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ENGINEERING PHYSICS 1 QUESTION BANKS/PAPERS HUGE COLLECTIONS

ANNA UNIVERSITY QUESTION BANKS FOR ECE, DOWNLOADS, ECE THIRD, OTHERS, QUESTION BANKS, QUESTION PAPERS, 

B.E./B.Tech. DEGREE EXAMINATIONS, MAY/JUNE 2010

Regulations 2008

First Semester

Common to all branches

PH2111 Engineering Physics I

Time: Three Hours Maximum: 100 Marks

Answer ALL Questions

Part A - (10 x 2 = 20 Marks)

1. Mention any two properties of ultrasonic waves.
2. What i s the principle of pulse echo system?
3. What are the conditions needed for laser action?
4. What is holography?.
5. De¯ne acceptance angle of a ¯bre.
6. Give the applications of the ¯bre optical system.
7. Calculate de Broglie wavelength of an electron moving with velocity 107 m/s.
8. Explain degenerate and non-degenerate states.
9. Calculate the ¯rst and second nearest neighbor distances in the body centered cubic unit
cell of sodium, which has lattice constant of 4:3 £ 10¡10 m.
10. What is meant by Frenkel imperfection?

Part B - (5 x 16 = 80 Marks)

11. (a) (i) What are magnetostriction and piezoelectric e®ect? (4)
(ii) Write down the complete experimental procedure with a neat circuit diagram of
producing ultrasonic waves by piezoelectric e®ect. (12)
OR
11. (b) (i) What is an acoustic grating? How is it used in determining the velocity of
ultrasound?
(2 + 6)
(ii) Explain the process of non-destructive testing of materials using ultrasonic waves
by pulse-echo method. (8)
12. (a) (i) Derive the relation between the probabilities of spontaneous emission and stimu-
lated emission in terms of Einstein's coe±cients. (8)
(ii) Explain the following pumping mechanisms.
(1) optical and
(2) electric discharge. (8)
OR
12. (b) (i) With a neat sketch, explain the construction, principle and working of CO2 laser.
(14)
(ii) Mention four applications of lasers in materials processing.
(2)
13. (a) (i) Explain the basic structure of an optical ¯bre and discuss the principle of trans-
mission of light through optical ¯bres. (5)
(ii) Derive an expression for numerical aperture. (5)
(iii) Brie°y discuss a technique of optical ¯bre drawing. (6)
OR
13. (b) (i) With a neat diagram, give an account on displacement sensors. (8)
(ii) Give an elaborate account on losses in optical ¯bres. (8)
14. (a) (i) What is Compton e®ect? Derive an expression for the change in wavelength
su®ered by an X-ray photon, when it collides with an electron. (2 + 12)
(ii) An electron at rest is accelerated through a potential of 2 kV. Calculate the de
Broglie wavelength of matter wave associated with it. (2)
OR
14. (b) (i) Solve the Schrodinger's wave equation for a free particle of mass m moving within
a one dimensional potential box of width L to obtain eigenvalues of energy and
eigenfunctions.
(11)
(ii) Find the eigenvalues of energies and eigenfunctions of an electron moving in a
one dimensional potential box of in¯nite height and 1 ºA of width. Given that m
= 9.11 £10¡31 kg and h = 6.63 £10¡34 J. (5)
15. (a) (i) What are Miller indices? Mention the steps involved to determine the Miller
indices with example. (2 + 4)
(ii) The material zinc has HCP structure. If the radius of the atom is
1
4
th of the
diagonal of hexagon, calculate the height of the unit cell in terms of atomic
radius.
(2)
(iii) Show that the packing factor for HCP i s 74% . (8)
OR
15. (b) (i) What i s Burger vector? (2)
(ii) Draw Burger circuit indicating Burger vector for edge dislocation and screw dis-
location. (4+4)
(iii) Certain defects in crystals improve the properties of crystals. Explain. (6)

THIS FOR YOUR REFERENCE PURPOSE ONLY

B.E./B.Tech.Degree Examinations, November/December 2010

Regulations 2008

First Semester

Common to all branches

PH2111 Engineering Physics I

Time: Three Hours Maximum: 100 Marks

Answer ALL Questions

Part A - (10 x 2 = 20 Marks)

1. List down the disadvantages of piezoelectric e®ect.
2. Compare transmission and re°ection modes of non destructive testing using ultra-
sonics.
3. Why is population inversion necessary to achieve lasing?
4. How is electrons and holes con¯nement made in the active region of heterojunction
laser?
5. What are the di®erent types of classi¯cation of optical ¯bres?
6. Explain dispersion in an optical ¯ber.
7. De¯ne blackbody and blackbody radiation.
8. What i s meant by wave function?
9. An element has a HCP structure. If the radius of the atom is 1.605ºA, ¯nd the height
of unit cell.
10. What i s meant by a `Schottky pair'?

