Gate (CS) 2014 Paper with Solution (Set C) | GATE Computer Science Engineering(CSE) 2025 Mock Test Series - Computer Science Engineering (CSE) PDF Download

 Page 1


                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 1 
Q. No. 1 – 5 Carry One Mark Each 
 
1.  While trying to collect an envelope from under  the table, Mr. X fell down and  
  I         II  III  
 was losing consciousness.  
  IV  
 Which one of the above underlined parts of the sentence is NOT appropriate?  
 (A) I  (B) II  (C) III  (D) IV  
Answer:   (D) 
 
2.  If she _______________ how to calibrate the instrument, she _______________ done the 
experiment.  
 (A) knows, will have    (B) knew, had  
 (C) had known, could have  (D) should have known, would have  
Answer:   (C) 
 
3. Choose the word that is opposite in meaning to the word “coherent”.  
 (A) sticky  (B) well-connected  (C) rambling  (D) friendly  
Answer:   (C) 
 
4.  Which number does not belong in the series below?  
 2, 5, 10, 17, 26, 37, 50, 64  
 (A) 17  (B) 37  (C) 64  (D) 26  
Answer:   (C) 
 
5. The table below has question-wise data on the performance of students in an examination. 
The marks for each question are also listed. There is no negative or partial marking in the 
examination. 
  
Q.No Marks Answered 
Correctly 
Answered 
Wrongly 
Not      
Attempted 
1 2 21 17 6 
2 3 15 27 2 
3 2 23 18 3 
 What is the average of the marks obtained by the class in the examination?  
 (A) 1.34  (B) 1.74  (C) 3.02  (D) 3.91 
Page 2


                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 1 
Q. No. 1 – 5 Carry One Mark Each 
 
1.  While trying to collect an envelope from under  the table, Mr. X fell down and  
  I         II  III  
 was losing consciousness.  
  IV  
 Which one of the above underlined parts of the sentence is NOT appropriate?  
 (A) I  (B) II  (C) III  (D) IV  
Answer:   (D) 
 
2.  If she _______________ how to calibrate the instrument, she _______________ done the 
experiment.  
 (A) knows, will have    (B) knew, had  
 (C) had known, could have  (D) should have known, would have  
Answer:   (C) 
 
3. Choose the word that is opposite in meaning to the word “coherent”.  
 (A) sticky  (B) well-connected  (C) rambling  (D) friendly  
Answer:   (C) 
 
4.  Which number does not belong in the series below?  
 2, 5, 10, 17, 26, 37, 50, 64  
 (A) 17  (B) 37  (C) 64  (D) 26  
Answer:   (C) 
 
5. The table below has question-wise data on the performance of students in an examination. 
The marks for each question are also listed. There is no negative or partial marking in the 
examination. 
  
Q.No Marks Answered 
Correctly 
Answered 
Wrongly 
Not      
Attempted 
1 2 21 17 6 
2 3 15 27 2 
3 2 23 18 3 
 What is the average of the marks obtained by the class in the examination?  
 (A) 1.34  (B) 1.74  (C) 3.02  (D) 3.91 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 2 
Answer:   (C) 
Exp: Total question  
  44×2=88 
  44×3=132 
  144 = 88 
  132     308 
  Total marks obtained= (21×2) + (15×3) + (23×2) =133 
  Total Number of students=44 
  Average
133
3.02
44
= =  
 
Q. No. 6 – 10 Carry One Mark Each 
 
6. A dance programme is scheduled for 10.00 a.m. Some students are participating in the 
programme and they need to come an hour earlier than the start of the event. These students 
should be accompanied by a parent. Other students and parents should come in time for the 
programme. The instruction you think that is appropriate for this is  
 (A) Students should come at 9.00 a.m. and parents should come at 10.00 a.m.  
 (B) Participating students should come at 9.00 a.m. accompanied by a parent, and other 
parents and students should come by 10.00 a.m.  
 (C) Students who are not participating should come by 10.00 a.m. and they should not bring 
their parents. Participating students should come at 9.00 a.m.  
 (D) Participating students should come before 9.00 a.m. Parents who accompany them should 
come at 9.00 a.m. All others should come at 10.00 a.m.  
Answer:   (B)      
 
7.  By the beginning of the 20th century, several hypotheses were being proposed, suggesting a 
paradigm shift in our understanding of the universe. However, the clinching evidence was 
provided by experimental measurements of the position of a star which was directly behind 
our sun. 
  Which of the following inference(s) may be drawn from the above passage?  
 
