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Continuity and Differentiability — NDA Mathematics PYQs

94 previous year questions from Continuity and Differentiability with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

94 questionsMathematicsSolutions on every page
1

What is the derivative of x |x| with respect to x , where x<0 ?

NDA 2026 Solution View → 2

Passage: Let f(x)= cases ax(x-1), & x 3 cases Given that f(x) is continuous for all x but not differentiable at x=1 . Further f'(x) is continuous at x=3 . Question: What is the val

NDA 2026 Solution View → 3

Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If q(x)=f g(x) , then which of the following statements is/are correct ? I. q(x) is continuous at x=0 . II. q(x) is differe

NDA 2026 Solution View → 4

Passage: Let f(x)= (x^2) and g(x)=x|x| for |x| Question: If p(x)=f(x)g(x) , then which of the following statements is/are correct ? I. p(x) is continuous at x=0 . II. p(x) is diffe

NDA 2026 Solution View → 5

If f(x) is differentiable at x=a , then consider the following statements : I. f(x) is continuous at x=a II. _ x a f(x)=f(a) Which of the statements given above is/are correct ?

NDA 2026 Solution View → 6

Passage: Let f(x)= cases ax(x-1), & x 3 cases Given that f(x) is continuous for all x but not differentiable at x=1 . Further f'(x) is continuous at x=3 . Question: What is the val

NDA 2026 Solution View → 7

Consider the following for the two (02) items that follow: Let f(x) = cases x^3, & x^2 What is _ x 0 f'(x) equal to?

NDA 2025 Solution View → 8

For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5] . What is dy dx at x = 3.5 equal to?

NDA 2025 Solution View → 9

For the following two (02) items: Consider the function f(x) = cases 4(5^x), & x If the function is continuous, then what is the value of k ?

NDA 2025 Solution View → 10

For the following two (02) items: Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5] . Consider the following statements: I. The function is differentiable

NDA 2025 Solution View → 11

Consider the following for the two (02) items that follow: Let f(x) = cases x^3, & x^2 Consider the following statements: I. The function is continuous at x = -1 . II. The function

NDA 2025 Solution View → 12

Paragraph: Let f(x)=[x]^2- [x^2 ] . Question: Consider the following statements: A. f(x) is continuous at x=0 . B. f(x) is continuous at x=1 Which of the statements given above is/

NDA 2024 Solution View → 13

Let f(x)=|x|+1 and g(x)=[x]-1 , where [.] is the greatest integer function. Let h(x)= f(x) g(x) Consider the following statements: 1. f(x) is differentiable for all x 0 2. g(x) is

NDA 2024 Solution View → 14

Let f(x)= array cl x-3 |x-3| +a ; & x 3 a-b ; & x=3 and x-3 |x-3| +b ; & x 3 array .f(x) be continuous at x=3 . What is the value of a ?

NDA 2023 Solution View → 15

Let f(x)= array cl x-3 |x-3| +a ; & x 3 a-b ; & x=3 and x-3 |x-3| +b ; & x 3 array .f(x) be continuous at x=3 . What is the value of b?

NDA 2023 Solution View → 16

Let f(x)= array cc a x(x+1)+b, & x 1 x-1 & 1 x 2 array . If the function f(x) is differentiable at x=1 , then what is the value of (a+b) ?

NDA 2023 Solution View → 17

Let f(x)= array cc x+6, & x 1 p x+q, & 1 x 2 5 x, & x 2 array . and f(x) is continuous. Question: What is the value of q ?

NDA 2023 Solution View → 18

Paragraph: Let f(x)= array cc x+6, & x 1 p x+q, & 1 x 2 5 x, & x 2 array . and f(x) is continuous. Question: What is the value of p ?

NDA 2023 Solution View → 19

Let y=[x+1],-4 x -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x=-3.5 ?

NDA 2022 Solution View → 20

Consider the following statements in respect of f(x)=|x|-1 : 1. f(x) is continuous at x=1 . 2. f(x) is differentiable at x=0 . Which of the above statements is/are correct?

NDA 2021 Solution View → 21

Let f(x)= array cc 1+ x 2 k , & 0 x 2 k x, & 2 x 4 array . If _ x 2 f(x) exists, then what is the value of k ?

NDA 2021 Solution View → 22

Consider the following statements in respect of the function f(x)= ( 1 x^2 ), x 0 1. It is continuous at x=0 , if f(0)=0 . 2. It is continuous at x= 2 . Which of the above statemen

NDA 2021 Solution View → 23

If the function f(x)= array ll a+b x, & x 1 5, & x=1 b-a x, & x 1 array . is continuous, then what is the value of (a+b) ?