Part B - (5 x 16 = 80 Marks)

11. (a) With a neat diagram explain the method of production of ultrasound using
piezoelectric crystal. (16)
OR
11. (b) (i) With a neat diagram, discuss the principle and inspection method of ul-
trasonic °aw detector. (12)
(ii) Calculate the velocity of ultrasonic waves in water using acoustic grat-
ing. The frequency of the ultrasonic wave generated by transducer is 2
MHz. The di®raction angle measured i s 8'45" in the third order di®rac-
tion pattern. Wavelength of laser light used in this experiment is 6328ºA.
(4)
12. (a) Discuss the Einstein's theory of stimulated absorption, spontaneous and stim-
ulated emission of radiation. What are the conditions for light ampli¯cation?
(16)
OR
12. (b) (i) What is hologram? What are the necessary steps involved in recording
and reconstruction of a hologram? (12)
(ii) Compare holography with photography. (4)
13. (a) (i) Explain the basic structure of an optical ¯bre and discuss the principle of
transmission of light through optical ¯bres. (5)
(ii) Derive an expression for numerical aperture. (5)
(iii) Brie°y discuss a technique of optical ¯bre drawing. (6)
OR
13. (b) Explain the concept of optical ¯ber sensors and describe in detail about any
two sensors. (6 + 10)
14. (a) (i) What is meant by matter wave? (2)
(ii) Derive an expression for the wavelength of matter wave. (8)
(iii) Describe an experiment that veri¯es the existance of matter waves. (6)
OR
14. (b) Solve Schrodinger's wave equation for a particle in a box (one dimensional)
and obtain the energy and eigen function. (16)
15. (a) (i) De¯ne atomic packing factor. (2)


(ii) Calculate the number of atoms, atomic radius, coordination number and
atomic packing fraction for BCC and FCC structures. (10)
(iii) The distance between successive planes of Miller indices (111) i s 2.078ºA
for a metal having FCC structure. Find the atomic radius and volume of
its unit cell. (4)
OR
15. (b) (i) De¯ne the terms polymorphism and allotropy. (2)
(ii) Explain in detail the crystal defects and their types. (14)

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ELECTRONIC CIRCUITS — I QUESTION BANKS PREVIOUS YEAR COLLECTIONS WITH SOLUTIONS ANSWER KEY

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

Third Semester

Electronics and Communication Engineering

MOST IMPORTANT QUESTION PAPERS

ECE THIRD, QUESTION PAPERS, QUESTION BANKS, OTHERS, ANNA UNIVERSITY QUESTION BANKS FOR ECE, DOWNLOADS, 

EC2205 — ELECTRONIC CIRCUITS — I

(Regulation 2008)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Define stability factor.2. Calculate the value of feedback resistor (Rs) required to self bias an N-channelJFET with IDSS = 40 mA, Vp = -10 v and VGSQ = -5V.3. Define Miller's theorem.4. What is the coupling schemes used in multistage amplifiers?5. Give the expressions for gain bandwidth product for voltage and current.6. What do you mean by amplifier rise time?7. What is cross over distortion?8. Draw the circuit of Class-D amplifier.9. Compare the half-wave and full-wave rectifiers.10. What are the advantages of SMPS?PART B — (5 × 16 = 80 Marks)11. (a) (i) Explain the fixed bias method and derive an expression for thestability factor. (8)(ii) Explain the voltage divider bias method and derive an expressionfor the stability factor. (8)Or(b) (i) Explain the circuit which uses a diode to compensate for changes inVBE and in ICO. (12)(ii) Discuss the operation of thermistor compensation. (4)12. (a) (i) Derive the expressions for the following of a small signal transistoramplifier in terms of the h-parameters(1) current gain(2) voltage gain(3) input impedance(4) output admittance. (12)(ii) Compare CB, CE and CC amplifiers. (4)Or(b) (i) Explain the operation of emitter coupled differential amplifier. (12)(ii) Discuss the transfer characteristics of the differential amplifier. (4)13. (a) Discuss the low frequency response and the high frequency response ofan amplifier. (16)Or(b) Explain the operation of high frequency common source FET amplifierwith neat diagram. Derive the expression for (i) voltage gain (ii) inputadmittance (iii) input capacitance (iv) output admittance. (16)14. (a) (i) Explain the operation of the transformer coupled class A audiopower amplifier. (12)(ii) Explain the terms conversion efficiency and maximum value ofefficiency used in audio power amplifiers. (4)Or(b) Explain the operation of the class-B push pull power amplifier with neatdiagram and list its advantages. (16)15. (a) Derive the expressions for the rectification efficiency, ripple factor,transformer utilization factor, form factor and peak factor of(i) half wave rectifier(ii) full wave rectifier. (16)Or(b) Explain the operation of(i) Voltage multiplier (8)(ii) Switched mode power supply. (8)

