GATE Mathematics 2024
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MATHEMATICS - MA
General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each
Q.1 If ‘→’ denotes increasing order of intensity, then the meaning of the words
[drizzle → rain → downpour] is analogous to [ ______ → quarrel → feud].
Which one of the given options is appropriate to fill the blank?
(A) bicker
(B) bog
(C) dither
(D) dodge
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MATHEMATICS - MA
Q.2 Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?
(A) Only I and II
(B) Only II and III
(C) Only I and III
(D) Only III
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MATHEMATICS - MA
Q.3 A student was supposed to multiply a positive real number p with another positive
real number q. Instead, the student divided p by q. If the percentage error in the
student’s answer is 80%, the value of q is
(A) 5
(B) √2
(C) 2
(D) √5
Q.4 If the sum of the first 20 consecutive positive odd numbers is divided by 202 , the
result is
(A) 1
(B) 20
(C) 2
(D) 1/2
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MATHEMATICS - MA
Q.5 The ratio of the number of girls to boys in class VIII is the same as the ratio of the
number of boys to girls in class IX. The total number of students (boys and girls) in
classes VIII and IX is 450 and 360, respectively. If the number of girls in classes
VIII and IX is the same, then the number of girls in each class is
(A) 150
(B) 200
(C) 250
(D) 175
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MATHEMATICS - MA
Q.6 – Q.10 Carry TWO marks Each
Q.6 In the given text, the blanks are numbered (i)−(iv). Select the best match for
all the blanks.
Yoko Roi stands (i) as an author for standing (ii) as an honorary
fellow, after she stood (iii) her writings that stand (iv) the freedom of
speech.
(A) (i) out (ii) down (iii) in (iv) for
(B) (i) down (ii) out (iii) by (iv) in
(C) (i) down (ii) out (iii) for (iv) in
(D) (i) out (ii) down (iii) by (iv) for
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MATHEMATICS - MA
Q.7 Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container.
The figure below shows the arrangement of the chalk-sticks inside the cylinder.
The length of the container is equal to the length of the chalk-sticks. The ratio of
the occupied space to the empty space of the container is
(A) 5/2
(B) 7/2
(C) 9/2
(D) 3
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MATHEMATICS - MA
Q.8 The plot below shows the relationship between the mortality risk of cardiovascula r
disease and the number of steps a person walks per day. Based on the data, which
one of the following options is true?
1
Mortality Risk of Cardiovascular Disease
0.8
0.6
0.4
0.2
0
0 5000 10000 15000 20000
Steps/Day
(A) The risk reduction on increasing the steps/day from 0 to 10000 is less than the risk
reduction on increasing the steps/day from 10000 to 20000.
(B) The risk reduction on increasing the steps/day from 0 to 5000 is less than the risk
reduction on increasing the steps/day from 15000 to 20000.
(C) For any 5000 increment in steps/day the largest risk reduction occurs on going from
0 to 5000.
(D) For any 5000 increment in steps/day the largest risk reduction occurs on going from
15000 to 20000.
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MATHEMATICS - MA
Q.9 Five cubes of identical size and another smaller cube are assembled as shown in
Figure A. If viewed from direction X, the planar image of the assembly appears as
Figure B.
Figure A Figure B
If viewed from direction Y, the planar image of the assembly (Figure A) will
appear as
(A)
(B)
(C)
(D)
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MATHEMATICS - MA
Q.10 Visualize a cube that is held with one of the four body diagonals aligned to the
vertical axis. Rotate the cube about this axis such that its view remains unchanged.
The magnitude of the minimum angle of rotation is
(A) 120°
(B) 60°
(C) 90°
(D) 180°
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GATE 2024 MATHEMATICS - MA
Useful data
Z - the set of integers.
R - the set of real numbers.
C - the set of complex numbers.
Im(z) - imaginary part of the complex number z.
Re(z) - real part of the complex number z.
sup(A) - supremum of the set A ⊆ R.
inf(A) - infimum of the set A ⊆ R.
Mn (F) - the set of n × n matrices over a field F.
GLn (F) - the set of n × n invertible matrices over a field F.
Fn - the n-dimensional vector space over a field F.
In - the n × n identity matrix.
Sn - the symmetric group on n elements.
