GATE Mathematics 2023
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Mathematics (MA)
General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each
Q.1 The village was nestled in a green spot, _______ the ocean and the hills.
(A) through
(B) in
(C) at
(D) between
Q.2 Disagree : Protest : : Agree : _______
(By word meaning)
(A) Refuse
(B) Pretext
(C) Recommend
(D) Refute
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Mathematics (MA)
Q.3 A ‘frabjous’ number is defined as a 3 digit number with all digits odd, and no two
adjacent digits being the same. For example, 137 is a frabjous number, while 133 is
not. How many such frabjous numbers exist?
(A) 125
(B) 720
(C) 60
(D) 80
Q.4 Which one among the following statements must be TRUE about the mean and the
median of the scores of all candidates appearing for GATE 2023?
(A) The median is at least as large as the mean.
(B) The mean is at least as large as the median.
(C) At most half the candidates have a score that is larger than the median.
(D) At most half the candidates have a score that is larger than the mean.
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Mathematics (MA)
Q.5 In the given diagram, ovals are marked at different heights (h) of a hill. Which one
of the following options P, Q, R, and S depicts the top view of the hill?
(A) P
(B) Q
(C) R
(D) S
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Mathematics (MA)
Q.6 – Q.10 Carry TWO marks Each
Q.6 Residency is a famous housing complex with many well-established individuals
among its residents. A recent survey conducted among the residents of the complex
revealed that all of those residents who are well established in their respective fields
happen to be academicians. The survey also revealed that most of these
academicians are authors of some best-selling books.
Based only on the information provided above, which one of the following
statements can be logically inferred with certainty?
(A) Some residents of the complex who are well established in their fields are also
authors of some best-selling books.
(B) All academicians residing in the complex are well established in their fields.
(C) Some authors of best-selling books are residents of the complex who are well
established in their fields.
(D) Some academicians residing in the complex are well established in their fields.
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Mathematics (MA)
Q.7 Ankita has to climb 5 stairs starting at the ground, while respecting the following
rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let F(N) denote the number of possible ways in which Ankita can reach the N tℎ
stair. For example, F(1) = 1, F(2) = 2, F(3) = 3.
The value of F(5) is _______.
(A) 8
(B) 7
(C) 6
(D) 5
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Mathematics (MA)
Q.8 The information contained in DNA is used to synthesize proteins that are necessary
for the functioning of life. DNA is composed of four nucleotides: Adenine (A),
Thymine (T), Cytosine (C), and Guanine (G). The information contained in DNA
can then be thought of as a sequence of these four nucleotides: A, T, C, and G. DNA
has coding and non-coding regions. Coding regions—where the sequence of these
nucleotides are read in groups of three to produce individual amino
acids—constitute only about 2% of human DNA. For example, the triplet of
nucleotides CCG codes for the amino acid glycine, while the triplet GGA codes for
the amino acid proline. Multiple amino acids are then assembled to form a protein.
Based only on the information provided above, which of the following statements
can be logically inferred with certainty?
(i) The majority of human DNA has no role in the synthesis of proteins.
(ii) The function of about 98% of human DNA is not understood.
(A) only (i)
(B) only (ii)
(C) both (i) and (ii)
(D) neither (i) nor (ii)
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Mathematics (MA)
Q.9 Which one of the given figures P, Q, R and S represents the graph of the following
function?
f(x) = | |x + 2| − |x − 1| |
(A) P
(B) Q
(C) R
(D) S
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Mathematics (MA)
Q.10 An opaque cylinder (shown below) is suspended in the path of a parallel beam of
light, such that its shadow is cast on a screen oriented perpendicular to the direction
of the light beam. The cylinder can be reoriented in any direction within the light
beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT
possible?
(A) P
(B) Q
(C) R
(D) S
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Mathematics (MA)
USEFUL DATA
ℕ Set of all positive integers
ℤ Set of all integers
ℚ Set of all rational numbers
ℝ Set of all real numbers
ℂ Set of all complex numbers
ℝn {(x1 , x2 , … , xn ) ∶ xi ∈ ℝ, i = 1,2, … , n}
ℝ×ℝ Cartesian product of ℝ with ℝ
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Q.11 – Q.35 Carry ONE mark Each
Q.11 Let f, g ∶ ℝ2 → ℝ be defined by
3
f(x, y) = x 2 − 2 xy 2 and g(x, y) = 4x 4 − 5x 2 y + y 2
for all (x, y) ∈ ℝ2 .
Consider the following statements:
P: f has a saddle point at (0,0).
Q: g has a saddle point at (0,0).
