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NDA Continuity and Differentiability PYQs with Solutions

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

94 questionsMathematics2010–2026Answers + solutions

About Continuity and Differentiability PYQs

Continuity and Differentiability has 94 previous year questions in the Quantrex Academy NDA Mathematics bank. The 94 questions in this list come from NDA (94), spanning 2010 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

NDA Continuity and Differentiability PYQs with Solutions

About Continuity and Differentiability PYQs

Questions by year (this list)

Continuity and Differentiability questions with solutions

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

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

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

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

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 ?

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

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?

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?

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 ?

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

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

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

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.

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

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

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) ?

Let f(x)=\ array cc x+6, & x ≤ 1 p x+q, & 1 Question: What is the value of q ?

Paragraph: Let f(x)=\ array cc x+6, & x ≤ 1 p x+q, & 1 Question: What is the value of p ?

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 ?

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?

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

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 π . Which of the above

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

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

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?

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

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

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

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

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?

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

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

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

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?

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

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

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

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

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

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

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

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

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

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?

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?

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

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

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

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

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

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

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

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?

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

Paragraph: Let f(x)=\ array ll -2, & -3 ≤ x ≤ 0 x-2, & 0 Question: Which of the following statements is/are correct? 1. g(x) is differentiable at x=0 . 2. g(x) is differentiable

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

Paragraph: Let f(x)=\ array ll -2, & -3 ≤ x ≤ 0 x-2, & 0 Question: Which of the following statements are correct? 1. g(x) is continuous at x=0 . 2. g(x) is continuous at x=2 . 3.

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

A function f(x) is defined as follows: f(x)=\ array ccc x+π & for & x ∈[-π, 0) π x & for & x ∈[0, (π)/(2)] (x-(π)/(2))^2 & for & x ∈((π)/(2), π] array . Consider the following

A function f(x) is defined as follows: f(x)=\ array ccc x+π & for & x ∈[-π, 0) π x & for & x ∈[0, (π)/(2)] (x-(π)/(2))^2 & for & x ∈((π)/(2), π] array . Consider the following

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

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

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

Consider the function f(x)=\ array cc -2 x & if x ≤-(π)/(2) A x+B & if -(π)/(2) < x < (π)/(2) x & if x ≥ (π)/(2) array . The value of B is

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

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

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

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

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

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 π , where n

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 ?

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 ?

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

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

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

Paragraph : Let f(x) be a function defined in 1 ≤ x 2 array . . Question : Consider the following statements: 1. The function is continuous at every point in the interval [1, ∞] .