A \ B = {a ∈ A : a ∈
/ B}.
Z/nZ - the cyclic group of order n.
ln - the natural logarithm.
f ′ , f ′′ , f ′′′ - the first-order, second-order, and third-order derivative, respectively, of a one variable
function f
MA 1 / 33
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GATE 2024 MATHEMATICS - MA
MCQ - 1 Mark
11. Consider the following condition on a function f : C → C
|f (z)| = 1 for all z ∈ C such that Im(z) = 0. (P)
Which one of the following is correct?
(A) There is a non-constant analytic polynomial f satisfying (P)
(B) Every entire function f satisfying (P) is a constant function
(C) Every entire function f satisfying (P) has no zeroes in C
(D) There is an entire function f satisfying (P) with infinitely many zeroes in C
12. Let C be the ellipse {z ∈ C : |z − 2| + |z + 2| = 8} traversed counter-clockwise. The
value of the contour integral
z 2 dz
I
2
C z − 2z + 2
is equal to
(A) 0
(B) 2πi
(C) 4πi
(D) −πi
MA 2 / 33
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GATE 2024 MATHEMATICS - MA
13. Let X be a topological space and A ⊆ X. Given a subset S of X, let int(S), ∂S, and S
denote the interior, boundary, and closure, respectively, of the set S. Which one of the
following is NOT necessarily true?
(A) int(X \ A) ⊆ X \ A
(B) A⊆A
(C) ∂A ⊆ ∂ (int(A))
(D) ∂ A ⊆ ∂A
14. Consider the following limit
1 ∞ −x/ε √
Z
lim e cos(3x) + x2 + x + 4 dx.
ε→0 ε 0
Which one of the following is correct?
(A) The limit does not exist
(B) The limit exists and is equal to 0
(C) The limit exists and is equal to 3
(D) The limit exists and is equal to π
MA 3 / 33
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15. Let R[X 2 , X 3 ] be the subring of R[X] generated by X 2 and X 3 . Consider the following
statements.
I. The ring R[X 2 , X 3 ] is a unique factorization domain.
II. The ring R[X 2 , X 3 ] is a principal ideal domain.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
16. Given a prime number p, let np (G) denote the number of p–Sylow subgroups of a finite
group G. Which one of the following is TRUE for every group G of order 2024?
(A) n11 (G) = 1 and n23 (G) = 11
(B) n11 (G) ∈ {1, 23} and n23 (G) = 1
(C) n11 (G) = 23 and n23 (G) ∈ {1, 88}
(D) n11 (G) = 23 and n23 (G) = 11
MA 4 / 33
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17. Consider the following statements.
I. Every compact Hausdorff space is normal.
II. Every metric space is normal.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
18. Consider the topology on Z with basis {S(a, b) : a, b ∈ Z and a ̸= 0}, where
S(a, b) = {an + b : n ∈ Z}.
Consider the following statements.
I. S(a, b) is both open and closed for each a, b ∈ Z with a ̸= 0.
II. The only connected set containing x ∈ Z is {x}.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
MA 5 / 33
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0 2
19. Let A = and T : M2 (C) → M2 (C) be the linear transformation given by
2 0
T (B) = AB. The characteristic polynomial of T is
(A) X 4 − 8X 2 + 16
(B) X2 − 4
(C) X2 − 2
(D) X 4 − 16
20. Let A ∈ Mn (C) be a normal matrix. Consider the following statements.
I. If all the eigenvalues of A are real, then A is Hermitian.
II. If all the eigenvalues of A have absolute value 1, then A is unitary.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
MA 6 / 33
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21. Let
2 −1 1
A= 1 −1
2
−1 1 2
and b be a 3 × 1 real column vector. Consider the statements.
I. The Jacobi iteration method for the system (A + εI3 )x = b converges for any
initial approximation and ε > 0.
II. The Gauss–Seidel iteration method for the system (A + εI3 )x = b converges for
any initial approximation and ε > 0.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
MA 7 / 33
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22. For the initial value problem
y ′ = f (x, y), y(x0 ) = y0 ,
generate approximations yn to y(xn ), xn = x0 + nh, for a fixed h > 0 and n =
1, 2, 3, . . ., using the recursion formula
yn = yn−1 + ak1 + bk2 , where
k1 = hf (xn−1 , yn−1 ) and k2 = hf (xn−1 + αh, yn−1 + βk1 ).
