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KARAIKUDI INSTITUTE OF TECHNOLOGY KARAIKUDI INSTITUTE OF TECHNOLOGY KIT & KIM TECHNICAL CAMPUS KIT & KIM TECHNICAL CAMPUS Keeranipatti, Thalakkavur, Karaikudi – 630 307 Keeranipatti, Thalakkavur, Karaikudi – 630 307

UNIT TEST-III MAR 2014(I YEAR B)(Answer All the

Questions)PART A -

2 Mark Qs (5x2=10)1.State Gauss divergence theorem. 2.Define periodic function..3.Write down any two properties of Laplace transform .4.State sufficient conditions for existence of Laplace transform. .5.Give two examples for a function for which Laplace transform does not exist.

PART B – (1x8=8 Mark)(2x16=32 Mark)6 Find the Laplace transform for the following functions. (a).sin2 3 t (b).sin 2 tcost (c).cos2 tcos 5t (d).(a+bt )2

7.(i).Verify Stokes theorem for For F⃗=− y i⃗+2 yz j⃗+ y2 k⃗ where S is the upper half

of the sphere x2+ y2+z2=a2 and C is the boundary on the xoy plane. (OR)

(ii)Verify divergence theorem for F⃗=x2 i⃗+z j⃗+ yz k⃗ over the cube formed by

x=±1 , y=±1 , z=±1. 9.(i).Find the Laplace transform for the following functions.

(a).eat−cosbtt

(b).∫0

∞

t e−2 t cos2t

(ii).Find (a) L(tsin3 t) (b)L(e−5 t cos 4 t) (c) L(t e−4 t sin 3 t)

10.(i). Find the Laplace transform of f ( t )=kt ,0<t<1 , suchthatf ( t+1 )=f (t )

(ii). Find the Laplace transform of f (t )={ t∈0<t<a2a−t∈a<t<2a .

and

f ( t+2a )= f (t ) (OR)

11.(i). Find the Laplace transform of f ( t )={asinωt∈0<t< πω

0∈ πω

< t< 2πω.

and

f (t+ 2 πω )=f (t)

(ii). Find the Laplace transform of f (t )=et ,0< t<2π , suchthatf ( t+2π )=f (t ) .

UNIT TEST-III(I YEAR B)

(Answer All the Questions)PART A - 2 Mark Qs (5x2=10)

1.State Gauss divergence theorem. 2.Define periodic function..3.Write down any two properties of Laplace transform .4.State sufficient conditions for existence of Laplace transform. .5.Give two examples for a function for which Laplace transform does not exist.

PART B – (1x8=8 Mark)(2x16=32 Mark)6 Find the Laplace transform for the following functions. (a).sin2 3 t (b).sin 2 tcost (c).cos2 tcos 5t (d).(a+bt )2

7.(i).Verify Stokes theorem for For F⃗=− y i⃗+2 yz j⃗+ y2 k⃗ where S is the upper half

of the sphere x2+ y2+z2=a2 and C is the boundary on the xoy plane. (OR)

(ii)Verify divergence theorem for F⃗=x2 i⃗+z j⃗+ yz k⃗ over the cube formed by

x=±1 , y=±1 , z=±1. 9.(i).Find the Laplace transform for the following functions.

(a).eat−cosbtt

(b).∫0

∞

t e−2 t cos2t

ENGINEERING MATHEMATICS-II 50 Marks

ENGINEERING MATHEMATICS-II 50 Marks


(ii).Find (a) L(tsin3 t) (b)L(e−5 t cos 4 t) (c) L(t e−4 t sin 3 t)

10.(i). Find the Laplace transform of f ( t )=kt ,0<t<1 , suchthatf ( t+1 )=f (t )

(ii). Find the Laplace transform of f (t )={ t∈0<t<a2a−t∈a<t<2a .

and

f ( t+2a )= f (t ) (OR)

11.(i). Find the Laplace transform of f (t )={asinωt∈0<t< πω

0∈ πω

< t< 2πω.

and

f (t+ 2 πω )=f (t)

(ii). Find the Laplace transform of f (t )=et ,0< t<2π , suchthatf ( t+2π )=f (t ) .

PART A - 2 Mark Qs (5x2=10)

1.State Newton’s formula to find f '( x) using forward differences. 2.What is the restriction on number of intervals in Simpson’s 3/8 rule. .3.State two point Gaussian quadrature formula .

4.Solve dydx

=1− y , y (o )=0 for x=0.1 by Euler’s method.

5.State Trapezoidal rule to evaluate ∫x0

xn

f ( x )dx .

PART B – (1x8=8 Mark)(2x16=32 Mark)

6.Compute the value of ∫4

5.2

logx dx by dividing the range into 6 equal parts by using

Trapezoidal rule.

7.(i).For the given data, find dydx

and d2 yd x2 at x=1.1.

x 1 1.1 1.2 1.3 1.4 1.5 1.6 f(x) 7.989 8.403 8.781 9.129 9.451 9.750 10.031

(ii)Evaluate ∫0

2 dxx2+4

using Romberg’s method.

