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I B.Tech (Sem-I) QUESTION BANK
Sub: ENGINEERING DRAWING
UNIT-1
1 (a).Describe a regular pentagon about a circle of 100mm diameter.
(b).A fixed point F is 7.5cm from a fixed straight line.Draw the locus of a point P moving in such a way
that its distance from a fixed straight line is equal to its distance from F .Name the curve .Draw normal
and tangent at a point 6cm from F.
2.(a).Construct an ellipse when the distance between the focus and directrix is 30mm and the eccentricity
is 3/4.Draw the tangent and normal at any point P on the curve using directrix.
(b).Construct a regular polygon of any number of sides ,given the length of its sides equal to 25mm.
3.(a). A ball thrown up in the air reaches a maximum height of 45 meters and travels a horizontal
distance of 75 meters. Trace the path of the ball, assuming it to be parabolic.
(b).Construct a regular octagon in a square of 75mm side.
4.(a).Inscribe a regular heptagon inside the given circle of 68mm diameter.
(b). The distance between two fixed points is equal to 75mm. Point P moves such that the sum of its
distances from the two fixed points is always a constant and is equal to 90mm. Draw the locus of P and
determine the minor axis.
5.(a). The major axis of an ellipse is 150 mm long and the minor axis is 100 mm long. Find the foci and
draw the ellipse. Draw a tangent to the ellipse at a point on it 25 mm above the major axis.
(b). Construct a regular hexagon of 40 mm side and draw in it, six equal circles, each touching one side of
the hexagon and two other circles.
6.(a). A circle 50 mm diameter rolls along a straight line without slipping. Draw the curve traced out by a
point P on the circumference, for one complete revolution of the circle. Name the curve. Draw a tangent
to the curve at a point on it 40 mm from the line.
(b). Outside a circle of 25 mm diameter, draw five equal circles, each touching the given circle and other
two circles.
Branch : ECE
UNIT-II
1 a). Draw the projections of the following points on the same ground line keeping the projectors 25mm
apart.
(a).A in HP and 20mm behind VP.
(b).B,40mm above the HP, and 25mm infront of VP.
(c).C,25mm below the HP ,and 50mm behind VP.
(d).D,15mm above HP,and 50mm behind VP.
(b). Draw the projections of the following points, keeping the distance between the projectors as 25mm
on the same reference line.
I. P- 25mm above HP and 45mm in front of VP.
II. Q- 35mm above HP and 50mm behind VP.
III. R- 45mm below HP and on VP.
IV. S- 30mm below HP and 40 mm in front of VP.
2(a).Construct a plain scale of 1:50000 to show kilometers and hectometers and long enough to measure
upto 7km measure a distance of 54hectometers on your scale.
(b).Construct a diagonal scale of RF 1:320000 to show kilometers and long enough upto 400km .Show
distance of 257km and 333km on scale.
3(a).Construct a vernier scaleto measure meters,decimeters and centimeters and long enough to measure
upto 4m. RF of scale is 1/20 mark on your scale a distance of 2.28m.
(b).A point is 20mm below HP and lies in the third quadrant its shortest distance from XY is 40mm
.Draw its projections.
4(a).Two pegs fixed on a wall are 4.5m apart the distance between the pegs measured parallel to the floor
is 3.5m if one peg is 1.5m above the floor,find the height of
the second peg and inclination of the line joining the two pegs with the floor.
(b).Two points A and B are in HP.The point A is 30mm infront of VP while B is behind the VP
distance between their projectors is 60mm.,line joining their topviews makes an angle 45' with XY.Find
the distance of the point B from VP.
5(a).Draw the projections of a 75mm long straight line in the following positions:
(a). Parallel to and 30mm above the HP and in the VP.
(b). Perpendicular to the VP,25mm above the HP and its one end in the VP.
(c).Inclined at 30 * to the HP and its one end 20mm above it,parallel to and 30mm in front of VP.
(b).The front view of a line inclined at 30* to the VP is 65mm long.Draw the projections of the line
when it is parallel to and 40mm above HP ,its one end being 30mm in front of VP.
6(a). Draw a vernier scale of R.F=1/25, to read centimeters up to 4 meters and on it, show lengths
representing 2.39m and 0.91m
(b). Draw the projections of a 60mm long straight line, in the following positions.
(i) Perpendicular to the HP, in the VP and its one end in the HP.
(ii) Inclined at 450 to the VP, in the HP and its one end in the VP.
UNIT-III
+
1. The top view of a 75mm long line AB measures 65 mm , while the length of its front view is
50mm ITS one end A is in the HP AND 12 mm infront of the vp . draw the projections of AB and
determine the inclinations with the HP AND the VP
2. A line CD measuring 80 mm is inclined at an angle of 300 to HP and 45
0 TO VP . the point C is
20 mm above HP and 30mm in front of VP. Draw the projections of the straight line
3. A line AB is 75mm long . A is 50 mm infront of VP and 15 mm above HP .B is 15mm in front of
vp and is above HP. TOP VIEW OF AB is 50 mm long. Find the front view length and the true
inclinations
4. A line AB 65 mm long has its end A 20 mm above HP AND 25mm in front of VP. END B is 40
mmabove HP and 65mm in front of VP. Draw the projections of AB. find its inclinations with HP
and VP
5. A 100 mm long line is inclined at 45o to the HP and 30
o to the VP. Its mid point is 25mm above
the HP and 35 mm in front of the VP. Draw the projections and locate the traces.
6. A line AB 65 mm long has its end A .15 mm above HP and 15mm in front of VP. It is inclined
AT 550 to HP and 35
0 to VP draw its projections.
UNIT-IV
1. A regular pentagonal plate of side 28mm is placed with one side on HP such that the surface is
inclined at 450 to HP and perpendicular to VP . draw its projections and traces
2. A thin circular metal plate of 48 mm diameter , having its plane vertical and inclined at 400 to VP
Its center is 33mm above HP and 25mm infront of VP. DRAW ITS projections and locate its
traces.
3. A circular plate of negligible thickness and 50mm diameter appears as an ellipse in the front
view, having its major axis 50mm long and minor axis 30mm long. Draw its top view when the
major axis of the ellipse is horizontal.
4. A thin 300-60
0 set square has its longest edge in the V.P and inclined at 30
0 to the H.P. its surface
makes an angle of 450 with the V.P. Draw the projections.
5. A) An equilateral triangle of 40mm side is parallel to VP perpendicular to HP. Draw its projections
when one of the sides is (i) Perpendicular to HP. (ii) Parallel to HP. (iii) Inclined 450 to HP.
b) Draw the projections of a circle of 55 mm diameter having the end A of a diameter AB in the
HP., the end B in the VP, and the surface inclined at 300 to the HP and one of its diameter at
600 to the VP.
6. a) A plate having shape of an isosceles triangle has base 50 mm long and altitude 70 mm. It is
so placed that in the front view it is seen as an equilateral triangle of 50 mm sides one side
inclined at 450 to xy. Draw its top view.
b) Draw the projections of a circle of 5 cm diameter, having its plane vertical and inclined at 300 to
VP. Its center is 3 cm above the HP and 4 cm in front of the VP.
UNIT-V
1. Draw the projections of a cylinder,base 30mm diameter and axis 40mm long resting with a point
of its base circle on HP such that the axis is making an angle of 300 with HP and parallel to VP.