Which one of the following choices of a, b, α, β for the above recursion formula gives
the Runge–Kutta method of order 2 ?
(A) a = 1, b = 1, α = 0.5, β = 0.5
(B) a = 0.5, b = 0.5, α = 2, β = 2
(C) a = 0.25, b = 0.75, α = 2/3, β = 2/3
(D) a = 0.5, b = 0.5, α = 1, β = 2
MA 8 / 33
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23. Let u = u(x, t) be the solution of
∂u ∂ 2u
− 4 2 = 0, 0 < x < 1, t > 0,
∂t ∂x
u(0, t) = u(1, t) = 0, t ≥ 0,
u(x, 0) = sin(πx), 0 ≤ x ≤ 1.
R1
Define g(t) = (u(x, t))2 dx, for t > 0. Which one of the following is correct?
0
(A) g is decreasing on (0, ∞) and lim g(t) = 0
t→∞
(B) g is decreasing on (0, ∞) and lim g(t) = 1/4
t→∞
(C) g is increasing on (0, ∞) and lim g(t) does not exist
t→∞
(D) g is increasing on (0, ∞) and lim g(t) = 3
t→∞
24. If y1 and y2 are two different solutions of the ordinary differential equation
y ′ + sin(ex )y = cos(ex+1 ), 0 ≤ x ≤ 1,
then which one of the following is its general solution on [0, 1]?
(A) c1 y1 + c2 y2 , c1 , c2 ∈ R
(B) y1 + c(y1 − y2 ), c∈R
(C) cy1 + (y1 − y2 ), c∈R
(D) c1 (y1 + y2 ) + c2 (y1 − y2 ), c1 , c2 ∈ R
MA 9 / 33
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25. Consider the following Linear Programming Problem P
minimize 5x1 + 2x2
subject to 2x1 + x2 ≤ 2,
x1 + x2 ≥ 1,
x1 , x2 ≥ 0.
The optimal value of the problem P is equal to
(A) 5
(B) 0
(C) 4
(D) 2
MA 10 / 33
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NAT - 1 Mark
Let p = 1, 12 , 31 , 14
26. ∈ R4 and f : R4 → R be a differentiable function such that
f (p) = 6 and f (λx) = λ3 f (x), for every λ ∈ (0, ∞) and x ∈ R4 . The value of
∂f ∂f ∂f ∂f
12 (p) + 6 (p) + 4 (p) + 3 (p)
∂x1 ∂x2 ∂x3 ∂x4
is equal to (answer in integer)
27. The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal
to (answer in integer)
28. Let f (x, y) = (x2 − y 2 , 2xy), where x > 0, y > 0. Let g be the inverse of f in a
neighborhood of f (2, 1). Then the determinant of the Jacobian matrix of g at f (2, 1) is
equal to (round off to TWO decimal places)
29. Let F3 be the field with exactly 3 elements. The number of elements in GL2 (F3 ) is
equal to (answer in integer)
30. Given a real subspace W of R4 , let W ⊥ denote its orthogonal complement with re-
spect to the standard inner product on R4 . Let W1 = Span{(1, 0, 0, −1)} and W2 =
Span{(2, 1, 0, −1)} be real subspaces of R4 . The dimension of W1⊥ ∩ W2⊥ over R is
equal to (answer in integer)
31. The number of group homomorphisms from Z/4Z to S4 is equal to
(answer in integer)
MA 11 / 33
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32. Let a ∈ R and h be a positive real number. For any twice-differentiable function
f : R → R, let Pf (x) be the interpolating polynomial of degree at most two that
interpolates f at the points a − h, a, a + h. Define d to be the largest integer such that
any polynomial g of degree d satisfies
g ′′ (a) = Pg′′ (a).
