Sardar Patel College of Engineering III.pdf · below. Find the area by Trapezoidal rule and Simpson's rule. Chainage (m) 0 10 20 30 40 50 60 Offset (m) 3.6 4.9 6.8 7.2 5.1 2.9 4.7

Bharatiya Vidya Bhavan's

Sardar Patel College of Engineering (A Government Aided Autonomous Institute)

Munshi Nagar, Andheri (West), Mumbai — 400058. Re-Exam jan,2020

Max. Marks: 100 Class: S.Y.B.Tech Semester: III Name of the Course: Building Drawing with CAD

Instructions: 1. All questions are compulsory 2. Make suitable assum tio

Q. P. Code: Duration: 3 Hrs

Program: Civil Engineering Course Code : ES-BTC-304

' _ _ _ - - J µ V i•ll le%, izat..iii t-iipai ty

Questio n No Max.

Marks

Course Outcome Number

BL/PI

liji

Q.1

It is proposed to construct G+1 RCC BUNGLOW for Electrical Engineer. Plot size is 25 M X 20 M. Requirements as below:

1. Habitable room. 2. Kitchen 3. Master Bed room - 4. Children bed room 5. Guest room 6. Office & drawing room .

Provide staircase, toilets and other facilities like parking as per standards, a) Draw a ground floor plan b) Draw a line plan for first floor

15 05

01 2/2.1.3

Q.2 Draw a sectional elevation passing through a stair and sanitary unit for given data in Q.1 20 01 2/2.1.3

Q.3 a) Draw a site plan for given data in Q.1 I 10

IP b) Draw a foundation plan for given data in Q.1 10 01 2/2.1.3

Q.4

Write a short note on the following points:

1. One & two point perspective ' 2. Building bylaws for Fire Safety 3. Planning principle= Aspect 4. Planning principle= Economy

20 01/02 1/1.2.3

(2.5 a) Explain Real estate regulation act in detail. 10 b)Draw a Location plan for given data in Q.1 10 01/02 2/2.1.3



Munshi Nagar, Andheri (West), Mumbai — 400058. Re- Examinations, January-2020

Max. Marks: 100 Class: S.Y.B.Tech. Semester: III Name of the Course: Engineering Materials

Instructions: 1. Question No 1 is compulsory 2. Attempt any four questions out of remaining six. 3. Draw neat diagrams

Q. P. Code: Duration: 3 hour Program: Civil

Course Code : PC -1}T-e-30-7

T-

Ques tion No

Points CO BL PI

(a) Discuss in detail Fiber glass. 4 1 2 1.3.1 (b) Discuss the applications of Plastics in construction industry. 4 3 3 2.4.2

Qi (c) What do you mean by ready mixed concrete? 4 2 2 1.3.1

(d) Explain Portland pozzolona cement. 1.3.1 4 3 1

(e) Highlight the features of smart concrete. 4 2.3.1 1 4

(a) Explain the functions of different Geosynthetic materials used in various Civil Engineering applications.

10 2 1 2.5.1

(b) Differentiate between: 6 2 2 2.2.3

Q2 (0 Earthenware and Terracotta (ii) First class and second class brick (c) Where would you recommend the packing mortar and fire 4 3 3 2.4.2 Resistant mortar?

Q3 (a) Draw the flow chart and explain procedure for the manufacturing of cement by wet process.

10 1 1 2.3.2

(b) Discuss efflorescence test and compression tests performed on bricks for its suitability.

6 2 2 1.2.1

(c) Write note on Light weight concrete. 2.5.1 4 2 1

(a) Explain chemical and electrical method of seasoning of timber. 6 1 2 2.4.2

(b) Describe the ingredients along with function for manufacturing of the glass. 2.2.3 6 2 1

Q4 (c) write note on (i) Cement paints 8 3 3 1.2.1 (ii) Hydrophobic cement

Q5 6

(a) Explain with neat sketch (i) Perforated Brick (ii) Hollow Brick (iii) Coping Brick (iv) Queen closer (b) Give the comparison of Tar and Asphalt in Tabular form. (c) Explain creosoting and ASCU treatment to timber.

8

6

1

3

3

2

4

3

1.2.2

2.2.3

1.6.1

Q6

(a) Explain the properties and uses of materials used for thermal and sound insulation. (b) Explain with neat sketch Ridge tile and Mangalore tiles. (c) Write note on Fiber reinforced plastic (d) What is plywood? State its application in construction.

8

4 4 4 1.3.1

3

3 1 2

3

1 2

1,3

2.4.2

1.3.1 2.2.3

Q7

(a) Describe the properties and applications of any two type of (i) ferrous metals (ii) Non-ferrous metals (b) Discuss the various applications of clay products in the building industry. (c) Explain Cutback and Blown bitumen. (d) State advantages and disadvantages of timber construction.

6

5

4 5

2

3

2 1

1,3

3

1 2

2.2.3

2.4.2

1.3.1 3.5.5







Instructions: 1. All questions are compulsory 2. Make suitable assumptions where necessary and state them clearly

Questio n No Max.

Marks


BL/PI

Q.1

It is proposed to construct G-fr 1 RCC BUN. GLOW for Electrical Engineer. Plot size is 25 M X 20 M. Requirements as below:

1. Habitable room. 2. Kitchen 3. Master Bed room 4. Children bed room 5. Guest room 6. Office & drawing room


15 05

01 2/2.1.3


Q.3 a) Draw a site plan for given data in Q.1 10

2/2.1.3

1/1.2.3

01 b) Draw a foundation plan for given data in Q.1 10

Q.4


1. One & two point perspective 2. Building bylaws for Fire Safety 3. Planning principle= Aspect 4. Planning principle= Economy

20 01/02

Q.5 Explain Real a) estate regulation act in detail. 10

01/02 2/2.1.3 b)Draw a Location plan for given data in Q.1 10

haratiya V idya I3 h


Munshi Nagar, Andheri (West), Mumbai /100058. Re-examinations, Jan‘ u ary-2020


Course Code : PC-IITC3 03

Max. Marks: 100 Class: S.Y.B.Tech. Semester: Ill Name of the Course: Basics of Surveying Instructions: 1. Question No 1 is compulsory 2. Attempt any four questions out of remaining six. 3. Draw neat diagrams 4. Assume suitable data if necessary

Question No

Points CO BL PI _

Q1

(a) Explain the fundamental principle of surveying. 4 (b) Write a note on Local attraction. 4 (c) What do you mean by balancing of sight? 4 (d) Explain effect of curvature and refraction on staff readings. 4 (e) What is Gale's table? How it helps in traverse computation? 4

1 3 3 1 3

2 4 2 1 2

1.2.1 2.1.2 1.2.1 1.2.1 1.3.1

(a) The bearings of the lines of a traverse arc given below. Find 10 2 4 2.1.2 the included angles and correct the bearings for local attraction, if any.