2. A pentagonal prism side of base 25mm and axis 50mm long resting with one of its edges on HP
such that the base containing that edge makes an angle of 300 to HP and its axis is parallel to
VP.Draw its projections.
3. Draw the projections of a cylinder 75mm diameter and 100 mm long, lying on the ground with its
axis inclined at 300 to the V.P and parallel to the ground.
4. Draw the projections of a cone, base 75 mm diameter and axis 100 mm long, lying on the H.P
on one of its generators with the axis parallel to the V.P.
5. Draw the projections of a pentagonal prism, base 25mm, side and axis 50mm long. Resting on
one of its rectangular faces on the H.P. with the axis inclined at 450 to the V.P
6. A hexagonal pyramid, base 25 mm side and axis 50 mm long has an edge of its base on the
ground. Its axis is inclined at 30° to the ground and parallel to the VP. Draw its projections.
DEPARTMENT OF COMPUTER SCIENCE
QUESTION BANK ( Academic Year 2018-19)
---------------------------------------------------------------------------------------------------------------------
UNIT – I
1. (a) Explain about various input and output devices of a computer (5m)
(b). What is a Computer? Explain the various Computer Parts (5m)
2. (a) Explain about algorithm and its characteristics (5m)
(b). Discuss the steps in program development (5m)
3. (a) Explain about Application and System Software (5m)
(b). Explain about Procedure oriented programming languages in detail (5m)
4. (a) Explain about Object Oriented Programming languages in detail(5m)
(b) Discuss about different computer languages with examples (5m)
UNIT – II
1. (a) Write the various rules for an identifier in C-language with suitable examples? (5m)
(b) Explain the various data types available in C-language (5m)
2. (a) Explain all arithmetic operations available in C-language with examples (5m)
(b) Explain about type conversion and casting with suitable examples. Also write the type
conversion rules in C-language (5m)
3. (a) Explain about Assignment operator in C-language with suitable examples (5m)
(b) Explain about conditional operator in C- language with suitable examples (5m)
4. (a) Explain about the various Unary operators available in C-language with suitable examples
(5m)
(b) Explain about the various logical operators available in C-language with suitable examples
(5m)
5. (a) Explain about the various relational and equality operators available in C-language with
suitable examples (5m)
(b) Write a constant? Explain the different constants available in C-language with suitable
Examples (5m)
Course: B.Tech Branch: ECE
Subject: COMPUTER PROGRAMMING Regulation: R16
DEPARTMENT OF COMPUTER SCIENCE
UNIT – III
1. (a) Write a C program that performs all arithmetic operations based on user choice using switch
case? (5m)
(b). Explain different looping statements with syntax and examples (5m)
2. (a) Differentiate between else-if and switch? Explain with an example? (5m)
(b) Write a C-program to swap the given two numbers without using a third variable(5m)
3. (a) What is the purpose of do-while and while loops? Discuss about their usage. Distinguish
between both of them (5m)
(b) Write a C program to find the sum of following series: 12+2
2+3
2+4
2+…+n
2 (5m)
4. (a) Write a program to verify whether the given number is prime number or not? (5m)
(b) Write a program to find the single digit sum of a given number (5m)
5. Explain the switch –case-default control statements with various options along with suitable
examples (10m)
UNIT – IV
1. (a) Explain the terms user defined functions and predefined functions(5m)
(b) Differentiate between iteration and recursion (5m)
2. (a) Explain the auto and register storage classes with suitable examples(5m)
(b) Explain the static and extern storage classes with suitable examples(5m)
3. (a) Explain about recursive functions with suitable examples (5m)
(b) Write a C program to find factorial of a given number using recursive functions(5m)
4. (a) Write a program to for Tower’s of Hanoi using recursion? (5m)
(b) Define a function for determining whether a given character is a vowel or not(5m)
5. (a) Write a program to find largest of three given numbers using functions? (5m)
(b) Explain about function prototypes and function scope rules with suitable examples? (5m)
UNIT – V
1. (a) Write a program to check whether the given string is palindrome or not? (5m)
(b) What is an array? How to initialize 1D and 2D arrays? Discuss about the advantage and
disadvantages of arrays (5m)
2. (a) Write a program to find the biggest and smallest elements of an array with their positions?
(5m)
(b) Explain about 2-Dim array initialization in C-language with suitable examples(5m)
3. (a) Write a program to traverse a single dimensional array (5m)
(b) Write a C program for matrix multiplication with sufficient conditions(5m)
4. (a) Write a program to create an array of 10 cells. Accept the data into the first 9 cells and store
the sum in the 10th
cell using functions (5m)
(b). Write a program to traverse a two dimensional array (5m)
DEPARTMENT OF COMPUTER SCIENCE
5. Explain the following string functions with suitable examples or programs :
strcat () , strcmp(), strcpy(), strlen(), strrev(). (10 m)
UNIT – VI
1 .(a) Write about call by value mechanism with suitable example? (5m)
(b). Write about call by reference mechanism with suitable example? (5m)
2. (a) Explain about malloc() and calloc() functions with suitable example(5m)
(b). Explain about realloc() and free() functions with suitable example(5m)
3. (a) Explain about dangling memory and memory leak with suitable examples (5m)
(b). Write a C program to explain the concept of pointer arithmetic 5m)
4. (a) Define a structure in C language? Explain the storage of structure elements in memory(5m)
(b). Define a union in C language? Explain the storage of union elements in memory (5m)
5. (a) Explain about fread() and fwrite() functions with suitable example? (5m)
(b).Write a program to merge any two files? (5m)
6. (a). Write a program to display the examination results of a student by using bit fields? (5m)
(b). Explain about typedef with suitable example (5m)
XXXXXXXXX
1
English QUESTION BANK FOR I B.TECH ECE
Unit-I
1) What is meant by “Human Resource”? Give examples from various professions. (5M)
2) What exactly is the problem with Mr. Neave‟s ideal family? (3M)
3) Fill the blank with the suitable verb.(2M)
a) The train had________(leave) before he reached the railway station.
b) She ______ (read) since morining
1) Write in detail how Srinivasa Ramanujan is an invaluable human resource, particularly to the
field of Mathematics. (5M)
2) What is the underlying irony in the story, „An Ideal Family‟? (3M)
3) Fill the blank with the suitable verb. (2M)
a) Summer________(come) after winter.
b) Malathi ______ (pay) the fee before the teacher announced.
1) Why human resource is considered invaluable? (5M)
2) Do you think Mr. Neave‟s family is an ideal family? Why/why not? (3M)
3) Fill the blank with the suitable verb. (2M)
a) They______ (watch) T.V when the postman came.
b) She_____ (see) the movie many times.
1) How is human resource the backbone of every industry?5M)
2) What is the shadowy meaning Mr.Neave has at the end of the story? (3M)
3) Fill the blank with the suitable verb. (2M)
a) She_____(write) a letter when her father came.
b) Kiran_______ (go) to the canteen just now.
1) Define the role of human resources in any industry. How can this resource strengthen the
industry? (5M)
2) Why do people call Mr.Neave‟s family an ideal family? (3M)
3) Fill the blank with the suitable verb. (2M)
a) She______ (wait) for the principal for two hours when he came.
b) Kiran_____ (come) to the college on foot.
2
1) Justify Indian government‟s decision of celebrating 22 December as National Mathematics Day.
