TIFR GS 2022 Question Paper Mathematics

Notation and Conventions
ˆ N denotes the set of natural numbers {0, 1, . . . }, Z the set of integers, Q the set of
rational numbers, R the set of real numbers, and C the set of complex numbers. These
sets are assumed to carry the usual algebraic and metric structures.

ˆ Rn denotes the Euclidean space of dimension n. Subsets of Rn are viewed as metric
spaces using the standard Euclidean distance on Rn .

ˆ Mn (R) denotes the real vector space of n × n real matrices, and Mn (C) the complex
vector space of n × n complex matrices. Mn (R) gets the topology transferred from any
R-linear isomorphism Mn (R) ∼
2
= Rn , and its subsets get the subspace topology.

ˆ Id denotes the identity matrix in Mn (R) ⊂ Mn (C).

ˆ For a metric space X, C(X, R) denotes the set of continuous functions from X to R,
viewed as a ring under pointwise addition and multiplication. Similarly, C(X, C) will
denote the ring of continuous functions from X to C, again with pointwise addition and
multiplication.

ˆ A matrix T ∈ Mn (C), or a linear transformation T : V → V from a vector space V to
itself, is called idempotent if T 2 = T , and nilpotent if T m = 0 for some positive integer
m.

ˆ All rings are associative, with a multiplicative identity. A subring of a ring R is assumed
by definition to contain the multiplicative identity of R.

ˆ If q is a power of a prime number, Fq will stand for the finite field with q elements.

ˆ For a ring R, R[x1 , . . . , xn ] denotes the polynomial ring in n variables x1 , . . . , xn over
R, and R× denotes the multiplicative group of units of R.

ˆ All logarithms are natural logarithms.

ˆ If B is a subset of a set A, we write A \ B for the set {a ∈ A | a 6∈ B}.

ˆ By the group of isometries of a metric space (X, d), we mean the group whose elements
are bijective maps f : X → X satisfying d(x, y) = d(f (x), f (y)) for all x, y ∈ X, and
whose multiplication is defined by composition.

ˆ If f is a real or complex valued function defined on an open interval (a, b) of R, then f
is said to be continuously differentiable on (a, b) if its derivative exists and is continuous
on (a, b).

ˆ If f is a real or complex valued function defined on an open interval of R, then f 0 will
stand for the first derivative of f (wherever it exists), and f 00 for the second derivative
of f (wherever it exists).

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PART A
Answer the following multiple choice questions.

1. Consider the following properties of a metric space (X, d):

(I) (X, d) is complete as a metric space.
(II) For any sequence {Zn }n∈N of closed nonempty subsets of X, such that Z1 ⊇ Z2 ⊇
. . . and !
lim sup d(x, y) = 0,
n→∞ x,y∈Zn
T∞
n=1 Zn is a singleton set.

Which of the following sentences is true?

(a) (I) implies (II) and (II) implies (I).
(b) (I) implies (II) but (II) does not imply (I).
(c) (I) does not imply (II) but (II) implies (I).
(d) (I) does not imply (II) and (II) does not imply (I).

2. Consider the following assertions:

(I) {(x, y) ∈ R2 | xy = 1} is connected.
(II) {(x, y) ∈ C2 | xy = 1} is connected.

Which of the following sentences is true?

(a) Both (I) and (II) are true.
(b) (I) is true but (II) is false.
(c) (I) is false but (II) is true.
(d) Both (I) and (II) are false.

3. What is the number of solutions of:
x2 x
x= − cos + 2
50 2
in [0, 10]?

(a) 0
(b) 1
(c) 2
(d) ∞

4. Let A be an element of M4 (R) with characteristic polynomial t4 − t. What is the
characteristic polynomial of A2 ?

(a) t4 − t

2

(b) t4 − 2t3 + t2
(c) t4 − t2
(d) None of the other three options

5. Let n be a positive integer, and let V = {f ∈ R[x] | deg f ≤ n} be the real vector space
of real polynomials of degree at most n. Let EndR (V ) denote the real vector space of
linear transformations from V to itself. For m ∈ Z, let Tm ∈ EndR (V ) be such that
(Tm f )(x) = f (x + m) for all f ∈ V . Then the dimension of the vector subspace of
EndR (V ) given by
Span ({Tm | m ∈ Z})
is

(a) 1
(b) n
(c) n + 1
(d) n2

6. Let T be the linear transformation from the real vector space R[x] to itself, given by
T (f ) = f 0 , where f 0 is the derivative of f . Consider the following statements about T :

(I) T is nilpotent.
(II) The only eigenvalue of T is 0.