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EC2204 SIGNALS AND SYSTEMS ANNA UNIVERSITY PREVIOUS YEAR QUESTION PAPER APRIL/MAY 2010 FOR ECE DEPARTMENT

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

EC2204 SIGNALS AND SYSTEMS ANNA UNIVERSITY PREVIOUS YEAR QUESTION PAPER   APRIL/MAY 2010 FOR ECE DEPARTMENT 

Third Semester

Electronics and Communication Engineering

EC2204 — SIGNALS AND SYSTEMS

MOST IMPORTANT QUESTION PAPERS

(Regulation 2008)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Define unit impulse and unit step signals.
2. When is a system said to be memoryless? Give an example.
3. State any two properties of Continuous – Time Fourier Transform.
4. Find the Laplace transform of the signal ) ( ) ( t u e t x at − = .
5. State the convolution integral for continuous time LTI systems.
6. What is the impulse response of two LTI systems connected in parallel?
7. State the Sampling theorem.
8. State the sufficient condition for the existence of DTFT for an aperiodic
sequence ) (n x .
9. Define one sided Z-transform and Two-sided Z-transform.
10. Define the shifting property of the discrete time unit Impulse function.

PART B — (5 × 16 = 80 Marks)

11. (a) Distinguish between the following:
(i) Continuous Time Signal and Discrete Time Signal. (4)
(ii) Unit step and Unit Ramp functions. (4)
(iii) Periodic and Aperiodic signals. (4)
(iv) Deterministic and Random signals. (4)
Or
(b) (i) Find whether the signal ) 1 4 ( sin ) 1 10 ( cos 2 ) ( − − + = t t t x is periodic
or not. (4)
(ii) Find the summation ∑∞
− =
−
8
2 ) 2 (
n
n n e δ . (4)
(iii) Explain the properties of unit impulse function. (4)
(iv) Find the fundamental period T of the continuous time signal




+ =
6
10 cos 20 ) (
π
πt t x . (4)
12. (a) (i) Find the trigonometric Fourier series for the periodic signal ) (t x
shown in the figure given below : (10)
(ii) Explain the Fourier spectrum of a periodic signal ) (t x . (6)
Or
(b) (i) Find the Laplace transform of the signal
) ( ) ( ) ( t u e t u e t x bt t a − + = − − . (8)
(ii) Find the Fourier transform of
t e t x
− = ) ( for 1 1 ≤ ≤ − t
otherwise 0 = . (8)
 132  132  132 
E 3074 3
13. (a) (i) Explain the steps to compute the convolution integral. (8)
(ii) Find the convolution of the following signals: (8)
) ( ) ( 2 t u e t x t − =
) 2 ( ) ( + = t u t h .
Or
(b) (i) Using Laplace transform, find the impulse response of an LTI
system described by the differential equation
) ( ) ( 2
) ( ) (
2
2
t x t y
dt
t dy
t d
t y d = − − . (8)
(ii) Explain the properties of convolution integral. (8)
14. (a) (i) Find the Fourier Transform of
N n
N n A n x
> =
≤ =
0
) (
. (8)
(ii) Explain any four properties of DTFT. (8)
Or
(b) (i) Find the Z-transform of the given signal ) (n x and find ROC.
( ) ( ) sin ( ) o
x n w n u n =     . (10)
(ii) Describe the sampling operation and explain how aliasing error can
be prevented. (6)
15. (a) (i) Find the impulse response of the discrete time system described by
the difference equation
) 1 ( ) ( 2 ) 1 ( 3 ) 2 ( − = + − − − n x n y n y n y . (8)
(ii) Discuss the block diagram representation for LTI discrete time
systems. (8)
Or
(b) (i) Describe the state variable model for discrete time systems. (8)
(ii) Find the state variable matrices A, B, C, D for the equation
) 2 ( 6 ) 1 ( 5 ) ( ) 2 ( 2 ) 1 ( 3 ) ( − + − + = − − − − n x n x n x n y n y n y . (8)

