Continuity & Differentiability Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. Let  be a function such that
,  and .  Then
(a) f(x) is a quadratic function
(b) f(x) is continuous but not differentiable
(c) f(x) is  differentiable in R
(d) f(x) is bounded in R
2. If x
2
 + y
2
 = a
2
 and k = , then k is equal to
(a)
(b)
(c)
Page 2


PART-I (Single Correct MCQs)
1. Let  be a function such that
,  and .  Then
(a) f(x) is a quadratic function
(b) f(x) is continuous but not differentiable
(c) f(x) is  differentiable in R
(d) f(x) is bounded in R
2. If x
2
 + y
2
 = a
2
 and k = , then k is equal to
(a)
(b)
(c)
(d)
3. Let a function  satisfy the equation
f (x + y) = f(x) + f(y) for all x, y, If the function f(x) is continuous at x = 0,
then
(a) f(x) = 0  for all x
(b) f(x) is continuous for all positive real x
(c) f(x) is continuous for all x
(d) None of these
4. Differential coefficient of  with respect to 
will be
(a) 1
(b) – 1
(c) – 1/2
(d) x
5. The values of a, b and c which make the function
continuous at x = 0 are
(a) , , b = 0
(b) , , 
Page 3


PART-I (Single Correct MCQs)
1. Let  be a function such that
,  and .  Then
(a) f(x) is a quadratic function
(b) f(x) is continuous but not differentiable
(c) f(x) is  differentiable in R
(d) f(x) is bounded in R
2. If x
2
 + y
2
 = a
2
 and k = , then k is equal to
(a)
(b)
(c)
(d)
3. Let a function  satisfy the equation
f (x + y) = f(x) + f(y) for all x, y, If the function f(x) is continuous at x = 0,
then
(a) f(x) = 0  for all x
(b) f(x) is continuous for all positive real x
(c) f(x) is continuous for all x
(d) None of these
4. Differential coefficient of  with respect to 
will be
(a) 1
(b) – 1
(c) – 1/2
(d) x
5. The values of a, b and c which make the function
continuous at x = 0 are
(a) , , b = 0
(b) , , 
(c) , , 
(d) None of these
6. The function f(x) = [x]
2
 – [x
2
] (where [y] is the greatest integer function
less than or equal to y), is discontinuous at :
(a) all integers
(b) all integers except 0 and 1
(c) all integers except 0
(d) all integers except 1
7. The value of p for which the function
 
may be continuous at x = 0, is
(a) 1
(b) 2
(c) 3
(d) None of these
8. In the mean value theorem , if a = 0,
b = 1/2 and f (x) = x (x – 1) (x – 2), the value of c is –
(a)
(b)
(c)
(d)
9. If  then at x = 0, f(x)
Page 4


PART-I (Single Correct MCQs)
1. Let  be a function such that
,  and .  Then
(a) f(x) is a quadratic function
(b) f(x) is continuous but not differentiable
(c) f(x) is  differentiable in R
(d) f(x) is bounded in R
2. If x
2
 + y
2
 = a
2
 and k = , then k is equal to
(a)
(b)
(c)
(d)
3. Let a function  satisfy the equation
f (x + y) = f(x) + f(y) for all x, y, If the function f(x) is continuous at x = 0,
then
(a) f(x) = 0  for all x
(b) f(x) is continuous for all positive real x
(c) f(x) is continuous for all x
(d) None of these
4. Differential coefficient of  with respect to 
will be
(a) 1
(b) – 1
(c) – 1/2
(d) x
5. The values of a, b and c which make the function
continuous at x = 0 are
(a) , , b = 0
(b) , , 
(c) , , 
(d) None of these
6. The function f(x) = [x]
2
 – [x
2
] (where [y] is the greatest integer function
less than or equal to y), is discontinuous at :
(a) all integers
(b) all integers except 0 and 1
(c) all integers except 0
(d) all integers except 1
7. The value of p for which the function
 