Which of the following sentences is true?

(a) Both (I) and (II) are true.
(b) (I) is true but (II) is false.
(c) (I) is false but (II) is true.
(d) Both (I) and (II) are false.

7. What is the cardinality of the set of θ ∈ [0, 2π) such that the linear map R2 → R2 given
by the matrix:  
cos θ − sin θ
sin θ cos θ
has an eigenvector in R2 ?

(a) 1
(b) 2
(c) 4
(d) ∞
 
2 −1
8. Let p be a prime number, and let A equal , viewed as a 2 × 2 matrix with
1 0
integer entries. What is the smallest positive integer n such that the matrix An is
congruent to the 2 × 2 identity matrix modulo p?

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(a) p2 − 1
(b) p − 1
(c) p
(d) p + 1

9. What is the largest value of n for which there exists a set {A1 , . . . , An } of (distinct)
nonzero matrices in M2 (C) such that A∗i Aj has trace zero for all 1 ≤ i < j ≤ n?

(a) 1
(b) Greater than 1 but at most 4
(c) Greater than 4 but finite
(d) ∞

10. Let p be a prime number. What is the number of elements in the group Q/Z that have
order exactly p?

(a) 0
(b) p − 1
(c) p
(d) ∞

11. Consider the real polynomial

f (x) = x11 − x7 + x2 − 1.

Which of the following sentences is correct?

(a) f (x) has exactly one positive root.
(b) f (x) has exactly two positive roots.
(c) f (x) has at least three positive roots.
(d) None of the other three options.

12. Consider polynomials
∞
X ∞
X
i j
f1 (x, y) = aij x y and f2 (x, y) = bij xi y j ∈ R[x, y]
i,j=0 i,j=0

(where aij = bij = 0 for all but finitely many (i, j) ∈ N2 ), such that f1 (p, q) = f2 (p, q)
for all (p, q) ∈ R2 satisfying p2 = q 2 . Which of the following sentences is true for all
such f1 and f2 ?

(a) a00 = b00 , but we may not have aij = bij for all (i, j) with i + j = 1.
(b) aij = bij if i + j ≤ 1, but we may not have aij = bij for all (i, j) with i + j = 2.
(c) aij = bij if i + j ≤ 2, but we may not have aij = bij for all (i, j) with i + j = 3.
(d) aij = bij if i + j ≤ 3, but we may not have f1 = f2 .

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13. What is the number of bijections f : {1, 2, . . . , 9} → {1, 2, . . . , 9} such that, for all
distinct i, j ∈ {1, . . . , 9}, whenever the squares labelled i and j in the diagram below
share an edge, the squares labelled f (i) and f (j) share an edge too?

1 2 3
4 5 6
7 8 9

(a) 8
(b) 9
(c) 4
(d) 24

14. Let m be the number of positive integers n such that 1 ≤ n ≤ 2022 and such that n has
an odd number of (positive integer) divisors. Then m is

(a) 22
(b) 33
(c) 44
(d) 55

15. Let S be the set of nonnegative continuous functions f on [0, 1] satisfying
Z 1 Z 1 Z 1
2
sin (x)f (x) dx = sin(x) cos(x)f (x) dx = cos2 (x)f (x) dx = 1.
0 0 0

Then S is:

(a) an uncountable set
(b) a countably infinite set
(c) a finite and nonempty set
(d) the empty set

16. Let f (x) = 1 − sin x for x ∈ R. Define
s    
n 1 2 n
an = f f ...f .
n n n

Then

(a) {an }n converges to 0
(b) {an }n diverges to ∞
(c) {an }n converges and lim an > 0
n→∞
(d) none of the other three options is correct

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17. Which of the following is true for every function u : R → R which is continuously
differentiable on R (i.e., u is diffferentiable on R and its derivative u0 is continuous on
R), and satisfies
u(y) ≥ u(x) + u0 (x)(y − x)
for all x, y ∈ R?

