TIFR GS 2020 Question Paper Mathematics

GS-2020 Mathematics

Notation and Conventions
• 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.

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

• Mn (R) denotes the real vector space of n×n real matrices, and Mn (C) the complex vector
space of n × n complex matrices. I denotes the identity matrix in Mn (R) ⊂ Mn (C).

• For any A ∈ Mn (C), we denote by tr(A) the trace of A and by det(A) the determinant
of A.

• All rings are associative, with a multiplicative identity.

• For a ring R, R[x] denotes the polynomial ring in one variable over R, and R× denotes
the multiplicative group of units of R.

• All logarithms are natural logarithms.

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

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PART A
Answer the following multiple choice questions.

1. Consider the sequences {an }∞ ∞
n=1 and {bn }n=1 defined by
n
an = (2n + 3n )1/n and bn = Pn 1 .
i=1 ai

What is the limit of {bn }∞
n=1 ?

(a) 2.
(b) 3.
(c) 5.
(d) The limit does not exist.

2. Consider the set of continuous functions f : [0, 1] → R that satisfy:
Z 1
1
f (x)(1 − f (x)) dx = .
0 4
Then the cardinality of this set is:

(a) 0.
(b) 1.
(c) 2.
(d) more than 2.

3. Let f : R → R be defined as:

x2 sin 1 , if x 6= 0, and
f (x) = x
0, if x = 0.

Which of the following statements is correct?

(a) f is a surjective function.
(b) f is bounded.
(c) The derivative of f exists and is continuous on R.
(d) {x ∈ R | f (x) = 0} is a finite set.

4. Let {an }∞
n=1 be a strictly increasing bounded sequence of real numbers such that
lim an = A. Let f : [a1 , A] → R be a continuous function such that for each positive
n→∞
integer i, f |[ai ,ai+1 ] : [ai , ai+1 ] → R is either strictly increasing or strictly decreasing.
Consider the set

B = {M ∈ R | there exist infinitely many x ∈ [a1 , A] such that f (x) = M }.

Then the cardinality of B is:

2

(a) necessarily 0.
(b) at most 1.
(c) possibly greater than 1, but finite.
(d) possibly infinite.

5. Let f : R → R be a function that satisfies:

|f (x) − f (y)| ≤ |x − y|| sin(x − y)|, for all x, y ∈ R.

Which of the following statements is correct?

(a) f is continuous but need not be uniformly continuous.
(b) f is uniformly continuous but not necessarily differentiable.
(c) f is differentiable, but its derivative may not be continuous.
(d) f is constant.

6. Let n o
C = f : R → R | f is differentiable, and lim (2f (x) + f 0 (x)) = 0 .
x→∞
Which of the following statements is correct?

(a) For each L with 0 6= L < ∞, there exists f ∈ C such that lim f (x) = L.
x→∞
(b) For all f ∈ C, lim f (x) = 0.
x→∞
(c) There exists f ∈ C such that lim f (x) does not exist.
x→∞
1
(d) There exists f ∈ C such that lim f (x) = .
x→∞ 2
log(2 + x) 1 Rm
7. Let f (x) = √ for x ≥ 0, and am = f (t) dt for every positive integer m.
1+x m 0
Then the sequence {am }∞m=1

(a) diverges to +∞.
(b) has more than one limit point.
1
(c) converges and satisfies lim am = log 2.
m→∞ 2
(d) converges and satisfies lim am = 0.
m→∞

8. Let f : R → R be a continuous function such that:

|f (x) − f (y)| ≥ log(1 + |x − y|), for all x, y ∈ R.

Then:

(a) f is injective but not surjective.
(b) f is surjective but not injective.
(c) f is neither injective nor surjective.
(d) f is bijective.

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9. What is the greatest integer less than or equal to
9999
X 1
√
4
?
n=1
n

(a) 1332
(b) 1352
(c) 1372
(d) 1392

10. Consider the following sentences:

(I) For every connected subset Y of a metric space X, its interior Y ◦ is connected.
(II) For every connected subset Y of a metric space X, its boundary ∂Y is connected.

Which of the following options is correct?

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

11. Consider a set {A1 , . . . , An } of events, n > 1. Suppose that one of the events in
{A1 , . . . , An } is certain to occur, but that no more than two of them can occur. Suppose
that for each 1 ≤ r, s ≤ n such that r 6= s, the probability of Ar occurring is p, while
the probability of both Ar and As occurring is q. Then:

(a) p ≤ 1/n and q ≤ 2/n.
(b) p ≤ 1/n and q ≥ 2/n.
(c) p ≥ 1/n and q ≤ 2/n.
(d) p ≥ 1/n and q ≥ 2/n.

12. Let {z1 , z2 , . . . , z7 } be a set of seven complex numbers with unit modulus. Assume that
they form the vertices of a regular heptagon in the complex plane. Define
X
w= zi z j .
i<j

Then:

(a) w = 0.
√
(b) |w| = 7.
(c) |w| = 7.
(d) |w| = 1.

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13. Consider R3 as the space of 3 × 1 real matrices. The multiplicative group GL3 (R) of
invertible 3 × 3 real matrices acts on this space by left multiplication. What is the
number of orbits for this action?

(a) 1.
(b) 2.
(c) 4.
(d) ∞.

