BIRLA INSTITUTE OF TECHNOLOGY, MESRA, RANCHI
DEPARTMENT OF APPLIED MATHEMATICS
MA 1201
Engineering Mathematics
2015 -16
Tutorial Sheet No. 2
Module IV:
1. Define Gamma function, and establish a Reduction formula for (n) .
2. Define Beta function and Prove that it is symmetric about m & n.
 m  1  n  1

 

2   2 

m
n
0 sin  cos  d   m  n  2 
2 

2


 /2
3. Prove that:
and hence find the value of
1
  .
2
4. Show that :
i)

2

0

2
tan  d  
0

iii)
x
0
1
v)
y
0
2

e  x dx   e  x dx 
4
4
0
q 1

1
 log 
y

1

cot  d 
2
p 1
ii)

0

iv)
8 2
1 x

2

0
( p)
dy  p where p>0 ,q>0
q
1
x 2 dx
4
dx

1 x
0
d
2


2

8. Prove that n, m   
0
y m 1 dy
1  y m n
1
4 2
0
2
vi)
 (8  x
0
(2m) where m is positive
7. Prove that (m, n)  (m  1, n)  (m, n  1)

  sin  d  
sin 
 ( m)  ( n )
2 m 1


2
5. Prove that (m, n)  (m  n)
1

6. Show that (m) m   
4
) dx 
3 3
1
60

x p 1


dx 
; 0  p  1, prove that (n) (1  n) 
; 0  n 1
9. Assuming 
1 x
sinp
sin n
0
10. Prove that
π
2
0

sin 2p-1θcos 2q -1θdθ 
 ( p ) ( q )
2 ( p  q )
1
 2
11. Prove that   = 
12. Prove that the integrals;

(a)
e
 ax
cos bx . x m -1dx =
 ax
sin bx . x m -1dx =
0

(b)
e
0
b
 ( m)
cos m where r  a 2  b 2 , tan 
m
a
r
m
b
sin m , where r  a 2  b 2 , tan 
m
r
a
5 1
    
6 2
13. Using Beta & Gamma function , Prove that   x 3  dx =
4
0 1 x 
3 
3
1
3

14. Using Beta & Gamma function , Prove that
x
1
4
1
2
e  x dx =
0
1
3

2
5
3
15. Evaluate,  x 1  x  dx using Beta & Gamma function.
10
0
16. Prove that l , m .l  m, n .l  m  n, p  
c  1
xc
; c 1
17. Prove that,  x dx 
log c c 1
0 c

1
x p 1  x q 1
18. Prove that , ( p, q)  
p  q dx
0 (1  x )
2
(l ) (m) ( n )( p )
l  m  n  p 
x2
1
  (x
19. Evaluate: a)
0
x
2
2
 xy )dydx
x y
  e
20. Evaluate: a)
0
0
4
x
x y
0
0
0
3a  y
0
y 2 / 4a
  dxdy
b)
3
x
2a
2
x y z
dzdydx
1 x x  y
   e dzdydx
z
b)
0
0
0
0
   z dzdydx
21. Evaluate
22. Change the order of integration of the following double integral:
a
a)
 
 ( y)
dydx
(a  x)( x  y )
2a
2 ax
0
c)
x

0
0


b)

0
lx
 
d)
0
2
f ( x, y ) dydx b
a2  x2
a
f ( x, y )dydx
2 ax  x
x  2a
a
Vdydx
mx
23. Change the order of integration of the following double integrals and hence evaluate
the
same:
a
a)
c)
a
0
x
y x 2  y 2 dydx

x

  xe
0
 x2 / y
dydx
b)
d)
0
a
a2  y2
0
 a2  y2


0
x

 dydx
 
ey
dydx
y
24. Transform from Cartesian to Polar form and then evaluate:
2
a)
2 x x2
 
0
0
x
dydx
x2  y2
a

b)
0
3
a
x2
y ( x 2  y 2 ) 3 / 2 dydx
25. Evaluate

r
a r
2
2
drd
over the loop of the lemniscate r2 = a 2 cos 2
Find the area that lies inside the circle r = a cos  and outside the Cardioid r= a(1  cos  ) .
29.
30. Find by triple
integration the
volume of the
ellipsoid
x2 y 2 z 2


 1.
a 2 b2 c 2
31.
Find the volume common to the two cylinders y2+z2= a2 and x2+z2= a2
32.
Find the volume bounded by the cylinder x2+y2 = 4 and the planes x+z = 4 and z = 0.
33.
Find the volume of the portion of the sphere x2+ y2 +z2 = a2 lying inside the cylinder
x2+y2 - ax =0.
Find the volume bounded by the paraboloid x2+y2 = az, the cylinder x2+y2 = 2ay and
34.
the plane z = 0.
Prove that the volume enclosed by the cylinders x2+y2 = 2ax and z 2 =2ax is
35.
128 3
a
15
36.
Find the area of the portion of the cylinder x2+z2=4 lying inside the cylinder x2+y2= 4.
37.
Find the area of the portion of the sphere x2+y2+z2=9 lying inside the cylinder x2 + y2 =3y
38.
Evaluate

