TIFR GS 2012 Question Paper Mathematics
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Page 1
Correct Answers are ticked in green
on the answer sheet at the end of this file.
MTH
GS-2012 (Mathematics)
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Written Test in MATHEMATICS ‐ December 11, 2011
Duration : Two hours (2 hours)
Name : _______________________________________________ Ref. Code : ____________
Please read all instructions carefully before you attempt the questions.
1. Please fill‐in details about name, reference code etc. on the answer sheet. The Answer Sheet
is machine‐readable. Read the instructions given on the reverse of the answer sheet before
you start filling it up. Use only HB pencils to fill‐in the answer sheet.
2. Indicate your ANSWER ON THE ANSWER SHEET by blackening the appropriate circle for each
question. Each corect answer will get 1 mark; each wrong answer will get a ‐1 mark, and a
question not answered will not get you any mark. Do not mark more than one circle for any
question : this will be treated as a wrong answer.
3. There are forty (40) questions divided into four parts, Part‐A, Part‐B, Part‐C and Part‐D. Each
Part consists of 10 True‐False questions.
4. We advise you to first mark the correct answers on the QUESTION PAPER and then to
TRANSFER these to the ANSWER SHEET only when you are sure of your choice.
5. Rough work may be done on blank pages of the question paper. If needed, you may ask for
extra rough sheets from an Invigilator.
6. Use of calculators is NOT permitted.
7. Do NOT ask for clarifications from the invigilators regarding the questions. They have been
instructed not to respond to any such inquiries from candidates. In case a
correction/clarification is deemed necessary, the invigilator(s) will announce it publicly.
8. See the back of this page for Notation and Conventions used in this test.
on the answer sheet at the end of this file.
MTH
GS-2012 (Mathematics)
TATA INSTITUTE OF FUNDAMENTAL RESEARCH
Written Test in MATHEMATICS ‐ December 11, 2011
Duration : Two hours (2 hours)
Name : _______________________________________________ Ref. Code : ____________
Please read all instructions carefully before you attempt the questions.
1. Please fill‐in details about name, reference code etc. on the answer sheet. The Answer Sheet
is machine‐readable. Read the instructions given on the reverse of the answer sheet before
you start filling it up. Use only HB pencils to fill‐in the answer sheet.
2. Indicate your ANSWER ON THE ANSWER SHEET by blackening the appropriate circle for each
question. Each corect answer will get 1 mark; each wrong answer will get a ‐1 mark, and a
question not answered will not get you any mark. Do not mark more than one circle for any
question : this will be treated as a wrong answer.
3. There are forty (40) questions divided into four parts, Part‐A, Part‐B, Part‐C and Part‐D. Each
Part consists of 10 True‐False questions.
4. We advise you to first mark the correct answers on the QUESTION PAPER and then to
TRANSFER these to the ANSWER SHEET only when you are sure of your choice.
5. Rough work may be done on blank pages of the question paper. If needed, you may ask for
extra rough sheets from an Invigilator.
6. Use of calculators is NOT permitted.
7. Do NOT ask for clarifications from the invigilators regarding the questions. They have been
instructed not to respond to any such inquiries from candidates. In case a
correction/clarification is deemed necessary, the invigilator(s) will announce it publicly.
8. See the back of this page for Notation and Conventions used in this test.
Page 2
NOTATION AND CONVENTIONS
N := Set of natural numbers
Z := Set of integers
Q := Set of rational numbers
R := Set of real numbers
C := Set of complex numbers
Rn := n-dimensional vector space over R
(a, b) := {x ∈ R | a < x < b}
For a differentiable real valued function f : R → R f 0 denotes its derivative
and f (k) means the k th derivative.
Subsets of Rn are assumed to carry the induced topology and the metric.
INSTRUCTIONS
THERE ARE 4 PARTS AND 40 QUESTIONS IN TOTAL, CONSISTING
OF 10 QUESTIONS IN EACH PART.
EVERY CORRECT ANSWER CARRIES +1 MARK AND
EVERY WRONG ANSWER CARRIES −1 MARK.
Page 2 of 6
N := Set of natural numbers
Z := Set of integers
Q := Set of rational numbers
R := Set of real numbers
C := Set of complex numbers
Rn := n-dimensional vector space over R
(a, b) := {x ∈ R | a < x < b}
For a differentiable real valued function f : R → R f 0 denotes its derivative
and f (k) means the k th derivative.
Subsets of Rn are assumed to carry the induced topology and the metric.
INSTRUCTIONS
THERE ARE 4 PARTS AND 40 QUESTIONS IN TOTAL, CONSISTING
OF 10 QUESTIONS IN EACH PART.