14. Let V be a finite dimensional vector space over R, and W ⊂ V a subspace. Then
W ∩ T (W ) 6= {0} for every linear automorphism T : V → V if and only if:

(a) W = V .
1
(b) dim W < dim V .
2
1
(c) dim W = dim V .
2
1
(d) dim W > dim V .
2
 
A A
15. Let A ∈ Mn (C). Then is diagonalizable if and only if:
0 A

(a) A = 0.
(b) A = I.
(c) n = 2.
(d) None of the other three options.

16. Let T : C → R be the map defined by T (z) = z + z̄. For a C-vector space V , consider
the map

ϕ : {f : V → C | f is C-linear} → {g : V → R | g is R-linear},

defined by ϕ(f ) = T ◦ f . Then this map is

(a) injective, but not surjective.
(b) surjective, but not injective.
(c) bijective.
(d) neither injective nor surjective.

17. Which of the following statements is correct for every linear transformation T : R3 → R3
such that T 3 − T 2 − T + I = 0?

(a) T is invertible as well as diagonalizable.
(b) T is invertible, but not necessarily diagonalizable.
(c) T is diagonalizable, but not necessarily invertible.
(d) None of the other three statements.

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18. Let n ≥ 2. Which of the following statements is true for every n × n real matrix A of
rank one?

(a) There exist matrices P, Q ∈ Mn (R) such that all the entries of the matrix P AQ are
equal to 1.
(b) There exists an invertible matrix P ∈ Mn (R) such that P AP −1 is a diagonal matrix.
(c) A has a nonzero eigenvalue.
(d) The vector (1, 1, . . . , 1) ∈ Rn is an eigenvector for A.

19. Let m, n be positive integers. Then the greatest common divisor (gcd) of the polynomials
xm − 1 and xn − 1 in the ring C[x] equals

(a) xmin(m,n) − 1.
(b) x − 1.
(c) xgcd(m,n) − 1.
(d) None of the other three options.

20. Let A4 denote the group of even permutations of {1, 2, 3, 4}. Consider the following
statements:

(I) There exists a surjective group homomorphism A4 → Z/4Z.
(II) There exists a surjective group homomorphism A4 → Z/3Z.

Which of the following statements is correct?

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

PART B
True/False Questions.

F 1. There exists no monotone function f : R → R which is discontinuous at every rational
number.

T 2. Let C([0, 1]) denote the set of continuous real valued functions on [0, 1], and RN the set
of all sequences of real numbers. Then there exists an injective map from C([0, 1]) to
RN .

T 3. Let {an }∞
n=1 be a bounded sequence of positive real numbers. Then:

1 1
lim sup = .
n→∞ an lim inf an
n→∞

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T 4. Let C([0, 1]) denote the metric space of continuous real valued functions on [0, 1] under
the supremum metric - i.e., the distance between f and g in C([0, 1]) equals

sup{|f (x) − g(x)| | x ∈ [0, 1]}.

Let Q ⊂ C([0, 1]) be the set of all polynomials in R[x] in which the coefficient of x2 is
0. Then Q is dense in C([0, 1]).

F 5. If X is a metric space such that every continuous function f : X → R is uniformly
continuous, then X is compact.

T 6. Let X be a metric space, and let C(X) denote the R-vector space of continuous real
valued functions on X. Then X is infinite if and only if dimR C(X) = ∞.

T 7. Let A be a countable union of lines in R3 . Then R3 \ A is connected.

T 8. An invertible linear map from R2 to itself takes parallel lines to parallel lines.

F 9. For any matrix C with entries in C, let m(C) denote the minimal polynomial of C, and
p(C) its characteristic polynomial. Then for any n ∈ N, two matrices A, B ∈ Mn (C)
are similar if and only if m(A) = m(B) and p(A) = p(B).

T 10. Let A, B ∈ M3 (R). Then
tr[(AB − BA)3 ]
det(AB − BA) = .
3

F 11. There exist an integer r ≥ 1 and a symmetric matrix A ∈ Mr (R) such that for all n ∈ N,
we have: √ √
2 n ≤ |tr(An )| ≤ 2020 · 2 n .

T 12. The polynomial 1 + x + x2! + · · · + x101! is irreducible in Q[x].
2 101

F 13. There exists an integer n > 3 such that the group of units of the ring Z/2n Z is cyclic.
F 14. For every surjective ring homomorphism ϕ : R → S, we have ϕ(R× ) = S × .
F 15. Let G be a finite group and P a p-Sylow subgroup of G, where p is a prime number.
Then for every subgroup H of G, H ∩ P is a p-Sylow subgroup of H.

T 16. Let G be an abelian group, with identity element e. If
{g ∈ G | g = e or g has infinite order}

is a subgroup of G, then either all elements of G \ {e} have infinite order, or all elements
of G have finite order.

F 17. There exists a natural number n, with 1 < n ≤ 10, such that xn and x are conjugate
for every element x of S7 , the group of permutations of {1, . . . , 7}.

F 18. Every noncommutative ring has at least 10 elements.

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T 19. Let {an }∞
n=1 be a sequence of elements in {0, 1} such that for all positive integers n,
n+9
X
ai is divisible by 3. Then there exists a positive integer k such that an+k = an for
i=n
all positive integers n.

T 20. The interior of any strip bounded by two parallel lines in R2 , of width strictly greater
than 1, contains a point with integer coordinates.

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