(a) u0 is nonnegative.
(b) u attains a minimum at some x ∈ R.
(c) u0 is nondecreasing.
(d) u0 is nonincreasing.

18. Let {xn } be a sequence of positive numbers such that lim xn = x. Define
n→∞
"  n−1 #
1  x n x
zn = x1 1 + + x2 1 + + · · · + xn (1 + x) .
n n n−1

Then

(a) {zn } converges to xe
(b) {zn } converges to ex
(c) {zn } does not have a limit
(d) {zn } converges to xex

19. The value of Z 1
nex
lim dx
n→∞ 0 1 + n2 x2
is

(a) πe
(b) π2
(c) π2 e
(d) π

20. What is the number of real solutions of the equation

esin x = π?

(a) 0
(b) 1
(c) Countably infinite
(d) Uncountable

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PART B
Answer whether the following statements are True or False.

1. R2 \ Q2 is connected but not path-connected. False

2. If X is a connected metric space, and F is a subring of C(X, R) that is a field, then
every element of C(X, R) that belongs to F is a constant function. True

3. Let K ⊆ [0, 1] be the Cantor set. Then there exists no injective ring homomorphism
C([0, 1], R) → C(K, R). False

4. There exists a metric space (X, d) such that the group of isometries of X is isomorphic
to Z. True

5. Let A ⊂ R2 be a nonempty subset such that any continuous function f : A → R is
constant. Then A is a singleton set.
True
6. For a nilpotent matrix A ∈ Mn (R), let
∞
X An A A2
exp(A) := = Id + + + · · · ∈ Mn (R).
n! 1! 2!
n=0

If A is a nilpotent matrix such that exp(A) = Id, then A is the zero matrix.
 
True
a b
7. There exists A = ∈ M2 (R), with A2 = A 6= 0, such that
c d

|a| + |b| < 1 and |c| + |d| < 1. False

8. If A ∈ M3 (C) is such that Ai has trace zero for all positive integers i, then A is nilpotent. True

9. For any finite cyclic group G, there exists a prime power q such that G is a subgroup
of F×
q . True
10. There are only finitely many isomorphism classes of finite nonabelian groups, all of
whose proper subgroups are abelian. False
11. Every subring of a unique factorization domain is a unique factorization domain.
False
12. Let f1 , f2 , f3 , f4 ∈ R[x] be monic polynomials each of degree exactly two. Then there
exist a real polynomial p ∈ R[x] and a subset {i, j} ⊂ {1, 2, 3, 4} with i 6= j, such that
fi ◦ p = cfj for some c ∈ R.
True
13. There exists a finite abelian group G such that the group Aut(G) of automorphisms of
G is isomorphic to Z/7Z.
False
14. There exists an integral domain R and a surjective homomorphism R → R of rings that
is not injective. True
15. There exists f ∈ C([0, 1], R) satisfying the following two conditions:

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R1
(i) 0 f (x) dx = 1; and
R1
(ii) lim 0 f (x)n dx = 0.
n→∞ False
P∞ √
16. Let an ≥ 0Pfor each positive integer n. If the series n=1 an converges, then so does
the series ∞ an
n=1 n1/4 . True
17. There exists a differentiable function f : R → R such that

lim f (x) = 2 and lim f 0 (x) = 1.
x→∞ x→∞ False
18. Let f : [0, 1] → [0, ∞) be continuous on [0, 1] and twice differentiable in (0, 1). If
f 00 (x) = 7f (x) for all x ∈ (0, 1), then f (x) ≤ max{f (0), f (1)} for all x ∈ [0, 1].
True
19. There are N balls in a box, out of which n are blue (1 < n < N ) and the rest are
red. Balls are drawn from the box one by one at random, and discarded. Then the
probability of picking all the blue balls in the first n draws is the same as the probability
of picking all the red balls in the first (N − n) draws.
True
20. The set {f (x) ∈ R[x] | f (n) ∈ Z for all n ∈ Z} is uncountable.
False

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