GATE Overflow Book

576 Pages·2017·18.89 MB·English

by  Phyllis K. Bankson

2017 Vol. I
etworks
cture
www.gateoverflow.in
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This book was created programmatically by GATE Overflow on Jun 27, 2017. If you feel any doubt regarding
the answer, click on the question link and give a comment on the site. Studying all these questions might get
you 60 marks in GATE but that might not be enough for an IIT. So, read standard books, solve exercise
questions and use these questions for cementing the concepts and aim 85+. At least if you are not getting the
solution to a given problem first refer standard book. If any error is found on any question it shall be updated
at http://gateoverflow.in/corrections.
PDFs for the remaining subjects will be made available at http://classroom.gateoverflow.in and you can
enroll in this course to get notification for the same. Enrollment is free and account details are of GATE
Overflow with a new password which have been sent to all registered emails on GATE Overflow. New users will
receive this email within a few minutes of confirming their email address.
You can now join our Facebook group for GATE CSE discussions.
You can visit http://playlists.gatecse.in for high quality videos for GATE CSE and how to use GO
site/ebook.
This book consists of only previous year GATE, TIFR, ISI and CMI questions (CS from 1987 and all 5 years of
IT) all of which are relevant for GATE. Out of syllabus subjects as of GATE 2017 are removed from this book
except in rare cases.
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Since GATE Overflow started in August 2014, a lot of people have dedicated their time and effort in bringing
this book now. Initiated by Omesh Pandita and Arjun Suresh as a Q/A platform for CSE students, Kathleen
Bankson was instrumental in getting all previous year GATE questions here. Then experts like Praven Saini,
Happy Mittal, Sankaranarayanan P.N., Suraj Kumar etc. have contributed a lot to the answers here.
Pragy Agarwal even after topping GATE has continuously contributed here with his knowledge as well as in
making the contents beautiful with fine latex skills. We also have to thank the work by Jothee, Misbah,
Ishrat and Nataliyah who are continuously adding and keeping the contents here neat and clean. There are
also many toppers of GATE 2015, 2016, 2017 and probably 2018 who are contributing a lot here. The list of all
the contributors can be found here but even that does not include the contributions of some like Arif Ali
Anapparakkal in helping design this book, Arvind Devaraj and others who have provided guidance and help
etc. Last but not the least, we thank all the users of GATE Overflow.
We thank the contributions of Silpa V.S., Rahul Kumar Yadav and others for getting the GATECSE
Lastrank page maintained. Bikram Ballav is behind most of the exams on GO (http://mockgate.com) and
Arindam Sarkar made the interface for it. Pragy Agarwal is also behind the rank and score predictor tool,
(http://mymarks.gatecse.in) used by GO which has 99-100% accuracy over the last 2 years.
Special thanks to Sachin Mittal for making the How to Use GO vidoes, Silpa V.S. for classifying the
questions topicwise for the book, Pooja Palod for making the GATE 2018 schedule and Debashish Deka
for GO classroom contributions.
Also thanks to all toppers who took time to write a review for GO.
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Table of Contents
1. Discrete Mathematics: Combinatory (51)
1. Generating Functions (4)
2. Permutations And Combinations (38)
3. Recurrence (6)
4. Summation (3)
2. Discrete Mathematics: Graph Theory (62)
1. Counting (7)
2. Degree Of Graph (13)
3. Euler Graph (1)
4. Graph Coloring (9)
5. Graph Connectivity (20)
6. Graph Isomorphism (2)
7. Graph Matching (1)
8. Graph Planarity (1)
9. Line Graph (1)
10. Regular Graph (1)
11. Spanning Tree (2)
12. Trees (3)
13. Vertex Cover (1)
3. Discrete Mathematics: Mathematical Logic (79)
1. First Order Logic (33)