The value of d is equal to (answer in integer)
33. Let Pf (x) be the interpolating polynomial of degree at most two that interpolates the
function f (x) = x2 |x| at the points x = −1, 0, 1. Then
sup |f (x) − Pf (x)| =
x∈[−1,1]
(round off to TWO decimal places)
34. The maximum of the function f (x, y, z) = xyz subject to the constraints
xy + yz + zx = 12, x > 0, y > 0, z > 0,
is equal to (round off to TWO decimal places)
35. If the outward flux of F(x, y, z) = x3 , y 3 , z 3 through the unit sphere x2 + y 2 + z 2 = 1
is απ, then α is equal to (round off to TWO decimal places)
MA 12 / 33
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MCQ - 2 Mark
36. Let H = {z ∈ C : Im(z) > 0} and D = {z ∈ C : |z| < 1}. Then
′ i
sup | f (0)| : f is an analytic function from D to H and f (0) =
2
is equal to
1
(A)
4
1
(B)
2
(C) 1
(D) 100
37. Let S 1 = {z ∈ C : |z| = 1}. For which one of the following functions f does there
exist a sequence of polynomials in z that uniformly converges to f on S 1 ?
(A) f (z) = z
(B) f (z) = Re(z)
(C) f (z) = ez
(D) f (z) = |z + 1|2
MA 13 / 33
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38. Let f : [0, 1] → R be a function. Which one of the following is a sufficient condition
for f to be Lebesgue measurable?
(A) |f | is a Lebesgue measurable function
(B) There exist continuous functions g, h : [0, 1] → R such that g ≤ f ≤ h on [0, 1]
(C) f is continuous almost everywhere on [0, 1]
(D) For each c ∈ R, the set {x ∈ [0, 1] : f (x) = c} is Lebesgue measurable
39. Let g : M2 (R) → R be given by g(A) = Trace(A2 ). Let 0 be the 2 × 2 zero matrix.
The space M2 (R) may be identified with R4 in the usual manner. Which one of the
following is correct?
(A) 0 is a point of local minimum of g
(B) 0 is a point of local maximum of g
(C) 0 is a saddle point of g
(D) 0 is not a critical point of g
MA 14 / 33
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40. Consider the following statements.
I. There exists a proper subgroup G of (Q, +) such that Q/G is a finite group.
II. There exists a subgroup G of (Q, +) such that Q/G is isomorphic to (Z, +).
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
41. Let X be the space R/Z with the quotient topology induced from the usual topology
on R. Consider the following statements.
I. X is compact.
II. X \ {x} is connected for any x ∈ X.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
MA 15 / 33
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42. Let ⟨·, ·⟩ denote the standard inner product on R7 . Let Σ = {v1 , . . . , v5 } ⊆ R7 be a
set of unit vectors such that ⟨vi , vj ⟩ is a non-positive integer for all 1 ≤ i ̸= j ≤ 5.
Define N (Σ) to be the number of pairs (r, s), 1 ≤ r, s ≤ 5, such that ⟨vr , vs ⟩ =
̸ 0.
The maximum possible value of N (Σ) is equal to
(A) 9
(B) 10
(C) 14
(D) 5
43. Let f (x) = |x| + |x − 1| + |x − 2|, x ∈ [−1, 2]. Which one of the following numerical
integration rules gives the exact value of the integral
Z 2
f (x)dx
−1
(A) The Simpson’s rule
(B) The trapezoidal rule
(C) The composite Simpson’s rule by dividing [−1, 2] into 4 equal subintervals
(D) The composite trapezoidal rule by dividing [−1, 2] into 3 equal subintervals
MA 16 / 33
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44. Consider the initial value problem (IVP)
2
y ′ = e−y + 1,
y(0) = 0.
I. IVP has a unique solution on R.
II. Every solution of IVP is bounded on its maximal interval of existence.
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
45. Let A be a 2 × 2 non-diagonalizable real matrix with a real eigenvalue λ and v be an
eigenvector of A corresponding to λ. Which one of the following is the general solution
of the system y′ = Ay of first-order linear differential equations?
(A) c1 eλt v + c2 teλt v, where c1 , c2 ∈ R
(B) c1 eλt v + c2 t2 eλt v, where c1 , c2 ∈ R
(C) c1 eλt v + c2 eλt (tv + u) , where c1 , c2 ∈ R and u is a 2 × 1 real column vector
such that (A − λI2 )u = v
(D) c1 eλt v + c2 teλt (v + u) , where c1 , c2 ∈ R and u is a 2 × 1 real column vector
such that (A − λI2 )u = v
MA 17 / 33
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46. Let D = {(x, y) ∈ R2 : x > 0 and y > 0}. If the following second-order linear partial
differential equation
∂ 2u 2
2∂ u ∂u
y2 2
− x 2
+y = 0 on D
∂x ∂y ∂y
is transformed to
2
∂ u ∂ 2u
∂u ∂u 1 ∂u ∂u 1
− 2 + + + a +b = 0 on D
∂η 2 ∂ξ ∂η ∂ξ 2η ∂η ∂ξ 2ξ
x2 y2
for some a, b ∈ R, via the co-ordinate transform η = and ξ = , then which one
2 2
of the following is correct?
