Unit-4 Set -A 1. The partial differential equation corresponding to the equation z f ( x 2 y 2 ) is
z z y 0 x y z z b) y x 0 x y z z c) y x 0 x y a)
x
d)
z z 0 x y
2z 2z 2 z z z 3 2 0 is 2. The partial differential equation 2 2 6 xy x y x x a) b) c) d)
Parabolic Hyperbolic Elliptic None of these
3. The initial solution of the partial differential equation
2 z z z 0 by separation of x 2 x y
variable is a) z X ( x) Y ( y) b) c)
z X ( x) Y ( y ) z X ( x).Y ( y)
d) None of these 4. The solution of the partial differential equation a)
Ae kx e kt / 3
b)
Ae kx e 3kt
c)
Ae k / 3 x e kt / 3
d) None of these
z z 3 0 by separation of variables is x y
5. The general solution of the equation
2 2u u 2 u c , u (0, t ) 0, u (l , t ) 0, 0, is 2 2 t x t t 0
2 2 nx c3 e c k t l 2 2 2 2 nx b) c 2 sin c3 e c n t / l l nx nct c) c 2 sin c3 cos l l
a) c 2 sin
d) None of these 6. The value of bn of the solution of the equation 2 2u u 2 u c , u (0, t ) 0, u (l , t ) 0, 0, u ( x,0) x is 2 2 t x t t 0
a)
2 (1) n n
2 (1) n n 1 c) (1) n n 1 d) (1) n n b)
7. The solution of one dimensional heat equation 2 2u u 2 u c , u (0, t ) 0, u (3, t ) 0, 0, u ( x,0) 5 sin x 2 sin 2x 2 2 t t 0 t x a) u( x, t ) 5 sin(x). cos ct 2 sin(2x). cos 2 ct b) u( x, t ) 5 sin(x). cos ct 2 sin(x). cos ct c) u( x, t ) 5 sin(x). cos ct 2 sin(2x). cos 2 ct
d) None of these
2u 2u 8. The second order P.D.E. x 2 y 2 0 is x y a) b) c) d)
Elliptic if x<0,y>0 Hyperbolic if x<0,y<0 Elliptic if x>0,y>0 Hyperbolic if x>0,y>0
9. Let u(x, t) satisfy
u 2 u , u (0, t ) 0, u (1, t ) 0, , u ( x,0) sin 2x than t x 2
a) u( x, t ) sin(2x) e
2
t
sin(2x) e u( x, t ) sin(2x) e
b) u ( x, t ) c)
n 2 2t n 2t / l 2
d) None of these 10. The general solution of the equation 2 2u u 2 u c , u (0, t ) 0, u (l , t ) 0, 0, u ( x,0) f ( x) is 2 2 t x t t 0
nx c 2n 2 2t / l 2 e l nx c 2n 2 2t / l 2 b) bn sin e l nx n 2 2t / l 2 c) bn sin e l a)
b
n
sin
d) None of these 11. The P.D.E. corresponding to the equation z ( x a) 2 ( y b) 2 ,where a and b are arbitrary constants is
z z a) 4 z x x
2
z z b) 4 z x x
2
2
2
z z z x x
2
z z d) z x x
2
2
c)
2
12. Which one of the following is correct? a) Wave equation is elliptic and Laplace equation is Hyperbolic b) Wave equation is Hyperbolic and Laplace equation is elliptic c) Wave equation and Laplace equation are elliptic d) Wave equation and Laplace equation are Hyperbolic 13. Which of the following is the solution of a) u( x, y) e y 2 x b) u( x, y) (2 x y) 3 c)
u( x, y) ( y 2 x)e y 2 x
d) u( x, y) ( y 2 x)e y 2 x
u u 2 0 x y
14. The solution of the Laplace equation by separation of variables is
(c1 cos kx c2 sin kx)(c3 cos ckt c4 sin ckt)
a)
