GATE Mathematics 2025
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Mathematics (MA)
General Aptitude
Q.1 – Q.5 Carry ONE mark Each
Q.1 Ravi had ______ younger brother who taught at ______ university. He was widely
regarded as ______ honorable man.
Select the option with the correct sequence of articles to fill in the blanks.
(A) a; a; an
(B) the; an; a
(C) a; an; a
(D) an; an; a
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Mathematics (MA)
Q.2 The CEO’s decision to downsize the workforce was considered myopic because it
sacrificed long-term stability to accommodate short-term gains.
Select the most appropriate option that can replace the word “myopic” without
changing the meaning of the sentence.
(A) visionary
(B) shortsighted
(C) progressive
(D) innovative
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Mathematics (MA)
Q.3 The average marks obtained by a class in an examination were calculated as 30.8.
However, while checking the marks entered, the teacher found that the marks of one
student were entered incorrectly as 24 instead of 42. After correcting the marks, the
average becomes 31.4. How many students does the class have?
(A) 25
(B) 28
(C) 30
(D) 32
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Mathematics (MA)
Q.4 Consider the relationships among P, Q, R, S, and T:
• P is the brother of Q.
• S is the daughter of Q.
• T is the sister of S.
• R is the mother of Q.
The following statements are made based on the relationships given above.
(1) R is the grandmother of S.
(2) P is the uncle of S and T.
(3) R has only one son.
(4) Q has only one daughter.
Which one of the following options is correct?
(A) Both (1) and (2) are true.
(B) Both (1) and (3) are true.
(C) Only (3) is true.
(D) Only (4) is true.
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Mathematics (MA)
Q.5 According to the map shown in the figure, which one of the following statements is
correct?
Note: The figure shown is representative.
1st Main Road
Library Canteen
Physics Lab Hospital
5th Cross Road
Hostels Chemistry
Lab
N
Classrooms
W E
S
(A) The library is located to the northwest of the canteen.
(B) The hospital is located to the east of the chemistry lab.
(C) The chemistry lab is to the southeast of physics lab.
(D) The classrooms and canteen are next to each other.
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Mathematics (MA)
Q.6 – Q.10 Carry TWO marks Each
Q.6 “I put the brown paper in my pocket along with the chalks, and possibly other things.
I suppose every one must have reflected how primeval and how poetical are the
things that one carries in one’s pocket: the pocket-knife, for instance the type of all
human tools, the infant of the sword. Once I planned to write a book of poems
entirely about the things in my pocket. But I found it would be too long: and the age
of the great epics is past.”
(From G.K. Chesterton’s “A Piece of Chalk”)
Based only on the information provided in the above passage, which one of the
following statements is true?
(A) The author of the passage carries a mirror in his pocket to reflect upon things.
(B) The author of the passage had decided to write a poem on epics.
(C) The pocket-knife is described as the infant of the sword.
(D) Epics are described as too inconvenient to write.
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Mathematics (MA)
Q.7 In the diagram, the lines QR and ST are parallel to each other. The shortest distance
between these two lines is half the shortest distance between the point P and line
QR. What is the ratio of the area of the triangle PST to the area of the trapezium
SQRT?
Note: The figure shown is representative.
P
S T
Q R
(A) 1
3
(B) 1
4
(C) 2
5
(D) 1
2
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Mathematics (MA)
Q.8 A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled
thrice. What is the probability of rolling ‘6’ exactly once?
(A) 75
216
(B) 1
6
(C) 1
18
(D) 25
216
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Mathematics (MA)
Q.9 A square paper, shown in figure (I), is folded along the dotted lines as shown in the
figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one
of the following patterns will be obtained when the paper is unfolded?
Note: The figures shown are representative.
(I) (II) (III) (IV)
(A)
(B)
(C)
(D)
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Mathematics (MA)
Q.10 A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops
of any flavor. The order in which the scoops are purchased is inconsequential.
If one wants to purchase 3 scoops of ice-cream, in how many ways can one make
that purchase?
(A) 4
(B) 20
(C) 24
(D) 48
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Mathematics (MA)
Q.11 – Q.35 Carry ONE mark Each
Q.11 Let
w1 1
S = {w = [w2 ] ∈ ℝ3 ∶ [ −3 ] [w1 w2 w3 ] is diagonalizable and ‖w‖ = 1},
w3 2
1
where ‖w‖ = (w12 + w22 + w32 )2 . Then, which one of the following is TRUE?
(A) S is compact and connected
(B) S is neither compact nor connected
(C) S is compact but not connected
(D) S is connected but not compact
Q.12 Given that the Laplace transforms of J0 (x), J0′ (x) and J0′′ (x) exist, where J0 (x) is
the Bessel function. Let Y = Y(s) be the Laplace transform of the Bessel function
J0 (x). Then, which one of the following is TRUE?
