MATHEMATICS-II

UTTARAKHAND TECHnical UNIVERSITY

B.TECH (FIRST YEAR)

2012

time:- 3 hr

Total Marks:100

Section A

Note:- Attempt all questions. All question carry equal marks.

Q1 Attempt any four parts of the following:-

5x4=20

1. Solve:
   [y²e^xy² + 4x³]dx +[2xye^xy²- 3y²]dy =0
2. Solve:(1+y²)dx =(tan^_1y –x)dy
3. Solve:
   (D²-2D+1)y = xsinx
4. Solve:
   
   dx/dt + 2x -3y = t, dy/dt -3x + 2y =e^2t
5. Solve:
   
   d²y/dx² + cotx dy/dx + 4y cosec²x =0
   by changing the independent variable.
6. Apply the method of variation of parameter to solve:
   d²y/dx² +n²₄ = secnx
   
   Section B
   

Q2:- Attempt any two parts of the following :

10x2=20

1. Find Laplace transform of
   (a)  te^at sin at
   (b) (cos at – cos bt)/t
2. Find
   (a) L^-1[log S+1/S-1]
   (b) L^-1 [1/S(s+a)³]
3. Using laplace transform solve the following equation:
   d²y/dt² + x = t cos2t
   given x(0) =x^’(0)=0

Section C

Q3:-Attempt any two parts of the following

10x2=20

1. (a) Test for convergence the series 

 

(b)Test for convergence the series

    

     2.  Discuss the convergence of the series

 

     3. Find the geometric series.

Section D

Q4:- Answer any two parts of the following:

10x2=20

1. Find the fourier series expansion for f(x) ,if
   
     
2. f(x)= expand this as Fourier sine series.
3. Solve (D³-4D²D + 4DD^’2) = 6 sin(3x+2y) 

Section E

Q5:- Attempt any two parts of the following :

10x2=20

1. A tightly stretched string with fixed end points x=0 and x=1 is initially in a position given by y(x,0) =  y₀ sin(πx/l). it is released from rest from this position, find the displacement y at any distance x from one end at any time t.
2. A homogeneous rod of conducting material of length 'l' has its ends kept at zero temperature. The temperature at the center is T and falls uniformly to zero at the two ends. Find the temperature distribution.
3. Solve     given u(0,y) = 4e^-y – e^-5y, by the method of separation of variables.

Mathematics-1

Uttarakhand Technical University

B.tech (first year)

2011-12

sub :- Mathematics-1

time : 3hr

Total marks :100

SECTION A

Q1:- Attempt any four of the following :

4x5=20

1. Reduce the matrix
   to normal form and hence find its rank.
2. Show that the vectors :
   X₂  = (1,2,4), X₂ = (2,-1,3), X₃ = (0,1,2), X₄ =(-3,7,2) are linearly dependent. Find the relation between them.
3. Show that the matrix
   is Hermitian and iA is skew Hermitian.
4. Verify Cayley-Hamilton theorem for
   and hence find A^-1.
5. Prove that the eigenvalues of a unitary matrix are of unit modulus.
6. Test the consistency for the following system of equation and if system is consistent solve them:
   x + y + z =6,
   x + y + 3z= 14,
   x + 4y + 7z =30.

SECTION B

Q2:- Attempt any four of the following: 

4x5=20

1. If  y = e ^asin-1x   , show that
   (1-x²)y_n+2 – (2n+1)xy_n+1 – ( n²+a²)y_n =0  
2. If  y = sin(msin^-1x), find y_n at x =0.
3. If u = x²tan^-1(y/x) – y² tan^-1 (x/y), show that
   
4. If 
   
5. If u = f(r,s,t) and r =x/y, s = y/z , t = z/x,
   show that
   
6. Expand f (x,y) = tan^-1y/x  in the neighborhood of (1,1) up to third degree term.

SECTION C

Q3:- Attempt any two of the following:

10X2=20

1. If u= xyz, v=x² + y² + z², w =x + y + z
   Find the Jacobian
   
2. The power P required to propel a steamer of length l at a speed u is given by p =λu³l³, where λ is constant .If u is increased by 3% and l is decreased by 1%. Find the corresponding increase in  p.
3. Using the Lagrange method of undetermined multiplier find the point upon the plane                 ax + by + cz=p qt which the function f = x² + y² + z²,has a minimum value and find this minimum f.