TAGS:QUESTION BANKS, QUESTION PAPERS, ECE THIRD, EEE THIRD, Digital Signal Processing(dsp), ANNA UNIVERSITY QUESTION BANKS FOR ECE, anna university question bank for eee, anna university question bank for CSE, 

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DIGITAL ELECTRONICS Electronics Question Papers and question papers huge collection with solutions download

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009.

Third Semester

 DIGITAL ELECTRONICS Electronics Question Papers and question papers huge collection with solutions download

Electronics and Communication Engineering

EC 2203 — DIGITAL ELECTRONICS

(Regulation 2008)

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Prove that the logical sum of all minterms of a Boolean function of 2 variables is 1.

2. Show that a positive logic NAND gate is a negative logic NOR gate.

3. Suggest a solution to overcome the limitation on the speed of an adder.

4. Differentiate a decoder from a demultiplexer.

5. Write down the characteristic equation for JK flipflop.

6. Distinguish between synchronous and asynchronous sequential circuits.

7. Compare and contrast static RAM and dynamic RAM.

8. What is PAL? How does it differ from PLA?

9. What are Hazards?

10. Compare the ASM chart with a conventional flow chart.

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Express the Boolean function Z X XY F + = in product of Maxterm.

(6)

(ii) Reduce the following function using K-map technique

( ) ( ) ( ) 6 , 2 14 , 12 , 10 , 8 , 7 , 4 , 3 , 0 , , , d D C B A f + = π . (10)

Or

(b) Simplify the following Boolean function by using Quine-Mcclusky method

( ) ( ) ∑ = 13 , 12 , 10 , 8 , 7 , 6 , 3 , 2 , 0 , , , D C B A F . (16)

12. (a) Design a carry look ahead adder with necessary diagrams. (16)

Or

(b) (i) Implement full subtractor using demultiplexer. (10)

(ii) Implement the given Boolean function using 8 : 1 multiplexer

( ) ( ) ∑ = 6 , 5 , 3 , 1 , , C B A F . (6)

13. (a) (i) How will you convert a D flipflop into JK flipflop? (8)

(ii) Explain the operation of a JK master slave flipflop. (8)

Or

(b) Explain in detail the operation of a 4 bit binary ripple counter. (16)

14. (a) Implement the following Boolean functions with a PLA

( ) ( ) ∑ = 4 , 2 , 1 , 0 , , 1 C B A F

( ) ( ) ∑ = 7 , 6 , 5 , 0 , , 2 C B A F

( ) ( ) ∑ = 7 , 5 , 3 , 0 , , 3 C B A F . (16)

Or

(b) Design a combinational circuit using a ROM. The circuit accepts a three

bit number and outputs a binary number equal to the square of the input

number. (16)

15. (a) Design a three bit binary counter using T flipflops. (16)

Or

(b) Design a negative-edge triggered ‘T flipflop’. (16)

Tag: Digital Communication, ECE THIRD, EEE THIRD, anna university question bank for eee, ANNA UNIVERSITY QUESTION BANKS FOR ECE, QUESTION BANKS, QUESTION PAPERS, ALL DEPT, 

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MEASUREMENTS AND INSTRUMENTATION important modal question papers Anna university

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009

 MEASUREMENTS AND INSTRUMENTATION important modal question papers Anna university 

Third Semester

Electricaland Electronics Engineering

EE2201 — MEASUREMENTS AND INSTRUMENTATION

(Regulation 2008)

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. Differentiate Resolution from Threshold.
2. How are the absolute and relative errors expressed mathematically?
3. What are the essential torques required for operating an instrument?
4. What is phase meter? Mention the types of phase meter.
5. What is an isolation amplifier? Where is it used?
6. State the condition for balance in a Wheatstone bridge.
7. What are the types of printers according to printing methodology?
8. What are the main parts of the cathode ray tube?
9. Differentiate sensor from transducer.
10. Draw the block diagram for 4 bit Analog to Digital Converter.

