1. Suppose the statement::j P and Q are false and the statements Rand 8 are true. Then the truth value of the statement V

is True.

is False.

is sometimes True and sometimes False.

cannot be determined.

2. The negation of the statement "There are apples that cost Rs or more" is

No apple costs less than Rs.

Some apples cost less than Rs.

There is an apple that costs Rs.

All apples cost less than Rs.

3. Consider the statements:
All bats are cats.
No cat is a rat.
All rats are tigers.
Some tigers are fighters.

Which of the following conclusions are true?
S1 Some tigers are cats.
S2 At least some fighters being rats is a possibility.
S3 At least some cats are definitely bats.
S4 None of the fighters is a rat.

Both S1 and S2 are true.

Both S3 and S4 are true.

Both S2 and S3 are true.

Both S1 and S4 are true.

4. Let X 0 x 0 y 4 For i,j EN U define Ai,j x j Y j 2 The number of elements in the set Ai,j C is

16.

12.

9.

4.

5. Let X 2,... Y and Z 2,... If F C Y C A and A C then the number of elements in F is equal to

64.

60.

56.

52.

6. Let denote the power set of A. Let cP be the empty set. Then the number of elements in is

0.

1.

2.

4.

7. In a hexagon ABCDEF, each vertex is joined with the remaining vertices with line segments. The number of line segments which are inside the hexagon is

8.

9.

12.

15.

8. Let C be the vertices of a triangular field of sides 10m, 10m and 15m. Assume that at each vertex a cow is tied with a rope of length 4m. The area that these three cows can graze together, in square meters, is

147f.

127f.

107f.

87f.

9. Raghu is at a point A. He walks 4 km to the North and then he turns to his left. He walks 4 km in this direction. He turns left again and walks 6 km. If he wishes to reach point A again, then the direction he should be walking and the distance he will have to cover are

South-East direction and 5 km.

North-East direction and 5 km.

South-East direction and 4 km.

North-East direction and 4 km.

10. The missing term in the following figure

C5 F3 I15
L1 R4
U5 X12 A60

B 5.

O 5.

B 4.

O 4.

Instructions for Questions 11 and 12: Read the following information carefully and answer the following questions.

Six people E and F are sitting on the ground at the vertices of a regular hexagon. A is not adjacent to B or D is not adjacent to Cor Band C are adjacent; F is in the middle of D and C.

11. Which of the following is not a possible neighbour pair?

A and F.

D and F.

B and E.

C and F.

12. Which of the following is in the right sequence for the above data?

A,F,B.

F,A,E.

B,C,F.

D,A,B.

13. In a certain code language MATH is written as MBTI and ABEL is written as ACEM. Then the possible word for JUNE in that code language is .

JWKP

JVNF

JPNQ

BAST

14. In a certain code language FOOD is written as HQQF and RAVI is written as TCXK. Then the possible word for JOIN in that code language is

LQKP

NQPK

JPKQ

BAST

15. In a certain code language WONDER is written as VPMEDS, Then the possible word for MASTER in that code language is

LBRWDW.

LCRVDV.

LBSVDV.

LBRUDS.

Instructions for Questions 16-20: Read the following information carefully and answer the following questions.

In a certain code language: "politics is not money" is written as "sa ri ga ma"; "media driven state politics" is written as "pa da ni sa"; "money controls many things" is written as "ok ri oo ha"; "media controls state politics" is written as "ok pa sa da" .

16. What may be the possible code for "money controls politics" in the code language?

ni pa sa.

ri ok sa.

ri panL

ga pa rna.

17. What may be the possible code for "media controls many things" in the code language?

ok da oo ha

ri ok oo ha

ok ri oo ni

ok da pa sa

18. What may be the possible code for "state driven politics" in the code language?

ga sa da

pa ni sa

ga ma ni

pa ni ga

19. What may be the possible code for "media is powerful" in the given code language?

xi ga da.

xi ma sa.

xi da rio

xi ni ha.

20. What may be the possible code for "state and controls" in the code language?

pa zz ok.

sa da zz.

00 ri zz.

ha ga pa.

21. Let x(x Consider the statements
S1 has 3 real roots.
S2: P" has 2 real roots.

Then

both S1 and S2 are true.

S1 is true but S2 is false.

