GATE 2022 Question Paper MA Mathematics
Download the GATE 2022 Question Paper MA Mathematics PDF for free at AglaSem. Solving this previous year question paper helps you understand the real GATE exam pattern, question types, difficulty level and marking scheme, and reveals important repeated topics — practise it to build speed, accuracy and exam confidence.
More Detail
Latest News
Finished viewing? Save it for later —
Download GATE 2022 Question Paper MA Mathematics (PDF · 66 pages)Downloaded 18 times
About GATE 2022 Question Paper MA Mathematics
GATE 2022 Question Paper MA Mathematics is available here for free download. Published by IIT for Graduate Aptitude Test in Engineering, this question paper can be viewed online or downloaded as a PDF (66 pages). Candidates preparing for Graduate Aptitude Test in Engineering can use GATE 2022 Question Paper MA Mathematics to understand the exam pattern, the type of questions asked, and the overall difficulty level.
Frequently Asked Questions
How can I download GATE 2022 Question Paper MA Mathematics?
Open this page and click the Download button to save GATE 2022 Question Paper MA Mathematics as a PDF. It is completely free on AglaSem Docs.
Is GATE 2022 Question Paper MA Mathematics free to download?
Yes. GATE 2022 Question Paper MA Mathematics can be viewed online and downloaded as a PDF free of cost on AglaSem Docs.
How many pages does GATE 2022 Question Paper MA Mathematics have?
GATE 2022 Question Paper MA Mathematics contains 66 pages, which you can read online or download together as a single PDF.
Where can I find more Graduate Aptitude Test in Engineering study material?
You can find more Graduate Aptitude Test in Engineering question papers, sample papers, syllabus, and answer keys on AglaSem Docs.
GATE 2022 Question Paper MA Mathematics – Text
Read the full text of this question paper below — useful to quickly search, copy and reference the content online without downloading the PDF.
📄 View text version (66 pages)
Page 1
Useful data
A\B {a ∈ A : a ∈ / B}
C Set of all complex numbers
Cm×n Set of all matrices of order m × n with complex entries
C∞ (Ω) Collection of all infinitely differentiable functions on the open domain Ω
√
i −1
I Identity matrix of appropriate order
L2 (R) := L2 (R, dx)
L2 [a, b] := L2 ([a, b], dx)
N Set of all positive integers
Q Set of all rational numbers
R Set of all real numbers
Rm×n Set of all matrices of order m × n with real entries
S1 {(x1 , x2 ) ∈ R2 : x21 + x22 = 1}
S2 {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 = 1}
Z Set of all integers
MA Page 1 of 66
A\B {a ∈ A : a ∈ / B}
C Set of all complex numbers
Cm×n Set of all matrices of order m × n with complex entries
C∞ (Ω) Collection of all infinitely differentiable functions on the open domain Ω
√
i −1
I Identity matrix of appropriate order
L2 (R) := L2 (R, dx)
L2 [a, b] := L2 ([a, b], dx)
N Set of all positive integers
Q Set of all rational numbers
R Set of all real numbers
Rm×n Set of all matrices of order m × n with real entries
S1 {(x1 , x2 ) ∈ R2 : x21 + x22 = 1}
S2 {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 = 1}
Z Set of all integers
MA Page 1 of 66
Page 2
GATE 2022 General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark each.
Q.1 As you grow older, an injury to your _________ may take longer to _________.
(A) heel / heel
(B) heal / heel
(C) heal / heal
(D) heel / heal
Page 2 of 66
Q.1 – Q.5 Carry ONE mark each.
Q.1 As you grow older, an injury to your _________ may take longer to _________.
(A) heel / heel
(B) heal / heel
(C) heal / heal
(D) heel / heal
Page 2 of 66
Page 3
Q.2 In a 500 m race, P and Q have speeds in the ratio of 3 ∶ 4. Q starts the race
when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?
(A) 20
(B) 40
(C) 60
(D) 140
Page 3 of 66
when P has already covered 140 m.
What is the distance between P and Q (in m) when P wins the race?
(A) 20
(B) 40
(C) 60
(D) 140
Page 3 of 66
Page 4
Q.3 Three bells P, Q, and R are rung periodically in a school. P is rung every 20
minutes; Q is rung every 30 minutes and R is rung every 50 minutes.
If all the three bells are rung at 12:00 PM, when will the three bells ring
together again the next time?
(A) 5:00 PM
(B) 5:30 PM
(C) 6:00 PM
(D) 6:30 PM
Page 4 of 66
minutes; Q is rung every 30 minutes and R is rung every 50 minutes.
If all the three bells are rung at 12:00 PM, when will the three bells ring
together again the next time?
(A) 5:00 PM
(B) 5:30 PM
(C) 6:00 PM
(D) 6:30 PM
Page 4 of 66
Page 5
Q.4 Given below are two statements and four conclusions drawn based on the
statements.
Statement 1: Some bottles are cups.
Statement 2: All cups are knives.
Conclusion I: Some bottles are knives.
Conclusion II: Some knives are cups.
Conclusion III: All cups are bottles.
Conclusion IV: All knives are cups.
Which one of the following options can be logically inferred?
(A) Only conclusion I and conclusion II are correct
(B) Only conclusion II and conclusion III are correct
(C) Only conclusion II and conclusion IV are correct
(D) Only conclusion III and conclusion IV are correct
Page 5 of 66
statements.
Statement 1: Some bottles are cups.
Statement 2: All cups are knives.
Conclusion I: Some bottles are knives.
Conclusion II: Some knives are cups.
Conclusion III: All cups are bottles.
Conclusion IV: All knives are cups.
Which one of the following options can be logically inferred?
(A) Only conclusion I and conclusion II are correct
(B) Only conclusion II and conclusion III are correct
(C) Only conclusion II and conclusion IV are correct
(D) Only conclusion III and conclusion IV are correct
Page 5 of 66
Page 6
Q.5 The figure below shows the front and rear view of a disc, which is shaded with
identical patterns. The disc is flipped once with respect to any one of the fixed
axes 1-1, 2-2 or 3-3 chosen uniformly at random.
What is the probability that the disc DOES NOT retain the same front and rear
views after the flipping operation?
Front View Rear View
(A) 0
1
(B)
3
2
(C)
3
(D) 1
Page 6 of 66
identical patterns. The disc is flipped once with respect to any one of the fixed
axes 1-1, 2-2 or 3-3 chosen uniformly at random.