(5M)
2) Who do you think is the cause for conditions in Mr. Neave‟s family? Justify your answer with
suitable instances from the text. (3M)
3) Fill the blank with the suitable verb. (2M)
a) They _____ (visit) Taj Mahal one year ago.
b) Krishna_______ (loss) the key just now.
Unit -II
1) You are an official from the Finance Department and are not very enthusiastic about
spending money on road safety schemes. You feel that a few road safety posters on the
main roads are sufficient. Explain your views to support your statement. (5M)
2) What is the central theme of the story, „War‟? (3M)
3) Write the meanings of the following phrases. (a) backdrop of pines (b) mists of the fall
morning. (2M)
1) You are an official from the Transport Department and want to spend money on
improving road intersections and on a new bypass. Explain your views to support your
statement. (5M)
2) What are the different views that passengers articulate regarding war? (3M)
3) Write the meanings of the following phrases. (a) checkerboard of farms (b) white clouds
of bloom (2M)
1) You are an official from the police department. You want to double the number of traffic
policemen so that laws can be enforced with on-the-spot fines. Explain your views to
support your statement. (5M)
2) What is the message that the author wishes to convey through this story? (3M)
3) Write the meanings of the following phrases. (a) backdrop of pines (b) mists of the fall
morning.(2M)
3
1) You are a representative of the Citizen‟s Welfare Association, and would like to
introduce a road safety training week in all school, colleges, factories and offices.
Explain your views to support your statement. (5M)
2) What are the fat man‟s feelings towards sending children to war? (3M)
3) Write the meanings of the following phrases. (a) checkerboard of farms (b) white clouds
of bloom(2M)
1) Explain the reasons why there is a need to enforce traffic rules and regulations
strictly.(5M)
2) Summarize the story, „War‟. (3M)
3) Write the meanings of the following phrases. (a) backdrop of pines (b) mists of the fall
morning. (2M)
1) Traffic hazards are increasing day after day. Suggest some ways by which these may be
countered. (5M)
2) Why was the woman who entered the carriage upset? How are the other passengers
affected by war? (3M)
3) Write the meanings of the following phrases. (a) checkerboard of farms (b) white clouds
of bloom(2M)
Unit-III
1) Does consumption of bio-mass affect forest resources? How? (5M)
2) What is the unexpected twist in the story, „Verger‟? (3M)
3) Write the synonyms for (a) Stringent (b) Hazard (2M)
1) It is sometime towards the end of the twenty-first century. Imagine you are living in an
Indian village. Write a paragraph describing what the village looks like under the impact
of technology over the years. (5M)
2) Explain the character of Albert Foreman. (3M)
3) Write the synonyms for (a) change (b) Hazard (2M)
4
1) What is the advantage of the new „Print and copy online‟ service? (5M)
2) Narrate the discussion between bank manager and Foreman. (3M)
3) Write the synonyms for (a) educate (b) mysterious (2M)
1) Elaborate the statement- „Mass production or production by the masses. (5M)
2) How does Mr. Foreman overcome all his obstacles in life? (3M)
3) Write the synonyms for (a) danger (b) advantage (2M)
1) Write the benefits of technology or on the problems created by it. (5M)
2) What is the central theme of the story, „Verger‟? (3M)
3) Write the synonyms for (a) direct (b) wander (2M)
1) Modern technology is a friend or foe. Explain with reasons? (5M)
2) How does Foreman expand his business? (3M)
3) Write the synonyms for (a) capital (b) urge (2M)
Unit-IV
1) Write a short note on Pedal Power. (5M)
2) Describe Mriganko Babu‟s reaction to the scarecrow? (3M)
3) Write the noun forms of (i) electricity (II) ferment.
1) Why is such intensive research being carried out to discover viable alternative sources of
energy? (5M)
2) Who was Mriganko Shekhar Mukhopadhyay? (3M)
3) Write the noun forms of (i) compose (II) protect.
1) List the problems involved in producing and using electricity in India. (5M)
2) Who was the scarecrow? (3M)
3) Write the noun forms of (i) construct (II) ferment.
5
1) You are a journalist writing a feature article on a village that makes use of alternative
energy sources. The people of the village are proud of their self-reliance. Prepare a list
of questions to ask the sarpanch of the village as well as a few other villagers. (5M)
2) Justify the title of the story „The Scarecrow”. (3M)
3) Write the noun forms of (i) pollute (II) destroy
1) What are DRE mini-grids? Describe some of its features and explain how they are
different from commercial grids. (5M)
2) Narrate the incident that occurred between Abhiram and Mriganko Babu? (3M)
3) Write the noun forms of (i) conserve (II) irrigate.
1) Write a short note on Solar Energy. (5M)
2) How did Mriganko Babu pass time while waiting for his driver? (3M)
3) Write the noun forms of (i) absorb (II) stimulate.
Unit- V
1) You and Student B live next door to each other. B is fond of dogs and has an Alsatian as
a pet. You want to tell B that it is not good to have pets in the house. Think of the points
you can make. (5M)
2) What is the message that the author conveys through the text- A Village Lost to the
Nation‟? (3M)
3) Write the synonyms for (i) annoying (ii) incessantly (2M)
1) Write a paragraph on Global warming. (5M)
2) What were the feelings of the author‟s parents regarding their village? (3M)
3) Write the synonyms for (i) catalyze (ii) resist(2M)
1) Humans are responsible for the destruction of animal species, both directly and
indirectly. Explain the statement. (5M)
2) How are the villages affected by the Hirakud Dam? (3M)
3) Write the synonyms for (i) invade (ii) umpteen(2M)
6
1) Write a paragraph on deforestation. (5M)
2) What were the people forced to do because their homes were submerged? (3M)
3) Write the synonyms for (i) yell (ii) whisper(2M)
1) In what ways are birds useful to humans? In what ways are they harmful?
2) What were the author‟s feelings for his lost home? (3M)
3) Write the synonyms for (i) teach (ii) danger(2M)
1) What are the efforts being made at present to preserve wildlife? (5M)
2) What is the theme of the passage- „A Village Lost to the Nation‟? (3M)
3) Write the synonyms for (i) build (ii) copious(2M)
Unit -VI
1) Imagine that you are the chief engineer of a company. Write a set of safety measures to be
adopted to avert industrial accidents. (5M)
2) What is Martin Luther King‟s second achievement according to the author? (3M)
3) Write the antonyms for (i) aggressive (ii) recluse (2M)
1) What are the objectives of training to employees? (5M)
2) What are the author‟s views on racism? (3M)
3) Write the antonyms for (i) collect (ii) diurnal (2M)
1) Write a paragraph about the importance of computer training today. (5M)
2) What is James Baldwin‟s view about African history? (3M)
3) Write the antonyms for (i) gather (ii) vanish (2M)
1) Describe the process of sedimentation. (5M)
2) According to author, Martin Luther King achieved two things. What is his first
achievement? (3M)
7
3) Write the antonyms for (i) tamed (ii) destroy (2M)
1) What is the need for training (i) a new employee, and (ii) an employee already in service?
