FLUID MECHANICS Civil Engineering 2nd year

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Set No. 1Code No: W0104/R05

I I B.Tech I Semester Supplementary Examinations, April/May 2011

FLUID MECHANICS

(Civil Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks

1. (a) Distinguish between

i. Ideal and Read Fluids

ii. Newtonian and Non- Newtonian Fluids

iii. Gases and Vapours.

iv. Adhesion and cohesion

(b) The velocity distribution in a uid is give by u = 40000 y (1-2y) where u is

the velocity in m/sec at a distance of y meters normal to the boundary. If thedynamic viscosity of uid is 1.8 × 10 poise, determine the shear stress at y-4

= 0.2m. [8+8]

2. (a) What do you mean by Hydrostatic pressure.

(b) Define Total pressure and centre of pressure

(c) A circular plate 2.5m in diameter is submerged in water as shown in figure

2c. Its greatest and least depths below free surface of water are 3m and 2m

respectively. Find

i. Total pressure on front face of the plate and

ii. the position of centre of pressure [3+4+9]

Figure 2c

3. (a) State and explain equation of continuity for incompressible uid and com-

pressible uid.

(b) Give examples of stream line ow, turbulent ow, steady ow, unsteady ow,

uniform ow and non-uniform ow.

(c) Oil ows through a pipeline which contracts from 45 cm diameter at A to 30

cm diameter at B and then branches into two pipes C and D. The diameter

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of the pipe C is 15 cm and diameter of the pipe D is 20 cm. If the velocity at

A be 1.8 m/sec and that at D be 3.6 m/sec Determine (figure3c)

Figure 3c

i. Velocity at section B

ii. Discharge at D

iii. Discharge at C

iv. Velocity at C. [3+4+9]

4. (a) What is a ow net. Draw a typical ow net and explain its applications. What are the limitations of ow nets.

(b) A pipe 50 cm in diameter branches into two pipes of diameters 25 cm and 20

cm respectively as shown in figure 4. It the average velocity in 50 cm diameter

pipe is 4m/sec find

i. Discharge through 50 cm diameter pipe and

ii. velocity in 20 cm diameter pipe if the average velocity in 25 cm pipe is 3

m/sec

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Figure 4

5. (a) Explain the phenomenon of boundary layer separation and its in uence on the

drag of an immersed body.

(b) In a at plate of 2m length and 1m wide, experiments were conducted in a

wind tunnel with a wind speed of 50 Km/hr. The plate is kept at such an angle

that the coe cients of drag and lift are 0.18 and 0.9 respectively. Determine

drag force, lift force, resultant force and power exerted by the air stream on

the plate. Take density of air as 1.15Kg/m . [7+9]3

6. (a) Prove that the velocity distribution for viscous ow between two parallel plates

when both plates are fixed across a section parabolic in nature. Also phone

that maximum velocity is equal to one and half times the average velocity.

(b) Water is owing between two large parallel plates which are 2 mm apart.Determine maximum velocity, pressure drop per unit length and the shear

stress at walls of the plate if the average velocity is 0.4 m/sec. Take viscosity

of water as 0.01 poise. [8+8]

7. (a) Describe Moody’s chart and explain the use of it.

(b) Discuss how friction factor varies with Reynolds Number.

(c) Two pipes one of 10 cm diameter, 200m long and another of 15cm diameter,400 m long are connected in parallel. The friction factors are 0.0075 for the

smaller pipe and 0.006 for the large pipe. The total discharge through the

system is 50 lit/sec. Find the discharge and head loss in each pipe. Neglectminor losses. Calculate the equivalent length of a 20cm diameter having f =

0.005. [4+3+9]

8. (a) Derive an equation for discharge of a venturimeter.

(b) Explain why C of a venturimeter is more than that of orifice meter.d

(c) A horizontal venturimeter with inlet and throat diameters 300 mm and 150mm

respectively is used to measure the ow of water. The pressure intensity atinlet is 130 KN/m while the vacuum pressure head at the throat is 350 mm 2

of mercury. Assuming that 3% of head is lost in between inlet and throat, findC of venturimeter and rate of ow. [6+2+8]

d

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FLUID MECHANICS

(Civil Engineering)




i. Specific weight and specific volume

ii. Density and Relative Density

iii. Adhesion and cohesion

iv. Dynamic and Kinematic viscosities

(b) An object having an area of 1.2m is sliding down an inclined plane at 40 to 2 0

the horizontal with a velocity of 0.42m/sec. There is a thin film of uid 2.2

mm thick between the sliding object and the plane. If the weight of the object

is given by 320 N, find the viscosity of the uid. [8+8]

2. (a) Find the total pressure force and the depth of centre of pressure on an inclined

plane surface submerged in a liquid.

(b) A trapezoidal plate of top width 6m, bottom width 5m and height 3.5m isimmersed vertically in water with its parallel sides parallel to the water level

and its top edge is at a depth of 2.5m below the water level. Find the water

thrust an one side of the plate and depth of centre of pressure. [8+8]

3. (a) Define the equation of continuity. Obtain an expression for continuity equation

for a three dimensional ow.

(b) In a two dimensional incompressible ow, the uid velocity components are

given by U = x-4y and V = -y-4x. Show that velocity potential exists and

determine its form as well as stream function. [8+8]

4. (a) Starting with Euler’s equation of motion along stream line, obtain Bernoullis

equation by its integration. List all the assumptions made.

(b) A pipe line carrying oil ( specific gravity 0.8) changes in diameter from 300

mm at section 1 to 600 mm diameter at section 2 which is 5 m at higher level.

If the pressures at section 1 and 2 are 100k N/m and 60KN/m respectively2 2

and the discharge is 300 lit/sec, determine

i. loss of head and

ii. Direction of ow. [7+9]

5. (a) What are the boundary conditions that must be satisfied by a given velocity

profile in laminar boundary layer ows.

(b) Describe von-karman momentum integral equation.

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(c) A smooth at plate of length 5m and width 2m is moving with a velocity

of 4m/sec in stationary air of density as 1.25KG/m and kinematic viscosity3

1.5 × 10 m / sec Determine thickness of the boundary layer at the trailing-5 2

edge of the smooth plate. Find the total drag on one side of the plate assumingthat the boundary layer is turbulent from the very beginning. [8+8]

6. (a) Derive Hagen poiseuille equation and state the assumptions made.

