MCQs For NCERT Class 12 Mathematics Chapter 5 Continuity and Differentiability

Please refer to the MCQ Questions for Class 12 Mathematics Chapter 5 Continuity and Differentiability with Answers. The following Continuity and Differentiability Class 12 Mathematics MCQ Questions have been designed based on the latest syllabus and examination pattern for Class 12. Our experts have designed MCQ Questions for Class 12 Mathematics with Answers for all chapters in your NCERT Class 12 Mathematics book.

Continuity and Differentiability Class 12 MCQ Questions with Answers

See below Continuity and Differentiability Class 12 Mathematics MCQ Questions, solve the questions and compare your answers with the solutions provided below.

Question. If f (x) = 2x and g(x) = (x²/2)+1 then which of the following can be a discontinuous function?
(a) f(x) + g (x)
(b) f(x) – g(x)
(c) f(x) · g (x)
(d) g(x)/f(x)

Question.

(a) k=e(1-1/a)
(b) k= e (1+a)
(c) k=e(2-a)
(d) the equality is not possible

Question. Let f: R→ R be a differentiable function having/

(a) 12
(b) 18
(c) 24
(d) 36 

Question. is equal to
(a) 0
(b) 1/2
(c) log2
(d) e⁴ 

Question. Let f (x)

points of discontinuity of f g (x)= sin x + cos x, then points of discontinuity of f{g(x)} in (0,2π) is
(a) {π/2,3π/4}
(b) {3π/4,7π/4}
(c) {2π/3, 5π/3}
(d) {5π/4,7π/3}

Question. The domain of the derivative of the function

(a) R −{-1 }
(b) R
(c) R − {-3}
(d) None of these 

Question.

is
(a) 1
(b) sin x/x
(c) x/sin x
(d) None of these       

Question. f (x)= [ x] +√{x}, where [.] and {.} denotes the greatest integer function and fractional part respectively, then
(a) f (x) is continuous but non-differentiable at x = 1
(b) f(x) is differentiable at x = 1
(c) f (x) is discontinuous at x = 1
(d) None of the above

Question. The value of the constant α and β such that lim

(a) (1,1 )
(b) (-1,1) 
(c) (1,-1)
(d) (0,1) 

Question. The value of

(a) (abc)³
(b) abc
(c) (abc)^1/3
(d) None of these

Question.

(a) 8/πf( 2)
(b) 2/πf( 2)
(c) 2/π f(1/2)
 (d) 4f(2 ) 

Question. f (x)= [sin x] +[cos x]x ∈ [0,2π], where [.] denotes the greatest integer function. Total number of points, where f x( ) is non-differentiable is equal to
(a) 2
(b) 3
(c) 5
(d) 4 

Question. 

 non-zero real number, then a is equal to
(a) 0
(b) n+1/n
(c) n
(d) n+1/n 

Question. Let f be twice differentiable function satisfying f(1)=1, f(2) = 4, f(3)=9, then
(a)f’ (x) = 2∀ x∈ (R)
(b) f'(x)=5,f(x) for some x∈(1,3)
(c) there exists atleast one x ∈( 1,3) such that f'(x)=2
(d) None of the above

Question.

Question. Let f : R→ R be any function. Define g: R→ by g (x)|(x)| for all x. Then, g is
(a) onto, if f is onto
(b) one-one, if f is one-one
(c) continuous, if f is continuous
(d) differentiable, if f is differentiable

Question. The limit of the following is

(a) − (2)^3/2 
(b) (2)^1/2
(c) (2) ^-3/2
 (d) 3

Question. If {x } and [x ] x are the fractional part function and greatest integer functions of x respectively, then

(a) 0
(b) 1/2
(c) e − 2
(d) does not exist 

Question.

(a) 10
(b) 10²
(c) 10³
(d) 10⁴ 

Question.

(a) 0
(b) 1/2
(c) π/2
(d) 5/6

Question.

Question. If f (x)=

(a) f(x) is continuous in R~ I 
(b) f(x) is continuous in R~ Q 
(c) f x) is continuous in R but not differentiable in R
(d) f(x) is neither continuous nor differentiable in R

Question. 

(a) f(x)/x  is differentiable in R
(b) f (x)/x is continuous but not differentiable in R
(c) f (x) is continuous in R
(d) f (x) is bounded in R

Question. if

denotes the greatest integer function), then
(a) f(x ) is continuous in R
(b) f (x) is continuous in R but not differentiable in R
(c) f ‘(x) exists everywhere but f’ (x) does not exist at some x ∈R
(d) None of the above

Question. If

exists and has non-zero value, then
(a) a, b, c are in AP
(b) a, b, c are in GP
(c) a, b, c are in HP
(d) None of these

Question. The values of p and q so that are

(a) p = 3 and q = 5
(b) p = 5 and q = 3
(c) p = 2 and q = 4
(d) p = 4 and q = 2

Question. If the function

is continuous in the interval[ , ], 0 π then the values of
(a ,b) are
(a) (-1,-1)
(b) (0, 0)
(c) (-1,1)
(d) (1, 1)

Question. If f (x) is a twice differentiable function, then between two consecutive roots of the equation
f (x)  = 0, there exists
(a) at least one root of f(x) = 0
(b) at most one root of f(x) = 0
(c) exactly one root of f(x) = 0
(d) at most one root of f'(x)= 0

Question.