(5M)
2) In what way did Mahatma Gandhi influence Martin Luther King? (3M)
3) Write the antonyms for (i) slimy (ii) detail (2M)
1) Write the unspoken rules of civility at the workplace. (5M)
2) How did Martin Luther King embrace his African roots? (3M)
3) Write the antonyms for (i) emit (ii) accessible (2M)
QUESTION BANK
SUBJECT: MATHEMATICS-I
UNIT- I
1.(a) Solve the D.E 𝒙𝟐 + 𝒚𝟐 𝒅𝒙 + 𝟐𝒙𝒚 𝒅𝒚 = 𝟎. 𝟓𝑴
(b) Find the Orthogonal trajectories of the family of circles 𝒙𝟐 + 𝒚𝟐 + 𝟐𝒇𝒚 + 𝟏 = 𝟎, 𝒇 being
the parameter. 𝟓𝑴
2.(a) Solve the D.E 𝒚 𝒙𝟒𝒚𝟒 + 𝒙𝟐𝒚𝟐 + 𝒙𝒚 𝒅𝒙 + 𝒙 𝒙𝟒𝒚𝟒 − 𝒙𝟐𝒚𝟐 + 𝒙𝒚 𝒅𝒚 = 𝟎 𝟓𝑴
(b) Find the orthogonal trajectory of 𝒓 = 𝒂 𝒔𝒆𝒄𝜽 + 𝒕𝒂𝒏𝜽 𝟓𝑴
3.(a) Solve 𝒅𝒚
𝒅𝒙+ 𝒙𝒔𝒊𝒏𝟐𝒚 = 𝒙𝟑𝒄𝒐𝒔𝟐𝒚 . 𝟓𝑴
(b) Find the orthogonal trajectory of 𝒓𝟐 = 𝒂𝟐𝒔𝒊𝒏𝟐𝜽 𝟓𝑴
4.(a) The temperature of a cup of coffee is 𝟗𝟐°𝑪, when freshly poured the room temperature
being 𝟐𝟒°𝑪. In one minute it was coaled to 𝟖𝟎°𝑪. How long a period
must elapse, before the temperature of the cup becomes 𝟔𝟓°𝑪. 𝟓𝑴
(b) The number of N of bacteria in a culture grew at a rate proportional to N. The value of N was initially 100 and increased to 332 in one hour. What was the value of N after 3/2 hours? 𝟓𝑴
5.(a) Find the orthogonal trajectory of 𝒓 =𝟐𝒂
𝟏+𝒄𝒐𝒔𝜽 . 𝟓𝑴
(b) Suppose that an object is heated to 𝟑𝟎𝟎° 𝑭 and allowed to cool in a room maintained at
𝟖𝟎° 𝑭. If after 10 minutes, the temperature of the object is 𝟐𝟓𝟎° 𝑭, what will be its temperature after 20 minutes? 𝟓𝑴
UNIT-II
1. (a) Solve 𝑫𝟐 + 𝟗 𝒚 = 𝒄𝒐𝒔𝒆𝒄 𝟑𝒙 by the method Variation of parameters. 𝟓𝑴
(b) Solve 𝑫𝟐 + 𝟏 𝒚 = 𝒔𝒆𝒄𝟐𝒙 by the method Variation of parameters. 𝟓𝑴 2. S olve 𝑫𝟑 + 𝟏 𝒚 = 𝒄𝒐𝒔 𝟐𝒙 − 𝟏 + 𝒙𝟐𝒆−𝒙 𝟏𝟎𝑴 3. (a) Solve 𝑫𝟐 + 𝟐𝑫 + 𝟏 𝒚 = 𝒙𝒄𝒐𝒔𝒙. 𝟓𝑴
(b) Solve 𝑫𝟐 + 𝟐 𝒚 = 𝒆𝒙𝒄𝒐𝒔𝟐𝒙 + 𝒙𝟐𝒆𝟑𝒙. 𝟓𝑴 4. (a) Solve 𝒚′′ − 𝟐𝒚′ + 𝟐𝒚 = 𝒆𝒙 + 𝒄𝒐𝒔𝒙 + 𝒙𝟐. 𝟓𝑴 (b) Solve 𝒚′′ − 𝟐𝒚′ + 𝒚 = 𝒆𝒙 𝒙 𝒔𝒊𝒏𝒙. 𝟓𝑴 5 .(a) Solve 𝑫𝟐 + 𝟒 𝒚 = 𝒄𝒐𝒔𝟐𝒙 + 𝒄𝒐𝒔𝒉𝟑𝒙. 𝟓𝑴
(b) Solve 𝑫𝟐 + 𝟐𝑫 + 𝟏 𝒚 = 𝒆−𝒙 + 𝒙 + 𝒔𝒊𝒏𝟐𝒕. 𝟓𝑴
6. (a) The charge q(t) on the capacitor is giving by D.E 𝟏𝟎 𝒅𝟐𝒒
𝒅𝒕𝟐+ 𝟏𝟐
𝒅𝒒
𝒅𝒕+ 𝟏𝟎𝟎𝟎𝒒 = 𝟏𝟕 𝒔𝒊𝒏𝟐𝒕
.At time zero the current in zero and the charge on the capacitor is 1/2000 coulomb. Find the charge on the capacitor for t >0 . 𝟓𝑴 (b) In an L-C-R circuit, the charge q on a plate of the condenser is given by
𝑳 𝒅𝟐𝒒
𝒅𝒕𝟐+ 𝑹
𝒅𝒒
𝒅𝒕+
𝒒
𝑪= 𝑬 𝒔𝒊𝒏𝝎𝒕, 𝒘𝒉𝒆𝒓𝒆 𝒊 =
𝒅𝒒
𝒅𝒕. The circuit is tuned to resonance so that
𝝎𝟐 = 𝟏
𝑳𝑪 .If 𝑹𝟐 <
𝟒𝑳
𝑪 𝒂𝒏𝒅 𝒒 = 𝟎, 𝒊 = 𝟎 𝒘𝒉𝒆𝒏 𝒕 = 𝟎, show that
𝒒 =𝑬
𝑹𝑾 −𝒄𝒐𝒔𝝎𝒕 + 𝒆
−𝑹𝒕
𝟐𝑳 𝒄𝒐𝒔𝒑𝒕 +𝑹
𝟐𝑳𝑷 𝒔𝒊𝒏𝒑𝒕 . 𝟓𝑴
UNIT-III
1. (a) Using the expression 𝒔𝒊𝒏 𝒙 = 𝒙 − 𝒙𝟑
𝟑!+
𝒙𝟓
𝟓!−
𝒙𝟒
𝟕!+ ⋯… .. show that
𝑳 𝒔𝒊𝒏 𝒕 = 𝝅
𝟐 𝒔𝟑/𝟐 𝒆−𝟏/𝟒𝒔 𝟓𝑴
(b) (i) Show that the function 𝒇 𝒕 = 𝒕𝟐 is of exponential order 3. 𝟐𝑴
(ii) Find the Inverse Laplace Transform of 𝒔−𝟏
𝒔𝟐+𝟓𝟐 𝟑𝑴
2. (a) Show that 𝒔𝒊𝒏𝟐𝒕+𝒔𝒊𝒏𝟑𝒕
𝒕𝒆𝒕
∞
𝟎 𝒅𝒕 =
𝟑𝝅
𝟒 . 𝟓𝑴
(b) Evaluate 𝒆−𝟓𝒕 𝜹 𝒕 − 𝟐 𝒅𝒕.∞
𝟎 𝟓𝑴
