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Case Study Questions for Class 12 Maths Chapter 5 Continuity and Differentiability

• Last modified on: 1 year ago
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Here we are providing case study questions for class 12 maths.

Case Study Questions for Class 12 Maths

Case study questions are a type of question that is commonly used in academic and professional settings to evaluate a person’s ability to analyze, interpret, and solve problems based on a given scenario or case study.

Typically, a case study question presents a real-world situation or problem that requires the individual to apply their knowledge and skills to identify the issues, consider various solutions, and recommend a course of action.

Case study questions are designed to test critical thinking skills, problem-solving abilities, and the capacity to work through complex and ambiguous situations.

Preparing for case study questions involves developing a deep understanding of the subject matter, being able to analyze and synthesize information quickly, and being able to communicate ideas clearly and effectively.

Importance of Solving Case Study Questions for Class 12 Maths

Solving case study questions for Class 12 Maths is extremely important as it provides students with an opportunity to apply the mathematical concepts they have learned to real-world scenarios. These questions present a situation or problem that requires students to use their problem-solving skills and critical thinking abilities to arrive at a solution.

The importance of solving case study questions for Class 12 Maths can be summarized as follows:

• Enhances problem-solving skills: Case study questions challenge students to think beyond textbook examples and apply their knowledge to real-world situations. This enhances their problem-solving skills and helps them develop a deeper understanding of the mathematical concepts.
• Improves critical thinking abilities: Case study questions require students to analyze and evaluate information, and draw conclusions based on their understanding of the situation. This helps them develop their critical thinking abilities, which are essential for success in many areas of life.
• Helps in retaining concepts: Solving case study questions helps students retain the concepts they have learned for a longer period of time. This is because they are more likely to remember the concepts when they have applied them to a real-world situation.
• Better preparation for exams: Many competitive exams, including the Class 12 Maths board exam, contain case study questions. Solving these questions helps students become familiar with the format of the questions and the skills required to solve them, which can improve their performance in exams.

In conclusion, solving case study questions for Class 12 Maths is important as it helps students develop problem-solving and critical thinking skills, retain concepts better, and prepare for exams.

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CBSE 12th Standard Maths Subject Continuity and Differentiability Case Study Questions 2021

By QB365 on 21 May, 2021

QB365 Provides the updated CASE Study Questions for Class 12 Maths, and also provide the detail solution for each and every case study questions . Case study questions are latest updated question pattern from NCERT, QB365 will helps to get  more marks in Exams

QB365 - Question Bank Software

Cbse 12th standard maths continuity and differentiability case study questions 2021.

12th Standard CBSE

Final Semester - June 2015

Case Study Questions

Let f(x) be a real valued function, then its  Left Hand Derivative (L.H.D.) :  \(\begin{equation} \mathrm{L} f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h} \end{equation}\)   Right Hand Derivative (R.H.D.) :  \(\begin{equation} \mathrm{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \end{equation}\)   Also, a function jfx) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal  For the function  \(\begin{equation} f(x)=\left\{\begin{array}{l} |x-3|, x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, x<1 \end{array}\right. \end{equation}\)  answer the following questions (i) R.H.D. of f(x) at x = 1is

(ii) L.H.D. of f(x) at x = 1 is

(iii) f(x) is non-differentiable at

(iv) Find the value of f'(2).

(v) The value of f'( -1) is

The function f(x) will be discontinuous at x = a if f(x) has  (a) Discontinuity of first kind  \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\)  both exist but are not equal. If is also known as irremovable discontinuity. (b)  Discontinuity of second kind: If none of the limits  \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\)  exist. (c) Removable discontinuity:  \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\)  both exist and equal but not equal to f(a).  Based on the above information, answer the following questions. (i) If  \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3}, & \text { for } x \neq 3 \\ 4, & \text { for } x=3 \end{array}\right. \end{equation}\)  ,then at x= 3

(ii) Let  \(\begin{equation} f(x)=\left\{\begin{array}{ll} x+2, & \text { if } x \leq 4 \\ x+4, & \text { if } x>4 \end{array}\right. \end{equation}\)  ,then at x = 4

(iii) Consider the function f(x) defined  \(\begin{equation} f(x)=\left\{\begin{array}{l} \frac{x^{2}-4}{x-2} \\ 5 \end{array}\right. \end{equation}\) , for  \(\begin{equation} x \neq 2 \end{equation}\)  

 (iv) If  \(\begin{equation} f(x)=\left\{\begin{array}{cc} \frac{x-|x|}{x}, & if\ x \neq 0 \\ 2, & if\ x=0 \end{array}\right. \end{equation}\)  ,then x = 0

(v) If  \(\begin{equation} f^{\prime}(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log (1+2 x)}, & \text { if } x \neq 0 \\ 7, & \text { if } x=0 \end{array}\right. \end{equation}\) , then at x = 0

(a) A function f(x) is said to be continuous in an open interval (a, b), if it is continuous at every point in this interval. (b) A function f(x) is said to be continuous in the closed interval [a, b], if f(x) is continuous in (a, b) and  \(\begin{equation} \lim _{h \rightarrow 0} f(a+h)=f(a) \text { and } \lim _{h \rightarrow 0} f(b-h)=f(b) \end{equation}\)   If function   \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{\sin (a+1) x+\sin x}{x} & , x<0 \\ c & , x=0 \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}} & , x>0 \end{array}\right. \end{equation}\)  is continuous at x = 0, then answer the following questions. (i) The value of a is

(ii) The value of b is

(iii) The value of c is

(iv) The value of a + c is

(v) The value oi c - a is

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f[g(x)] is a differentiable function of x and  \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x} \end{equation}\) . This rule is also known as CHAIN RULE. Based on the above information, find the derivative of functions w.r.t. x in the following questions (i)  \(\begin{equation} \cos \sqrt{x} \end{equation}\)  

(ii)  \(\begin{equation} 7^{x+\frac{1}{x}} \end{equation}\)  

(iii)  \(\begin{equation} \sqrt{\frac{1-\cos x}{1+\cos x}} \end{equation}\)  

(v) (d) : \(\begin{equation} \sec ^{-1} x+\operatorname{cosec}^{-1} \frac{x}{\sqrt{x^{2}-1}} \end{equation}\)   

If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x. When a given relation expresses y as an implicit function of x and we want to find  \(\begin{equation} \frac{d y}{d x} \end{equation}\) .then we differentiate every term of the given relation w.r.t. x. remembering that a term in y is first differentiated w.r.t. y and then multiplied by  \(\begin{equation} \frac{d y}{d x} \end{equation}\) . Based on the above information, find the value of  \(\begin{equation} \frac{d y}{d x} \end{equation}\)  in each of the following questions  (i) x 3 +x 2 y+xy 2 +y3=81

(ii) x y = c- y

(iii) e siny = xy

(iv) sin 2 x + cos 2 y = 1

(v)  \(\begin{equation} y=(\sqrt{x})^{\sqrt{x}} \end{equation}\)  

*****************************************

Cbse 12th standard maths continuity and differentiability case study questions 2021 answer keys.

we have, \(\begin{equation} f(x)=\left\{\begin{array}{ll} x-3 & , x \geq 3 \\ 3-x & , 1 \leq x<3 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4} & , x<1 \end{array}\right. \end{equation}\)   (i) (b) :  \(\begin{equation} \mathrm{R} f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h} \end{equation}\)   \(\begin{equation} =\lim _{h \rightarrow 0} \frac{3-(1+h)-2}{h}=\lim _{h \rightarrow 0}-\frac{h}{h}=-1 \end{equation}\)   (ii) (b) :   \(\begin{equation} \mathrm{L}_{\mathrm{s}}^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{-h} \end{equation}\)   \(\begin{equation} =\lim _{h \rightarrow 0} \frac{-1}{h}\left[\frac{(1-h)^{2}}{4}-\frac{3(1-h)}{2}+\frac{13}{4}-2\right] \end{equation}\)   \(\begin{equation} =\lim _{h \rightarrow 0}\left(\frac{1+h^{2}-2 h-6+6 h+13-8}{-4 h}\right) \end{equation}\)   \(\begin{equation} =\lim _{h \rightarrow 0}\left(\frac{h^{2}+4 h}{-4 h}\right)=-1 \end{equation}\)   (iii) (c) : Since, R.H.D. at x = 3 is 1 and L.H.D. at x = 3 is-1 \(\therefore\)   f(x) is non-differentiable at x = 3. (iv) (d) (v) (c) : From above, we have \(\begin{equation} f^{\prime}(x)=\frac{x}{2}-\frac{3}{2}, x<1 \end{equation}\)   \(\begin{equation} \therefore f^{\prime}(-1)=\frac{-1}{2}-\frac{3}{2}=-2 \end{equation}\)  