S2 is true but S1 is false

both S1 and S2 are false.

22. Let for n E N. Then which of the following statements is true?

fn converges uniformly on

fn converges uniformly on for €E

L fn converges uniformly on

L fn converges uniformly on € E .

23. Let f R. f is continuous}. Consider with the norm sup If is a bounded sequence in then has a convergent subsequence with respect to 00 provided

each fn is a polynomial.

f1 f2 ...0

f1 f2 f3 ...

fn: nEN} is equicontinuous.

24. Let D be the unit open disc. Which of the following statements need not imply that an analytic function f is a constant function?

Range(f) ib: b 3a, a E R.}.

C V z E C.

C is bounded.

D V nEN.

25. The nature of the singular points of is

a pole of order 1 and 2 poles of order 2 each.

a pole of order 2 and 2 poles of order 1 each.

3 poles of order 1.

4 poles of order 1.

26. Consider the following statements.

S1 R R is a measurable function if and only if is a measurable set for every r E Q.

S2 R R is a measurable function if and only if is a measurable set for every open set U in R. Then

both S1 and S2 are true.

S1 is true but S2 is false.

S2 is true but S1 is false

both S1 and S2 are false.

27. Pick up a true statement from the following.

If f R R is Lebesgue integrable then f^2 is Lebesgue integrable.

If R R is Lebesgue integrable then 2 is Lebesgue integrable.

If f R R is Lebesgue integrable then is Lebesgue integrable.

If f R R is Lebesgue integrable then f is Riemann integrable in every compact interval.

28. Let X be any topological space, and Y a Hausdorff topological space. Let g X Y be continuous maps. Consider the following two sets:
p y E c Y x
Q EX: c X.
Then pick up a statement which is always true from the following.

P is open in Y x Y and Q is open in X.

P is open in Y x Y and Q is closed in X.

P is closed in Y x Y and Q is open in X.

P is closed in Y x Y and Q is closed in X.

29. Let denote the dual of the normed linear space X. Consider the following statements.

81: l^oe •

82: l^1. Then

both S1 and S2 are true.

S1 is true but S2 is false.

S2 is true but S1 is false

both S1 and S2 are false.

30. Let X be a Hilbert space and T be a bounded linear operator satisfying TT I. Then T is

a normal operator.

a self-adjoint operator.

an isometry.

a compact operator.

31. Consider the following statements

S1 Every complex n x n matrix of finite order A^m In for some is diagonalizable.

S2 Every real and symmetric matrix is diagonalizable. Then

both S1 and S2 are true.

S1 is true but S2 is false.

S2 is true but S1 is false

both S1 and S2 are false.

32. Let A be a complex matrix of order 5 x 5 with minimal polynomial 1). Then the number of possible Jordan canonical forms for the matrix A is 1.

2.

3.

4.

33. The number of distinct 3-cycles in S6 is

120.

40.

3!.

6!.

34. The degree of the field extension Q (sqrt2, sqrt3, sqrt6) over Q is

2.

4.

6.

8.

35. Consider the ring R x with pointwise addition and multiplication. Then the number of units in R is

21.

42.

84.

168.

36. The group generated by two elements, a and b with relations a^4 and a^3 b ab^3 ba is isomorphic to

Z/8Z.

S3.

the group of quaternions.

Q/Z.

37. Consider the regular Sturm-Liouville eigenvalue problem AS)Y 0 where p S O. Which of the following statements is true?

There exists a sequence An of eigenvalues such that An 0 as n 00.

There exists a sequence An of eigenvalues such that An 00 as n 00.

It is possible to have a purely imaginary eigenvalue.

Eigenfunctions are always of the form sin(ax) for some a E R.


38. Consider the system
dx/dt 2x
dy/dt 5x 6y. Then

there are no critical points for the given system.

there exists a critical point which is stable and there exists a critical point which is unstable.

all critical points are stable.

all critical points are unstable.

39. The solution of the problem
uxx(x, x
0 x
t 0. is

sin(2x) sin(x).

sin(2x) sin(x).

sin(x).

e^t sin(x).

40. Pick up the partial differential equation which is parabolic from the following

Uxx Uyy x^2.

Ut -Uxx 0.

Utx u^2 0.

Utt Uxx u^2 0.