(A) dY 2sY
+ 2 = 0, s>0
ds s +1
(B) dY 2sY
− 2 = 0, s>0
ds s +1
(C) dY sY
− 2 = 0, s>0
ds s +1
(D) dY sY
+ 2 = 0, s>0
ds s +1
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Mathematics (MA)
Q.13 3
To find a real root of the equation x 3 + 4x 2 − 10 = 0 in the interval (1, 2 ) by
using the fixed-point iteration scheme, consider the following two statements:
10
S1: The iteration scheme xk+1 = √4+x , k = 0, 1, 2, …, converges for any initial
k
3
guess x0 ∈ (1, 2 ).
1
S2: The iteration scheme xk+1 = √10 − xk3 , k = 0, 1, 2, …, diverges for some
2
3
initial guess x0 ∈ (1, 2 ).
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.14 For the linear programming problem:
Maximize Z = 2x1 + 4x2 + 4x3 − 3x4
Subject to αx1 + x2 + x3 = 4,
x1 + βx2 + x4 = 8,
x1 , x2 , x3 , x4 ≥ 0,
consider the following two statements:
S1: If α = 2 and β = 1, then (x1 , x2 )T forms an optimal basis.
S2: If α = 1 and β = 4, then (x3 , x2 )T forms an optimal basis.
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.15 Consider the following subsets of the Euclidean space ℝ4 :
S = {(x1 , x2 , x3 , x4 ) ∈ ℝ4 ∶ x12 + x22 + x32 − x42 = 0 },
T = {(x1 , x2 , x3 , x4 ) ∈ ℝ4 ∶ x12 + x22 + x32 − x42 = 1 }, and
U = {(x1 , x2 , x3 , x4 ) ∈ ℝ4 ∶ x12 + x22 + x32 − x42 = −1 }.
Then, which one of the following is TRUE?
(A) S is connected, but T and U are not connected
(B) T and U are connected, but S is not connected
(C) S and U are connected, but T is not connected
(D) S and T are connected, but U is not connected
Consider the system of ordinary differential equations
Q.16
dX
= MX,
dt
where M is a 6 × 6 skew-symmetric matrix with entries in ℝ. Then, for this system,
the origin is a stable critical point for
(A) any such matrix M
(B) only such matrices M whose rank is 2
(C) only such matrices M whose rank is 4
(D) only such matrices M whose rank is 6
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Mathematics (MA)
Q.17 Let X = { f ∈ C[0, 1] ∶ f(0) = 0 = f(1)} with the norm ‖f‖∞ = sup |f(t)|,
0≤t≤1
where C[0, 1] is the space of all real-valued continuous functions on [0, 1].
1
1 2
Let Y = C[0, 1] with the norm ‖f‖2 = (∫0 |f(t)|2 dt) . Let UX and UY be the
closed unit balls in X and Y centred at the origin, respectively. Consider T: X → ℝ
and S: Y → ℝ given by
1 1
Tf = ∫0 f(t) dt and Sf = ∫0 f(t) dt.
Consider the following statements:
S1: sup |Tf| is attained at a point of UX .
f∈UX
S2: sup |Sf| is attained at a point of UY .
f∈UY
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.18 Let g(x, y) = f(x, y)e 2x+3y be defined in ℝ2 , where f(x, y) is a continuously
differentiable non-zero homogeneous function of degree 4. Then,
∂g ∂g
x ∂x + y ∂y = 0 holds for
(A) all points (x, y) in ℝ2
(B) all points (x, y) on the line given by 2x + 3y + 4 = 0
(C) all points (x, y) in the region of ℝ2 except on the line given by 2x + 3y + 4 = 0
(D) all points (x, y) on the line given by 2x + 3y = 0
Q.19 The partial differential equation
∂2 u ∂2 u ∂2 u ∂u ∂u
(1 + x 2 ) 2
+ 2x(1 − y 2 ) + (1 − y 2 ) +x + (1 − y 2 ) =0
∂x ∂y∂x ∂y2 ∂x ∂y
is
(A) elliptic in the region {(x, y) ∈ ℝ2 : |y| ≤ 1}
(B) hyperbolic in the region {(x, y) ∈ ℝ2 : |y| > 1}
(C) elliptic in the region {(x, y) ∈ ℝ2 : |y| > 1}
(D) hyperbolic in the region {(x, y) ∈ ℝ2 : |y| < 1}
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Mathematics (MA)
Q.20 Let u(x, t) be the solution of the following initial-boundary value problem
∂u ∂2 u
− = 0, x ∈ (0, π), t > 0,
∂t ∂x2
u(0, t) = u(π, t) = 0, u(x, 0) = sin 4x cos 3x .