 (i)  Our understanding of the universe changes based on the positions of stars  
 (ii)  Paradigm shifts usually occur at the beginning of centuries  
 (iii) Stars are important objects in the universe  
 (iv) Experimental evidence was important in confirming this paradigm shift  
  
   (A) (i), (ii) and (iv)     (B) (iii) only  (C) (i) and (iv)  (D) (iv) only  
Answer:   (D)      
Page 3


                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 1 
Q. No. 1 – 5 Carry One Mark Each 
 
1.  While trying to collect an envelope from under  the table, Mr. X fell down and  
  I         II  III  
 was losing consciousness.  
  IV  
 Which one of the above underlined parts of the sentence is NOT appropriate?  
 (A) I  (B) II  (C) III  (D) IV  
Answer:   (D) 
 
2.  If she _______________ how to calibrate the instrument, she _______________ done the 
experiment.  
 (A) knows, will have    (B) knew, had  
 (C) had known, could have  (D) should have known, would have  
Answer:   (C) 
 
3. Choose the word that is opposite in meaning to the word “coherent”.  
 (A) sticky  (B) well-connected  (C) rambling  (D) friendly  
Answer:   (C) 
 
4.  Which number does not belong in the series below?  
 2, 5, 10, 17, 26, 37, 50, 64  
 (A) 17  (B) 37  (C) 64  (D) 26  
Answer:   (C) 
 
5. The table below has question-wise data on the performance of students in an examination. 
The marks for each question are also listed. There is no negative or partial marking in the 
examination. 
  
Q.No Marks Answered 
Correctly 
Answered 
Wrongly 
Not      
Attempted 
1 2 21 17 6 
2 3 15 27 2 
3 2 23 18 3 
 What is the average of the marks obtained by the class in the examination?  
 (A) 1.34  (B) 1.74  (C) 3.02  (D) 3.91 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 2 
Answer:   (C) 
Exp: Total question  
  44×2=88 
  44×3=132 
  144 = 88 
  132     308 
  Total marks obtained= (21×2) + (15×3) + (23×2) =133 
  Total Number of students=44 
  Average
133
3.02
44
= =  
 
Q. No. 6 – 10 Carry One Mark Each 
 
6. A dance programme is scheduled for 10.00 a.m. Some students are participating in the 
programme and they need to come an hour earlier than the start of the event. These students 
should be accompanied by a parent. Other students and parents should come in time for the 
programme. The instruction you think that is appropriate for this is  
 (A) Students should come at 9.00 a.m. and parents should come at 10.00 a.m.  
 (B) Participating students should come at 9.00 a.m. accompanied by a parent, and other 
parents and students should come by 10.00 a.m.  
 (C) Students who are not participating should come by 10.00 a.m. and they should not bring 
their parents. Participating students should come at 9.00 a.m.  
 (D) Participating students should come before 9.00 a.m. Parents who accompany them should 
come at 9.00 a.m. All others should come at 10.00 a.m.  
Answer:   (B)      
 
7.  By the beginning of the 20th century, several hypotheses were being proposed, suggesting a 
paradigm shift in our understanding of the universe. However, the clinching evidence was 
provided by experimental measurements of the position of a star which was directly behind 
our sun. 
  Which of the following inference(s) may be drawn from the above passage?  
 
 (i)  Our understanding of the universe changes based on the positions of stars  
 (ii)  Paradigm shifts usually occur at the beginning of centuries  
 (iii) Stars are important objects in the universe  
 (iv) Experimental evidence was important in confirming this paradigm shift  
  
   (A) (i), (ii) and (iv)     (B) (iii) only  (C) (i) and (iv)  (D) (iv) only  
Answer:   (D)      
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 3 
8.  The Gross Domestic Product (GDP) in Rupees grew at 7% during 2012-2013. For 
international comparison, the GDP is compared in US Dollars (USD) after conversion based 
on the market exchange rate. During the period 2012-2013 the exchange rate for the USD 
increased from Rs. 50/ USD to Rs. 60/ USD. India’s GDP in USD during the period 2012-
2013  
 (A) increased by 5 %    (B) decreased by 13%  
 (C) decreased by 20%   (D) decreased by 11%  
Answer:   (D)      
Exp: Per 100 Rs final value 107 Rs 
 
100
Per Dollars
50
? final value 
107
60
  
 for 100 dollars____? 
 
100 50 107
89.16
100 60
×
= × =  
 Decreased by 11%.  
 