Line AB BC CD DE EA Q2

FB 73°40' 113°50' 164° 20' 223° 40' 303' 50'

BB 252°30' 295° 20' 344° 20' 430 123 45'

(b) Describe in detail radial method of contouring. 10 1 2 2.4.2

Q3 (a) A 20 M chain was found to be 4 cm short after chaining 1760 OS 3 4 2.1.2

M. It was 8 cm short at the end of days' work after chaining a total distance of 2880 M distance. If the chain was correct before commencement of work, find true distance.

(b) Derive an expression for the curvature and refraction correction. 08 1 3 2.4.2

(c) Explain in detail the sources of error in plane table survey. 07 2 2 2.3.1

(a) Derive an expression for zero circle of planimeter. 06 3 2 2.4.2 (b) Following readings were observed during fly levelling. Rule out a page of field book and apply necessary checks using rise 09 1 4 2.1.2

Q4 and fall method. B.S.: 0.980 (BM, RL= 125.360m), 1.185, 0.335, 2.615, 1.395, 0.765, 2.915. F.S.: 2.725, 3.480, 1.815, 3.810, 2.415, 0.915, 1.665.

(c) Explain with neat sketch reciprocal ranging. 05 2 2 2.3.1

2_

Q5

(a) Explain the Reiteration method for measurement o

horizontal angles. (b) Explain in detail intersection method of plane table survey.

(c) The offset taken from a survey line to a boundary are given

below. Find the area by Trapezoidal rule and Simpson's rule.

0 10 20 30 40 50 60 Chainage (m)

3.6 4.9 6.8 7.2 5.1 2.9 4.7 Offset (m)

(b)Write short notes on (i) Two peg method (ii) Orientation of Plane Table

(a) A traverse Al3CDEFA was run usimdiOtalllieock)hte.

11:dance the traverse using Bowditch rule. Also calculate

n)dcpendentco-ordinales.

rinc Length corrected Wt'l t 01

Q6

(a Write short notes on (i) Optical square (ii) Variation in declination

(iii) Auto level (iv) Capacity of reservoir using contour

(v) Testing of chain

183°9' 4" FA 30.80

AU 13.88 80° 24' 07"

In' 18.19

12.70

1Th 13.12

44° 57' 42"

80° 55' 28"

9° 17' 28"

El; 39.40 271°4' 46"

12 2.1.2

08 1,2 9 1.2.1

9 3 1.3.1

7 1 3 1.3.1

6 3 4 2.1.2

20 1,2 2,3 2. I .2

PI BL CO Points Q.No. Questions

'Bharatiya 'Vidya Mayan's

SARDAR PATEL COLLEGE OF ENGINEERING (Government Aided Autonomous Institute)

Mtmshi Nagar, Andheri (W) Iviumbai — 400058

TEST ODD SEM JANUARY 2020

Program: S.Y. B.Tech. Course Code: BS-BTC305 Course Name: Engineering Geology Notes: Answer any 5 questions. Draw neat labeled diagrams where needed..

Q.No. Questions Points I CO BL PI

7 CO2 L3

L3

L2

Describe the deepest layer of the Earth's structure in detail

Write a short note on the different types of chemical weathering with suitable examples

How do we know that the Earths outer core is liquid in nature? Write a short note on the mineral group which is known for its vitreous luster and absence of cleavage

List some of the physical properties of mineral with

2b suitable examples 7

What are the 5 key criteria a substance should meet in order to be called a mineral? What are the key

2c ,phoical Properties of asbestos

3a Explain the different types of metamorphism What inferences can be made about the environment of deposition from the physical

3b appearance of a sedimentary rock How do rocks develop porphyritic texture? Is there any relationship between grain size and rate of

3c cooling?

rloenrihR Any 4 mass extinctions in detail

_ Cal 1.3.1 L1 8

L4_ CO3 5

L3 Cal- 8

L2 Col 5

CO3 8

7 CO1 L2 1.2.1

2.3.1

1.2.1

CO1 L1 1.1.2

1.3.1

L2 2.1.2

1a

lb

1c

2a

5 CO2

8 CO1

Duration: 3 hours Maximum Points: 100 Semester: Ill

Bharatiya Vida Bhavads


.Munshi Nagar, Andlteri (W) Mumbai — 400058


4b Write a short note on the different types of folds 7 CO2 L1 2.3.1

4c

What is the nature of the bed if the contour lines intersect the bed boundary? Write about contour lines 5 CO2 L3 1.2.1

5a Write a detailed geological case study on the Koyna Dam 8 CO3 L2 2.3.1

5b

Describe in detail use of aerial photographs, satellite imagery, seismic and gravity survey for site investigation 7 CO3 L2 2.1.2

5c Define: Density, Specific gravity, Unit Weight, Porosity and Absorption of a rock specimen 5 CO2 L1 1.3.1

6a Briefly explain the zones of the water table 7 CO1 L2_ 2.1.2

6b Write a short note on the types of concrete dams 8 CO2 L1 1.2.1

6c State the importance of geological conditions while selecting site of dam or type of dam 5 CO2 L3 2.3.1

7a Describe briefly the components and types of tunnels 8 CO1 L1

1 1.3.1

7b

What is the effect of the dip and strike of beds, of faults and folds on the stability of the tunnel

7 CO2 L3 2.3.1

7c

List some methods to overcome the difficulties faced during tunneling

5 CO3 L3 2.3.1



Munshi Nagar, Andheri (West), Mumbai — 400058. Re- Examinations, January-2020

Max. Marks: 100 Class: S.Y.B.Tech. Semester: III Name of the Course: Engineering Materials

Instructions: 1. Question No 1 is compulsory 2. Attempt any four questions out of remaining six. 3. Draw neat diagrams


Course Code : Pc-B-Teao7

13T— 2-0 I.

Ques tion No

Points CO BL PI

(a) Discuss in detail Fiber glass. 4 1 2 1.3.1 (b) Discuss the applications of Plastics in construction industry. 4 3 3 2.4.2

Qi (c) What do you mean by ready mixed concrete? 4 2 2 1.3.1

(d) Explain Portland pozzolona cement. 4 3 1 1.3.1

(e) Highlight the features of smart concrete. 4 2.3.1 1 4

(a) Explain the functions of different Geosynthetic materials used in various Civil Engineering applications.

10 2 1 2.5.1

(b) Differentiate between: 6 2 2 2.2.3

Q2 (i) Earthenware and Terracotta (ii) First class and second class brick (c) Where would you recommend the packing mortar and fire 4 3 3 2.4.2 Resistant mortar?

Q3 (a) Draw the flow chart and explain procedure for the manufacturing of cement by wet process.

10 1 1 2.3.2

(b) Discuss efflorescence test and compression tests performed on bricks for its suitability.