(a) discontinuous at only one point
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of the above

Question. The set of points where the function f given by f (x) =|2x – 1| sin x is differentiable is
(a) R
(b) R – (1/2)
(c) (0, ∞)
(d) None of these

Question. The function f (x) = cot x is discontinuous on the set

Question. The function f (x) = e^|x| is
(a) continuous everywhere but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) None of the above

Question. If f (x) = x² sin 1/x where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is
(a) 0
(b) -1
(c) 1
(d) None of these

Question.

Question. If f (x) =|sin x|, then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
(c) f is everywhere continuous but not differentiable at x = (2n + 1) π/2. n ∈ Z
(d) None of the above

Question.

Question.

Question. The derivative of cos^-1 (2x – 1 ) w.r.t. cos^-1 x is

Question. If x = t² and y = t³, then d²y/dx² is equal to
(a) 3/2
(b) 3/4t
(c) 3/2t
(d) 3/2t

Question. The value of c in Rolle’s theorem for the function f (x) = x³ – 3x in the
interval [0, √3] is
(a) 1
(b) -1
(c) 3/2
(d) 1/3

Question. For the function f (x) = x + (1/x), x ∈ [1, 3], the value of c for mean value theorem is
(a) 1
(b) √3
(c) 2
(d) None of these

Question. 

(a) differentiable at x = 2
(b) not differentiable at x = 2
(c) continuous at x = 2
(d) None of these

Question. 

(a) 1
(b) – 1
(c) 2
(d) None of these

Question. If x² + y² = 1, then
(a) yy’– (2y’)² +1 = 0
(b) yy’ – (y’)² +1 = 0
(c) yy’ – (y’)² –1 = 0
(d) yy’ – 2(y’)² +1 = 0

Question. If y = a^x . b^2x–1 , then d²y/dx² is
(a) y².log ab²
(b) y.log ab²
(c) y. (log ab²)²
(d) y. (log a²b)²

Question. The number of points at which the function f (x) = 1/log | x | is discontinuous is
(a) 1
(b) 2
(c) 3
(d) 4

Question. 

If f(x) is continuous at x = 0, then f(0) =
(a) a – b
(b) a + b
(c) b – a
(d) ln a + ln b

Question. A real function f is said to be continuous, if it is continuous at every point in the
(a) domain of f
(b) codomain of f
(c) range of f
(d) None of these

Question. Let f(x) satisfy the requirements of Lagrange’s mean value theorem in [0, 2]. If f(0) = 0 and f’ (x) ≤ 1/2 for all x in [0, 2], then
(a) f (x) ≤ 2
(b) f(x) ≤ 1
(c) f(x) = 2x
(d) f(x) = 3 for atleast one x in [0, 2]

Question. If f (x) = ae ^|x| + b|x|² , a b ∈ R and f(x) is differentiable at x = 0. Then, a and b are
(a) a = 0, b ∈ R
(b) a = 1, b = 2
(c) b = 0, a ∈ R
(d) a = 4, b = 5

Question. If y = a cos x – b sin x and dⁿy/dxⁿ = a cos x + b sin x , then n =
(a) 2
(b) 4
(c) 6
(d) 8

Question. If a function f(x) is defined as

(a) f(x) is continuous at x = 0 but not differentiable at x = 0
(b) f(x) is continuous as well as differentiable at x = 0
(c) f(x) is discontinuous at x = 0
(d) None of these.

Question. If x^x = y^y, then dy/dx is equal to
(a) – y/x
(b) – x/y
(c) 1+log (x/y)
(d) 1+log x/1+log y

Question. 

Question. Match the terms given in column-I with the terms given in column-II and choose the correct option from the codes given below. 

Codes
A B C D E
(a) 2 3 4 1 5
(b) 1 2 4 3 5
(c) 3 1 4 2 5
(d) 4 5 1 2 3

Question. The set of the points where f(x) = x | x | is twice differentiable, will be
(a) R
(b) R₀
(c) R⁺
(d) R^–

Question. 

is continuous at x = 3, then the value of λ is equal to :
(a) 1
(b) – 1
(c) 0
(d) does not exist

Question. 

is continuous at π/4, then a is equal to
(a) 4
(b) 2
(c) 1
(d) 1/4

Question. If y^x = e^{y – x}, then dy/dx is equal to 
(a) 1 + log y / y log y
(b) (1+ log y)²/y log y
(c) 1+ log y / (log y)²
(d) (1 + log y)² / log y

Question. 

Question. The 2^nd derivative of a sin³t with respect to a cos³t at t = π/4 is
(a) 4√2/3a
(b) 2
(c) 1/12a
(d) None of these

True/False

Question. Rolle’s theorem is applicable for the function f (x) =|x – 1| in [ 0, 2 ].

Question. If f is continuous on its domain D, then | f | is also continuous on D.

Question. The composition of two continuous function is a continuous function.

Question. Trigonometric and inverse trigonometric functions are differentiable in their respective domain.

Question. If f × g is continuous at x = a, then f and g are separately continuous at x = a.