SECTION D

sub (Mathematics-1) b tech 1st year  utu

Q4:- Attempt any two of the following:

10X2=20

1. Evaluate the integral
   by changing the order of integration.
2. Show that
   , where symbols have their usual meaning.
3. Evaluate
   ,over the area between y= x²  and y=x

SECTION E

Q5:- Attempt any two of the following:

10X2=20

1. Find the angles between the normal surface xy=  z²  at the point (4,1,2) and (3,3,-3).
2. Prove that div (grad rⁿ) =n(n+1) r^n-2  ,where r²= x² + y² + z²   hence show that 
   
3. Explain the Stoke's theorem .using stoke theorem evaluate the integral  where F=y²i^{^}+x²j^{^}-(x+z)k^{^},and c is the boundary of the triangle with vertices (0,0,0), (1,0,0) and (1,1,0).

BASIC ELECTRICAL ENGINEERING 1ST YEAR

UTTARAKHAND TECHnical UNIVERSITY

B.TECH FIRST YEAR, 2012

SUB :- BASIC ELECTRICAL ENGINEERING

time : 3hr

Total marks :100

SECTION A

Q1:- Attempt any four of the following :

1. Define the following
   (i)  Resistance
   (ii) Specific Resistance
   (iii) Potential difference
   (iv) Unilateral Element
   (v)  Ammeter
2. In the circuit shown 100 V dc voltage is applied across terminal A-B, calculate the power dissipated in each resistor and the reading of a voltmeter connected across the 5 ohm resistor.
   
3. State and prove the maximum Power Transfer theorem.
4. Using nodal method, find current through 100 ohm resistor.
5. Explain the conversion of current source into equivalent voltage source  for solving a problem.

SECTION B

Q2:- Attempt any two of the following :

1. Find the equivalent impedance of the following impedances  connected in parallel
   Z₁ = 8 + j6, Z₂ = 8 -j6, Z₃ = 8.66 + j5.
2. For the following impedance   Z₁ = 10 + j20, find its conductance and susceptance.
3. What do you mean by resonance in a series ac circuit? What will be the power factor of series resonating circuit?
4. A balance star connected load of 8 + j6 ohms per phase is connected to a 3-phase, 230 V supply. Find the line current, power factor , power, reactive volt amperes and total volt-amperes.
5. A voltage of 250 V at 50 Hz is applied to the circuit shown below, find current drawn from the source.

   

SECTION C

 Q3:- Attempt any two of the following :

1. Draw the phasor diagram for a practical transformer under the condition that, a lagging power factor load is connected across its terminals.
2. An iron ring of mean circumference 80 cm is made of iron of area 8 Cm² .It has a cut of 2 mm wide and is wound with 500 turns. Find the current required to produce a flux of 0.8 mWb across the air-gap. Relative permeability of the iron is 1000.
3. A transformer is rated at 100 kVA. at full load its copper loss is 1400 W and iron losses are 940 W. Calculate
   (i) The efficiency at full load, unity power factor.
   (ii) The efficiency at half load, same power factor.
   (iii) The load kVA at which maximum efficiency will occur.

SECTION D

 Q4:- Attempt any two of the following :

1.  Explain the principle and construction of attraction type moving iron instruments. Discuss their merits and demerits?
2. Draw the characteristics of a compound D C generator? An 8-pole lap connected armature has 40 slots generates a voltage of 500 V. Determine the speed at which it is running if the flux per pole is 50 mWb?
3. Explain the principal of operation of a three phase induction motor? Draw its speed-torque characteristics.  

SECTION E

Q5:- Attempt any two of the following :

1. Write short note on any two of the following?
   (i) Operation of stepper motor with diagram?
   (ii) Construction diagram and the working of Auto-transformer starter? 
2. A 3 phase, 6 pole 50 Hz induction motor has a slip of 2% at no load and 3% at full load. Determine
   (i) Synchronous speed
   (ii) No load speed
   (iii) full load speed
   (iv) frequency of rotor current at standstill.
   (v) frequency of rotor current at full load.
3. Is Synchronous machine a self starting motor? If not then explain the methods of starting of a synchronous motor?

MECHANICAL 1ST YEAR

UTTARAKHAND TECH. UNIVERSITY

UTU

B.TECH FIRST YEAR, 2012

SUB :- MECHANICAL

time : 3hr]  

Total marks :100

SECTION A

Q1:- Attempt any four of the following :

1. Define the following terms:
   (i) Quasi-static process
   (ii) Thermodynamic equilibrium
   (iii) Thermal reservoirs
2. Explain the concept of free expansion with zero work transfer.
3. The properties of a closed system change following the relation between pressure and volume as pV = 3.0, where p is in bar and V is in m³. Calculate the work done when the pressure increase from 1.5 bar to 7.5 bar.
4. Define 'internal energy' and prove that it is a property of a system.
5. Explain heat transfer is a path function?
6. Define and derive steady flow energy equation, andapply this equation to nozzle.

