Download GATE (Graduate Aptitude Test in Engineering) Last 10 Years 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011 and 2010 MA Mathematics Question Paper With Solution And Answer Key
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
GATE 2018 MATHEMATICS
Q.31 For an odd prime p; consider the ringZ[
p
p] =fa +b
p
p : a;b2 Zg C: Then the
element 2 inZ[
p
p] is
(A) a unit (B) a square (C) a prime (D) irreducible
Q.32 Consider the following two statements:
P: The matrix
0 5
0 7
has in?nitely many LU factorizations, where L is lower triangular
with each diagonal entry 1 andU is upper triangular.
Q: The matrix
0 0
2 5
has no LU factorization, whereL is lower triangular with each diag-
onal entry 1 andU is upper triangular.
Then which one of the following options is correct?
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) P is FALSE and Q is TRUE
(D) Both P and Q are FALSE
Q.33 If the characteristic curves of the partial differential equation xu
xx
+ 2x
2
u
xy
= u
x
1 are
(x;y) =c
1
and (x;y) =c
2
; wherec
1
andc
2
are constants, then
(A) (x;y) =x
2
y; (x;y) =y
(B) (x;y) =x
2
+y; (x;y) =y
(C) (x;y) =x
2
+y; (x;y) =x
2
(D) (x;y) =x
2
y; (x;y) =x
2
MA 6/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
GATE 2018 MATHEMATICS
Q.31 For an odd prime p; consider the ringZ[
p
p] =fa +b
p
p : a;b2 Zg C: Then the
element 2 inZ[
p
p] is
(A) a unit (B) a square (C) a prime (D) irreducible
Q.32 Consider the following two statements:
P: The matrix
0 5
0 7
has in?nitely many LU factorizations, where L is lower triangular
with each diagonal entry 1 andU is upper triangular.
Q: The matrix
0 0
2 5
has no LU factorization, whereL is lower triangular with each diag-
onal entry 1 andU is upper triangular.
Then which one of the following options is correct?
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) P is FALSE and Q is TRUE
(D) Both P and Q are FALSE
Q.33 If the characteristic curves of the partial differential equation xu
xx
+ 2x
2
u
xy
= u
x
1 are
(x;y) =c
1
and (x;y) =c
2
; wherec
1
andc
2
are constants, then
(A) (x;y) =x
2
y; (x;y) =y
(B) (x;y) =x
2
+y; (x;y) =y
(C) (x;y) =x
2
+y; (x;y) =x
2
(D) (x;y) =x
2
y; (x;y) =x
2
MA 6/10
GATE 2018 MATHEMATICS
Q.34 Letf :X!Y be a continuous map from a Hausdorff topological spaceX to a metric space
Y . Consider the following two statements:
P: f is a closed map and the inverse imagef
1
(y) =fx2 X : f(x) = yg is compact for
eachy2Y:
Q: For every compact subsetKY; the inverse imagef
1
(K) is a compact subset ofX:
Which one of the following is true?
(A) Q implies P but P does NOT imply Q
(B) P implies Q but Q does NOT imply P
(C) P and Q are equivalent
(D) neither P implies Q nor Q implies P
Q.35 LetX denoteR
2
endowed with the usual topology. LetY denoteR endowed with the co-?nite
topology. IfZ is the product topological spaceYY; then
(A) the topology ofX is the same as the topology ofZ
(B) the topology ofX is strictly coarser (weaker) than that ofZ
(C) the topology ofZ is strictly coarser (weaker) than that ofX
(D) the topology ofX cannot be compared with that ofZ
Q.36 Consider R
n
with the usual topology for n = 1; 2; 3. Each of the following options gives
topological spacesX andY with respective induced topologies. In which option isX home-
omorphic toY ?
(A) X =f(x;y;z)2R
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:z = 0;x
2
+y
2
6= 0g
(B) X =f(x;y)2R
2
:y = sin(1=x); 0
:x = 0; 1y 1g;
Y = [0; 1]R
(C) X =f(x;y)2R
2
:y =x sin(1=x); 0
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:x
2
+y
2
=z
2
6= 0g
Q.37 LetfX
i
g be a sequence of independent Poisson() variables and letW
n
=
1
n
P
n
i=1
X
i
. Then
the limiting distribution of
p
n(W
n
) is the normal distribution with zero mean and variance
given by
(A) 1 (B)
p
(C) (D)
2
MA 7/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
GATE 2018 MATHEMATICS
Q.31 For an odd prime p; consider the ringZ[
p
p] =fa +b
p
p : a;b2 Zg C: Then the
element 2 inZ[
p
p] is
(A) a unit (B) a square (C) a prime (D) irreducible
Q.32 Consider the following two statements:
P: The matrix
0 5
0 7
has in?nitely many LU factorizations, where L is lower triangular
with each diagonal entry 1 andU is upper triangular.