3.(a) Solve the D.E 𝒚′′ − 𝟔𝒚′ + 𝟗𝒚 = 𝒕𝟐𝒆𝟑𝒕 if 𝒚 𝟎 = 𝟐, 𝒚′ 𝟎 = 𝟔 using Laplace transforms method. 𝟓𝑴
(b) Solve the D.E 𝒚′′ + 𝟐𝒚′ + 𝟓𝒚 = 𝟖𝒔𝒊𝒏𝒕 + 𝟒𝒄𝒐𝒔𝒕, if 𝒚 𝟎 = 𝟏, 𝒚′ 𝝅
𝟒 = 𝟐 using
Laplace transforms method. 𝟓𝑴
4. (a) Find 𝒊 𝑳−𝟏 𝟏
(𝒔𝟐+𝟏)(𝒔𝟐+𝟗) 𝒊𝒊 𝑳−𝟏
𝟑
(𝒔−𝝅
𝟐)𝟒 𝟓𝑴
(b) Find 𝒊 𝑳−𝟏 𝒔
𝒔𝟒+𝒔𝟐+𝟏 𝒊𝒊 𝑳−𝟏 ( 𝒕 −
𝟏
𝒕)𝟑 𝟓𝑴
5. (a) Solve the following differential equation by the transform method
𝑫𝟐 + 𝒏𝟐 𝒙 = 𝒂 𝒔𝒊𝒏 𝒏𝒕 + 𝜶 , 𝒙 = 𝟎 𝒂𝒕 𝒕 = 𝟎 . 𝟓𝑴
(b) Find 𝑳[𝒇 𝒕 ] where 𝒇 𝒕 = 𝒆𝒕 𝒊𝒇 𝟎 < 𝒕 < 1
𝟎 𝒊𝒇 𝒕 > 1 . 𝟓𝑴
6. (a) Find 𝑳−𝟏 𝟏
(𝒔𝟐(𝒔𝟐+𝟏)𝟐 using convolution theorem. 𝟓𝑴
(b) State convolution theorem and use it to evaluate 𝑳−𝟏 𝟏
𝒔𝟐+𝟒𝒔+𝟏𝟑 𝟐 𝟓𝑴
UNIT-IV
1.(a) Find 𝒙𝝏𝒖
𝝏𝒙+ 𝒚
𝝏𝒖
𝝏𝒚 if 𝒖 =
𝒙𝟑𝒚𝟑
𝒙𝟑+𝒚𝟑 𝟓𝑴
(b) Find the extreme values of 𝒇 𝒙,𝒚 = 𝒙𝟑 + 𝟑𝒙𝒚𝟐 − 𝟑𝒙𝟐 − 𝟑𝒚𝟐 + 𝟕. 𝟓𝑴
2. (a) Expand 𝒇 𝒙,𝒚 = 𝒙𝒚𝟐 + 𝒄𝒐𝒔 𝒙𝒚 in powers of 𝒙 − 𝟏 , 𝒚 −𝝅
𝟐 upto second
degree term. 𝟓𝑴 (b) Discuss the Maxima and Minima of 𝒇 𝒙,𝒚 = 𝒙𝟑𝒚𝟐 𝟏 − 𝒙 − 𝒚 . 𝟓𝑴
3. (a) Show that the functions u= 𝒙
𝒚, 𝒗 =
𝒙+𝒚
𝒙−𝒚 are functinally dependent and
find the relation between them. 𝟓𝑴 (b) Find the dimensions of a rectangular parallelopipid box open at the top of max capacity whose surface area is 108 sq inches. 𝟓𝑴
4.(a)Prove that 𝐽𝐽′ = 1 If 𝒖 =𝒚𝒛
𝒙, 𝒗 =
𝒛𝒙
𝒚, 𝒘 =
𝒙𝒚
𝒛. 𝟓𝑴
(b) Find the point in the plane 𝟐𝒙 + 𝟑𝒚 − 𝒛 = 𝟓 which is nearest to the origin. 𝟓𝑴
5. (a) If u = 𝒇 𝒓 and x= r𝐜𝐨𝐬 𝜽, 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 prove that 𝝏𝟐𝒖
𝝏𝒙𝟐 +𝝏𝟐𝒖
𝝏𝒚𝟐 = 𝒇′′ 𝒓 +𝟏
𝒓 𝒇′(𝒓).
𝟓𝑴
(b) Find the maximum and minimum values of 𝒇 = 𝟑𝒙𝟒 − 𝟐𝒙𝟑 − 𝟔𝒙𝟐 + 𝟔𝒙 + 𝟏.
𝟓𝑴
6. (a) If 𝒙 = 𝒗𝒘, 𝒚 = 𝒘𝒖 ,𝒛 = 𝒖𝒗 ,𝒙 = 𝒓𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝝋, 𝒚 = 𝒓𝒔𝒊𝒏𝜽𝒔𝒊𝒏𝝋, 𝒛 = 𝒓𝒄𝒐𝒔𝜽
Then find 𝑱 𝒙,𝒚,𝒛
𝒓,𝜽,𝝋 . 𝟓𝑴
(b) Expand 𝒆𝒙 𝐬𝐢𝐧 𝒚 in powers of 𝒙,𝒚 . 𝟓𝑴
UNIT-V
1. (a) Form the Partial differential equation by eliminating arbitrary constants from
𝒊 𝒛 = 𝒂𝒙 + 𝒃𝒚 + 𝒂𝒃, 𝒊𝒊 𝒛 = 𝒂𝒙 + 𝒃𝒚 + 𝒂𝟐 + 𝒃𝟐 𝟓𝑴
(b) Form the Partial differential equation by eliminating 𝒇 & 𝑔 from
𝒛 = 𝒇 𝒚 + 𝒈(𝒙 + 𝒚). 𝟓𝑴
2. (a) Solve the PDE 𝒙 𝒚 − 𝒛 𝒑 + 𝒚 𝒛 − 𝒙 𝒒 = 𝒛(𝒙 − 𝒚). 𝟓𝑴
(b) Solve the PDE (𝒑
𝟐+ 𝒙)𝟐 + (
𝒒
𝟐+ 𝒚)𝟐 = 𝟏. 𝟓𝑴
3. (a) Solve 𝒙 + 𝟐𝒛 𝒑 + 𝟒𝒛 − 𝒚 𝒒 = 𝟐𝒙 + 𝒚 . 𝟓𝑴
(b) Solve the PDE 𝒛𝟐 𝒑𝟐 + 𝒒𝟐 = 𝟏. 𝟓𝑴
4. (a) Solve 𝒙 𝒚𝟐 + 𝒛 𝒑 − 𝒚 𝒙𝟐 + 𝒛 𝒒 = 𝒛 𝒙𝟐 − 𝒚𝟐 . 𝟓𝑴
(b) Solve the PDE 𝒑𝟐𝒒𝟐 + 𝒙𝟐𝒚𝟐 = 𝒙𝟐𝒒𝟐 𝒙𝟐 + 𝒚𝟐 . 𝟓𝑴
5. (a) Solve the PDE 𝒙𝟐𝒑𝟐 + 𝒚𝟐𝒒𝟐 = 𝟏. 𝟓𝑴
(b) Solve the PDE 𝒚 + 𝒛 𝒑 − 𝒛 + 𝒙 𝒒 = 𝒙 − 𝒚 𝟓𝑴
6. (a) Solve the PDE 𝒑𝐜𝐨𝐬 𝒙 + 𝒚 + 𝒒𝒔𝒊𝒏 𝒙 + 𝒚 = 𝒛. 𝟓𝑴
(b) Solve the PDE 𝒙𝟐
𝒑+
𝒚𝟐
𝒒= 𝒛. 𝟓𝑴
UNIT-VI
1. (a) Solve 𝑫𝟐 + 𝟐𝑫𝑫′ − 𝟖𝑫′𝟐 𝒛 = 𝟐𝒙 + 𝟑𝒚. 𝟓𝑴
(b) Solve 𝑫𝟐 + 𝑫𝑫′ − 𝟔𝑫′𝟐 𝒛 = 𝒙𝟐𝒔𝒊𝒏 𝒙 + 𝒚 . 𝟓𝑴
2. (a) Solve 𝑫𝟐 − 𝑫𝑫′ 𝒛 = 𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝟐𝒚. 𝟓𝑴
(b) Solve 𝑫𝟐 − 𝑫𝑫′ − 𝟐𝑫 𝒛 = 𝒔𝒊𝒏 𝟑𝒙 + 𝟒𝒚 . 𝟓𝑴
3. (a) Solve 𝝏𝟐𝒛
𝝏𝒙𝟐 − 𝟔𝝏𝟐𝒛
𝝏𝒙𝝏𝒚+ 𝟗
𝝏𝟐𝒛
𝝏𝒚𝟐 = 𝟏𝟐𝒙𝟐 + 𝟑𝟔𝒙𝒚 + 𝒆𝒙+𝒚 𝟓𝑴
(b) Solve 𝝏𝟐𝒛
𝝏𝒙𝟐 −𝝏𝟐𝒛
𝝏𝒙𝝏𝒚− 𝟐
𝝏𝟐𝒛
𝝏𝒚𝟐 = (𝒚 − 𝟏)𝒆𝒙 𝟓𝑴
4.(a) Solve 𝑫𝟑 − 𝑫′𝟑 𝒛 = 𝒙𝟑𝒚𝟑. 𝟓𝑴