(b) A uid of viscosity 0.5 poise and specific gravity 1.2 is owing through acircular pipe of diameter 10 cm. The maximum shear stress at the pipe wall is

given as 147.15 N/m Find pressure gradient, average velocity and Reynolds 2

number of the ow. [8+8]

7. (a) Derive the Darcy - Weisbach equation for friction head loss in a pipe .

(b) Water is owing through a horizantal pipe line 1500m long and 200 mm indiameter. Pressures at the two ends of the pipe line are respectively 12 kpa

and 2 kpa. If f = 0.015, determine the discharge through the pipe in litres per minute. Consider only frictional loss. [8+8]

8. (a) Derive an equation for discharg of an orifice meter.

(b) An orifice meter with orifice diameter 15cm is inserted in a pipe of 30cm

diameter. The pressure gauges fitted upstream and downstream of the orifice

meter give readings of 14.715 N/cm and 9.81 N/cm respectively. Find the 2 2

rate of ow of water through the pipe in lit/sec. Take C = 0.6. [8+8]d

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FLUID MECHANICS

(Civil Engineering)




i. Specific weight and specific volume

ii. Density and Relative Density

iii. Adhesion and cohesion

iv. Dynamic and Kinematic viscosities

(b) An object having an area of 1.2m is sliding down an inclined plane at 40 to 2 0

the horizontal with a velocity of 0.42m/sec. There is a thin film of uid 2.2

mm thick between the sliding object and the plane. If the weight of the object

is given by 320 N, find the viscosity of the uid. [8+8]

2. (a) Explain how you would find the resultant pressure on a curved surface im-

mersed in a liquid.

(b) A plane area in the form of a right angle triangle of height ‘h’ is immersed

vertically in water with its vertex at the water surface. Calculate the total

force on one side of the triangular plane and the location of centre of pressure.

[8+8]

3. (a) Explain one, two and three dimensional ows.

(b) What is a stream tube, what are its characteristics?

(c) Given the following two dimensional velocity field U = 3 by + 4 ax and V =

3bx - 4ay Evaluate the stream and velocity potential functions. [3+4+9]

4. (a) Define potential head, velocity head and datum head.

(b) List out the assumptions and limitations of Bernoulli?s equation.

(c) 360 liters per second of water is owing in a pipe. The pipe is bent by 120 0

The diameters at the inlet and outlet of the bend being 360 mm 240 mmrespectively and volume of the bend is 0.14m . The pressure at the entrance3

is 72KN/m and the exit is 2.4m above the entrance section. Find the force 2

exerted by water on the bend. [3+3+10]

5. (a) What is meant by Magnus e ect. Explain.

(b) Describe with the help of a sketch, the variation of drag coe cient for acylinder over a wide range of Reynolds number.

(c) A kite 0.8m×0.8m weighing 3.924N assumes an angle of 12 to the horizontal.0

The string attached to the kite makes an angle of 45 to the horizontal. The 0

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pull on the string is 24.525N when the wind is owing at a speed of 30 Km/hr.

Find the corresponding coe cient of drag and lift. Take mass density of air as 1.25Kg/m . [3+4+9]3

6. (a) Draw a neat sketch of Reynolds apparatus and explain how the laminar ow

can be demonstrated with the help of the apparatus.

(b) Two parallel plates kept 100 mm apart have laminar ow of oil between them

with a maximum velocity of 1.5 m/sec. Calculate discharge per metre width,shear stress at the plates and the di erence in pressure between two points

20m apart. Assume viscosity of oil to be 24.5 poise. [8+8]

7. (a) Derive and expression for head lost due to sudden contraction of a pipe.

(b) A pipe increases in diameter suddenly from 10 cm to 20 cm. If the discharge

of water through the pipe is 100 lit/sec., determine the loss of head due tosudden enlargement of cross sectional area. Also determine the di erence of

pressure between two sections of the pipe line. [8+8]

8. (a) Explain the principal and working of venturimeter with the help of a neat sketch.

(b) Water ows through a horizontal venturimeter of inlet diameter 15 cm andinlet pressure 215 kpa (absolute). Find the minimum throat diameter for the

meter to pass a discharge of 150 lps without causing cavitations. Assume

saturation vapour pressure of water = -80 kpa (gange). Assume atmospheric

pressure = 76cm of mercury and Cd of the meter is 0.978. [8+8]

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FLUID MECHANICS

(Civil Engineering)




i. Ideal and Read Fluids

ii. Newtonian and Non- Newtonian Fluids

iii. Gases and Vapours.

iv. Adhesion and cohesion

(b) The velocity distribution in a uid is give by u = 40000 y (1-2y) where u is

the velocity in m/sec at a distance of y meters normal to the boundary. If thedynamic viscosity of uid is 1.8 × 10 poise, determine the shear stress at y-4

= 0.2m. [8+8]

2. (a) Explain how you would find the resultant pressure on a curved surface im-

mersed in a liquid.

(b) A plane area in the form of a right angle triangle of height ‘h’ is immersed

vertically in water with its vertex at the water surface. Calculate the total

force on one side of the triangular plane and the location of centre of pressure.

[8+8]

3. (a) State and explain equation of continuity for incompressible uid and com- pressible uid.

(b) Give examples of stream line ow, turbulent ow, steady ow, unsteady ow,

uniform ow and non-uniform ow.

(c) Oil ows through a pipeline which contracts from 45 cm diameter at A to 30

cm diameter at B and then branches into two pipes C and D. The diameterof the pipe C is 15 cm and diameter of the pipe D is 20 cm. If the velocity at

A be 1.8 m/sec and that at D be 3.6 m/sec Determine (figure3c)

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Figure 3c

i. Velocity at section B

ii. Discharge at D

iii. Discharge at C

iv. Velocity at C. [3+4+9]

4. (a) Starting with Euler’s equation of motion along stream line, obtain Bernoullis

equation by its integration. List all the assumptions made.

(b) A pipe line carrying oil ( specific gravity 0.8) changes in diameter from 300mm at section 1 to 600 mm diameter at section 2 which is 5 m at higher level.

If the pressures at section 1 and 2 are 100k N/m and 60KN/m respectively2 2

and the discharge is 300 lit/sec, determine

i. loss of head and

ii. Direction of ow. [7+9]

5. (a) Explain the phenomenon of boundary layer separation and its in uence on thedrag of an immersed body.