(x+y)2 dxdy, where R is the parallelogram in the xy-plane with vertices
(1,0), (3.1), (2,2), (0.1) using the transformation u = x+y and v = x-2y.
Module V:
39.
Show that the polar equation of the ellipse
x2 y2

 1, if the pole be at its centre and the
9
4
positive direction of the x  axis be the direction of the polar axis is
r2 
40.
36
9  5 cos 2 
Write down the polar equation of the circle of radius 2 units with centre on the initial line
at a distance of 2 units from the pole on the positive side.
4
41.
Show that the condition that the straight line
r  2k cos  is
42.
1
 a cos   b sin  may touch the circle
r
b 2 k 2  2ak  1.
Show that the equation of chord of the conic
l
 1  e cos  joining the points A and B
r
whose polar angles are A(   ) and B (   ) is
43.
l
 e cos   sec  cos(   )
r
Show that the equation of tangent to the conic
l
 1  e cos 
r
at
A( ) is
l
 e cos   cos(   )
r
44.
Show that the straight line
l
l
 A cos   B sin  touches the conic
 1  e cos  if
r
r
( A  e) 2  B 2  1.
45.
Verify whether the line r cos(   )  2 5 , is a tangent to the conic
6
 2  cos  ,
r
where tan   2 . If so, find the point of tangency.
46.
l1
l
 1  e1 cos  and 2  1  e2 cos(   ) will touch each other if
r
r
Show that the conics
l1 (1  e2 )  l 2 (1  e1 )  2l1l 2 (1  e1e2 cos  ).
2
47.
2
2
Show that the condition that the straight line
l
 1  e cos 
r
48.
2
is
1
 a cos   b sin  may touch the conic
r
(al  e) 2  b 2 l 2  1.
Show that the equation of chord of contact of the tangents to the conic

l
 l
drawn from the point (  ,  ) is   e cos    e cos    cos(   )
r
 

5
l
 1  e cos  ,
r
49.
Show
that
the
director
circle
of
the
conic
l
 1  e cos  is
r
r 2 (e 2  1)  2ler cos   2l 2  0.
50.
Show that the locus of the point of intersection of a pair of perpendicular tangents to the
conic l  1  e cos represents a circle (director circle).
r
51.
Prove that the equation of pair of asymptotes to the conic
2

l
 1  e cos  is
r

l

1
2
2
  (e  e ) cos    1  e sin  .
r

Module VI:


52.
If P  5t 2i  t 3 j  tk and Q  2 sin ti  cos tj  5tk , find
(i)
53.
d  
( P.Q )
dt
(ii)

d
( P  Q) .
dt
Consider the curve given by the parametric equations: x  cos t , y  sin t , z  t / 3 for
 4  t  4 . Find the tangent vector and length of the tangent vector.
54.
Compute the curvature of the curve having parametric representation:
x  cos t  t sin t , x  sin t  t cos t , z  t 2 for t  0.
55.
Consider the curve with position function:
F (t )  (cos t  t sin t )iˆ  (sin t  t cos t ) ˆj  t 2 kˆ for t  0 .
Compute the unit tangent T (s) and the unit normal N (s )
56.
For the curve C with position function: F (t )  (cos t  t sin t )iˆ  (sin t  t cos t ) ˆj  t 2 kˆ
for t  0 . Compute the tangential and normal components of
acceleration.
57.
A particle moves along the curve r  t 2 iˆ  t 3 ˆj  t 4 kˆ, where t is the time. Find the
magnitude of tangential and normal components of the acceleration when t  1.
6
58.
The position of a moving particle is given by r  2 cos tiˆ  2 sin tˆj  3tkˆ, Find the vectors
ˆ , ˆ . Also find the curvature.
ˆ , 
59.
Find the curvature and torsion for the following curve: r (t )  e t cos tiˆ  e t sin tˆj  2kˆ
60.
Find a unit vector normal to the surface xy2 = 4 at the point (-1,-1.2).
61.
Find a unit vector normal to the surface xy3 = 4 at the point (-1,1,2).
62.
Find the constants a & b so that the surface ax 2 – byz = (a+2) x is orthogonal to the
surface 4x2-yz+z3=4 at the point (l,l,-2).
63.
Find the directional derivative of f(x,y,z) = 2xy+z 2 at the point (l,-l,3) in the direction of
the vector i+2j+2k
64.
In what direction from (3,1,-2) is the directional derivative of  = x2 y - 2x4 maximum?
Find also the magnitude of this maximum.
65.
Evaluate: i) div F and Curl F, where
F= grad (x3+y3+z3-3xyz).
ii) If  = (x+y+z) i+j-(x+y) k show that Curl  = 0.
66.
Prove that
(i)
div grad f = 2f
(ii)
curl grad f = X f = 0
(iii)
div curl F =. X F = 0
(iv)
 X ( X F) =(.F)- 2F
(v)
(.)= X (  X )+2
where f and ( X )
being vector point functions j
67.
Show that 2 (rn) = n(n+1) rn-2
68.
If u =v, where u and v are scalar fields and  is a vector field, show that  curl=0
69.
If r1, r2 be the vectors joining the fixed points (x1, y1, z1) and (x2, y2, z2 ) respectively
to a variable point (x,y, z) prove that
(i)
div(r1 X r2 ) = 0
(ii)
grad (r1. r2) = r1 + r2
(iii)
curl(r1 X r2) = 2( r1- r2)
Module VII:
70.
Find the work done in moving a particle once around a circle C in the XY plane, of
7
the circle has center at the origin and radius 2 and if the force field is given by
F = (2x-y+2z) i+ (x+y+z)j + (3x-2y-5z)k .
71.
If F = 3xy i – y2j, evaluate
 F.dr where C is the curve in the xy plane y = 2x2
C
from (0,0) to (1,2).
72.
Find the circulation of F around the curve C where F = yi + zj+xk and and C is the
circle x2+y2 = 1, z=0
73.

