Mathematics Times – July 2019

CONTINUITY & DIFFERENTIABILITY
[ONLINE QUESTIONS]

1. b 2. b 3. c 4. b 5. b


2. d 7. a 8. d 9. c 10. b


3. d 12. b 13. b 14. c 15. a


4. c 17. a 18. b 19. a 20. b


5. d 22. b 23. d

CONTINUITY & DIFFERENTIABILITY

[ONLINE QUESTIONS]

1. 
   
   
   
   2. 
      
      
      
      3. 
         
         
         
         4. 
            
            
            
            5. 
               
            
            
            
            
         
         
         
         
      
      
      
      
   
   
   
   


2. 
   
   
   
   7. 
      
      
      
      8. 
         
         
         
         9. 
            
            
            
            10. 
                
            
            
            
            
         
         
         
         
      
      
      
      
   
   
   
   


3. 
   
   
   
   12. 
       
       
       
       13. 
           
           
           
           14. 
               
               
               
               15. 
                   
               
               
               
               
           
           
           
           
       
       
       
       
   
   
   
   


4. 
   
   
   
   17. 
       
       
       
       18. 
           
           
           
           19. 
               
               
               
               20. 
                   
               
               
               
               
           
           
           
           
       
       
       
       
   
   
   
   


5. 
   
   
   
   22. 
       
       
       
       23. 
           
       
       
       
       
   
   
   
   

DETERMINANTS [ONLINE QUESTIONS]
1.Sol: For no solution

2 10

3 1

k k

k k k

  

 

( 2)( 3) 10k k   k

k k^2     5 6 0 k 2,3

k 2 for k 2 both lines identical

so k 3 only

 number of values of k is 1

2.Sol: Given^2

cos 1

( ) 2sin 2

tan 1

x x

f x x x x

x x



x x x^2 tan cos 

lim '  lim2 tan cos ^2 (sec sin )^2

x x

f x x x x x x x

 x  x

  

 

=limx2 tan cosx x x x   (sec sin )^2 x x

=-2

3.Sol: Given

0 cos sin

sin 0 cos 0

cos sin 0

x x

x x

x x





i.e., 0 0 sin cos cos 0 cos  x x   ^2 x

sin sinx x^2  0 0

 cos sin^3 x^3 x 0

i.e.,tan 1^3 x

Now

tan tan

tan^3

(^3) 1 tan tan
3
x
x
x



  
  
  
3 1
1 3



3 1 2 3
x s1 3
    




4.Sol: We know adj adjA A A  n^2

Given n 3

i.e.,adj adjA A A 

5. Sol: Given the system of linear equations has
   infinitely many solutions.

i.e.,

6. Sol:

2

2 4

4 2 0

2 2









 



 ^3 20 40 0 

  has only one root and it lies between 3.05

and 3.

4 1 4 1 4 1

3 1 3 1 3 1

A A

       

     

     

13 3

9 2

 

 

  and A1.

Now, A^2016 ^2 A A^2015 ^2014 A A A I^2014 ^2 ^2  

A^2016  2 A A^2015 ^2014  A A A I^20142  2 

2014 20 5 25

15 5

A  

 

7.Sol: Given

DETERMINANTS [ONLINE QUESTIONS]
1.Sol:

2.Sol:

3.Sol:

4.Sol:

5. Sol:


6. Sol:

7.Sol:

cos sin sin

sin cos sin 0

sin sin cos

x x x

x x x

x x x



C C C C 1    2 3 ,

i.e.,

cos 2sin sin sin

cos 2sin cos sin 0

cos 2sin sin cos

x x x x

x x x x

x x x x



 



Now  

1 sin sin

cos 2sin 1 cos sin 0

1 sin cos

x x

x x x x

x x

 

R R R 1   2 1

R R R 2   3 2

i.e.,  

0 sin cos 0

cos 2sin 0 cos sin sin cos 0

1 sin cos

x x

x x x x x x

x x



   