(b) In a at plate of 2m length and 1m wide, experiments were conducted in a

wind tunnel with a wind speed of 50 Km/hr. The plate is kept at such an angle

that the coe cients of drag and lift are 0.18 and 0.9 respectively. Determine

drag force, lift force, resultant force and power exerted by the air stream on

the plate. Take density of air as 1.15Kg/m . [7+9]3

6. (a) Derive the expressions for discharge per unit width and shear stress for ow ofviscous uid between two parallel plates when one plate is moving and other

at rest.

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(b) Two parallel plates kept 75 mm apart have laminar ow of glycerin between

them with a maximum velocity of 1 m/sec. Calculate the di erence in pressurebetween two points 25 m apart and the velocity gradients at the plates and

velocity at 15 mm from the plate. Take viscosity of glycerine as 8.35 poise. [8+8]

7. (a) Define major energy loss and minor energy loss.

(b) Brie y explain Hydraulic Gradient Line and total energy Line.

(c) The rate of ow of water through a horizontal pipe is 0.3 m /sec. The di-3

ameter of the pipe is suddenly enlarged from 25 cm to 50 cm. The pressure

intensity in the smaller pipe is 1.4 kgf/cm . Determine loss of head due to2

sudden enlargement, pressure intensity in the large pipe and power lost dueto enlargement. [4+4+8]

8. (a) What is meant by an orifice. Give the complete classification of orifices.

(b) What is vena contract. Explain.

(c) Water issues from an orifice 80 mm diameter under a head of 10m. Determinethe velocity of the jet of water and discharge through the orifice. Also calculate

coe cient of contraction. Take C = 0.6 and C = 0.9. [6+2+8]d V

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Set No. 1Code No: X0201/R05


MATHEMATICS-III( Common to Electrical & Electronic Engineering, Electronics &

Communication Engineering, Electronics & Instrumentation Engineering,Electronics & Control Engineering, Electronics & Telematics, Electronics &

Computer Engineering and Instrumentation & Control Engineering)



1. Evaluate the following using ß - G functions.

v2 - x dx2

(a) x3

0

8

(b) x4 (1+x 5 )dx (1+x) 15

0

8

(c) 2 x e dx. [5+6+5]11 -x 4

0

2. (a) Prove that (xJ J ) = x(J - J ). d 2 2n n+1 n n+1dx

(b) Express x +2x –x–3 in terms of Legende polynomials. [8+8]3 2

3. (a) Determine the analytic function whose real part u = e (x cos2y – y sin 2y)2x

and prove that u is harmonic.

(b) Separate the real and imaginary pants of ( 1+2i) i. [8+8] 1+v3

4. (a) Evaluate where C is | z | = 1/2 using Cauchy’s integral formula. (ez sin 2z-1) dz z (z+2)2 2

C

(b) Evaluate where C is | z + 2 | = 1 using Cauchy’s integral formula. (e-2 z) z 2 dz

(z-1) (z+2)3 C

[8+8]

5. (a) State and prove Taylor’s theorem.

(b) Find the Laurent series expansion of the function

in the region 3< |z+2| <5. [8+8] z 2 -6z-1(z-1)(z-3)(z+2)

6. (a) Find the poles of f(z) and the residues at the poles which lie on imaginary axis

if f(z) = .(z 2 +2z)

(z+1) (z +4)2 2

(b) Evaluate where C is the circle |z| = 2 by using residue theorem.(z 2 -z+2)dz z +10z +94 2

C[8+8]

7. (a) Evaluate 2p d using residue theorem.Cos20 5+4Cos

(b) Evaluate 8 using residue theorem. [8+8] x2 dx -8 (x +1)(x +4)2 2

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8. (a) Find and plot the image of triangular region with vertices (0,0), (0,1), (1,0)

under the transformation W= (1-i) z+3.

(b) Determine the image of the region 0<y<2, under the transformation in w =

1/z

(c) Find the image of the rectangular region –1= x= 3, –p =y= p in the z-plane

under the transformation w = e . [6+5+5]z

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1. (a) Show that ß(m,n) = (G (m) G (n))/G (m + n ).

1

(b) Show that dx = where n is an odd integer. 2.4.6....(n-1)x nv1-x 1.3.5....n2

0

vtan d =p/2 (c) Show that . [6+5+5]G(1/4) (3/4)

G 2

0

p

2. (a) When n is a positive integer prove that P (x) = (x ± vx - 1 cos ) d . 1 n2n p

0

1 2

(b) Prove that xJ (ax)J (ßx)dx = 0 if a = ß = J if a = ß(a)1 n n n+12

0

where a and ß are the roots of J (x) = 0. [8+8]n

3. (a) Show that the real and imaginary parts of an analytic function

f(z) = u(r, ) + i v(r, ) satisfy the Laplace equation in polar form

+ + = 0 and + + = 0 respectively. 2 u 1 u 1 2 u 2 v 1 v 1 2 v r r r r r r r r 2 2 2 2 2 2

(b) If u is a harmonic function, show that w = u is not a harmonic function2

unless u is a constant. [8+8]

4. (a) Evaluate using Cauchy’s integral formula (z+1)dz where C :| z + 1 + i | = 2.(z2 +2z+4)

C

-(b) Evaluate z dz f rom z = 0 to 4 + 2i along the curve C given by

C

i. z=t +it 2

ii. Along the line z=0 to z=2 and then from z=2 to 4+2i. [8+8]

5. (a) State and prove Laurent’s theorem.

(b) Obtain all the Laurent series of the function about z= -2. [8+8]7z-2 (z+1)(z)(z -2)

6. (a) Find the poles and the corresponding residues of the function . (z+2) (z-2)(z+1) 2

(b) Evaluate where C is |z+1| = 1 by residue theorem. [8+8]zdz(z +1) 2

C

2p

7. (a) Evaluate using residue theorem. d (5-3 sin ) 2

0

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8

(b) Evaluate using residue theorem. [8+8] x sin mx dxx +164

0

8. (a) Find the image of unit circle under the bilinear transformation

w=(iz+1)/(z +1).

(b) Find the image of the region bounded by the lines y=2 and y=4 under the

transformation w= z . [8+8]2

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1 1. (a) Evaluate interms of ß function.x 2 dxv1-x

5

0

[ ( )] 1 2G 1

(b) Prove that (1 - x ) dx = .n 1/n 1 n

n 2G(2/n) 0

n -1

(c) Prove that G G G .........G = . [5+5+6](2 )1 2 3 n-1 2 n n n n n 1/ 2

2. (a) Prove that J (x) = 1 - + - +...........x x x2 4 6 0 2 2 22 .42 2 2 .42 .6 2

1

(b) x P (x)dx = [8+8]2 n +1 .(n!) 2n n (2n+1)!

