TS Inter Maths1B

TS Inter 1st Year Maths 1B QP 2025

TS Inter 1st Year Maths 1B 2025 Question Papers with Solutions & Practice Set

Education, TS Inter Maths, TS Inter Maths1B /

TS Inter 1st Year Maths 1B Question Papers 2025 Inter 1st Year Maths 1B – Are you preparing for the Telangana State Intermediate 1st Year exams? Strengthen your preparation with Maths 1B Question Papers! Access previous years’ papers, model papers, and practice questions to build confidence and improve your problem-solving skills. These papers cover important topics like Coordinate Geometry, Calculus, and Trigonometry, ensuring you are well-prepared for every type of question. Download now and start practicing to excel in your TS Inter 1st Year Maths exams! ⇒ Stay ahead in your studies and achieve your academic goals! TS Inter 1st Year Maths 1B Question Papers: Your Key to Scoring High Are you gearing up for the Telangana State Intermediate 1st Year Maths 1B exams? To help you prepare effectively, we’ve curated a collection of question papers that cover all essential topics, helping you master the syllabus and perform with confidence. Why Practice Maths 1B Question Papers? Practicing question papers is the secret to success. Here’s why: Familiarize yourself with the exam pattern and question formats. Identify repeated questions and important topics like Coordinate Geometry, Calculus, and Trigonometry. Build problem-solving speed and accuracy with regular practice. Strengthen your confidence and reduce exam stress by simulating real exam conditions. What’s Included? . ♦Model Papers: Get a glimpse of potential questions for the upcoming exams. How to Get Started? Download the TS Inter 1st Year Maths 1B Question Papers today. Dedicate time daily to solve a set of questions. Analyze your answers using the provided solutions and improve your weak areas. Practice, revise, and repeat! Inter 1st Year Maths 1B Pro Tip: Combine these question papers with textbook exercises and online tutorials for a comprehensive preparation strategy. Download Now and share this post with your friends! Let’s prepare together and aim for top scores in Maths 1B!       TS INTER MATERIAL MATHS 1A QUESTION PAPERS MATHS 2A QUESTION PAPERS MATHS 2B QUESTION PAPERS TS 10th Maths Concept Quantitative Aptitude 6th to 9 maths Content Polytechnic  Visit my YouTube channel: Click on the Logo

TS Inter 1st Year Maths 1B 2025 Question Papers with Solutions & Practice Set Read More »

Maths 1B Model Papers Cover Page

TS Inter Maths 1b Model Papers – 2024

Qestion Papers, TS Inter Maths1B /

 Maths 1b Model Papers – 2024 maths1b model papers – “Prepare for Success: TS Inter Maths 1B Model Papers – 2024” Eager to excel in your Telangana State Intermediate Mathematics 1B exam? Look no further! Our meticulously crafted model papers for 2024 are designed to enhance your understanding, boost confidence, and hone your problem-solving skills. Dive into a comprehensive collection of practice questions, meticulously curated to align with the latest syllabus and exam pattern. Whether you’re revising key concepts or striving for perfection, our model papers provide the perfect roadmap to excel in your exams. Get ready to conquer the TS Inter Maths 1B exam with flying colors!   maths 1b model papers – 2024   [flippingbook width=”100%” height=”auto” ratio=”16:9″ title=”Maths 1B Model Papers 2024″]https://online.flippingbook.com/view/395330620/[/flippingbook] “Unlock Your Potential: TS Inter Maths 1B Model Papers – 2024” Step into success with our exclusive collection of TS Inter Maths 1B model papers for 2024. Tailored to perfection, these papers offer a strategic blend of theory and application-based questions, meticulously crafted by seasoned educators. Elevate your preparation with in-depth solutions and insightful explanations that illuminate every concept, ensuring a deeper understanding and mastery of the subject. maths1b model papers – 2024     Whether you’re aiming to ace your exams or seeking to strengthen your foundation, our model papers provide the perfect platform to unleash your full potential. Embrace excellence and embark on your journey toward academic triumph with confidence and clarity.. Prepare to conquer the TS Inter Maths 1B exam with our cutting-edge model papers for 2024. Crafted with precision and expertise, these papers offer a comprehensive array of questions spanning every topic, skillfully aligned with the latest syllabus and exam format. Dive into a treasure trove of practice problems, each designed to challenge and inspire, paving the way for unparalleled growth and understanding..   With detailed solutions and insightful explanations at your fingertips, navigate through complex concepts with ease and confidence. Whether you’re aiming to top the charts or aiming for steady improvement, our model papers provide the perfect blend of rigor and guidance to propel you toward academic excellence. Embrace the journey, empower your potential, and emerge victorious in the TS Inter Maths 1B exam! Maths 1B model papers Visit my YouTube channel: Click on the Logo

TS Inter Maths 1b Model Papers – 2024 Read More »

