
2
2 2 y
1 y
is x=0, x=1 and y=0, y=1..
2.
evaluate
Find the area between the
curve
y 2 x  4a 2  2a  x  &
its
asymptote.
14. Find the area inside the circle
r  a sin 
and
outside
the
cardiode r  a 1  cos   .
2
xdxdy
1 1 x 2
15. Evaluate
 
0
0
1  x2  y 2
dxdy
1  x2  y 2
[5]
11
16. Find the volume of paraboloid
of revolution
x 2  y 2  4 z cut
off by the plane z = 4.
23. Find the moment of inertia about
the line  

2
of the area enclosed
by r  a 1  cos  in the upper half.
24.
Find the centroid of the
region
bounded
by
z  x 2  y 2 , z  0, x  a, x  a, y  a, y  a
17.
Find the volume of the
cylinder
x 2  y 2  2ax
intersected
between
the
paraboloid
x 2  y 2  2az & the XY-plane.
25. Find the position of centre of
gravity of the area of the cardioids
r  a 1  cos 
initial
which
lies
above
line.
18. Find the volume enclosed by the
cylinders x 2  y 2  2ax &
z 2  2ax
19. Find the volume bounded by
x 2  y 2  4 and
cylinder
plane
y  z  4 and z  0 .
20. Evaluate

V
1
x2 y 2 z 2
 
dxdydz
a 2 b2 c 2
where V is volume of ellipsoid
x2 y 2 z 2
   1.
a 2 b2 c 2
21. Find the root mean square value
of
an
electric
current
given
by
 2 t

 4 t

I  I o  I1 sin 
 1   I 2 sin 
 2  .
 T

 T

22. Find moment of inertia of one
loop of lemniscate
r 2  a 2 cos 2
bout initial line.
12
13