Then
(A) both P and Q are TRUE
(B) P is FALSE but Q is TRUE
(C) P is TRUE but Q is FALSE
(D) both P and Q are FALSE
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Mathematics (MA)
Q.12 Let ℝ3 be a topological space with the usual topology and ℚ denote the set of
rational numbers. Define the subspaces X, Y, Z and W of ℝ3 as follows:
X = {(x, y, z) ∈ ℝ3 ∶ |x| + |y| + |z| ∈ ℚ}
Y = {(x, y, z) ∈ ℝ3 ∶ xyz = 1}
Z = {(x, y, z) ∈ ℝ3 ∶ x 2 + y 2 + z 2 = 1}
W = {(x, y, z) ∈ ℝ3 ∶ xyz = 0 }
Which of the following statements is correct?
(A) X is homeomorphic to Y
(B) Z is homeomorphic to W
(C) Y is homeomorphic to W
(D) X is NOT homeomorphic to W
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Mathematics (MA)
Q.13 Let P(x) = 1 + e 2πix + 2 e 3πix , x ∈ ℝ, i = √−1. Then
N−1
1
lim ∑ P( k√2 )
N→∞ N
k=0
is equal to
(A) 0
(B) 1
(C) 3
(D) 4
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Mathematics (MA)
Q.14 Let T ∶ ℝ3 → ℝ3 be a linear transformation satisfying
T(1, 0, 0) = (0, 1, 1), T(1, 1, 0) = (1, 0, 1) and T(1, 1, 1) = (1, 1, 2).
Then
(A) T is one-one but T is NOT onto
(B) T is one-one and onto
(C) T is NEITHER one-one NOR onto
(D) T is NOT one-one but T is onto
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Mathematics (MA)
Q.15 Let D = { z ∈ ℂ ∶ |z| < 1} and f: D → ℂ be defined by
z 5 z 7 z 9 z11
f(z) = z – 25z 3 + − + −
5! 7! 9! 11!
Consider the following statements:
P: f has three zeros (counting multiplicity) in D .
1
Q: f has one zero in U = { z ∈ ℂ ∶ 2 < |z| < 1}.
Then
(A) P is TRUE but Q is FALSE
(B) P is FALSE but Q is TRUE
(C) both P and Q are TRUE
(D) both P and Q are FALSE
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Mathematics (MA)
Q.16 Let N ⊆ ℝ be a non-measurable set with respect to the Lebesgue measure on ℝ.
Consider the following statements:
P: If M = { x ∈ N ∶ x is irrational }, then M is Lebesgue measurable.
Q: The boundary of N has positive Lebesgue outer measure.
Then
(A) both P and Q are TRUE
(B) P is FALSE and Q is TRUE
(C) P is TRUE and Q is FALSE
(D) both P and Q are FALSE
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Mathematics (MA)
Q.17 1
For k ∈ ℕ, let Ek be a measurable subset of [0,1] with Lebesgue measure k 2 .
Define
∞ ∞
E = ∩∞ ∞
n=1 ∪k=n Ek and F = ∪n=1 ∩k=n Ek
Consider the following statements:
P: Lebesgue measure of E is equal to zero.
Q: Lebesgue measure of F is equal to zero.
Then
(A) both P and Q are TRUE
(B) both P and Q are FALSE
(C) P is TRUE but Q is FALSE
(D) Q is TRUE but P is FALSE
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Mathematics (MA)
Q.18 Consider ℝ2 with the usual Euclidean metric. Let
1
X = {(x, x sin x) ∈ ℝ2 ∶ x ∈ (0,1]} ⋃ {(0, y) ∈ ℝ2 : −∞ < y < ∞} and
1
Y = {(x, sin x) ∈ ℝ2 : x ∈ (0,1]} ⋃ {(0, y) ∈ ℝ2 : −∞ < y < ∞}.
Consider the following statements:
P: X is a connected subset of ℝ2 .
Q: Y is a connected subset of ℝ2 .
Then
(A) both P and Q are TRUE
(B) P is FALSE and Q is TRUE
(C) P is TRUE and Q is FALSE
(D) both P and Q are FALSE
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Q.19 4 −3
Let M = [ ].
1 0
Consider the following statements:
P: M8 + M12 is diagonalizable.
Q: M7 + M9 is diagonalizable.
Which of the following statements is correct?
(A) P is TRUE and Q is FALSE
(B) P is FALSE and Q is TRUE
(C) Both P and Q are FALSE
(D) Both P and Q are TRUE
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Q.20 Let C[0, 1] = { f ∶ [0, 1] → ℝ ∶ f is continuous}.
Consider the metric space (C[0,1], d∞ ), where
d∞ (f, g) = sup{ |f(x) − g(x)| ∶ x ∈ [0, 1] } for f, g ∈ C[0,1].
Let f0 (x) = 0 for all x ∈ [0,1] and
1
X = {f ∈ (C[0, 1], d∞ ) ∶ d∞ (f0 , f) ≥ } .