What is the probability that the disc DOES NOT retain the same front and rear
views after the flipping operation?
Front View Rear View
(A) 0
1
(B)
3
2
(C)
3
(D) 1
Page 6 of 66
Page 7
Q. 6 – Q. 10 Carry TWO marks each.
Q.6 Altruism is the human concern for the wellbeing of others. Altruism has been
shown to be motivated more by social bonding, familiarity and identification of
belongingness to a group. The notion that altruism may be attributed to empathy
or guilt has now been rejected.
Which one of the following is the CORRECT logical inference based on the
information in the above passage?
(A) Humans engage in altruism due to guilt but not empathy
(B) Humans engage in altruism due to empathy but not guilt
(C) Humans engage in altruism due to group identification but not empathy
(D) Humans engage in altruism due to empathy but not familiarity
Page 7 of 66
Q.6 Altruism is the human concern for the wellbeing of others. Altruism has been
shown to be motivated more by social bonding, familiarity and identification of
belongingness to a group. The notion that altruism may be attributed to empathy
or guilt has now been rejected.
Which one of the following is the CORRECT logical inference based on the
information in the above passage?
(A) Humans engage in altruism due to guilt but not empathy
(B) Humans engage in altruism due to empathy but not guilt
(C) Humans engage in altruism due to group identification but not empathy
(D) Humans engage in altruism due to empathy but not familiarity
Page 7 of 66
Page 8
There are two identical dice with a single letter on each of the faces. The
Q.7
following six letters: Q, R, S, T, U, and V, one on each of the faces. Any of the
six outcomes are equally likely.
The two dice are thrown once independently at random.
What is the probability that the outcomes on the dice were composed only of
any combination of the following possible outcomes: Q, U and V?
(A) 1
4
(B) 3
4
(C) 1
6
(D) 5
36
Page 8 of 66
Q.7
following six letters: Q, R, S, T, U, and V, one on each of the faces. Any of the
six outcomes are equally likely.
The two dice are thrown once independently at random.
What is the probability that the outcomes on the dice were composed only of
any combination of the following possible outcomes: Q, U and V?
(A) 1
4
(B) 3
4
(C) 1
6
(D) 5
36
Page 8 of 66
Page 9
Q.8 The price of an item is 10% cheaper in an online store S compared to the price
at another online store M. Store S charges ₹ 150 for delivery. There are no
delivery charges for orders from the store M. A person bought the item from the
store S and saved ₹ 100.
What is the price of the item at the online store S (in ₹) if there are no other
charges than what is described above?
(A) 2500
(B) 2250
(C) 1750
(D) 1500
Page 9 of 66
at another online store M. Store S charges ₹ 150 for delivery. There are no
delivery charges for orders from the store M. A person bought the item from the
store S and saved ₹ 100.
What is the price of the item at the online store S (in ₹) if there are no other
charges than what is described above?
(A) 2500
(B) 2250
(C) 1750
(D) 1500
Page 9 of 66
Page 10
Q.9 The letters P, Q, R, S, T and U are to be placed one per vertex on a regular convex
hexagon, but not necessarily in the same order.
Consider the following statements:
The line segment joining R and S is longer than the line segment joining
P and Q.
The line segment joining R and S is perpendicular to the line segment
joining P and Q.
The line segment joining R and U is parallel to the line segment joining T
and Q.
Based on the above statements, which one of the following options is
CORRECT?
(A) The line segment joining R and T is parallel to the line segment joining Q and S
(B) The line segment joining T and Q is parallel to the line joining P and U
(C) The line segment joining R and P is perpendicular to the line segment joining U
and Q
(D) The line segment joining Q and S is perpendicular to the line segment joining R
and P
Page 10 of 66
hexagon, but not necessarily in the same order.
Consider the following statements:
The line segment joining R and S is longer than the line segment joining
P and Q.
The line segment joining R and S is perpendicular to the line segment
joining P and Q.
The line segment joining R and U is parallel to the line segment joining T
and Q.
Based on the above statements, which one of the following options is
CORRECT?
(A) The line segment joining R and T is parallel to the line segment joining Q and S
(B) The line segment joining T and Q is parallel to the line joining P and U
(C) The line segment joining R and P is perpendicular to the line segment joining U
and Q
(D) The line segment joining Q and S is perpendicular to the line segment joining R
and P
Page 10 of 66
Page 11
Q.10
An ant is at the bottom-left corner of a grid (point P) as shown above. It aims to
move to the top-right corner of the grid. The ant moves only along the lines
marked in the grid such that the current distance to the top-right corner strictly
decreases.
Which one of the following is a part of a possible trajectory of the ant during
the movement?
(A)
(B)
(C)
(D)
Page 11 of 66
An ant is at the bottom-left corner of a grid (point P) as shown above. It aims to
move to the top-right corner of the grid. The ant moves only along the lines
marked in the grid such that the current distance to the top-right corner strictly
decreases.
Which one of the following is a part of a possible trajectory of the ant during
the movement?
(A)
(B)
(C)
(D)
Page 11 of 66
Page 12
Q.11 – Q.35 Carry ONE mark each.
Q.11 Suppose that the characteristic equation of M ∈ C3×3 is
λ3 + αλ2 + βλ − 1 = 0,
where α, β ∈ C with α + β ̸= 0.
Which of the following statements is TRUE?
(A) M (I − βM ) = M −1 (M + αI)
(B) M (I + βM ) = M −1 (M − αI)
(C) M −1 (M −1 + βI) = M − αI
(D) M −1 (M −1 − βI) = M + αI
MA Page 12 of 66
Q.11 Suppose that the characteristic equation of M ∈ C3×3 is
λ3 + αλ2 + βλ − 1 = 0,
where α, β ∈ C with α + β ̸= 0.
Which of the following statements is TRUE?
(A) M (I − βM ) = M −1 (M + αI)
(B) M (I + βM ) = M −1 (M − αI)
(C) M −1 (M −1 + βI) = M − αI
(D) M −1 (M −1 − βI) = M + αI
MA Page 12 of 66
Page 13
Q.12 Consider
P: Let M ∈ Rm×n with m > n ≥ 2. If rank(M ) = n, then the system of
linear equations M x = 0 has x = 0 as the only solution.
Q: Let E ∈ Rn×n , n ≥ 2 be a non-zero matrix such that E 3 = 0. Then
I + E 2 is a singular matrix.
Which of the following statements is TRUE?