6 2 2 1.2.1

(c) Write note on Light weight concrete. 2.5.1 4 2 1

(a) Explain chemical and electrical method of seasoning of timber. 6 1 2 2.4.2

(b) Describe the ingredients along with function for manufacturing of the glass. 2.2.3 6 2 1

Q4 (c) write note on (i) Cement paints 8 3 3 1.2.1 (ii) Hydrophobic cement

Q5 6

(a) Explain with neat sketch (i) Perforated Brick (ii) Hollow Brick (iii) Coping Brick (iv) Queen closer (b) Give the comparison of Tar and Asphalt in Tabular form. (c) Explain creosoting and ASCU treatment to timber.

8

6

1

3

3

2

4

3

1.2.2

2.2.3

1.6.1

Q6

(a) Explain the properties and uses of materials used for thermal and sound insulation. (b) Explain with neat sketch Ridge tile and Mangalore tiles. (c) Write note on Fiber reinforced plastic (d) What is plywood? State its application in construction.

8

4 4 4 1.3.1

3

3 1 2

3

1 2

1,3

2.4.2

1.3.1 2.2.3

Q7

(a) Describe the properties and applications of any two type of (i) ferrous metals (ii) Non-ferrous metals (b) Discuss the various applications of clay products in the building industry. (c) Explain Cutback and Blown bitumen. (d) State advantages and disadvantages of timber construction.

1.3.1

6

5

4 5

2

3

2 1

1,3

3

1 2

2.2.3

2.4.2

3.5.5

•




)




Instructions: 1. Al) questions are compulsory 2. Make suitable assumptions where necessary and state them clear)

Questio n No

..

Max. Marks


BL/PI

11°

Q.1

It is proposed to construct G+1 RCC BUNGLOW for Electrical Engineer. Plot size is 25 M X 20 M. Requirements as below:

1. Habitable room. 2. Kitchen 3. Master Bed room 4. Children bed room 5. Guest room 6. Office & drawing room .


15 05

01 2/2.1.3


Q.3 a) Draw a site plan for given data in Q.1 10

b) Draw a foundation plan for given data in Q.1 10 01 2/2.1.3

MI I i

Q.4


1. One & two point perspective 2. Building bylaws for Fire Safety 3. Planning principle= Aspect 4. Planning principle= Economy

20 01/02 1/1.2.3

Q.5 1 a) Explain Real estate regulation act in detail. 10 b)Draw a Location plan for given data in Q.1 10 01/02 2/2.1.3 I

OF ENGINEERING nous Institute) Mumbai - 400058

; - January 2020

Program: S.Y. B.Tech (Chi!: Duration: 03 hours

Course Code: ES-BTC 302 Maximum Points: 100 marks

Course Name: Mechanics of Ma .erials Semester: III

Notes: 1) Attempt any FIVE questions out of seven questions

2) Assume suitable data wherever required and state it clearly.

3) Figures to the right indicate full marks.

Q.No. Questions Points CO BL PI

Ql. a) With the help of stress-strain curve for mild steel explain the

following terms: 1. Proportional limit 2. Elastic limit 3. Yield stress 4. Strain hardening region 5. Ultimate stress

Also, draw the stress-strain curve for brittle materials and explain the difference between ductile and brittle materials.

10 CO2 L4 1.3.1

b) A compound tube consists of a steel tube of 140 mm internal diameter and 160 mm external diameter and an outer brass tube of 160 mm internal diameter and 180 mm external diameter. Both the tubes are of 1.5 m length. If the compound tube carries an axial compressive load of 900 kN, find its reduction in length. Also find the stresses and the loads carried by each tube. Es = 2 x 105 Nimm2, Eb = 1 x 105 N/mm2.

10 CO2 Li 1.3.1, 2. 1. 1, 2.1.2, 2 .1.3

Q2. a) Draw axial force, shear force and bending moment

for the beam shown in figure below. 15N

c A 60 c 20 Nm -..olaggg

gr_A‘il im

diagram

5 Nfin

8

20 CO1 Li, L3

1.3.1, 2.1.2, 2.1.3, 1.1.1

4.' D ... 5 m *I 1 4-- 2.5 m --414-- 2.5 m 014

Figure 1.

Page


Munshi Nagar, Andheri (W) Mumbai — 400058

Re- Examination - January 2020

-13

EG,

Q3.

a) M oE Derive the bending equation, — = — = — I y R Also state the assumptions made in the theory of pure bending

(any 2).

12 CO2 L3, L4

1.3.1

b) Prove that the maximum shear stress in triangular cross section of base 'b' and height 'h' is 1.5 times the average shear

08 CO2 Li, L4

1.3.1, 2. 1. 1 , stress of the section. Also draw the shear stress

distribution. 2. 1 . 21k 2. 1. 3 6 Q4.

a) A plane element is subjected to the stresses as shown in the figure 2 below. Determine analytically:

i) The

12 CO2 Li, L3

1.3.1, 2.1.1,

principal stresses and their directions 2.1.2, ii) The maximum shearing stresses and the directions

of the plane in which they act. iii) Normal and shearing stresses on the inclined

2.2.2, 2.2.3

plane.

100 MPa

50 MPa

150 MPa 150 MPa

° 30 50 MPa

100 MPa

Figure 2. b) Solve Q.4 (a) by Mohr's Circle Method. 08 CO2 Li, 1.3.1,

L3 2.1.1, 2.1.2, 2.2.2, 2.2.3 QS.

a) Find the diameter of the shaft required to transmit 60 kW at 150

10 CO2 Li, 1.3.1, rpm if the maximum torque exceeds 25 % of the mean L2 2.1.1,

torque for a maximum permissible shear stress of 60 Nimm2. Find 2.1.2,

also the angle of twist for a length of 4 m. Take G = 80 GPa. 2.1.3,

2.2.2 b)

[

In a tensile test on mild steel bar of 20 mm diameter, the 07 CO2 L3, 1.3.1, elongation in a gauge length of 100 mm was 0.072 mm when L4 2. 1. 1,

Page I 2

LUCt .t

3AWR PATEL COLLEGE OF ENGINEERING (Government Aided Autonomous Institute)

Munshi Nagar. Andheri (W) Mumbai — 400058


we mac was LI•J kN. lhe reduction in diameter was 0.0036 nr 1. Find the elastic constants 'E', 'G' and `K.'.

2.1.2, 2.1.3, 2.2.2 c) State any three assumptions made while deriving the torsional

formula. 03 CO1 L4 1. 3 . 1

Q6.

A a) rod of steel 16.5 m in length is at a temperature of 27°C. Find:

i) the free expansion and the corresponding stress when it is subjected to a rise in temperature and raised to 110° C.

ii) stress if no expansion is allowed iii) stress when the expansion of 7 mm is allowed.