b) (c1 cos kx c2 sin kx)(c3 e ky c4 e ky )
(c1 cos kx c2 sin kx)c3 e k
c)
2
t
d) None of these 15. The solution of Laplace equation
2u 2u 0, u ( x,0) 0, u ( x, ) 0, u (, y) 0, u (0, y) 3 sin 2 y, is x 2 y 2 a) 3 sin 2 x e 2 y b) 3 sin 2 y e 2 x
2 sin 3x e 3 y
c)
d) 2 sin 3 y e 3 x 16. The solution of the P.D.E. U tt U yy is a) sin( x t ) b) sin( x t )
sin( x t ) d) sin( x t ) d2A w 2 t is 17. If A i cos wt j sin wt than the value of 2 dt c)
a) 1 b) 0
A
c)
d) w 2 A
1. If A (t 3 1) iˆ t 2 ˆj (2t 5) kˆ , than the magnitude of velocity at t=1 sec is a) 17
17 b) c) Both (a) and (b) d) None of these 2. The normal vector of a surface is also known as a) Gradient of the surface b) Curl of the surface c) Divergence of the surface d) None of these 3. The gradient of the surface (xy+yz+zx) at the point (1,-1,1) is
a)
2 iˆ 2 ˆj 2 kˆ
b) 2 iˆ 2 ˆj
2 ˆj 2 kˆ d) 2 ˆj c)
4. The unit normal vector of the surface xy 3 z 2 at the point (1,1,2) is a)
1
(iˆ 3 ˆj kˆ)
11 1 ˆ b) (i 3 ˆj kˆ) 11 1 ˆ c) (i 3 ˆj kˆ) 11 1 d) (iˆ 3 ˆj kˆ) 11
5. The directional derivative of a surface in direction of a vector d is
a)
b) ( ).dˆ c)
( ).d
d) None of these 6. A vector is said to be rotational if its a) Curl is not zero b) Curl is zero c) Divergence is zero d) Divergence is not zero
7. The vector v 3 y 4 z 2 iˆ 4 x 3 z 2 ˆj 3x 2 y 2 kˆ is a) Not solenoidal b) Not rotational c) Solenoidal d) None of these 8. The value of a for which the vector
v (2 xy 3 yz )iˆ ( x 2 axz 4 z 2 ) ˆj (3xy byz )kˆ is irrotational is
a) b) c) d)
3 -3 8 -8
9. The vector potential of the vector v ( x 2 y 2 x)iˆ (2 xy y) ˆj is
x2 3 x2 3 x2 3 x3 3
x y2 xy 2 C 2 3 x y2 b) xy 2 C 3 3 x y3 c) xy 2 C 3 3 x2 y2 d) xy 2 C 2 3 10. The limit of the vector A (t 3 2t ) iˆ t 2 ˆj (2t 7) kˆ as t 1 is a) (2iˆ ˆj 2kˆ) a)
b) (2iˆ ˆj 7kˆ)
(3iˆ ˆj 9kˆ)
c)
d) (3iˆ ˆj 2kˆ)
11. The vector function V (t ) is said to be continuous at a point t=a if
V ( t ) V (a) limit
a)
t a
limit V (t ) V (a)
b)
t a
V ( t ) V (a) limit
c)
t a
limit V (t ) V (a)
d)
t a
12. The length of space curve for the vector A acost ˆi asint ˆj kˆ , 0 t 2 is a) b) c) d) a)
2 2a 2 2a
13. If F is a conservative force field, than the value of Curl F is a) b) c) d)
0 1 -1 Can’t say
14. The value of a for which the vector A a( x y)iˆ 4 yˆj 3k is solenoidal is a) 4 b) -4
c) 3 d) -3
15. The value of x( ) is, where is any surface (Where x is vector cross product). a) b) c) d)
1 0 -1 None of these
16. The value Div(Curl A) is a) b) c) d)
1 -1 0 None of these
17. The limit of the vector F sin t iˆ cos 2t ˆj kˆ as t is
(iˆ ˆj kˆ)
a)
b) (iˆ ˆj 5kˆ)
( ˆj kˆ)
c)
d) ( ˆj kˆ)
Unit 6 Set-A