NDA 2021 Solution View → 24

If f(x)=e^ |x| , then which one of the following is correct?

NDA 2021 Solution View → 25

Consider the following statements for f(x)=e^ -|x| : 1. The function is continuous at x=0 . 2. The function is differentiable at x=0 . Which of the above statements is/are correct?

NDA 2020 Solution View → 26

If f(x)= x x , where x R , is to be continuous at x=0 , then the value of the function at x=0

NDA 2020 Solution View → 27

The value of k which makes f ( x )= array cc x & x 0 k & x=0 array . continuous at x =0 , is

NDA 2019 Solution View → 28

Consider the following statements in respect of the function f(x)= ( 1 x ) for x^1 0 and f(0)=0 ; 1. _ x 0 f(x) exists 2. f(x) is continuous at x=0 Which of the above statements is

NDA 2019 Solution View → 29

For what value of k is the function f(x)= array ccc 2 x+ 1 4 & , & x 0 k & , & x=0 continuous? (x+ 1 2 )^2 & , & x 0 array .

NDA 2019 Solution View → 30

Consider the function f(x)= array l 2 x 5 x if x 0 2 15 if x=0 array . Which one of the following is correct in respect of the function?

NDA 2018 Solution View → 31

For the function f(x)=|x-3| , which of the following is not correct?

NDA 2018 Solution View → 32

If f ( x )= 25- x ^2 , then what is _ x 1 f ( x )- f (1) x -1 equal to

NDA 2018 Solution View → 33

Consider the function f ( x )= array cl x ^2 ( x ) & x 0 0 & x =0 array . . What is f ^ (0) equal to?

NDA 2018 Solution View → 34

If f(x)= x^2-9 x^2-2 x-3 , x 3 is continuous at x=3 , then which one of the following is correct?

NDA 2018 Solution View → 35

Let f(x+y)=f(x) f(y) and f(x)=1+x g(x) (x) , where _ x 0 g(x)=a and _ x 0 (x)=b . Then what is f^ (x) equal to?

NDA 2018 Solution View → 36

If f(x)=|x|+|x-1| , then which one of the following is correct?

NDA 2018 Solution View → 37

If f(x)= x 2 -1 , then on the interval [0, ] which one of the following is correct?

NDA 2017 Solution View → 38

The set of all points, where the function f( x )= 1- e ^ - x ^2 is differentiable, is

NDA 2017 Solution View → 39

If f(x)=x( x - x+1 ) , then f(x) is

NDA 2017 Solution View → 40

Let g be the greatest integer function. Then the function f( x )=( g ( x ))^2- g ( x ) is discontinuous at

NDA 2017 Solution View → 41

If F(x)= 9-x^2 , then what is _ x 1 F(x)-F(1) x-1 equal to?

NDA 2017 Solution View → 42

Let f( x ) be defined as follows : f(x)= cases 2 x +1, & -3 x -2 x -1, & -2 x 0 x +2, & 0 x 1 cases Which one of the following statements is correct in respect of the above functio

NDA 2017 Solution View → 43

Suppose the function f( x )= x ^ n , n 0 is differentiable for all x . Then n can be any element of the interval

NDA 2017 Solution View → 44

A function is defined as follows: f( x )= array cc - x x ^2 , & x 0 0, & x =0 array . Which one of the following is correct in respect of the above function?

NDA 2017 Solution View → 45

Consider the following : 1. x + x ^2 is continuous at x =0 2. x + 1 x is discontinuous at x =0 3. x^2+ 1 x is continuous at x=0 Which of the above are correct?

NDA 2017 Solution View → 46

Consider the following statements : 1. Derivative of f( x ) may not exist at some point. 2. Derivative of f( x ) may exist finitely at some point. 3. Derivative of f( x ) may be in

NDA 2017 Solution View → 47

A function is defined in (0, ) by f(x)= array ccc 1-x^2 & for & 0 x 1 In & x & for 1 x 2 2-1+0.5 x & for & 2 x array . Which one of the following is correct in respect of the deriv

NDA 2017 Solution View → 48

The left-hand derivative of f( x )=[ x ] ( x ) at x = k where k is an integer and [ x ] is the greatest integer function, is

NDA 2017 Solution View → 49

Paragraph: f( x )= array lr 3 x^2+12 x-1 & -1 x 2 37-x, & 2 x 3 array . Question: Which of the following statements are correct? 1. f(x) is continuous at x=2 . 2. f(x) attains grea

NDA 2016 Solution View → 50

Let f: A R , where A=R 0 is such that f(x)= x+|x| x . On which one of the following sets is f(x) continuous?