(i) (a) :  f(3) = 4 \(\begin{equation} \lim _{x \rightarrow 3} f(x)=\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}=\lim _{x \rightarrow 3} \frac{(x+3)(x-3)}{(x-3)} \end{equation}\)   \(\begin{equation} =\lim _{x \rightarrow 3}(x+3)=6 \because \lim _{x \rightarrow 3} f(x) \neq f(3) \end{equation}\)   \(\therefore\)  f(x) has removable discontinuity at x = 3. (ii) (c) :   \(\begin{equation} \lim _{x \rightarrow 4^{-}} f(x)=\lim _{x \rightarrow 4}(x+2)=4+2=6 \end{equation}\)   \(\begin{equation} \lim _{x \rightarrow 4^{+}} f(x)=\lim _{x \rightarrow 4}(x+4)=4+4=8 \end{equation}\)   \(\begin{equation} \therefore \quad \lim _{x \rightarrow 4^{-}} f(x) \neq \lim _{x \rightarrow 4^{+}} f(x) \end{equation}\)   \(\begin{equation} \therefore f(x) \end{equation}\)  has an irremovable discontinuity at x = 4. (iii) (a) :    \(\begin{equation} \lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2} \frac{\left(x^{-4}-4\right)}{(x-2)}=\lim _{x \rightarrow 2}(x+2)=4 \end{equation}\)   \(\begin{equation} \therefore f(x) \end{equation}\)  has removable discontinuity at x = 2. (iv) (c) : f(0)=2 \(\begin{equation} \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} \frac{x+x}{x}=2 \end{equation}\)   \(\begin{equation} \lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} \frac{x-x}{x}=0 \end{equation}\)   \(\begin{equation} \because \lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x) \end{equation}\) \(\begin{equation} \therefore f(x) \end{equation}\)  has an irremovable discontinuity at x = 0. (v) (d) : f(0) = 7 \(\begin{equation} \lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{e^{x}-1}{\log (1+2 x)}=\lim _{x \rightarrow 0} \frac{\frac{\left(\frac{e^{x}-1}{x}\right)}{\log (1+2 x)}{2 x} \cdot 2}=\frac{1}{2} \end{equation}\)   \(\begin{equation} \because \ \lim _{x \rightarrow 0} f(x) \neq f(0) \end{equation}\)   \(\begin{equation} \therefore f(x) \end{equation}\)  has removable discontinuity at x = 0.

L.H.L (at x = 0) =  \(\begin{equation} \lim _{x \rightarrow 0} \frac{\sin (a+1) x+\sin x}{x}\left(\frac{0}{0} \text { form }\right) \end{equation}\)   Using L' Hospital rule, we get L.H.L.(at x = 0) \(\begin{equation} =\lim _{x \rightarrow 0}(a+1) \cos (a+1) x+\cos x=a+2 \end{equation}\)   R.H.L  \(\begin{equation} \text { (at } x=0)=\lim _{x \rightarrow 0} \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}}=\lim _{x \rightarrow 0} \frac{\sqrt{1+b x}-1}{b x} \end{equation}\)   \(\begin{equation} =\lim _{x \rightarrow 0} \frac{1}{\sqrt{1+b x}+1}=\frac{1}{2} \end{equation}\)   Since,f(x) is continuous at x = 0. \(\therefore\)  From (i) and (ii), we get \(\begin{equation} a+2=c=\frac{1}{2} \Rightarrow a=-\frac{3}{2}, c=\frac{1}{2} \end{equation}\)   Also, value of b does not affect the continuity of f(x), so b can be any real number. (i) (a) (ii) (d)  (iii) (b) (iv) (c) :   \(\begin{equation} a+c=-\frac{3}{2}+\frac{1}{2}=-1 \end{equation}\)   (v) (d) :   \(\begin{equation} c-a=\frac{1}{2}+\frac{3}{2}=2 \end{equation}\)

(i) (a) : Let  \(\begin{equation} y=\cos \sqrt{x} \end{equation}\)   \(\begin{equation} \therefore \quad \frac{d y}{d x}=\frac{d}{d x}(\cos \sqrt{x})=-\sin \sqrt{x} \cdot \frac{d}{d x}(\sqrt{x}) \end{equation}\)   \(\begin{equation} =-\sin \sqrt{x} \times \frac{1}{2 \sqrt{x}}=\frac{-\sin \sqrt{x}}{2 \sqrt{x}} \end{equation}\)   (ii) (a) : Let  \(\begin{equation} y=7^{x+\frac{1}{x}} \quad \therefore \quad \frac{d y}{d x}=\frac{d}{d x}\left(7^{x+\frac{1}{x}}\right) \end{equation}\)   \(\begin{equation} =7^{x+\frac{1}{x}} \cdot \log 7 \cdot \frac{d}{d x}\left(x+\frac{1}{x}\right)=7^{x+\frac{1}{x}} \cdot \log 7 \cdot\left(1-\frac{1}{x^{2}}\right) \end{equation}\)   \(\begin{equation} =\left(\frac{x^{2}-1}{x^{2}}\right) \cdot 7^{x+\frac{1}{x}} \cdot \log 7 \end{equation}\)   (iii) (a) : Let  \(\begin{equation} y=\sqrt{\frac{1-\cos x}{1+\cos x}}=\sqrt{\frac{1-1+2 \sin ^{2} \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}-1+1}}=\tan \left(\frac{x}{2}\right) \end{equation}\)   \(\begin{equation} \therefore \frac{d y}{d x}=\sec ^{2} \frac{x}{2} \cdot \frac{1}{2}=\frac{1}{2} \sec ^{2} \frac{x}{2} \end{equation}\)   (iv) (b) :  Let  \(\begin{equation} y=\frac{1}{b} \tan ^{-1}\left(\frac{x}{b}\right)+\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right) \end{equation}\)   \(\begin{equation} \therefore \quad \frac{d y}{d x}=\frac{1}{b} \times \frac{1}{1+\frac{x^{2}}{b^{2}}} \times \frac{1}{b}+\frac{1}{a} \times \frac{1}{1+\frac{x^{2}}{a^{2}}} \times \frac{1}{a} \end{equation}\)   \(\begin{equation} =\frac{1}{b^{2}+x^{2}}+\frac{1}{a^{2}+x^{2}} \end{equation}\)   (v) (d) :  Let  \(\begin{equation} y=\sec ^{-1} x+\operatorname{cosec}^{-1} \frac{x}{\sqrt{x^{2}-1}} \end{equation}\)   Put   \(\begin{equation} x=\sec \theta \Rightarrow \theta=\sec ^{-1} x \end{equation}\)   \(\begin{equation} \therefore \quad y=\sec ^{-1}(\sec \theta)+\operatorname{cosec}^{-1}\left(\frac{\sec \theta}{\sqrt{\sec ^{2} \theta-1}}\right) \end{equation}\)   \(\begin{equation} =\theta+\sin ^{-1}\left[\sqrt{1-\cos ^{2} \theta}\right] \end{equation}\)   \(\begin{equation} =\theta+\sin ^{-1}(\sin \theta)=\theta+\theta=2 \theta=2 \sec ^{-1} x \end{equation}\)   \(\begin{equation} \therefore \quad \frac{d y}{d x}=2 \frac{d}{d x}\left(\sec ^{-1} x\right)=2 \times \frac{1}{|x| \sqrt{x^{2}-1}}=\frac{2}{|x| \sqrt{x^{2}-1}} \end{equation}\)

(i) (b) :  x 3 + x 2 y+ xy 2 +y 3 = 81 \(\begin{equation} \Rightarrow 3 x^{2}+x^{2} \frac{d y}{d x}+2 x y+2 x y \frac{d y}{d x}+y^{2}+3 y^{2} \frac{d y}{d x}=0 \end{equation}\)   \(\begin{equation} \Rightarrow \left(x^{2}+2 x y+3 y^{2}\right) \frac{d y}{d x}=-3 x^{2}-2 x y-y^{2} \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}=\frac{-\left(3 x^{2}+2 x y+y^{2}\right)}{x^{2}+2 x y+3 y^{2}} \end{equation}\)   (ii) (c) :  x y = e x-y ⇒y log x = x - y \(\begin{equation} \Rightarrow y \times \frac{1}{x}+\log x \cdot \frac{d y}{d x}=1-\frac{d y}{d x} \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}[\log x+1]=1-\frac{y}{x} \Rightarrow \frac{d y}{d x}=\frac{x-y}{x[1+\log x]} \end{equation}\)   (iii) (d) :  \(\begin{equation} e^{\sin y}=x y \Rightarrow \sin y=\log x+\log y \end{equation}\)   \(\begin{equation} \Rightarrow \cos y \frac{d y}{d x}=\frac{1}{x}+\frac{1}{y} \frac{d y}{d x} \Rightarrow \frac{d y}{d x}\left[\cos y-\frac{1}{y}\right]=\frac{1}{x} \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}=\frac{y}{x(y \cos y-1)} \end{equation}\)   (iv) (d) : sin 2 x + cos 2 y = 1 \(\begin{equation} \Rightarrow \quad 2 \sin x \cos x+2 \cos y\left(-\sin y \frac{d y}{d x}\right)=0 \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}=\frac{-\sin 2 x}{-\sin 2 y}=\frac{\sin 2 x}{\sin 2 y} \end{equation}\)   (v) (d) :  \(\begin{equation} y=(\sqrt{x})^{\sqrt{x}} \quad \Rightarrow y=(\sqrt{x})^{y} \end{equation}\)   \(\begin{equation} \Rightarrow \log y=y(\log \sqrt{x}) \Rightarrow \log y=\frac{1}{2}(y \log x) \end{equation}\)   \(\begin{equation} \Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{1}{2}\left[y \times \frac{1}{x}+\log x\left(\frac{d y}{d x}\right)\right] \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}\left\{\frac{1}{y}-\frac{1}{2} \log x\right\}=\frac{1}{2} \frac{y}{x} \end{equation}\)   \(\begin{equation} \Rightarrow \frac{d y}{d x}=\frac{y}{2 x} \times \frac{2 y}{(2-y \log x)}=\frac{y^{2}}{x(2-y \log x)} \end{equation}\)

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Case Study on Continuity And Differentiability Class 12 Maths PDF

The passage-based questions are commonly known as case study questions. Students looking for Case Study on Continuity And Differentiability Class 12 Maths can use this page to download the PDF file. 