1. If a force F 2 x 2 yi 3xyj displace a particle in xy-plane from (0, 0) to (1, 4) along the curve
y 4x 2 . Than the value of work done is 104 6 104 b) 5 114 c) 5 114 d) 6 2. If ( F .n )ds 0 than F is said to be a)
s
a) b) c) d)
Irrotational vector Solenoidal vector Both (a) and (b) None of these
3. If F 2 xi 3 yj 4 zk than the value of
F dv is, (where v is the region bounded by v
x=0,y=0,z=0,x=1,y=1,z=1)
1 3 i 2 2 1 3 f) i 2 2 1 3 g) i 2 2 1 3 h) i 2 2 e)
j 2k
j 2k j 2k j 2k
4. If ( x, y ), ( x, y ),
are continuous function over a region R bounded by simple closed , y x
curve C in x-y plane, then according to Greens theorem
(dx dy) x
a)
C
b)
(dx dy) x
dxdy y
dxdy y
dxdy y
R
(dx dy) x
C
d)
dxdy y
R
C
c)
R
(dx dy) x
C
R
5. The value of line integral {sin y i x(1 cos y ) j }.dr , where C is the circular path given by C
x y a a) a b) 2a c) a 2 d) 2a 2 2
2
2
6. The value of line integral
(3x
2
8 y 2 )dx (4 y 6 xy )dy) , where C is the boundary of the
C
region bounded by x 0, y 0 and 2 x 3 y 6 {By using of green’s theorem} is a) b) c) d)
20 -20 21 -21
7. Which of the following is the mathematical expression of stock’s theorem
F . d r Curl F ds
a)
S
b) F .dr Curl F .n ds S
F . n d r Curl F ds
c)
S
d) None of these 8. By using stock’s theorem the value of the
xdx ydy zdz is, where c is x
2
y2 1
c
a) b) c) d)
0 1 -1 None of these
9. By using stock’s theorem the value of the
(2 x y)dx yz
2
dy y 2 zdz is, where c is
c
x y 1 a) 2
2
b) c) / 2 d) None of these 10. The
ˆ zˆj xk and s is the surface z 1 x 2 y 2 , z 0 , is curl v . n ds , where v y i s
a) b) c) / 2 d) None of these 11. The mathematical form of Gauss divergence theorem is a)
F .dr Curl F ds S
b) F .dr Curl F .n ds S
c) F .n ds div F dv v
d) None of these 12. The value of
Div F dv , where F 3 x i 4 y j 5 z k over the region bounded by the v
sphere x y 2 z 2 1 is 2
a)
3584 3
3584 4 3584 c) 3 b)
d) None of these 13. By using Gauss divergence theorem the value of
F.n ds,
where F 4 yi 6 zj 7 xk is
s
a) b) c) d)
0 1 -1 None of these
14. Gauss divergence gives a relationship between a) Surface and volume integral b) Line and volume integral c) Line and surface integral d) None of these 15. Green’s Theorem is a special case of ; a) Stock’s Theorem b) Gauss Theorem c) All of the above d) None of these 16. The value of line integral
xydx xydy ) , where C is the region bounded by C
0 x 1,0 y 1{By using of green’s theorem} is a) b) c) d)
1 -1 0 None of these
17. The value of
x
2
zdx 3xdy y 3 dz, where c is x 2 y 2 1
c
a) b) c) d)
3 3
None of these
18. The value of line integral
xdx ydy ) , where C is the region bounded by 0 x 1,0 y 1 C
{By using of green’s theorem} is e) 1 f) -1 g) 0
h) None of these 19.