may be continuous at x = 0, is
(a) 1
(b) 2
(c) 3
(d) None of these
8. In the mean value theorem , if a = 0,
b = 1/2 and f (x) = x (x – 1) (x – 2), the value of c is –
(a)
(b)
(c)
(d)
9. If  then at x = 0, f(x)
(a) has no limit
(b) is discontinuous
(c) is continuous but not differentiable
(d) is differentiable
10. where g is a continuous function then 
 does not exist if
(a) g(x) is any constant function
(b) g(x) = x
(c) g(x) = x
2
(d) g(x) = x h (x), where h(x) is a polynomial
11. Let f : R ? R be a function defined by f (x) = max {x, x
3
}. The set of all
points where f (x) is NOT differentiable is
(a) {-1, 1}
(b) {-1, 0}
(c) {0, 1}
(d) {-1, 0, 1}
12. The function  is not defined atx = .  The value of
f  so that f is continuous at x =  is
(a)
(b)
(c) 2
(d) None of these
13. If g is the inverse function of f and  = sin x, then 
is
(a)
(b)
Page 5


PART-I (Single Correct MCQs)
1. Let  be a function such that
,  and .  Then
(a) f(x) is a quadratic function
(b) f(x) is continuous but not differentiable
(c) f(x) is  differentiable in R
(d) f(x) is bounded in R
2. If x
2
 + y
2
 = a
2
 and k = , then k is equal to
(a)
(b)
(c)
(d)
3. Let a function  satisfy the equation
f (x + y) = f(x) + f(y) for all x, y, If the function f(x) is continuous at x = 0,
then
(a) f(x) = 0  for all x
(b) f(x) is continuous for all positive real x
(c) f(x) is continuous for all x
(d) None of these
4. Differential coefficient of  with respect to 
will be
(a) 1
(b) – 1
(c) – 1/2
(d) x
5. The values of a, b and c which make the function
continuous at x = 0 are
(a) , , b = 0
(b) , , 
(c) , , 
(d) None of these
6. The function f(x) = [x]
2
 – [x
2
] (where [y] is the greatest integer function
less than or equal to y), is discontinuous at :
(a) all integers
(b) all integers except 0 and 1
(c) all integers except 0
(d) all integers except 1
7. The value of p for which the function
 
may be continuous at x = 0, is
(a) 1
(b) 2
(c) 3
(d) None of these
8. In the mean value theorem , if a = 0,
b = 1/2 and f (x) = x (x – 1) (x – 2), the value of c is –
(a)
(b)
(c)
(d)
9. If  then at x = 0, f(x)
(a) has no limit
(b) is discontinuous
(c) is continuous but not differentiable
(d) is differentiable
10. where g is a continuous function then 
 does not exist if
(a) g(x) is any constant function
(b) g(x) = x
(c) g(x) = x
2
(d) g(x) = x h (x), where h(x) is a polynomial
11. Let f : R ? R be a function defined by f (x) = max {x, x
3
}. The set of all
points where f (x) is NOT differentiable is
(a) {-1, 1}
(b) {-1, 0}
(c) {0, 1}
(d) {-1, 0, 1}
12. The function  is not defined atx = .  The value of
f  so that f is continuous at x =  is
(a)
(b)
(c) 2
(d) None of these
13. If g is the inverse function of f and  = sin x, then 
is
(a)
(b)
(c)
(d)
14. Which of the following functions is differentiable at x = 0?
(a)
(b)
(c)
(d)
15. If the equation  ............. +  = 0
  0, n  2, has a positive root x = , then the equation
 + (n – 1) + ......... +  = 0 has a positive root,
which is
(a) greater than 
(b) smaller than 
(c) greater than or equal to 
(d) equal to 
16. If f (x) = 
greatest integer less than or equal to x, then in order that f be continuous at x
=0, the value of k is
(a) equal to 0
(b) equal to 1
(c) equal to –1
(d) indeterminate