tangent f'(a) LHD = RHD at a
d/dx
Chapter 5  ·  Class XII Mathematics  ·  MCQ Practice

MCQ Practice Arena

Continuity & Differentiability

From Limits to Logarithmic Differentiation — Build an Unbreakable Base

📋 50 MCQs ⭐ 30 PYQs ⏱ 100 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
14Easy
22Medium
14Hard
30PYQs
100 secAvg Time/Q
6Topics
Easy 28% Medium 44% Hard 28%

Why Practise These MCQs?

JEE MainJEE AdvancedCBSEBITSAT

This is the calculus chapter that decides whether the rest of Class XII goes smoothly. JEE Main and Advanced both draw heavily from logarithmic differentiation, parametric derivatives and continuity-at-a-point checks. CBSE boards focus on LHD=RHD proofs. This bank front-loads exactly those repeat patterns.

Topic-wise MCQ Breakdown

Continuity at a Point / Interval8 Q
Differentiability & Chain Rule8 Q
Exponential & Logarithmic Derivatives8 Q
Logarithmic Differentiation8 Q
Parametric Derivatives8 Q
Second Order Derivatives10 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\dfrac{d}{dx}(\log x) = \dfrac{1}{x}$
$\dfrac{d}{dx}(e^x) = e^x$
$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} \text{ (parametric)}$
$\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)$

MCQ Solving Strategy

For differentiability MCQs, always check LHD = RHD at the point in question — never assume continuity guarantees it. Use logarithmic differentiation the moment you see a variable base raised to a variable power. For parametric derivatives, differentiate y and x separately with respect to the parameter before dividing — never differentiate y directly with respect to x.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Check continuity by direct substitution, apply basic derivative rules

② Medium

LHD/RHD checks at a point, chain rule on composite functions, log differentiation

③ Hard

Parametric and second-order derivatives, differentiability of piecewise functions

★ PYQ

JEE Main — log differentiation of complex expressions; CBSE — LHD=RHD proofs

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank Previous year questions ✔️❌ True / False Concept-based True & False questions 📘✔️ Exercises Step by step Solutions to textbook exercises ➡ Next Chapter Application of Derivatives MCQs
🎯 Knowledge Check

