1(a)
Prove by mathematical induction:\[ \frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{\left ( 2n-1 \right )\left ( 2n+1 \right )}=\frac{n}{2n+1}\] for n≥

3 M

Solve any one question.Q1(a,b) Q2(a,b)

1(a)
A four cylinder vertical engine has cranks 150mm long. The cylinders are spaced 200 mm apart, Mass of reciprocating parts of 1st,

2nd and 4th cylinders are 50kg,

60kg and 50kg respectively. Find the reciprocating mass of the 3rd cylinder and relative angular positions of the cranks to achieve complete primary balance

2nd and 4th cylinders are 50kg,

60kg and 50kg respectively. Find the reciprocating mass of the 3rd cylinder and relative angular positions of the cranks to achieve complete primary balance

6 M

1(b)
Define the followin with examples:

i) uncountable infinite set

ii) Countable infinite set.

i) uncountable infinite set

ii) Countable infinite set.

3 M

1(b)
Determine the expression for natural frequency of the system shown in Fig.1

!mage

!mage

4 M

1(c)
If R=[(a,b),

(b,a),

(b,c),

(c, d),

(d,o)] be a relation on the set A = (a,

b,

c,

d). Find the transitive closer of R using Waeshall's algorithm .

(b,a),

(b,c),

(c, d),

(d,o)] be a relation on the set A = (a,

b,

c,

d). Find the transitive closer of R using Waeshall's algorithm .

6 M

2(a)
Let, A=(a,

b,

c,

d) and B= (1,

2,

3). Determine whether the relation R from A to B is a function. Justify. If it is function give the range:

i) R=[(a,

1),

(b,

2),

(c,1),

(d,2)]

R=[(a,

1), (b,

2), (a,

2),

(c,

1),

(d,

2)].

b,

c,

d) and B= (1,

2,

3). Determine whether the relation R from A to B is a function. Justify. If it is function give the range:

i) R=[(a,

1),

(b,

2),

(c,1),

(d,2)]

R=[(a,

1), (b,

2), (a,

2),

(c,

1),

(d,

2)].

4 M

2(a)
A shock absorber is to be designed so that is overshoot is 10% of the initital displacement when released. Determine the damping factor. Also find the overshoot if the damping factor is reduced to 50%.

6 M

2(b)
The following is the Hasse diagram of the poset: (a,

, b,

c,

d,

e),<). Is it a lattice? Justify.

!mage

, b,

c,

d,

e),<). Is it a lattice? Justify.

!mage

2 M

2(b)
Explain the terms Static Balancing and Dynamic Balancing.

4 M

2(c)
In a class of 80 students, 50 students know English, 55 know French and 46 know German language, 37 students know English and French, 28 students know French and German, 25 know English and German,7 students know none of the languages. Findout:

i) How many students know all the 3 languages?

ii) How many students know exactly 2 languages ?

iii) How many students know only one language?

i) How many students know all the 3 languages?

ii) How many students know exactly 2 languages ?

iii) How many students know only one language?

6 M

Solve any one question.Q5(a,b,c) Q4(a,b,c)

3(a)
A bag contains 6 red and green 8 balls.

i) If on ball is drawn at random, them what is the probability of the ball being green?

ii) If two balls are drawn at random, then what is the probability that one is red and the other is green?

i) If on ball is drawn at random, them what is the probability of the ball being green?

ii) If two balls are drawn at random, then what is the probability that one is red and the other is green?

6 M

Solve any one question.Q3(a,b) Q4(a,b)

3(a)
A single cylinder vertical petrol engine of total mass 320kg is mounted on steel chassis and causes a vertical static deflection of 2mm. The reciprocating parts of the engine have a mass of 24kg and move through a vertical stroke of 150mm with SHM. A dashpot attached to the system offers resistance of 490N at a velocity of 0.3 m/s. Determine:

i) the speed of driving shaft at reasonance

ii) the amplitude of steady state vibrations when the driving shaft of the engine roatates at 480 rpm.

i) the speed of driving shaft at reasonance

ii) the amplitude of steady state vibrations when the driving shaft of the engine roatates at 480 rpm.

6 M

3(b)
Determine which of the graphs below represents Eulerian circuit,

Elulerian path,

Hmailtonian circuit and Hamiltonian path. Justify

!mage

Elulerian path,

Hmailtonian circuit and Hamiltonian path. Justify

!mage

4 M

3(b)
Define the following terms:

i) Damping coefficient

ii) Critical damping coefficient

iii) Damping factor

iv) Logarithmic decrement

i) Damping coefficient

ii) Critical damping coefficient

iii) Damping factor

iv) Logarithmic decrement

4 M

3(c)
Define the graph K

_{a}and K_{mn}
2 M

4(a)
In how many ways can 6 men and 5 women be seated a line so that no two women sit together? In how many ways can 6 men and 5 women sit in a line so that women occupy the even places.

6 M

4(a)
A horiziontal spring mass system with coulomb damping has a mass of 5 kg attached to a spring of stifffness 980N/m. If the coefficient of friction is 0.25, calculate:

i) The frequency of free oscillations

ii) The number of cycles corresponding to 50% reduction in amplitude if the initial amplitude is 5 cm

iii) Time taken to achieve this 50% reduction

i) The frequency of free oscillations

ii) The number of cycles corresponding to 50% reduction in amplitude if the initial amplitude is 5 cm

iii) Time taken to achieve this 50% reduction

6 M

4(b)
Use Djikstra algorithm to find the shortest pathe between a and z

!mage

!mage

6 M

4(b)
Write short note on Forced vibrations due to reciprocating unbalance.