-1

3. (a) Derive Cauchy Riemann equations in polar coordinates.

(b) Prove that the function f(z) = ¯ z is not analytic at any point.

(c) Find the general and the principal values of (i) log (1+v3i) (ii) log (–1). e e

[5+5+6]

4. (a) Show that (z + 1) dz = 0 where C is the boundary of the square whose

Cvertices at the points z = 0, z = 1, z = 1+i, z = i.

(b) If F(a)= using Cauchy’s integral formula where c is |z| = 2 find3z2 +7z+1)dz

(z-a)C

F(1) F(3) F (1-i). [8+8]

5. (a) Expand f(z) = about z=1 as a Laurent series. Also find the region ofe 2z

(z-1) 3 convergence.

(b) Find the Taylor series for about z=1, also find the region of convergence.z z+2

[8+8]

6. (a) Find the residue of f(z) = at each pole.Z 2 -2Z(Z+1) (Z +1) 2 2

(b) Evaluate dz where c is the circle | z | = using residue theorem. 4-3z 3z(z-1)(z-2) 2

c

[8+8]

2p

7. (a) Evaluate using residue theorem.d

(5-3cos ) 2

0

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8

(b) Evaluate dx using residue theorem. [8+8] sin mx x

0

8. (a) Find and plot the map of rectangular region 0=x=1; 0=y=2, under the transformation w =v 2 e z +(1-2i). ip/4

(b) Find the bilinear transformation that maps the points 0,i,1 into the points

–1,0,1. [8+8]

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vcot d .p/2

1. (a) Evaluate0

vp

(b) prove that G(n + ) = . G(2n+1)12 4 G(n+1)n

(c) If m>0, n>0, then prove that ß(m, n + 1) = ß(m + 1, n) = ß(m,n) .[5+5+6]1 1

n m (m+n)

2. (a) Show that J (x) + J (x) = J (x).2n n-1 n+1 nx

(b) Prove that J (x) = cos x.2-1/2 px

(c) Show that (n+1) P (x) - (2n+1) x P (x) + n P (x) = 0. [5+5+6]n+1 n n-1

3. (a) Find the analytic function whose imaginary part is

f(x,y) = x y – xy + xy +x +y where z = x+iy.3 3

(b) Prove that + |Real f(z)| = 2|f (z)| where w =f(z) is analytic. 2 2 2 2 x y2 2

[8+8]

4. (a) Evaluate where c: |z - 1| = , using Caucy’s integral Formula.log z dz 1

(z-1) 3 2c

(b) State and prove Cauchy’s Theorem. [8+8]

5. Expand f (z) = in the region.(z-2)(z+2)

(z+1)(z+4)

(a) 1 < |z| < 4

(b) | z | < 1. [8+8]

6. (a) Determine the poles of the function and the corresponding residues of f(z)=.z+1

z (z-2)2

(b) Evaluate , where c is the circle | z | = 4 using residue theorem. [8+8]dzC sinhz

2p

7. (a) Use method of contour integration to prove that = , d 2p 1+a -2acos 1-a2 2

00< a<1.

8

(b) Evaluate using residue theorem. [8+8]dx

(x +9)(x +4)2 2 2 0

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8. (a) Prove that the transformation w=sinz maps the families of lines x=constant

and y=constant in to two families of confocal conics.

(b) Find the bilinear transformation which maps the points (i, -i, 1) of the z-plane

into (0,1, 8) of the w-plane. [8+8]

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PROBABILITY AND STATISTICS( Common to Computer Science & Engineering, Information Technology

and Computer Science & Systems Engineering)Time: 3 hours Max Marks: 80

Answer any FIVE Questions


1. (a) If A and B are independent, prove that

i. A and B are independent c c

ii. A and B are independent [2x4=8]c

(b) Three machines produces 40%, 30% and 30% of the total number of items

of a factory the percentages of defective items of these machines are 4%, 2%and 3%. If an item is selected at random find the probability that the item is

defective if the item is defective what is the probability that it is from machine

i. I

ii. II. [8]

2. (a) Suppose a continuous random variable X has the probability density

f(x)=k(1 - x ) for 0 < x < 1 and f(x) = 0 otherwise. Find 2

i. k

ii. mean

iii. variance.

(b) The probability that John hits a target is , He fires 6 times. Find the prob-1

2ability that he hits the target

i. exactly 2 times

ii. more than 4 times

iii. atleast once. [8+8]

3. (a) Define Poisson distribution and find its variance and the mean.

(b) Find the mean and standard deviation of a normal distribution in which 7%of items are under 35 and 89% are under 63. [8+8]

4. Take 30 slips of paper and label, 5 each -4 and 4, four each - 3 and 3, three each -

2 and 2 and each-1, 0 and 1, if each slip of the paper has the same probability of

being drown find the probabilities of getting - 4,- 3, - 2, - 1, 0, 1, 2, 3, 4 and findthe mean and varience of this distribution of means. [16]

5. (a) In a random sample of 160 workers exposed to a certain amount of radiation,

24 experienced some ill e ects. Construct a 99% confidence interval for thecorresponding ture percrntage.

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(b) What is the size of the sample required to estimate an unknown proportion

to within a maximum error of 0.06 with at least 5% confidence given thatstandard deviation is 2?

(c) The performance of a computer is observed over a period of 2 years to checkthe claim that the probability is 0.20 that its downtime will exceed 5 hours

in any given week. Testing the null hypothesis P = 0.20 against the alternate

hypothesis P = 0.20, what can we conclude at the level of significance a = 0.05,

if there were only 11 weeks in which the downtime of the computer exceeded5 hours? [5+5+6]

6. (a) To examine the hypothesis that the husbands are more intelligent than the

wives, an investigator took a sample of 10 couples and administered them a

test which measures the IQ as follows:

Test the hypothesis with a reasonable test at the level of significance of 0.05?