Q: The matrix
0 0
2 5
has no LU factorization, whereL is lower triangular with each diag-
onal entry 1 andU is upper triangular.
Then which one of the following options is correct?
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) P is FALSE and Q is TRUE
(D) Both P and Q are FALSE
Q.33 If the characteristic curves of the partial differential equation xu
xx
+ 2x
2
u
xy
= u
x
1 are
(x;y) =c
1
and (x;y) =c
2
; wherec
1
andc
2
are constants, then
(A) (x;y) =x
2
y; (x;y) =y
(B) (x;y) =x
2
+y; (x;y) =y
(C) (x;y) =x
2
+y; (x;y) =x
2
(D) (x;y) =x
2
y; (x;y) =x
2
MA 6/10
GATE 2018 MATHEMATICS
Q.34 Letf :X!Y be a continuous map from a Hausdorff topological spaceX to a metric space
Y . Consider the following two statements:
P: f is a closed map and the inverse imagef
1
(y) =fx2 X : f(x) = yg is compact for
eachy2Y:
Q: For every compact subsetKY; the inverse imagef
1
(K) is a compact subset ofX:
Which one of the following is true?
(A) Q implies P but P does NOT imply Q
(B) P implies Q but Q does NOT imply P
(C) P and Q are equivalent
(D) neither P implies Q nor Q implies P
Q.35 LetX denoteR
2
endowed with the usual topology. LetY denoteR endowed with the co-?nite
topology. IfZ is the product topological spaceYY; then
(A) the topology ofX is the same as the topology ofZ
(B) the topology ofX is strictly coarser (weaker) than that ofZ
(C) the topology ofZ is strictly coarser (weaker) than that ofX
(D) the topology ofX cannot be compared with that ofZ
Q.36 Consider R
n
with the usual topology for n = 1; 2; 3. Each of the following options gives
topological spacesX andY with respective induced topologies. In which option isX home-
omorphic toY ?
(A) X =f(x;y;z)2R
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:z = 0;x
2
+y
2
6= 0g
(B) X =f(x;y)2R
2
:y = sin(1=x); 0
:x = 0; 1y 1g;
Y = [0; 1]R
(C) X =f(x;y)2R
2
:y =x sin(1=x); 0
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:x
2
+y
2
=z
2
6= 0g
Q.37 LetfX
i
g be a sequence of independent Poisson() variables and letW
n
=
1
n
P
n
i=1
X
i
. Then
the limiting distribution of
p
n(W
n
) is the normal distribution with zero mean and variance
given by
(A) 1 (B)
p
(C) (D)
2
MA 7/10
GATE 2018 MATHEMATICS
Q.38 LetX
1
;X
2
;:::;X
n
be independent and identically distributed random variables with proba-
bility density function given by
f
X
(x;) =
e
(x 1)
; x 1;
0 otherwise:
Also, let X =
1
n
P
n
i=1
X
i
. Then the maximum likelihood estimator of is
(A) 1=X (B)
1=X
1 (C) 1=
X 1
(D) X
Q.39 Consider the Linear Programming Problem (LPP):
Maximize x
1
+x
2
Subject to 2x
1
+x
2
6,
x
1
+x
2
1,
x
1
+x
2
4,
x
1
0;x
2
0,
where is a constant. If (3; 0) is the only optimal solution, then
(A) < 2 (B) 2< < 1 (C) 1< < 2 (D) > 2
Q.40 Let M
2
(R) be the vector space of all 2 2 real matrices over the ?eldR: De?ne the linear
transformationS :M
2
(R)!M
2
(R) byS(X) = 2X +X
T
; whereX
T
denotes the transpose
of the matrixX: Then the trace ofS equals .
Q.41 ConsiderR
3
with the usual inner product. Ifd is the distance from (1; 1; 1) to the subspace
spanf(1; 1; 0); (0; 1; 1)g ofR
3
; then 3d
2
= .
