PSSOU MA/MSC (Mathematics) Previous Real Analysis (L–292) Solved Assignment 2025

(Assignment—1) 
Section—A 
1. If p is prime, then : 
(a) p is rational. 
(b) p is irrational. 
(c) p is complex. 
(d) None of the above 
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 2. Function 
(a) 
f
 x
 f x x x
 ( ) , 0
  
 (0) 1
 
 (b) 
f
 (0)
 
 1
 may be continuous at origin, if : 
  
(c) (0) 0 f   
(d) cannot be continuous for any value of (0) f 
3. If 1
 2 1
 1 I ax
 x 
  , then I is : 
(a) convergent 
(b) divergent 
(c) equal to 2 
(d) None of the above 
4. 1 lim 1
 n
 n n 
       
 is equal to : 
(a) 1
 e  
(b) e 
(c) 1 
(d) 0 
5. Sequence     1 n n   is : 
(a) convergent 
(b) divergent 
(c) oscillatory 
(d) None of the above 
6. The distance between points of X and Y is : 
(a) X Y  
(b) X Y  
(c) 2 2 X Y  
(d) None of the above 
  
7. If 1 (A B)  exists, then : 
(a) 1A and 1B both exist. 
(b) 1A and 1B do not exist. 
(c) at least one of 1A and 1B exist. 
(d) nothing can be said. 
8. The value of 1 1
 0 0
 xydxdy   is : 
(a) 1
 4  
(b) 1
 2  
(c) 1 
(d) None of the above 
Section—B 
9. Define supremum of a set. 
10. Find the value of 
0
 ( ) lim ( ) x
 f x
 g x 
 , where 2 1 ( ) sin f x x x x   and ( ) sin gx x  . 
11. Show that if f is monotonic on [ , ] ab , then f is integrable on [ , ] ab . 
12. Show that the limit of a convergent sequence is unique. 
13. Find the value of x and y, if 2 6 A 3 9
      
 , 3 B 2
 x
 y
      
 and AB = 0. 
14. Find the value of 3 2
 0 1
 ( 1 ) xy x y dxdy      .