π
Then, for each t > 0, the value of u ( 4 , t) is
(A) e −49t
(e 48t − 1)
2√2
(B) e −49t
(1 − e 48t )
2√2
(C) e −49t
(1 + e 48t )
2√2
(D) e −49t
(1 − e 48t )
4√2
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Mathematics (MA)
Q.21 Consider the function F: ℝ2 ⟶ ℝ2 given by
F(x, y) = (x 3 − 3xy 2 − 3x , 3x 2 y − y 3 − 3y).
Then, for the function F, the inverse function theorem is
(A) applicable at all points of ℝ2
(B) not applicable at exactly one point of ℝ2
(C) not applicable at exactly two points of ℝ2
(D) not applicable at exactly three points of ℝ2
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Mathematics (MA)
Q.22 Let the functions f: ℝ2 → ℝ and g: ℝ2 → ℝ be given by
f(x1 , x2 ) = x12 + x22 − 2x1 x2 , and
g(x1 , x2 ) = 2x12 + 2x22 − x1 x2 .
Consider the following statements:
S1: For every compact subset K of ℝ, f −1 (K) is compact.
S2: For every compact subset K of ℝ, g−1 (K) is compact.
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.23 Let pA (x) denote the characteristic polynomial of a square matrix A. Then, for
which of the following invertible matrices M, the polynomial pM (x) − pM−1 (x) is
constant?
(A) 5 7
M=[ ]
2 3
(B) 3 1
M=[ ]
4 2
(C) 1 3
M=[ ]
2 −1
(D) 5 −8
M=[ ]
2 −3
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Mathematics (MA)
Q.24 Consider the balanced transportation problem with three sources S1 , S2 , S3 , and four
destinations D1 , D2 , D3 , D4 , for minimizing the total transportation cost whose cost
matrix is as follows:
D1 D2 D3 D4 Supply
S1 2 6 20 11 α + 10
S2 12 7 4 10 α + λ + 10
S3 8 14 16 11 5
Demand α+5 10 λ+5 α+λ
where α, λ > 0. If the associated cost to the starting basic feasible solution obtained
by using the North-West corner rule is 290, then which of the following is/are
correct?
(A) α 2 + λ2 = 100
(B) α 2 + αλ = 150
(C) The optimal cost of the transportation problem is 260
(D) The optimal cost of the transportation problem is 290
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Mathematics (MA)
Q.25 Consider the following regions:
S1 = {(x1 , x2 ) ∈ ℝ2 ∶ 2x1 + x2 ≤ 4, x1 + 2x2 ≤ 5, x1 , x2 ≥ 0},
S2 = {(x1 , x2 ) ∈ ℝ2 ∶ 2x1 − x2 ≤ 5, x1 + 2x2 ≤ 5, x1 , x2 ≥ 0}.
Then, which of the following is/are TRUE?
(A) The maximum value of x1 + x2 is 3 on the region S2
(B) The maximum value of x1 + x2 is 5 on the region S2 − S1
(C) The maximum value of x1 + x2 is 3 on the region S1 ∩ S2
(D) The maximum value of x1 + x2 is 4 on the region S1 ∪ S2
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Mathematics (MA)
Q.26 Let f: ℝ2 − {(0,0)} ⟶ ℝ be a function defined by
x2 − y2 1
f(x, y) = ( 2 2
) + x sin ( 2 ).
x +y x + y2
Consider the following three statements:
S1: lim lim f(x, y) exists.
x→0 y→0
S2: lim lim f(x, y) exists.
y→0 x→0
S3: lim f(x, y) exists.
(x,y)→(0,0)
Then, which of the following is/are correct?
(A) S2 and S3 are TRUE and S1 is FALSE
(B) S1 and S2 are TRUE and S3 is FALSE
(C) S1 and S3 are TRUE and S2 is FALSE
(D) S1, S2 and S3 all are TRUE
Q.27 Let M be a 7 × 7 matrix with entries in ℝ and having the characteristic polynomial
cM (x) = (x − 1)α (x − 2)β (x − 3)2 , where α > β.
Let rank(M − I7 ) = rank(M − 2I7 ) = rank(M − 3I7 ) = 5, where I7 is the 7 × 7
identity matrix. If mM (x) is the minimal polynomial of M, then mM (5) is equal to
_____ (in integer)
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Mathematics (MA)
Q.28 Let y = Pn (x) be the unique polynomial of degree n satisfying the Legendre
differential equation
(1 − x 2 )y ′′ − 2xy ′ + n(n + 1)y = 0 and y(1) = 1.