Q.42 Consider the matrixA = I
9
2u
T
u withu =
1
3
[1; 1; 1; 1; 1; 1; 1; 1; 1]; whereI
9
is the 9 9
identity matrix andu
T
is the transpose ofu: If and are two distinct eigenvalues ofA; then
j j = .
Q.43 Letf(z) =z
3
e
z
2
forz2C and let be the circlez =e
i
; where varies from 0 to 4: Then
1
2i
I
f
0
(z)
f(z)
dz = :
MA 8/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
GATE 2018 MATHEMATICS
Q.31 For an odd prime p; consider the ringZ[
p
p] =fa +b
p
p : a;b2 Zg C: Then the
element 2 inZ[
p
p] is
(A) a unit (B) a square (C) a prime (D) irreducible
Q.32 Consider the following two statements:
P: The matrix
0 5
0 7
has in?nitely many LU factorizations, where L is lower triangular
with each diagonal entry 1 andU is upper triangular.
Q: The matrix
0 0
2 5
has no LU factorization, whereL is lower triangular with each diag-
onal entry 1 andU is upper triangular.
Then which one of the following options is correct?
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) P is FALSE and Q is TRUE
(D) Both P and Q are FALSE
Q.33 If the characteristic curves of the partial differential equation xu
xx
+ 2x
2
u
xy
= u
x
1 are
(x;y) =c
1
and (x;y) =c
2
; wherec
1
andc
2
are constants, then
(A) (x;y) =x
2
y; (x;y) =y
(B) (x;y) =x
2
+y; (x;y) =y
(C) (x;y) =x
2
+y; (x;y) =x
2
(D) (x;y) =x
2
y; (x;y) =x
2
MA 6/10
GATE 2018 MATHEMATICS
Q.34 Letf :X!Y be a continuous map from a Hausdorff topological spaceX to a metric space
Y . Consider the following two statements:
P: f is a closed map and the inverse imagef
1
(y) =fx2 X : f(x) = yg is compact for
eachy2Y:
Q: For every compact subsetKY; the inverse imagef
1
(K) is a compact subset ofX:
Which one of the following is true?
(A) Q implies P but P does NOT imply Q
(B) P implies Q but Q does NOT imply P
(C) P and Q are equivalent
(D) neither P implies Q nor Q implies P
Q.35 LetX denoteR
2
endowed with the usual topology. LetY denoteR endowed with the co-?nite
topology. IfZ is the product topological spaceYY; then
(A) the topology ofX is the same as the topology ofZ
(B) the topology ofX is strictly coarser (weaker) than that ofZ
(C) the topology ofZ is strictly coarser (weaker) than that ofX
(D) the topology ofX cannot be compared with that ofZ
Q.36 Consider R
n
with the usual topology for n = 1; 2; 3. Each of the following options gives
topological spacesX andY with respective induced topologies. In which option isX home-
omorphic toY ?
(A) X =f(x;y;z)2R
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:z = 0;x
2
+y
2
6= 0g
(B) X =f(x;y)2R
2
:y = sin(1=x); 0
:x = 0; 1y 1g;
Y = [0; 1]R
(C) X =f(x;y)2R
2
:y =x sin(1=x); 0
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:x
2
+y
2
=z
2
6= 0g
Q.37 LetfX
i
g be a sequence of independent Poisson() variables and letW
n
=
1
n
P
n
i=1
X
i
. Then
the limiting distribution of
p
n(W
n
) is the normal distribution with zero mean and variance
given by
(A) 1 (B)
p
(C) (D)
2
MA 7/10
GATE 2018 MATHEMATICS
Q.38 LetX
1
;X
2
;:::;X
n
be independent and identically distributed random variables with proba-
bility density function given by
f
X
(x;) =
e
(x 1)
; x 1;
0 otherwise:
Also, let X =
1
n
P
n
i=1
X
i
. Then the maximum likelihood estimator of is
(A) 1=X (B)
1=X
1 (C) 1=
X 1
(D) X
Q.39 Consider the Linear Programming Problem (LPP):
Maximize x
1
+x
2
Subject to 2x
1
+x
2
6,
x
1
+x
2
1,
x
1
+x
2
4,
x
1
0;x
2
0,
where is a constant. If (3; 0) is the only optimal solution, then
(A) < 2 (B) 2< < 1 (C) 1< < 2 (D) > 2
Q.40 Let M
2
(R) be the vector space of all 2 2 real matrices over the ?eldR: De?ne the linear
transformationS :M
2
(R)!M
2
(R) byS(X) = 2X +X
T
; whereX
T
denotes the transpose
of the matrixX: Then the trace ofS equals .