Take a = 12x 10-6/° C, E = 220 GN/m2.

08 CO2 Li 1.3.1, 2 . 1. 1 , 2.1.2, 2.1.3

b) A steel 1-section shown in figure 3 below is placed to a bending moment of 24 kN-m sagging. Find:

i) Location of neutral axis. ii) Moment of inertia about neutral axis iii) Maximum tensile and compressive stresses in

bending. iv) Moment shared by the two flanges.

14- 80 mm +I i

12 CO2 Li, L2

1.3.1, 2. 1. 1, 2.1.2, 2.1.3

20 mm

20 mm 260

i

r mm

40 mm 14- 160 f mm -44

Figure 3. Q7. a) A brass bar having cross sectional area of 1000 mm2 is

subjected to axial forces as shown in figure 4 below. Find the total elongation of the bar if E = 1.05 x 105 Nimm2.

A B C I)

10 CO2 L3, L4

1.3.1, 2 . 1. 1, 2.1.2,

50 k-N 80 01 10 kN 2.1.3, 2.2.2 I 0 4-

20 kN

1 0

600 mm 04.4_ 1 m

I, 4 H 1,20 m —01

Figure 4.

Page 13




cloaca d`

b) Figure 5 below shows of 18 kN. Sketch section.

T E E

R

1

a 'C' section subjected to a shear force the shear stress distribution across the

10 CO2 Li, L2

1.3.1 2.1.1 2.1.2 2.1.3

20 m m

T

4-25 rim,

k , iisrnm

120 mm 14-- Figure 5.

************************************GOOD LuCK****************************

SARDAR PATEL COLLEGE OF ENGINEERING

Andhc„ (,):ilioary 2020)

( .6

Program: tiC.1 Civa ErigineeJring Duration; 3 hour

Course Code: PC-BTC-306 Maximum Points: 100

Course Name: Flu41 Scri.:.A.er

; i\11 '.111P1 .1007

• (11 I ..:1.1ril.(1 .111

111L'11(1011 HcH if!

aLxeinrtadtaura

Points „

I (a) vAplain of fluids liw.:;++,-(1 or+.• +.+++,,+(,+++, 13i v

(h) 1(..ixplain.1(.+1.7++,.+1 stirf+:1++,...c its.nsion ;++.+H i+.:+p•iliai if,,v. VvHn. +

. 1 would 1.-y- !hi+. + +,n 1liiary risc iv: i !,+ .;+. -.+.i++,+++. +,+.H.:+c +..)1+ 2.5 min

, 1 • . diarnete+',+,.,]+,i+ i. .iinnersed yertici.i.1.1y +1+! y i.:+ ..+1+. if surface H+ • 1 ; tension i.+1i \++...++,+++:+1 0.0135 N/1171 an;1 th,c 0.- ++,...+1+ (..oritact for 1

+ wat.(.+r Lco+.1 ++;1;.+..++;:$ is xero? , ,

1 1 .-! (a)Define ;•iiii(.1 i.+2.xplain the terins absoi+,41{..., g:It++,++„?...i. ,... ..+1.tinosplieric

+ d vacutun pressure. , '2 • • _• . .. (b) Find the equivi,:ilent pressure head (.)+f :...j(...) meter of vk,Tater in + 4

1110 1 terms o: (i) Kerosene of specific gra+++/1.1.2./ of 0.82; and fii) 1 f K 1.0 1 1 ; 1 ,

r • , _ L(.3riVettrirt of specific gravity of 1.26. + , . +-

'3 1 (alState and _prove Pascal law's laY. 10 + i I 1 . , _ .. - —1— —1------ 1 (b) A circuiar plate 4.5 meter diameter is immersed in water 1in such a w61y- th.at it gTeatest . and least !ilepth below the free 1

1 + surface is 3 meter and. 2,5 meter resp(ctively. 1;)(+.-ti-r:rnine the 1+,)+ i ! ++- 3 2.4.1 total pressure on one face of the .:plate and position of centre 1

1 of pressure. 1 , + i+ 1 —1- .• 1.... ._____±.____ r___i _!..- __ ;,:i7.. ..... . ,

1. I (a) Discuss equilibrium conditions of a floating 10 body. 1 -1—• i

1,

1 i -1-- ; (b) A. wooden block of rectangular section. 1.30 meter wide, , 1 2.25 meter deep and 4.50 long .floats horizontally in • sea. 1

, , 1 1 Water. if the specific . gravity of wood is•0,65 and water WeighS 10 1

+ 2

1 1 .3,1

. I I 1025 kR it rin3; find the volume of liquid displaced and the 1 1 11 3 1

\ 4 1 i

1 . ..Lposaion Oi .-1. ..Ci ...!: cenirt .0): iopoyancy,, + 1 ,

1 (PTO)

I

SARDAR

?ATEL COLLEGE OF ENGINEERiNG

((!(;\ ,\l..(1 Atlimn()111()11.-, Irp.iittoc)

(W) N/1(11111)11 ,if)()()‘,s

1

(a) St.ok. the

ti)Stenclv riov, f2 yr

11()%k JI• N(A) UllIt • Ili lil)\`

(5) State het \yio, c()Ini)(H:'( I (,

(y/t) and w (z !').,•eoici,:.;tits

I irrotation;Al

lj

.?(I\ (.111c;

, Derive Ei -dcr''. o r 0101,(10,

(b) A 30 (in.

oil of specihc 'f..i.a\H", 1i3c throat

section for an a 1. Ior 2 : 41:

, (r,:0Exptairl with 1-.1c;..1. 1-3(.41mdary ;

(1-.)J Derive 11 Poiseuille equa.tiori lot a lit r

Pipes.

• I :


Munshi Nagar, A ndheri (MI) Mumbai 400058

Re Examinations- January 2020

Program: Civil Engineering Duration: 3 hours

Course Code: BS-BTC301 Maximum Points: 100

Course Name: Engineering Mathematics III Semester: III

Instructions:

1. Question No 1 is compulsory. 2. Attempt any four queNtions out of remainin,..

Q.NO. Q11(11)11% 1 •111111, 4 4 ) i I I 11

1(a) 'r Prove that f

sin 2t + sin 3t 3R- dt = 6 1 ii, 1.1

te' ) 4 0 iii .1

(b) _

Show that the transformation 2z+3 the w , maps circle

6 2 iv, 2.4 z — 4 v .1

.X72 + y2 - 4x 0 into the straight line 4u +3 = 0 in the w-

plane

(c) Let A be a square matrix of order 3 x 3 with 1A1=1. 8 3 ii, 2.4

If = —1+ i-Nh . the /1 is one of eigen values of A,

V .1

2

(i) Find all the eigen values of A

(ii) If Am° = pA2 +qA + rl , find p,q and r

6 I i, ii 2.4 2(a) If L {erf -J. } = 1 find , L {te3t erf (21)}

sVs +1 .1

(b) If function f (z) is analytic and f (z)1 is constant, prove 6 2 ii, 1.1

that .f (z) is constant iii .1

1, (c) Find Figen

matrix A=

Values and corresponding

—2 —8 —12

1 4 4

0 0 1 _

Eigen Vectors of the 83 ii,

in 1.1

.1

3(a) Reduce the following matrix to normal form and hence 6 3 i, ii 2.4 find its rank.

