# Home / GATE 2017-2018 / GATE Mathematics :: Practice Test Paper 1

### GATE 2017-2018 :: GATE Mathematics

1. In a topological space, which of the following statements is NOT always true :
2.  A. Union of any finite family of compact sets is compact. B. Union of any family of closed sets is closed. C. Union of any family of connected sets having a non empty intersection is connected. D. Union of any family of dense subsets is dense.   3. Consider the following statements:
P: The family of subsets {An = (-1/n, 1/n), n = 1, 2, ...} satisfies the finite intersection property.
Q: On an infinite set X , a metric d : X * X --> R is defined as d(x,y) = [0 , x = y and 1, x â‰  y] The metric space (X,d) is compact.
R: In a Frechet (T1) topological space, every finite set is closed.
S: If f : R --> X is continuous, where R is given the usual topology and (X, t) is a Hausdorff (T2) space, then f is a one-one function.
Which of the above statements are correct?
4.  A. P and R B. P and S C. R and S D. Q and S   5. A simple random sample of size 10 from 2 N(Î¼,Ïƒ2) gives 98% confidence interval (20.49, 23.51). Then the null hypothesis H0 : Î¼ = 20.5 against HA : Î¼ â‰  20.5
6.  A. can be rejected at 2% level of significance B. cannot be rejected at 5% level of significance C. can be rejected at 10% level of significance D. cannot be rejected at any level of significance   7. For the linear programming problem
Maximize z = x1 + 2x2 + 3x3 - 4x4
Subject to      2x1 + 3x2 - x3 - x4 = 15
6x1 + x2 + x3 - 3x4 = 21
8x1 + 2x2 + 3x3 - 4x4 = 30
x1, x2, x3, x4 â‰¥ 0,
x1 = 4, x2 = 3, x3 = 0, x4 = 2 is
8.  A. an optimal solution B. a degenerate basic feasible solution C. a non-degenerate basic feasible solution D. a non-basic feasible solution   9. Which one of the following statements is TRUE?
10.  A. A convex set cannot have infinite many extreme points. B. A linear programming problem can have infinite many extreme points. C. A linear programming problem can have exactly two different optimal solutions. D. A linear programming problem can have a non-basic optimal solution.   11. Let V = â„‚2 be the vector space over the field of complex numbers and Bï€½{(1, i), (i,1)}be a given ordered basis of V. Then for which of the following, B* = {f1, f2}is a dual basis of B over â„‚?
12.  A. f1(z1, z2) = 1/2 (z1 - iz2), f2(z1, z2) = 1/2 (z1 + iz2) B. f1(z1, z2) = 1/2 (z1 + iz2), f2(z1, z2) = 1/2 (iz1 + z2) C. f1(z1, z2) = 1/2 (z1 - iz2), f2(z1, z2) = 1/2 (-iz1 + z2) D. f1(z1, z2) = 1/2 (z1 + iz2), f2(z1, z2) = 1/2 (-iz1 - z2)   13. Let R = â„¤*â„¤*â„¤ and I = â„¤*â„¤*{0}. Then which of the following statement is correct?
14.  A. I is a maximal ideal but not a prime ideal of R . B. I is a prime ideal but not a maximal ideal of R . C. I is both maximal ideal as well as a prime ideal of R . D. I is neither a maximal ideal nor a prime ideal of R .   15. The function u(r, Î¸) satisfying the Laplace equation subject to the conditions u(e, Î¸) = 1, u(e2 ,Î¸) = 0 is
16.  A. ln(e/r) B. ln(e/r2) C. ln(e2/r) D. .   17. The functional is path independent if k equals
18.  A. 1 B. 2 C. 3 D. 4   19. If a transformation y = uv transforms the given differential equation
f(x)y'' - 4f'(x)y' + g(x)y = 0 into the equation of the form v'' + h(x)v = 0, then u must be
20.  A. 1/f2 B. xf C. 1/2f D. f2   