′
Then, the value of P11 (1) is equal to _____ (in integer)
Q.29 ̂ be a unit vector parallel to the tangent at the point P(1, 1, √2 ) to the curve
Let a
of intersection of the surfaces 2x 2 + 3y 2 − z 2 = 3 and x 2 + y 2 = z 2 .
Then, the absolute value of the directional derivative of
f(x, y, z) = x 2 + 2y 2 − 2√11 z
̂ is _____ (in integer)
at P in the direction of a
Q.30 The volume of the region bounded by the cylinders x 2 + y 2 = 4 and x 2 + z 2 = 4
is _____ (rounded off to TWO decimal places)
Q.31 Let W be the vector space (over ℝ) consisting of all bounded real-valued solutions
of the differential equation
d4 y d2y
+2 2+y =0.
dx 4 dx
Then, the dimension of W is _____ (in integer)
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Mathematics (MA)
Q.32 ̂ be a vector field, and let S be the surface
Let F⃗ = (y − z)î + (z − x)ĵ + (x − y)k
2 2 2
x + y + (z − 1) = 9, 1 ≤ z ≤ 4. If n ̂ denotes the unit outward normal vector
to S, then the value of
1
⃗⃗ × F⃗ ) ⋅ n
|∬(∇ ̂ dS|
π
S
is equal to _____ (in integer)
Q.33 Consider
1 sin z
I= ∮ dz,
2πi C 1 − cos(z 3 )
where C = {z ∈ ℂ ∶ z = x + iy, |x| + |y| = 1, x, y ∈ ℝ} is oriented positively
as a simple closed curve. Then, the value of 120I is equal to _____ (in integer)
Q.34 Let α, β, γ, δ ∈ ℝ be such that the quadrature formula
1
∫ f(x) dx = αf(−1) + βf(1) + γf ′ (−1) + δf′(1)
−1
is exact for all polynomials of degree less than or equal to 3.
Then, 9(α 2 + β 2 + γ 2 + δ 2 ) is equal to _____ (in integer)
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Mathematics (MA)
Q.35 Let y(x) be the solution of the initial value problem
dy
= sin(π(x + y)), y(0) = 0.
dx
Using Euler’s method, with the step-size ℎ = 0.5, the approximate value of
y(1.5) + 2y(1) is equal to _____ (in integer)
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Mathematics (MA)
Q.36 – Q.65 Carry TWO marks Each
Q.36 Consider the linear system Ax = b, where A = [aij ], i, j = 1, 2, 3, and aii ≠ 0 for
a11 0 0
i = 1, 2, 3, is a matrix with entries in ℝ. For D = [ 0 a22 0 ], let
0 0 a33
1 1 −2 4
D−1 A = [ 3 1 2] and D −1 b = [ 4 ].
1 1 1 1
Consider the following two statements:
S1: The approximation of x after one iteration of the Jacobi scheme with initial
vector x0 = [1 1 1]T is x1 = [5 − 1 − 1]T .
S2: There exists an initial vector x0 for which Jacobi iterative scheme diverges.
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.37 Let y(x) be the solution of the differential equation
x 2 y ′′ + 7xy ′ + 9y = x −3 log e x, x > 0,
satisfying y(1) = 0 and y ′ (1) = 0. Then, the value of y(e) is equal to
(A)
1 −3
e
3
(B)
1 −3
e
6
(C)
2 −3
e
3
(D)
1 −3
e
2
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Mathematics (MA)
Q.38 Let y1 (x) and y2 (x) be the two linearly independent solutions of the differential
equation
(1 + x 2 )y ′′ − xy ′ + (cos2 x)y = 0,
satisfying the initial conditions
y1 (0) = 3, y1′ (0) = −1 and y2 (0) = −5, y2′ (0) = 2.
y1 (x) y2 (x) 1
Define W(x) = | |. Then, the value of W (2) is
y1′ (x) y2′ (x)
(A) √5
4
(B) √5
2
(C) 2
√5
(D) 4
√5
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Mathematics (MA)
Q.39 Let C be the curve of intersection of the surfaces z 2 = x 2 + y 2 and
4x + z = 7. If P is a point on C at a minimum distance from the xy-plane, then the
distance of P from the origin is
(A) 7
5
(B) 7√2
5
(C) 14
5
(D) 14√2
5
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Mathematics (MA)
Q.40 Let u(x, t) be the solution of the initial-value problem
∂2 u ∂2 u ∂u
2
−9 = 0, x ∈ ℝ, t > 0, u(x, 0) = e x , (x, 0) = sin x.