9.  he ratio of male to female students in a college for five years is plotted in the following line 
graph. If the number of female students in 2011 and 2012 is equal, what is the ratio of male 
students in 2012 to male students in 2011?  
 
 
 
 
 
 
 
 
 
 
 
 (A) 1:1  (B) 2:1  (C) 1.5:1  (D) 2.5:1  
Answer:   (C)    
Exp: Take number of female students in 2011=100 
 ?  Number of male in 2011=100 
 No. of female in 2012=100 
 No. of male in 2012=150 
 
150
Ratio
100
=   = 1.5: 1 
 
10.  Consider the equation: (7526)
8
 - (Y)
8
 = (4364)
8
 , where (X)
 N
 stands for X to the base N. Find 
Y.  
 (A) 1634  (B) 1737  (C) 3142  (D) 3162 
Answer:   (C) 
3.5
3
2.5
2
1.5
1
0.5
0
2008 2009 2010 2011 2012
Ra
tio 
of 
m
ale 
to 
fe
m
ale 
stu
de
nts 
Page 4


                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 1 
Q. No. 1 – 5 Carry One Mark Each 
 
1.  While trying to collect an envelope from under  the table, Mr. X fell down and  
  I         II  III  
 was losing consciousness.  
  IV  
 Which one of the above underlined parts of the sentence is NOT appropriate?  
 (A) I  (B) II  (C) III  (D) IV  
Answer:   (D) 
 
2.  If she _______________ how to calibrate the instrument, she _______________ done the 
experiment.  
 (A) knows, will have    (B) knew, had  
 (C) had known, could have  (D) should have known, would have  
Answer:   (C) 
 
3. Choose the word that is opposite in meaning to the word “coherent”.  
 (A) sticky  (B) well-connected  (C) rambling  (D) friendly  
Answer:   (C) 
 
4.  Which number does not belong in the series below?  
 2, 5, 10, 17, 26, 37, 50, 64  
 (A) 17  (B) 37  (C) 64  (D) 26  
Answer:   (C) 
 
5. The table below has question-wise data on the performance of students in an examination. 
The marks for each question are also listed. There is no negative or partial marking in the 
examination. 
  
Q.No Marks Answered 
Correctly 
Answered 
Wrongly 
Not      
Attempted 
1 2 21 17 6 
2 3 15 27 2 
3 2 23 18 3 
 What is the average of the marks obtained by the class in the examination?  
 (A) 1.34  (B) 1.74  (C) 3.02  (D) 3.91 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 2 
Answer:   (C) 
Exp: Total question  
  44×2=88 
  44×3=132 
  144 = 88 
  132     308 
  Total marks obtained= (21×2) + (15×3) + (23×2) =133 
  Total Number of students=44 
  Average
133
3.02
44
= =  
 
Q. No. 6 – 10 Carry One Mark Each 
 
6. A dance programme is scheduled for 10.00 a.m. Some students are participating in the 
programme and they need to come an hour earlier than the start of the event. These students 
should be accompanied by a parent. Other students and parents should come in time for the 
programme. The instruction you think that is appropriate for this is  
 (A) Students should come at 9.00 a.m. and parents should come at 10.00 a.m.  
 (B) Participating students should come at 9.00 a.m. accompanied by a parent, and other 
parents and students should come by 10.00 a.m.  
 (C) Students who are not participating should come by 10.00 a.m. and they should not bring 
their parents. Participating students should come at 9.00 a.m.  
 (D) Participating students should come before 9.00 a.m. Parents who accompany them should 
come at 9.00 a.m. All others should come at 10.00 a.m.  
Answer:   (B)      
 
7.  By the beginning of the 20th century, several hypotheses were being proposed, suggesting a 
paradigm shift in our understanding of the universe. However, the clinching evidence was 
provided by experimental measurements of the position of a star which was directly behind 
our sun. 
  Which of the following inference(s) may be drawn from the above passage?  
 
 (i)  Our understanding of the universe changes based on the positions of stars  
 (ii)  Paradigm shifts usually occur at the beginning of centuries  
 (iii) Stars are important objects in the universe  
 (iv) Experimental evidence was important in confirming this paradigm shift  
  
   (A) (i), (ii) and (iv)     (B) (iii) only  (C) (i) and (iv)  (D) (iv) only  
Answer:   (D)      
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 3 
8.  The Gross Domestic Product (GDP) in Rupees grew at 7% during 2012-2013. For 
international comparison, the GDP is compared in US Dollars (USD) after conversion based 
on the market exchange rate. During the period 2012-2013 the exchange rate for the USD 
increased from Rs. 50/ USD to Rs. 60/ USD. India’s GDP in USD during the period 2012-
2013  
 (A) increased by 5 %    (B) decreased by 13%  
 (C) decreased by 20%   (D) decreased by 11%  
Answer:   (D)      
Exp: Per 100 Rs final value 107 Rs 
 
100
Per Dollars
50
? final value 
107
60
  
 for 100 dollars____? 
 