SECTION B

Q2:- Attempt any four of the following :

1. What is the otto cycle? Plot the cycle on P-V and T-S diagram and derive the efficiency for the same, 
2.  State the Clausius and Kelvin-Plank statements being used for second law of thermodynamics.
3. Differentiate between two stroke and four stroke I.C. engine.
4. 0.05 m³ of air at a pressure of 8 bar and 280^oC expands to eight times its original volume and the final temp after expansion is 25^oC. Calculate change of entropy of air during the process. Assume C_p = 1.005kJ/kgK; C_v= 0.172 kJ/kgK.
5. Air enters at a condition of 1 bar and  30^o to an air standard Diesel cycle and compressed to 20 bar, cut-off takes places 6% of stroke. Draw P-V and T-S diagram for the cycle and calculate :
   (i) Power output
   (ii) Heat input
   (iii) Air cycle efficiency
6. Describe the different processes of Rankine cycle. Plot the different process of Rankine cycle on P-V, h-s and T-S diagram and derive the expression for its efficiency.

SECTION C

Q3:- Attempt any two of the following :

1. The extremities A and D of a light inextensible string ABCD are tied to two points in the same horizontal line. Weights W and 3W are tied to a string at the points B and C respectively. If AB and CD are inclined to the vertical at angle 60^oC and  30^o respectively, show that BC is horizontal and find the vensions in the various parts of the strings.
   

   

   FIG 1
   

   
   

   
   

   
   

   
   

2. A ladder of length 'L' rests against a wall, the angle of inclination be  45^o. If the co-efficient of friction between the ladder and the ground and that between the ladder and the wall be 0.5 each. What will be the maximum distance along ladder to which is 1.5 times the weight is 1.5 times the weight of ladder may ascend before the ladder beings to slip?
3. A beam AB 10 m long has supports at its ends A and B . It carries a point load of 2.5 kN at 3m from A and a point load of 0.5 kN/m between the point loads . Draw the shear force and bending moment diagram for the beam.

SECTION D

 Q4:- Attempt any two of the following : 

1. A rectangular beam of 200 mm in width and 400 mm in depth is simply supported over a span of 4m and carriers a distributed load of 10kN/m. Determine maximumbending stress in the beam.
2. A plane stress condition exists at a point on the surface of a loaded structure. Where the stress have the magnitudes and direction shown on the stress elements. Determine
   (a) The normal stress and the shear stress and he resultants stress on a plane whose normal is inclined at an angle 45^o to the axis of major principle stress.
   (b) The principle stresses and the position of the principle
   (c) Magnitude and position of the greatest shear stress
   (d) Angle of obliquity
   

   

   fig 2

3. Define the term torsion. Write the assumptions used in torsion equation and derive the torsion equation.

SECTION E

Q5:- Attempt any two of the following :

1. Derive the relation between three elastic moduli.
2. A steel bar is subjected to loads as in figure. If Young's modulus for the bar material is 200 kN/mm², determine the change in length of bar. The bar is 200 mm diameter.
   

   

   FIG 3

3. A truss is shown below. Find the forces in all the members of the truss and indicate of the truss and indicate whether it is in tension or compression.

   

   fig 4

ENGINEERING PHYSICS

ENGINEERING PHYSICS
B.TECH EXAMINATION
SEM III, 2012-13
UTTARAKHAND TECHNICAL UNIVERSITY(UTU)

Time: 3 hours

Total marks: 100

Unit -I

Attempt any four of the following:

1. What do you understand by absolute motion; define inertial and non inertial frames? Explain the Newtonian principle of relativity.
2. Discuss the non existence of aether using Michelson Morley experiment.
3. An event occurs at x=100 m, y =10 m, z=5 m and t =100μ sec in frame S. Find the coordinates of this event in a frame S' which is moving with velocity 2.7 x 10⁸ m/s w.r.t the frame S along the common XX' axis using 1) Galilean transformation 2) Lorentz transformation.
4. State the fundamental postulate of the special theory of realtivity and write down Lorentz's direct and inverse transformation equations. Discuss how these accounts for the phenomenon of length contraction by explaining the concept of proper length.
5. An iron furnace radiates 1.53 x 10⁵ calories per hour through an opening of the cross section 10^-4 sq meter. If the radiation emittance of the furnace is 0.80, calculate the temperature of the furnace. Given σ =1.36 x 10^-8cal/m²-s-K⁴
6. State Stefan's law of heat radiation. Describe an experiment to determine Stefan's law.