2. Logical Reasoning (10)
3. Propositional Logic (36)
4. Discrete Mathematics: Set Theory & Algebra (173)
1. Binary Operation (8)
2. Counting (1)
3. Fields (1)
4. Functions (33)
5. Generating Functions (1)
6. Groups (22)
7. Inequality (1)
8. Lattice (9)
9. Lines Curves (1)
10. Mathematical Induction (2)
11. Number Theory (9)
12. Partial Order (12)
13. Polynomials (8)
14. Relations (31)
15. Ring (1)
16. Sets (33)
5. Engineering Mathematics: Calculus (51)
1. Continuity (4)
2. Differentiability (8)
3. Functions (4)
4. Integration (11)
5. Limits (11)
6. Maxima Minima (12)
7. Polynomials (1)
6. Engineering Mathematics: Linear Algebra (69)
1. Determinant (6)
2. Eigen Value (22)
3. Matrices (25)
4. System Of Equations (11)
5. Vector Space (5)
7. Engineering Mathematics: Probability (101)
1. Bayes Theorem (4)
2. Binomial Distribution (5)
3. Conditional Probability (6)
4. Expectation (11)
5. Exponential Distribution (1)
6. Normal Distribution (1)
7. Poisson Distribution (4)
8. Probability (59)
9. Random Variable (8)
10. Uniform Distribution (2)
8. General Aptitude: Numerical Ability (214)
1. Absolute Value (5)
2. Algebra (1)
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3. Arithmetic Series (1)
4. Bar Charts (2)
5. Bayes Theorem (3)
6. Cartesian Coordinates (3)
7. Circle (1)
8. Clock Time (3)
9. Complex Number (1)
10. Compound Interest (1)
11. Conditional Probability (2)
12. Cost Market Price (3)
13. Counting (1)
14. Currency Money (1)
15. Data Interpretation (17)
16. Direction Sense (3)
17. Factors (6)
18. Fractions (4)
19. Functions (3)
20. Geometry (9)
21. Inference (1)
22. Limits (1)
23. Logarithms (2)
24. Logical Reasoning (22)
25. Maxima Minima (3)
26. Mean (1)
27. Modular Arithmetic (1)
28. No Of Digits (1)
29. Number Representation (2)
30. Number Series (11)
31. Numerical Computation (14)
32. Odd One (3)
33. Percentage (10)
34. Permutations And Combinations (11)
35. Pie Chart (4)
36. Pigeonhole (1)
37. Polynomials (1)
38. Probability (13)
39. Proportions (1)
40. Quadratic Equations (5)
41. Ratios (5)
42. Sequence (4)
43. Sequence Series (1)
44. Sets (1)
45. Speed Time Distance (11)
46. Statement Argument (1)
47. Statement Sufficiency (1)
48. Statistics (1)
49. Summation (2)
50. System Of Equations (1)
51. Triangles (2)
52. Variance (1)
53. Venn Diagrams (2)
54. Work Time (3)
9. General Aptitude: Verbal Ability (180)
1. Closest Word (3)
2. English Grammar (26)
3. Geometry (1)
4. Grammatically Incorrect Sentence (2)
5. Inference (2)
6. Logical Reasoning (15)
7. Meaning (20)
8. Median (1)
9. Most Appropriate Alternative (4)
10. Most Appropriate Word (33)
11. Noun Verb Adjective (1)
12. Odd One (3)
13. Opposite (3)
14. Passage Reading (23)
15. Percentage (1)
16. Phrasal Verbs (1)
17. Probability (1)
18. Speed Time Distance (1)
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19. Statements Follow (1)
20. Synonym (5)
21. Tenses (3)
22. Venn Diagrams (1)
23. Verbal Reasoning (22)
24. Word Pairs (7)
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1 Discrete Mathematics: Combinatory (51)
top
1.1 Generating Functions(4)
top
1.1.1 Generating Functions: GATE2017-2-47
top http://gateoverflow.in/118392
∞ 1+z
If the ordinary generating function of a sequence {an}}n=0 is (1−z)3 , then a3−a0 is equal to ___________ .
gate2017-2 permutations-and-combinations generating-functions numerical-answers
Answer
1.1.2 Generating Functions: GATE1987-10b
top http://gateoverflow.in/82451
What is the generating function G(z) for the sequence of Fibonacci numbers?
gate1987 permutations-and-combinations generating-functions
Answer
1.1.3 Generating Functions: GATE 2016-1-26
top http://gateoverflow.in/39693
The coefficient of x12 in (x3+x4+x5+x6+…)3 is ___________.
gate2016-1 permutations-and-combinations generating-functions normal numerical-answers
Answer
1.1.4 Generating Functions: TIFR2010-A-12
top http://gateoverflow.in/18391
The coefficient of x3 in the expansion of (1+x)3(2+x2)10 is.
a. 214
b. 31
c. (3)+(10)
3 1
d. (3)+2(10)
3 1
e. (3)(10)29
3 1