100 50 107
89.16
100 60
×
= × =  
 Decreased by 11%.  
 
9.  he ratio of male to female students in a college for five years is plotted in the following line 
graph. If the number of female students in 2011 and 2012 is equal, what is the ratio of male 
students in 2012 to male students in 2011?  
 
 
 
 
 
 
 
 
 
 
 
 (A) 1:1  (B) 2:1  (C) 1.5:1  (D) 2.5:1  
Answer:   (C)    
Exp: Take number of female students in 2011=100 
 ?  Number of male in 2011=100 
 No. of female in 2012=100 
 No. of male in 2012=150 
 
150
Ratio
100
=   = 1.5: 1 
 
10.  Consider the equation: (7526)
8
 - (Y)
8
 = (4364)
8
 , where (X)
 N
 stands for X to the base N. Find 
Y.  
 (A) 1634  (B) 1737  (C) 3142  (D) 3162 
Answer:   (C) 
3.5
3
2.5
2
1.5
1
0.5
0
2008 2009 2010 2011 2012
Ra
tio 
of 
m
ale 
to 
fe
m
ale 
stu
de
nts 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 4 
Q. No. 1 – 25 Carry One Mark Each 
 
1. Consider the following statements: 
 P: Good mobile phones are not cheap 
 Q: Cheap mobile phones are not good 
 
 L: P implies Q 
 M: Q implies P 
 N: P is equivalent to Q 
 
 Which one of the following about L, M, and N is CORRECT? 
 
 (A) Only L is TRUE.   (B) Only M is TRUE. 
 (C) Only N is TRUE.   (D) L, M and N are TRUE. 
Answer:  (D) 
Exp:   
( ) [ ]
( )
g : mobile is good c : mobile is cheap
P : Good mobile phones are not cheap g c g c a b a b
Q : Cheap mobile phones are not good c g c g
Both P and Q are equivalent which means P and Q imply each other
? ? ¬ ? ¬ ? ¬ ? = ¬ ?
? ? ¬ ? ¬ ? ¬
?
?
 
 
2. Let X and Y be finite sets and f : X Y ? be a function. Which one of the following 
statements is TRUE? 
 (A)  For any subsets A and B of X, ( ) ( ) ( ) f A B f A f B ? = + 
 (B)  For any subsets A and B of X, ( ) ( ) ( ) f A B f A f B n = n 
 (C)  For any subsets A and B of X, ( ) ( ) ( )
{ }
f A B min f A , f B n = 
 (D)  For any subsets S and T of Y, ( ) ( ) ( )
1 1 1
f S T f S f T
- - -
n = n 
Answer:  (D) 
Exp: ( ) ( ) ( ) f : X Y defined by f a 1,f b 1,f c 2 where ? = = = 
 
{ } { }
{ } { }
( ) ( ) ( )
( ) { } ( ) { } ( ) { }
( ) ( ) { }
( )
X a,b,c Y 1,2
Let A a,c ,B b,c be subsets of X
then f A B 2 ; f A 2 ; f B 2
f A B 2 ; f A 1,2 ;f B 1,2
f A f B 1,2
f A B 1
= =
= =
? = = =
n = = =
n =
n =
 
 ? Options (A), (B), (C) are not true 
 Hence, option (D) is true 
Page 5


                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 1 
Q. No. 1 – 5 Carry One Mark Each 
 
1.  While trying to collect an envelope from under  the table, Mr. X fell down and  
  I         II  III  
 was losing consciousness.  
  IV  
 Which one of the above underlined parts of the sentence is NOT appropriate?  
 (A) I  (B) II  (C) III  (D) IV  
Answer:   (D) 
 
2.  If she _______________ how to calibrate the instrument, she _______________ done the 
experiment.  
 (A) knows, will have    (B) knew, had  
 (C) had known, could have  (D) should have known, would have  
Answer:   (C) 
 
3. Choose the word that is opposite in meaning to the word “coherent”.  
 (A) sticky  (B) well-connected  (C) rambling  (D) friendly  
Answer:   (C) 
 
4.  Which number does not belong in the series below?  
 2, 5, 10, 17, 26, 37, 50, 64  
 (A) 17  (B) 37  (C) 64  (D) 26  
Answer:   (C) 
 
5. The table below has question-wise data on the performance of students in an examination. 
The marks for each question are also listed. There is no negative or partial marking in the 
examination. 
  