Q.41 ConsiderR
3
with the usual inner product. Ifd is the distance from (1; 1; 1) to the subspace
spanf(1; 1; 0); (0; 1; 1)g ofR
3
; then 3d
2
= .
Q.42 Consider the matrixA = I
9
2u
T
u withu =
1
3
[1; 1; 1; 1; 1; 1; 1; 1; 1]; whereI
9
is the 9 9
identity matrix andu
T
is the transpose ofu: If and are two distinct eigenvalues ofA; then
j j = .
Q.43 Letf(z) =z
3
e
z
2
forz2C and let be the circlez =e
i
; where varies from 0 to 4: Then
1
2i
I
f
0
(z)
f(z)
dz = :
MA 8/10
GATE 2018 MATHEMATICS
Q.44 LetS be the surface of the solid
V =f(x;y;z) : 0x 1; 0y 2; 0z 3g:
Let ^ n denote the unit outward normal toS and let
~
F (x;y;z) =x
^
i +y
^
j +z
^
k; (x;y;z)2V:
Then the surface integral
ZZ
S
~
F ^ ndS equals .
Q.45 LetA be a 3 3 matrix with real entries. If three solutions of the linear system of differential
equations _ x(t) =Ax(t) are given by
2
4
e
t
e
2t
e
t
+e
2t
e
t
+e
2t
3
5
;
2
4
e
2t
e
t
e
2t
e
t
e
2t
+e
t
3
5
and
2
4
e
t
+ 2e
t
e
t
2e
t
e
t
+ 2e
t
3
5
;
then the sum of the diagonal entries ofA is equal to .
Q.46 Ify
1
(x) =e
x
2
is a solution of the differential equation
xy
00
+ y
0
+ x
3
y = 0
for some real numbers and ; then = .
Q.47 LetL
2
([0; 1]) be the Hilbert space of all real valued square integrable functions on [0; 1] with
the usual inner product. Let be the linear functional onL
2
([0; 1]) de?ned by
(f) =
Z
3=4
1=4
3
p
2f d;
where denotes the Lebesgue measure on [0; 1]. Thenk k = .
Q.48 Let U be an orthonormal set in a Hilbert space H and let x 2 H be such thatkxk = 2.
Consider the set
E =
u2U :jhx;uij
1
4
:
Then the maximum possible number of elements inE is .
MA 9/10
FirstRanker.com - FirstRanker's Choice
GATE 2018 General Aptitude (GA) Set-2
GA 1/3
Q. 1 ? Q. 5 carry one mark each.
Q.1 ?The dress _________ her so well that they all immediately _________ her on her
appearance.?
The words that best fill the blanks in the above sentence are
(A) complemented, complemented (B) complimented, complemented
(C) complimented, complimented (D) complemented, complimented
Q.2 ?The judge?s standing in the legal community, though shaken by false allegations of
wrongdoing, remained _________.?
The word that best fills the blank in the above sentence is
(A) undiminished (B) damaged (C) illegal (D) uncertain
Q.3 Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
(A) UVWXY (B) TUVWX (C) STUVW (D) RSTUV
Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the
following statements is true?
(A) The circle has the largest area.
(B) The square has the largest area.
(C) The equilateral triangle has the largest area.
(D) All the three shapes have the same area.
Q.5
The value of the expression
1
1 + log
?? ????
+
1
1 + log
?? ????
+
1
1 + log
?? ????
is _________.
(A) -1 (B) 0 (C) 1 (D) 3
Q. 6 ? Q. 10 carry two marks each.
Q.6 Forty students watched films A, B and C over a week. Each student watched either only
one film or all three. Thirteen students watched film A, sixteen students watched film B
and nineteen students watched film C. How many students watched all three films?
(A) 0 (B) 2 (C) 4 (D) 8
GATE 2018 General Aptitude (GA) Set-2
GA 2/3
Q.7 A wire would enclose an area of 1936 m
2
, if it is bent into a square. The wire is cut into
two pieces. The longer piece is thrice as long as the shorter piece. The long and the short
pieces are bent into a square and a circle, respectively. Which of the following choices is
closest to the sum of the areas enclosed by the two pieces in square meters?
(A) 1096 (B) 1111 (C) 1243 (D) 2486
Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each
operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How
many additional robots would be required to complete the work on time, if each robot is
now operational for 8 hours a day?