(A) Both P and Q are TRUE
(B) Both P and Q are FALSE
(C) P is TRUE and Q is FALSE
(D) P is FALSE and Q is TRUE
MA Page 13 of 66
P: Let M ∈ Rm×n with m > n ≥ 2. If rank(M ) = n, then the system of
linear equations M x = 0 has x = 0 as the only solution.
Q: Let E ∈ Rn×n , n ≥ 2 be a non-zero matrix such that E 3 = 0. Then
I + E 2 is a singular matrix.
Which of the following statements is TRUE?
(A) Both P and Q are TRUE
(B) Both P and Q are FALSE
(C) P is TRUE and Q is FALSE
(D) P is FALSE and Q is TRUE
MA Page 13 of 66
Page 14
Q.13 Consider the real function of two real variables given by
u(x, y) = e2x [sin 3x cos 2y cosh 3y − cos 3x sin 2y sinh 3y].
Let v(x, y) be the harmonic conjugate of u(x, y) such that v(0, 0) = 2. Let z = x+iy
and f (z) = u(x, y) + iv(x, y), then the value of 4 + 2if (iπ) is
(A) e3π + e−3π
(B) e3π − e−3π
(C) −e3π + e−3π
(D) −e3π − e−3π
MA Page 14 of 66
u(x, y) = e2x [sin 3x cos 2y cosh 3y − cos 3x sin 2y sinh 3y].
Let v(x, y) be the harmonic conjugate of u(x, y) such that v(0, 0) = 2. Let z = x+iy
and f (z) = u(x, y) + iv(x, y), then the value of 4 + 2if (iπ) is
(A) e3π + e−3π
(B) e3π − e−3π
(C) −e3π + e−3π
(D) −e3π − e−3π
MA Page 14 of 66
Page 15
Q.14 The value of the integral
z 100
Z
101 + 1
dz
C z
where C is the circle of radius 2 centred at the origin taken in the anti-clockwise
direction is
(A) −2πi
(B) 2π
(C) 0
(D) 2πi
MA Page 15 of 66
z 100
Z
101 + 1
dz
C z
where C is the circle of radius 2 centred at the origin taken in the anti-clockwise
direction is
(A) −2πi
(B) 2π
(C) 0
(D) 2πi
MA Page 15 of 66
Page 16
Q.15 Let X be a real normed linear space. Let X0 = {x ∈ X : ∥x∥ = 1}. If X0 contains
two distinct points x and y and the line segment joining them, then, which of the
following statements is TRUE?
(A) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly independent
(B) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly dependent
(C) ∥x + y∥2 = ∥x∥2 + ∥y∥2 and x, y are linearly independent
(D) ∥x + y∥ = 2∥x∥∥y∥ and x, y are linearly dependent
MA Page 16 of 66
two distinct points x and y and the line segment joining them, then, which of the
following statements is TRUE?
(A) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly independent
(B) ∥x + y∥ = ∥x∥ + ∥y∥ and x, y are linearly dependent
(C) ∥x + y∥2 = ∥x∥2 + ∥y∥2 and x, y are linearly independent
(D) ∥x + y∥ = 2∥x∥∥y∥ and x, y are linearly dependent
MA Page 16 of 66
Page 17
Q.16 Let {ek : k ∈ N} be an orthonormal basis for a Hilbert space H.
j
(−1)n+1 en , j ∈ N.
P
Define fk = ek + ek+1 , k ∈ N and gj =
n=1
∞
2
P
Then |⟨gj , fk ⟩| =
k=1
(A) 0
(B) j 2
(C) 4j 2
(D) 1
MA Page 17 of 66
j
(−1)n+1 en , j ∈ N.
P
Define fk = ek + ek+1 , k ∈ N and gj =
n=1
∞
2
P
Then |⟨gj , fk ⟩| =
k=1
(A) 0
(B) j 2
(C) 4j 2
(D) 1
MA Page 17 of 66
Page 18
Q.17 Consider R2 with the usual metric. Let A = {(x, y) ∈ R2 : x2 + y 2 ≤ 1} and B =
{(x, y) ∈ R2 : (x−2)2 +y 2 ≤ 1}. Let M = A∪B and N = interior(A) ∪ interior(B ).
Then, which of the following statements is TRUE?
(A) M and N are connected
(B) Neither M nor N is connected
(C) M is connected and N is not connected
(D) M is not connected and N is connected
MA Page 18 of 66
{(x, y) ∈ R2 : (x−2)2 +y 2 ≤ 1}. Let M = A∪B and N = interior(A) ∪ interior(B ).
Then, which of the following statements is TRUE?
(A) M and N are connected
(B) Neither M nor N is connected
(C) M is connected and N is not connected
(D) M is not connected and N is connected
MA Page 18 of 66
Page 19
Q.18 The real sequence generated by the iterative scheme
xn−1 1
xn = + , n≥1
2 xn−1
√
(A) converges to 2, for all x0 ∈ R \ {0}
√ q
(B) converges to 2, whenever x0 > 23
√
(C) converges to 2, whenever x0 ∈ (−1, 1) \ {0}
(D) diverges for any x0 ̸= 0
MA Page 19 of 66
xn−1 1
xn = + , n≥1
2 xn−1
√
(A) converges to 2, for all x0 ∈ R \ {0}
√ q
(B) converges to 2, whenever x0 > 23
√
(C) converges to 2, whenever x0 ∈ (−1, 1) \ {0}
(D) diverges for any x0 ̸= 0
MA Page 19 of 66
Page 20
Q.19 The initial value problem
dy
= cos (xy), x ∈ R, y(0) = y0 ,
dx
where y0 is a real constant, has
(A) a unique solution
(B) exactly two solutions
(C) infinitely many solutions
(D) no solution
MA Page 20 of 66
dy
= cos (xy), x ∈ R, y(0) = y0 ,
dx
where y0 is a real constant, has
(A) a unique solution
(B) exactly two solutions
(C) infinitely many solutions
(D) no solution
MA Page 20 of 66
Page 21
Q.20 If eigenfunctions corresponding to distinct eigenvalues λ of the Sturm-Liouville
problem
d2 y dy
− 3 = λy, 0 < x < π,
dx2 dx
y(0) = y(π) = 0
are orthogonal with respect to the weight function w(x), then w(x) is
(A) e−3x
(B) e−2x
(C) e2x
(D) e3x
MA Page 21 of 66
problem
d2 y dy
− 3 = λy, 0 < x < π,
dx2 dx
y(0) = y(π) = 0
are orthogonal with respect to the weight function w(x), then w(x) is
(A) e−3x
(B) e−2x
(C) e2x
(D) e3x
MA Page 21 of 66
Page 22
Q.21 The steady state solution for the heat equation
∂u ∂ 2 u
− 2 = 0, 0 < x < 2, t > 0,
∂t ∂x
with the initial condition u(x, 0) = 0, 0 < x < 2 and the boundary conditions
u(0, t) = 1 and u(2, t) = 3, t > 0, at x = 1 is
(A) 1
(B) 2
(C) 3
(D) 4
MA Page 22 of 66
∂u ∂ 2 u
− 2 = 0, 0 < x < 2, t > 0,
∂t ∂x
with the initial condition u(x, 0) = 0, 0 < x < 2 and the boundary conditions
u(0, t) = 1 and u(2, t) = 3, t > 0, at x = 1 is
(A) 1
(B) 2
(C) 3
(D) 4
MA Page 22 of 66
Page 23
Q.22 Consider ([0, 1], T1 ), where T1 is the subspace topology induced by the Euclidean
topology on R, and let T2 be any topology on [0, 1]. Consider the following state-
ments:
P : If T1 is a proper subset of T2 , then ([0, 1], T2 ) is not compact.