Husbands: 117 105 97 105 123 109 86 78 103 107

Wives 106 98 87 104 116 95 90 69 108 85

(b) In an investigation on the machine performance the following results were

obtained:

No.of Units inspected No. of defectives

Machine 1 375 17

Machine 2 450 22

Test whether there is any significant performance of two machines at a=0.05

[8+8]

7. (a) Derive normal equations to fit y = ax b

(b) Fit a parabola of the form y = a + bx + cx for the following data [6+10]2

x 2 4 6 8 10

y 3.07 12.85 31.47 57.38 91.29

8. The marks obtained by 10 students in mathematics and statistics are given below.

Find the coe cient of correlation between the two subjects and the two lines ofregression. [16]

Marks in maths 75 30 60 80 53 35 15 40 38 48

Marks in statistics 85 45 54 91 58 63 35 43 45 44

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1. (a) Define a random experiment, sample space, event and mutually exclusive

events. Give examples of each.

(b) Box A contains 5 red and 3 white marbles and box B contains 2 red and 6

white marbles.

i. If a marble is drawn from each box, what is the probability that they are

both of the same colour? [8+8]

2. (a) Find the variance of the binomial distribution .

(b) Determine the probability distribution of the number of bad eggs in a basket

containing 6 eggs given that 10% of eggs are bad in a large consignment[8+8]

3. (a) A Poisson distribution has a double mode at x = 2 and x = 3, find the

maximum probability and also find p(x=2).

(b) The weekly wages of 1000 workers are normally distributed around a mean of

Rs.70 and S.D of Rs.5/- Estimate the number of workers whose weekly wages

will be

i. between Rs.70 and Rs.72

ii. between 69 and 72 [8+8]

4. A population consists of 5,10,14,18,13,24 consider all possible samples of size twowhich can be drawn without replacement from the population. Find

(a) The mean of the population.

(b) The standard deviation of the population.

(c) The mean of the sampling distribution of means

(d) The standard deviation of sampling distribution of means. [4×4]

5. (a) A sample of 400 items is taken from a population whose standard deviation

is 10. The mean of the sample is 40. Test whether the sample has come froma population with mean 38. Also calculate 95% confidence interval for the

proportion.

(b) A social worker believes that greater than 25% of the couples in a certain area

are ever used any form of birth control. A random sample of 120 couples was

contacted. Twenty of them said that they have used. Test the belief if social

worker at 0.05 level. [8+8]

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6. Measurements of the fat content of two kinds of ice creams brand A and brand B

yielded the following sample data.

Brand A 13.5 14.0 13.6 12.9 13.0

Brand B 12.9 13.0 12.4 13.5 12.7

Test the significant between the means at 0.05 level. [16]

7. (a) Derive normal equations to fit y = ax b

(b) Fit a parabola of the form y = a + bx + cx for the following data [6+10]2

x 2 4 6 8 10

y 3.07 12.85 31.47 57.38 91.29

8. Find the least squares regression equation of X on X and X from the following 1 2 3

data. [16]

X 3 5 6 8 12 141

X 16 10 7 4 3 22

X 90 72 54 42 30 143

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1. (a) There are 12 cards numbered 1 to 12 in a box. If two cards are selected what

is the probability that the sum in odd

i. With replacement

ii. Without replacement

(b) Suppose colored balls are distributed in three indistinguishable boxes as fol-lows:

Box-I Box-I I Box-I II

Red 2 4 3

White 3 1 4

Blue 5 3 3

A box is selected at random from which a ball is selected at a random. What is

the probability that the ball is colored

(a) red

(b) blue [6+10]

2. (a) If X is a continuous random variable and K is a constant then prove that

i. Var (X+K) = Var (X)

ii. Var(kX) = k Var (X)2

(b) Determine the probability of getting 9 exactly twice in 3 throws with a pair

of fair dice. [8+8]

3. (a) Suppose 300 misprints are distributed randomly through out a book of 500pages. Find the probability that a given page contains

i. exactly 2 misprints

ii. 2 or more misprints.

(b) Suppose the diameter d of bolts manufactured by a company are normally

distributed with mean .25 inches and standard deviation .02 inches. A bolt isconsidered defective if d =.20 inches or d = .28 inches. Find the percentage

of defective bolts manufactured by the company. [8+8]

4. A population consists of five numbers 2,3,6,8,11. Consider all possible samples of size two which can be drown without replacement from the population. Find

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(a) The mean of the population

(b) Standard deviation of the population.

(c) The mean of the sampling distribution of means

(d) The standard deviation of the sampling distribution of means. [4+4+4+4]

5. (a) In a random sample of 160 workers exposed to a certain amount of radiation,

24 experienced some ill e ects. Construct a 99% confidence interval for thecorresponding ture percrntage.

(b) What is the size of the sample required to estimate an unknown proportionto within a maximum error of 0.06 with at least 5% confidence given that

standard deviation is 2?

(c) The performance of a computer is observed over a period of 2 years to check

the claim that the probability is 0.20 that its downtime will exceed 5 hours

in any given week. Testing the null hypothesis P = 0.20 against the alternate

hypothesis P = 0.20, what can we conclude at the level of significance a = 0.05,

if there were only 11 weeks in which the downtime of the computer exceeded5 hours? [5+5+6]

6. Four methods are under development for making discs of a super conducting ma-

terial. Fifty discs are made by each method and they are checked for super con-

ductivity when cooled with liquid.

1 Method 2 Method 3 Method 4 Methodst nd rd th

Super Conductors 31 42 22 25

Failures 19 8 28 25

Test the significant di erence between the proportions of Superconductors at .05

level. [16]

7. (a) Fit the curve y = ae to the following data bx

x: 0 1 2 3 4 5 6 7 8

y: 20 30 52 77 135 211 326 550 1052

(b) Fit a second degree polynomial to the following data, taking x as independentvariable: [8+8]

x: 1 2 3 4 5 6 7 8 9

y: 2 6 7 8 10 11 11 10 15

8. The marks obtained by 10 students in mathematics and statistics are given below.

Find the coe cient of correlation between the two subjects and the two lines of

regression. [16]

Marks in maths 75 30 60 80 53 35 15 40 38 48

Marks in statistics 85 45 54 91 58 63 35 43 45 44

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1. (a) For any three arbitrary events A, B, C prove thatP (A B C ) = P(A) + P(B) + P(C) - P(A n B) - P(B n C ) - P (C n

A) + P(A n B n C)

(b) In a certain town 40% have brown hair, 25% have brown eyes and 15% have

both brown hair and brown eyes. A person is select at random from the town

i. If he has brown hair, what is the probability that he has brown eyes also

ii. If he has brown eyes, determine the probability that he does not have

brown hair [8+8]

2. (a) Let F(x) be the distribution function of a random variable X given by

F(x) = cx 3 when 0 = x < 3

= 1 when x = 3= 0 when x < 0.