(A) 50 (B) 89 (C) 146 (D) 175
Q.9 A house has a number which needs to be identified. The following three statements are
given that can help in identifying the house number.
i. If the house number is a multiple of 3, then it is a number from 50 to 59.
ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.
iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.
What is the house number?
(A) 54 (B) 65 (C) 66 (D) 76
GATE 2018 General Aptitude (GA) Set-2
GA 3/3
Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted.
One trial implies six tosses of the coin. If H stands for head and T stands for tail, the
following are the observations from the four trials:
(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.
Which statement describing the last two coin tosses of the fourth trial has the highest
probability of being correct?
(A) Two T will occur.
(B) One H and one T will occur.
(C) Two H will occur.
(D) One H will be followed by one T.
END OF THE QUESTION PAPER
GATE 2018 MATHEMATICS
Q.1 Q.25carryonemarkeach
Q.1 The principal value of ( 1)
( 2i=)
is
(A) e
2
(B) e
2i
(C) e
2i
(D) e
2
Q.2 Letf :C!C be an entire function withf(0) = 1,f(1) = 2 andf
0
(0) = 0. If there exists
M > 0 such thatjf
00
(z)jM for allz2C, thenf(2) =
(A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i
Q.3 In the Laurent series expansion of f(z) =
1
z(z 1)
valid forjz 1j> 1; the coef?cient of
1
z 1
is
(A) 2 (B) 1 (C) 0 (D) 1
Q.4 LetX andY be metric spaces, and letf :X!Y be a continuous map. For any subsetS of
X; which one of the following statements is true?
(A) IfS is open, thenf(S) is open
(B) IfS is connected, thenf(S) is connected
(C) IfS is closed, thenf(S) is closed
(D) IfS is bounded, thenf(S) is bounded
Q.5 The general solution of the differential equation
xy
0
=y +
p
x
2
+y
2
forx> 0
is given by (with an arbitrary positive constantk)
(A) ky
2
=x +
p
x
2
+y
2
(B) kx
2
=x +
p
x
2
+y
2
(C) kx
2
=y +
p
x
2
+y
2
(D) ky
2
=y +
p
x
2
+y
2
MA 1/10
GATE 2018 MATHEMATICS
Q.6 Letp
n
(x) be the polynomial solution of the differential equation
d
dx
[(1 x
2
)y
0
] +n(n + 1)y = 0
withp
n
(1) = 1 forn = 1; 2; 3;:::: If
d
dx
[p
n+2
(x) p
n
(x)] =
n
p
n+1
(x);
then
n
is
(A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3
Q.7 In the permutation groupS
6
; the number of elements of order 8 is
(A) 0 (B) 1 (C) 2 (D) 4
Q.8 LetR be a commutative ring with 1 (unity) which is not a ?eld. LetI R be a proper ideal
such that every element ofR not inI is invertible inR: Then the number of maximal ideals of
R is
(A) 1 (B) 2 (C) 3 (D) in?nite
Q.9 Letf :R!R be a twice continuously differentiable function. The order of convergence of
the secant method for ?nding root of the equationf(x) = 0 is
(A)
1 +
p
5
2
(B)
2
1 +
p
5
(C)
1 +
p
5
3
(D)
3
1 +
p
5
Q.10 The Cauchy problem uu
x
+yu
y
=x withu(x; 1) = 2x; when solved using its characteristic
equations with an independent variablet; is found to admit of a solution in the form
x =
3
2
se
t
1
2
se
t
; y =e
t
; u =f(s;t):
Thenf(s;t) =
(A)
3
2
se
t
+
1
2
se
t
(B)
1
2
se
t
+
3
2
se
t
(C)
1
2
se
t
3
2
se
t
(D)
3
2
se
t
1
2
se
t
Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three
black balls are added to the urn. The probability that ?ve balls in the urn are black is
(A) 2=7 (B) 3=8 (C) 1=2 (D) 5=7
MA 2/10
GATE 2018 MATHEMATICS
Q.12 For a linear programming problem, which one of the following statements is FALSE?
(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign
(B) Both primal and its dual can be infeasible
(C) If primal is unbounded, then its dual is infeasible
(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual
can differ
Q.13 LetA =
2
4
a 2f 0
2f b 3f
0 3f c
3
5
; wherea;b;c;f are real numbers andf6= 0: The geometric multi-
plicity of the largest eigenvalue ofA equals .