(b) Solve 𝑫𝟐 − 𝟐𝑫𝑫′ 𝒛 = 𝒆𝟐𝒙 + 𝒙𝟑𝒚. 𝟓𝑴
5.(a) Solve 𝑫𝟐 − 𝟐𝑫𝑫′ + 𝑫′𝟐 𝒛 = 𝟐𝒙𝒄𝒐𝒔𝒚. 𝟓𝑴
(b) Solve 𝑫𝟐 − 𝑫′𝟐 𝒛 = 𝒄𝒐𝒔 𝒙 + 𝒚 . 𝟓𝑴
6.(a) Solve 𝝏𝟐𝒛
𝝏𝒙𝟐 + 𝟐𝝏𝟐𝒛
𝝏𝒙𝝏𝒚+
𝝏𝟐𝒛
𝝏𝒚𝟐 = 𝟐𝒔𝒊𝒏𝒚 − 𝒙𝒄𝒐𝒔𝒚. 𝟓𝑴
(b) Solve 𝝏𝟑𝒛
𝝏𝒙𝟑 − 𝟑𝝏𝟑𝒛
𝝏𝒙𝟐𝝏𝒚+ 𝟒
𝝏𝟑𝒛
𝝏𝒚𝟑 = 𝒆𝒙+𝟐𝒚. 𝟓𝑴
QUESTION BANK
SUBJECT: NUMERICAL METHODS & COMPLEX VARIABLES
UNIT-I
1. a) Find the Real root of the equation 𝒙𝟐 − 𝒙 − 𝟒 = 𝟎 using iteration method (5M)
b) Find the Real root of the equation 𝒆𝟐𝒙 − 𝒆𝒙 − 𝟐 = 𝟎 using Newton Raphson method (5M)
2 (a) Using Newton-Rapson method find the root of the equation 2x-log10 x =7 correct to
four decimal places. (5M)
b) Find the Real root of the equation 𝒙𝒆𝒙 = 𝟐 using Bisection method. (5M) 3 (a) Find the Real root of the equation 𝐱 + 𝐥𝐨𝐠𝟏𝟎 𝒙 − 𝟐 = 𝟎 using false position method (5M) (b) Solve 𝐱𝟑 − 𝟐𝐱 − 𝟓 = 𝟎,for a positive root by iteration method. (5M) 4 (a) Using Newton – Raphson method, find a root of the equation 2x- 3sinx =5 near x=5 correct to three decimal places. (5M) (b) Find reciprocal of a real number 19 using Regula falsi method (5M) 5(a) Using Regula-Falsi method, find the root of 𝒙𝟑 − 𝒙 − 𝟐 = 𝟎, over (1, 2) (5M)
(b) By using Newton-Raphson method, find the root of 𝒙𝟒 − 𝒙 − 𝟏𝟎 = 𝟎, correct to three decimal places. (5M)
UNIT-II
CLASS: I B.TECH BRANCH: ECE
1(a) Using Lagrange’s interpolation formulae find the value of y (12) from the data (5M) (b) Find 𝒚 𝟐. 𝟒 using Newton’s backward difference formula the table (5M) 2 (a) Find f (2.5) using Newton’s forward formula for the following table
(5M)
(b) Calculate f(4) from the following table. (5M)
X 0 1 2 5
f(x) 2 3 12 147
3(a) Find 𝒇 𝟏. 𝟕𝟓 𝒊𝒇 𝒇 𝟏. 𝟕 = 𝟓. 𝟒𝟕𝟒, 𝒇 𝟏. 𝟖 = 𝟔. 𝟎𝟓𝟎, 𝒇 𝟏. 𝟗 = 𝟔. 𝟔𝟖𝟔, 𝒇 𝟐 = 𝟕. 𝟑𝟖𝟗. (5M) (b) Using a forward difference formula, find y(5)from the given table (5M)
x 1 6 11 16 21 26
y 5 10 14 18 24 32
4 (a) Using Gauss Backward difference polynomial, find y(5) given that (5M)
X 2 4 6 8 10
Y 5 11 13 15 17
(b) Use Gauss backward interpolation formula to find f(32) given that f (25) = 0.2707, f (30) = 0.3027, f (35) = 0.3386, f (40) = 0.3794. (5M) 5(a) Prove that (𝟏 + ∆)(𝟏 − 𝛁) = 𝟏. (5M) (b) Given that y(3) = 6, y(5) = 24, y(7) = 58, y(9) = 108, y(11) = 174, find x when y = 100 using Lagrange’s formula. (5M)
6(a) Prove that (i) 𝑬𝛁 = 𝛁𝑬 = ∆ (ii) 𝜹𝑬𝟏
𝟐 = ∆ (iii) ∆𝛁 = 𝛁∆ . (5M) (b) If the interval of differencing is unity, find ∆𝒕𝒂𝒏𝒙 𝒕𝒂𝒌𝒆 𝒉 = 𝟏. (5M)
UNIT-III
x 5 7 9 13
y 11 13 18 27
x 1 1.4 1.8 2.2
f(x) 3.49 4.82 5.91 6.5
x 0 1 2 3 4 5
y 0 1 8 21 72 94
1(a) Using modified Euler method solve numerically the equation 𝒅𝒚
𝒅𝒙= 𝒙 − 𝒚 ,y(0)=1 at
𝒙 = 𝟎. 𝟏 , 𝟎. 𝟐 (5M)
(b)Using Euler’s method, solve for y (0.6) from 𝒅𝒚
𝒅𝒙= −𝟐𝒙𝒚 , y(0) =1 using step size 0.2. (5M)
2(a) Given 𝒅𝒚
𝒅𝒙= 𝒙 + 𝒔𝒊𝒏𝒚, y(0) = 1, compute y(0.2) and y(0.4) using Euler’s modified method.