tifr2010 generating-functions
Answer
Answers: Generating Functions
1.1.1 Generating Functions: GATE2017-2-47
top http://gateoverflow.in/118392

Selected Answer
1+z =(1+z)(1−z)−3
(1−z)3
(1−z)−3=1+(3)z+(4)z2 +(5)z3 +…∞
1 2 3
(1+z)(1−z)−3=(1+z)∗(1+(3)z+(4)z2 +(5)z3 +…∞)
1 2 3
a0 is the first term in the expansion of above series and a3 is the fourth term (or) coefficient of z3
a0 = coefficient of z0 =1
a3 = coefficient of z3 =(53)+(42)=10+6
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⇒a −a =16−1=15
3 0
 14 votes -- Manish Joshi ( 25k points)
1.1.2 Generating Functions: GATE1987-10b
top http://gateoverflow.in/82451

Selected Answer
The general form is G(x)= x
1−x−x2
and after solving this using partial fraction, we will get
f = 1 ((1+√5)n−(1−√5)n)
n √5 2 2
 3 votes -- Manu Madhavan ( 1.2k points)
1.1.3 Generating Functions: GATE 2016-1-26
top http://gateoverflow.in/39693

Selected Answer
we will get x12 as
1. (x4)3 having coefficient 3C0 =1
2. (x3)2(x6) having coefficient 3C1 =3
3. (x3)(x4)(x5) having coefficient 3C2 ×2C1 =6
So it is 10
Second Method:
[x12](x3+x4+x5+x6+…)3
[x12][x3(1+x1+x2+x3+…)]3
[x12] x9(1+x1+x2+x3+…)3
[ ]
[x3] (1+x1+x2+x3+…)3
[ ]
1 3
[x3]
(1−x)
[ ]
∞ 3+k−1
[x3] xk
∑( k )
[k=0 ]
Now , put k = 3
2+3
Coefficient of [x3]= =5C3=5C2=10
( 3 )
[x12](x3+x4+x5+x6+…)3 ⇒10
 24 votes -- Praveen Saini ( 53k points)
1.1.4 Generating Functions: TIFR2010-A-12
top http://gateoverflow.in/18391

Selected Answer
(1+x)3 =(1+x3+3x+3x2)
(2+ 2 10 10 ∗ 0∗( 2 10 10 ∗ 1∗( 2 9+................ 10 ∗ 9 ∗( 2 1 10 ∗ 10 ∗( 2 0
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and (2+x2)10 =10 C∗20∗(x2)10 +10 C∗21∗(x2)9+................+10C∗29 ∗(x2)1+10 C∗210 ∗(x2)0
0 1 9 10

So , coefficient of x3 =10 C∗210 +3∗10 C∗29 =29(32)=214
10 9
As here we need to multiply last term of second expansion with first term of first coefficient ( x3 ) and 3x with x2 in the
second expansion.
 7 votes -- Shounak Kundu ( 5.4k points)
1.2 Permutations And Combinations(38)
top
1.2.1 Permutations And Combinations: CMI2010-A-02
top http://gateoverflow.in/46132
We need to choose a team of 11 from a pool of 15 players and also select a captain. The number of different ways this can
be done is
15
A.
(11)
15
B. 11 .
(11)
C. 15 . 14 . 13 . 12 . 11 .10 . 9 . 8 . 7 . 6 . 5
D. (15 . 14 . 13 . 12 . 11 .10 . 9 . 8 . 7 . 6 . 5) . 11
cmi2010 permutations-and-combinations
Answer
1.2.2 Permutations And Combinations: TIFR2012-A-10
top http://gateoverflow.in/25014
In how many different ways can r elements be picked from a set of n elements if
(i) Repetition is not allowed and the order of picking matters?
(ii) Repetition is allowed and the order of picking does not matter?
n! (n+r−1)!
a. and , respectively.
(n−r)! r!(n−1)!
n! n!
b. and , respectively.
(n−r)! r!(n−1)!
n! (n−r+1)!
c. and , respectively.
r!(n−r)! r!(n−1)!
n! n!
d. and , respectively.
r!(n−r)! (n−r)!
n! r!
e. and , respectively.
r! n!
tifr2012 permutations-and-combinations
Answer
1.2.3 Permutations And Combinations: TIFR2013-A-9
top http://gateoverflow.in/25431
There are n kingdoms and 2n champions. Each kingdom gets 2 champions. The number of ways in which this can be done
is:
(2n)!
a. 2n
(2n)!
b.
n!
(2n)!
c. 2n.n!
d. n!/2
e. None of the above.
tifr2013 permutations-and-combinations
Answer
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