If P(X=3) = 0 Determine

i. c

ii. mean

iii. P( x > 1)

(b) A student takes a true false examination consisting of 8 questions. He guesses

each answer. The guesses are made at random. Find the smallest value of n

that the probability of guessing atleast n correct answers is less that . [8+8]12

3. (a) Given that p(x=2)=45. p(x=6)-3. p(x=4) for a Poisson variate X, find the

probability that

i. x = 1

ii. x < 2

(b) The marks obtained in statistics in a certain examination are found to be

normally distributed. If 15% of the candidates get = 60 marks, 40% < 30 0

marks find the mean and the standard deviation of marks. [8+8]

4. (a) A random sample of size 144 is taken from an infinite population having the

mean 75 and varience 225. What is the probability that x will be between 72and 77.

(b) A normal population has a mean of 0.1 and standard deviation of 2:1. Find the

probability that the mean of simple sample of 900 members will be negative. [8+8]

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5. (a) A die is thrown 256 times an even digit turns up 150 times. Can we say that

the die is unbiased.

(b) If we can assert with 95% that the maximum error is 0.05 and p=0.2, find the

sample size.

(c) Write about null hypothesis and testing of null hypothesis . [5+5+6]

6. A pair of dice are thrown 360 times and the frequency of each sum is indicated

below:

Sum 2 3 4 5 6 7 8 9 10 11 12

Frequency 8 24 35 37 44 65 51 42 26 14 14

Would you say that the dice are fair on the basis of the chi-square test at .05 levelof significance. [16]

7. (a) Fit a straight line for the following data

x 1 2 3 4 5 6 7 8 9 10

y 52.5 58.7 65 70.2 75.4 81.1 87.2 95.5 102.2 108.4

(b) Fit a curve of the form y = ae by the method of least squares for the following bx

data and estimate the value of y when x = 300. [8+8]

x 77 100 185 239 285

y 2.4 3.4 7.0 11.1 19.6

8. (a) The regression equations of two variables x and y arex = 0.7 y + 5.2, y = 0.3x + 2.8. Find the mean of the variables and the

coe cient of correlation between them

(b) Consider the following data:

x -4 -3 -2 -1 0 1 2 3 4

y 0.1 2.5 3.4 3.9 4.1 3.8 3.5 2.8 0.3

Find the correlation coe cient ‘r’. [6+10]

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ENGINEERING MECHANICS(Chemical Engineering)



1. A mast AB supported by a spherical socket at A and guy wires BC and BD carriesa vertical load P at B as shown in figure 1. Point B is 0.3 m vertically below the

xy plane. Find the axial force induced in each of the three members of this system.

[16]

Figure 1

2. (a) Define the following:

i. Friction

ii. Angle of friction

iii. Limiting friction

iv. Cone of friction

(b) A ladder 5 m long and of 250 N weight is placed against a vertical wall in a

position where its inclination to the vertical is 30 . A man weighing 800 N0

climbs the ladder. At what position will he induce slipping? The co-e cient of friction for both the contact surfaces of the ladder viz. with the wall and

the oor is 0.2. [8+8]

3. (a) Derive an expression for length of an open belt in standard form.

(b) A belt is running over a pulley of diameter 1200 mm at 200 r.p.m. The angle

of contact is 165 and coe cient of friction between the belt and pulley is 0.30

If the maximum tension in the belt is 3000 N, find the power transmitted by

the belt. [6+10]

4. (a) Di erentiate between ‘polar moment of inertia’ and ‘product of inertia’

(b) Find the moment of inertia and radius of gyration about the horizontal cen-

troidal axis. shown in Figure 4b. [6+10]

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Figure 4b

5. Compute the mass moment of inertia about the x – axis of the steel link shown in

figure5. [16]

Figure 5

6. (a) Ram and Rahim are sitting in cars A and B respectively. The cars are 300 m

apart and at rest. Ram starts the car and moves towards B with an accelera-

tion of 0.5m/s . After three seconds , Rahim starts his car towards A with an2

acceleration of 1m/s . Calculate the time and point at which two cars meet2

with respect to A.

(b) A projectile is fired at a speed of 800 m/s at an angle of elevation of 50 from0

the horizontal. Neglecting the resistance of air, calculate the distance of the

point along the inclined surface at which the projectile will strike the inclined

surface which makes an angle of 15 with the horizontal. [8+8]0

7. (a) A homogeneous sphere of radius of a=100 mm and weight W=100 N can

rotate freely about a diameter. If it starts from rest and gains, with constantangular acceleration, an angular speed n=180rpm, in 12 revolutions, find the

acting moment. .

(b) A block starts from rest from‘A’. If the coe cient of friction between all sur-

faces of contact is 0.3, find the distance at which the block stop on the horizontal plane. Assume the magnitude of velocity at the end of slope is same as

that at the beginning of the horizontal plane.

As shown in the Figure7b. [8+8]

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Figure 7b

8. The vertical shaft shown in the figure 8 is fixed at the top and carries a ywheel of weight 5000N welded at its bottom. The radius of gyration of ywheel is 250 mm.

The diameter of the shaft is 100 mm and its length is 1000 mm. The modulus of

elasticity E=2×10 N/mm and the modulus of rigidity is 8.16x10 N/m . Deter-5 2 5 2

mine the frequencies for free torsional vibrations and longitudinal vibrations.

[16]

Figure 8

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ENGINEERING MECHANICS(Chemical Engineering)



1. A mast AB supported by a spherical socket at A and horizontal guy wires BC and

BD carries a vertical load P at B as shown in Figure 1. Find the axial force induced

in each of the three members of this system. [16]

Figure 1

2. (a) Define the following:

i. Friction

ii. Angle of friction

iii. Limiting friction

iv. Cone of friction

(b) A ladder 5 m long and of 250 N weight is placed against a vertical wall in a

position where its inclination to the vertical is 30 . A man weighing 800 N0

climbs the ladder. At what position will he induce slipping? The co-e cient

of friction for both the contact surfaces of the ladder viz. with the wall and

the oor is 0.2. [8+8]

3. A belt transmits 15 kW from a pulley of 900 mm diameter running at 300 r.p.m.The angle of lap is 160 and coe cient of friction is 0.25, thickness of the belt is0

6mm and its density is 1000kg/m . Determine minimum width of the belt required3

if stress in belt is limited to 2 N/mm . [16] 2

4. (a) State and prove parallel axis theorem.