∂t ∂x2 ∂t
π π
Then, the value of u ( 2 , 6 ) is
(A) 1 π 1
(e − )
2 3
(B) 1 π 1
(e + )
2 3
(C) 1 π 5
(e + )
2 3
(D) 1 π 5
(e − )
2 3
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Mathematics (MA)
Q.41 1
Let T be the Mö bius transformation that maps the points 0, 2 and 1 conformally
onto the points −3, ∞ and 2, respectively, in the extended complex plane.
If T maps the circle centred at 1 with radius k onto a straight line given by the
2k(α+β)+γ
equation αx + βy + γ = 0, then the value of is equal to
α+β−2kγ
(A) 1
7
(B) 2
7
(C) 1
3
(D) 2
3
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Mathematics (MA)
Q.42 Let U = {z ∈ ℂ ∶ Im(z) > 0} and D = {z ∈ ℂ ∶ |z| < 1}, where Im(z) denotes
the imaginary part of z. Let S be the set of all bijective analytic functions
f: U → D such that f(i) = 0. Then, the value of sup |f(4i)| is
f∈S
(A) 0
(B) 1
4
(C) 1
2
(D) 3
5
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Mathematics (MA)
Q.43 Let Ω be a non-empty open connected subset of ℂ and f: Ω → ℂ be a non-constant
function. Let the functions f 2 : Ω → ℂ and f 3 : Ω → ℂ be defined by
2 3
f 2 (z) = (f(z)) and f 3 (z) = (f(z)) , z ∈ Ω.
Consider the following two statements:
S1: If f is continuous in Ω and f 2 is analytic in Ω, then f is analytic in Ω.
S2: If f 2 and f 3 are analytic in Ω, then f is analytic in Ω.
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.44 In the following, all subsets of Euclidean spaces are considered with the respective
subspace topologies.
Define an equivalence relation ~ on the sphere
S = {(x1 , x2 , x3 ) ∈ ℝ3 ∶ x12 + x22 + x32 = 1}
by (x1 , x2 , x3 ) ~ (y1 , y2 , y3 ) if x3 = y3 , for (x1 , x2 , x3 ), (y1 , y2 , y3 ) ∈ S.
Let [x1 , x2 , x3 ] denote the equivalence class of (x1 , x2 , x3 ), and let X denote the set
of all such equivalence classes. Let ℒ ∶ S → X be given by
ℒ((x1 , x2 , x3 )) = [x1 , x2 , x3 ]. If X is provided with the quotient topology induced
by the map ℒ, then which one of the following is TRUE?
(A) X is homeomorphic to {x ∈ ℝ ∶ −1 ≤ x ≤ 1}
(B) X is homeomorphic to {(x1 , x2 ) ∈ ℝ2 ∶ x12 + x22 = 1}
(C) X is homeomorphic to {(x1 , x2 ) ∈ ℝ2 ∶ x12 + x22 ≤ 1}
(D) X is homeomorphic to {(x1 , x2 , x3 ) ∈ ℝ3 ∶ x12 + x22 = 1 and − 1 ≤ x3 ≤ 1}
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Mathematics (MA)
Q.45 Consider the following two spaces:
X = (C[−1, 1], ∥⋅∥∞ ), the space of all real-valued continuous functions defined on
[−1, 1] equipped with the norm ∥ f ∥∞ = sup |f(t)|.
t∈[−1,1]
Y = (C[−1, 1], ∥⋅∥2 ), the space of all real-valued continuous functions defined on
1 1/2
[−1, 1] equipped with the norm ∥ f ∥2 = (∫−1|f(t)|2 dt) .
Let W be the linear span over ℝ of all the Legendre polynomials. Then, which one
of the following is correct?
(A) W is dense in X but not in Y
(B) W is dense in Y but not in X
(C) W is dense in both X and Y
(D) W is dense neither in X nor in Y
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Mathematics (MA)
Q.46 Consider the metric spaces X = (ℝ, d1 ) and Y = ([0, 1], d2 ) with the metrics
defined by d1 (x, y) = |x − y|, x, y ∈ ℝ and d2 (x, y) = |x − y|, x, y ∈ [0,1],
respectively. Then, which one of the following is TRUE?
(A) 1
[0, 4) is open in X but not in Y
(B) 1
[0, 4) is open in Y but not in X
(C) 1
[0, 4) is open in both X and Y
(D) 1
[0, 4) is open neither in X nor in Y
Q.47 Let K be an algebraically closed field containing a finite field F. Let L be the
subfield of K consisting of elements of K that are algebraic over F.
Consider the following statements:
S1: L is algebraically closed.
S2: L is infinite.
Then, which one of the following is correct?