i

. I

A = 1 6 1

7 9 10 3

(b) Using method of Laplace Transforms solve following 6 1 ii, 2.4 differential equation iii .1

(D2 — D —2)y = sin 2t where y(0) = 1, y1(0) = 2

(c) Prove that the function u = ex (x cos y — y sin y) is 8 2 iv, 1.1

harmonic. find its harmonic conjugate and corresponding .1 v

analytic function

4(a) Find the image of the circle lz —21= 2 under the 6 2 i, ii 1.1

.1 transformation I

z

(1)) Prove that 1, \/;t I

{sill Nit ---c , I Iv . ' , I

1 v 2s ' .1

(c) For the following matrix A, find two non-singular matrices 8 3 ii, 2.4 P and Q such that PAQ is in the normal form where iii .1

3 1 1

A= —1 5 —1 . Hence find A-1

1 —1 3 - -

and v are conjugate is also harmonic.

harmonic functions, prove that 6 2 i, ii 2.4 .1

ig Cayley Hamilton Theorem, Find inverse of the 6 3 ii, 2.4 7 2 —2 iii .1

.ix A = —6 —1 2

6 2 —1_

I 1 \ I I

6(a)

Find an analytic function f (z) = u(x, y)+iv(x, y) if

v = e-x [2xy cos y + (y2 — x2 )sin y]

(b) cos° sin° 0 Find Eigen values of the matrix A = —sin 0 cos° 0

0 0 1

(c) Find the bilinear transformation which maps the points 2, i, —2 of z — plane onto 1, i, —1 of w— plane respectively

5(a)

(b)

(c)

u

mat

6 2

6 3 iv, 2.4

8 2

7(a) Sho‘v that the tranforniation II

/17

1.7

circle lz1 =1 into a circle in the w-plane.

ita.,1(,ii tlit-

Test the consistency of the following system of equations 6 and solve them if they are consistent

4x —2y +6z =8 x+y-3z =-1 15x-3y+9z=21

Evaluate L 1 s4

s

s 4 + 4 8

(b) 3 2.4 .1

(c) 1

Bharatiya .Vidya Bhavan's

SARDAR PATEL COLLEGE OF ENGINEERING (Government Aided Autonomous institute)

Munshi Nagar, Andheri (W) Mumbai — 400058


Duration: 3 hours Maximum Points: 100 Semester: Ill

•

Program: S.Y. B.Tech. Course Code: BS-BTC305 Course Name: Engineering Geology Notes: Answer any 5 questions. Draw neat labeled diagrams where needed..

1 Points Q.No. Questions CO BL

PI

la

lb

4a

Describe the deepest layer of the Earth's structure in detail

Write a short note on the different types of chemical weathering with suitable examples

How do we know that the Earths outer core is liquid in nature? Write a short note on the mineral group which is known for its vitreous luster and absence of cleavage

List some of the physical properties of mineral with suitable examples What are the 5 key criteria a substance should meet in order to be called a mineral? What are the key Dhvsicallproperties of asbestos

Explain the different types of metamorphism What inferences can be made about the environment of deposition from the physical a••earance of a sedimentar rock How do rocks develop porphyritic texture? Is there any relationship between grain size and rate of coolin ?

Describe any 4 MRSS extinctions in detail

2b

2

3a

3b

7 CO1 L2 1.2.1

CO1 L1 1.3.1

CO3 L4_ 2.3.1

CO1 L3 1.2.1

CO1

CO1

Li

1.1.2

L2

1.3.1

CO3 L2 2.1.2

CO2 L3

2.3.1

•


Bharatiya Vidya .Bhavan's


.Munshi Nagar, Andheri (W) Mumbai -400058


4b Write a short note on the different types of folds 7 CO2 L1 2.3.1

4c

What is the nature of the bed if the contour lines intersect the bed boundary? Write about contour lines 5 CO2 L3 1.2.1

5a Write a detailed geological case study on the Koyna Dam 8 CO3 L2 2.3.1

5b

Describe in detail use of aerial photographs, satellite imagery, seismic and gravity survey for site investigation 7 CO3 L2 2.1.2

5c Define: Density, Specific gravity, Unit Weight, Porosity and Absorption of a rock specimen 5 CO2 L1 1.3.1

6a Briefly explain the zones of the water table 7 CO1 L2_ 2.1.2

6b Write a short note on the types of concrete dams 8 CO2 L1 1.2.1

6c State the importance of geological conditions while selecting site of dam or type of dam 5 CO2 L3 2.3.1

7a Describe briefly the components and types of tunnels 8 CO1 L1

i

1.3.1

7b

What is the effect of the dip and strike of beds, of faults and folds on the stability of the tunnel

7 CO2 L3 2.3.1

7c

List some methods to overcome the difficulties faced during tunneling

5 CO3 L3 2.3.1

Points CO I BL I PI I Question No

Answer the followings:

(i) Explain clearly, the difference between static and dynamic

analysis of structure

(ii) What is an earthquake? How the earthquakes are classified

based on their causes?

(iii) Explain the different types of seismic waves and their

characteristics

(i)

A uniform rigid slab of total mass 25 t is supported by four columns of height 8.0 m. rigidly connected to the top of slab and fixed at bottom. Each column is rectangular section of 750 mm x 300 mm as shown in figure. If the system is subjected to harmonic ground motion of amplitude 0.3g at frequency of 10 rad/sec in X direction only, calculate the maximum lateral displacement of slab in X direction and maximum stress in each column = 5% and E = 20, 000 MPa.

(ii) In the above problem, If the columns are hinged at bottom, then calculate the maximum lateral displacement of slab in X direction and maximum stress in each column. Comment on the effect of fixity of column on these parameters

Q1 (b)

3 3 1,2 1.2.1

2

1.2.1

1,2 3

4

3 3 3

Q1 (a)

Yi Coo 1----1

2 0 Tr)

tO r()



Munshi Nagar, Andheri (West), Mumbai — 400058. RE Examination January - 2020

CL)

Max. Marks: 100 Class: M.Tech. Semester: Ill Name of the Course: Earthquake Engineering

Instructions: • Attempt any FIVE questions out of SEVEN questions. • Answers to all sub questions should be grouped together. • Figures to the right indicate full marks. • Assume suitable data if necessary and state the same clearly

Duration: 3 Hours Program: Civil Engineering Course Code: EC-MST 301

Q1 (c).

