1. The principal value for equation (−1)^(−2i/π) is

e²
eⁱ
e²ⁱ
e⁻²

2. Let X and Y be metric spaces, and let f : X → Y be a continuous map. For any subset S of X, choose the correct statement?

f(S) is open when S is open
f(S) is connected when S is connected
f(S) is closed when S is closed
f(S) is bounded when S is bounded

3. Let f :R² →R be defined by

f (x, y)= x⁶-2x²y-x⁴y+2y².

(R is the set of all real numbers and R2 ={(x, y) : x, y ϵ R)}

Choose the correct statement?

f has a local maximum at origin
f has a local minimum at origin
f has a saddle point at origin
The origin is not a critical point of f

4. In the Laurent series expansion of f(z) = 1/(z(z − 1)) , where |z − 1| > 1, what is the coefficient of 1/(z − 1)?

-2
-1
0
1

5. In the permutation group S6, for order 8 what is the number of elements?

0
1
3
5

6. An urn contains four balls, each ball having equal probability of being white or black. Three black balls are added to the urn. The probability that five balls in the urn are black is

2/7
3/8
1/2
5/7

7. Let I = [2, 3), J be the set of all rational numbers in the interval [4, 6], K be the Cantor (ternary) set, and let L = {7 + x : x ∈ K}. Then the Lebesgue measure of the set I ∪ J ∪ L Equals 

1
2
3
4

8. Let F be a field with 7⁶elements and let K be a sub field of F with 49 elements. Then the dimension of F as a vector space over K is 

1
2
3
4

9. Let X denote R² endowed with the usual topology. Let Y denote R endowed with the co-finite topology. If Z is the product topological space Y × Y, then

The topology of Z is same as the topology of X
The topology of X is strictly stronger than that of Z
The topology of Z is strictly stronger than that of X
The topology of Z cannot be compared with that of X

10. A certain commodity is produced by the manufacturing plants P1 and P2 whose capacities are 6 and 5 units, respectively. The commodity is shipped to markets M1, M2, M3 and M4 whose requirements are 1, 2, 3 and 5 units, respectively. The cost for transporting a single unit from plant Pi to market Mj is given below:

Then the optimal cost of transportation is

54
55
56
57

11. If y1(x) = e^{−x^2} is a solution of the differential equation xy”+ay’+bx³y= 0

for some real numbers a and b, then ab =

3
4
5
6

12. Consider the Linear Programming Problem (LPP):

Maximize αx1 + x2

Subject to 2x1 + x2 ≤ 6,

− x1 + x2 ≤ 1,

x1 + x2 ≤ 4,

x1 ≥ 0, x2 ≥ 0,

where α is a constant. Let (3, 0) be only optimal solution, then

α < < 2
2 < α < 1
1 < α < 2
α > 2

13. Choose the correct statement?

Every group of order 12 has a non-trivial proper normal subgroup
Some group of order 12 does not have a non-trivial proper normal subgroup 
Every group of order 12 has a subgroup of order 6
Every group of order 12 has an element of order 12

14. Consider the polynomial p(X) = X⁴ + 4 in the ring Q[X] of polynomials in the variable X with coefficients in the field Q of rational numbers. Then

The set of zeroes of p(X) in C forms a group under multiplication
p(X) is reducible in the ring Q[X]
The splitting field of p(X) has degree 3 over Q
The splitting field of p(X) has degree 4 over Q

15. For a linear programming problem, which one of the following statements is FALSE?

The corresponding dual variable is unrestricted in sign when a constraint is an equality
Primal and dual of primal both can be infeasible
Primal’s dual is infeasible when it is unbounded
The optimal values of the primal and the dual can differ, even if both primal and dual are feasible