The case study questions on Continuity And Differentiability are based on the CBSE Class 12 Maths Syllabus, and therefore, referring to the Continuity And Differentiability case study questions enable students to gain the appropriate knowledge and prepare better for the Class 12 Maths board examination. Continue reading to know how should students answer it and why it is essential to solve it, etc.

Case Study on Continuity And Differentiability Class 12 Maths with Solutions in PDF

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The case study on Continuity And Differentiability Class 12 Maths with solutions in PDF helps students tackle questions that appear confusing or difficult to answer. The answers to the Continuity And Differentiability case study questions are very easy to grasp from the PDF - download links are given on this page.

Why Solve Continuity And Differentiability Case Study Questions on Class 12 Maths?

There are three major reasons why one should solve Continuity And Differentiability case study questions on Class 12 Maths - all those major reasons are discussed below:

• To Prepare for the Board Examination: For many years CBSE board is asking case-based questions to the Class 12 Maths students, therefore, it is important to solve Continuity And Differentiability Case study questions as it will help better prepare for the Class 12 board exam preparation.
• Develop Problem-Solving Skills: Class 12 Maths Continuity And Differentiability case study questions require students to analyze a given situation, identify the key issues, and apply relevant concepts to find out a solution. This can help CBSE Class 12 students develop their problem-solving skills, which are essential for success in any profession rather than Class 12 board exam preparation.
• Understand Real-Life Applications: Several Continuity And Differentiability Class 12 Maths Case Study questions are linked with real-life applications, therefore, solving them enables students to gain the theoretical knowledge of Continuity And Differentiability as well as real-life implications of those learnings too.

How to Answer Case Study Questions on Continuity And Differentiability?

Students can choose their own way to answer Case Study on Continuity And Differentiability Class 12 Maths, however, we believe following these three steps would help a lot in answering Class 12 Maths Continuity And Differentiability Case Study questions.

• Read Question Properly: Many make mistakes in the first step which is not reading the questions properly, therefore, it is important to read the question properly and answer questions accordingly.
• Highlight Important Points Discussed in the Clause: While reading the paragraph, highlight the important points discussed as it will help you save your time and answer Continuity And Differentiability questions quickly.
• Go Through Each Question One-By-One: Ideally, going through each question gradually is advised so, that a sync between each question and the answer can be maintained. When you are solving Continuity And Differentiability Class 12 Maths case study questions make sure you are approaching each question in a step-wise manner.

What to Know to Solve Case Study Questions on Class 12 Continuity And Differentiability?

 A few essential things to know to solve Case Study Questions on Class 12 Continuity And Differentiability are -

• Basic Formulas of Continuity And Differentiability: One of the most important things to know to solve Case Study Questions on Class 12 Continuity And Differentiability is to learn about the basic formulas or revise them before solving the case-based questions on Continuity And Differentiability.
• To Think Analytically: Analytical thinkers have the ability to detect patterns and that is why it is an essential skill to learn to solve the CBSE Class 12 Maths Continuity And Differentiability case study questions.
• Strong Command of Calculations: Another important thing to do is to build a strong command of calculations especially, mental Maths calculations.

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NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

September 13, 2019 by phani

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability is designed and prepared by the best teachers across India. All the important topics are covered in the exercises and each answer comes with a detailed explanation to help students understand concepts better. These NCERT solutions play a crucial role in your preparation for all exams conducted by the CBSE, including the JEE.

Did you try solving RD Sharma Class 12 Maths Continuity and Differtiability ?

Chapter 5 Continuity and Differentiability Maths NCERT Solutions cover eight exercises. The answer to each question in every exercise is provided along with complete, step-wise solutions for your better understanding. This will prove to be most helpful to you in your home assignments as well as practice sessions.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second-Order Derivative
• Mean Value Theorem

There are total eight exercises and one misc exercise( 144 Questions fully solved ) in the class 12th maths chapter 5 Continuity and Differentiability.

• Class 12 Maths Chapter 5 Exercise 5.1 (34 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.2 (10 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.3 (15 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.4 (10 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.5 (18 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.6 (11 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.7 (17 Questions fully solved)
• Class 12 Maths Chapter 5 Exercise 5.8 (6 Questions fully solved)
• Class 12 Maths Chapter 5 Continuity and Differentiability Miscellaneous Exercise (23 Questions fully solved)

Class 12 Maths Continuity and Differentiability Exercise 5.1

• Class 12 Maths Chapter 5 Exercise 5.2
• Class 12 Maths Chapter 5 Exercise 5.3

Class 12 Maths Continuity and Differentiability Exercise 5.2 and 5.3

• Class 12 Maths Chapter 5 Exercise 5.4
• Class 12 Maths Chapter 5 Exercise 5.5

Class 12 Maths Continuity and Differentiability Exercise 5.4

Class 12 Maths Continuity and Differentiability Exercise 5.5

• Class 12 Maths Chapter 5 Exercise 5.6

Class 12 Maths Continuity and Differentiability Exercise 5.6

• Class 12 Maths Chapter 5 Exercise 5.7

Class 12 Maths Continuity and Differentiability Exercise 5.7

• Class 12 Maths Chapter 5 Exercise 5.8

Class 12 Maths Continuity and Differentiability Exercise 5.8

• Class 12 Maths Chapter 5 Miscellaneous

Class 12 Maths Continuity and Differentiability Miscellaneous Exercise

• Class 12 Maths Chapter 5 Exercise 5.1

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.1

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

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NCERT Solutions Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability explains the fundamental concepts of continuity of functions, their differentiability, and the relation between both of them. There are various aspects of nature that exhibit the property of continuity like the continuously flowing rivers, time in human life, and so on. Likewise, in mathematics, the notion of the continuity of a function is applied in various situations. Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain. The concept of continuity and differentiability is studied in advanced mathematics and has multiple applications in engineering, science, economics, etc. Thus, it is crucial to gain a deep understanding of this topic.

NCERT Solutions Class 12 Maths Chapter 5 is helpful in building a clear understanding of the various aspects of limits, continuity of functions, their derivatives, properties, and methods to find derivatives. Some of the important topics explained in this chapter are derivatives of the specific standard function, logarithmic differentiation, derivatives of functions in parametric forms, and second-order derivatives. With the regular practice of Class 12 Maths NCERT Solutions Chapter 5, students will quickly gain a deep understanding of this topic. To learn and practice with NCERT Solutions Chapter 5 Continuity and Differentiability, download the exercises provided in the links below.

• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.1
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.2
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.3
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.4
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.5
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.6
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.7
• NCERT Solutions Class 12 Maths Chapter 5 Ex 5.8
• NCERT Solutions Class 12 Maths Chapter 5 Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 5 PDF

NCERT Solutions for Class 12 Maths Chapter 5 are meticulously formulated guides that promote the fundamental understanding of the representation of continuity and differentiability of functions along with their applications. To practice with these solutions, click on the links of the pdf files given below.

☛ Download Class 12 Maths NCERT Solutions Chapter 5 Continuity and Differentiability

NCERT Class 12 Maths Chapter 5   Download PDF

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT solutions Class 12 Maths Chapter 5 Continuity and Differentiability are highly competent guides that impart the foundational knowledge of calculus. These reliable learning resources deliver a clear sense of geometric interpretation of continuity and the differentiability of functions. The pattern followed by these solutions is quite amiable for performing a deep study of all topics. To practice the exercise-wise NCERT Solutions Class 12 Maths Continuity and Differentiability, try the exercise-wise links given below.

• Class 12 Maths Chapter 5 Ex 5.1 - 34 Questions
• Class 12 Maths Chapter 5 Ex 5.2 - 10 Questions
• Class 12 Maths Chapter 5 Ex 5.3 - 15 Questions
• Class 12 Maths Chapter 5 Ex 5.4 - 10 Questions
• Class 12 Maths Chapter 5 Ex 5.5 - 18 Questions
• Class 12 Maths Chapter 5 Ex 5.6 - 11 Questions
• Class 12 Maths Chapter 5 Ex 5.7 - 17 Questions
• Class 12 Maths Chapter 5 Ex 5.8 - 6 Questions
• Class 12 Maths Chapter 5 Miscellaneous Ex - 23 Questions

☛ Download CBSE Class 12 Maths Chapter 5 NCERT Book

Topics Covered: The main topics covered in NCERT Solutions Class 12 Maths Chapter 5 are continuity, differentiability , exponential , logarithmic functions , logarithmic differentiation, derivatives of functions in parametric forms, second-order derivative, and mean value theorem. Each of these topics is elaborated efficiently with interactive illustrations to help students grasp them better.