EVERY CORRECT ANSWER CARRIES +1 MARK AND
EVERY WRONG ANSWER CARRIES −1 MARK.
Page 2 of 6
Page 3
PART A
1. If H1 & H2 are subgroups of a group G then H1 .H2 = {h1 h2 ∈ G|h1 ∈
H1 , h2 ∈ H2 } is a subgroup of G.
2. There exist polynomials f (x) and g(x), with complex coefficients, such
2
that fg(x)
(x)
= x.
3. Let f be real valued, differentiable on (a, b) and f 0 (x) 6= 0 for all x ∈
(a, b). Then f is 1 − 1.
P∞ (log log 2)n
4. The inequality n=0 n!
> 35 holds.
5. Every subgroup of order 74 in a group of order 148 is normal.
6. Let u1 , u2 , u3 , u4 be vectors in R2 and
4
X 4
X
u= tj uj ; tj > 0 and tj = 1.
j=1 j=1
2
Then three vectors v1 , v2 , v3 ∈ R may be chosen from {u1 , u2 , u3 , u4 }
such that
3
X 3
X
u= sj vj , sj ≥ 0, sj = 1.
j=1 j=1
7. The inequality
√
1 + x < 1 + x/2
for x ∈ (−1, 10) is true
8. If n is not a multiple of 23 then the remainder when n11 is divided by
23 is ±1( mod 23).
9. Suppose A is a nilpotent matrix and I is the identity matrix. Then
(I + A) is invertible.
10. The equations
1 1
x1 + x2 + x 3 = 1
2 3
1 1
x1 + x2 + x 3 = 1
4 9
1 1
x1 + x2 + x 3 = 1
8 27 Page 3 of 6
has no solution.
1. If H1 & H2 are subgroups of a group G then H1 .H2 = {h1 h2 ∈ G|h1 ∈
H1 , h2 ∈ H2 } is a subgroup of G.
2. There exist polynomials f (x) and g(x), with complex coefficients, such
2
that fg(x)
(x)
= x.
3. Let f be real valued, differentiable on (a, b) and f 0 (x) 6= 0 for all x ∈
(a, b). Then f is 1 − 1.
P∞ (log log 2)n
4. The inequality n=0 n!
> 35 holds.
5. Every subgroup of order 74 in a group of order 148 is normal.
6. Let u1 , u2 , u3 , u4 be vectors in R2 and
4
X 4
X
u= tj uj ; tj > 0 and tj = 1.
j=1 j=1
2
Then three vectors v1 , v2 , v3 ∈ R may be chosen from {u1 , u2 , u3 , u4 }
such that
3
X 3
X
u= sj vj , sj ≥ 0, sj = 1.
j=1 j=1
7. The inequality
√
1 + x < 1 + x/2
for x ∈ (−1, 10) is true
8. If n is not a multiple of 23 then the remainder when n11 is divided by
23 is ±1( mod 23).
9. Suppose A is a nilpotent matrix and I is the identity matrix. Then
(I + A) is invertible.
10. The equations
1 1
x1 + x2 + x 3 = 1
2 3
1 1
x1 + x2 + x 3 = 1
4 9
1 1
x1 + x2 + x 3 = 1
8 27 Page 3 of 6
has no solution.
Page 4
PART B
11. The automorphism group Aut (Z/2 × Z/2) is abelian
12. Let V be the vector space of consisting of polynomials of R[t] of deg
0
≤ 2. The map T : V → V sending f (t) to f (t) + f (t) is invertible.
13. The polynomials (t−1)(t−2), (t−2)(t−3), (t−3)(t−4), (t−4)(t−6) ∈
R[t] are linearly independent.
14. A ∈ M2 (C) and A is nilpotent then A2 = 0.
15. Let P be an n × n matrix whose row sums equal 1. Then for any
positive integer m the row sums of the matrix P m equal 1.
16. There is a non trivial group homomorphism from C to R.
17. If the equation
xyz = 1
holds in a group G, does it follow that
yzx = 1.
18. Any 3 × 3 and 5 × 5 skew-symmetric matrices have always zero deter-
minants.
19. The rank of the matrix
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
is 2.
20. The number 2 is a prime in Z[i]
Page 4 of 6
11. The automorphism group Aut (Z/2 × Z/2) is abelian
12. Let V be the vector space of consisting of polynomials of R[t] of deg
0
≤ 2. The map T : V → V sending f (t) to f (t) + f (t) is invertible.
13. The polynomials (t−1)(t−2), (t−2)(t−3), (t−3)(t−4), (t−4)(t−6) ∈
R[t] are linearly independent.