(A) S1 is TRUE and S2 is FALSE
(B) S2 is TRUE and S1 is FALSE
(C) both S1 and S2 are TRUE
(D) neither S1 nor S2 is TRUE
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Mathematics (MA)
Q.48 Let M2 (ℝ) be the vector space (over ℝ) of all 2 × 2 matrices with entries in ℝ.
Consider the linear transformation T: M2 (ℝ) → M2 (ℝ) defined by T(X) = AXB,
1 −2 6 5
where A = [ ] and B = [ ]. If P is the matrix representation of
1 4 −2 −1
T with respect to the standard basis of M2 (ℝ), then which of the following is/are
TRUE?
(A) P is an invertible matrix
(B) The trace of P is 25
(C) The rank of (P2 − 4I4 ) is 4, where I4 is the 4 × 4 identity matrix
(D) The nullity of (P − 2I4 ) is 0, where I4 is the 4 × 4 identity matrix
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Mathematics (MA)
Q.49 Consider the linear programming problem (LPP):
Maximize Z = 3x1 + 5x2
Subject to x1 + x3 = 4 ,
2x2 + x4 = 12,
3x1 + 2x2 + x5 = 18,
x1 , x2 , x3 , x4 , x5 ≥ 0.
Given that xB = (x3 , x2 , x1 )T forms the optimal basis of the LPP with basis matrix
α β −β
−1
B and respective B = [ 0 γ 0 ] . If ( p, q, r) is the optimal solution of the
0 −β β
dual of the LPP, then which of the following is/are TRUE?
(A) α + 3β + 2γ = 3
(B) α − 3β + 4γ = 1
(C) 5
p+q+r =2
(D) 17
p2 + q 2 + r 2 = 4
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Mathematics (MA)
Q.50 Let 0 < α < 1. Define
|f(t) − f(s)|
C α [0, 1] = {f: [0, 1] → ℝ ∶ sup < ∞}.
s≠t |t − s|α
s,t∈[0,1]
It is given that C α [0, 1] is a Banach space with respect to the norm ‖⋅‖α given by
|f(t) − f(s)|
‖f‖α = |f(0)| + sup .
s≠t |t − s|α
s,t∈[0,1]
Let C[0, 1] be the space of all real-valued continuous functions on [0, 1] with the
norm ‖f‖∞ = sup |f(t)|. If T: C α [0, 1] → C[0,1] is the map Tf = f,
0≤t≤1
f ∈ C α [0, 1], then which of the following is/are TRUE?
(A) T is a compact linear map
(B) Image of T is closed in C[0, 1]
(C) Image of T is dense in C[0, 1]
(D) T is not a bounded linear map
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Mathematics (MA)
Q.51 Let u(x, t) be the solution of the initial value problem
∂u ∂u
+3 = u, x ∈ ℝ, t > 0, u(x, 0) = cos x,
∂t ∂x
and let v(x, t) be the solution of the initial value problem
∂v ∂v
+3 = v 2 , x ∈ ℝ, t > 0, v(x, 0) = cos x.
∂t ∂x
Then, which of the following is/are TRUE?
(A) |u(x, t)| ≤ e t for all x ∈ ℝ and for all t > 0
(B) v(x, 1) is not defined for certain values of x ∈ ℝ
(C) v(x, 1) is not defined for any x ∈ ℝ
(D) u(2π, π) = −e π
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Mathematics (MA)
Q.52 Let u(x, t) be the solution of the initial-boundary value problem
∂u ∂2 u
=2 , 0 < x < 1, t > 0,
∂t ∂x2
u(0, t) = u(1, t) = 0, u(x, 0) = 2x(1 − x).
Then, which of the following is/are TRUE?
(A) 1
0 ≤ u(x, t) ≤ 4 for all t ≥ 0 and x ∈ [0, 1]
(B) u(x, t) = u(1 − x, t) for all t ≥ 0 and x ∈ [0, 1]
(C) 1 2
∫0 (u(x, t)) dx is a decreasing function of t
(D) 1 2
∫0 (u(x, t)) dx is not a decreasing function of t
Q.53 Consider the function f: ℝ2 → ℝ2 given by
f(x, y) = (e 2πx cos 2πy, e 2πx sin 2πy).
Then, which of the following is/are TRUE?
(A) If G is open in ℝ2 , then f(G) is open in ℝ2
(B) If G is closed in ℝ2 , then f(G) is closed in ℝ2
(C) If G is dense in ℝ2 , then f(G) is dense in ℝ2
(D) f is surjective
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Mathematics (MA)
Q.54 Let {xk }∞
k=1 be an orthonormal set of vectors in a real Hilbert space X with inner
product <⋅ , ⋅>. Let n ∈ ℕ, and let Y be the linear span of {xk }nk=1 over ℝ.