Q2 (a)

Explain the characteristics of ground motions

A one story RCC building is idealized as plane frame as shown in figure. The cross section of columns is 250 mm x 250 mm and E= 20,000 Mpa. If the building is to be designed

for ground motion, the response spectrum of which is shown

in figure! I. Determine the design values of lateral deformation

and bending moments in the columns for the following two

conditions: (i) Supports of columns are fixed.

(ii) If the columns of the frame are hinged at base. Comment on the influence of base fixity on the design

deformation and bending moments

4 3 3

4 3 3

A A free vibration test is conducted to determine the dynamic

properties of a one storey building. The mass of the building is ilk Initial displacement of the building is 60 mm. Maximum displacement on the first cycle is 40 mm and period of

Q2 (b) displacement cycle is 1.5 sec. Determine: (i) Un damped frequency (ii) Logarithmic decrement (iii) Damping ratio (iv) Damping coefficient

(v) Amplitude after 6 cycles.

5 3 1.4.1

1.3.1, 1.4.1

3 1.3.1,

• 7 3 2.4.1

Q2 (c)

A two storey frame has the following free vibration

characteristics. The frame is subjected to a harmonic force of 100 Kn at 2" floor level with frequency of 20 rad/sec. Assume damping ratio = 5%. Calculate the upper bound on

response of each floor.

Floor Mass Mode I co, Mode shapes

No. (t) No. rad/sec

(1)it Cpi2

1 20 1 14.58 1.0 1.481

2 115 2 38.07 1.0 -0.822

3

Q3

The

(i)

(ii) x

mm

7e-

structure supported roof kg/m2.

frequencies

(iii)As

motion shown deformation,

plan of

weight The

Derive

If the structure direction,

a special x 600

only in

consists on

one

four

and modes

write

mm, and in X

figurel base

is uniformly plan dimensions

the stiffness

case,

S ot

story building is as shown in figure. The of a roof idealized as a rigid diaphragm, I corner columns as shown in figure. The

distributed and has magnitude 200 are b---- 30 m d=20m

matrix and determine the natural shapes of vibrations of the structure

is subjected to ground motion only in down the equations of motion for the system

if all columns are of the same size, 300 if the system is subjected to the ground

direction. the response spectrum of which is . Determine the design value of lateral shear and bending moment for the system.

4-----f

10

2

8

C. S.

.7. 2

3

5

5

2 n7

o Coo i

3

2

3

MPA-1

'

1.3.1, 1.4.1

-, V 01

et--

3 vo I 7 A 1----f

6 o 0 4. b t 900

1

Q4 (a) What is response

spectrum characteristics.

spectrum? Explain briefly, the response I 5 4

Q4 (b) Explain the

for a single

procedure to construct elastic response

ground motion record. .•

spectrum 6 4 3

A two story characteristics. motion characterized figure 1 but Calculate the

frame has the following The frame is to be designed

by the design spectrum scaled to peak ground acceleration design values of lateral deformation

for the given

of

free vibration ground in the

of 0.2g. floors.

9 5 3 2.3.1 2.4.1

r Floor

No.

Mass (t) Mode co,

No. radisec

Mode shapes1

(Dii (1)12

1 70 1 14.58 1.0 1.481

,..i 2 15 2 138.07 1.0 1 -0.822

1 1

i

Q5 (a) Explain the various types of Irregular Buildings as per IS

1 893-20 1 6

4 I 2 2.12.

Q5 (b)

As per IS 1893-2016, how many mode need to be considered in the earthquake force calculation by Response Spectrum

Method

2 6 i 2 2.12

Q5 (c)

State the limitation of Equivalent static Method. As per IS

1893-2016, under what conditions the Equivalent static

Method is permitted to use to calculate the earthquake forces.

2 6 2 2.4.1

Using response spectrum method, calculate the seismic force on each floor of the frame whose pre vibration properties are given below. Use the following additional data: Z=0.24, I =1.5, R=5.0 and 4 = 5%. Assume foundation strata

as soft and use response spectrum given in figure 2.

12 5,6 4 2.12

Q5 (d) Story No.

Mass No.

Mass (t)

co rad/sec

Mode shapes

0,1 012 i

013

1 1 25 15.73 0.399 0.747 1.0

2 2 25 49.85 1.0 0.727 -0.471

3 3 25 77.82 -0.908 1.0 -0.192

Q 6(a)

Explain the following with reference to SDOF systems:

(i) Allowable Ductility (ii) Ductility Demand

1 4 6 2 2.4.1

Q 6(b)

Define the followings: (i) Joint probability distribution (ii) Stationary random

process (iii) Power spectral density function (iv) Auto

correlation function.

4 .

1 1 2.4.1

, Q 6(c)

Briefly explain the following:

(i) Structure of Earth

(ii) Magnitude and Intensity of an earthquake

6 2 2 2.4.1

LAP I— A single story frame shown in figure is subjected to ground

motion as shown. Determine the maximum displacement at

girder level, base shear and bending moments in column.

0- .

6 3 3 2.4.1

rAi 4 IMPAFICIWAVAIIIIIIIAN u a-)

Q 6(d) ; 3

CIn 21

t--- dr---,--,--x> —7-7,--7"

(.1 cicl-) 2xto101,,,-2—

r 2 O°0 Cm4-

I

Q 70) What is ductility of a structure? Explain the importance o ductility in seismic resistant structures.

2.4.1

Q 7 (b) What is shear Wall? Explain the advantages of shear walls for earthquake resistant structure.

3 6 2.4.1

Q 7(c)

Explain the provisions of IS 13920 for (i)Beams: General provisions, longitudinal reinforcement and web reinforcement (ii)Shear wall: General requirements, longitudinal and transverse reinforcement.

12 6 2 2.4.1

,

Briefly explain the different types of structural systems used Q 7(d)

in a building structure to resist lateral loads due earthquake 2 6 2 1 2.2.1 1

I

2 3 6 6 NATURAL PERIOD T. , s

28 SPECTRA FOR RESPONSE SPECTRUM METHOD

FIG. 2 DESIGN ACCELERATION COEFFICIENT (S.4) (CORRKSPUNDING TO 5 PERCENT DAMPING)

3.0

2.5

2.0

1.5

1,0

0.5

0 0

Type I ROCK OR HARD SOIL

-- Type H MEDIUM SOIL

- Type III SOFT SOIL

.D T

OU B

RO

CON

STRU

CTIO

N -

MAN

APAK

KAM

, CH

ENNA

I ON

• ...... - DISPLACEMENT IN NMTERS .0 o p o

Jo.

cn0 ts" o I

ob 8

• 4.. ' 44,,f4• -4 .