Total Questions: Class 12 Maths Chapter 5 Continuity and Differentiability has 144 questions in 9 exercises which are mostly short answer type questions. The sums provided in each exercise of these solutions competently cover all important topics from an exam point of view.

List of Formulas in NCERT Solutions Class 12 Maths Chapter 5

Continuity and Differentiability are the basic topics in studying calculus. Building a precise understanding of these concepts will help in learning a higher branch of mathematics called “Analysis”. To learn the continuity of functions and their derivatives it is required to have a clear knowledge of important theorems, definitions, formulas , and rules. NCERT Solutions Class 12 Maths Chapter 5 elaborates each of them with the help of examples. Some of the important terms and formulas explained in these NCERT Solutions for Class 12 Maths Chapter 5 are given below.

• Continuous Function: A function is said to be a continuous function if it is continuous on the whole of its domain. A continuous function f(x) is continuous on a given interval if it is continuous at every point of the interval .
• Arithmetic of Continuous Functions: The result of arithmetic operations performed on continuous functions is continuous. i.e., if f and g are continuous functions, then

(f ± g) (x) = f(x) ± g(x) is continuous.

(f . g) (x) = f(x) . g (x) is continuous.

( f / g ) (x) = f(x ) / g(x) (wherever g(x) ≠ 0) is continuous.

• Differentiable: A function is said to be differentiable if the derivative of the function exists at all points in its domain . Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain. Some derivative rules used to find the derivatives of a function are:

i) ( f ± g )′ = f′ ± g′

ii) ( fg )′ = f′g +f g′

iii) ( f/g )′ = (f′g−fg′) /g 2

• Logarithmic functions : Logarithmic functions are the inverses of exponential functions. The logarithmic function y = log a x is defined to be equivalent to the exponential equation x = a y .

FAQs on NCERT Solutions for Class 12 Maths Chapter 5

What is the importance of ncert solutions for class 12 maths chapter 5 continuity and differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability are highly reliable resources that provide accurate knowledge of all topics included in this chapter. A strategic study of these resources will enable students to grasp all complex concepts easily. These well-framed guides are prominent in enhancing the mathematical quotient of students to score well in annual exams. They are also a promising tool to build self-confidence for various competitive exams.

What are the Important Topics Covered in NCERT Solutions Class 12 Maths Chapter 5?

Chapter 5 Continuity and Differentiability of class 12 Maths precisely describes all necessary concepts of continuity and differentiability of functions. The important topics covered in the NCERT Solutions Class 12 Maths Chapter 5 are continuity of functions, algebra of continuous functions, differentiability, Exponential and Logarithmic Functions, logarithmic differentiation, derivatives of functions in parametric forms, second-order derivative, and mean value theorem. The problems provided in these solutions are pretty efficient in delivering the fundamentals related to the above-mentioned topics. These purposefully created solutions are excellent to improve the math proficiency of students in an easy way.

Do I Need to Practice all Questions Provided in NCERT Solutions Class 12 Maths Continuity and Differentiability?

Practicing all the questions included in NCERT Solutions Class 12 Maths Continuity and Differentiability gives a step-by-step knowledge of each topic. The gradually paced problems in these solutions will assist students to simply adapt to the rising complexity level with ease. Students can go through the detailed solution of each question to build the fundamental knowledge of basics related to the continuity and differentiability of functions. Regularly practicing with these solutions will surely prepare students to appear in competitive exams.

How Many Questions are there in Class 12 Maths NCERT Solutions Chapter 5 Continuity and Differentiability?

Class 12 Maths Chapter 5 Continuity and Differentiability has 144 questions in 9 exercises. With the efficient practice of all the sums provided in these solutions, students can learn to accurately answer the various patterns and types of questions. They can also apply these resources to build a firm mathematical foundation for excelling in exams. The intuitive format of these solutions is reliable to gain a sound knowledge of core fundamentals.

What are the Important Formulas in NCERT Solutions Class 12 Maths Chapter 5?

NCERT Solutions Class 12 Maths Chapter 5 mainly focuses on the Continuity of functions, their differentiability, properties, and applications. Some of the important concepts explained in these solutions are the differentiability of special functions and the theorems related to them. All these basics are covered in detail with suitable examples to enhance mathematical learnability in students. The objective of these solutions is to devise a precise skillset for mastering this topic.

Why Should I Practice NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability?

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability is a competent learning resource that gives a deep insight into exam preparation. By solving all the problems present in this chapter, students can assess their practice and knowledge for the annual exams. It is also an easy approach for covering the entire syllabus of CBSE Class 12 Maths Chapter 5. Additionally, it is a convenient way to clear all doubts and learn complex concepts.

CBSE Class 12 Maths –Chapter 5 Continuity and Differentiability- Study Materials

NCERT Solutions Class 12 All Subjects Sample Papers Past Years Papers

Notes and Study Materials

• Concepts of Continuity and Differentiability
• Master File for Continuity and Differentiability
• Continuity and Differentiability Note
• NCERT Solutions for – Continuity and Differentiability
• NCERT Exemplar Solutions for – Continuity and Differentiability
• R D Sharma Solution of Algebra of Differentiability
• R D Sharma Solution of Algebra of Continuity
• Past Many Years CBSE Questions and Answer Of Relation and Function
• Continuity and Differentiability Mind Map

Examples and Exercise

• Continuity and Differentiability : Practice Paper 1
• Continuity and Differentiability : Practice Paper 2
• Continuity and Differentiability : Practice Paper 3
• Continuity and Differentiability : Practice Paper 4

CBSE Revision Notes for CBSE Class 12 Mathematics Continuity and Differentiability Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

CONTINUITY DEFINITION A function f (x) is said to be continuous at x = a; where a ∈ domain of f(x), if

i.e., LHL=RHL = value of a function at x= a

Reasons of discontinuity

If f (x) is not continuous at x = a, we say that f (x) is discontinuous at x = a. There are following possibilities of discontinuity

PROPERTIES OF CONTINUOUS FUNCTIONS

THE INTERMEDIATE VALUE THEOREM

Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y₀ is a number between f(a) and f(b), their exits a number c between a and b such that f(c)=y₀.

That a function f which is continuous in [a, b] possesses the following properties:

( i ) If f (a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x) =0 in the open interval (a, b).

(ii) If K is any real number between f(a) and f(b), then there exists at least one solution of the equation f (x) = K in the open interval (a, b).

CONTINUITY IN AN INTERVAL

(a) A function f is said to be continuous in (a, b) if f is continuous at each and every point ∈ (a, b).

(b) A function f is said to be continuous in a closed interval [a, b] if:

(1) f is continuous in the open interval (a, b) and

(2) f is right continuous at ‘a’ i.e. limit ₓ→ₐ⁺ f(x)=f(a) = a finite quantity.

(3) f is left continuous at ‘b’; i.e. limit ₓ→b⁻f(x) = f(b) a finite quantity.

A LIST OF CONTINUOUS FUNCTIONS

TYPES OF DISCONTINUITIES

Type-1: (Removable type of discontinuities)

In case, limit ₓ→c f(x) exists but is not equal to f (c) then the function is said to have a removable discontnuity or discontinuity of the first kind . In this case, we can redefine the function such that limit ₓ→c f(x) = f (c) and make it continuous at x=c. Removable type of discontinuity can be further classified as:

(a) Missing Point Discontinuity :

Where limit ₓ→ₐ f(x) exists finitely but f(a) is not defined.

From the adjacent graph note that

– f is continuous at x =-1 – f has isolated discontinuity at x = 1 – f has missing point discontinuity at x = 2 – f has non-removable (finite type) discontinity at the origin.

(a) In case of dis-continuity of the second kind the non- negative difference between the value of the RHL at x=a and LHL at x=a is called the jump of discontinuity. A function having a finite number of jumps in a given interval I is called a piece wise continuous or sectionally continuous function in this interval.

(b) All Polynomials, Trigonometrical functions, exponential and Logarithmic functions are continuous in their domains.

(c) If f (x) is continuous and g (x) is discontinuous at x = a then the product function ∅ (x)=f(x). g (x) is not necessarily be discontinuous at x = a. e.g.

DIFFERENTIABILITY

DIFFERENTIABILITY IN A SET

1. A function f (x) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b), if it is differentiable at each point of (a, b).

2. A function f (x) defined on closed interval [a, b] is said to be differentiable or derivable. “If f is derivable in the open interval (a, b) and also the end points a and b, then f is said to be derivable in the closed interval [a, b]”.

A function f is said to be a differentiable function if it is differentiable at every point of its domain.

1. If f (x) and g (x) are derivable at x = a then the functions f(x)+g (x), f(x)-g(x), f(x). g (x) will also be derivable at x=a and if g (a) ≠ 0 then the function f (x)/g(x) will also be derivable at x=a.

2. If f (x) is differentiable at x = a and g (x) is not differentiable at x = a, then the product function F (x) = f (x). g (x) can still be differentiable at x=a. E.g. f (x) =x and g (x) = |x|.

3. If f (x) and g (x) both are not differentiable at x = a then the product function; F (x) = f(x). g (x) can still be differentiable at x = a. E.g., f(x) = |x| and g (x) = |x|.

4. If f (x) and g (x) both are not differentiable at x=a then the sum function F (x) =f (x)+g (x) may be a differentiable function. E.g., f (x) = |x| and g (x)=-|xl.