(A) a = 2, b = 0
(B) a = 0, b = −1
(C) a = 1, b = −1
(D) a = 1, b = 0
MA 18 / 33
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!1/p
∞
X
47. Let ℓp = x = (xn )n≥1 : xn ∈ R, ∥x∥p = |xn |p < ∞ for p = 1, 2. Let
n=1
C00 = {(xn )n≥1 : xn = 0 for all but finitely many n ≥ 1} .
∞
X x
For x = (xn )n≥1 ∈ C00 , define f (x) = √n . Consider the following statements.
n=1
n
I. There exists a continuous linear functional F on (ℓ1 , ∥ · ∥1 ) such that F = f on C00 .
II. There exists a continuous linear functional G on (ℓ2 , ∥ · ∥2 ) such that G = f on C00 .
Which one of the following is correct?
(A) Both I and II are TRUE
(B) I is TRUE and II is FALSE
(C) I is FALSE and II is TRUE
(D) Both I and II are FALSE
MA 19 / 33
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MSQ - 2 Mark
( )
∞
Let ℓ2Z = x2j < ∞
P
48. (xj )j∈Z : xj ∈ R and endowed with the inner product
j=−∞
∞
X
⟨x, y⟩ = xj y j , x = (xj )j∈Z , y = (yj )j∈Z ∈ ℓ2Z .
j=−∞
Let T : ℓ2Z → ℓ2Z be given by T ((xj )j∈Z ) = (yj )j∈Z , where
xj + x−j
yj = , j ∈ Z.
2
Which of the following is/are correct?
(A) T is a compact operator
(B) The operator norm of T is 1
(C) T is a self-adjoint operator
(D) Range(T ) is closed
49. Let X be the normed space (R2 , ∥ · ∥), where
∥(x, y)∥ = |x| + |y|, (x, y) ∈ R2 .
Let S = {(x, 0) : x ∈ R} and f : S → R be given by f ((x, 0)) = 2x for all x ∈ R.
Recall that a Hahn–Banach extension of f to X is a continuous linear functional F on
X such that F |S = f and ∥F ∥ = ∥f ∥, where ∥F ∥ and ∥f ∥ are the norms of F and f
on X and S, respectively. Which of the following is/are true?
(A) F (x, y) = 2x + 3y is a Hahn–Banach extension of f to X
(B) F (x, y) = 2x + y is a Hahn–Banach extension of f to X
(C) f admits infinitely many Hahn–Banach extensions to X
(D) f admits exactly two distinct Hahn–Banach extensions to X
MA 20 / 33
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50. Let {[a, b) : a, b ∈ R, a < b} be a basis for a topology τ on R. Which of the following
is/are correct?
(A) Every (a, b) with a < b is an open set in (R, τ )
(B) Every [a, b] with a < b is a compact set in (R, τ )
(C) (R, τ ) is a first-countable space
(D) (R, τ ) is a second-countable space
51. Let T, S : R4 → R4 be two non–zero, non–identity R-linear transformations. Assume
T 2 = T . Which of the following is/are TRUE?
(A) T is necessarily invertible
(B) T and S are similar if S 2 = S and Rank(T ) = Rank(S)
(C) T and S are similar if S has only 0 and 1 as eigenvalues
(D) T is necessarily diagonalizable
52. Let p1 < p2 be the two fixed points of the function g(x) = ex − 2, where x ∈ R. For
x0 ∈ R, let the sequence (xn )n≥1 be generated by the fixed point iteration
xn = g(xn−1 ), n ≥ 1.
Which one of the following is/are correct?