Q.14 Consider the subspaces
W
1
= f(x
1
;x
2
;x
3
)2R
3
:x
1
=x
2
+ 2x
3
g
W
2
= f(x
1
;x
2
;x
3
)2R
3
:x
1
= 3x
2
+ 2x
3
g
ofR
3
: Then the dimension of W
1
+W
2
equals .
Q.15 LetV be the real vector space of all polynomials of degree less than or equal to 2 with real
coef?cients. LetT :V!V be the linear transformation given by
T (p) = 2p +p
0
forp2V;
where p
0
is the derivative of p: Then the number of nonzero entries in the Jordan canonical
form of a matrix ofT equals .
Q.16 Let I = [2; 3), J be the set of all rational numbers in the interval [4; 6], K be the Cantor
(ternary) set, and letL =f7 +x :x2Kg: Then the Lebesgue measure of the setI[J[L
equals .
Q.17 Letu(x;y;z) =x
2
2y + 4z
2
for (x;y;z)2R
3
. Then the directional derivative ofu in the
direction
3
5
^
i
4
5
^
k at the point (5; 1; 0) is .
Q.18 If the Laplace transform ofy(t) is given by Y (s) =L(y(t)) =
5
2(s 1)
2
s 2
+
1
2(s 3)
;
theny(0) +y
0
(0) = .
MA 3/10
GATE 2018 MATHEMATICS
Q.19 The number of regular singular points of the differential equation
[(x 1)
2
sinx]y
00
+ [cosx sin(x 1)]y
0
+ (x 1)y = 0
in the interval
0;
2
is equal to .
Q.20 LetF be a ?eld with 7
6
elements and letK be a sub?eld ofF with 49 elements. Then the
dimension ofF as a vector space overK is .
Q.21 LetC([0; 1]) be the real vector space of all continuous real valued functions on [0; 1]; and let
T be the linear operator onC([0; 1]) given by
(Tf)(x) =
Z
1
0
sin(x +y)f(y)dy; x2 [0; 1]:
Then the dimension of the range space ofT equals .
Q.22 Leta2 ( 1; 1) be such that the quadrature rule
Z
1
1
f(x)dx'f( a) +f(a)
is exact for all polynomials of degree less than or equal to 3: Then 3a
2
= .
Q.23 LetX andY have joint probability density function given by
f
X;Y
(x;y) =
2; 0x 1 y; 0y 1
0; otherwise:
Iff
Y
denotes the marginal probability density function ofY , thenf
Y
(1=2) = .
Q.24 Let the cumulative distribution function of the random variableX be given by
F
X
(x) =
8
>
>
>
<
>
>
>
:
0; x< 0;
x; 0x< 1=2;
(1 +x)=2; 1=2x< 1;
1; x 1:
ThenP(X = 1=2) = .
Q.25 LetfX
j
g be a sequence of independent Bernoulli random variables withP(X
j
= 1) = 1=4
and let Y
n
=
1
n
P
n
j=1
X
2
j
: ThenY
n
converges, in probability, to .
MA 4/10
GATE 2018 MATHEMATICS
Q.26 Q.55carrytwomarkseach
Q.26 Let be the circle given byz = 4e
i
; where varies from 0 to 2: Then
I
e
z
z
2
2z
dz =
(A) 2i(e
2
1) (B) i(1 e
2
) (C) i(e
2
1) (D) 2i(1 e
2
)
Q.27 The image of the half plane Re(z) + Im(z)> 0 under the map w =
z 1
z +i
is given by
(A) Re(w)> 0 (B) Im(w)> 0 (C) jwj> 1 (D) jwj< 1
Q.28 LetDR
2
denote the closed disc with center at the origin and radius 2. Then
ZZ
D
e
(x
2
+y
2
)
dxdy =
(A) (1 e
4
) (B)
2
(1 e
4
) (C) (1 e
2
) (D)
2
(1 e
2
)
Q.29 Consider the polynomialp(X) = X
4
+ 4 in the ringQ[X] of polynomials in the variableX
with coef?cients in the ?eldQ of rational numbers. Then
(A) the set of zeros ofp(X) inC forms a group under multiplication
(B) p(X) is reducible in the ringQ[X]
(C) the splitting ?eld ofp(X) has degree 3 overQ
(D) the splitting ?eld ofp(X) has degree 4 overQ
Q.30 Which one of the following statements is true?