5. If f(x) is derivable at x= a • F’ (x) is continuous at x=a.

RELATION B/W CONTINUITY & DIFFERENNINBILINY

In the previous section we have discussed that if a function is differentiable at a point, then it should be continuous at that point and a discontinuous function cannot be differentiable. This fact is proved in the following theorem.

Theorem : If a function is differentiable at a point, it is necessarily continuous at that point. But the converse is not necessarily true,

Or f(x) is differentiable at x=c f(x) is continuous at x = c.

Converse : The converse of the above theorem is not necessarily true i.e., a function may be continuous at a point but may not be differentiable at that point.

E.g., The function f (x) = |x| is continuous at x = 0 but it is not differentiable at x = 0, as shown in the figure.

The figure shows that sharp edge at x = 0 hence, function is not differentiable but continuous at x = 0.

(a) Let f´⁺(a) = p & f´⁻(a)=q where p & q are finite then

(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at X=a.

Theorem 2 : Let fand g be real functions such that fog is defined if g is continuous at x = a and f is continuous at g (a), show that fog is continuous at x = a.

DIFFERENTIATION

(a) Let us consider a function y=f(x) defined in a certain interval. It has a definite value for each value of the independent variable x in this interval.

Now, the ratio of the increment of the function to the increment in the independent variable

DERIVATIVE OF STANDARD FUNCTION

THEOREMS ON DERIVATIVES

METHODS OF DIFFERENTIATION 4.1 Derivative by using Trigonometrical Substitution

4.2 Logarithmic Differentiatio

DERIVATIVE OF ORDER TWO & THREE

Let a function y =f (x) be defined on an open interval (a, b). It’s derivative, if it exists on (a, b), is a certain function f'(x) [or (dy/dx) or y’] & is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of y w.r.t. x & is denoted by f”(x) or (d²y/dx²) or y”.

Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is

Important Questions for CBSE Class 12 Maths Continuity

Previous year examination questions, 4 marks questions, cbse class 12 maths differntiability, continuity and differentiability class 12 mcqs questions with answers.

Question 1. If f (x) = 2x and g (x) = \(\frac{x^2}{2}\) + 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) \(\frac{g(x)}{f(x)}\)

Answer: (d) \(\frac{g(x)}{f(x)}\)

Question 2. The function f(x) = \(\frac{4-x^2}{4x-x^3}\) is (a) discontinuous at only one point at x = 0 (b) discontinuous at exactly two points (c) discontinuous at exactly three points (d) None of these

Answer: (a) discontinuous at only one point at x = 0

Question 3. The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is (a) R (b) R = {\(\frac{1}{2}\)} (c) (0, ∞) (d) None of these

Answer: (b) R = {\(\frac{1}{2}\)}

Question 4. The function f(x) = cot x is discontinuous on the set (a) {x = nπ, n ∈ Z} (b) {x = 2nπ, n ∈ Z} (c) {x = (2n + 1) \(\frac{π}{2}\) n ∈ Z} (d) {x – \(\frac{nπ}{2}\) n ∈ Z}

Answer: (a) {x = nπ, n ∈ Z}

Question 5. 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 these

Answer: (a) continuous everywhere but not differentiable at x = 0

Question 6. If f(x) = x² sin\(\frac{1}{x}\), where x ≠ 0, then the value of the function f(x) at x = 0, so that the function is continuous at x = 0 is (a) 0 (b) -1 (c) 1 (d) None of these

Answer: (a) 0

Answer: (c) n = \(\frac{mπ}{2}\)

Question 8. If y = log(\(\frac{1-x^2}{1+x^2}\)), then \(\frac{dy}{dx}\) is equal to (a) \(\frac{4x^3}{1-x^4}\) (b) \(\frac{-4x}{1-x^4}\) (c) \(\frac{1}{4-x^4}\) (d) \(\frac{-4x^3}{1-x^4}\)

Answer: (b) \(\frac{-4x}{1-x^4}\)

Question 9. Let 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 no differentiable at x = (2n + 1) \(\frac{π}{2}\) n ∈ Z (d) None of these

Answer: (b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

Question 10. If y = \(\sqrt{sin x+y}\) then \(\frac{dy}{dx}\) is equal to (a) \(\frac{cosx}{2y-1}\) (b) \(\frac{cosx}{1-2y}\) (c) \(\frac{sinx}{1-xy}\) (d) \(\frac{sinx}{2y-1}\)

Answer: (a) \(\frac{cosx}{2y-1}\)

Question 11. The derivative of cos -1 (2x² – 1) w.r.t cos -1 x is (a) 2 (b) \(\frac{-1}{2\sqrt{1-x^2}}\) (c) \(\frac{2}{x}\) (d) 1 – x²

Answer: (a) 2

Question 12. If x = t², y = t³, then \(\frac{d^2y}{dx^2}\) (a) \(\frac{3}{2}\) (b) \(\frac{3}{4t}\) (c) \(\frac{3}{2t}\) (d) \(\frac{3}{4t}\)

Answer: (b) \(\frac{3}{4t}\)

Question 13. The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is (a) 1 (b) -1 (c) \(\frac{3}{2}\) (d) \(\frac{1}{3}\)

Answer: (a) 1

Question 14. For the function f(x) = x + \(\frac{1}{x}\), x ∈ [1, 3] the value of c for mean value theorem is (a) 1 (b) √3 (c) 2 (d) None of these

Answer: (b) √3

Question 15. Let f be defined on [-5, 5] as f(x) = {\(_{-x, if x is irrational}^{x, if x is rational}\) Then f(x) is (a) continuous at every x except x = 0 (b) discontinuous at everyx except x = 0 (c) continuous everywhere (d) discontinuous everywhere

Answer: (b) discontinuous at everyx except x = 0

Question 16. Let function f (x) =  (a) continuous at x = 1 (b) differentiable at x = 1 (c) continuous at x = -3 (d) All of these

Answer: (d) All of these

Question 17. If f(x) = \(\frac{\sqrt{4+x}-2}{x}\) x ≠ 0 be continuous at x = 0, then f(o) = (a) \(\frac{1}{2}\) (b) \(\frac{1}{4}\) (c) 2 (d) \(\frac{3}{2}\)

Answer: (b) \(\frac{1}{4}\)

Question 18. let f(2) = 4 then f”(2) = 4 then \(_{x→2}^{lim}\) \(\frac{xf(2)-2f(x)}{x-2}\) is given by (a) 2 (b) -2 (c) -4 (d) 3

Answer: (c) -4

Question 19. It is given that f'(a) exists, then \(_{x→2}^{lim}\) [/latex] \(\frac{xf(a)-af(x)}{(x-a)}\) is equal to (a) f(a) – af'(a) (b) f'(a) (c) -f’(a) (d) f (a) + af'(a)

Answer: (a) f(a) – af'(a)

Question 20. If f(x) = \(\sqrt{25-x^2}\), then \(_{x→2}^{lim}\)\(\frac{f(x)-f(1)}{x-1}\) is equal to (a) \(\frac{1}{24}\) (b) \(\frac{1}{5}\) (c) –\(\sqrt{24}\) (d) \(\frac{1}{\sqrt{24}}\)

Answer: (d) \(\frac{1}{\sqrt{24}}\)

Question 21. If y = ax² + b, then \(\frac{dy}{dx}\) at x = 2 is equal to ax (a) 4a (b) 3a (c) 2a (d) None of these

Answer: (a) 4a

Question 22. If x sin (a + y) = sin y, then \(\frac{dy}{dx}\) is equal to (a) \(\frac{sin^2(a+y)}{sin a}\) (b) \(\frac{sin a}{sin^2(a+y)}\) (c) \(\frac{sin(a+y)}{sin a}\) (d) \(\frac{sin a}{sin(a+y)}\)

Answer: (a) \(\frac{sin^2(a+y)}{sin a}\)

Question 23. If x \(\sqrt{1+y}+y\sqrt{1+x}\) = 0, then \(\frac{dy}{dx}\) = (a) \(\frac{x+1}{x}\) (b) \(\frac{1}{1+x}\) (c) \(\frac{-1}{(1+x)^2}\) (d) \(\frac{x}{1+x}\)

Answer: (c) \(\frac{-1}{(1+x)^2}\)

Question 24. If y = x tan y, then \(\frac{dy}{dx}\) = (a) \(\frac{tan x}{x-x^2-y^2}\) (b) \(\frac{y}{x-x^2-y^2}\) (c) \(\frac{tan y}{y-x}\) (d) \(\frac{tan x}{x-y^2}\)

Answer: (b) \(\frac{y}{x-x^2-y^2}\)

Question 25. If y = (1 + x) (1 + x²) (1 + x 4 ) …….. (1 + x 2n ), then the value of \(\frac{dy}{dx}\) at x = 0 is (a) 0 (b) -1 (c) 1 (d) None of these

Answer: (c) 1

Question 26. If f(x) = \(\frac{5x}{(1-x)^{2/3}}\) + cos² (2x + 1), then f'(0) = (a) 5 + 2 sin 2 (b) 5 + 2 cos 2 (c) 5 – 2 sin 2 (d) 5 – 2 cos 2

