Code No: R161202 R16 SET - 1

I B. Tech II Semester Regular Examinations, April/May - 2017

MATHEMATICS-II (MM)
(Com. to CE, EEE, ME, CHEM, AE, BIO, AME, MM, PE, PCE, MET)

Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)
2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART –A

1. a) Explain the Bisection method. (2M)

b) Prove that ∆ = E − 1 . (2M)
c) Write Newton’s forward interpolation formula. (2M)
d) Write Trapezoidal rule and Simpson’s 3/8th rule. (2M)
e) Write the Fourier series for f ( x ) in the interval ( 0, 2π ) . (2M)

f) Write One dimensional wave equation with boundary and initial conditions. (2M)
g) If F ( s ) is the complex Fourier transform of f ( x ) , then prove that (2M)

1 s
F { f ( ax )} = F .
a a
PART –B

2. a) Using bisection method, obtain an approximate root of the equation x 3 − x − 1 = 0 . (7M)

b) Develop an Iterative formula to find the square root of a positive number N using (7M)
Newton-Raphson method.

3. a) Evaluate ∆ 2 ( tan −1 x ) . (6M)

b) Using Newton’s forward formula, find the value of f (1.6) , if (8M)

x 1 1.4 1.8 2.2
f ( x) 3.49 4.82 5.96 6.5

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Code No: R161202 R16 SET - 1

1.4
3
∫ ( sin x − log x + e ) dx using Simpson’s 8
x th
Compute the value of rule.
4. a) 0.2 (7M)
b) Using the fourth order Runge – Kutta formula, find y (0.2) and y (0.4) given that (7M)
2 2
dy y − x
= 2 , y(0) = 1.
dx y + x 2

5. a) Find a Fourier series to represent f ( x) = x − x 2 in −π ≤ x ≤ π . Hence show that (7M)

1 1 1 1 π2
− + − + ⋯⋯ = .
12 22 32 42 12

b) Obtain the half range sine series for f ( x) = e x in 0 < x <1 . (7M)

6. a) Solve by the method of separation of variables (7M)

4u x + u y = 3u and u ( 0, y ) = e−5 y .
b) A tightly stretched string with fixed end points x = 0 and x = L is initially in a (7M)

π x
position given by y = y0 sin 3   if it is released from rest from this position,
 L 
find the displacement y ( x, t ) .

1 for x ≤ 1
Express the function f ( x ) =  as a Fourier integral. Hence
7. a) 0 for x ≥ 1 (7M)

∞
sin λ cos λ x
evaluate ∫ dλ .
0
λ

1 − x
2
for x ≤ 1 (7M)
Find the Fourier transform of f ( x) =  hence evaluate
b) 0 for x > 1
∞
 x cos x − sin x  x
∫ 
0 x 3  cos dx .
 2

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Code No: R161202 R16 SET - 2

I B. Tech II Semester Regular Examinations, April/May - 2017

MATHEMATICS-II (MM)
(Com. to CE, EEE, ME, CHEM, AE, BIO, AME, MM, PE, PCE, MET)

Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)
2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART –A

1. a) Explain the Method of false position. (2M)

b) Prove that ∇ = 1 − E −1 . (2M)
c) Write Newton’s backward interpolation formula. (2M)
d) Write Simpson’s 1/3rd and 3/8th rule. (2M)
e) Write the Fourier series for f ( x) in the interval ( 0, 2 L ) . (2M)
f) Write the suitable solution of one dimensional wave equation. (2M)
g) If F ( s ) is the complex Fourier transform of f ( x) , then prove that (2M)
F { f ( x − a )} = ei a s F ( s ) .
PART -B
2. a) Using bisection method, compute the real root of the equation x3 − 4 x + 1 = 0 . (7M)
b) Develop an Iterative formula to find the cube root of a positive number N using (7M)
Newton-Raphson method.

3. a) Evaluate ∆ ( e x log 2 x ) . (6M)

b) Using Newton’s forward formula compute f (142) from the following table: (8M)
x 140 150 160 170 180
f ( x) 3.685 4.854 6.302 8.076 10.225

2
1
4. a) Evaluate, ∫ e dx by using Trapezoidal rule and Simpson’s 3
− x2 rd
rule taking (7M)
0

h = 0.25.
b) Find the value of y at x = 0.1 by Picard’s method, given that (7M)
dy y − x
= , y ( 0 ) = 1.
dx y + x

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Code No: R161202 R16 SET - 2

− π , − π < x < 0
5. a) Given that f ( x) =  . Find the Fourier series for f ( x) . (7M)
 x, 0 < x <π
1 1 1 1
Also deduce that
π2
+ + + + ⋯ ⋯ = .
12 32 52 7 2 8
b) Express f ( x) = x as a half-range cosine series in 0 < x < 2 . (7M)

6. a) Solve by the method of separation of variables (7M)

∂u ∂u
= 2 + u, given u ( x,0 ) = 6e−3 x .
∂x ∂t
b) A string of length L is initially at rest in equilibrium position and each of its points (7M)
 ∂y  πx
is given the velocity   = b sin 3   . Find displacement y ( x, t ) .
 ∂t t =0  L 

1 for 0 ≤ x ≤ π
7. a) Express f ( x) = 0 for x > π
as a Fourier sine integral and hence evaluate (7M)

∞
1 − cos(πλ )
∫0
λ
sin( xλ ) d λ .

b) Find the Fourier sine and cosine transform of (7M)

−a x
f ( x) = e , a > 0, x > 0 .