(A) Every group of order 12 has a non-trivial proper normal subgroup
(B) Some group of order 12 does not have a non-trivial proper normal subgroup
(C) Every group of order 12 has a subgroup of order 6
(D) Every group of order 12 has an element of order 12
MA 5/10
GATE 2018 MATHEMATICS
Q.31 For an odd prime p; consider the ringZ[
p
p] =fa +b
p
p : a;b2 Zg C: Then the
element 2 inZ[
p
p] is
(A) a unit (B) a square (C) a prime (D) irreducible
Q.32 Consider the following two statements:
P: The matrix
0 5
0 7
has in?nitely many LU factorizations, where L is lower triangular
with each diagonal entry 1 andU is upper triangular.
Q: The matrix
0 0
2 5
has no LU factorization, whereL is lower triangular with each diag-
onal entry 1 andU is upper triangular.
Then which one of the following options is correct?
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) P is FALSE and Q is TRUE
(D) Both P and Q are FALSE
Q.33 If the characteristic curves of the partial differential equation xu
xx
+ 2x
2
u
xy
= u
x
1 are
(x;y) =c
1
and (x;y) =c
2
; wherec
1
andc
2
are constants, then
(A) (x;y) =x
2
y; (x;y) =y
(B) (x;y) =x
2
+y; (x;y) =y
(C) (x;y) =x
2
+y; (x;y) =x
2
(D) (x;y) =x
2
y; (x;y) =x
2
MA 6/10
GATE 2018 MATHEMATICS
Q.34 Letf :X!Y be a continuous map from a Hausdorff topological spaceX to a metric space
Y . Consider the following two statements:
P: f is a closed map and the inverse imagef
1
(y) =fx2 X : f(x) = yg is compact for
eachy2Y:
Q: For every compact subsetKY; the inverse imagef
1
(K) is a compact subset ofX:
Which one of the following is true?
(A) Q implies P but P does NOT imply Q
(B) P implies Q but Q does NOT imply P
(C) P and Q are equivalent
(D) neither P implies Q nor Q implies P
Q.35 LetX denoteR
2
endowed with the usual topology. LetY denoteR endowed with the co-?nite
topology. IfZ is the product topological spaceYY; then
(A) the topology ofX is the same as the topology ofZ
(B) the topology ofX is strictly coarser (weaker) than that ofZ
(C) the topology ofZ is strictly coarser (weaker) than that ofX
(D) the topology ofX cannot be compared with that ofZ
Q.36 Consider R
n
with the usual topology for n = 1; 2; 3. Each of the following options gives
topological spacesX andY with respective induced topologies. In which option isX home-
omorphic toY ?
(A) X =f(x;y;z)2R
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:z = 0;x
2
+y
2
6= 0g
(B) X =f(x;y)2R
2
:y = sin(1=x); 0
:x = 0; 1y 1g;
Y = [0; 1]R
(C) X =f(x;y)2R
2
:y =x sin(1=x); 0
3
:x
2
+y
2
= 1g; Y =f(x;y;z)2R
3
:x
2
+y
2
=z
2
6= 0g
Q.37 LetfX
i
g be a sequence of independent Poisson() variables and letW
n
=
1
n
P
n
i=1
X
i
. Then
the limiting distribution of
p
n(W
n
) is the normal distribution with zero mean and variance
given by
(A) 1 (B)
p
(C) (D)
2
MA 7/10
GATE 2018 MATHEMATICS
Q.38 LetX
1
;X
2
;:::;X
n
be independent and identically distributed random variables with proba-
bility density function given by
f
X
(x;) =
e
(x 1)
; x 1;
0 otherwise:
Also, let X =
1
n
P
n
i=1
X
i
. Then the maximum likelihood estimator of is
(A) 1=X (B)
1=X
1 (C) 1=
X 1
(D) X
Q.39 Consider the Linear Programming Problem (LPP):
Maximize x
1
+x
2
Subject to 2x
1
+x
2
6,
x
1
+x
2
1,
x
1
+x
2
4,
x
1
0;x
2
0,
where is a constant. If (3; 0) is the only optimal solution, then
(A) < 2 (B) 2< < 1 (C) 1< < 2 (D) > 2
Q.40 Let M
2
(R) be the vector space of all 2 2 real matrices over the ?eldR: De?ne the linear
transformationS :M
2
(R)!M
2
(R) byS(X) = 2X +X
T
; whereX
T
denotes the transpose
of the matrixX: Then the trace ofS equals .