Mathematics — CONTINUITY AND DIFFERENTIABILITY

50 Questions Class 12 MCQs
1
The function \(f(x)=x^2+3x+5\) is continuous at
2
If \(f(x)=\dfrac{1}{x-2}\), then \(f\) is continuous at
3
The function \(f(x)=|x|\) is
4
If \(f(x)=\sin x\), then \(f'(x)\) is
5
The derivative of \(x^5\) is
6
If \(f(x)=3x^2-5x+7\), then \(f'(2)\) is
7
The function \(f(x)=\dfrac{x^2-1}{x-1}\), \(x\ne1\), has limit at \(x=1\) equal to
8
For continuity of \(f(x)\) at \(x=a\), which condition is necessary?
9
The function \(f(x)=\dfrac{x^2-4}{x-2}\), \(x\ne2\), can be continuously extended at \(x=2\) by defining \(f(2)\) as
10
If \(f(x)=\begin{cases}x+1,&x<2\\5,&x=2\\x+3,&x>2\end{cases}\), then \(f\) is
11
If \(f(x)=x^3-4x+1\), then \(f'(x)\) is
12
The derivative of \(e^x\) is
13
The derivative of \(\log x\) is
14
If \(y=\sin^{-1}x\), then \(\dfrac{dy}{dx}\) is
15
If \(y=\tan^{-1}x\), then \(\dfrac{dy}{dx}\) is
16
If \(y=x^2\sin x\), then \(y'\) is
17
If \(y=\dfrac{x^2+1}{x}\), then \(\dfrac{dy}{dx}\) is
18
If \(y=\sin(x^2)\), then \(\dfrac{dy}{dx}\) is
19
If \(y=e^{2x+1}\), then \(\dfrac{dy}{dx}\) is
20
If \(f(x)=|x-3|\), then \(f\) is not differentiable at
21
The function \(f(x)=|x|\) is
22
If \(f\) is differentiable at \(x=a\), then \(f\) must be
23
If \(f(x)=\begin{cases}x^2,&x\le1\\ax+b,&x>1\end{cases}\) is continuous at \(x=1\), then
24
For the function \(f(x)=\begin{cases}kx+1,&x<2\\5,&x=2\\3x+k,&x>2\end{cases}\), continuity at \(x=2\) requires
25
If \(f(x)=\begin{cases}\dfrac{\sin x}{x},&x\ne0\\k,&x=0\end{cases}\), then \(f\) is continuous at \(x=0\) when
26
If \(f(x)=\begin{cases}\dfrac{1-\cos x}{x^2},&x\ne0\\k,&x=0\end{cases}\), then continuity at \(x=0\) requires
27
If \(y=(x^2+1)^5\), then \(\dfrac{dy}{dx}\) is
28
If \(y=\log(\sin x)\), then \(\dfrac{dy}{dx}\) is
29
If \(y=x^x\), \(x>0\), then \(\dfrac{dy}{dx}\) is
30
If \(y=(\sin x)^x\), then \(\dfrac{dy}{dx}\) is
31
If \(y=\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)\), for values of \(x\) where the expression is valid, then \(\dfrac{dy}{dx}\) can be simplified to
32
If \(f(x)=x^2\sin\left(\dfrac{1}{x}\right)\) for \(x\ne0\) and \(f(0)=0\), then \(f\) is
33
If \(f(x)=|x|^3\), then \(f'(0)\) is
34
If \(f(x)=|x|^\alpha\), then \(f\) is differentiable at \(x=0\) for
35
If \(f(x)=\begin{cases}ax+b,&x\le1\\x^2,&x>1\end{cases}\) is differentiable at \(x=1\), then \(a\) and \(b\) satisfy
36
If \(f(x)=\begin{cases}x^2+ax+b,&x<1\\3x+2,&x\ge1\end{cases}\) is continuous at \(x=1\), then
37
If \(y=\dfrac{\sin x+\cos x}{\sin x-\cos x}\), then \(\dfrac{dy}{dx}\) is
38
If \(y=\dfrac{x+\sqrt{x^2+1}}{x-\sqrt{x^2+1}}\), then \(y'\) is
39
If \(x=a\cos t\) and \(y=a\sin t\), then \(\dfrac{dy}{dx}\) is
40
If \(x=a\cos^3t\) and \(y=a\sin^3t\), then \(\dfrac{dy}{dx}\) is
41
If \(y=x\sin^{-1}x+\sqrt{1-x^2}\), then \(\dfrac{dy}{dx}\) is
42
If \(y=\tan^{-1}\left(\dfrac{2x}{1-x^2}\right)\), then for \(|x|<1\), \(\dfrac{dy}{dx}\) is
43
If \(y=\log\left(\dfrac{1+x}{1-x}\right)\), then \(\dfrac{dy}{dx}\) is
44
If \(y=x^n\log x\), where \(x>0\), then \(\dfrac{dy}{dx}\) is
45
If \(f(x)=\begin{cases}x^2\sin(1/x),&x\ne0\\0,&x=0\end{cases}\), then \(f'(0)\) equals
46
If \(f(x)=\begin{cases}x\sin(1/x),&x\ne0\\0,&x=0\end{cases}\), then \(f\) at \(x=0\) is
47
If \(f(x)=\begin{cases}\dfrac{x^2\sin(1/x)}{|x|},&x\ne0\\0,&x=0\end{cases}\), then \(f'(0)\) is
48
If \(f(x)=\begin{cases}\dfrac{x^2-1}{x-1},&x\ne1\\k,&x=1\end{cases}\), then \(f\) is differentiable at \(x=1\) when
49
If \(f(x)=\begin{cases}ax^2+b,&x\le1\\x^3+2x,&x>1\end{cases}\) is differentiable at \(x=1\), then \(a\) and \(b\) satisfy
50
If \(f(x)=\begin{cases}x^2\sin(1/x^2),&x\ne0\\0,&x=0\end{cases}\), then at \(x=0\), \(f\) is
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ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025
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NCERT Class 12 Continuity & Differentiability MCQs
NCERT Class 12 Continuity & Differentiability MCQs — Complete Notes & Solutions · academia-aeternum.com
Prepare effectively for NCERT Class 12 Mathematics Chapter 5: Continuity and Differentiability with these 50 carefully designed multiple-choice questions (MCQs). The questions are arranged in increasing order of difficulty, beginning with fundamental concepts and progressing towards application-based and higher-order problems. This MCQ practice set covers important topics such as continuity of functions, limits, differentiability, derivatives of standard functions, chain rule, product rule,…
🎓 Class 12 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
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Frequently Asked Questions

A function f(x) is continuous at x = c if f(c) is defined and lim(x?c) f(x) = f(c). Equivalently, the left-hand limit, right-hand limit and function value must be equal.

The necessary condition is lim(x?c-) f(x) = lim(x?c+) f(x) = f(c). If any one of these conditions fails, the function is discontinuous at c.

Continuity means the function has no break at a point, while differentiability means the derivative exists at that point. Every differentiable function is continuous, but every continuous function need not be differentiable.

Yes. If a function is differentiable at x = c, then it is necessarily continuous at x = c. This is an important theorem in Class 12 Mathematics.

First check continuity on each individual interval, then examine every point where the definition changes. At a boundary point c, verify that the left-hand limit, right-hand limit and f(c) are equal.

If x = f(t) and y = g(t), then the derivative is dy/dx = (dy/dt)/(dx/dt), provided dx/dt is not zero.

Logarithmic differentiation is a method used to differentiate complicated products, quotients or functions in which the variable occurs in both the base and exponent. Taking logarithms first simplifies the differentiation.

If y = f(g(x)), then the chain rule gives dy/dx = f'(g(x))g'(x). It is used to differentiate composite functions.

The standard derivatives include d/dx(sin?¹x) = 1/v(1-x²), d/dx(cos?¹x) = -1/v(1-x²), and d/dx(tan?¹x) = 1/(1+x²).

The chapter develops essential calculus skills used in CBSE Board, JEE Main, JEE Advanced and other entrance examinations, particularly for continuity tests, derivatives, composite functions, implicit differentiation, logarithmic differentiation and parametric differentiation.

These MCQs cover continuity, differentiability, limits, derivatives, standard derivatives, chain rule, product rule, quotient rule, inverse trigonometric functions, logarithmic differentiation, and piecewise functions.

The practice set contains 50 multiple-choice questions with four options, correct answers, and concise explanations.

Yes, the questions are designed around the concepts and topics covered in NCERT Class 12 Mathematics Chapter 5, Continuity and Differentiability.

Yes, these MCQs are useful for revising key concepts and practising objective-type questions relevant to CBSE Class 12 Mathematics examinations.

Yes, every question includes the correct answer and a concise explanation to help students understand the underlying concept.

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