4ir

•

CL

SARDAR PA TEL COLLEGE OF ENGINEERING iovernment Aided Autonomous Institute)

Munshi .Nau,ar‘ Andheri (V) Mumbai —4000.58

Re Examinations ( For Academic Year 2018-19)- January 2020

Program: Civil Engineering Duration: 3 hours

Course Code: BS-BTC301

Maximum Points: 100

Course Name: Engineering Mathematics III

Semester: III

Instructions:

1. Question No 1 is compulsory. 2. Attempt any four questions out of remaining six.

0.No. , .h.,•.,f,, , .... 1.,..,,I. '

1(a) ''''r Prove that f

sin 2/ + sin 3t \ 37z- dt

:= 6

0 tel i 4

(b) Show that the transformation 2z + 36

maps the circle x2 + — 4x = w = y2 0 z — 4

into the straight line 4u +3 = 0 in the w-plane

(c) Let A be a square matrix of order 3 x 3 with lAl =1 . 8

If A. —1 + iNri

is one = of the eigen values of A, 2

0 Find all the eigen values of A (ii) If Alm _____ p •A

2 ± qA + rI , find p, q and r

led. V t (i - I , tilid I , .

(I t( • ' ( '1.1 (2,11)1,

i s s + I

() 'ILa; \ f.., k II L

(b) If function f (z) is analytic and If (z)I is constant, prove that f(z) is 6

constant

(c) Find Eigen Values and corresponding Eigen Vectors of the matrix 8

—2 —8 —12

A— I 4 4

0 0 1 _ -

,

3(a) Reduce

A=

the following

8 3 6 I -1 6 4 2 7 9 I 0 3 _ _

matrix to normal form and hence find its rank. 6

(b)

(i )

Using method of Laplace Transforms solve following differential equation

(D2 - D -2)y = sin 2/ where v(0) I v'(0) 2

141''\ r (Hi ilic fun , II ,, i) H ( '( k I gg I 1 111 1 1 i 11 11 ,, I , i „i H

iM111101iic coiiitIgalc Mid ( 41114.-41141114,111W ,111,11.\ 11( 111114 114111

6

4(a) Find the image of the circle lz -21= 2 under the transformation1 z

6

(b)

-\/71- - 4s Prove that L {sin Vi} = 6

3 e 3 2s/2

(c) For the following matrix A, find two non-singular

that PAQ is in the normal form where A =

matrices

- 3 1 1

-1 5 -1

1 -1 3 _ _

P and Q such

. Hence find Al

8

5(a) If u and v are conjugate harmonic functions, prove that uv is also harmonic,

6

(b) Using

A=

Cayley Hamilton

7 2 -2

-6 -1 2

6 2 -1 - -

Theorem, Find inverse of the matrix 6

(c) Evaluate (i) E

l {2s2 +5s +2} (I

8

(s-lf (ii) I:1 {10g +

s

6

6 (b)

(c)

cos() sine 0

—sine cost9 0 0 0 1

Find Eigen values of the matrix A =

Find the 1)iline;tr iransli)rnuilicwi which iimpt, Hie p()iiiv. ' ,wi I , / sp,

6

6

8

6(a) Find an analytic function f (z) = u(x, y)+ iv(x , y) if

v= e-x [2xy cos y + (y2 —xlsin y ]

Show that the transformation w= 5-4z transforms the circle Id =1 into a 4z-2

circle in the w-plane.

Test the consistency of the following system of equations and solve them if they are consistent

4x — 2y +6z =8 x+y-3z=-1

15x-3y+9z =21

Evaluate s s4 +4 •

7(a)

(b)

(c)

\ix I

SARDAR PATEL COLLEGE (Government Aided Autonomous !rt., ' .

MlitiStli -Nagar, A ndheri (W.) Muinbai -

Re- Examination - January 2,C:.?-0

zogram: S.Y. B.Tech (Civil) ration: 03 hours

..;ourse Code: ES-BTC 302

Maximum points: 100 marks

Course Name: Mechanics of Materials Semester: III

Notes: 1) Attempt any FIVE questions out of seven questions

2) Assume suitable data wherever required and state it clearly.

3) Figures to the right indicate full marks.


Ql. a) With the help of stress-strain curve for mild steel explain the

following terms: 1. Proportional limit 2. Elastic limit 3. Yield stress 4. Strain hardening region 5. Ultimate stress

Also, draw the stress-strain curve for brittle materials and explain the difference between ductile and brittle materials.

10 CO2 L4 1.3.1

b) A compound tube consists of a steel tube of 140 mm internal diameter and 160 mm external diameter and an outer brass tube of 160 mm internal diameter and 180 mm external diameter. Both the tubes are of 1.5 m length. If the compound tube carries an- axial compressive load of 900 kN, find its reduction in length. Also find the stresses and the loads carried by each tube. Es = 2 x 105 N/rnm2, Eb = 1 x 105 Nimm2.

10 CO2 Li 1.3.1 2.1.1 2 . 1 .(4-2. 1 ..':

Q2.

a) Draw axial force, shear force for the beam shown in figure

s N ,

A 604 c 20 Nm .

and bending moment below. 15N

mooring

diagram

5 Nurn

AB

....

*I

20 CO1 Li, L3

1.3.] 2.1.c 2. i.1 1 . 1. "

C. air D

5 m 14-- 2.5 r --ii4- 2.5 m ol4

Figure 1.

Pagel I


Munshi Nagar. Andheri (W) Mumbai - 400058


-

Q3.

a) M a E Derive the bending equation, —

I = —

y -= —R

Also state the assumptions made in the theory of pure bending (any 2).

12 CO2 L3, L4

1.3.1

b) Prove that the maximum shear stress in triangular cross section of base 'b' and height 'h' is 1.5 times the average shear stress of the section. Also draw the shear stress distribution.

08 CO2 Li, L4

1.3.1, 2. 1. 1, 2.1.2 2.1.3 Q4.

a) A plane element is subjected to the stresses as shown in the 12 CO2 Li, 1.3.1, figure 2 below. Determine analytically:

L3 2. 1. 1, i) The principal stresses and their directions 2.1.2, ii) The maximum shearing stresses and the directions

of the plane in which they act. iii) Normal and shearing stresses on the inclined

2.2.2, 2.2.3

plane. 100 MPa

50 MPa

150 MPa 150 MPa

30°

50 MPa

100 MPa

Figure 2. b) Solve Q.4 (a) by Mohr's Circle Method. 08 CO2 Li, 1.3.1,

L3 2.1.1, 2.1.2, 2.2.2, 2.2.3

Q5. a) Find the diameter of the shaft required to transmit 60 kW at 10 CO2 Li, 1.3.1,

150 rpm if the maximum torque exceeds 25% of the mean L2 2.1.1, torque for a maximum permissible shear stress of 60 N/mm2. 2.1.2, Find also the angle of twist for a length of 4 m. Take G = 80 GPa. 2.1.3,

2.2.2 b) In a tensile test on mild steel bar of 20 mm diameter, the 07 CO2 L3, 1.3.1, elongation in a gauge length of 100 mm was 0.072 mm when L4 2.1.1,

Page I 2

CO1 L4

A rod of steel 16.5 m in length is at a temperature of 27°C. Find:

i) the free expansion and the corresponding stress when it is subjected to a rise in temperature and raised to 110°C.

ii) stress if no expansion is allowed iii) stress when the expansion of 7 mm is allowed.