14. A ∈ M2 (C) and A is nilpotent then A2 = 0.
15. Let P be an n × n matrix whose row sums equal 1. Then for any
positive integer m the row sums of the matrix P m equal 1.
16. There is a non trivial group homomorphism from C to R.
17. If the equation
xyz = 1
holds in a group G, does it follow that
yzx = 1.
18. Any 3 × 3 and 5 × 5 skew-symmetric matrices have always zero deter-
minants.
19. The rank of the matrix
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
is 2.
20. The number 2 is a prime in Z[i]
Page 4 of 6
Page 5
PART C
∂ f 2
21. Let f : R2 → R be a continuous function. Then the derivative ∂x∂y can
exist without ∂f
∂x
existing.
R1
22. If f is continuous on [0, 1] and if 0
f (x)xn dx = 0 for n = 0, 1, 2, 3, · · · .
R1
Then 0 f 2 (x) dx = 0.
23. Suppose that f ∈ L2 (R). Then f ∈ L1 (R).
24. The integral
Z +∞
e−x
dx
−∞ 1 + x2
is convergent.
0
25. If A ⊂ R and open then the interior of the closure A is A.
26. If f ∈ C ∞ and f (k) (0) = 0 for all integer k ≥ 0, then f ≡ 0.
R1
27. Let f : [0, 1] → [0, 1] be continuous then f assumes the value f 2 (t)dt
0
somewhere in [0, 1].
28. Let f : R → R be a function such that
f (x + h) − f (x − h)
lim
h→0 h
exists for all x ∈ R. Then f is differentiable in R.
29. The functions f (x) = x|x| and x| sin x| are not differentiable at x = 0.
30. The composition of two uniformly continuous functions need not always
be uniformly continuous.
Page 5 of 6
∂ f 2
21. Let f : R2 → R be a continuous function. Then the derivative ∂x∂y can
exist without ∂f
∂x
existing.
R1
22. If f is continuous on [0, 1] and if 0
f (x)xn dx = 0 for n = 0, 1, 2, 3, · · · .
R1
Then 0 f 2 (x) dx = 0.
23. Suppose that f ∈ L2 (R). Then f ∈ L1 (R).
24. The integral
Z +∞
e−x
dx
−∞ 1 + x2
is convergent.
0
25. If A ⊂ R and open then the interior of the closure A is A.
26. If f ∈ C ∞ and f (k) (0) = 0 for all integer k ≥ 0, then f ≡ 0.
R1
27. Let f : [0, 1] → [0, 1] be continuous then f assumes the value f 2 (t)dt
0
somewhere in [0, 1].
28. Let f : R → R be a function such that
f (x + h) − f (x − h)
lim
h→0 h
exists for all x ∈ R. Then f is differentiable in R.
29. The functions f (x) = x|x| and x| sin x| are not differentiable at x = 0.
30. The composition of two uniformly continuous functions need not always
be uniformly continuous.
Page 5 of 6
Page 6
PART D
31. f : [0, ∞] → [0, ∞] is continuous and bounded then f has a fixed point.
32. The polynomial X 8 + 1 is irreducible in R[X].
33.
1 π 3
The matrix is diagonalisable
0 2 4
0 0 3
34. If a rectangle R := {(x, y) ∈ R2 | A ≤ x ≤ B, C ≤ y ≤ D} can be
covered (allowing overlaps) by 25 discs of radius 1 then it can also be
covered by 101 discs of radius 12 .
35. Given any integer n ≥ 2, we can always find an integer m such that
each of the n − 1 consecutive integers m + 2, m + 3,..., m + n are
composite.
36.
v1 w1 ··· v1 w10
The 10×10 matrix v2 w1 · · · v2 w10 has rank 2, where vi , wi ∈ C.
v10 w1 · · · v10 w10
37. If every continuos function on X ⊂ R2 is bounded, then X is compact.
38. The graph of xy = 1 is C2 is connected.
39. If z1 , z2 , z3 , z4 ∈ C satisfy z1 + z2 + z3 + z4 = 0 and
|z1 |2 + |z2 |2 + |z3 |2 + |z4 |2 = 1, then the least value of
|z1 − z2 |2 + |z1 − z4 |2 + |z2 − z3 |2 + |z3 − z4 |2 is 2.
40. Consider the differential equations (with y is a function of x)
dy dy 1
dx
= y dx
= |y| 3
(1) (2)
y(0) = 0 y(0) = 0.