2
Let f1 , f2 ∈ C[0, 1] be defined by f1 (x) = x and f2 (x) = 1 − x for all x ∈ [0,1].
Consider the following statements:
P: f1 is in the interior of X.
Q: f2 is in the interior of X.
Which of the following statements is correct?
(A) P is TRUE and Q is FALSE
(B) P is FALSE and Q is TRUE
(C) Both P and Q are FALSE
(D) Both P and Q are TRUE
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Mathematics (MA)
Q.21 Consider the metrics ρ1 and ρ2 on ℝ, defined by
0, if x = y
ρ1 (x, y) = |x − y| and ρ2 (x, y) = {
1, if x ≠ y
1
Let X = {n ∈ ℕ ∶ n ≥ 3} and Y = { n + n ∶ n ∈ ℕ}.
2, if x ∈ X
Define f: X ∪ Y → ℝ by f(x) = {
3, if x ∈ Y
Consider the following statements:
P: The function f: (X ∪ Y, ρ1 ) → (ℝ, ρ1 ) is uniformly continuous.
Q: The function f: (X ∪ Y, ρ2 ) → (ℝ, ρ1 ) is uniformly continuous.
Then
(A) P is TRUE and Q is FALSE
(B) P is FALSE and Q is TRUE
(C) both P and Q are FALSE
(D) both P and Q are TRUE
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Mathematics (MA)
Q.22 Let T ∶ ℝ4 → ℝ4 be a linear transformation and the null space of T be the
subspace of ℝ4 given by
{ (x1 , x2 , x3 , x4 ) ∈ ℝ4 ∶ 4x1 + 3x2 + 2x3 + x4 = 0}.
If Rank(T − 3I) = 3, where I is the identity map on ℝ4 , then the minimal
polynomial of T is
(A) x(x − 3)
(B) x(x − 3)3
(C) x 3 (x − 3)
(D) x 2 (x − 3)2
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Mathematics (MA)
Let C[0,1] denote the set of all real valued continuous functions defined on [0,1]
Q.23
and ‖f‖∞ = sup{|f(x)| ∶ x ∈ [0,1]} for all f ∈ C[0,1]. Let
X = { f ∈ C[0,1] ∶ f(0) = f(1) = 0 }.
1
Define F ∶ (C[0,1], ‖⋅‖∞ ) → ℝ by F(f) = ∫0 f(t)dt for all f ∈ C[0,1].
Denote SX = {f ∈ X ∶ ‖f‖∞ = 1}.
Then the set {f ∈ X ∶ F(f) = ‖F‖} ∩ SX has
(A) NO element
(B) exactly one element
(C) exactly two elements
(D) an infinite number of elements
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Mathematics (MA)
Q.24 Let X and Y be two topological spaces. A continuous map f ∶ X → Y is said to be
proper if f −1 (K) is compact in X for every compact subset K of Y, where f −1 (K)
is defined by f −1 (K) = {x ∈ X ∶ f(x) ∈ K} .
Consider ℝ with the usual topology. If ℝ ∖ {0} has the subspace topology induced
from ℝ and ℝ × ℝ has the product topology, then which of the following maps is
proper?
(A) f: ℝ ∖ {0} → ℝ defined by f(x) = x
(B) f: ℝ × ℝ → ℝ × ℝ defined by f(x, y) = (x + y, y)
(C) f: ℝ × ℝ → ℝ defined by f(x, y) = x
(D) f: ℝ × ℝ → ℝ defined by f(x, y) = x 2 − y 2
Q.25 Consider the following Linear Programming Problem P:
Minimize 3x1 + 4x2
subject to x1 − x2 ≤ 1,
x1 + x2 ≥ 3,
x1 ≥ 0, x2 ≥ 0.
The optimal value of the problem P is _____________.
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Mathematics (MA)
Q.26 Let u(x, t) be the solution of
∂ 2u 1 ∂ 2u
− = 0, x ∈ (−∞, ∞), t > 0,
∂x 2 c 2 ∂t 2
u(x, 0) = sin x, x ∈ (−∞, ∞),
∂u
(x, 0) = cos x, x ∈ (−∞, ∞),
∂t
for some positive real number c.
Let the domain of dependence of the solution u at the point P(3,2) be the line
segment on the x-axis with end points Q and R.
If the area of the triangle PQR is 8 square units, then the value of c 2 is ___________.
Q.27 Let
∞
z
= ∑ an z n , an ∈ ℝ
1 − z − z2
n=0
for all z in some neighbourhood of 0 in ℂ.
Then the value of a6 + a5 is equal to _________.
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Mathematics (MA)
Q.28 Let p(x) = x 3 − 2x + 2. If q(x) is the interpolating polynomial of degree less than
or equal to 4 for the data
x −2 −1 0 1 3
,
q(x) p(−2) p(−1) 2.5 p(1) p(3)
d4q
then the value of at x = 0 is ______________.
dx 4
Q.29 2
For a fixed c ∈ ℝ, let α = ∫0 (9x 2 − 5cx 4 )dx.