Take a = 12 x 10-6/0 C, E = 220 GN/m2.

a)

SARDAR rEL COLL. ' ENGINEERING (tio\,erincni i(ied u ious Institute)

Must li Ancilwri (') Mumbai —400058

Re- Examjna ion - January 2020

C)

Q6.

the load was 45 kN. The reduction in diameter was 0.0036 mm. Find the elastic constants `E., *G' and `Ic.

State any three assumptions made while deriving the torsional formula.

2.1.2, 2.1.3, 2.2.2 1.3.1 03

b)

Li

A steel 1-section shown in figure 3 below is placed to a bending moment of 24 kN-m sagging. Find:

i) Location of neutral axis. ii) Moment of inertia about neutral axis iii) Maximum tensile and compressive stresses in

bending. iv) Moment shared by the two flanges.

14— 80 mm

08 CO2

12 CO2 Li, L2

20 rnm

20 mm 260 mm

40 mm 14— 160 mm f Figure 3.

(27- a)

A brass bar having cross sectional area of 1000 mm2 is subjected to axial forces as shown in figure 4 below. Find the total elongation of the bar if E = 1.05 x 105 Nimm2.

A G 0 50 kN 80 kll 10 kN 4

20 kt4

600 mm • 14-- —+4---— 1 Erl 1.20 in

Figure 4.

10 CO2 L3, L4

1.3.1, 2. 1. 1, 2.1.2, 2.1.3, 2.2.2

Page I 3

SARDAR PATEL COLLEGE OF ENGINEERING (Government -Aided Autonomous Institute)



b) Figure 5 below shows of 18 kN. Sketch section.

T E E R ,...

I_

a 'C' section subjected the shear stress distribution

to a shear force across the

k

10 CO2 Li, L2

1.3.1, 2.1.1, 2.1.2, 2.1.3

i

20 mm

T

25 mm 4-

I 5 mm

I4--- 120 mm

Figure 5.

************************************GooD LuCK****************************

Page I 4

(t)


Munshi Nagar, Andheri (Ws) Mumbai -40005X

• Re Ekaminations- January 2020 .

Program: Civil Engineering

Course Code: BS-BTC301

Course Name: Engineering Mathematics III

Instructions: 1. Question No 1 is compulsory. 2. Attempt any lour questions out al-cm:61nm'. six.

Duration: 3 hours

Maximum Points: 100

Semester: III

().No. (.)Iic,iiffir, r.,,,il, t 41 Ili 11

1(a) ' Prove that f

(sin 2t + sin 3t 3 7r dt =

6 1 ii, 1.1

ter o .1

(b) Show that the transformation

= 2z+3 the w maps circle 6 2 iv, 2.4

z — 4

= 0 into the straight line 4u +3 = 0 in the w- x2 + y2 - 4x

plane

v .1

(c) Let A be a square matrix of order 3 x 3 with IA 1 = 1. 8 3 ii, 2.4

—I + i'Nh is 1f 2 the

V .1 = one of eigen values of A,

2

(i) Find all the eigen values of A (ii) If Aim = p A 2 ±qA+rI , find p,q and r

6 24 1 i, ii If L lerf 2(a) . = 1 find L Ite3terf (2,101 , s V s +1 .1

(b) if function f (z) is analytic and If (z)1 is constant, prove 6 2 ii, 1.1

that f (z) is constant iii .1

(c) Find Eigen

matrix A =

Values and corresponding

—2 —8 —12

1 4 4

0 0 1 _ _

Eigen Vectors of the S3 ii,

iii 1.1

.1

3(a) Reduce find

A -

the following its rank.

(, 1

1 6 -1 2

7 9 10 3 _ _

matrix to normal form and hence 6 3 i, ii 2.4 .1

(b) Using method of Laplace Transforms solve following differential equation

(D2 — D — 2)y = sin 2t where y(0) = 1, AO) = 2

6 1 ii,

iii

2.4

.1

(c) Prove that the function u = ex (x cos y— y sin y) is

harmonic, find its harmonic conjugate and corresponding analytic function

8 2 iv, V

1.1

.1

4(a) Find the image of the

transformation 1 z

circle

,7 3

2s''

/

lz —21= 2 under the

c

6

()

2

I

i, ii

k

v

1.1

.1

) -I (b) / Prove that L {sin vt '.

i

(c) For the

P and

A=

following

Q such that

3 1 1

—1 5 —1

1 —1 3 _ _

matrix A, find two non-singular matrices

PAQ is in the normal form where

. Hence find A-1

8 3 ii,

iii

2.4

.1

5(a') Ii i, and v are conjugate harmonic functions, prove that /iv is also harmonic.

6 2 i, ii 2.4 .1

(b)

(c)

Using Cayley Hamilton

matrix A --=

-

7 2 —2

—6 —1 2

6 2 —1

Theorem, Find inverse of the 6 3 ii, iii

2.4 .1

1 ‘ahi,11(- tit / 1 \ I) '

( Ii, , I . 1\ 1 1

1

6(a) Find an

v =- e- x

analytic function f (z) = u(x,

[2xy cos y + (y2 _ x2)sin y]

y)+iv(x, y) if

_

6 2 ii, v

1.1 .1

(b)

Find Eigen values of the matrix A =

cos 0 sin 0 0 —sin 0 cos 0 0

0 0 1

6 3 iv, v

2.4 .1

(c) Find the bilinear transformation which maps the points 2, i, —2 of z — plane onto 1, i, —1 of w — plane respectively

8 2 i, ii 1.1 .1

,

7(a) ra n Show that the tiiskftimit io H. ti:iii!,k Hill. thi.

circle Id =1 into a circle in the w-plane.

i, ii i I

(13) Test the consistency of the following system of equations and solve them if they are consistent

4x — 2y + 6z = 8 x+y-3z=-1

15x-3y+9z = 21

6 3 ii, iii

2.4 .1

(c) Evaluate L-I s 8 1 ii, 1.1

{ s' + 4} v .1

Sardar Patel College of Engineering III.pdf · below. Find the area by Trapezoidal rule and Simpson's rule. Chainage (m) 0 10 20 30 40 50 60 Offset (m) 3.6 4.9 6.8 7.2 5.1 2.9 4.7

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