PART B — (5 × 16 = 80 Marks)

11. (a) Define and explain the static characteristics of an instrument.
Or
(b) (i) Draw and explain the general block diagram of measurement
system.
(ii) Write a note on different types of errors.
12. (a) Describe the construction and working principle of single phase
induction type energy meter. Write a short note on any two adjustments
required in energy meters.
Or
(b) Explain with neat sketch the classification of Instrument Transformers.
Write a note on the errors affecting the characteristics of an Instrument
Transformer.
13. (a) With a neat sketch describe a bridge to determine the unknown
inductance and a bridge to determine the unknown capacitance.
Or
(b) Explain the grounding techniques in detail to reduce the ground loop
interference signal.
14. (a) Describe the LED and LCD display devices.
Or
(b) Describe the direct and frequency modulation magnetic tape recording
types. Give its merits and demerits.
15. (a) What are the selection criteria for the transducer? Explain the working
principle of LVDT with neat sketch and characteristics. Give
advantages, disadvantages and applications of LVDT.
Or
(b) What are the performance parameters of Analog to Digital Converter?
Explain any two basic A/D conversion techniques in detail.

Tags:QUESTION BANKS, QUESTION PAPERS, ECE THIRD, EEE THIRD, anna university question bank for eee, ALL DEPT, 

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EE2254 — LINEAR INTEGRATED CIRCUITS AND APPLICATIONS question papers with solution four years huge collection

B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

Fourth Semester

Electrical and Electronics Engineering

EE2254 — LINEAR INTEGRATED CIRCUITS AND APPLICATIONS previous year collections

(Regulation 2008)

(Common to Instrumentation and Control Engineering and Electronics and

Instrumentation Engineering)

Time: Three hours Maximum: 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. What is the purpose of oxidation process in IC fabrication?

2. What is parasitic capacitance?

3. List any four characteristics of an ideal OP-Amp.

4. Design an amplifier with a gain of –10 and input resistance of 10 k .

5. Define slew rate and state its significance.

6. An 8 bit DAC has a resolution of 20mV/bit. What is the analog output voltage

for the digital input code 00010110 (the MSB is the left most bit)?

7. Draw the pin diagram of IC 555 timer.

8. Mention any two application of multiplier IC.

9. List the important parts of regulated power supply.

10. What are the advantages of switch mode power supplies?

PART B — (5 × 16 = 80 Marks)

11. (a) Explain the basic processes used in silicon planar technology with neat

diagram.

Or

(b) Discuss the various methods used for fabricating IC resistors and

compare their performance.

12. (a) (i) Explain the functions of all the basic building blocks of an Op-Amp.

(8)

(ii) Explain the application of OPAMP as (1) integrator

(2) differentiator. (8)

Or

(b) Find 0V of the following circuit.

13. (a) Design and explain triangular wave generator using Schmitt trigger and

integrator circuit.

Or

(b) (i) Explain the operation of dual slope ADC. (8)

(ii) Explain the following characteristics of ADC resolution, accuracy,

settling time, linearity. (8)

14. (a) With neat block diagram, explain IC566 VCO operation and discuss any

two applications.

Or

(b) What are the modes of operation of IC555? Derive the expression of time

delay of a monostable multivibrator.

15. (a) With a neat diagram, explain working principle of switch mode lower

supply.

Or

(b) Write brief notes on:

(i) IC MA 78 40

(ii) Optocoupler.

Tags:LINEAR INTEGRATED  CIRCUITS, ECE FOURTH, ANNA UNIVERSITY QUESTION BANKS FOR ECE, EEE, EEE FOURTH, EEE THIRD, anna university question bank for eee, QUESTION PAPERS, QUESTION BANKS,  

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MATHS, Anna University Question Bank Engineering Mathematics -1, CSE FIRST, ECE FIRST, EEE FIRST, EEE FIRST, FIRST SEMESTER, IT FIRST,

MATHS, Anna University Question Bank Engineering Mathematics -1, CSE FIRST, ECE FIRST, EEE FIRST, EEE FIRST, FIRST SEMESTER, IT FIRST, 

