Mathematics(Gate > 2017-2018 ) Questions and Answers
Question 1. The straight lines L1 : x = 0, L2 : y = 0 and L3 : x + y = 1 are mapped by the transformation w = z2 into the curves C1 , C2 and C3 respectively. The angle of intersection between the curves at w = 0 is
0
Ï€/4
Ï€/2
Ï€
Explanation:-
Answer: Option D. -> π
-NA-
Question 2. A simple random sample of size 10 from 2 N(μ,σ2) gives 98% confidence interval (20.49, 23.51). Then the null hypothesis H0 : μ = 20.5 against HA : μ ≠20.5
can be rejected at 2% level of significance
cannot be rejected at 5% level of significance
can be rejected at 10% level of significance
cannot be rejected at any level of significance
Explanation:-
Answer: Option C. -> can be rejected at 10% level of significance
-NA-
Question 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 ≠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?
P and R
P and S
R and S
Q and S
Explanation:-
Answer: Option A. -> P and R
-NA-
Question 4. In a topological space, which of the following statements is NOT always true :
Union of any finite family of compact sets is compact.
Union of any family of closed sets is closed.
Union of any family of connected sets having a non empty intersection is connected.
Union of any family of dense subsets is dense.
Explanation:-
Answer: Option B. -> Union of any family of closed sets is closed.
-NA-
Question 5. 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 ≥ 0, x1 = 4, x2 = 3, x3 = 0, x4 = 2 is
an optimal solution
a degenerate basic feasible solution
a non-degenerate basic feasible solution
a non-basic feasible solution
Explanation:-
Answer: Option D. -> a non-basic feasible solution
-NA-
Question 6. Which one of the following statements is TRUE?
A convex set cannot have infinite many extreme points.
A linear programming problem can have infinite many extreme points.
A linear programming problem can have exactly two different optimal solutions.
A linear programming problem can have a non-basic optimal solution.
Explanation:-
Answer: Option D. -> A linear programming problem can have a non-basic optimal solution.
-NA-
Question 7. Let V = ℂ2 be the vector space over the field of complex numbers and B{(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 ℂ?