1. The integrating factor of the DE 2 coshx cosy dx = sinhx siny dy is (a) sinhx (b) coshx (c) tanhx (d) cothx 2. Which of the following is false? (a) The topological space that are finite are always connected (b) The set of real numbers R with usual topology is not compact (c) The diameter of A ̅ is always same as the diameter of A, where A⊂ R (d) There exist topological space with countably many points that have uncountably many open set 3. Let f(x) = |x|, for -1 ≤ x ≤ 2 and the partition of P = ( -1, -1/2, 0, 1/2, 1, 3/2, 2 ) then the value of L(f,p)= (a) 7/4 (b) 13/4 (c) 9/4 (d) 10/4 4. Find the functions that are not uniformly continuous on (0,1) (a) 1/x^2 (b) x^2 (c) sinx (d) sinx/x 5. Let the bounded set S contains a sequence S_n of real numbers such that liminf S_n≠ limsup S_n then which one of the following options is false? (a) The limit of the sequence S_n does not exist (b) The sequence S_n is not Cauchy (c) There exists an infinite number of domainant terms for S_n (d) There exists a convergent subsequence 6. Which of the following are Lebesgve integrable functions on [0,1]? (i) (x ln⁡x)/(1+x^2 ) (ii) sin⁡〖π x〗/ln⁡x (iii) ln (x) [ln (1-x)] (iv) ln⁡x/√(1-x^2 ) (a) Both (i),(ii) (b) All (i),(ii) and (iv) (c) All (ii),(iii) and (iv) (d) All (i),(ii),(iii) and (iv) 7. The solution curves to the DE 〖2xyy〗^1=y^2-x^2 are (a) the circles passing through the origin with centers on x-axis (b) The ellipse (c ) the hyperbolas (d) the parabolas 8. An example of a set A ▁(⊂) R such that A^0 = ∅ and ¯A=R (a) The set of integers Z (b) The set of rational number Q (c) The set of Natural number N (d) The set of complex number C 9. The radius of convergence of the series ∑_(n=1)^∞▒x^n is (a) (-1,1) (b) [-1,1] (c) (-∞,∞) (d) [-1/2, 1/2] 10. Let the n^th term of the series be 〖(-1)〗^n 〖(n+1)〗^n/n^n then the series is (a) convergent (b) divergent (c) inconclusive (d) None of the above

1. The integrating factor of the DE 2 coshx cosy dx = sinhx siny dy is (a) sinhx (b) coshx (c) tanhx (d) cothx

 

2. Which of the following is false? (a) The topological space that are finite are always connected (b) The set of real numbers R with usual topology is not compact (c) The diameter of A ̅ is always same as the diameter of A, where A⊂ R (d) There exist topological space with countably many points that have uncountably many open set

 

3. Let f(x) = |x|, for -1 ≤ x ≤ 2 and the partition of P = ( -1, -1/2, 0, 1/2, 1, 3/2, 2 ) then the value of L(f,p)= (a) 7/4 (b) 13/4 (c) 9/4 (d) 10/4

 

4. Find the functions that are not uniformly continuous on (0,1) (a) 1/x^2 (b) x^2 (c) sinx (d) sinx/x

 

5. Let the bounded set S contains a sequence S_n of real numbers such that liminf S_n≠ limsup S_n then which one of the following options is false? (a) The limit of the sequence S_n does not exist (b) The sequence S_n is not Cauchy (c) There exists an infinite number of domainant terms for S_n (d) There exists a convergent subsequence

 

6. Which of the following are Lebesgve integrable functions on [0,1]? (i) (x ln⁡x)/(1+x^2 ) (ii) sin⁡〖π x〗/ln⁡x (iii) ln (x) [ln (1-x)] (iv) ln⁡x/√(1-x^2 ) (a) Both (i),(ii) (b) All (i),(ii) and (iv) (c) All (ii),(iii) and (iv) (d) All (i),(ii),(iii) and (iv)

 

7. The solution curves to the DE 〖2xyy〗^1=y^2-x^2 are (a) the circles passing through the origin with centers on x-axis (b) The ellipse (c ) the hyperbolas (d) the parabolas

 

8. An example of a set A ▁(⊂) R such that A^0 = ∅ and ¯A=R (a) The set of integers Z (b) The set of rational number Q (c) The set of Natural number N (d) The set of complex number C

 

9. The radius of convergence of the series ∑_(n=1)^∞▒x^n is (a) (-1,1) (b) [-1,1] (c) (-∞,∞) (d) [-1/2, 1/2]

 

10. Let the n^th term of the series be 〖(-1)〗^n 〖(n+1)〗^n/n^n then the series is (a) convergent (b) divergent (c) inconclusive (d) None of the above