Q.41 ConsiderR
3
with the usual inner product. Ifd is the distance from (1; 1; 1) to the subspace
spanf(1; 1; 0); (0; 1; 1)g ofR
3
; then 3d
2
= .
Q.42 Consider the matrixA = I
9
2u
T
u withu =
1
3
[1; 1; 1; 1; 1; 1; 1; 1; 1]; whereI
9
is the 9 9
identity matrix andu
T
is the transpose ofu: If and are two distinct eigenvalues ofA; then
j j = .
Q.43 Letf(z) =z
3
e
z
2
forz2C and let be the circlez =e
i
; where varies from 0 to 4: Then
1
2i
I
f
0
(z)
f(z)
dz = :
MA 8/10
GATE 2018 MATHEMATICS
Q.44 LetS be the surface of the solid
V =f(x;y;z) : 0x 1; 0y 2; 0z 3g:
Let ^ n denote the unit outward normal toS and let
~
F (x;y;z) =x
^
i +y
^
j +z
^
k; (x;y;z)2V:
Then the surface integral
ZZ
S
~
F ^ ndS equals .
Q.45 LetA be a 3 3 matrix with real entries. If three solutions of the linear system of differential
equations _ x(t) =Ax(t) are given by
2
4
e
t
e
2t
e
t
+e
2t
e
t
+e
2t
3
5
;
2
4
e
2t
e
t
e
2t
e
t
e
2t
+e
t
3
5
and
2
4
e
t
+ 2e
t
e
t
2e
t
e
t
+ 2e
t
3
5
;
then the sum of the diagonal entries ofA is equal to .
Q.46 Ify
1
(x) =e
x
2
is a solution of the differential equation
xy
00
+ y
0
+ x
3
y = 0
for some real numbers and ; then = .
Q.47 LetL
2
([0; 1]) be the Hilbert space of all real valued square integrable functions on [0; 1] with
the usual inner product. Let be the linear functional onL
2
([0; 1]) de?ned by
(f) =
Z
3=4
1=4
3
p
2f d;
where denotes the Lebesgue measure on [0; 1]. Thenk k = .
Q.48 Let U be an orthonormal set in a Hilbert space H and let x 2 H be such thatkxk = 2.
Consider the set
E =
u2U :jhx;uij
1
4
:
Then the maximum possible number of elements inE is .
MA 9/10
GATE 2018 MATHEMATICS
Q.49 Ifp(x) = 2 (x + 1) +x(x + 1) x(x + 1)(x ) interpolates the points (x;y) in the table
x 1 0 1 2
y 2 1 2 7
then + = .
Q.50 If sin(x) =a
0
+
P
1
n=1
a
n
cos(nx) for 0
+a
1
) = .
Q.51 Forn = 1; 2;:::; letf
n
(x) =
2nx
n 1
1 +x
,x2 [0; 1]. Then lim
n!1
Z
1
0
f
n
(x)dx = .
Q.52 LetX
1
;X
2
;X
3
;X
4
be independent exponential random variables with mean 1; 1=2; 1=3; 1=4;
respectively. ThenY = min(X
1
;X
2
;X
3
;X
4
) has exponential distribution with mean equal to
.
Q.53 LetX be the number of heads in 4 tosses of a fair coin by Person 1 and letY be the number of
heads in 4 tosses of a fair coin by Person 2. Assume that all the tosses are independent. Then
the value ofP(X =Y ) correct up to three decimal places is .
Q.54 Let X
1
and X
2
be independent geometric random variables with the same probability
mass function given by P(X = k) = p(1 p)
k 1
, k = 1; 2;:::. Then the value of
P(X
1
= 2jX
1
+X
2
= 4) correct up to three decimal places is .
Q.55 A certain commodity is produced by the manufacturing plantsP
1
andP
2
whose capacities are
6 and 5 units, respectively. The commodity is shipped to marketsM
1
,M
2
,M
3
andM
4
whose
requirements are 1, 2, 3 and 5 units, respectively. The transportation cost per unit from plant
P
i
to marketM
j
is as follows:
M
1
M
2
M
3
M
4
P
1
1 3 5 8 6
P
2
2 5 6 7 5
1 2 3 5
Then the optimal cost of transportation is .
ENDOFTHEQUESTIONPAPER
MA 10/10
FirstRanker.com - FirstRanker's Choice
This post was last modified on 18 December 2019