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Term End Examination - November 2012

Course : MAT101 - ultivariable Calculus and Differential

quations

Slot: F1+TF1

Class NBR : 4330 / 4969

Time : Three Hours Max.Marks:100

PART – A (10 X 3 = 30 Marks)

Answer ALL the Questions

1. If  ),( y x f  z = andvu yvu x

−=+= , then find .v

 z

u

 z

∂

∂

+∂

∂

 

2. If  xy

 y xu

−

+=

1and  y xv

11 tantan −− += then find .),(

),(

 y x

vu

∂

∂ 

3.Calculate ∫ ∫

−

+

2

0

2

022

2 x x

 y x

 xdydxby changing in to polar coordinates.

4. Evaluate dxe xx

4

0

2 −∞

∫  

5. Find the directional derivative of  22 z xy +=φ at (1,-1,3) in the direction of 

.22 k  jir

rr

++  

6. Evaluate ∫ ++−

c

dy y xdx x xy )()2( 222 where C is the closed curve of the region

bounded by 2 x y = and .

2 x y =  

7. Solve  xe ydx

dy

dx

 yd  3

2

2

23 =+−  

8. Reduce  x x y

dx

dy x

dx

 yd  x

1242

2

2 +=++ in to a linear differential equation with constant

coefficients.

9. Find the Laplace transform of  t tet cos−

. 

10. Determine )0( f  and )(∞ f  if  .)1(

4

)1(

4

)1(

2)(

23 ++

++

+=

ssssF   

PART – B (5 X 14 = 70 Marks)

Answer any FIVE Questions 

11. a) Expand )1log( ye x + at ) ,( 00 as the Taylor’s series up to third degree. [7]

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b) A rectangular box open at the top is to have a volume of 32 cubic feet. Find the

dimensions of the box requiring least material for its construction.

[7]

12. a) Change the order of integration in ∫ ∫−1

0

22

 x

 x xydydx and hence evaluate it.

b) Evaluate ∫∫∫ ++

 D

dxdydz z y x )( by transforming in to spherical coordinates where D

is the volume bounded by the positive octant of the sphere 2222a z y x =++ .

[7]

[7]

13. a) Verify Green’s theorem in the XY- plane for ∫ −+−

C 

dy xy ydx y x )64()83( 22  

where C is the boundary of the region given by 100 =+== y x , y , x  

b) Evaluate ∫∫∧

S 

dsnF .r

where k  z j yi xF r

rrr

2224 +−= and S is the surface bounding

the region 0,422 ==+ z y x and 3= z . 

[7]

[7]

14. a) Solve  x ydx

 yd 2tan44

2

2

=+ by using the method of variation of parameters.

b) Solve  xe y D Dx cos10)( 23 =+ by the method of undetermined coefficients.

[7]

[7]

15. a) Find the Laplace transform of the triangular wave function

≤≤−

≤≤

= at a for t a

at  for t 

t  f  22

0

)(  

b) Using Convolution theorem, find

+

−222

1

)( as

s L

.

 

[7]

[7]

16.a) Show that k  z y z x j z xyi xz yF 

rrrr

)22()2()2( 222 +−+−++= is irrotational and

also find its scalar potential.

b) Using Laplace transform solve the integral equation ∫ =++

t 

dt t  y ydt dy

0

0)(2 , given

that .1)0( = y  

[7]

[7]

17. a) Using Laplace transform solve t et t  yt  yt  y 24)(2)(3)( +=+′−′′ where

1)0(,1)0( −=′= y y .

b) Solve the differential equation

 x x ydx

dy

 xdx

 yd 

 x log242

22

=++  

[7]

[7]

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