2
If the value of ∫0 (9x 2 − 5cx 4 )dx obtained by using the Trapezoidal rule is equal
to α, then the value of c is __________ (rounded off to 2 decimal places).
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Mathematics (MA)
Q.30 If for some α ∈ ℝ,
π
4 x α sec θ
1 4 1
∫ ∫ 2 2
dydx = ∫ ∫ drdθ,
1 −x x + y r
π
− sec θ
4
then the value of α equals _______.
Q.31 Let S be the portion of the plane z = 2x + 2y − 100 which lies inside the cylinder
x 2 + y 2 = 1. If the surface area of S is απ, then the value of α is equal to
___________.
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Mathematics (MA)
Q.32 Let
1
L2 [−1, 1] = {f: [−1, 1] → ℝ ∶ f is Lebesgue measurable and ∫−1|f(x)|2 dx < ∞}
1
1 2
and the norm ‖f‖2 = (∫−1|f(x)|2 dx) for f ∈ L2 [−1,1].
Let F: (L2 [−1, 1], ‖⋅‖2 ) → ℝ be defined by
1
F(f) = ∫−1 f(x)x 2 dx for all f ∈ L2 [−1,1] .
If ‖F‖ denotes the norm of the linear functional F, then 5‖F‖2 is equal to
_________.
Q.33 Let y(t) be the solution of the initial value problem
t, 0 ≤ t ≤ 2,
y ′′ + 4y = { and y(0) = y ′ (0) = 0.
2, 2 < t < ∞,
π 4
If α = y ( 2 ), then the value of π α is ________(rounded off to 2 decimal places).
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Mathematics (MA)
Q.34 Consider ℝ4 with the inner product < x, y > = ∑4i=1 xi yi , for x = (x1 , x2 , x3 , x4 )
and y = (y1 , y2 , y3 , y4 ).
Let M = {(x1 , x2 , x3 , x4 ) ∈ ℝ4 ∶ x1 = x3 } and M⊥ denote the orthogonal
complement of M. The dimension of M⊥ is equal to ________.
Q.35 3 −1 −2 1 0 0
Let M = [ 0 2 4 ] and I = [0 1 0]. If 6M −1 = M2 − 6M + αI for
0 0 1 0 0 1
some α ∈ ℝ, then the value of α is equal to ___________.
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Q.36 – Q.65 Carry TWO marks Each
Q.36 Let GL2 (ℂ) denote the group of 2 × 2 invertible complex matrices with usual
matrix multiplication. For S, T ∈ GL2 (ℂ), < S, T > denotes the subgroup
0 −1
generated by S and T. Let S = [ ] ∈ GL2 (ℂ) and G1 , G2 , G3 be three
1 0
subgroups of GL2 (ℂ) given by
i 0
G1 = < S, T1 >, where T1 = [ ],
0 i
i 0
G2 = < S, T2 >, where T2 = [ ],
0 −i
0 1
G3 = < S, T3 >, where T3 = [ ].
1 0
Let Z(Gi ) denote the center of Gi for i = 1, 2, 3.
Which of the following statements is correct?
(A) G1 is isomorphic to G3
(B) Z(G1 ) is isomorphic to Z(G2 )
(C) 1 0
Z(G3 ) = { [ ] }
0 1
(D) Z(G2 ) is isomorphic to Z(G3 )
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Q.37 Let ℓ2 = {(x1 , x2 , x3 , … ) ∶ xn ∈ ℝ for all n ∈ ℕ and ∑∞ 2
n=1 xn < ∞}.
1
For a sequence (x1 , x2 , x3 , … ) ∈ ℓ2 , define ‖(x1 , x2 , x3 , … )‖2 = (∑∞ 2
n=1 xn )2 .
Let S ∶ (ℓ2 , ‖⋅‖2 ) → (ℓ2 , ‖⋅‖2 ) and T ∶ (ℓ2 , ‖⋅‖2 ) → (ℓ2 , ‖⋅‖2 ) be defined by
0, n = 1
S(x1 , x2 , x3 , … ) = (y1 , y2 , y3 , … ), where yn = {
xn−1 , n ≥ 2
0, n is odd
T(x1 , x2 , x3 , … ) = (y1 , y2 , y3 , … ), where yn = {
xn , n is even
Then
(A) S is a compact linear map and T is NOT a compact linear map
(B) S is NOT a compact linear map and T is a compact linear map
(C) both S and T are compact linear maps
(D) NEITHER S NOR T is a compact linear map
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Q.38 Let
c00 = {(x1 , x2 , x3 , … ) ∶ xi ∈ ℝ, i ∈ ℕ, xi ≠ 0 only for finitely many indices i}.