Q : If T2 is a proper subset of T1 , then ([0, 1], T2 ) is not Hausdorff.
Then
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) Both P and Q are FALSE
(D) P is FALSE and Q is TRUE
MA Page 23 of 66
topology on R, and let T2 be any topology on [0, 1]. Consider the following state-
ments:
P : If T1 is a proper subset of T2 , then ([0, 1], T2 ) is not compact.
Q : If T2 is a proper subset of T1 , then ([0, 1], T2 ) is not Hausdorff.
Then
(A) P is TRUE and Q is FALSE
(B) Both P and Q are TRUE
(C) Both P and Q are FALSE
(D) P is FALSE and Q is TRUE
MA Page 23 of 66
Page 24
Q.23 Let p : ([0, 1], T1 ) → ({0, 1}, T2 ) be the quotient map, arising from the characteristic
function on [ 21 , 1], where T1 is the subspace topology induced by the Euclidean
topology on R. Which of the following statements is TRUE?
(A) p is an open map but not a closed map
(B) p is a closed map but not an open map
(C) p is a closed map as well as an open map
(D) p is neither an open map nor a closed map
MA Page 24 of 66
function on [ 21 , 1], where T1 is the subspace topology induced by the Euclidean
topology on R. Which of the following statements is TRUE?
(A) p is an open map but not a closed map
(B) p is a closed map but not an open map
(C) p is a closed map as well as an open map
(D) p is neither an open map nor a closed map
MA Page 24 of 66
Page 25
Y
Q.24 Set Xn := R for each n ∈ N. Define Y := Xn . Endow Y with the product
n∈N
topology, where the topology on each Xn is the Euclidean topology. Consider the
set
∆ = {(x, x, x, · · · ) | x ∈ R}
with the subspace topology induced from Y . Which of the following statements is
TRUE?
(A) ∆ is open in Y
(B) ∆ is locally compact
(C) ∆ is dense in Y
(D) ∆ is disconnected
MA Page 25 of 66
Q.24 Set Xn := R for each n ∈ N. Define Y := Xn . Endow Y with the product
n∈N
topology, where the topology on each Xn is the Euclidean topology. Consider the
set
∆ = {(x, x, x, · · · ) | x ∈ R}
with the subspace topology induced from Y . Which of the following statements is
TRUE?
(A) ∆ is open in Y
(B) ∆ is locally compact
(C) ∆ is dense in Y
(D) ∆ is disconnected
MA Page 25 of 66
Page 26
Q.25 Consider the linear sytem of equations Ax = b with
3 1 1 2
A= 1 4 1 and b = 3 .
2 0 3 4
Which of the following statements are TRUE?
0 1/4 1/3
(A) The Jacobi iterative matrix is 1/3 0 1/3
2/3 0 0
(B) The Jacobi iterative method converges for any initial vector
(C) The Gauss-Seidel iterative method converges for any initial vector
(D) The spectral radius of the Jacobi iterative matrix is less than 1
MA Page 26 of 66
3 1 1 2
A= 1 4 1 and b = 3 .
2 0 3 4
Which of the following statements are TRUE?
0 1/4 1/3
(A) The Jacobi iterative matrix is 1/3 0 1/3
2/3 0 0
(B) The Jacobi iterative method converges for any initial vector
(C) The Gauss-Seidel iterative method converges for any initial vector
(D) The spectral radius of the Jacobi iterative matrix is less than 1
MA Page 26 of 66
Page 27
Q.26 The number of non-isomorphic abelian groups of order 22 .33 .54 is .
MA Page 27 of 66
MA Page 27 of 66
Page 28
Q.27 The number of subgroups of a cyclic group of order 12 is .
MA Page 28 of 66
MA Page 28 of 66
Page 29
Q.28 The radius of convergence of the series
X
3n+1 z 2n , z ∈ C
n≥0
is (round off to TWO decimal places).
MA Page 29 of 66
X
3n+1 z 2n , z ∈ C
n≥0
is (round off to TWO decimal places).
MA Page 29 of 66
Page 30
Q.29 The number of zeros of the polynomial
2z 7 − 7z 5 + 2z 3 − z + 1
in the unit disc {z ∈ C : |z| < 1} is .
MA Page 30 of 66
2z 7 − 7z 5 + 2z 3 − z + 1
in the unit disc {z ∈ C : |z| < 1} is .
MA Page 30 of 66
Page 31
Q.30 If P (x) is a polynomial of degree 5 and
6 6
!
X Y
α= P (xi ) (xi − xj )−1 ,
i=0 j=0, j̸=i
where x0 , x1 , · · · , x6 are distinct points in the interval [2, 3], then the value of
α2 − α + 1 is .
MA Page 31 of 66
6 6
!
X Y
α= P (xi ) (xi − xj )−1 ,
i=0 j=0, j̸=i
where x0 , x1 , · · · , x6 are distinct points in the interval [2, 3], then the value of
α2 − α + 1 is .
MA Page 31 of 66
Page 32
Q.31 The maximum value of f (x, y) = 49−x2 −y 2 on the line x+3y = 10 is .
MA Page 32 of 66
MA Page 32 of 66
Page 33
1 1
Q.32 If the function f (x, y) = x2 +xy +y 2 + + , x ̸= 0, y ̸= 0 attains its local minimum
x y
value at the point (a, b), then the value of a3 + b3 is (round off to
TWO decimal places).