Then (1) has infinitely many solutions but (2) has finite number of
solutions. Page 6 of 6
31. f : [0, ∞] → [0, ∞] is continuous and bounded then f has a fixed point.
32. The polynomial X 8 + 1 is irreducible in R[X].
33.
1 π 3
The matrix is diagonalisable
0 2 4
0 0 3
34. If a rectangle R := {(x, y) ∈ R2 | A ≤ x ≤ B, C ≤ y ≤ D} can be
covered (allowing overlaps) by 25 discs of radius 1 then it can also be
covered by 101 discs of radius 12 .
35. Given any integer n ≥ 2, we can always find an integer m such that
each of the n − 1 consecutive integers m + 2, m + 3,..., m + n are
composite.
36.
v1 w1 ··· v1 w10
The 10×10 matrix v2 w1 · · · v2 w10 has rank 2, where vi , wi ∈ C.
v10 w1 · · · v10 w10
37. If every continuos function on X ⊂ R2 is bounded, then X is compact.
38. The graph of xy = 1 is C2 is connected.
39. If z1 , z2 , z3 , z4 ∈ C satisfy z1 + z2 + z3 + z4 = 0 and
|z1 |2 + |z2 |2 + |z3 |2 + |z4 |2 = 1, then the least value of
|z1 − z2 |2 + |z1 − z4 |2 + |z2 − z3 |2 + |z3 − z4 |2 is 2.
40. Consider the differential equations (with y is a function of x)
dy dy 1
dx
= y dx
= |y| 3
(1) (2)
y(0) = 0 y(0) = 0.
Then (1) has infinitely many solutions but (2) has finite number of
solutions. Page 6 of 6
Page 7
GS-2012 (MATHEMATICS)
ANSWER SHEET
Please see reverse for instructions on filling of answer sheet.
Name Reference Code :
Ref Code 1 { { { { {
Address 2 { { { { {
3 { { { { {
4 { { { { {
5 { { { { {
Phone 6 { { { { {
Email 7 { { { { {
8 { { { { {
9 { { { { {
0 { { { { {
PART-A PART-B PART-C PART-D
True False True False True False True False
1 { { 1 { { 1 { { 1 { {
2 { { 2 { { 2 { { 2 { {
3 { { 3 { { 3 { { 3 { {
4 { { 4 { { 4 { { 4 { {
5 { { 5 { { 5 { { 5 { {
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7 { { 7 { { 7 { { 7 { {
8 { { 8 { { 8 { { 8 { {
9 { { 9 { { 9 { { 9 { {
10 { { 10 { { 10 { { 10 { {
____________________
Signature of the Student
ANSWER SHEET
Please see reverse for instructions on filling of answer sheet.
Name Reference Code :
Ref Code 1 { { { { {
Address 2 { { { { {
3 { { { { {
4 { { { { {
5 { { { { {
Phone 6 { { { { {
Email 7 { { { { {
8 { { { { {
9 { { { { {
0 { { { { {
PART-A PART-B PART-C PART-D
True False True False True False True False
1 { { 1 { { 1 { { 1 { {
2 { { 2 { { 2 { { 2 { {
3 { { 3 { { 3 { { 3 { {
4 { { 4 { { 4 { { 4 { {
5 { { 5 { { 5 { { 5 { {
6 { { 6 { { 6 { { 6 { {
7 { { 7 { { 7 { { 7 { {
8 { { 8 { { 8 { { 8 { {
9 { { 9 { { 9 { { 9 { {
10 { { 10 { { 10 { { 10 { {
____________________
Signature of the Student
Page 8
INSTRUCTIONS
The Answer Sheet is machine-readable. Apart from filling in the details on the answer sheet,
please make sure that the Reference Code is filled by blackening the appropriate circles in
the box provided on the right-top corner. Only use HB pencils to fill-in the answer sheet.
e.g. if your reference code is 15207 :
Also, the multiple choice questions are to be answered by blackening the appropriate circles
as described below
e.g. if your answer to question 1 is (b)
and your answer to question 2 is (d)
then ..........
The Answer Sheet is machine-readable. Apart from filling in the details on the answer sheet,
please make sure that the Reference Code is filled by blackening the appropriate circles in
the box provided on the right-top corner. Only use HB pencils to fill-in the answer sheet.
e.g. if your reference code is 15207 :
Also, the multiple choice questions are to be answered by blackening the appropriate circles
as described below
e.g. if your answer to question 1 is (b)
and your answer to question 2 is (d)
then ..........
Document Details
| Board / Org | Default |
| Exam | TIFR GS |
| Type | Question Paper |
| Pages | 8 |
| Updated | 22 Jul 2026 |