For (x1 , x2 , x3 , … ) ∈ c00 , let ‖(x1 , x2 , x3 , … )‖∞ = sup{|xi | ∶ i ∈ ℕ}.
Define F, G ∶ (c00 , ‖⋅‖∞ ) → (c00 , ‖⋅‖∞ ) by
1 1
F((x1 , x2 , … , xn , … )) = ((1 + 1)x1 , (2 + 2) x2 , …, (n + n) xn , … ),
x1 x2 xn
G((x1 , x2 , … , xn , … )) = ( , , …, ,… ),
1+1 2+1 n +
1
2 n
for all (x1 , x2 , … , xn , … ) ∈ c00 .
Then
(A) F is continuous but G is NOT continuous
(B) F is NOT continuous but G is continuous
(C) both F and G are continuous
(D) NEITHER F NOR G is continuous
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Q.39 Consider the Cauchy problem
∂u ∂u
x +y = u;
∂x ∂y
u = f(t) on the initial curve Γ = (t, t); t > 0.
Consider the following statements:
P: If f(t) = 2t + 1, then there exists a unique solution to the Cauchy problem
in a neighbourhood of Γ.
Q: If f(t) = 2t − 1, then there exist infinitely many solutions to the Cauchy
problem in a neighbourhood of Γ.
Then
(A) both P and Q are TRUE
(B) P is FALSE and Q is TRUE
(C) P is TRUE and Q is FALSE
(D) both P and Q are FALSE
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Q.40 2 −1 −2
Consider the linear system Mx = b, where M = [ ] and b = [ ].
−4 3 5
Suppose M = LU, where L and U are lower triangular and upper triangular square
matrices, respectively. Consider the following statements:
P: If each element of the main diagonal of L is 1, then trace(U) = 3.
Q: For any choice of the initial vector x (0) , the Jacobi iterates x (k) , k = 1,2,3 …
converge to the unique solution of the linear system Mx = b.
Then
(A) both P and Q are TRUE
(B) P is FALSE and Q is TRUE
(C) P is TRUE and Q is FALSE
(D) both P and Q are FALSE
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Q.41 Let φ and ψ be two linearly independent solutions of the ordinary differential
equation
y ′′ + (2 − cos x) y = 0, x ∈ℝ.
Let α, β ∈ ℝ be such that α < β, φ(α) = φ(β) = 0 and φ(x) ≠ 0 for all
x ∈ (α, β).
Consider the following statements:
P: φ ′ (α)φ ′ (β) > 0.
Q: φ(x)ψ(x) ≠ 0 for all x ∈ (α, β).
Then
(A) P is TRUE and Q is FALSE
(B) P is FALSE and Q is TRUE
(C) both P and Q are FALSE
(D) both P and Q are TRUE
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Q.42 Let D = { z ∈ ℂ ∶ |z| < 1} and f: D → ℂ be an analytic function given by the
1
power series f(z) = ∑∞ n
n=0 an z , where a0 = a1 = 1 and an = 22n for n ≥ 2.
Consider the following statements:
P: If z0 ∈ D, then f is one-one in some neighbourhood of z0 .
1
Q: If E = { z ∈ ℂ ∶ |z| ≤ 2}, then f(E) is a closed subset of ℂ.
Which of the following statements is/are correct?
(A) P is TRUE
(B) Q is TRUE
(C) Q is FALSE
(D) P is FALSE
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Q.43 1
Let Ω be an open connected subset of ℂ containing U = { z ∈ ℂ ∶ |z| ≤ 2}.
Let J = { f ∶ Ω → ℂ ∶ f is analytic and sup |f(z) − f(w)| = 1 }.
z,w ∈ U
Consider the following statements:
P: There exists f ∈ ℑ such that |f ′ (0)| ≥ 2.
Q: |f (3) (0)| ≤ 48 for all f ∈ ℑ, where f (3) denotes the third derivative of f.
Then
(A) P is TRUE
(B) Q is FALSE
(C) P is FALSE
(D) Q is TRUE
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Q.44 Let (ℝ, τ) be a topological space, where the topology τ is defined as
τ = {U ⊆ ℝ ∶ U = ∅ or 1 ∈ U}.
Which of the following statements is/are correct?
(A) (ℝ, τ) is first countable
(B) (ℝ, τ) is Hausdorff
(C) (ℝ, τ) is separable
(D) The closure of (1,5) is [1,5]
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Mathematics (MA)
Q.45 Let ℛ = {p(x) ∈ ℚ[x] ∶ p(0) ∈ ℤ}, where ℚ denotes the set of rational numbers
and ℤ denotes the set of integers. For a ∈ ℛ, let ⟨a⟩ denote the ideal generated by
a in ℛ.
Which of the following statements is/are correct?