MA Page 33 of 66
Q.32 If the function f (x, y) = x2 +xy +y 2 + + , x ̸= 0, y ̸= 0 attains its local minimum
x y
value at the point (a, b), then the value of a3 + b3 is (round off to
TWO decimal places).
MA Page 33 of 66
Page 34
Q.33 If the ordinary differential equation
d2 ϕ dϕ
x2 2
+x + x2 ϕ = 0, x > 0
dx dx
∞
X
r
has a solution of the form ϕ(x) = x an xn , where an ’s are constants and a0 ̸= 0,
n=0
2
then the value of r + 1 is .
MA Page 34 of 66
d2 ϕ dϕ
x2 2
+x + x2 ϕ = 0, x > 0
dx dx
∞
X
r
has a solution of the form ϕ(x) = x an xn , where an ’s are constants and a0 ̸= 0,
n=0
2
then the value of r + 1 is .
MA Page 34 of 66
Page 35
2α
Q.34 The Bessel functions Jα (x), x > 0, α ∈ R satisfy Jα−1 (x) + Jα+1 (x) = Jα (x).
x
Then, the value of (πJ 3 (π))2 is .
2
MA Page 35 of 66
Q.34 The Bessel functions Jα (x), x > 0, α ∈ R satisfy Jα−1 (x) + Jα+1 (x) = Jα (x).
x
Then, the value of (πJ 3 (π))2 is .
2
MA Page 35 of 66
Page 36
Q.35 The partial differential equation
∂ 2u ∂ 2u ∂ 2u
7 + 16 + 4 =0
∂x2 ∂x∂y ∂y 2
is transformed to
∂ 2u ∂ 2u ∂ 2u
A + B + C = 0,
∂ξ 2 ∂ξ∂η ∂η 2
using ξ = y − 2x and η = 7y − 2x.
1
Then, the value of 3 (B 2 − 4AC) is .
12
MA Page 36 of 66
∂ 2u ∂ 2u ∂ 2u
7 + 16 + 4 =0
∂x2 ∂x∂y ∂y 2
is transformed to
∂ 2u ∂ 2u ∂ 2u
A + B + C = 0,
∂ξ 2 ∂ξ∂η ∂η 2
using ξ = y − 2x and η = 7y − 2x.
1
Then, the value of 3 (B 2 − 4AC) is .
12
MA Page 36 of 66
Page 37
Q.36 – Q.65 Carry TWO marks each.
Q.36 Let R[X] denote the ring of polynomials in X with real coefficients. Then, the
quotient ring R[X]/(X 4 + 4) is
(A) a field
(B) an integral domain, but not a field
(C) not an integral domain, but has 0 as the only nilpotent element
(D) a ring which contains non-zero nilpotent elements
MA Page 37 of 66
Q.36 Let R[X] denote the ring of polynomials in X with real coefficients. Then, the
quotient ring R[X]/(X 4 + 4) is
(A) a field
(B) an integral domain, but not a field
(C) not an integral domain, but has 0 as the only nilpotent element
(D) a ring which contains non-zero nilpotent elements
MA Page 37 of 66
Page 38
Q.37 Consider the following conditions on two proper non-zero ideals J1 and J2 of a
non-zero commutative ring R.
P: For any r1 , r2 ∈ R, there exists a unique r ∈ R such that r − r1 ∈ J1
and r − r2 ∈ J2 .
Q: J1 + J2 = R
Then, which of the following statements is TRUE?
(A) P implies Q but Q does not imply P
(B) Q implies P but P does not imply Q
(C) P implies Q and Q implies P
(D) P does not imply Q and Q does not imply P
MA Page 38 of 66
non-zero commutative ring R.
P: For any r1 , r2 ∈ R, there exists a unique r ∈ R such that r − r1 ∈ J1
and r − r2 ∈ J2 .
Q: J1 + J2 = R
Then, which of the following statements is TRUE?
(A) P implies Q but Q does not imply P
(B) Q implies P but P does not imply Q
(C) P implies Q and Q implies P
(D) P does not imply Q and Q does not imply P
MA Page 38 of 66
Page 39
Q.38 Let f : [−π, π] → R be a continuous function such that f (x) > f (0) 2
, |x| < δ for
n
some δ satisfying 0 < δ < π. Define Pn,δ (x) = (1+cos x−cos δ) , for n = 1, 2, 3, · · · .
Then, which of the following statements is TRUE?
Z2δ
(A) lim f (x)Pn,δ (x)dx = 0
n→∞
0
Z0
(B) lim f (x)Pn,δ (x)dx = 0
n→∞
−2δ
Zδ
(C) lim f (x)Pn,δ (x)dx = 0
n→∞
−δ
Z
(D) lim f (x)Pn,δ (x)dx = 0
n→∞
[−π,π]\[−δ,δ]
MA Page 39 of 66
, |x| < δ for
n
some δ satisfying 0 < δ < π. Define Pn,δ (x) = (1+cos x−cos δ) , for n = 1, 2, 3, · · · .
Then, which of the following statements is TRUE?
Z2δ
(A) lim f (x)Pn,δ (x)dx = 0
n→∞
0
Z0
(B) lim f (x)Pn,δ (x)dx = 0
n→∞
−2δ
Zδ
(C) lim f (x)Pn,δ (x)dx = 0
n→∞
−δ
Z
(D) lim f (x)Pn,δ (x)dx = 0
n→∞
[−π,π]\[−δ,δ]
MA Page 39 of 66
Page 40
∞
X
Q.39 P : Suppose that an xn converges at x = −3 and diverges at x = 6. Then
n=0
∞
(−1)n an converges.
P
n=0
∞
X (−1)n xn
Q: The interval of convergence of the series is [−4, 4].
n=2
4n loge n
Which of the following statements is TRUE?
(A) P is true and Q is true
(B) P is false and Q is false
(C) P is true and Q is false
(D) P is false and Q is true
MA Page 40 of 66
X
Q.39 P : Suppose that an xn converges at x = −3 and diverges at x = 6. Then
n=0
∞
(−1)n an converges.
P
n=0
∞
X (−1)n xn
Q: The interval of convergence of the series is [−4, 4].
n=2
4n loge n
Which of the following statements is TRUE?
(A) P is true and Q is true
(B) P is false and Q is false
(C) P is true and Q is false
(D) P is false and Q is true
MA Page 40 of 66
Page 41
Q.40 Let
x2
fn (x) = , x ∈ [0, 1], n = 1, 2, 3, · · · .
x2 + (1 − nx)2
Then, which of the following statements is TRUE?
