univ.tifr.res.in GS Admissions Exam Mathematics Question Paper 2019 : Tata Institute of Fundamental Research

Name of the Centre : Tata Institute of Fundamental Research
Document type : Previous Question Papers
Name Of The Exam : GS-2019 Graduate School Admissions Exam
Name Of The Subject : Mathematics Question Paper
Year : 2019
Website : http://univ.tifr.res.in/gs2019/Prev_QP/Prev_QP.htm

GS Admissions Exam Mathematics Question Paper

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Instructions

** 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 with the Euclidean metric, and I denotes the identity matrix in Mn(R).
** All rings are associative, with a multiplicative identity.

** For any prime number p, Fp denotes the finite field with p elements.
** If A and B are sets, then A – B refers to {x ? A | x ? B}.
** For a ring R, R[x] denotes the polynomial ring in one variable over R, and R[x, y] denotes the polynomial ring in two variables over R.

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2018 :
https://www.pdfquestion.in/uploads/pdf2019/33696-GS2018MTH.pdf
2019 :
https://www.pdfquestion.in/uploads/pdf2019/33696-GS2019MTH.pdf

Part A

Answer the following multiple choice questions.
1. The following sum of numbers (expressed in decimal notation)
1 + 11 + 111 + ··· + 11…1 } {{ }
is equal to
(a) (10n+1 – 10 – 9n)/81
(b) (10n+1 – 10 + 9n)/81
(c) (10n+1 – 10 – n)/81
(d) (10n+1 – 10 + n)/81

2. For n = 1, the sequence {xn}8n=1, where:
xn =1+ v12 + ··· + v1n – 2vn
is
(a) decreasing
(b) increasing
(c) constant
(d) oscillating

3. Define a function:
f(x) = {x + x2 cos(px), x = 0 0, x = 0.
Consider the following statements:
(i) f (0) exists and is equal to 1
(ii) f is not increasing in any neighborhood of 0
(iii) f (0) does not exist (iv) f is increasing on R.
How many of the above statements is/are true?
(a) 0
(b) 1
(c) 2
(d) 3

4. Consider differentiable functions f : R ? R with the property that for all a, b ? R we have: f(b) – f(a)=(b – a)f (a + b ). Then which one of the following sentences is true?
(a) Every such f is a polynomial of degree less than or equal to 2
(b) There exists such a function f which is a polynomial of degree bigger than 2
(c) There exists such a function f which is not a polynomial
(d) Every such f satisfies the condition f (a + b ) = f(a) + 2 f(b) for all a, b ? R

5. Let V be an n-dimensional vector space and let T : V ? V be a linear transformation such that Rank T = Rank T 3. Then which one of the following statements is necessarily true?
(a) Null space(T) = Range(T)
(b) Null space(T) n Range(T) = {0}
(c) There exists a nonzero subspace W of V such that Null space(T) n Range(T) = W
(d) Null space(T) ? Range(T)

6. The limit n?8lim n2 ? 0 1(1 + 1 x2)ndx is equal to
(a) 1
(b) 0
(c) +8
(d) 1/2

7. Let A be an n × n matrix with rank k. Consider the following statements:
(i) If A has real entries, then AAt necessarily has rank k (ii) If A has complex entries, then AAt necessarily has rank k. Then
(a) (i) and (ii) are true
(b) (i) and (ii) are false
(c) (i) is true and (ii) is false
(d) (i) is false and (ii) is true

8. Consider the following two statements: 3
(E) Continuous functions on [1,2] can be approximated uniformly by a sequence of even polynomials (i.e., polynomials p(x) ? R[x] such that p(-x) = p(x)). (O) Continuous functions on [1,2] can be approximated uniformly by a sequence of odd polynomials (i.e., polynomials p(x) ? R[x] such that p(-x) = -p(x)). Choose the correct option below.
(a) (E) and (O) are both false
(b) (E) and (O) are both true
(c) (E) is true but (O) is false
(d) (E) is false but (O) is true

9. Let f : (0,8) ? R be defined by f(x) = sin(x3)
x . Then f is
(a) bounded and uniformly continuous
(b) bounded but not uniformly continuous
(c) not bounded but uniformly continuous (d) not bounded and not uniformly continuous

10. Let S = {x ? R | x = Trace(A) for some A ? M4(R) such that A2 = A}. Then which of the following describes S?
(a) S = {0,2,4} (b) S = {0,1/2,1,3/2,2,5/2,3,7/2,4} (c) S = {0,1,2,3,4} (d) S = [0,4] 11. Let f be a continuous function on [0,1]. Then the limit lim n?8? 10 nxnf(x)dx is equal to
(a) f(0)
(b) f(1)
(c) sup x?[0,1]f(x)
(d) The limit need not exist

Part B

Answer whether the following statements are True or False.
F1. There exists a continuous function f : R ? R such that f(Q) ? R-Q and f(R-Q) ? Q.
T2. If A ? M10(R) satisfies A2 + A + I = 0, then A is invertible.
F3. Let X ? Q2. Suppose each continuous function f : X ? R2 is bounded. Then X is necessarily finite.
F4. If A is a 2 × 2 complex matrix that is invertible and diagonalizable, and such that A and A2 have the same characteristic polynomial, then A is the 2 × 2 identity matrix.
F5. Suppose A,B,C are 3 × 3 real matrices with Rank A = 2,Rank B = 1,Rank C = 2. Then Rank (ABC) = 1.
F6. For any n = 2, there exists an n × n real matrix A such that the set {Ap | p = 1} spans the R-vector space Mn(R).
T7. The matrices ?0 i 0

8. Consider the set A ? x3 – 3×2 + 2x – 1. Then M3(R) of 3 × 3 real A is a compact subset matrices of M3(R) with ~= Rcharacteristic 9. polynomial
9. There exists an injective ring homomorphism from the product ring R × R into C(R), where C(R) denotes the ring of all continuous functions R ? R under pointwise addition and multiplication.

10. R and R ? R are isomorphic as vector spaces over Q.
11. If 0 is a limit point of a set A ? (0,8), then the set of all x ? (0,8) that can be expressed as a sum of (not necessarily distinct) elements of A is dense in (0,8).

12. The only idempotents in the ring Z51 (i.e., Z/51Z) are 0 and 1. (An idempotent is an element x such that x2 = x).
13. Let A be a commutative ring with 1, and let a, b, c ? A. Suppose there exist x,y,z ? A such that ax+by+cz = 1. Then there exist x ,y ,z ? A such that a50x +b20y +c15z = 1.

14. The ring R[x]/(x5 + x – 3) is an integral domain.
15. Given any group G of order 12, and any n that divides 12, there exists a subgroup H of G of order n.

16. Let H, N be subgroups of a finite group G, with N a normal subgroup of G. If the orders of G/N and H are relatively prime, then H is necessarily contained in N
17. If every proper subgroup of an infinite group G is cyclic, then G is cyclic.
18. Each solution of the differential equation y + exy = 0 remains bounded as x ? 8.

19. There exists a uniformly continuous function f : (0,8) ? (0,8) such that ?8n=1 1 f(n) converges.
20. Let v : R ? R2 be C8 (i.e., has derivatives of all orders). such that v(1) – v(0) is a scalar multiple of dvdtThen there exists t0 ? (0,1)