Answer: (c) 5 – 2 sin 2

Question 27. If sec(\(\frac{x^2-2x}{x^2+1}\)) – y then \(\frac{dy}{dx}\) is equal to (a) \(\frac{y*2}{x^2}\) (b) \(\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}\) (c) \(\frac{(x^2+x-1)}{y\sqrt{y^2-1}}\) (d) \(\frac{x^2-y^2}{x^2+y^2}\)

Answer: (b) \(\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}\)

Question 28. If f(x) = \(\sqrt{1+cos^2(x^2)}\), then the value of f’ (\(\frac{√π}{2}\)) is (a) \(\frac{√π}{6}\) (b) –\(\frac{√π}{6}\) (c) \(\frac{1}{√6}\) (d) \(\frac{π}{√6}\)

Answer: (b) –\(\frac{√π}{6}\)

Question 29. Differential coefficient of \(\sqrt{sec√x}\) is (a) \(\frac{1}{4√x}\) = sec √x sin √x (b) \(\frac{1}{4√x}\) = (sec√x) 3/2 sin√x (c) \(\frac{1}{2}\) √x sec√x sin √x. (d) \(\frac{1}{2}\)√x (sec√x) 3/2 sin√x

Answer: (b) \(\frac{1}{4√x}\) = (sec√x) 3/2 sin√x

Question 30. Let f(x)={\(_{1-cos x, for x ≤ 0}^{sin x, for x > 0}\) and g (x) = e x . Then the value of (g o f)’ (0) is (a) 1 (b) -1 (c) 0 (d) None of these

Answer: (c) 0

Question 31. If x m y n = (x + y) m+n , then \(\frac{dy}{dx}\) is equal to (a) \(\frac{x+y}{xy}\) (b) xy (c) \(\frac{x}{y}\) (d) \(\frac{y}{x}\)

Answer: (d) \(\frac{y}{x}\)

Question 32. If \(\sqrt{(x+y)}\) + \(\sqrt{(y-x)}\) = a, then \(\frac{dy}{dx}\)

Answer: (a) \(\frac{\sqrt{(x+y)}-\sqrt{(y-x)}}{\sqrt{y-x}+\sqrt{x+y}}\)

Question 33. If ax² + 2hxy + by² = 1, then \(\frac{dy}{dx}\)equals (a) \(\frac{hx+by}{ax+by}\) (b) \(\frac{ax+by}{hx+by}\) (c) \(\frac{ax+hy}{hx+hy}\) (d) \(\frac{-(ax+hy)}{hx+by}\)

Answer: (d) \(\frac{-(ax+hy)}{hx+by}\)

Question 34. If sec (\(\frac{x-y}{x+y}\)) = a then \(\frac{dy}{dx}\) is (a) –\(\frac{y}{x}\) (b) \(\frac{x}{y}\) (c) –\(\frac{x}{y}\) (d) \(\frac{y}{x}\)

Question 35. If y = tan -1 (\(\frac{sinx+cosx}{cox-sinx}\)) then \(\frac{dy}{dx}\) is equal to (a) \(\frac{1}{2}\) (b) \(\frac{π}{4}\) (c) 0 (d) 1

Answer: (d) 1

Question 36. If y = tan -1 (\(\frac{√x-x}{1+x^{3/2}}\)), then y'(1) is equal to (a) 0 (b) (\(\frac{√x-x}{1+x^{3/2}}\)) (c) -1 (d) –\(\frac{1}{4}\)

Answer: (d) –\(\frac{1}{4}\)

Question 37. The differential coefficient of tan -1 (\(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\)) is (a) \(\sqrt{1-x^2}\) (b) \(\frac{1}{\sqrt{1-x^2}}\) (c) \(\frac{1}{2\sqrt{1-x^2}}\) (d) x

Answer: (c) \(\frac{1}{2\sqrt{1-x^2}}\)

Question 38. \(\frac{d}{dx}\)[tan -1 (\(\frac{a-x}{1+ax}\))] is equal to

Answer: (a) –\(\frac{1}{1+x^2}\)

Question 39. \(\frac{d}{dx}\)(x\(\sqrt{a^2-x^2}+a^2 sin^{-1}(\frac{x}{a})\)) is equal to (a) \(\sqrt{a^2-x^2}\) (b) 2\(\sqrt{a^2-x^2}\) (c) \(\frac{1}{\sqrt{a^2-x^2}}\) (d) None of these

Answer: (b) 2\(\sqrt{a^2-x^2}\)

Question 40. If f(x) = tan -1 (\(\sqrt{\frac{1+sinx}{1-sinx}}\)), 0 ≤ x ≤ \(\frac{π}{2}\), then f'(\(\frac{π}{6}\)) is (a) –\(\frac{1}{4}\) (b) –\(\frac{1}{2}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{2}\)

Answer: (d) \(\frac{1}{2}\)

Question 41. If y = sin -1 (\(\frac{√x-1}{√x+1}\)) + sec -1 (\(\frac{√x+1}{√x-1}\)), x > 0, then \(\frac{dy}{dx}\) is equal to (a) 1 (b) 0 (c) \(\frac{π}{2}\) (d) None of these

Answer: (b) 0

Question 42. If x = exp {tan -1 (\(\frac{y-x^2}{x^2}\))}, then \(\frac{dy}{dx}\) equals (a) 2x [1 + tan (log x)] + x sec² (log x) (b) x [1 + tan (log x)] + sec² (log x) (c) 2x [1 + tan (logx)] + x² sec² (log x) (d) 2x [1 + tan (log x)] + sec² (log x)

Answer: (a) 2x [1 + tan (log x)] + x sec² (log x)

Question 43. If y = e 3x+n , then the value of \(\frac{dy}{dx}\)| x=0 is (a) 1 (b) 0 (c) -1 (d) 3e 7

Answer: (d) 3e 7

Question 44. Let f (x) = e x , g (x) = sin -1 x and h (x) = f |g(x)|, then \(\frac{h'(x)}{h(x)}\) is equal to (a) e sin -1 x (b) \(\frac{1}{\sqrt{1-x^2}}\) (c) sin -1 x (d) \(\frac{1}{(1-x^2)}\)

Answer: (b) \(\frac{1}{\sqrt{1-x^2}}\)

Question 45. If y = ae x + be -x + c Where a, b, c are parameters, they y’ is equal to (a) ae x – be -x (b) ae x + be -x (c) -(ae x + be -x ) (d) ae x – be x

Answer: (a) ae x – be -x

Question 46. If sin y + e -xcos y = e, then \(\frac{dy}{dx}\) at (1, π) is equal to (a) sin y (b) -x cos y (c) e (d) sin y – x cos y

Answer: (c) e

Question 47. Derivative of the function f (x) = log 5 (Iog,x), x > 7 is (a) \(\frac{1}{x(log5)(log7)(log7-x)}\) (b) \(\frac{1}{x(log5)(log7)}\) (c) \(\frac{1}{x(logx)}\) (d) None of these

Answer: (a) \(\frac{1}{x(log5)(log7)(log7-x)}\)

Question 48. If y = log 10 x + log y, then \(\frac{dy}{dx}\) is equal to (a) \(\frac{y}{y-1}\) (b) \(\frac{y}{x}\) (c) \(\frac{log_{10}e}{x}\)(\(\frac{y}{y-1}\)) (d) None of these

Answer: (c) \(\frac{log_{10}e}{x}\)(\(\frac{y}{y-1}\))

Question 49. If y = log [e x (\(\frac{x-1}{x-2}\))\(^{1/2}\)], then \(\frac{dy}{dx}\) is equal to (a) 7 (b) \(\frac{3}{x-2}\) (c) \(\frac{3}{(x-1)}\) (d) None of these

Answer: (d) None of these

Question 50. If y = e \(\frac{1}{2}\) log(1+tan²x) , then \(\frac{dy}{dx}\) is equal to (a) \(\frac{1}{2}\) sec² x (b) sec² x (c) sec x tan x (d) e \(\frac{1}{2}\) log(1+tan²x)

Answer: (c) sec x tan x

Question 51. If y = 2 x 3 2x-1 then \(\frac{dy}{dx}\) is equal to dx (a) (log 2) (log 3) (b) (log lg) (c) (log 18²) y² (d) y (log 18)

Answer: (d) y (log 18)

Question 52. If x x = y y , then \(\frac{dy}{dx}\) is equal to (a) –\(\frac{y}{x}\) (b) –\(\frac{x}{y}\) (c) 1 + log (\(\frac{x}{y}\) ) (d) \(\frac{1+logx}{1+logy}\)

Answer: (d) \(\frac{1+logx}{1+logy}\)

Question 53. If y = (tan x) sin x , then \(\frac{dy}{dx}\) is equal to (a) sec x + cos x (b) sec x+ log tan x (c) (tan x) sin x (d) None of these

Question 54. If x y = e x-y then \(\frac{dy}{dx}\) is (a) \(\frac{1+x}{1+log x}\) (b) \(\frac{1-log x}{1+log y}\) (c) not defined (d) \(\frac{-y}{(1+log x)^2}\)

Answer: (d) \(\frac{-y}{(1+log x)^2}\)

Question 55. The derivative of y = (1 – x) (2 – x)…. (n – x) at x = 1 is equal to (a) 0 (b) (-1) (n – 1)! (c) n ! – 1 (d) (-1) n-1 (n – 1)!

Answer: (b) (-1) (n – 1)!