(b) Find the moment of inertia about the horizontal centroidal axis of shadedportion for the Figure 4b. [6+10]

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Figure 4b

5. Compute the mass moment of inertia about the x – axis of the steel link shown in

figure5. [16]

Figure 5

6. A roller of radius 0.1m rides between two horizontal bars moving in opposite direc-

tions as shown in figure6 Assuming no slip at the points of contact A and B, locate

the instantaneous center ‘I’ of the roller. Also locate the instantaneous center whenboth the bars are moving in the same directions. [16]

Figure 6

7. (a) A body weighing 20 N is projected up a 20 inclined plane with a velocity of 0

12 m/s, coe cient of friction is 0.15. Find

i. The maximum distance S, that the body will move up the inclined plane

ii. Velocity of the body when it returns to its original position.

(b) Find the acceleration of the moving loads as shown in figure 7b. Take mass

of P=120 kg and that of Q=80 Kg and coe cient of friction between surfaces

of contact is 0.3. Also find the tension in the connecting string. [8+8]

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Figure 7b

8. A weight of 10N attached to a spring oscillates at a frequency of 60 oscillations

per minute. If the maximum amplitude is 30mm, find the tension induced in the

spring. Also find the spring constant and the maximum velocity in the spring. [16]

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II B.Tech I Semester Sup plementar y Examinations, April/May 2011

ENGINEERING MECHANICS

(Chemical En gineering)Time: 3 hours Max Marks: 80



******

1. A mast AB supported by a spherical socket at A and guy wires BC and BD carries

a vertical load P at B as shown in figure 1. Point B is 0.3 m vertically below thexy plane. Find the axial force induced in each of the three memb ers of this system.

[16]

Figure 1

2. (a) Explain the types of friction with ex amples.(b) Two equal bodies A and B of weight `W' each are placed on a rough inclined

plane. The bodies are connected by a light string. If µ= 1/2 and µ = 1/3,A B

show that the bodies will be both on the point of motion when the plane is

inclined at tan 1 (5/12). [6+10]

3. (a) Derive an expression for length of an open belt in standard form.(b) A belt is running over a pulley of diameter 1200 mm at 200 r.p.m. The angle

of contact is 1650 and coefficient of friction between the belt and pulley is 0.3

If the maximum tension in the belt is 3000 N, find the power transmitted by

the belt. [6+10]

4. (a) State and p rove parallel axis theorem.(b) Find the centroid of the shaded lamina shown in Figure 4. [6+10]

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Figure 4

5. Determine the mass moment of inertia of a thin equilateral triangular plate of

mass `m' and thickness `t' about the axis perpendicular to the plane of the plate

and passing through the mass center. Base width= `b' and height of vertex above

base= `h'. Density of material is `w'. [16]

6. (a) A body moves along a straight line and its acceleration `a' which varies with

time `t' is given b y a =2-3t. Five seconds after th e start of observ ation, the

velocity is 20 m/s. The distance moved by the body 10sec after the start of

observation of motion from origin is 85 mDetermine

i. the acceleration, velo city and distance from the origin at the start of

observation.

ii. the time after the start of observation at which the velocity becomes zero

and the distance travelled from the origin.(b) A car is uniformly accelerated and passes sucessive kilometre-stones with ve-

locities of 20km/hour and 30km/hour respectively. Calculate its velocity when

it passes the next kilometre stone and the time taken for each of th ese two

intervals of one kilometre. [8+8]

7. (a) A body weighing 20 N is projected up a 200 inclined plane with a v elocity of12 m/s, coefficient of friction is 0.15.

Find

i. The maximum distance S, that the body will move up the inclined plane

ii. Velocity of the bod y when it returns to its original position.

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(b) Find the acceleration of the moving loads as shown in Figure 7(b). Take massof P=120 kg and that of Q=80 Kg and coefficient of friction between surfaces


Figure7(b).

8. Two springs of sti_ness 200 N/m are attached to a ball of weight 5 N as shown in

The Figure 8. If the b all is initially displaced by 2.5 cm to the left and released,find the period of oscillation of the ball. Find also the velocity of the ball when it

passes through the middle position. [16]

Figure 8

*******

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II B.Tech I Semester Sup plementar y Examinations, April/May 2011

ENGINEERING MECHANICS

(Chemical En gineering)Time: 3 hours Max Marks: 80



******

1. A mast AB supported by a spherical socket at A and guy wires BC and BD carries

a vertical load P at B as shown in figure 1. Point B is 0.3 m vertically below thexy plane. Find the axial force induced in each of the three members of this system.

[16]

Figure1.

2. (a) Explain the types o f friction with ex amples.

(b) Two equal bodies A and B of weight `W' each are placed on a rough inclined

plane. The bodies are connected by a light string. If µ = 1/2 and µ = 1/3,A B

show that the bodies will be both on the point of motion when the plane is-1 inclined at tan (5/12). [6+10]

3. Power transmitted between two shafts 3.5 m apart by a crossed belt drive round

two pulley, 600 mm and 300 mm in diameters is 6KW. The speed of the langerpulley is 220 r.p.m. The permissible load on the belt is 25 N/mm width of the belt

which is 5 mm thick. The coefficient of friction between the smaller pulley surface

and the belt is 0.35.

Determine:

(a) Necessary length of the belt(b) The width of the b elt, and

(c) The necessary initial tension in the belt. [16]

4. (a) Define the terms centroid, moment of inertia and radius of gyration.

(b) Compute moment of inertia of hemisphere about its diametral base of radius`R'. [6+10]

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5. (a) Define mass moment of inertia and ex plain Transfer formula for mass moment

of inertia.(b) Derive the ex pression for the moment of inertia of a homogeneous sphere of

radius `r' and mass density `w' with referen ce to its diameter. [8+8]

6. (a) A train is traveling at a speed of 60 km/hr. It has to slow down due to certain

repair wo rk on the track. Hence, it moves with a constant retardation of 1

km/hr/second until its speed is reduced to 15 km/hr. It then travels at aconstant speed of for 0.25 km/hr and accelerates at 0.5 km/hr per second

until its speed once more reaches 60 km/hr. Find the delay caused.

(b) The motion of a particle in rectilinear motion is defined by the relation3 2s = 2t -9t +12t-10 where s is expressed in metres and t in seconds. Find

i. the acceleration of the particle when the velocity is zeroii. the position and the total distance traveled when the acceleration is zero.