B.E./B.Tech.Degree Examinations, November/December 2010

Regulations 2008

First Semester

Common to all branches

MA2111 Mathematics I

Time: Three Hours Maximum: 100 Marks

Answer ALL Questions

Part A - (10 x 2 = 20 Marks)
1. If ¡1 i s an eigen value of the matrix A = µ 1 ¡2
¡3 2 ¶, ¯nd the eigen values of A4
using properties.
2. Use Cayley-Hamilton theorem to ¯nd A4 ¡ 8A3 ¡ 12A2 when A = · 5 3
1 3 ¸:
3. Find the centre and radius of the sphere 2(x2 + y2 + z2) + 6x ¡ 6y + 8z + 9 = 0.
4. State the conditions for the equation ax2 +by2 +cz2 +2fyz +2gzx+2hxy +2ux+
2vy + 2wz + d = 0 to represent a cone with vertex at the origin.
5. Find the radius of curvature of y = ex at x = 0.
6. Find the envelope of the family of straight lines y = mx +
1
m
, where m is a
parameter.
7. If u = f(x ¡ y; y ¡ z; z ¡ x), show that
@u
@x
+
@u
@y
+
@u
@z
= 0.
8. If u =
x + y
1 ¡ xy
and v = tan¡1 x + tan¡1 y, ¯nd
@(u; v)
@(x; y)
.
 132  132  132 
9. Evaluate
1 Z0
2 Z0
ex+y dx dy.
10. Evaluate
1 Z0
2 Z0
3 Z0
xyzdzdydx:
Part B - (5 x 16 = 80 Marks)
11. (a) (i) Using Cayley-Hamilton theorem, ¯nd the inverse of the matrix A = 24
¡1 0 3
8 1 7
¡3 0 8
35
.
(8)
(ii) Find the eigen values and eigen vectors of the matrix 24
2 2 1
1 3 1
1 2 2
35
.
(
8
)
OR
11. (b) Reduce the quadratic form
2x2
1 + x2
2 + x2
3 + 2x1x2 ¡ 2x1x3 ¡ 4x2x3
to canonical form by an orthogonal transformation. Also ¯nd the rank, index,
signature and nature of the quadratic form. (16)
12. (a) (i) Find the equation of the sphere having its centre on the plane 4x¡5y¡z =
3 and passing through the circle x2 + y2 + z2 ¡ 2x ¡ 3y + 4z + 8 = 0;
x ¡ 2y + z = 8. (8)
(ii) Find the equation of the right circular cone whose vertex i s the origin,
whose axis i s the line
x
1
=
y
2
=
z
3
and which has semi-vertical angle 30±.
(8)
OR
12. (b) (i) Find the equation of the sphere passing through the circle x2 + y2 + z2 +
x¡3y +2z ¡1 = 0, 2x+5y ¡z +7 = 0 and cuts orthogonally the sphere
x2 + y2 + z2 ¡ 3x + 5y ¡ 7z ¡ 6 = 0. (8)
(ii) Find the equation of the right circular cylinder whose axis is
x ¡ 1
2
=
y ¡ 2
1
=
z ¡ 3
2
and radius 2. (8)


13. (a) (i) Find the equation of the circle of curvature of the curve px + py = pa
at ³a
4
;
a
4´. (8)
(ii) Prove that the radius of curvature of the curve xy2 = a3 ¡x3 at the point
(a, 0) is
3a
2
: (8)
OR
13. (b) (i) Show that the evolute of the hyperbola
x2
a2 ¡
y2
b2 = 1 i s (ax)2=3 ¡(by)2=3 =
(a2 + b2)2=3. (10)
(ii) Find the envelope of
x
a
+
y
b
= 1, where a and b are connected by the
relation a2 + b2 = c2; c being constant. (6)
14. (a) (i) Find the Taylor's series expansion of ex cos y in the neighborhood of the
point ³1;
¼
4 ´ upto third degree terms. (8)
(ii) If u = log(x2 + y2) + tan¡1 ³y
x´, prove that uxx + uyy = 0. (8)
OR
14. (b) (i) Discuss the maxima and minima of the function f(x; y) = x4 +y4 ¡2x2 +
4xy ¡ 2y2. (8)
(ii) Find the Jacobian of y1; y2; y3 with respect to x1; x2; x3 if y1 =
x2x3
x1
; y2 =
x3x1
x2
; y3 =
x1x2
x3
. (8)
15. (a) (i) Evaluate
1 Z0
1 Zx
e¡y
y
dx dy by changing the order of integration. (8)
(ii) Evaluate
1 Z0
1 Z0
e¡(x2+y2) dx dy by converting to polar coordinates. Hence
deduce the value of
1 Z0
e¡x2
dx. (8)
OR
15. (b) (i) Using triple integration, ¯nd the volume of the sphere x2 + y2 + z2 = a2.
(8)
(ii) Find the area bounded by the parabolas y2 = 4 ¡ x and y2 = x by double
integration. (8)

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