For x ∈ X, let Sn (x) = ∑nk=1 < x, xk > xk . Then, which of the following is/are
TRUE?
(A) Sn (x) is the orthogonal projection of x onto Y
(B) Sn (x) is the orthogonal projection of x onto Y ⊥
(C) (x − Sn (x)) is orthogonal to Sn (x) for all x in X
(D) ∑nk=1|< x, xk >|2 = ‖x‖2 for all x in X
Q.55 Consider the sequence { fn } of continuous functions on [0, 1] defined by
x 1 2
f1 (x) = , fn+1 (x) = fn (x) − ( (fn (x)) − x) , n = 1, 2, 3, … .
2 2
Then, which of the following is/are TRUE?
(A) The sequence { fn } converges pointwise but not uniformly on [0, 1]
(B) The sequence { fn } converges uniformly on [0, 1]
(C) 2√x
√x − fn (x) > 2+n√x for all x ∈ [0, 1] and n = 1, 2, 3, …
(D) 0 ≤ fn (x) ≤ √x for all x ∈ [0, 1] and n = 1, 2, 3, …
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Mathematics (MA)
Q.56 sin nx
For x ∈ (0, π), let un (x) = , n = 1, 2, 3, … . Then, which of the
√n
following is/are TRUE?
(A) ∑∞
n=1 un (x) converges uniformly on (0, π)
(B) ∑∞ ′
n=1 un (x) converges uniformly on (0, π)
(C) ∑∞
n=1 un (x) converges pointwise but not uniformly on (0, π)
(D) ∑∞
n=1 un (x) converges uniformly on every compact subset of (0, π)
Q.57 Let ℝ1 and ℝ2 be provided with the respective Euclidean topologies, and let
S 1 = {(x1 , x2 ) ∈ ℝ2 ∶ x12 + x22 = 1}
be assigned the subspace topology induced from ℝ2 . If f: S 1 → ℝ1 is a non-constant
continuous function, then which of the following is/are TRUE?
(A) f maps closed sets to closed sets
(B) f is injective
(C) f is surjective
(D) There exists λ ∈ ℝ such that f(cos λ, sin λ) = f(− cos λ, − sin λ)
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Mathematics (MA)
Q.58 Let X be an uncountable set. Let the topology on X be defined by declaring a subset
U ⊆ X to be open if X − U is either empty or finite or countable, and the empty set
to be open. Then, which of the following is/are TRUE?
(A) Every compact subset of X is closed
(B) Every closed subset of X is compact
(C) X is T1 (singleton subsets are closed) but not T2 (Hausdorff)
(D) X is T2 (Hausdorff)
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Mathematics (MA)
Q.59 All rings considered below are assumed to be associative and commutative with
1 ≠ 0. Further, all ring homomorphisms map 1 to 1.
Consider the following statements about such a ring R:
P1: R is isomorphic to the product of two rings R1 and R2 .
P2: ∃ r1 , r2 ∈ R such that r12 = r1 ≠ 0 ≠ r2 = r22 , r1 r2 = 0 and r1 + r2 = 1.
P3: ∃ ideals I1 , I2 ⊆ R with R ≠ I1 ≠ (0) ≠ I2 ≠ R such that R = I1 + I2 and
I1 ⋂ I2 = (0).
P4: ∃ a, b ∈ R with a ≠ 0 ≠ b such that ab = 0.
Then, which of the following is/are TRUE?
(A) P1 ⟹ P2
(B) P2 ⟹ P3
(C) P3 ⟹ P4
(D) P4 ⟹ P1
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Mathematics (MA)
Q.60 Let E ⊂ F and F ⊂ K be field extensions which are not algebraic. Let α ∈ K be
algebraic over F and α ∉ F. Let L be the subfield of K generated over E by the
coefficients of the monic polynomial of minimal degree over F which has α as a
zero. Then, which of the following is/are TRUE?
(A) F(α) ⊃ L(α) is a finite extension if and only if F ⊃ L is a finite extension
(B) The dimension of L(α) over L is greater than the dimension of F(α) over F
(C) The dimension of L(α) over L is smaller than the dimension of F(α) over F
(D) F(α) ⊃ L(α) is an algebraic extension if and only if F ⊃ L is an algebraic
extension
Q.61 Consider the inner product space of all real-valued continuous functions defined on
[−1, 1] with the inner product
1
〈f, g〉 = ∫ f(x)g(x) dx.