NDA 2016 Solution View → 51

Paragraph: Let f(x)=[|x|-|x-1|]^2 Question: What is f ^ ( x ) equal to when x 1 ?

NDA 2016 Solution View → 52

Paragraph: Let f(x)=[|x|-|x-1|]^2 Question: What is f ^ ( x ) equal to when 0 x 1 ?

NDA 2016 Solution View → 53

Paragraph: Let f(x)=[x] , where [.] is the greatest integer function and g(x)= x be two real valued functions over R . Question: Which of the following statements is correct?

NDA 2016 Solution View → 54

Paragraph: Let f(x)= array ll e^x-1 x , & x 0 0, & x=0 array . be a real valued function. Question: Which of the following statements is/are correct? 1. f(x) is right continuous at

NDA 2016 Solution View → 55

Paragraph: Let f ( x )= array ll -2, & -3 x 0 x -2, & 0 x 3 array . and g ( x )= f (| x |)+| f ( x )| Question: Which of the following statements is/are correct? 1. g(x) is differe

NDA 2016 Solution View → 56

Paragraph: Let f ( x )= array ll -2, & -3 x 0 x -2, & 0 x 3 array . and g ( x )= f (| x |)+| f ( x )| Question: What is the value of the differential coefficient of g(x) at x=-2 ?

NDA 2016 Solution View → 57

Paragraph: Let f ( x )= array ll -2, & -3 x 0 x -2, & 0 x 3 array . and g ( x )= f (| x |)+| f ( x )| Question: Which of the following statements are correct? 1. g ( x ) is continu

NDA 2016 Solution View → 58

Consider the function f ( x )=| x -1|+ x ^2 , where x R . Which one of the following statements is correct?

NDA 2016 Solution View → 59

A function f(x) is defined as follows: f(x)= array ccc x+ & for & x [- , 0) x & for & x [0, 2 ] (x- 2 )^2 & for & x ( 2 , ] array . Consider the following statements: 1. The functi

NDA 2016 Solution View → 60

A function f ( x ) is defined as follows: f(x)= array ccc x+ & for & x [- , 0) x & for & x [0, 2 ] (x- 2 )^2 & for & x ( 2 , ] array . Consider the following statements: 1. The fun

NDA 2016 Solution View → 61

Consider the following in respect of the function f(x)= array l 2+x, x 0 2-x, x 0 array . 1. _ x 1 f (x) does not exist. 2. f(x) is differentiable at x=0 3. f(x) is continuous at x

NDA 2016 Solution View → 62

The function f ( x )= 1- x + x 1+ x + x is not defined at x = . The value of f( ) so that f(x) is continuous at x= is

NDA 2015 Solution View → 63

Consider the function f(x)= array ccc a x-2 & for & -2 x -1 -1 & for & -1 x 1 a+2(x-1)^2 & for & 1 x 2 array . What is the value of a for which f(x) is continuous at x=-1 and x =1

NDA 2015 Solution View → 64

Consider the function f(x)= array cc -2 x & if x - 2 A x+B & if - 2 x 2 x & if x 2 array . The value of B is

NDA 2015 Solution View → 65

Consider the following functions : 1. f(x)= array lll 1 x & if & x 0 0 & if & x=0 array . 2. f(x)= array ccc 2 x+5 & if & x 0 x^2+2 x+5 & if & x 0 array . Which of the above functi

NDA 2015 Solution View → 66

Given a function f(x)= array ccc -1 & If & x 0 a x+b & If & 0 x 1 1 & If & x 1 array . where a , b are constants. The function is continuous everywhere. What is the value of b?

NDA 2015 Solution View → 67

Consider the function f(x)= array cc -2 x & if x - 2 A x+B & if - 2 x 2 x & if x 2 array . which is continuous everywhere. The value of A is

NDA 2015 Solution View → 68

Consider the function f(x)= array ccc x -2 x & If & x 2 3 & If & x= 2 array . Which is continuous at x= 2 , where is a constant. What is the value of ?

NDA 2015 Solution View → 69

Given a function f(x)= array ccc -1 & If & x 0 a x+b & If & 0 x 1 1 & If & x 1 array . where a , b are constants. The function is continuous everywhere. What is the value of a?