(A) If p(x) is an irreducible element in ℛ, then ⟨p(x)⟩ is a prime ideal in ℛ
(B) ℛ is a unique factorization domain
(C) ⟨x⟩ is a prime ideal in ℛ
(D) ℛ is NOT a principal ideal domain
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Mathematics (MA)
Q.46 Consider the rings
ℤ2 [x]
S1 = ℤ[x]⁄⟨2, x 3 ⟩ and S2 = ⁄⟨x 2 ⟩
where ⟨2, x 3 ⟩ denotes the ideal generated by {2, x 3 } in ℤ[x] and ⟨x 2 ⟩ denotes the
ideal generated by x 2 in ℤ2 [x].
Which of the following statements is/are correct?
(A) Every prime ideal of S1 is a maximal ideal
(B) S2 has exactly one maximal ideal
(C) Every element of S1 is either nilpotent or a unit
(D) There exists an element in S2 which is NEITHER nilpotent NOR a unit
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Mathematics (MA)
Q.47 Consider the sequence of Lebesgue measurable functions fn : ℝ → ℝ given by
2 (x
1
fn (x) = {n − n), if x ∈ [n, n + ]
n2
0, otℎerwise
For a measurable subset E of ℝ, denote m(E) to be the Lebesgue measure of E.
Which of the following statements is/are correct?
(A) sup |fn (x)| → 0 as n → ∞
x∈ℝ
(B) ∫ℝ|fn (x)|dx → 0 as n → ∞
(C) 1
m({x ∈ ℝ ∶ |fn (x)| > 2}) → 0 as n → ∞
(D) m({x ∈ ℝ ∶ |fn (x)| > 0}) → 0 as n → ∞
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Mathematics (MA)
Q.48 Define the characteristic function χE of a subset E in ℝ by
1, if x ∈ E
χE (x) = {
0, if x ∉ E
For 1 ≤ p < 2, let
1
Lp [0, 1] = {f: [0, 1] → ℝ ∶ f is Lebesgue measurable and ∫0 |f(x)|p dx < ∞}.
Let f: [0, 1] → ℝ be defined by
∞
2n
f(x) = ∑ 3 χ[ 1 , 1 ] (x).
n 2n+1 2n
n=1
Consider the following two statements:
P: f ∈ Lp [0, 1] for every p ∈ (1, 2).
Q: f ∈ L1 [0, 1].
Then
(A) P is TRUE
(B) Q is TRUE
(C) Q is FALSE
(D) P is FALSE
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Mathematics (MA)
Q.49 Let x(t), y(t), t ∈ ℝ, be two functions satisfying the following system of
differential equations:
x ′ (t) = y(t),
y ′ (t) = x(t),
and x(0) = α, y(0) = β, where α, β are real numbers.
Which of the following statements is/are correct?
(A) If α = 1, β = −1, then |x(t)| + |y(t)| → 0 as t → ∞
(B) If α = 1, β = 1, then |x(t)| + |y(t)| → 0 as t → ∞
(C) If α = 1.01, β = −1, then |x(t)| + |y(t)| → 0 as t → ∞
(D) If α = 1, β = 1.01, then |x(t)| + |y(t)| → 0 as t → ∞
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Mathematics (MA)
Q.50 For ℎ > 0, and α, β, γ ∈ ℝ, let
α f(a − ℎ) + βf(a) + γf(a + 2ℎ)
Dℎ f(a) =
6ℎ
be a three-point formula to approximate f ′ (a) for any differentiable function
f: ℝ → ℝ and a ∈ ℝ.
If Dℎ f(a) = f ′ (a) for every polynomial f of degree less than or equal to 2 and
for all a ∈ ℝ, then
(A) α + 2γ = −2
(B) α + 2β − 2γ = 0
(C) α + 2γ = 2
(D) α + 2β − 2γ = 1
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Mathematics (MA)
Q.51 Let f be a twice continuously differentiable function on [a, b] such that f ′ (x) < 0
and f ′′ (x) < 0 for all x ∈ (a, b). Let f(ζ) = 0 for some ζ ∈ (a, b). The Newton-
Raphson method to compute ζ is given by
f(xk )
xk+1 = xk − , k = 0, 1, 2, …
f ′ (xk )
for an initial guess x0 .
If xk ∈ (ζ, b) for some k ≥ 0, then which of the following statements is/are
correct?
