Mathematics Times – July 2019

(c)

1

lim ( )

n 2

f n 

(d) If tan(cos (6)), then^1 f ^2 2 1

 0

5.Let f R R:  be a function we say that f

has

Property 1: If^0

( ) (0)

limh

f h f

 h



exist and is

finite, and

Property 2: If 0 2

( ) (0)

limh

f h f

 h



exist and is

finite.

Then which of the following options is/are

correct?

(a) f x x( ) has property 1

(b) f x x( ) 2/3 has property 1

(c) f x x x( ) has property 2

(d) f x( ) sin x has property 2

6.For a R a ,  1 , let

3 3

7/3

2 2 2

lim 1 2 .... 54

(^11) ... 1
( 1) ( 2) ( )
n
n
n
na na na n

   
    
    
 
Then possible value(s) a is/are –
(a) -9 (b) -6 (c) 7 (d) 8
7.Let^12
1 0 0 1 0 0
0 1 0 , 0 0 1 ,
0 0 1 0 1 0
P I P
   
    
   
   
3
0 1 0
1 0 0
0 0 1
P
 
 
 
 
,^4
0 1 0
0 0 1
1 0 0
P
 
 
 
 
,
5
0 0 1
1 0 0
0 1 0
P
 
 
 
 
,^6
0 0 1
0 1 0
1 0 0
P
 
 
 
 
, and
6
1
2 1 3
1 0 2
3 2 1
T
X Pk k Pk
 
  
 
 

.

Where PkT denotes the transpose of matrix Pk.

Then which of the following options is/are

correct?

(a) X-30I is an invertible matrix

(b) The sum of diagonal entries of X is 18

(c) If

1 1

1 1

1 1

X 

   

   

   

     

, then  30

(d) X is a symmetric matrix

8. Let

2

sin

( ) , 0

x

f x x

x



 

Let x x x x 1   2 3 .... n ....be all points of

local maximum of 1 and y y y 1  2  3 ....yn

...be all the points of local minimum of f.

Then which of the following options is/are

correct?

(a) x y 1  1

(b) x xn 1  n 2 for every n

(c)

1

2 ,2

n 2

x n n  

 

for every n

(d) x yn n 1 for every n

SECTION-II

9. Suppose

2

0 0

0 0

det 0

3

n n n

k k k

n n n n k

k k k k

k C k

C k C

 

 

 

 

 

 

 

 

 

holds for some positive integer n. Then

5. 
   

Property 1:

Property 2:

6. 
   

7. 
   

8.

SECTION-II

9.

10.

0 1

n n

k

k

C

 k

equals.

10. The value of
    1 10
    0

1 7 7 ( 1)

sec sec sec

(^4) k 12 2 12 2
  k  k 

     
      
    

