ENGINEERING MATHEMATICS

Applied Mathematics-II

UNIT-I

1. (a) If x + 1 = 2 cos ө and y + 1= 2cos ө

x                                x

Show that one of the value of

xmyn +        1        is 2cos(m ө + nф) 

                         xm + yn 

(b) If (ө + iф) e^a show that

ө = (n + 1/2)∏/2                    and

          ф=1 log tan (∏  + a)

2               4     2

(c) Sum the series by C+ is method:

xsin ө – 1  x² sin2 ө +1x³ sin 3ө ………∞

2                     3

 UNIT- II

2. (a) Solve the equation:

d²y – 2dy + y= xe^x sin x.

dx²      dx

(b) Solve by the method of variation of parameter:

d²y + y= x sin x

dx²

(c) Solve the equation

x²d²y – 3xdy + 4y= (1+x)²

dx²        dx

 UNIT- III

3. (a) Change the order of the integration in

2-x

^lʃ₀ ʃ xy dx dy

and hence evaluate same.

(b) Find the volume bounded by the cylinder x² + y² = 4 and the planes

y + z = 4 and z = 0

(c) Solve  ʃ ʃxy(x² + y²)^n/2 dxxy over the positive quadrant of

x² + y² = 4 sup posin g n + 3> 0

 UNIT- IV

4. (a) find the work done in moving a particle in the force field:

F= 3x³I + (2xz – y)j + zk along:

(i) the straight line form (0,0,0) to (2,1,3)

(ii) the curve defined by x² =4y, 3x² =8z from x = 0 x = 2

(b) Evaluate divergence & curl of

F = 3x² I + 5xy² j +xyz³ K

At the point (1, 2, 3)

(c) Use divergence theorem to evaluate ʃF ds where

F = 4xl – 2y² j + z² K and S is the surface bounding the region

x² + y² = 4, z = 0, z = 3

UNIT-V

5. (a) Solve the equation by Ferrari’s method:

x⁴  - 12x² + 41x² – 18x- 72 = 0

(b) Solve the equation by Cardin’s method:

x³ – 15x – 126 = 0

(c) Solve x³ – 4x² – 20x + 48 = 0 given that the roots a and β are connected by the relation a + 2β = 0Applied Mathematics-II

UNIT-I

1. (a) If x + 1 = 2 cos ө and y + 1= 2cos ө

x                                x

Show that one of the value of

xmyn +        1        is 2cos(m ө + nф) 

                         xm + yn 

(b) If (ө + iф) e^a show that

ө = (n + 1/2)∏/2                    and

          ф=1 log tan (∏  + a)

2               4     2

(c) Sum the series by C+ is method:

xsin ө – 1  x² sin2 ө +1x³ sin 3ө ………∞

2                     3

 UNIT- II

2. (a) Solve the equation:

d²y – 2dy + y= xe^x sin x.

dx²      dx

(b) Solve by the method of variation of parameter:

d²y + y= x sin x

dx²

(c) Solve the equation

x²d²y – 3xdy + 4y= (1+x)²

dx²        dx

 UNIT- III

3. (a) Change the order of the integration in

2-x

^lʃ₀ ʃ xy dx dy

and hence evaluate same.

(b) Find the volume bounded by the cylinder x² + y² = 4 and the planes

y + z = 4 and z = 0

(c) Solve  ʃ ʃxy(x² + y²)^n/2 dxxy over the positive quadrant of

x² + y² = 4 sup posin g n + 3> 0

 UNIT- IV

4. (a) find the work done in moving a particle in the force field:

F= 3x³I + (2xz – y)j + zk along:

(i) the straight line form (0,0,0) to (2,1,3)

(ii) the curve defined by x² =4y, 3x² =8z from x = 0 x = 2

(b) Evaluate divergence & curl of

F = 3x² I + 5xy² j +xyz³ K

At the point (1, 2, 3)

(c) Use divergence theorem to evaluate ʃF ds where

F = 4xl – 2y² j + z² K and S is the surface bounding the region

x² + y² = 4, z = 0, z = 3

UNIT-V

5. (a) Solve the equation by Ferrari’s method:

x⁴  - 12x² + 41x² – 18x- 72 = 0

(b) Solve the equation by Cardin’s method:

x³ – 15x – 126 = 0

(c) Solve x³ – 4x² – 20x + 48 = 0 given that the roots a and β are connected by the relation a + 2β = 0