Are these practice questions for Continuity and Differentiability based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Continuity and Differentiability free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Continuity and Differentiability mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Continuity and Differentiability for full coverage.

How many practice questions are there for Continuity and Differentiability?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Continuity and Differentiability Class 12 Mathematics Practice Questions with Answers

easy

Explain the geometric meaning of a function being not continuous at a point x=c.

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easy

Calculate the second derivative of

y = \log(\sin x)

.

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easy

Formulate a function that is continuous for all real numbers but fails to be differentiable at the specific point

x=a

.

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easy

Propose a modification to the function

f(x) = \frac{x^2 - 9}{x-3}

to make it continuous at

x=3

. Justify your proposal.

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easy

State the derivative of \cos^{-1} x with respect to x and specify its domain.

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easy

Analyze the function defined by

f(x) = \begin{cases} \frac{x^2 - 9}{x-3}, & \text{if } x \neq 3 \ k, & \text{if } x = 3 \end{cases}

and calculate the value of

k

that makes

f(x)

continuous at

x=3

.

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easy

Justify whether the converse of the theorem 'Every differentiable function is continuous' is true. If it is not true, provide a counterexample.

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easy

Solve for

\frac{dy}{dx}

if

y = \sin^{-1}(x) + \cos^{-1}(x)

.

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easy

Recall the derivatives of the following inverse trigonometric functions with respect to x: (a) \tan^{-1} x (b) \sin^{-1} x

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easy

State the product rule (also known as the Leibnitz rule) for the differentiation of two functions u(x) and v(x).

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easy

Identify the derivative of the natural exponential function e^x with respect to x.

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easy

Define the continuity of a real function f at a point c in its domain.

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medium

What is an implicit function? Provide one example.

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medium

If

x = a(t - \sin t)

and

y = a(1 - \cos t)

, calculate

\frac{dy}{dx}

.

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medium

Examine if the function

f(x) = |x-2| + |x+1|

is continuous at

x=2

.

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medium

Prove that the function

f(x) = |\cos x|

is continuous for all

x \in \mathbb{R}

but is not differentiable at

x = (2n+1)\frac{\pi}{2}

, where

n

is an integer.

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medium

State the Chain Rule for differentiating a composite function. If f is a composite of two functions u and v such that f = v \circ u, what is the derivative of f?

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medium

Critique the following statement and justify your conclusion: 'If two functions

f(x)

and

g(x)

are both discontinuous at a point

x=c

, then their sum,

(f+g)(x)

, must also be discontinuous at

x=c

.'

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medium

Recall the definition of the derivative of a function f at a point c using the limit definition.

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medium

Design a function that is continuous for all real numbers but is not differentiable at exactly two points,

x=-2

and

x=2

. Provide the function and prove that it satisfies these conditions.

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medium

Evaluate the values of

p

and

q

for which the function

f(x)

is differentiable at

x=1

.

f(x) = \begin{cases} x^2 + 3x + p, & x \leq 1 \ qx + 2, & x > 1 \end{cases}

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medium

If

y = A e^{2x} + B e^{-x}

, solve the differential equation

\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0

.

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medium

Explain why every polynomial function is continuous for all real numbers.

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medium

State the theorem regarding the algebra of continuous functions. If f and g are two real functions continuous at a real number c, list the continuity of their sum, difference, product, and quotient.

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medium

Evaluate the differentiability of the function

f(x) = |x| + |x-1|

at

x=0

and

x=1

by justifying the geometric interpretation of the graph.

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medium

Calculate the derivative of

f(x) = e^{\cos(x^2)}

with respect to

x

.

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medium

Derive a general formula for the derivative of the product of three differentiable functions,

h(x) = f(x)g(x)k(x)

, by applying the product rule twice. Justify each step.

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medium

Describe in detail the three conditions that must be satisfied for a function f to be continuous at a point x = c. Explain what each condition means.

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medium

Calculate

\frac{dy}{dx}

for the function

y = \tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)

.

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medium

Calculate the derivative of

y = x^{\cos x}

with respect to

x

.

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medium

Calculate

\frac{dy}{dx}

if

x^3 + y^3 = 3axy

.

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medium

Summarize the technique of logarithmic differentiation. Explain the type of functions for which this method is particularly useful and identify the necessary conditions for its application.

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hard

Demonstrate that the function

f(x) = |x-3|

is not differentiable at

x=3

.

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hard

Given the function

y = (x + \sqrt{x^2 + 1})^m

, prove the relation

(x^2+1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - m^2y = 0

.

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hard

Explain the relationship between differentiability and continuity at a point. State the relevant theorem and provide a well-known example to show that the converse of the theorem is not true.

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hard

Describe the concept of a second-order derivative. List at least three different notations used to represent it.

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hard

If

x^y = e^{x-y}

, calculate

\frac{dy}{dx}

.

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hard

Justify the relationship that must exist between the constants

a

and

b

for the function

f(x)

defined below to be continuous at

x=0

.

f(x) = \begin{cases} \frac{1 - \cos(ax)}{x \sin x}, & x \neq 0 \ b, & x=0 \end{cases}

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hard

If

f(x)

is a differentiable function with a differentiable inverse

g(x)

(so that

f(g(x)) = x

), derive a formula for the second derivative of the inverse function,

g''(x)

, in terms of the derivatives of the original function,

f'

and

f''

. Justify each step of the derivation.

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hard

Formulate the second derivative of

y

with respect to

x

, given the parametric equations

x = a(\theta - \sin \theta)

and

y = a(1 - \cos \theta)

.

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hard

Calculate the derivative of

y = (\log x)^x + x^{\sin x}

with respect to

x

.

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hard

Critique the following attempt to differentiate

y = x^{\sin x}

. A student takes the logarithm of both sides and proceeds as follows:

\log y = \log(x^{\sin x}) = (\sin x)(\log x)

. Then, the student incorrectly assumes that the derivative of a product of functions is the product of their derivatives, writing

\frac{1}{y}\frac{dy}{dx} = (\cos x)(\frac{1}{x})

. Identify the conceptual error and provide the correct, complete solution.

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hard

If

y = (\tan^{-1} x)^2

, demonstrate that

(x^2+1)^2 y_2 + 2x(x^2+1)y_1 - 2 = 0

, where

y_1 = \frac{dy}{dx}

and

y_2 = \frac{d^2y}{dx^2}

.

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hard

If

x = e^{\sin(2t)}

and

y = e^{\cos(2t)}

, prove that

\frac{dy}{dx} = -\frac{y \log x}{x \log y}

.

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Continuity and DifferentiabilityClass 12 Mathematics Practice Questions