(A) {fn } is not equicontinuous on [0, 1]
(B) {fn } is uniformly convergent on [0, 1]
(C) {fn } is equicontinuous on [0, 1]
(D) {fn } is uniformly bounded and has a subsequence converging uniformly on [0, 1]
MA Page 41 of 66
x2
fn (x) = , x ∈ [0, 1], n = 1, 2, 3, · · · .
x2 + (1 − nx)2
Then, which of the following statements is TRUE?
(A) {fn } is not equicontinuous on [0, 1]
(B) {fn } is uniformly convergent on [0, 1]
(C) {fn } is equicontinuous on [0, 1]
(D) {fn } is uniformly bounded and has a subsequence converging uniformly on [0, 1]
MA Page 41 of 66
Page 42
Q.41 Let (Q, d) be the metric space with d(x, y) = |x−y|. Let E = {p ∈ Q : 2 < p2 < 3}.
Then, the set E is
(A) closed but not compact
(B) not closed but compact
(C) compact
(D) neither closed nor compact
MA Page 42 of 66
Then, the set E is
(A) closed but not compact
(B) not closed but compact
(C) compact
(D) neither closed nor compact
MA Page 42 of 66
Page 43
Q.42 Let T : L2 [−1, 1] → L2 [−1, 1] be defined by T f = f˜, where f˜(x) = f (−x) almost
everywhere. If M is the kernel of I − T , then the distance between the function
ϕ(t) = et and M is
1
p
(A) 2
(e2 − e−2 + 4)
1
p
(B) 2
(e2 − e−2 − 2)
1
p
(C) 2
(e2 − 4)
1
p
(D) 2
(e2 − e−2 − 4)
MA Page 43 of 66
everywhere. If M is the kernel of I − T , then the distance between the function
ϕ(t) = et and M is
1
p
(A) 2
(e2 − e−2 + 4)
1
p
(B) 2
(e2 − e−2 − 2)
1
p
(C) 2
(e2 − 4)
1
p
(D) 2
(e2 − e−2 − 4)
MA Page 43 of 66
Page 44
Q.43 Let X, Y and Z be Banach spaces. Suppose that T : X → Y is linear and
S : Y → Z is linear, bounded and injective. In addition, if S ◦ T : X → Z is
bounded, then, which of the following statements is TRUE?
(A) T is surjective
(B) T is bounded but not continuous
(C) T is bounded
(D) T is not bounded
MA Page 44 of 66
S : Y → Z is linear, bounded and injective. In addition, if S ◦ T : X → Z is
bounded, then, which of the following statements is TRUE?
(A) T is surjective
(B) T is bounded but not continuous
(C) T is bounded
(D) T is not bounded
MA Page 44 of 66
Page 45
Q.44 The first derivative of a function f ∈ C ∞ (−3, 3) is approximated by an interpolating
polynomial of degree 2, using the data
(−1, f (−1)), (0, f (0)) and (2, f (2)).
It is found that
2
f ′ (0) ≈ − f (−1) + αf (0) + βf (2).
3
1
Then, the value of is
αβ
(A) 3
(B) 6
(C) 9
(D) 12
MA Page 45 of 66
polynomial of degree 2, using the data
(−1, f (−1)), (0, f (0)) and (2, f (2)).
It is found that
2
f ′ (0) ≈ − f (−1) + αf (0) + βf (2).
3
1
Then, the value of is
αβ
(A) 3
(B) 6
(C) 9
(D) 12
MA Page 45 of 66
Page 46
Q.45 The work done by the force F = (x + y)î − (x2 + y 2 )ĵ, where î and ĵ are unit
−−→ −−→
vectors in OX and OY directions, respectively, along the upper half of the circle
x2 + y 2 = 1 from (1, 0) to (−1, 0) in the xy-plane is
(A) −π
(B) − π2
π
(C) 2
(D) π
MA Page 46 of 66
−−→ −−→
vectors in OX and OY directions, respectively, along the upper half of the circle
x2 + y 2 = 1 from (1, 0) to (−1, 0) in the xy-plane is
(A) −π
(B) − π2
π
(C) 2
(D) π
MA Page 46 of 66
Page 47
Q.46 Let u(x, t) be the solution of the wave equation
∂ 2u ∂ 2u
− 2 = 0, 0 < x < π, t > 0,
∂t2 ∂x
with the initial conditions
∂u
u(x, 0) = sin x + sin 2x + sin 3x, (x, 0) = 0, 0 < x < π
∂t
π the boundary conditions u(0, t) = u(π, t) = 0, t ≥ 0. Then, the value of
and
u , π is
2
(A) −1/2
(B) 0
(C) 1/2
(D) 1
MA Page 47 of 66
∂ 2u ∂ 2u
− 2 = 0, 0 < x < π, t > 0,
∂t2 ∂x
with the initial conditions
∂u
u(x, 0) = sin x + sin 2x + sin 3x, (x, 0) = 0, 0 < x < π
∂t
π the boundary conditions u(0, t) = u(π, t) = 0, t ≥ 0. Then, the value of
and
u , π is
2
(A) −1/2
(B) 0
(C) 1/2
(D) 1
MA Page 47 of 66
Page 48
Q.47 Let T : R2 → R2 be a linear transformation defined by
T ((1, 2)) = (1, 0) and T((2, 1)) = (1, 1).
For p, q ∈ R, let T −1 ((p, q)) = (x, y).
Which of the following statements is TRUE?
(A) x = p − q; y = 2p − q
(B) x = p + q; y = 2p − q
(C) x = p + q; y = 2p + q
(D) x = p − q; y = 2p + q
MA Page 48 of 66
T ((1, 2)) = (1, 0) and T((2, 1)) = (1, 1).
For p, q ∈ R, let T −1 ((p, q)) = (x, y).
Which of the following statements is TRUE?
(A) x = p − q; y = 2p − q
(B) x = p + q; y = 2p − q
(C) x = p + q; y = 2p + q
(D) x = p − q; y = 2p + q
MA Page 48 of 66
Page 49
Q.48 Let y = (α, −1)T , α ∈ R be a feasible solution for the dual problem of the linear
programming problem
Maximize: 5x1 + 12x2
subject to: x1 + 2x2 + x3 ≤ 10
2x1 − x2 + 3x3 = 8
x1 , x2 , x3 ≥ 0.