8.(i).When a train is moving at 30 meters per second steam is shut off and breaks are applied. the speed of the train (V) in meters per second after t seconds is given by

t 0 5 10 15 20 25 30 35 40 v 30 24 19.5 165 13.6 11.7 10 8.5 7

(ii). Evaluate ∫0

1

∫1

2 2 xydxdy(1+x2 )(1+ y2)

by Trapezoidal rule with h=k=0.

9.(i).Using Simpson’s rule ,evaluate

∫4

4.4

∫2

2.6 dxdyxy

h=0.3 and k=0.2.

(ii).Using Runge- Kutta method of fourth order solve

y '= y2−x2

y2+x2 with y (0 )=1at x=0.2,0 .4 .

10.(i).Using improved Euler’s method find y(0.2) and y(0.4) from y '=x+ y , y (0 )=1with h=0.2. (ii). Consider the second order initial value problem

d2 yd x2 −2 dy

dx+2 y=e2x sinx with y (0 )=−0.4∧ y ' (0 )=−0.6, using Fourth

order RK method find y(0.2). (Answer All the Questions)PART A - 2 Mark Qs (5x2=10)1.Locate the negative root ofx3−2 x+5=02. What is the condition for convergence of Gauss-Jacobi method.3. Give Lagrange’s inverse interpolation formula.4. Write divided difference table for x 30 35 45 55y 148 96 68 34


5. Give Newton’s forward interpolation formula.

PART B – (1x8=8 Mark)(2x16=32 Mark)6. Solve x+ y+54 z=110 ;27 x+6 y−z=85 ;6 x+15 y+2 z=72 ,by Gauss Seidel method.

(i) Use Power method to find the dominant eigen value and eigen vector of [ 2 −1 0−1 2 −10 −1 2 ]

(ii) Solve10 x−2 y+3 z=23 ;2x+10 y−5 z=−33 ;3 x−4 y+10 z=41by Gauss elimination method.(i). Find the equation of the parabola passing through the points (0,0),(1,1)and (2,20) using Lagrange’s formula. (ii). Use Newton’s divided differences formula to find f(3)from the data

x 0 1 2 4 5 f(x) 1 14 15 5 6

Find the cubic spline approxi,ation and hence find f(1.5)x 1 2 3 4 f(x) 1 2 5 11

11.(i)Find the values of y at x=21 and x=28 from the data given below

(ii)Solve sinx=1+x3lies between (-2,-1) to 3 decimal places by Newton Raphson method.

INTERNAL TEST-I FEB 2014ME2026 – UNCONVENTIONAL MACHINING PROCESS 50

Time: 09:15 a.m. – 11:00 a.m. Marks


1.What are the industrial needs for UCM?2.List down the various mechanical energy based Unconventional Machining Processes.3.What are the different machining characteristics will respect to which the non-traditional machining processes can be analyzed?4.Distinguish traditional & non-traditional machining processes?5.How will you compare various non-traditional processes?

PART B – 1x8=8 Mark2x16=32 Mark

6. What is the need for the development of UCM process? Explain with examples. (8)7. Make a comparison between traditional and unconventional machining processes in terms of cost, application, scope, machining time, advantages & limitations. (16)

(OR)8.With a neat sketch explain the process of AJM? Write its advantages and applications.(16)

9. Make a comparison between traditional and unconventional machining processes in terms of cost, application, scope, machining time, advantages & limitations.(16)

(OR)10.Discuss the effects of the following parameters on the material removal and surface finish in ultrasonic machining:

a. Amplitude and frequencyb. Abrasive sizec. Concentration of abrasivesd. Material hardness

x 20 23 26 29y 0.3420 0.3907 0.4384 0.4848


INTERNAL TEST-I FEB 2014ME2026 – UNCONVENTIONAL MACHINING PROCESS

Time: 09:15 a.m. – 11:00 a.m.50

Marks


1.What are the industrial needs for UCM?2.List down the various mechanical energy based Unconventional Machining Processes.3.What are the different machining characteristics will respect to which the non-traditional machining processes can be analyzed?4.Distinguish traditional & non-traditional machining processes?5.How will you compare various non-traditional processes?

PART B – 1x8=8 Mark2x16=32 Mark

6. What is the need for the development of UCM process? Explain with examples. (8)7. Make a comparison between traditional and unconventional machining processes in terms of cost, application, scope, machining time, advantages & limitations. (16)

(OR)8.With a neat sketch explain the process of AJM? Write its advantages and applications.(16)

9. Make a comparison between traditional and unconventional machining processes in terms of cost, application, scope, machining time, advantages & limitations.(16)

(OR)10.Discuss the effects of the following parameters on the material removal and surface finish in ultrasonic machining:a. Amplitude and frequencyb. Abrasive sizec. Concentration of abrasives

d. Material hardness

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