NDA 2015 Solution View → 70

Consider the following statements: 1. f(x)=[x] , where [.] is the greatest integer function, is discontinuous at x=n , where n Z . 2. f(x)= x is discontinuous at x=n , where n Z .

NDA 2015 Solution View → 71

Consider the function f(x)= cases k x x , & x 0 3 x+2 k^2, & x 0 cases What is the non-zero value of k for which the function is continuous at x=0 ?

NDA 2014 Solution View → 72

Consider the function f(x)= aligned x^2-5, & x 3 x+13 , & x 3 aligned . What is the differential coefficient of f(x) at x=12 ?

NDA 2014 Solution View → 73

Consider the following statements: 1. The function f(x)=[x] , where [.] is the greatest integer function defined on R , is continuous at all points except at x=0 . 2. The function

NDA 2014 Solution View → 74

Consider the following statements: 1. The function f(x)= [3] x is continuous at all x except at x=0 . 2. The function f(x)=[x] is continuous at x=2.99 where [.] is the bracket func

NDA 2014 Solution View → 75

Consider the following statements: 1. The function f(x)=|x| is not differentiable at x=1 . 2. The function f(x)=e^x is differentiable at x=0 . Which of the above statements is/are

NDA 2014 Solution View → 76

Paragraph : Let f(x) be a function defined in 1 x by f(x)= array ll 2-x & for 1 x 2 3 x-x^2 & for x 2 array . . Question : Consider the following statements: 1. The function is con

NDA 2014 Solution View → 77

Paragraph : Let f(x) be a function defined in 1 x by f(x)= array ll 2-x & for 1 x 2 3 x-x^2 & for x 2 array . . Question : What is the differentiable coefficient of f(x) at x=3 ?

NDA 2014 Solution View → 78

Paragraph : Let f(x) be a function defined in 1 x by f(x)= array ll 2-x & for 1 x 2 3 x-x^2 & for x 2 array . . Question : Consider the following statements: 1. f^ (2+0) does not e

NDA 2014 Solution View → 79

Consider the function f(x)= aligned x^2-5, & x 3 x+13 , & x 3 aligned . Consider the following statements: 1. The function is discontinuous at x=3 . 2. The function is not differen

NDA 2014 Solution View → 80

Consider the following functions: 1. f(x)=e^x , where x 0 2. g(x)=|x-3| Which of the above functions is/are continuous?

NDA 2013 Solution View → 81

A function f: R R is defined as f(x)=x^2 for x 0 and f(x)=-x for x 0 . Consider the following statements in respect of the above function: 1. The function is continuous at x=0 . 2.

NDA 2013 Solution View → 82

The derivative of |x| at x=0

NDA 2013 Solution View → 83

Consider the following statements in respect of a function f(x) : 1. f(x) is continuous at x=a iff _ x a f(x) exists. 2. If f(x) is continuous at a point, then 1 f(x) is also conti

NDA 2013 Solution View → 84

Consider the function f(x)= array cc x^2, & x 2 3 x-2, & x 2 array . . Which one of the following statements is correct in respect of the above function?

NDA 2013 Solution View → 85

Which one of the following is correct in respect of the function f(x)= x^2 |x| for x 0 and f(0)=0 ?

NDA 2012 Solution View → 86

Which one of the following functions is differentiable for all real values of x ?

NDA 2012 Solution View → 87

Consider the following in respect of the function f(x)=|x-3|: 1. f(x) is continuous at x=3 2. f(x) is differentiable at x=0 . Which of the above statements is/are correct?

NDA 2012 Solution View → 88

Which one of the following statement is correct?

NDA 2012 Solution View → 89

What is the value of k for which the following function f(x) is continuous for all x ? f(x)= array cc x^3-3 x+2 (x-1)^2 & for x 1 k & for x=1 array .

NDA 2011 Solution View → 90

Which one of the following is correct in respect of the function f(x)=|x|+x^2 ?

NDA 2011 Solution View → 91

Consider the following statements: 1. f(x)=|x-3| is continuous at x=0 2. f(x)=|x-3| is differentiable at x=0 Which of the statements given above is/are correct?

NDA 2010 Solution View → 92

If the function f(x)= x(x-2) x^2-4 , x 2 is continuous at x=2 , then what is f(2) equal to?

NDA 2010 Solution View → 93

At how many points is the function f(x)=[x] discontinuous?

NDA 2010 Solution View → 94

The function f(x)=x cosec x is

NDA 2010 Solution View →
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