4 M

Solve any one question.Q5(a,b,c) Q6(a,b,c)

5(a)
Find maximum flow in the transport network using liabeling procedure. Determine the corresponding min cut.

!mage

!mage

7 M

Solve any one question.Q5(a,b) Q6(a,b)

5(a)
Find the natural frequencies of the system shown in fig.2

m

m

r=

r

k

k

k

!mage

m

_{1}=10 kg,m

_{2}= 12 kgr=

_{1}=0.10m,r

_{2}=0.11mk

_{1}=40×10^{3}N/mk

_{2}=50×10^{3}N/mk

_{3}=60×10^{3}N/m.!mage

12 M

5(b)
Define:

i) Forest

ii) Height pf a tree

iii) Ordered tree

iv) Preperties of tree.

i) Forest

ii) Height pf a tree

iii) Ordered tree

iv) Preperties of tree.

4 M

5(b)
Define the following terms:

i) Zero frequency

ii) Node point

i) Zero frequency

ii) Node point

4 M

5(c)
State whether the given code is prefix code. Justify:

000,

001,

01,

10, 111.

000,

001,

01,

10, 111.

2 M

6(a)
Give the stepaise construction of minimum spanning tree using Prim's algorithm for the following graph. Obtain the total cost of minimum spanning tree

!mage.

!mage.

6 M

6(a)
Find the natural frequencies and mode shapes for the tosional system shown in Fig.3. Assume J

J

!mage

_{1}=J_{0},J

_{2}=2J_{0}and stiffness for each spring as!mage

12 M

6(b)
Explain fundamental system of cutset with suitable examples.

4 M

6(b)
Explain the concept of torsionally equivalent shaft.

4 M

6(c)
Explain binary tree,

binary search tree and ordered tree with suitable exampls.

binary search tree and ordered tree with suitable exampls.

3 M

Solve any one question.Q7(a,b,c) Q8(a,b,c)

7(a)
Consider the bibary reation* defined onn the set: A=(a,

b,

c,

d) by the following table. Fill the empty cell.

b,

c,

d) by the following table. Fill the empty cell.

* | c | a | b | c |

c | c | a | b | c |

a | a | b | c | c |

c | ||||

c |

2 M

Solve any one question.Q7(a,b) Q8(a,b)

7(a)
An accelerometer has a suspended mass of 0.01kg with adamped natural frequency of vibration of 150 Hz. It is mounted on an engine running at 6000 rpm and undergoes an acceleration of 1 g. The instrument records an acceleration of 9.5 m/s

^{2}. Find the damping constant and the spring stiffness of the accelerometer.
8 M

7(b)
Prove that:\[ (\left ( a+b\sqrt{2} \right),+,*)/] where a,

b,

ϵ R is integral domain.

b,

ϵ R is integral domain.

6 M

7(b)
Write a short note prediction of vibration failure using time and frequency domain analysis of vibration signals.

8 M

7(c)
Define Normal Subgroup and rignd with example.

5 M

8(a)
Let R=()*,

60*,

120*,

180*,

240*,

300*) and * is a binary operation so that a and b in R, a is overall angular rotation corresponding to successive rotation by a and then b. Show that (R, *) is a group.

60*,

120*,

180*,

240*,

300*) and * is a binary operation so that a and b in R, a is overall angular rotation corresponding to successive rotation by a and then b. Show that (R, *) is a group.

2 M

8(a)
For finding vibraton parameter of a machine running at 260rpm, a seismic instrument is used. The natural frequency of the instrument is 7 Hz and the recorded displacement is 6mm. Determine the displacement, velocity and acceleration of the vibrating machine assuming no damping.

8 M

8(b)
Let A= [0,

1]. Is A closed under:

i) Multiplication

ii) Addition.

1]. Is A closed under:

i) Multiplication

ii) Addition.

6 M

Solve any one question.Q9(a,b) Q10(a,b,c)

8(b)
Write a short note on:

i) FFT analyzer

ii) Condition monitoring of machine

i) FFT analyzer

ii) Condition monitoring of machine

8 M

8(c)
Explain Isornorphism and Automarphism.

5 M

9(a)
Determine the sound power level of a source generating

i) 0.5W

ii) 1.5W

iii) 2.2W

iv) 3 W of sound power

i) 0.5W

ii) 1.5W

iii) 2.2W

iv) 3 W of sound power

8 M

9(b)
Explain the following term:

i) Wavelength

ii) Velocity of sound

iii) Decibel scale

iv) Sound power level

v) Sound pressure

i) Wavelength

ii) Velocity of sound

iii) Decibel scale

iv) Sound power level

v) Sound pressure

10 M

10(a)
Define the following terms:

i) Reflection coefficient

ii) Absorption coefficient

iii) Transmission coefficient

i) Reflection coefficient

ii) Absorption coefficient

iii) Transmission coefficient

6 M

10(b)
Draw and explain the main components of human hearing mechanism.

6 M

10(c)
Show that if the sound pressure is doubled, the sound pressure level increases by six decibels.

6 M

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