(A) (xn )n≥0 converges to p1 for any x0 ∈ (p1 , p2 )
(B) (xn )n≥0 converges to p2 for any x0 ∈ (p1 , p2 )
(C) (xn )n≥0 converges to p2 for any x0 > p2
(D) (xn )n≥0 converges to p1 for any x0 < p1
MA 21 / 33
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53. Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem
y ′′ + λy = 0, 0 ≤ x ≤ π,
y(0) = y ′ (0),
y(π) = y ′ (π)?
(A) λ=1
(B) λ=2
(C) λ=3
(D) λ=4
54. Let f : R2 → R be a function such that
2
x p
1 − cos x2 + y 2 , if y ̸= 0, x ∈ R,
f (x, y) = y2
0, otherwise.
Which of the following is/are correct?
(A) f is continuous at (0, 0), but not differentiable at (0, 0)
(B) f is differentiable at (0, 0)
(C) All the directional derivatives of f at (0, 0) exist and they are equal to zero
(D) Both the partial derivatives of f at (0, 0) exist and they are equal to zero
MA 22 / 33
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GATE 2024 MATHEMATICS - MA
55. For an integer n, let fn (x) = xe−nx , where x ∈ [0, 1]. Let S := {fn : n ≥ 1}.
Consider the metric space (C([0, 1]), d), where
d(f, g) = sup {|f (x) − g(x)|} , f, g ∈ C([0, 1]).
x∈[0,1]
Which of the following statement(s) is/are true?
(A) S is an equi-continuous family of continuous functions
(B) S is closed in (C([0, 1]), d)
(C) S is bounded in (C([0, 1]), d)
(D) S is compact in (C([0, 1]), d)
MA 23 / 33
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GATE 2024 MATHEMATICS - MA
56. Let T : R4 → R4 be an R-linear transformation such that 1 and 2 are the only eigen-
values of T . Suppose the dimensions of Kernel(T − I4 ) and Range(T − 2I4 ) are 1
and 2, respectively. Which of the following is/are possible (upper triangular) Jordan
canonical form(s) of T ?
(A)
1 0 0 0
0 2 0 0
0 0 2 1
0 0 0 2
(B)
1 0 0 0
0 2 1 0
0 0 2 1
0 0 0 2
(C)
1 1 0 0
0 1 0 0
0 0 2 1
0 0 0 2
(D)
1 1 0 0
0 1 0 0
0 0 2 0
0 0 0 2
MA 24 / 33
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GATE 2024 MATHEMATICS - MA
NAT - 2 Mark
57. Let L2 ([−1, 1]) denote the space of all real-valued Lebesgue square-integrable func-
tions on [−1, 1], with the usual norm ∥ · ∥. Let P1 be the subspace of L2 ([−1, 1])
consisting of all the polynomials of degree at most 1. Let f ∈ L2 ([−1, 1]) be such that
18 R1 R1
∥f ∥2 = , f (x)dx = 2, and xf (x)dx = 0. Then
5 −1 −1
inf ∥f − g∥2 =
g∈P1
(round off to TWO decimal places)
58. The maximum value of f (x, y, z) = 10x + 6y − 8z subject to the constraints
5x − 2y + 6z ≤ 20,
10x + 4y − 6z ≤ 30,
x, y, z ≥ 0,
is equal to (round off to TWO decimal places)
59. Let K ⊆ C be the field extension of Q obtained by adjoining all the roots of the
polynomial equation (X 2 − 2)(X 2 − 3) = 0. The number of distinct fields F such that
Q ⊆ F ⊆ K is equal to (answer in integer)
60. Let H be the subset of S3 consisting of all σ ∈ S3 such that
Trace(A1 A2 A3 ) = Trace(Aσ(1) Aσ(2) Aσ(3) ),
for all A1 , A2 , A3 ∈ M2 (C). The number of elements in H is equal to
(answer in integer)
MA 25 / 33
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GATE 2024 MATHEMATICS - MA
61. Let r : [0, 1] → R2 be a continuously differentiable path from (0, 2) to (3, 0) and let
F : R2 → R2 be defined by F(x, y) = (1 − 2y, 1 − 2x). The line integral of F along r
Z
F · dr
is equal to (round off to TWO decimal places)
62. Let u = u(x, t) be the solution of the initial value problem
∂ 2u ∂ 2u
− 2 = 0, x ∈ R, t > 0,
∂t2 ∂x
u(x, 0) = 0, x ∈ R,
∂u x4 (1 − x)4 ,
if 0 < x < 1,
(x, 0) =
∂t 0, otherwise.