(5M)
(b) Evaluate y(0.2) and y(0.4) correct to three decimals by Taylor’s method if y(x) satisfies
𝒚′ = 𝟏 − 𝟐𝒙𝒚 , 𝒚 𝟎 = 𝟎 (5M)
3(a) Evaluate 𝟏
𝟏+𝒙 𝒅𝒙 𝒘𝒊𝒕𝒉 𝒉 = 𝟎. 𝟏
𝟐
𝟎 by Trapezoidal rule. (5M)
(b) Solve , 𝒚′ = 𝒚 − 𝒙𝟐 𝒚 𝟎 = 𝟏 using Picard’s method up to 4th approx. (5M)
4(a) If 𝒚′ = 𝟐𝒙 − 𝒚, y(1) = 3, find the solution, up to third degree term, using Picard’s
method. (5M)
(b)Evaluate 𝐜𝐨𝐬 𝒙
𝟏+𝒙 𝒅𝒙
𝝅
𝟐𝟎
by (i) Trapezoidal rule (ii) Simpson’s 3/8th Rule (5M)
5(a) Find y (0.1) using 4th order Runge-Kutta method given that , 𝒚′ = 𝒙 + 𝒙𝟐 𝒚 , 𝒚 𝟎 = 𝟏.
(5M)
(b) Use Runge-Kutta 4th order to compute y(1.2) for the equation 𝒚′ =𝒙𝟐+𝒚
𝒙 , 𝒚 𝟏 = 𝟐 (5M)
UNIT – IV:
1(a) Prove that the function 𝒇(𝒛) defined by 𝒇 𝒛 = 𝒙𝟑 𝟏+𝒊 −𝒚𝟑 𝟏−𝒊
𝒙𝟐+𝒚𝟐 𝒛 ≠ 𝟎
𝟎, 𝒛 = 𝟎 is
continuous and the Cauchy-Riemann equations are satisfied at the origin, yet 𝒇′(𝟎) does not exist. (5M) (b) Show that the function 𝒇 𝒛 = 𝒛𝒛 or 𝒛 𝟐 is differentiable but not analytic at 𝒛 = 𝟎. (5M)
2(a) Prove that 𝝏𝟐
𝝏𝒙𝟐+
𝝏𝟐
𝝏𝒚𝟐 𝑹𝒆𝒂𝒍 𝒇(𝒛) 𝟐 = 𝟐 𝒇′(𝒛) 𝟐 where 𝒘 = 𝒇(𝒛) is analytic. (5M)
(b) Prove that 𝒛𝒏, n is a positive integer is analytic and hence find its derivative. ( 5M ) 3(a) Check 𝒖 𝒙, 𝒚 =𝒆−𝒙(𝒙 𝒔𝒊𝒏𝒚 − 𝒚 𝒄𝒐𝒔𝒚) is harmonic or not. If harmonic find its conjugate. (5M)
(b) Show that the function 𝒖 = 𝟐𝐥𝐨𝐠(𝒙𝟐 + 𝒚𝟐) is harmonic and find its harmonic conjugate. (5M)
4(a) If 𝒇 𝒛 = 𝒖 𝒓, 𝜽 + 𝒊 𝒗(𝒓, 𝜽) is differentiable at 𝒛 = 𝒓𝒆𝒊𝜽 ≠ 𝟎 then prove that 𝝏𝒖
𝝏𝒓=
𝟏
𝒓 𝝏𝒗
𝝏𝜽 𝒂𝒏𝒅
𝝏𝒗
𝝏𝒓= −
𝟏
𝒓 𝝏𝒖
𝝏𝜽. (5M)
(b) Show that the function 𝒇 𝒛 = 𝒙𝒚 is not analytic at the origin, although Cauchy-Riemann equations are satisfied at that point. (5M)
6(a) Show that 𝝏𝟐
𝝏𝒙𝟐 +𝝏𝟐
𝝏𝒚𝟐 𝒍𝒐𝒈 𝒇′(𝒛) = 𝟎, where 𝒇(𝒛) is an analytic function. (5M)
(b) Show that 𝒖 𝒙, 𝒚 =𝒆𝟐𝒙(𝒙 𝒄𝒐𝒔𝟐𝒚 − 𝒚 𝒔𝒊𝒏𝟐𝒚) is harmonic and find its harmonic conjugate. (5M)
UNIT – V: 1(a) Evaluate 𝒚𝟐 + 𝟐𝒙𝒚 𝒅𝒙 + 𝒙𝟐 − 𝟐𝒙𝒚 𝒅𝒚 where C is the boundary of the region by
𝒚 = 𝒙𝟐 𝒂𝒏𝒅 𝒙 = 𝒚𝟐. 𝟓𝐌
(b) Evaluate 𝒆𝟐𝒛
𝒛−𝟏 (𝒛−𝟐)𝒅𝒛 where c is the circle 𝒛 = 3. 𝟓𝐌
2(a) Evaluate 𝟑𝒙𝟐 + 𝟓𝒚 + 𝒊(𝒙𝟐 − 𝒚𝟐 𝒅𝒛(𝟏,𝟏)
(𝟎,𝟎) along 𝒚𝟐 = 𝒙 . 𝟓𝐌
(b) Evaluate 𝒛𝟒
𝒛+𝟏 (𝒛−𝒊)𝟐 𝒅𝒛 where c is the Ellipse 𝟗𝒙𝟐 + 𝟒𝒚𝟐 = 𝟑𝟔. 𝟓𝐌
3(a) Find the Laurent or Taylor series expansion of 𝟏
𝒛𝟐−𝟒𝒛+𝟑 for
(i) 1 < 𝒁 < 3 (ii) 𝒛 < 1 𝟓𝐌
(b) Evaluate 𝒛𝟐𝟏+𝒊
𝟎𝒅𝒛 along 𝒚 = 𝒙𝟐. 𝟓𝐌
4(a) Obtain Taylor’s or Laurent series to represent the function 𝒛𝟐−𝟏
𝒛+𝟐 (𝒛+𝟑) in the region
(i) 𝒛 < 2 (ii) 2 < 𝒁 < 3 𝟓𝐌
(b) Evaluate 𝒆𝟐𝒛
(𝒛+𝟏)𝟒𝒅𝒛 around 𝒄 ∶ 𝒛 − 𝟏 = 𝟑 𝟓𝐌
5(a) Evaluate 𝒛+_𝟒
𝒛𝟐+𝟐𝒛+𝟓 where c is the circle (i) 𝒛 = 𝟏 (ii) 𝒛 + 𝟏 − 𝒊 = 𝟐
(iii) 𝒛 + 𝟏 + 𝒊 = 𝟐 𝟓𝐌
(b) Evaluate 𝒍𝒐𝒈𝒛
(𝒛−𝟏)𝟑𝒅𝒛 where 𝒄 ∶ 𝒛 − 𝟏 =
𝟏
𝟐 using Cauchy’s integral formula. 𝟓𝐌
6(a) Evaluate 𝒆𝒛
𝒛𝟑 +𝒛𝟒
(𝒛+𝒊)𝟐 𝒅𝒛 where 𝒄 ∶ 𝒛 = 𝟐 using Cauchy’s integral formula . 𝟓𝐌
(b) Expand 𝒇 𝒛 =𝟏
𝒛𝟐−𝟑𝒛+𝟐 in the region (i) 0 < 𝒁 − 𝟏 < 1 (ii) 1 < 𝒁 < 2 𝟓𝐌
7(a) Obtain Laurent’s expansion for 𝒇 𝒛 =𝟏
𝒛+𝟐 𝟏+𝒛 in 1 < 𝒁 < 2 𝟓𝐌
(b) Find the Laurent series expansion of the function 𝒇 𝒛 =𝒛𝟐−𝟔𝒛−𝟏
𝒛−𝟏 𝒛−𝟑 (𝒛+𝟐) in the region
3 < 𝒁 + 𝟐 < 5. 𝟓𝐌
UNIT – VI:
1(a) Find the residue of 𝒛𝟐−𝟐𝒛
𝒛+𝟏 𝟐(𝒛𝟐+𝟏) at each pole. 𝟓𝐌
(b) Evaluate 𝟒−𝟑𝒛
𝒛 𝒛−𝟏 (𝒛−𝟐)𝒅𝒛 where c is the circle 𝒛 =
𝟑
𝟐 using residue theorem. 𝟓𝐌
2(a) Evaluate (𝟐𝒛+𝟏)𝟐
𝟒𝒛𝟑+𝒛𝒅𝒛 where C is the circle 𝒛 = 𝟏 using residue theorem. 𝟓𝐌
(b) Evaluate 𝟐𝒆𝒛
𝒛(𝒛−𝟑)𝒅𝒛 where C is the circle 𝒛 = 𝟐 using residue theorem. 𝟓𝐌
3(a) Show that 𝒅𝜽
𝒂+𝒃𝒔𝒊𝒏𝜽=
𝒅𝜽
𝒂+𝒃𝒄𝒐𝒔𝜽=
𝟐𝝅
𝟎
𝟐𝝅
𝟎
𝟐𝝅
𝒂𝟐+𝒃𝟐 , 𝒂 > 𝒃 > 0 𝟓𝐌
(b) Find the residue of tanz at each pole. 𝟓𝐌
4(a) Evaluate 𝒄𝒐𝒕𝒉𝒛
𝒛−𝟏 𝒅𝒛 where c: 𝒛 = 𝟐 using Cauchey’s residue theorem. 𝟓𝐌
(b) Evaluate 𝟏
𝟓−𝟒𝐬𝐢𝐧 𝜽
𝟐𝝅
𝟎 𝒅𝜽 𝟓𝐌
5(a) Prove that 𝒙𝟐−𝒙+𝟐
𝒙𝟒+𝟏𝟎𝒙𝟐+𝟗𝒅𝒙
∞
−∞=
𝟓𝝅
𝟏𝟐 𝟓𝐌
(b) Evaluate 𝒁
𝒁−𝟏 (𝒁−𝟐)𝟐𝒅𝒛 where C is the circle 𝒛 − 𝟏 =
𝟏
𝟐 using residue theorem 𝟓𝐌
6(a) Show that 𝒅𝜽
𝒂+𝒃𝒄𝒐𝒔𝜽 𝟐=
𝝅
𝟎
𝝅𝒂
(𝒂𝟐−𝒃𝟐)𝟑/𝟐 , 𝒂 > 𝒃 > 0 𝟓𝐌
(b) Evaluate 𝒙 𝒔𝒊𝒏𝒎𝒙
(𝟏𝟔+𝒙𝟒)
∞
𝟎𝒅𝒙 𝟓𝐌
UNIT-I
1 (a ) State and explain the Principle of superposition of waves.- 4M
(b Explain the formation of Newton’s rings and obtain an expression for the diameter of the