Question 56. If f(x) = cos x, cos 2 x, cos 4 x, cos 8 x, cos 16 x, then the value of'(\(\frac{π}{4}\)) is (a) 1 (b) √2 (c) \(\frac{1}{√2}\) (d) 0

Question 57. x y . y x = 16, then the value of \(\frac{dy}{dx}\) at (2, 2) is (a) -1 (b) 0 (c) -1 (d) None of these

Answer: (a) -1

Question 58. If y = e x+e x+e x+….to∞ find \(\frac{dy}{dx}\) = (a) \(\frac{y^2}{1-y}\) (b) \(\frac{y^2}{y-1}\) (c) \(\frac{y}{y-1}\) (d) \(\frac{-y}{y-1}\)

Answer: (c) \(\frac{y}{y-1}\)

Question 59. If x = \(\frac{1-t^2}{1+t^2}\) and y = \(\frac{2t}{1+t^2}\) then \(\frac{dy}{dx}\) is equal to dx (a) –\(\frac{y}{x}\) (b) \(\frac{y}{x}\) (c) –\(\frac{x}{y}\) (d) \(\frac{x}{y}\)

Answer: (c) –\(\frac{x}{y}\)

Question 60. If x = a cos 4 θ, y = a sin 4 θ. then \(\frac{dy}{dx}\) at θ = \(\frac{3π}{4}\) is (a) -1 (b) 1 (c) -a² (d) a²

Question 61. If x = sin -1 (3t – 4t³) and y = cos -1 (\(\sqrt{1-t^2}\)) then \(\frac{dy}{dx}\) is equal to (a) \(\frac{1}{2}\) (b) \(\frac{2}{5}\) (c) \(\frac{3}{2}\) (d) \(\frac{1}{3}\)

Answer: (d) \(\frac{1}{3}\)

Question 62. Let y = t 10 + 1 and x = t 8 + 1, then \(\frac{d^2y}{dx^2}\), is equal to (a) \(\frac{d^2y}{dx^2}\) (b) 20t 8 (c) \(\frac{5}{16t^6}\) (d) None of these

Question 63. The derivative of e x 3 with respect to log x is (a) e e 3 (b) 3x 2 2e x 3 (c) 3x 3 e x 3 (d) 3x 2 e x 3 + 3x 2

Answer: (c) 3x 3 e x 3

Question 64. If x = e t sin t, y = e t cos t, t is a parameter, then \(\frac{dy}{dx}\) at (1, 1) is equal to (a) –\(\frac{1}{2}\) (b) –\(\frac{1}{4}\) (c) 0 (d) \(\frac{1}{2}\)

Question 65. The derivative of sin -1 (\(\frac{2x}{1+x^2}\)) with respect to cos -1 (\(\frac{1-x^2}{1+x^2}\)) is (a) -1 (b) 1 (c) 2 (d) 4

Answer: (b) 1

Class 12 Maths: Case Study Based Questions PDF Download

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You must practice some good Case Study questions of Class 12 Maths to boost your preparation to score 95+% on Boards. In this post, you will get Case Study Questions of All Chapters which will come in CBSE Class 12 Maths Board Exams.

Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.

We have provided here Case Study questions for the Class 12 Maths exams. You can read these chapter-wise Case Study questions. Prepared by subject experts and experienced teachers. The answer key is also provided so that you can check the correct answer for each question. Practice these questions to score well in your Board Final exams.

We are providing Case Study questions for class 12 Biology based on the latest syllabi. There is a total of 13 chapters included in CBSE class 12 Maths exams. Students can practice these questions for concept clarity and score better marks in their exams.

Table of Contents

CBSE Class 12th – MATHS : Chapterwise Case Study Question & Solution

CBSE will ask two Case Study Questions in the CBSE class 12 maths questions paper. Question numbers 15 and 16 are case-based questions where 5 MCQs will be asked based on a paragraph. Each theme will have five questions and students will have a choice to attempt any four of them.

Case Study Based Questions for Class 12 Maths

Class 12 Physics Case Study Questions Class 12 Chemistry Case Study Questions Class 12 Biology Case Study Questions Class 12 Maths Case Study Questions

Books for Class 12 Maths

Strictly as per the new term-wise syllabus for Board Examinations to be held in the academic session 2022-23 for class 12 Multiple Choice Questions based on new typologies introduced by the board- Stand-Alone MCQs, MCQs based on Assertion-Reason Case-based MCQs. Include Questions from CBSE official Question Bank released in April 2022 Answer key with Explanations What are the updates in the book: Strictly as per the Term wise syllabus for Board Examinations to be held in the academic session 2022-23. Chapter-wise -Topic-wise Multiple choice questions based on the special scheme of assessment for Board Examination for Class 12th.

Class 12 Maths Syllabus 2022-23

Unit-I: Relations and Functions

1. Relations and Functions (15 Periods)

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions (15 Periods)

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices (25 Periods)

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Oncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants 25 Periods

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability (20 Periods)

Continuity and differentiability, chain rule, derivative of inverse trigonometric functions, 𝑙𝑖𝑘𝑒 sin −1  𝑥 , cos −1  𝑥 and tan −1  𝑥, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives (10 Periods)

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as reallife situations).

3. Integrals (20 Periods)

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals (15 Periods)

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations (15 Periods)

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors (15 Periods)

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry (15 Periods)

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming (20 Periods)

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability 30 (Periods)

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

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Case Based Questions (MCQ)

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Question 2 - Case Based Questions (MCQ) - Chapter 5 Class 12 Continuity and Differentiability

Last updated at April 16, 2024 by Teachoo

Rolle’s Theorem: Suppose following three  condition hold for function y = f (x):

1. function is defined and continuous on closed  interval [a, b];, 2. exists finite derivative f ‘(x) on interval (a, b);, 3. f (a) = f (b)., then there exists point c(a < c < b) such that f ‘(c)  = 0., based on the above information, answer any four  of the following questions. , googletag.cmd.push(function() { googletag.display('div-gpt-ad-1669298377854-0'); });, (i) rolle’s theorem is not applicable for the function f(x) = tan x in [0, π] because  _______., (a) it is not continuous in [0, π], (b) it is differentiable in (0, π), (c) f(0) ≠ f (π), (d) f(0) = f (π), (adsbygoogle = window.adsbygoogle || []).push({});, the value of c satisfying rolle’s theorem for the  function g(x) = sin x in [0, π] is  _______., (a) 0  , (c) π/2 , the value of c satisfying rolle’s theorem for the  function h(x) = cos x in [0, 2π] is  _______., (d) 3π/2 (adsbygoogle = window.adsbygoogle || []).push({});, the value of c satisfying rolle’s theorem for the  function p(x) = sin x + cos x in [0, 𝜋] is  _______., (c) π/4 , rolle’s theorem is not applicable for the function  f (x) = |x| in [–2, 2] because  _______., (a) f (–2)¹ f(2), (b) f (x) is not continuous in [–2, 2], (c) f (x) is not differentiable in (–2, 2), (d) none of these.

Question Rolle’s Theorem: Suppose following three condition hold for function y = f(x): 1. function is defined and continuous on closed interval [a, b]; 2. exists finite derivative f ‘(x) on interval (a, b); 3. f(a) = f(b). then there exists point c(a < c < b) such that f ‘(c) = 0. Based on the above information, answer any four of the following questions. Question 1 (i) Rolle’s theorem is not applicable for the function f(x) = tan x in [0, 𝜋] because _______. (a) it is not continuous in [0, 𝜋] (b) it is differentiable in (0, 𝜋) (c) f(0) ≠ f(𝜋) (d) f(0) = f(𝜋) Since tan 𝜋/2 is not defined, tan x is not continuous at x = 𝜋/2 So, the correct answer is (A) Question 2 The value of c satisfying Rolle’s theorem for the function g(x) = sin x in [0, 𝜋] is _______. (a) 0 (b) p (c) 𝜋/2 (d) 𝜋/4 According to Rolle’s theorem, There exists a c ∈ (0, 𝜋) such that g’(x) = 0 (sin x)’ = 0 cos x = 0 ∴ x = 𝝅/𝟐 Thus, value of c = 𝝅/𝟐 So, the correct answer is (C) Question 3 The value of c satisfying Rolle’s theorem for the function h(x) = cos x in [0, 2𝜋] is _______. (a) 0 (b) 𝜋 (c) 𝜋/2 (d) 3𝜋/2 According to Rolle’s theorem, There exists a c ∈ (0, 2𝜋) such that h’(x) = 0 (cos x)’ = 0 −sin x = 0 ∴ x = 𝜋 Thus, value of c = 𝜋 So, the correct answer is (B) Question 4 The value of c satisfying Rolle’s theorem for the function p(x) = sin x + cos x in [0, 𝜋] is _______. (a) 0 (b) 𝜋 (c) 𝜋/4 (d) 𝜋/2 According to Rolle’s theorem, There exists a c ∈ (0, 𝜋) such that p’(x) = 0 (sin x + cos x)’ = 0 cos x − sin x = 0 cos x = sin x ∴ x = 𝜋/4 Thus, value of c = 𝜋/4 So, the correct answer is (C) Question 5 Rolle’s theorem is not applicable for the function f (x) = |x| in [–2, 2] because _______. (a) f (–2) ¹ f(2) (b) f(x) is not continuous in [–2, 2] (c) f(x) is not differentiable in (–2, 2) (d) None of these We know that |𝑥| is not differentiable at x = 0. So, the correct answer is (C)

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myCBSEguide

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• Class 12 Maths Case...