Q.No Marks Answered 
Correctly 
Answered 
Wrongly 
Not      
Attempted 
1 2 21 17 6 
2 3 15 27 2 
3 2 23 18 3 
 What is the average of the marks obtained by the class in the examination?  
 (A) 1.34  (B) 1.74  (C) 3.02  (D) 3.91 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 2 
Answer:   (C) 
Exp: Total question  
  44×2=88 
  44×3=132 
  144 = 88 
  132     308 
  Total marks obtained= (21×2) + (15×3) + (23×2) =133 
  Total Number of students=44 
  Average
133
3.02
44
= =  
 
Q. No. 6 – 10 Carry One Mark Each 
 
6. A dance programme is scheduled for 10.00 a.m. Some students are participating in the 
programme and they need to come an hour earlier than the start of the event. These students 
should be accompanied by a parent. Other students and parents should come in time for the 
programme. The instruction you think that is appropriate for this is  
 (A) Students should come at 9.00 a.m. and parents should come at 10.00 a.m.  
 (B) Participating students should come at 9.00 a.m. accompanied by a parent, and other 
parents and students should come by 10.00 a.m.  
 (C) Students who are not participating should come by 10.00 a.m. and they should not bring 
their parents. Participating students should come at 9.00 a.m.  
 (D) Participating students should come before 9.00 a.m. Parents who accompany them should 
come at 9.00 a.m. All others should come at 10.00 a.m.  
Answer:   (B)      
 
7.  By the beginning of the 20th century, several hypotheses were being proposed, suggesting a 
paradigm shift in our understanding of the universe. However, the clinching evidence was 
provided by experimental measurements of the position of a star which was directly behind 
our sun. 
  Which of the following inference(s) may be drawn from the above passage?  
 
 (i)  Our understanding of the universe changes based on the positions of stars  
 (ii)  Paradigm shifts usually occur at the beginning of centuries  
 (iii) Stars are important objects in the universe  
 (iv) Experimental evidence was important in confirming this paradigm shift  
  
   (A) (i), (ii) and (iv)     (B) (iii) only  (C) (i) and (iv)  (D) (iv) only  
Answer:   (D)      
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 3 
8.  The Gross Domestic Product (GDP) in Rupees grew at 7% during 2012-2013. For 
international comparison, the GDP is compared in US Dollars (USD) after conversion based 
on the market exchange rate. During the period 2012-2013 the exchange rate for the USD 
increased from Rs. 50/ USD to Rs. 60/ USD. India’s GDP in USD during the period 2012-
2013  
 (A) increased by 5 %    (B) decreased by 13%  
 (C) decreased by 20%   (D) decreased by 11%  
Answer:   (D)      
Exp: Per 100 Rs final value 107 Rs 
 
100
Per Dollars
50
? final value 
107
60
  
 for 100 dollars____? 
 
100 50 107
89.16
100 60
×
= × =  
 Decreased by 11%.  
 
9.  he ratio of male to female students in a college for five years is plotted in the following line 
graph. If the number of female students in 2011 and 2012 is equal, what is the ratio of male 
students in 2012 to male students in 2011?  
 
 
 
 
 
 
 
 
 
 
 
 (A) 1:1  (B) 2:1  (C) 1.5:1  (D) 2.5:1  
Answer:   (C)    
Exp: Take number of female students in 2011=100 
 ?  Number of male in 2011=100 
 No. of female in 2012=100 
 No. of male in 2012=150 
 
150
Ratio
100
=   = 1.5: 1 
 
10.  Consider the equation: (7526)
8
 - (Y)
8
 = (4364)
8
 , where (X)
 N
 stands for X to the base N. Find 
Y.  
 (A) 1634  (B) 1737  (C) 3142  (D) 3162 
Answer:   (C) 
3.5
3
2.5
2
1.5
1
0.5
0
2008 2009 2010 2011 2012
Ra
tio 
of 
m
ale 
to 
fe
m
ale 
stu
de
nts 
                                        ?CS-GATE-2014 PAPER-03|                 www.gateforum.com 

 India’s No.1 institute for GATE Training  
 1 Lakh+ Students trained till date  
 65+ Centers across India 
 4 
Q. No. 1 – 25 Carry One Mark Each 
 
1. Consider the following statements: 
 P: Good mobile phones are not cheap 
 Q: Cheap mobile phones are not good 
 
 L: P implies Q 
 M: Q implies P 
 N: P is equivalent to Q 
 
 Which one of the following about L, M, and N is CORRECT? 
 