dark rings in reflected system..– 6M
2 (a) In Newton’s rings experiment, diameter of the tenth dark ring due to wavelength 6000Å
in air is 0.5 cm. Find the radius of curvature of the lens.– 4 M
(b) If the air film in the Newton’s rings apparatus is replaced by an oil film, then how does
the radius of the rings change? Explain.– 6M
3 (a) What are the necessary conditions to get clear and distinct interference fringes – 4M
( b ) Describe principle ,construction and working of Michelson Interferometer. - 6M
4( a ) Explain the colours in a thin film when exposed it to a sun light – 4M
( b ) Explain why the centre of Newton’s rings is dark in the reflected system. Why are they
circular – 6M
5 (a) Distinguish between Monochromatic and Polychromatic light sources, Give one
example for each –3M
(b) With a ray of diagram, discuss the theory of thin films and the condition constructive
and destructive interference in the case of reflected light.—7M
6 (a) ) Derive cosine law and write down the conditions for brightness & darkness in the
reflected system. - 6M
APPLIED PHYSICS QUESTION BANK
Class – I ECE - I Semester
(b) In Newton’s rings experiment, diameter of 10th
dark ring due to wavelength 6000 A in
air is 0.5 cm. Find the radius of curvature of lens.
UNIT—II
1 (a) What are the types of diffraction and give the difference between them ? 4 M
(b) Obtain the condition for primary maxima in Fraunhofer diffraction due to single slit
and derive an expression for width of the central maxima - 6 M
2 (a) What is the difference between interference and diffraction –4M
(b) Explain the diffraction due to two parallel slits and obtain the Intensity of light on the
screen.—6M
3 (a) Define the grating and Explain with necessary theory, the Fraunhofer diffraction due
to ‘N’parallel slits.– 6 M
(b) Calculate the maximum number of order possible for a tranmission grating - 4 M
4 (a) What happens to the diffraction fringes, if the slit width is reduced in single slit
experiment? Explain why?.- 6 M
(b) A grating has 6000 lines/cm.Find the angular separation between two wavelengths of 500
nm and 510 nm in 3rd
order – 4M
5 (a) What is meant by Diffraction of light? Explain it on the basis of Huygen’s wave theory ?
4 M
(b) Explain the theory of plane transmission grating abd derive equations for maxima and
minima.- 6M
6 (a) Define resolving power of grating and explain Rayliegh criterion for resolution and
determine the resolving power of the Telescope - 6 M
(b) How many orders will be visible ,if wave length of light is 5000 A ? Given that the
number of lines per centimeter on the grating is 6655. .- 4 M
UNIT – III
1(a) What is a half wave plate and Quarter wave plate? Deduce an expression for its
thickness-6 M
(b) Calculate the thickness of half wave plate of quartz for a wavelength 500nm.
Here μe= 1.553 and μo= 1.544..- 4 M
2 (a ) Write the difference between Spontaneous and Stimulated Emissions.– 4M
(b) Explain the working of Ruby laser with the help of neat energy level diagram..– 6M
3 (a) What is population inversion and how can it be achieved ?.- 4 M
(b) Explain the working of He-Ne gas laser with the help of neat energy level diagram.- 6 M
4 (a) Distinguish between polarized and unpolarized lights –3M
(b) State and explain Brewsters law? Discuss how to produce the plane,Circular and
Elliptical polarized lights?
5 (a) Explain Einstein’s coefficients. Derive the relation between them.-5M
(b) What are the characteristics and applicatios of LASER beam.-5M
6(a) Write a note on double refraction? 4M
(b) Explain the principle ,construction and working of a Nicol prism.-6M
polarized light can be produced-7M
UNIT 6
1 convert the following isometric view to ortho graphic
view
2 convert the following isometric view to ortho graphic
view
3)
4 convert the following orthographic view to isometric
view
5)
6)
7.
8.
9.