If α = inf{t > 0 : u(2, t) > 0}, then α is equal to (round off to TWO
decimal places)
63. The boundary value problem
x2 y ′′ − 2xy ′ + 2y = 0, 1 ≤ x ≤ 2,
y(1) − y ′ (1) = 1,
y(2) − ky ′ (2) = 4,
has infinitely many distinct solutions when k is equal to (round off to TWO
decimal places)
64. The global maximum of f (x, y) = (x2 + y 2 )e2−x−y on {(x, y) ∈ R2 : x ≥ 0, y ≥ 0} is
equal to (round off to TWO decimal places)
MA 26 / 33
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GATE 2024 MATHEMATICS - MA
65. Let k ∈ R and D = {(r, θ) : 0 < r < 2, 0 < θ < π}. Let u(r, θ) be the solution of the
following boundary value problem
∂ 2 u 1 ∂u 1 ∂ 2u
+ + = 0, (r, θ) ∈ D,
∂r2 r ∂r r2 ∂θ2
u(r, 0) = u(r, π) = 0, 0 ≤ r ≤ 2,
u(2, θ) = k sin(2θ), 0 < θ < π.
π
If u 1, = 2, then the value of k is equal to (round off to TWO decimal
4
places)
MA 27 / 33
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Mathematics (MA)
Final Answer Key
Question
Q. No. Session Section Key/Range Mark
Type
1 4 MCQ GA A 1
2 4 MCQ GA B 1
3 4 MCQ GA D 1
4 4 MCQ GA A 1
5 4 MCQ GA B 1
6 4 MCQ GA D 2
7 4 MCQ GA B 2
8 4 MCQ GA C 2
9 4 MCQ GA A 2
10 4 MCQ GA A 2
11 4 MCQ MA C 1
12 4 MCQ MA C 1
13 4 MCQ MA C 1
14 4 MCQ MA C 1
15 4 MCQ MA D 1
16 4 MCQ MA B 1
17 4 MCQ MA A 1
18 4 MCQ MA A 1
19 4 MCQ MA A 1
20 4 MCQ MA A 1
21 4 MCQ MA A 1
22 4 MCQ MA C 1
23 4 MCQ MA A 1
24 4 MCQ MA B 1
25 4 MCQ MA D 1
26 4 NAT MA 216 to 216 1
27 4 NAT MA 2 to 2 1
28 4 NAT MA 0.04 to 0.06 1
29 4 NAT MA 48 to 48 1
30 4 NAT MA 2 to 2 1
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31 4 NAT MA 16 to 16 1
32 4 NAT MA 3 to 3 1
33 4 NAT MA 0.14 to 0.16 1
34 4 NAT MA 7.99 to 8.01 1
35 4 NAT MA 2.39 to 2.41 1
36 4 MCQ MA C 2
37 4 MCQ MA C 2
38 4 MCQ MA C 2
39 4 MCQ MA C 2
40 4 MCQ MA D 2
41 4 MCQ MA A 2
42 4 MCQ MA A 2
43 4 MCQ MA D 2
44 4 MCQ MA B 2
45 4 MCQ MA C 2
46 4 MCQ MA B 2
47 4 MCQ MA B 2
48 4 MSQ MA B;C;D 2
49 4 MSQ MA B;C 2
50 4 MSQ MA A;C 2
51 4 MSQ MA B;D 2
52 4 MSQ MA A;D 2
53 4 MSQ MA A;D 2
54 4 MSQ MA A;D 2
55 4 MSQ MA A;C 2
56 4 MSQ MA A;D 2
57 4 NAT MA 1.59 to 1.61 2
58 4 NAT MA 56.66 to 56.68 2
59 4 NAT MA 5 to 5 2
60 4 NAT MA 3 to 3 2
61 4 NAT MA 0.99 to 1.01 2
62 4 NAT MA 0.99 to 1.01 2
63 4 NAT MA 1.99 to 2.01 2
64 4 NAT MA 3.99 to 4.01 2
65 4 NAT MA 7.99 to 8.01 2
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Source: GATE. Question paper published by GATE.