(A) xk+1 > ζ
(B) xk+1 < ζ
(C) xk+1 < xk
(D) f ′ (η)
For every η ∈ (ζ, xk ), f′ (x ) > 1
k
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Mathematics (MA)
Q.52 Let f: ℝ2 → ℝ be defined by
2x 2 y
, (x, y) ≠ (0, 0)
f(x, y) = { x 2 + y 2
0, (x, y) = (0, 0)
Then
(A) 1 1 1
the directional derivative of f at (0,0) in the direction of ( , ) is
√2 √2 √2
(B) the directional derivative of f at (0,0) in the direction of (0, 1 ) is 1
(C) the directional derivative of f at (0,0) in the direction of (1, 0) is 0
(D) f is NOT differentiable at (0,0)
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Mathematics (MA)
Q.53 Let C[0,1] = { f: [0, 1] → ℝ ∶ f is continuous} and
d∞ (f, g) = sup{ |f(x) − g(x)|: x ∈ [0, 1]} for f, g ∈ C[0,1].
For each n ∈ ℕ, define fn : [0,1] → ℝ by fn (x) = x n for all x ∈ [0, 1].
Let P = {fn : n ∈ ℕ}.
Which of the following statements is/are correct?
(A) P is totally bounded in (C[0, 1], d∞ )
(B) P is bounded in (C[0, 1], d∞ )
(C) P is closed in (C[0, 1], d∞ )
(D) P is open in (C[0, 1], d∞ )
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Mathematics (MA)
Q.54 Let G be an abelian group and Φ ∶ G → (ℤ, +) be a surjective group
homomorphism. Let 1 = Φ(a) for some a ∈ G.
Consider the following statements:
P: For every g ∈ G, there exists an n ∈ ℤ such that gan ∈ ker(Φ).
Q: Let e be the identity of G and < a > be the subgroup generated by a. Then
G = ker(Φ) < a > and ker(Φ) ∩ < a > = {e}.
Which of the following statements is/are correct?
(A) P is TRUE
(B) P is FALSE
(C) Q is TRUE
(D) Q is FALSE
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Mathematics (MA)
Q.55 Let C be the curve of intersection of the cylinder x 2 + y 2 = 4 and the plane
z − 2 = 0. Suppose C is oriented in the counterclockwise direction around the
z-axis, when viewed from above. If
|∫(sin x + e x ) dx + 4x dy + e z cos2 z dz| = απ,
C
then the value of α equals ____________.
Let ℓ2 = {(x1 , x2 , x3 , … ) ∶ xn ∈ ℝ for all n ∈ ℕ and ∑∞ 2
n=1 xn < ∞}.
Q.56
For a sequence (x1 , x2 , x3 , … ) ∈ ℓ2 , define
1
∞ 2
‖(x1 , x2 , x3 , … )‖2 = ( ∑ xn2 )
n=1
x
Consider the subspace M = {(x1 , x2 , x3 , … ) ∈ ℓ2 ∶ ∑∞ n
n=1 n = 0} . 4
Let M denote the orthogonal complement of M in the Hilbert space (ℓ2 , ‖⋅‖2 ).
⊥
1 1 1
Consider (1, 2 , 3 , 4 , … ) ∈ ℓ2 .
1 1 1
If the orthogonal projection of (1, 2 , 3 , 4 , … ) onto M⊥ is given by
1 1 1 1
α (∑∞
n=1 n4n ) (4 , 42 , 43 , … ) for some α ∈ ℝ, then α equals _________.
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Mathematics (MA)
Q.57 Consider the transportation problem between five sources and four destinations as
given in the cost table below. The supply and demand at each of the source and
destination are also provided:
DESTINATIONS Supply
P Q R S
1 13 8 12 9 20
SOURCES
2 10 7 5 20 10
3 3 19 5 12 50
4 4 9 7 15 30
5 14 0 1 7 40
Demand 60 10 20 60
Let CN and CL be the total cost of the initial basic feasible solution obtained from
the North-West corner method and the Least-Cost method, respectively. Then
CN − CL equals _________.
Q.58 Let σ ∈ S8 , where S8 is the permutation group on 8 elements. Suppose σ is the
product of σ1 and σ2 , where σ1 is a 4-cycle and σ2 is a 3-cycle in S8 . If σ1 and σ2
are disjoint cycles, then the number of elements in S8 which are conjugate to σ is
_________.
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Mathematics (MA)
Q.59 Let A be a 3 × 3 real matrix with det(A + i I) = 0, where i = √−1 and I is the
3 × 3 identity matrix. If det(A) = 3, then the trace of A2 is __________.
Q.60 Let A = [aij ] be a 3 × 3 real matrix such that
1 1 0 0 −1 −1
A [2] = 2 [2] , A [1] = 2 [1] and A [ 1 ] = 4 [ 1 ] .
1 1 1 1 0 0
If m is the degree of the minimal polynomial of A, then a11 + a21 + a31 + m
equals __________.
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Mathematics (MA)
Q.61 Let Ω be the disk x 2 + y 2 < 4 in ℝ2 with boundary ∂Ω. If u(x, y) is the solution
of the Dirichlet problem
∂ 2u ∂ 2u
+ = 0, (x, y) ∈ Ω,
∂x 2 ∂y 2
u(x, y) = 1 + 2 x 2 , (x, y) ∈ ∂Ω,
then the value of u(0,1) is _____________.