Which of the following statements is TRUE?
(A) α<3
(B) 3 ≤ α < 5.5
(C) 5.5 ≤ α < 7
(D) α ≥ 7
MA Page 49 of 66
programming problem
Maximize: 5x1 + 12x2
subject to: x1 + 2x2 + x3 ≤ 10
2x1 − x2 + 3x3 = 8
x1 , x2 , x3 ≥ 0.
Which of the following statements is TRUE?
(A) α<3
(B) 3 ≤ α < 5.5
(C) 5.5 ≤ α < 7
(D) α ≥ 7
MA Page 49 of 66
Page 50
Q.49 Let K denote the subset of C consisting of elements algebraic over Q. Then, which
of the following statements are TRUE?
(A) No element of C\K is algebraic over Q
(B) K is an algebraically closed field
(C) For any bijective ring homomorphism f : C −→ C, we have f (K) = K
(D) There is no bijection between K and Q
MA Page 50 of 66
of the following statements are TRUE?
(A) No element of C\K is algebraic over Q
(B) K is an algebraically closed field
(C) For any bijective ring homomorphism f : C −→ C, we have f (K) = K
(D) There is no bijection between K and Q
MA Page 50 of 66
Page 51
Q.50 Let T be a Möbius transformation
√ such that T (0) = α, T (α) = 0 and T (∞) = −α,
where α = (−1 + i)/ 2. Let L denote the straight line passing through the origin
with slope −1, and let C denote the circle of unit radius centred at the origin.
Then, which of the following statements are TRUE?
(A) T maps L to a straight line
(B) T maps L to a circle
(C) T −1 maps C to a straight line
(D) T −1 maps C to a circle
MA Page 51 of 66
√ such that T (0) = α, T (α) = 0 and T (∞) = −α,
where α = (−1 + i)/ 2. Let L denote the straight line passing through the origin
with slope −1, and let C denote the circle of unit radius centred at the origin.
Then, which of the following statements are TRUE?
(A) T maps L to a straight line
(B) T maps L to a circle
(C) T −1 maps C to a straight line
(D) T −1 maps C to a circle
MA Page 51 of 66
Page 52
1 x
Q.51 Let a > 0. Define Da : L2 (R) → L2 (R) by (Da f ) (x) = √ f , almost every-
a a
where, for f ∈ L2 (R). Then, which of the following statements are TRUE?
(A) Da is a linear isometry
(B) Da is a bijection
(C) Da ◦ Db = Da+b , b > 0
(D) Da is bounded from below
MA Page 52 of 66
Q.51 Let a > 0. Define Da : L2 (R) → L2 (R) by (Da f ) (x) = √ f , almost every-
a a
where, for f ∈ L2 (R). Then, which of the following statements are TRUE?
(A) Da is a linear isometry
(B) Da is a bijection
(C) Da ◦ Db = Da+b , b > 0
(D) Da is bounded from below
MA Page 52 of 66
Page 53
Q.52 Let {ϕ0 , ϕ1 , ϕ2 , · · · } be an orthonormal set in L2 [−1, 1] such that ϕn = Cn Pn , where
Cn is a constant and Pn is the Legendre polynomial of degree n, for each n ∈ N∪{0}.
Then, which of the following statements are TRUE?
(A) ϕ6 (1) = 1
(B) ϕ7 (−1) = 1
r
15
(C) ϕ7 (1) =
2
r
13
(D) ϕ6 (−1) =
2
MA Page 53 of 66
Cn is a constant and Pn is the Legendre polynomial of degree n, for each n ∈ N∪{0}.
Then, which of the following statements are TRUE?
(A) ϕ6 (1) = 1
(B) ϕ7 (−1) = 1
r
15
(C) ϕ7 (1) =
2
r
13
(D) ϕ6 (−1) =
2
MA Page 53 of 66
Page 54
Q.53 Let X = (R, T ), where T is the smallest topology on R in which all the singleton
sets are closed. Then, which of the following statements are TRUE?
(A) [0, 1) is compact in X
(B) X is not first countable
(C) X is second countable
(D) X is first countable
MA Page 54 of 66
sets are closed. Then, which of the following statements are TRUE?
(A) [0, 1) is compact in X
(B) X is not first countable
(C) X is second countable
(D) X is first countable
MA Page 54 of 66
Page 55
Q.54 Consider (Z, T ), where T is the topology generated by sets of the form
Am,n = {m + nk | k ∈ Z},
for m, n ∈ Z and n ̸= 0. Then, which of the following statements are TRUE?
(A) (Z, T ) is connected
(B) Each Am,n is a closed subset of (Z, T )
(C) (Z, T ) is Hausdorff
(D) (Z, T ) is metrizable
MA Page 55 of 66
Am,n = {m + nk | k ∈ Z},
for m, n ∈ Z and n ̸= 0. Then, which of the following statements are TRUE?
(A) (Z, T ) is connected
(B) Each Am,n is a closed subset of (Z, T )
(C) (Z, T ) is Hausdorff
(D) (Z, T ) is metrizable
MA Page 55 of 66
Page 56
Q.55 Let A ∈ Rm×n , c ∈ Rn and b ∈ Rm . Consider the linear programming primal
problem
Minimize: cT x
subject to: Ax = b
x ≥ 0.
Let x0 and y 0 be feasible solutions of the primal and its dual, respectively. Which
of the following statements are TRUE?
(A) c T x 0 ≥ bT y 0
(B) cT x0 = bT y 0
(C) If cT x0 = bT y 0 , then x0 is optimal for the primal
(D) If cT x0 = bT y 0 , then y 0 is optimal for the dual
MA Page 56 of 66
problem
Minimize: cT x
subject to: Ax = b
x ≥ 0.
Let x0 and y 0 be feasible solutions of the primal and its dual, respectively. Which
of the following statements are TRUE?
(A) c T x 0 ≥ bT y 0
(B) cT x0 = bT y 0
(C) If cT x0 = bT y 0 , then x0 is optimal for the primal
(D) If cT x0 = bT y 0 , then y 0 is optimal for the dual
MA Page 56 of 66
Page 57
Q.56 Consider R3 as a vector space with the usual operations of vector addition and
scalar multiplication. Let x ∈ R3 be denoted by x = (x1 , x2 , x3 ). Define subspaces
W1 and W2 by
W1 := {x ∈ R3 : x1 + 2x2 − x3 = 0}
and
W2 := {x ∈ R3 : 2x1 + 3x3 = 0}.
