TIFR GS 2011 Question Paper Mathematics

Notation and Conventions

Z = set of integers
N = set of natural numbers
Q = set of rational numbers
R = set of real numbers
C = set of complex numbers
Rn = Euclidean space of dimension n

For a natural number n, the product of all the natural numbers from 1 upto n is
denoted by n!

[a, b] = {x ∈ R : a ≤ x ≤ b} for real numbers a and b with a < b.

(a, b) = {x ∈ R : a < x < b} for real numbers a and b with a < b.

For a differentiable function f : R → R, f ′ denotes its derivative.

For any natural number n, Z/nZ denotes the ring of integers modulo n.

Subsets of Rn are assumed to carry the induced topology and metric.

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PART A
1. Consider the sequence {xn } defined by xn = [nx]
n
for x ∈ R where [·] denotes
the integer part. Then {xn }

(a) converges to x.
(b) converges but not to x.
(c) does not converge
(d) oscillates

2. lim x sin(1/x2 ) equals
x→0

(a) 1.
(b) 0.
(c) ∞.
(d) oscillates

3. Let A be a 5 × 5 matrix with real entries, then A has

(a) an eigenvalue which is purely imaginary.
(b) at least one real eigenvalue.
(c) at least two eigenvalues which are not real.
(d) at least 2 distinct real eigenvalues.

4. The groups Z9 and Z3 × Z3 are

(a) isomorphic
(b) abelian
(c) non abelian
(d) cyclic

5. The differential equation
dy 1
= y 3 , y(0) = 0
dx
has

(a) a unique solution
(b) no nontrivial solution
(c) finite number of solutions.
(d) infinite number of solutions.

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6. The function fn (x) = n sin(x/n)

(a) does not converge for any x as n → ∞.
(b) converges to the constant function 1 as n → ∞.
(c) converges to the function x as n → ∞.
(d) does not converge for all x as n → ∞.

7. The equation x22 ≡ 2 mod 23 has

(a) no solutions.
(b) 23 solutions.
(c) exactly one solution.
(d) 22 solutions.

8. The sum of the squares of the roots of the cubic equation x3 − 4x2 + 6x + 1 is

(a) 0.
(b) 4.
(c) 16.
(d) none of the above.

9. The function f (x) defined by

ax + b x ≥ 1,
f (x) =
x2 + 3x + 3 x ≤ 1
is differentiable

(a) for a unique value of a and infinitely many values of b.
(b) for a unique value of b and infinitely many values of a.
(c) for infinitely many values of a and b.
(d) none of the above.

10. Let m ≤ n be natural numbers. The number of injective maps from a set of
cardinality m to a set of cardinality n is

(a) m!
(b) n!
(c) (n − m)!
(d) none of the above.

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11. For any real number c, the polynomial x3 + x + c has exactly one real root.

12. √
e 2 > 3.

13. A is 3 × 4-matrix of rank 3. Then the system of equations,

Ax = b

has exactly one solution.

14. log x is uniformly continuous on ( 12 , ∞).

15. If A, B are closed subsets of [0, ∞), then

A + B = {x + y | x ∈ A, y ∈ B}

is closed in [0, ∞).

16. The polynomial x4 + 7x3 − 13x2 + 11x has exactly one real root.

17. The value of the infinite product
∞  
Y 1
1− 2
n=2
n

is 1.

18. Consider the map T from the vector space of polynomials of degree at most
5 over the reals to R × R, given by sending a polynomial P to the pair
(P (3), P ′ (3)) where P ′ is the derivative of P . Then the dimension of the
kernel is 3.

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19. The derivative of the function
Z √x
2
e−t dt
0

at x = 1 is e−1 .

20. The equation 63x + 70y + 15z = 2010 has an integral solution.

21. Any continuous function from the open unit interval (0, 1) to itself has a fixed
point.

22. There exists a group with a proper subgroup isomorphic to itself.

23. The space of solutions of infinitely differentiable functions satisfying the equa-
tion
y” + y = 0
is infinite dimensional.

24. The series ∞ √ √
X n+1− n
n=1
n
diverges.

25. The function 
0 if x is rational
f (x) =
x if x is irrational

is not continuous anywhere on the real line.

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PART B
1. Let A be a 2 × 2-matrix with complex entries. The number of 2 × 2-matrices
A with complex entries satisfying the equation A3 = A is infinite.

2. In the ring Z/8Z, the equation x2 = 1 has exactly 2 solutions.

3. There are n homomorphisms from the group Z/nZ to the additive group of
rationals Q.

4. A bounded continuous function on R is uniformly continuous.

5. The symmetric group S5 consisting of permutations on 5 symbols has an ele-
ment of order 6.

6. Suppose fn (x) is a sequence of continuous functions on the closed interval [0, 1]
converging to 0 pointwise. Then the integral
Z 1
fn (x)dx
0

converges to 0.

7. There is a non-trivial group homomorphism from S3 to Z/3Z.

8. If A and B are 3 × 3 matrices and A is invertible, then there exists an integer
n such that A + nB is invertible.

9. Let P be a degree 3 polynomial with complex coefficients such that the con-
stant term is 2010. Then P has a root α with |α| > 10.

10. Suppose a box contains three cards, one with both sides white, one with both
sides black, and one with one side white and the other side black. If you pick
a card at random, and the side facing you is white, then the probability that
the other side is white is 1/2.

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11. There exists a set A ⊂ {1, 2, · · · , 100} with 65 elements, such that 65 cannot
be expressed as a sum of two elements in A.

12. Let S be a finite subset of R3 such that any three elements in S span a two
dimensional subspace. Then S spans a two dimensional space.

13. Any non-singular k×k-matrix with real entries can be made singular by chang-
ing exactly one entry.

14. Let f be a continuous integrable function of R such that either f (x) > 0 or
R∞
f (x) + f (x + 1) > 0 for all x ∈ R. Then −∞ f (x)dx > 0.

15. A gardener throws 18 seeds onto an equilateral triangle shaped plot of land
with sides of length one metre. Then at least two seeds are within a distance
of 25 centimetres.

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