Mathematics Times – July 2019

6. If

(b) f'' 2 f' 2 0 

(c) f'' 2 f' 2 4 

(d) f^2 f' 2 f'' 2 0 

2 15 2 15

y x x   1   x x 1 , then

 

2

2

(^12)
x d y dyx
dx dx
  is equal to [2017]
(a)

7. Let

225 y^2 (b) 224 y^2 (c)^12 y (d)^225 y

a b R a,  , 0 . If the function f is defined

as

 

2

2

3

(^2) , 0 1
, 1 2
2 4 , 2
x x
a
f x a x
b b x
x

  

  
 
   

is
continuous in the interval [0, ), then an ordered
pair (a,b) is : [2016]
(a)  2,1 3  (b)  2,1 3 
(c)  2, 1 3   (d)  2,1 3 

8. If the function

 

(^1)  
, 1
cos , 1 2
x x
f x
a  x b x
 

     is
differentiable at x1, then
a
b equal to^ [2016]
(a)

9. Let k be a non-zero number. If

 

 

 1 cos 2^1   (b)

2

2

 

(c)

2

2



(d)

2

2



2

1

, 0

sin log 1

4

12 , 0

ex

x

f x x x

k

x

 

 

    

     

    

 

is a continuous function then the value of k is :

[2015]

(a) 1 (b) 2 (c) 3 (d) 4

10.If Rolle’s theorem holds for the function

f x x x cx x 2 6^3 ^2  ,   1,1 , at the point

(^1) ,
2
x then 2 b c equals [2015]
(a)  3 (b)  1
11.If the function
 
(c) 1 (d) 2
 
2
2 cos 1x ,x
f x x
k x



  
 
 
 

is continuous at
x, then k equals: [2014]
(a) 0 (b)
12.If
1
2
(c) 2 (d)
1
4
f x  is continuous and
9 2
,
2 9
f  
 
then
0 2
lim 1 cos3
x
f x
 x
  
 
 
is equal to [2014]
(a)
13.Let :
0 (b)
2
9 (c)
8
9 (d)
9
2
f R R be a function such that
f x x ^2 , for all x R. Then at x0, f is
[2014]
(a) Continuous but not differentiable
(b) Continuous as well as differentiable
(c) Neither continuous nor differentiable
(d) Differentiable but not continuous
14.Let f, R R be two functions defined by
sin , x 0^1
( )
0 , 0
x
f x x
x
   
  
  
  , and g(x) xf (x) :-
[2014]
Statement I : f is a continuous function at
Statement II : g is a differentiable function at
x 0
x 0
(a) Statement I is false and statement II is true
(b) Statement I is true and statement II is the false
(c) Both statement I and II are true
(d) Both statements I and II are false
15.If

( )^2 5,^1
2
f x x x x    , and g(x) is its inverse
function, then g'(7) equals :- [2014]
6.
[2017]
7.
[2016]
8.
[2016]
9.
[2015]
10.
[2015]
11.
[2014]
12.
[2014]
13.
[2014]
14.
[2014]
Statement I :
Statement II :
15.
[2014]