TS inter Rate Measure 4 marks important questions 2024

ts inter maths 1b Rate Measure 4m important questions

TS Inter Maths, TS Inter Maths1B /

  Rate Measure 4m questions—important questions Rate Measure 4m important questions In the realm of mathematics, understanding rates and measures holds paramount importance, especially in the intermediate-level curriculum. The topic of “TS Inter Maths 1B Rate Measure” delves into the intricate workings of rates, proportions, and various measures, constituting a fundamental aspect of mathematical comprehension. This curated collection of “4 Marks Important Questions” serves as a comprehensive guide for students navigating through this segment of their studies. These questions have been meticulously selected to encapsulate key concepts, principles, and problem-solving techniques, aiding students in honing their proficiency and confidence in tackling rate measure problems effectively. Rate Measure 4m important questions   Covering a spectrum of topics such as ratios, proportions, percentages, and unit conversions, these questions are designed to stimulate critical thinking, analytical skills, and mathematical reasoning. By engaging with these questions, students can deepen their understanding of rate measure principles while preparing themselves comprehensively for examinations and assessments. Rate Measure 4m important questions Each question presented within this collection has been crafted to challenge students’ comprehension and application of rate measure concepts, ensuring a robust grasp of the subject matter. Whether it’s calculating simple interest, solving for unknown variables in proportionality problems, or interpreting real-world scenarios involving rates, these questions offer a diverse array of challenges to bolster students’ mathematical prowess.     Emphasizing clarity, precision, and relevance, this compilation aims to provide students with a strategic roadmap for mastering the intricacies of TS Inter Maths 1B Rate Measure. Through diligent practice and engagement with these important questions, students can fortify their mathematical skills, instilling a solid foundation for academic success and lifelong learning. Within the realm of TS Inter Maths 1B, the study of Rate Measure serves as a cornerstone for various applications in fields ranging from economics to engineering. This specialized topic deals with the quantitative analysis of rates of change, proportions, and measurements, making it indispensable for understanding complex systems and phenomena.     The collection of “4 Marks Important Questions” is meticulously tailored to highlight the most crucial aspects of Rate Measure, ensuring students grasp not only the theoretical underpinnings but also the practical applications. From calculating rates of change in dynamic systems to interpreting graphical representations of proportions, each question is crafted to foster a deep understanding of the subject matter. Moreover, these questions are carefully structured to align with the examination format, preparing students to excel in assessments with confidence. By engaging with these questions, students can refine their problem-solving skills, enhance their mathematical reasoning abilities, and develop a systematic approach to tackling Rate Measure problems effectively. Furthermore, the diverse range of question types presented in this collection encourages students to think, critically and apply their knowledge in diverse contexts. Whether it’s analyzing data sets, solving word problems, or interpreting mathematical models, students are equipped with the necessary tools to navigate through complex scenarios with ease. Ultimately, this compilation of “4 Marks Imp Questions” serves as an invaluable resource for students aspiring to excel in TS Inter Maths 1B Rate Measure. By immersing themselves in these questions, students can not only conquer examinations but also cultivate a deep appreciation for the beauty and utility of mathematics in the world around them. Visit my YouTube channel: Click on the Logo    

ts inter maths 1b Rate Measure 4m important questions Read More »

TS inter Tangent and Normals 4 marks important questions 2024

ts inter maths 1b Tangents & Normals 4m important questions

Education, TS Inter Maths, TS Inter Maths1B /

Tangents & Normals 4m important questions tangents & normals –  In the realm of mathematics, the study of tangents and normals holds significant importance, especially within the Telangana State Intermediate (TS Inter) curriculum. This topic delves into the intricate relationship between curves and straight lines, exploring their points of contact and perpendicularity. In this context, the “TS Inter Maths 1B Tangents and Normals: 4M Important Questions” serves as a comprehensive resource aimed at aiding students in their preparation for exams and assessments. Tangents & Normals These 4M (4 Marks) questions are carefully selected to encapsulate the core concepts, theories, and problem-solving techniques essential for mastering this topic. Covering various aspects such as finding equations of tangents and normals to curves, determining points of tangency, understanding slopes, and applying calculus principles to solve related problems, these questions offer a structured approach to consolidating knowledge and honing problem-solving skills. With clear explanations, illustrative examples, and strategic hints, this collection of important questions serves as a valuable tool for students striving to excel in their TS Inter Maths 1B examinations. Whether used for self-assessment, revision or as a supplement to classroom learning, these questions provide a focused pathway toward achieving academic success in this fundamental area of mathematics. Tangents and Normals 4m important questions “TS Inter Maths 1B Tangents and Normals: 4M Important Questions” is designed to empower students with the confidence and proficiency needed to tackle complex problems with ease. Each question is meticulously crafted to not only assess understanding but also to foster critical thinking and problem-solving skills.     These questions are strategically chosen to cover a wide spectrum of topics within tangents and normals, ensuring comprehensive coverage of the curriculum. From basic concepts to advanced applications, students will encounter a diverse range of scenarios that challenge their analytical abilities and deepen their understanding of mathematical principles. Tangents and Normals 4m important questions   Moreover, this resource emphasizes the practical relevance of tangents and normals, illustrating their significance in real-world contexts such as engineering, physics, and geometry.   tangents & normals By contextualizing mathematical concepts in practical scenarios, students gain a deeper appreciation for the subject and develop the ability to apply their knowledge beyond the confines of the classroom.. Furthermore, the inclusion of 4M questions highlights the importance of mastering key concepts within this topic, as these questions typically carry significant weight in exams and assessments. By focusing on these questions, students can prioritize their study efforts effectively and maximize their chances of academic success.   tangents & normals In summary, “TS Inter Maths 1B Tangents and Normals: 4M Important Questions” is more than just a compilation of exercises; it’s a roadmap to excellence in mathematics. By engaging with these questions, students embark on a journey of discovery, mastery, and academic achievement in the fascinating realm of tangents and normals. Visit my YouTube channel: Click on the Logo  