If F= (3x2+6y) i – 14yzj+20xz2k, evaluate the line integral F.dr from (0,0,0) to
C
(1,1,1) along the following paths C given by
(i)
74.
x=t, y=t2, z=t3
(ii)
the straight line from (0,0,0) to (1,0,0) and then (1,1,1)
(iii)
the straight line joining (0,0,0) to 1,1,1).
Show that F = (2xy-z3)i+x2j+3x2 k is a conservative force field. Find the scalar potential.
Find also the work done in moving an objects in this field from (1,2,1) to (3,1,4).
75.
Evaluate
 ( yzi  zxj  xyk )ds where S is the surface of the sphere
S
x2+y2+z2 = a2 in the first octant.
76.
By Gauss’s Divergence theorem evaluate
{( x
2
dydz  y 2 dzdx  2 z ( xy  x  y )dxdy}
S
where S is the surface of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤z ≤ 1.
77.
Verify divergence theorem for F = x2 i+zj+yzk, taken over the cube bounded by
x = 0, x =1, y = 0, y =1, z = 0, z =1.
78.
By transforming to a triple integral evaluate,
 ( x dydz  x
3
2
zdxdy
S
where S is the closed surface bounded by the planes x=0, x=6 and cylinder x2+y2=a2.
79.
Apply divergence theorem to evaluate
{( x  z )dydz  ( y  z )dzdx  ( x  y )dxdy}
S
where S is the surface of the sphere x2+y2+z2=4.
8
80.
State and prove Green’s theorem in xy plane.
81.
State and prove stoke’s theorem.
82.
State and prove Gauss’s Divergence Theorem
83.
2
2
2
Applying Green’s theorem to evaluate  {(x  xy) dx  ( x  y ) dy}
C
where C is the square formed by the lines x =  1, y = 1.
84.
Applying Green’s theorem to evaluate
 {(y - Sinx) dx  Cosxdy} where C is the
C
plane triangle enclosed by the lines y=0, x= π/2,and y = 3x/2
85.
Verify Green’s theorem
 {(xy  y
C
2
)dx  x 2 dy} , where C is bounded by the curve
y =x and y = x2.
86.
Verify Stoke’s theorem for F = (x2+y2)i-2xy j taken around the rectangle bounded by the
lines x =  a, y=0, y=b.
87.
Verify stoke’s theorem for the vector field F = (x 2-y2) i + 2xyj integrated round the
rectangle in the plane z = 0 and bounded by the lines x=0, y=0, x = a and y=b.
88.
Verify stoke’s theorem for the vector field F = (2x-y) i-yz 2j – y2zk over the upper half
surface of x2+y2+z2=1, bounded by its projection on the xy plane.
89.
Verify Divergence theorem for F = (x2-yz) i + (y2-zx) j + (z2-xy) taken over the rectangular
parallelepiped 0 ≤ x ≤a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.
90.
Verify Gauss’s Divergence theorem for the function F = y i + x j + z2k
cylindrical region bounded by x2+y2=9, z=0 and z=2.
9
over
the