Unit -II

Attempt any two of the following:

1. (a) Obtain the condition of the interference of the light reflected by a thin parallel film. Explain how you device a non reflecting film.
   (b) If the angle of the wedge is 0.25^o and the wavelength of the sodium lines are 589 nm and 589.9 nm, find the distance from the apex at which the maxima due to two wavelengths first coincides when observed in reflected light.
2. (a) In the second order spectrum of a plane diffraction grating, a certain spectral line appears at an angle of 10^o, while other line of wavelength 0.05 nm greater appears at an angle 3" greater. Find the wavelength of the lines.
   (b) Explain the term coherence and write the method to produce coherent sources of light. Also write down the suitable method to find out the thickness of the thin filmby using the biprism.
3. (a) A grating with 1500 ruling per inch is illuminated normally with white light from 400-700 nm. Show that only the first order spectrum is isolated but the second and third orders overlap.
   (b) Derive the formula for the wavelength of the light used and refractive index of a liquid in Newton's ring experiment.

Unit -III

Attempt any two of the following:

1. (a) How can you produce plane, circularly and elliptically polarized light? Also explain how to distinguish between circularly polarized light and unpolarized light experimentally.
   (b) Explain the principle of LASER. Derive the relation between the Einstein's coefficients.
2. (a) Draw the schematic diagram of the He-Ne laser. Using energy level diagram explain its working.
   (b) A plane polarized light of wavelength 600 nm is incident on a thin quartz plate cut with faces parallel to the optic axes. Calculate the ratio of the intensity of ordinary and extra-ordinary light makes an angle 30 with the optic axis. Refractive index of the ordinary and extra-ordinary rays = 1.633
3. (a) A plane polarized light of wavelength 600 nm is incident on a thin quartz plate cut with faces parallel to the optic axes. Calculate
   (i) the minimum thickness of the plate which introduces a phase difference of 60^o between ordinary and extra-ordinary rays.
   (ii) The minimum thickness of the plate for which ordinary and extra-ordinary rays will combine to produce plane polarized beam. Refractive index of the ordinary and extra-ordinary rays=1.633.
   (b) What are optically active substances? Explain the working of a polarimeter to find the specific rotation of cane sugar solution with suitable diagram.

Unit -IV

Attempt any two of the following:

1. (a) A paramagnetic material has a magnetic field intensity of 10⁴ Am^-1. If the susceptibility of the material at room temperature is 3.7 x 10^-3 . Calculate the magnetization and flux density of the material.
   (b) A system of electron spins is placed in a magnetic field of 2 Wb/m² at a temperature T. The number of spins parallel to the magnetic field is twice as large as the number of antiparallel spins. Determine T.
2. (a) A magnetic material has a magnetization of 2300 Am^-1 and produces a flux density of 0.00314 Wb/m^-2. Calculate the magnetizing force and the relative permeability of the material.
   (b) A positive point charge Q is at the center of a spherical conducting shell of an inner radius R₁ and the counter radius R_o. Determine E and V as the function of the radial distance R.
3. (a) Find the poynting vector on the surface of a long, straight conducting wire (of radius b and conductivity σ) that carries a direct current I. Verify the poynting's theorem.
   (b) State and prove Poynting's theorem and explain each term.

Unit -V

Attempt any two of the following:

1. (a) Derive the London equation of superconductivity. Show that the second London equation explains Meissner effect.
   (b) Calculate the critical current density for 1 mm diameter wire of lead at
   (i) 4.2 K and (ii) 7 K.
   Given : T_c for lead is 7.18 K and H_o for lead is 6.5 x 10⁴A/m.
2. (a) The London penetration depths for Pb at 3K and 7.1 K are respectively 39.6 nm and 173 nm. Calculate its transition temperature as well as the depth 0K.
   (b) A nucleon is confined to a nucleus of diameter 5 x 10^-4 m. Calculate the minimum uncertainty in the momentum of nucleon. Also calculate the minimum kinetic energy of the nucleon.
3. (a) Derive the relation for energy levels of a particle confined within one dimensional rigid wall with infinitely high sides.
   (b) Calculate the uncertainty in the angle of emergence of 1 MeV electron passing through a hole of 20 micron.