 (A) Only L is TRUE.   (B) Only M is TRUE. 
 (C) Only N is TRUE.   (D) L, M and N are TRUE. 
Answer:  (D) 
Exp:   
( ) [ ]
( )
g : mobile is good c : mobile is cheap
P : Good mobile phones are not cheap g c g c a b a b
Q : Cheap mobile phones are not good c g c g
Both P and Q are equivalent which means P and Q imply each other
? ? ¬ ? ¬ ? ¬ ? = ¬ ?
? ? ¬ ? ¬ ? ¬
?
?
 
 
2. Let X and Y be finite sets and f : X Y ? be a function. Which one of the following 
statements is TRUE? 
 (A)  For any subsets A and B of X, ( ) ( ) ( ) f A B f A f B ? = + 
 (B)  For any subsets A and B of X, ( ) ( ) ( ) f A B f A f B n = n 
 (C)  For any subsets A and B of X, ( ) ( ) ( )
{ }
f A B min f A , f B n = 
 (D)  For any subsets S and T of Y, ( ) ( ) ( )
1 1 1
f S T f S f T
- - -
n = n 
Answer:  (D) 
Exp: ( ) ( ) ( ) f : X Y defined by f a 1,f b 1,f c 2 where ? = = = 
 
{ } { }
{ } { }
( ) ( ) ( )
( ) { } ( ) { } ( ) { }
( ) ( ) { }
( )
X a,b,c Y 1,2
Let A a,c ,B b,c be subsets of X
then f A B 2 ; f A 2 ; f B 2
f A B 2 ; f A 1,2 ;f B 1,2
f A f B 1,2
f A B 1
= =
= =
? = = =
n = = =
n =
n =
 
 ? Options (A), (B), (C) are not true 
 Hence, option (D) is true 
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 5 
3. Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L G ? and that 
the size of L is at least 4. The size of L is _______. 
Answer:  (5) 
Exp: Order of subgroup divides order of group (Lagrange’s theorem). 
 3, 5 and 15 can be the order of subgroup. As subgroup has atleast 4 elements and it is not 
equal to the given group, order of subgroup can’t be 3 and 15. Hence it is 5. 
 
4. Which one of the following statements is TRUE about every n × n matrix with only real 
eigenvalues? 
 (A) If the trace of the matrix is positive and the determinant of the matrix is negative, at least 
one of its eigenvalues is negative. 
 (B) If the trace of the matrix is positive, all its eigenvalues are positive. 
 (C) If the determinant of the matrix is positive, all its eigenvalues are positive. 
 (D) If the product of the trace and determinant of the matrix is positive, all its eigenvalues are 
positive. 
Answer:  (A) 
Exp: If the trace of the matrix is positive and the determinant of the matrix is negative then atleast 
one of its eigen values is negative.   
 Since determinant = product of eigen values. 
 
5. If V
1
 and V
2
 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest  
possible dimension of 
1 2
V V n is _______. 
Answer: (2) 
Exp:  Let the basis of 6-dimensional vector space be {e1, e2, e3,e4, e5, e6}. In order for V1 n V2 to 
have smallest possible dimension V1 and V2 could be, say, {e1, e2, e3,e4} and {e3, e4, e5, 
e6} respectively. The basis of V1 n V2 would then be {e3, e4}. => Smallest possible 
dimension = 2.  
 
6. If 
2
0
xsin x
p
?
 dx = kp, then the value of k is equal to _______. 
Answer:  ( 4 ) 
Exp:
 
( )
( ) ( ) ( )
( ) ( ) ( )
( )
2 2
0 0
2
0
sin x sin x
xsin x dx K xsin x dx xsin x dx K
x 2
x cos x 1 sin x x cos x sin x K
cos sin 0 2 cos2 sin 2 cos sin K
0 2 0 0 K 4 K k 4
p p p
p
p p
p
? ? =-
= p? + - = p
? ?
p< < p
? ?
? - - - - - + = p
? -p p + p - -? - p p+ p - -p p + p ? = p
? ?
? p + -?- p+ - p+ ? = p? p = p ? =
? ?
? ? ?
?