Let dim(U) denote the dimension of the subspace U .
Which of the following statements are TRUE?
(A) dim(W1 ) = dim(W2 )
(B) dim(W1 ) + dim(W2 ) − dim(R3 ) = 1
(C) dim(W1 + W2 ) = 2
(D) dim(W1 ∩ W2 ) = 1
MA Page 57 of 66
scalar multiplication. Let x ∈ R3 be denoted by x = (x1 , x2 , x3 ). Define subspaces
W1 and W2 by
W1 := {x ∈ R3 : x1 + 2x2 − x3 = 0}
and
W2 := {x ∈ R3 : 2x1 + 3x3 = 0}.
Let dim(U) denote the dimension of the subspace U .
Which of the following statements are TRUE?
(A) dim(W1 ) = dim(W2 )
(B) dim(W1 ) + dim(W2 ) − dim(R3 ) = 1
(C) dim(W1 + W2 ) = 2
(D) dim(W1 ∩ W2 ) = 1
MA Page 57 of 66
Page 58
Q.57 Three companies C1 , C2 and C3 submit bids for three jobs J1 , J2 and J3 . The costs
involved per unit are given in the table below:
J1 J2 J3
C1 10 12 8
C2 9 15 10
C3 15 10 9
Then, the cost of the optimal assignment is .
MA Page 58 of 66
involved per unit are given in the table below:
J1 J2 J3
C1 10 12 8
C2 9 15 10
C3 15 10 9
Then, the cost of the optimal assignment is .
MA Page 58 of 66
Page 59
dy
Q.58 The initial value problem = f (x, y), y(x0 ) = y0 is solved by using the following
dx
second order Runge-Kutta method:
K1 = hf (xi , yi )
K2 = hf (xi + αh, yi + βK1 )
1
yi+1 = yi + (K1 + 3K2 ) , i ≥ 0,
4
where h is the uniform step length between the points x0 , x1 , · · · , xn and yi =
y(xi ). The value of the product αβ is (round off to TWO decimal
places).
MA Page 59 of 66
Q.58 The initial value problem = f (x, y), y(x0 ) = y0 is solved by using the following
dx
second order Runge-Kutta method:
K1 = hf (xi , yi )
K2 = hf (xi + αh, yi + βK1 )
1
yi+1 = yi + (K1 + 3K2 ) , i ≥ 0,
4
where h is the uniform step length between the points x0 , x1 , · · · , xn and yi =
y(xi ). The value of the product αβ is (round off to TWO decimal
places).
MA Page 59 of 66
Page 60
Q.59 The surface area of the paraboloid z = x2 + y 2 between the planes z = 0 and z = 1
is (round off to ONE decimal place).
MA Page 60 of 66
is (round off to ONE decimal place).
MA Page 60 of 66
Page 61
Q.60 The rate of change of f (x, y, z) = x + x cos z − y sin z + y at P0 in the direction from
P0 (2, −1, 0) to P1 (0, 1, 2) is .
MA Page 61 of 66
P0 (2, −1, 0) to P1 (0, 1, 2) is .
MA Page 61 of 66
Page 62
Q.61 If the Laplace equation
∂ 2u ∂ 2u
+ = 0, 1 < x < 2, 1 < y < 2
∂x2 ∂y 2
with the boundary conditions
∂u ∂u
(1, y) = y, (2, y) = 5, 1 < y < 2
∂x ∂x
and
∂u αx2 ∂u
(x, 1) = , (x, 2) = x, 1 < x < 2
∂y 7 ∂y
has a solution, then the constant α is .
MA Page 62 of 66
∂ 2u ∂ 2u
+ = 0, 1 < x < 2, 1 < y < 2
∂x2 ∂y 2
with the boundary conditions
∂u ∂u
(1, y) = y, (2, y) = 5, 1 < y < 2
∂x ∂x
and
∂u αx2 ∂u
(x, 1) = , (x, 2) = x, 1 < x < 2
∂y 7 ∂y
has a solution, then the constant α is .
MA Page 62 of 66
Page 63
Q.62 Let u(x, y) be the solution of the first order partial differential equation
∂u ∂u
x + (x2 + y) = u, for all x, y ∈ R
∂x ∂y
satisfying u(2, y) = y − 4, y ∈ R. Then, the value of u(1, 2) is .
MA Page 63 of 66
∂u ∂u
x + (x2 + y) = u, for all x, y ∈ R
∂x ∂y
satisfying u(2, y) = y − 4, y ∈ R. Then, the value of u(1, 2) is .
MA Page 63 of 66
Page 64
Q.63 The optimal value for the linear programming problem
Maximize: 6x1 + 5x2
subject to: 3x1 + 2x2 ≤ 12
−x1 + x2 ≤ 1
x1 , x2 ≥ 0
is .
MA Page 64 of 66
Maximize: 6x1 + 5x2
subject to: 3x1 + 2x2 ≤ 12
−x1 + x2 ≤ 1
x1 , x2 ≥ 0
is .
MA Page 64 of 66
Page 65
Q.64 A certain product is manufactured by plants P1 , P2 and P3 whose capacities are
15, 25 and 10 units, respectively. The product is shipped to markets M1 , M2 , M3
and M4 , whose requirements are 10, 10, 10 and 20, respectively. The transportation
costs per unit are given in the table below.
M1 M2 M3 M4
P1 1 3 1 3 15
P2 2 2 4 1 25
P3 2 1 1 2 10
10 10 10 20
Then the cost corresponding to the starting basic solution by the Northwest-corner
method is .
MA Page 65 of 66
15, 25 and 10 units, respectively. The product is shipped to markets M1 , M2 , M3
and M4 , whose requirements are 10, 10, 10 and 20, respectively. The transportation
costs per unit are given in the table below.
M1 M2 M3 M4
P1 1 3 1 3 15
P2 2 2 4 1 25
P3 2 1 1 2 10
10 10 10 20
Then the cost corresponding to the starting basic solution by the Northwest-corner
method is .
MA Page 65 of 66
Page 66
Q.65 Let M be a 3 × 3 real matrix such that M 2 = 2M + 3I. If the determinant of M
is −9, then the trace of M equals .
MA Page 66 of 66
is −9, then the trace of M equals .
MA Page 66 of 66
Document Details
| Board / Org | IIT |
| Exam | Graduate Aptitude Test in Engineering |
| Type | Question Paper |
| Pages | 66 |
| Language | english |
| Updated | 22 Jul 2026 |