−1
If p(x) = α + βx 2 − 30x 4 , α, β ∈ ℝ is orthogonal to all the polynomials having
degree less than or equal to 3, with respect to this inner product, then α + 5 β is
equal to _____ (in integer)
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Mathematics (MA)
Q.62 For X = (x1 , x2 , x3 )T ∈ ℝ3 , consider the quadratic form:
Q(X) = 2x12 + 2x22 + 3x32 + 4x1 x2 + 2x1 x3 + 2x2 x3 .
Let M be the symmetric matrix associated with the quadratic form Q(X) with
respect to the standard basis of ℝ3 . Let Y = (y1 , y2 , y3 )T ∈ ℝ3 be a non-zero vector,
and let
Y T (M + I3 )n+1 Y
an = , n = 1, 2, 3, … ,
Y T (M + I3 )n Y
where I3 is the 3 × 3 identity matrix. Then, the value of lim an is _____
n→∞
(in integer)
Q.63 Let α, β be distinct non-zero real numbers, and let Q(z) be a polynomial of degree
less than 5. If the function
α 6 sin βz − β 6 (e 2αz − Q(z))
f(z) =
z6
α
satisfies Morera’s theorem in ℂ\{0}, then the value of is equal to
4β
____ (in integer)
Q.64 Let G be a group with identity element e, and let g, ℎ ∈ G be such that the following
hold:
(i) g ≠ e, g2 = e,
(ii) ℎ ≠ e, ℎ2 ≠ e, and gℎg−1 = ℎ2 .
Then, the least positive integer n for which ℎn = e is _____ (in integer)
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Mathematics (MA)
Q.65 Let (ℝ2 , d1 ) and (ℝ2 , d2 ) be two metric spaces with
d1 ((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 |
and
d1 ((x1 , x2 ), (y1 , y2 ))
d2 ((x1 , x2 ), (y1 , y2 )) = .
1 + d1 ((x1 , x2 ), (y1 , y2 ))
1
If the open ball centred at (0,0) with radius in (ℝ2 , d1 ) is equal to the open ball
7
1
centred at (0,0) with radius in (ℝ2 , d2 ), then the value of α
α
is _____ (in integer)
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Answer Key for Mathematics (MA)
Q. No. Session Q. Type Section Key/Range Marks
1 1 MCQ GA A 1
2 1 MCQ GA B 1
3 1 MCQ GA C 1
4 1 MCQ GA A 1
5 1 MCQ GA C 1
6 1 MCQ GA C 2
7 1 MCQ GA A 2
8 1 MCQ GA A 2
9 1 MCQ GA A 2
10 1 MCQ GA B 2
11 1 MCQ MA B 1
12 1 MCQ MA D 1
13 1 MCQ MA A 1
14 1 MCQ MA B 1
15 1 MCQ MA D 1
16 1 MCQ MA A 1
17 1 MCQ MA B 1
18 1 MCQ MA B 1
19 1 MCQ MA B 1
20 1 MCQ MA A 1
21 1 MCQ MA C 1
22 1 MCQ MA B 1
23 1 MSQ MA A;C;D 1
24 1 MSQ MA B;D 1
25 1 MSQ MA C;D 1
26 1 MSQ MA B 1
27 1 NAT MA 96 to 96 1
28 1 NAT MA 66 to 66 1
29 1 NAT MA 2 to 2 1
30 1 NAT MA 42.50 to 42.70 1
Page 1 of 2
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31 1 NAT MA 2 to 2 1
32 1 NAT MA 18 to 18 1
33 1 NAT MA 2 to 2 1
34 1 NAT MA 20 to 20 1
35 1 NAT MA 1 to 1 1
36 1 MCQ MA C 2
37 1 MCQ MA B 2
38 1 MCQ MA B 2
39 1 MCQ MA B 2
40 1 MCQ MA C 2
41 1 MCQ MA A 2
42 1 MCQ MA D 2
43 1 MCQ MA C 2
44 1 MCQ MA A 2
45 1 MCQ MA C 2
46 1 MCQ MA B 2
47 1 MCQ MA C 2
48 1 MSQ MA A;B 2
49 1 MSQ MA A;C 2
50 1 MSQ MA A;C 2
51 1 MSQ MA A;B;D 2
52 1 MSQ MA B;C 2
53 1 MSQ MA A;C 2
54 1 MSQ MA A;C 2
55 1 MSQ MA B;D 2
56 1 MSQ MA C;D 2
57 1 MSQ MA A;D 2
58 1 MSQ MA A;C 2
59 1 MSQ MA A;B;C 2
60 1 MSQ MA A;D 2
61 1 NAT MA 126 to 126 2
62 1 NAT MA 6 to 6 2
63 1 NAT MA 8 to 8 2
64 1 NAT MA 3 to 3 2
65 1 NAT MA 8 to 8 2
Page 2 of 2
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Source: GATE. Question paper published by GATE.