Class 12 Maths Case Study Questions

Table of Contents

myCBSEguide App

Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.

Class 12 Maths question paper will have 1-2 Case Study Questions. These questions will carry 5 MCQs and students will attempt any four of them. As all of these are only MCQs, it is easy to score good marks with a little practice. Class 12 Maths Case Study Questions are available on the myCBSEguide App and Student Dashboard .

Why Case Studies in CBSE Syllabus?

CBSE has introduced case study questions in the CBSE curriculum recently. The purpose was to make students ready to face real-life challenges with the knowledge acquired in their classrooms. It means, there was a need to connect theories with practicals. Whatsoever the students are learning, they must know how to apply it in their day-to-day life. That’s why CBSE is emphasizing case studies and competency-based education .

Case Study Questions in Maths

Let’s have a look over the class 12 Mathematics sample question paper issued by CBSE, New Delhi. Question numbers 17 and 18 are case study questions.

Focus on concepts

If you go through each MCQ there, you will find that the theme/case study is common but the questions are based on different concepts related to the theme. It means, that if you have done ample practice on the various concepts, you can solve all these MCQs in minutes.

Easy Questions with a Practical Approach

The difficulty level of the questions is average or say easy in some cases. On the other hand, you get four options to choose from. So, you get two levels of support to get full marks with very little effort.

Practice Questions Regularly

Most of the time we feel that it’s easy and neglect it. But in the end, we have to pay for this negligence. This may happen here too. Although it’s easy to score good marks on the case study questions if you don’t practice such questions, you may lose your marks. So, we suggest students should practice at least 30-40 such questions before writing the board exam.

12 Maths Case-Based Questions

We are giving you some examples of case study questions here. We have arranged hundreds of such questions chapter-wise on the myCBSEguide App. It is the complete guide for CBSE students. You can download the myCBSEguide App and get more case study questions there.

Case Study Question – 1

• A is a diagonal matrix
• A is a scalar matrix
• A is a zero matrix
• A is a square matrix
• If A and B are two matrices such that AB = B and BA = A, then B 2 is equal to

Case Study Question – 2

• 4(x 3  – 24x 2   + 144x)
• 4(x 3 – 34x 2   + 244x)
• x 3  – 24x 2   + 144x
• 4x 3  – 24x 2   + 144x
• Local maxima at x = c 1
• Local minima at x = c 1
• Neither maxima nor minima at x = c 1
• None of these

Case Study Questions Matrices -1

Answer Key:

Case Study Questions Matrices – 2

Read the case study carefully and answer any four out of the following questions: Once a mathematics teacher drew a triangle ABC on the blackboard. Now he asked Jose,” If I increase AB by 11 cm and decrease the side BC by 11 cm, then what type of triangle it would be?” Jose said, “It will become an equilateral triangle.”

Again teacher asked Suraj,” If I multiply the side AB by 4 then what will be the relation of this with side AC?” Suraj said it will be 10 cm more than the three times AC.

Find the sides of the triangle using the matrix method and  answer the following questions:

• (a) 3  ×  3

Case Study Questions Determinants – 01

DETERMINANTS:  A determinant is a square array of numbers (written within a pair of vertical lines) that represents a certain sum of products. We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants. Using the properties of determinants solve the problem given below and answer the questions that follow:

Three shopkeepers Ram Lal, Shyam Lal, and Ghansham are using polythene bags, handmade bags (prepared by prisoners), and newspaper envelopes as carrying bags. It is found that the shopkeepers Ram Lal, Shyam Lal, and Ghansham are using (20,30,40), (30,40,20), and (40,20,30) polythene bags, handmade bags, and newspapers envelopes respectively. The shopkeeper’s Ram Lal, Shyam Lal, and Ghansham spent ₹250, ₹270, and ₹200 on these carry bags respectively.

• (b) Shyam Lal
• (a) Ram Lal

Case Study Questions Determinants – 02

Case study questions application of derivatives.

• R(x) = -x 2  + 200x + 150000
• R(x) = x 2  – 200x – 140000
• R(x) = 200x 2  + x + 150000
• R(x) = -x 2  + 100 x + 100000
• R'(x) > 0
• R'(x) < 0
• R”(x) = 0
• (a) -x 2  + 200x + 150000
• (a) R'(x) = 0
• (c) 257, -63

Case Study Questions Vector Algebra

• tan−1⁡(5/12)
• tan−1⁡(12/3)
• (b) 130 m/s
• (a)  tan−1⁡(5/12)
• (b) 170 m/s

More Case Study Questions

These are only some samples. If you wish to get more case study questions for CBSE class 12 maths, install the myCBSEguide App. It has class 12 Maths chapter-wise case studies with solutions.

12 Maths Exam pattern

Question Paper Design of CBSE class 12 maths is as below. It clearly shows that 20% weightage will be given to HOTS questions. Whereas 55% of questions will be easy to solve.

• No. chapter-wise weightage. Care to be taken to cover all the chapters
• Suitable internal variations may be made for generating various templates keeping the overall weightage to different forms of questions and typology of questions the same.

Choice(s): There will be no overall choice in the question paper. However, 33% of internal choices will be given in all the sections

12 Maths Prescribed Books

• Mathematics Part I – Textbook for Class XII, NCERT Publication
• Mathematics Part II – Textbook for Class XII, NCERT Publication
• Mathematics Exemplar Problem for Class XII, Published by NCERT
• Mathematics Lab Manual class XII, published by NCERT

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Case Study Questions for Class 12 Maths PDF Download

We have provided here Case Study questions for the Class 12 Maths for board exams. You can read these chapter-wise Case Study questions. These questions are prepared by subject experts and experienced teachers. The answer key is also provided so that you can check the correct answer for each question. Practice these questions to score well in your final exams.

CBSE 12th Standard CBSE Maths question papers, important notes, study materials, Previous Year questions, Syllabus, and exam patterns. Free 12th Standard CBSE Maths books and syllabus online. Important keywords, Case Study Questions, and Solutions.

Class 12 Maths Case Study Questions

CBSE Class 12 Maths question paper will have case study questions too. These case-based questions will be objective type in nature. So, Class 12 Maths students must prepare themselves for such questions. First of all, you should study NCERT Textbooks line by line, and then you should practice as many questions as possible.

Chapter-wise Solved Case Study Questions for Class 12 Maths

Class 12 students should go through important Case Study problems for Maths before the exams. This will help them to understand the type of Case Study questions that can be asked in Grade 12 Maths examinations. Our expert faculty for standard 12 Maths have designed these questions based on the trend of questions that have been asked in last year’s exams. The solutions have been designed in a manner to help the grade 12 students understand the concepts and also easy to learn solutions.

Books for Class 12 Maths

Strictly as per the new term-wise syllabus for Board Examinations to be held in the academic session 2022-23 for class 12 Multiple Choice Questions based on new typologies introduced by the board- Stand-Alone MCQs, MCQs based on Assertion-Reason Case-based MCQs. Include Questions from CBSE official Question Bank released in April 2022 Answer key with Explanations What are the updates in the book: Strictly as per the Term wise syllabus for Board Examinations to be held in the academic session 2022-23. Chapter-wise -Topic-wise Multiple choice questions based on the special scheme of assessment for Board Examination for Class 12th.

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• NCERT Solutions
• NCERT Class 12
• NCERT 12 Maths
• Chapter 5: Continuity And Differentiability
• Exercise 5.5

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.5

Ncert solutions for class 12 maths chapter 5 continuity and differentiability exercise 5.5 – free pdf download.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability , contains solutions for all Exercise 5.5 questions. These NCERT solutions are based on the latest CBSE syllabus and are extremely helpful for quick revisions. Download the NCERT Solutions of Class 12 Maths and practise to score well.

Download PDF of NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.5

Access answers of maths ncert class 12 chapter 5 continuity and differentiability exercise 5.5 page number 178.

Access other exercise solutions of Class 12 Maths Chapter 5 Continuity and Differentiability

Exercise 5.1 Solutions : 34 Questions (Short Answers)

Exercise 5.2 Solutions : 10 Questions(Short Answers)

Exercise 5.3 Solutions : 15 Questions ( Short Answers)

Exercise 5.4 Solutions : 10 Questions (Short Answers)

Exercise 5.6 Solutions : 11 Questions (Short Answers)

Exercise 5.7 Solutions : 17 Questions (Short Answers)

Exercise 5.8 Solutions : 6 Questions (Short Answers)

Miscellaneous Exercise Solutions : 23 Questions (6 Long, 17 Short)

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.5, based on the following topic:

• Logarithmic Differentiation

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NCERT Solutions for Class 12

NCERT Solutions 

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NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous

Ncert solutions for class 12 maths chapter 5 continuity and differentiability miscellaneous examples.

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 5 miscellaneous prepared by expert Mathematics teacher at Mathongo.com as per CBSE (NCERT) books guidelines. Download our Class 12 Maths Chapter 5 Continuity and Differentiability miscellaneous Questions with Solutions to help you to revise complete Syllabus and Score More marks in your exams.

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