Q.62 For every k ∈ ℕ ∪ {0}, let yk (x) be a polynomial of degree k with yk (1) = 5.
Further, let yk (x) satisfy the Legendre equation
(1 − x 2 )y ′′ − 2xy ′ + k(k + 1)y = 0.
If
1 n 1 n
1 2 2
∫ ∑(yk (x) − yk−1 (x)) dx − ∫ ∑(yk (x)) dx = 24,
2
−1 k=1 −1 k=1
for some positive integer n, then the value of n is ____________.
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Mathematics (MA)
Q.63 Consider the ordinary differential equation (ODE)
4 (ln x) y ′′ + 3 y ′ + y = 0, x > 1.
If r1 and r2 are the roots of the indicial equation of the above ODE at the regular
singular point x = 1, then |r1 − r2 | is equal to _________ (rounded off to 2 decimal
places).
Q.64 Let u(x, t) be the solution of the non-homogeneous wave equation
∂ 2u ∂ 2u
− = sin x sin(2t), 0 < x < π, t > 0
∂x 2 ∂t 2
∂u
u(x, 0) = 0, and (x, 0) = 0, for 0 ≤ x ≤ π,
∂t
u(0, t) = 0, u(π, t) = 0, for t ≥ 0.
π 3π
Then the value of u ( 2 , 2 ) is ____________ (rounded off to 2 decimal places).
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Mathematics (MA)
Q.65 Consider the Linear Programming Problem P:
Maximize 3x1 + 2x2 + 5x3
subject to
x1 + 2x2 + x3 ≤ 44,
x1 + 2x3 ≤ 48,
x1 + 4x2 ≤ 52,
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
The optimal value of the problem P is equal to ____________.
END OF QUESTION PAPER
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GATE 2023 Mathematics (MA)
Question Type (QT)
Q. No. Session Subject Name (SN) Key/Range (KY) Mark (MK)
MCQ/MSQ/NAT
1 5 MCQ GA D 1
2 5 MCQ GA C 1
3 5 MCQ GA D 1
4 5 MCQ GA C 1
5 5 MCQ GA B 1
6 5 MCQ GA MTA 2
7 5 MCQ GA A 2
8 5 MCQ GA D 2
9 5 MCQ GA A 2
10 5 MCQ GA D 2
11 5 MCQ MA A 1
12 5 MCQ MA D 1
13 5 MCQ MA B 1
14 5 MCQ MA C 1
15 5 MCQ MA A 1
16 5 MCQ MA B 1
17 5 MCQ MA A 1
18 5 MCQ MA A 1
19 5 MCQ MA D 1
20 5 MCQ MA D 1
21 5 MCQ MA B 1
22 5 MCQ MA A 1
23 5 MCQ MA A 1
24 5 MCQ MA B 1
25 5 NAT MA 10 to 10 1
26 5 NAT MA 4 to 4 1
27 5 NAT MA 13 to 13 1
28 5 NAT MA 2 to 2 1
29 5 NAT MA 0.24 to 0.26 1
30 5 NAT MA 4 to 4 1
31 5 NAT MA 3 to 3 1
32 5 NAT MA 2 to 2 1
33 5 NAT MA 0.49 to 0.51 1
34 5 NAT MA 1 to 1 1
35 5 NAT MA 11 to 11 1
36 5 MCQ MA D 2
37 5 MCQ MA D 2
38 5 MCQ MA B 2
39 5 MCQ MA D 2
40 5 MCQ MA A 2
41 5 MCQ MA C 2
42 5 MSQ MA A, B 2
43 5 MSQ MA C, D 2
44 5 MSQ MA A, C 2
45 5 MSQ MA A 2
Text of page 2
46 5 MSQ MA A, B, C 2
47 5 MSQ MA B, C, D 2
48 5 MSQ MA B, D 2
49 5 MSQ MA A 2
50 5 MSQ MA A, B 2
51 5 MSQ MA A, C 2
52 5 MSQ MA A, C, D 2
53 5 MSQ MA B, C 2
54 5 MSQ MA A, C 2
55 5 NAT MA 16 to 16 2
56 5 NAT MA 15 to 15 2
57 5 NAT MA 380 to 380 2
58 5 NAT MA 3360 to 3360 2
59 5 NAT MA 7 to 7 2
60 5 NAT MA 4 to 4 2
61 5 NAT MA 4 to 4 2
62 5 NAT MA 12 to 12 2
63 5 NAT MA 0.24 to 0.26 2
64 5 NAT MA 0.64 to 0.70 2
65 5 NAT MA 140 to 140 2
MTA = Marks to All
Page 1 of 53
Source: GATE. Question paper published by GATE.