[8+8]

7. (a) A body weighing 2 0 N is projected up a 200 inclined plane with a velocity of

12 m/s, coefficient of friction is 0.15. Find

i. The maximum distance S, that the body will move up the inclined planeii. Velocity of the bod y when it returns to its original position.

(b) Find the acceleration of the moving loads as shown in figure 7(b). Take mass

of P=120 kg and that of Q=80 Kg and coefficient of friction between surfaces


Figure 7(b ).

8. Determine the frequ ency of torsional vibrations of the disc shown in figure 8. ifboth the ends of the shaft are fixed and diameter of the shaft is 40mm. The disc

has a mass of 600Kg, and a radius of gyration of 0.4m.Taking modulus of rigidity2for the shaft material as 85GN/m .l =1m, and l = 0.8m. [16]1 2

Figure 8.******

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APPLIED CHEMISTRY AND BIOCHEMISTRY

(Bio-Medical Engineering)



1. (a) Define the terms, specific conductivity and molecular conductivity. What are

its units. What is electrolytic cell.

(b) Describe the westar standard cell with a neat diagram. [8+8]

2. (a) How is Vinyl Chloride manufactured commercially?

(b) Describe the preparation, properties and applications on novolac resin. [8+8]

3. (a) What do you understand by the term hardness’ of a sample of water? Definethe degree of hardness and discuss the various units of its Expression.

(b) Distinguish between carbonate hardness and non-carbonate hardness of a sample of water.

(c) Calculate the temporary and permanent chard ness of water, in pp on units.Using the chemical analysis. [5+5+6]

4. (a) Define a living cell and draw a neat diagram of a plant cell along with the

organelles.

(b) How the fractionation of cell organelles is carried out by centrifugation method?

[8+8]

5. (a) Explain the kinetics of multisubstrate reactions.

(b) Explain the e ect of temperature and PH as enzymic reactions. [8+8]

6. What is oxidative deamination? Describe the fate of aminoacids after their absorp-tion. [16]

7. (a) What are ketone bodies?

(b) Explain how ketone bodies are formed.

(c) What is the clinical significance of ketone bodies formation? [5+6+5]

8. Mention the applications of isotopes in biochemistry. Explain the important types

of Chromatography. [16]

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1. (a) Define electrolyte and non-electrolyte with suitable examples.

(b) Define strong and weak electrolytes

(c) Explain Arrhenius ionic theory along with postulates. [5+5+6]

2. (a) Explain the di erence between addition and condensation polymerization with

examples.

(b) What is polyethylene? How is it made? Mention the di erent grades of

polythene and their uses. [8+8]

3. (a) What do you understand by the term hardness’ of a sample of water? Define

the degree of hardness and discuss the various units of its Expression.

(b) Distinguish between carbonate hardness and non-carbonate hardness of a sample of water.


4. How the transport of metabolites takes place across bio membranes of cells? Explain

them. [16]

5. (a) How an enzyme is extracted and purified from a living cell?

(b) Define the unit of enzyme activity. How can you explain the purification ofan enzyme? [8+8]

6. Write a short note on:

(a) Transcription.

(b) Translation.

(c) Replication. [5+5+6]

7. Describe the electrophoretic representation of normal human plasma proteins and

add a note on the functions of various plasma proteins. [16]

8. Discuss about High performance liquid Chromatography (HPLC) and Thin layer

Chromatography (TLC) with the help of required diagrams. [16]

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1. (a) Define specific conductivity and molecular conductivity and what are the units

the express them.

(b) What are the di erences between strong and weak electrolytes?

(c) Calculate the cell constant and equivalent conductivity of an electrolyte, when

a conductivity cell has two parallel electrodes set apart , when it is filled with

normal solution of an electrolyte at some temperature and o ered by some

resistance. [5+5+6]

2. (a) What is the Composition of natural rubbers? How is it obtained from its

natural source?

(b) Write a note on vulcanization of rubber. [8+8]

3. (a) What do you understand by the term hardness’ of a sample of water? Define

the degree of hardness and discuss the various units of its Expression.

(b) Distinguish between carbonate hardness and non-carbonate hardness of a sam-

ple of water.


4. (a) Di erentiate between prokaryotic and eukouyotic cells with suitable examples.

(b) Explain the composition or cell walls of prokaryotic cells. [8+8]

5. (a) What are immobilized enzymes? Explain their role in industry.

(b) Define isoelectric point. How does it help in the extraction of enzyme fromsolution?

(c) Give a general procedure for the extraction of an enzyme from tissues. [6+5+5]

6. What are the di erent types of Immunological techniques or Immuno Assay. Ex-

plain each of the type with principle and applications. [16]

7. Explain the following in brief

(a) Hypoglycermine

(b) Serum proteins

(c) Platelets

(d) Ketone bodies. [4+4+4+4]

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8. Write short notes on:

(a) Serum albumin and Globulin

(b) Creatine and Creatinine

(c) Blood glucose. [6+5+5]

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1. (a) Define electrolyte and non-electrolyte with suitable examples.

(b) Define strong and weak electrolytes

(c) Explain Arrhenius ionic theory along with postulates. [5+5+6]

2. (a) Di erentiate between addition and condensation polymerization with suitable

examples.

(b) Describe the manufacture of polyethylene by the free-radical process. [8+8]

3. (a) What causes hardness to water? What are the advantages and disadvantages

of hard water?

(b) How is the hardness of water removed by ion-exchanged method? Explain.

[8+8]

4. (a) Describe Oxidative Phosphorilation.

(b) Write the di erences between Prokaryotes and Eukaryotes. Give suitable ex-amples on each of the above. [8+8]

5. (a) Define co-enzymic, apoenzyme and holoenzyme.

(b) Define inhibitory and activators

(c) Explain di erence types of enzyme inhibitory and their di erences in Km

&V values. [3+3+10]max

6. Describe the formation of Ammonia and fate of Ammonia in the body. [16]

7. (a) Describe the chemical composition of urine under normal conditions. Add a

note on tubular maximum ( T ). m

(b) What are the normal and abnormal constituents of urine? Explain. [8+8]

8. Write short notes on:

(a) Serum albumin and Globulin

(b) Creatine and Creatinine

(c) Blood glucose. [6+5+5]

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Documents

FLUID MECHANICS Civil Engineering 2nd year