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Code No: R161202 R16 SET - 3
I B. Tech II Semester Regular Examinations, April/May - 2017
MATHEMATICS-II (MM)
(Com. to CE, EEE, ME, CHEM, AE, BIO, AME, MM, PE, PCE, MET)

Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)
2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART –A

1. a) Explain the Newton-Raphson method. (2M)

b) Prove that δ = E1 2 − E −1 2 . (2M)
c) Write Lagrange’s interpolation formula for unequal intervals. (2M)
dy (2M)
Explain Taylor’s series method for solving IVP = f ( x, y ) with y ( x0 ) = y0 .
d) dx
e) Write the Fourier series for f ( x ) in the interval ( −π , π ) . (2M)
f) Write the suitable solution of one dimensional heat equation. (2M)
g) If F ( s ) is the complex Fourier transform of f ( x ) , then prove that (2M)
1
F { f ( x ) cos ax } =  F ( s + a ) + F ( s − a )  .
2
PART –B

2. a) Using Regula-Falsi method, compute the real root of the equation x 3 − 4 x − 9 = 0 . (7M)
b) 1 (7M)
Develop an Iterative formula to find . Using Newton-Raphson method.
N

 x2 
3. a) Evaluate ∆  . (6M)
 cos 2 x 
b) Compute f ( 27 ) Using Lagrange’s formula from the following table: (8M)
x 14 17 31 35
f ( x) 68.7 64.0 44.0 39.1

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Code No: R161202 R16 SET - 3

0.6
1
∫0 e
− x2 rd
4. a) Evaluate dx by using Simpson’s rule taking seven ordinates. (7M)
3
b) dy (7M)
Given that = 2 + xy , y (1) = 1 .
dx
Find y ( 2 ) in steps of 0.2 using the Euler’s method.

x , 0 ≤ x ≤π (7M)
5. a) Find the Fourier series for the function f ( x ) =  .
 2π − x , π ≤ x ≤ 2π
1 1 1 1
Also deduce that
π2
+ + + + ⋯ ⋯ = .
12 32 52 7 2 8
b) Obtain the Fourier expansion of f ( x ) = x sin x as a cosine series in (0, π ) . (7M)

6. ∂ 2u ∂ 2u
Solve the Laplace’s equation + = 0 in a rectangle in the xy -plane, (14M)
∂x 2 ∂y 2
0 ≤ x ≤ a and 0 ≤ y ≤ b satisfying the following boundary condition
u (0, y ) = 0 , u (a, y ) = 0 , u ( x, b) = 0 and u ( x, 0) = f ( x ).

7. a) Find the Fourier sine transform of the function (7M)

x , 0 < x <1

f ( x) = 2 − x , 1 < x < 2 .
0 , x>2

b) Find the Fourier cosine integral and Fourier sine integral of (7M)
−k x
f ( x) = e ,k > 0.

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Code No: R161202 R16 SET - 4

I B. Tech II Semester Regular Examinations, April/May - 2017

MATHEMATICS-II (MM)
(Com. to CE, EEE, ME, CHEM, AE, BIO, AME, MM, PE, PCE, MET)

Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B)
2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART –A

1. a) Explain Iteration method. (2M)

b) 1 (2M)
Prove that µ = ( E1 2 + E −1 2 ) .
2
c) Prove that ∆ y2 = ∇ 3 y5 .
3 (2M)
d) Explain Runge-Kutta method of fourth order for solving IVP (2M)
dy
= f ( x, y ) with y ( x0 ) = y0 .
dx
e) Write the Fourier series for f ( x ) in the interval ( − L, L ) . (2M)
f) Write the various possible solutions of two-dimensional Laplace equation. (2M)
g) If F ( s ) and G ( s ) are the complex Fourier transform of f ( x ) and g ( x ) , then (2M)
prove that F { a f ( x ) + b g ( x ) } = a F ( s ) + b G ( s ) .

PART –B

2. a) Find a positive real root of the equation x 4 − x − 10 = 0 using Newton-Raphson’s (7M)

method.
b) Explain the bisection method. (7M)

3. a) Evaluate ∆ 2 ( cos 2x ) . (6M)

b) Using Newton’s backward formula compute f (84) from the following table: (8M)
x 40 50 60 70 80 90
f ( x) 184 204 226 250 276 304

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Code No: R161202 R16 SET - 4

∫0 e
− x2
4. a) Evaluate dx by using Trapezoidal rule with n = 10 . (7M)

b) Obtain Picard’s second approximate solution of the initial value problem (7M)
2
dy x
= 2 , y (0) = 0 .
dx y + 1

2
π − x 
5. a) Obtain the Fourier series f ( x) =  2  in the interval 0 < x < 2π. Deduce that (8M)
 
1 1 1 1 π2
+ + + + ⋯ ⋯ = .
12 22 32 42 6
b) Express f ( x ) = x as a half-range cosine series in 0 < x < 2 . (6M)

6. a) Solve by the method of separation of variables (7M)

∂u ∂u
=4 and u ( 0, y ) = 8e−3 y .
∂x ∂y
∂ 2u ∂ 2u (7M)
b) Solve the Laplace’s equation + = 0 in a rectangle in the xy -plane,
∂x 2 ∂y 2
0 ≤ x ≤ a and 0 ≤ y ≤ b satisfying the following boundary condition
u ( x, 0) = 0 , u ( x, b) = 0 , u (0, y ) = 0 and u ( a, y ) = f ( y ).

7. a) Find the Fourier cosine integral and Fourier sine integral of (7M)
−a x −b x
f ( x) = e −e , a > 0 , b > 0.
x2
− a2 x2 −
b) Find the Fourier transform of e , a > 0. Hence deduce that e 2 is self (7M)
reciprocal in respect of Fourier transform.

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