ts inter maths 1b Tangents & Normals 4m important questions Read More »

TS inter Differentiation marks important questions 2024

TS Inter Maths 1B Differentiation 4Marks Important Questions

Education, TS Inter Maths, TS Inter Maths1B /

Differentiation 4Marks Important Differentiation 4Marks Important – In the dynamic landscape of mathematics education, staying ahead with crucial topics and key questions is essential for success. As the academic year progresses, students of the Telangana State Intermediate Education Board (TS Inter) delve into the intricate realm of calculus, specifically focusing on differentiation in Mathematics 1B.   Differentiation 4 Marks Questions This curated collection of 4 marks important questions for the year 2024 caters to the discerning needs of TS Inter Maths 1B students. As students embark on their journey through this mathematical territory, mastering these crucial questions can solidify their understanding and enhance their problem-solving skills. Each question in this selection has been meticulously crafted to encompass various aspects of differentiation, ranging from basic principles to more complex applications. These questions aim to challenge students, encouraging them to think critically and apply their knowledge to solve mathematical problems effectively.     Whether preparing for examinations or seeking a deeper comprehension of calculus, these 4 marks important questions serve as invaluable tools for TS Inter Maths 1B students in 2024. With diligent practice and a thorough understanding of differentiation, students can navigate through mathematical challenges with confidence and precision.     Differentiation 4 Marks Questions   In the realm of TS Inter Maths 1B, differentiation emerges as a cornerstone topic, bridging theoretical concepts with real-world applications. As students engage with these 4 marks important questions for 2024, they embark on a journey of discovery and mastery, unlocking the intricacies of calculus and its profound implications.     Differentiation 4 marks are important. From exploring the fundamental principles of differentiation to delving into advanced techniques such as implicit differentiation and related rates, each question encapsulates a unique facet of this mathematical discipline. Students are encouraged to not only grasp the mechanics of differentiation but also to cultivate a deeper intuition for its significance in modelling dynamic systems and analyzing change. Moreover, these questions are designed to foster critical thinking and problem-solving skills essential for academic success and beyond. By tackling these challenges head-on, students cultivate resilience and adaptability, attributes that are invaluable in navigating the complexities of the modern world. Differentiation 4marks important   Furthermore, the emphasis on 4 marks important questions underscores the importance of precision and clarity in mathematical reasoning. Students are encouraged to communicate their solutions concisely and effectively, honing their ability to articulate mathematical concepts with confidence. Ultimately, as students engage with these carefully curated questions, they not only deepen their understanding of differentiation but also cultivate a mindset of curiosity and inquiry. Armed with this knowledge and skillset, TS Inter Maths 1B students are poised to excel academically and embark on a lifelong journey of mathematical exploration and discovery. Visit my YouTube channel: Click on the Logo  

TS Inter Maths 1B Differentiation 4Marks Important Questions Read More »

TS inter Continuity 4 marks important questions 2024

TS Inter Maths 1b Continuity 4 Marks Important Questions 2024

Education, TS Inter Maths, TS Inter Maths1B /

Continuity 4 Marks Important questions 2024 Continuity 4 Marks Important Questions – Are you preparing for the Telangana State Intermediate Mathematics 1B examination? Continuity is a crucial concept that forms the foundation of calculus and its applications. In the 2024 examination, understanding continuity deeply can be the key to scoring well.         To help you excel in your preparation, we have compiled a set of important questions specifically focused on continuity, each carrying four marks. These questions are meticulously selected to cover various aspects of continuity, including the definition of continuity at a point, continuity over an interval, the intermediate value theorem, and the properties of continuous functions. By practising these questions, you’ll not only strengthen your understanding of continuity but also enhance your problem-solving skills, ultimately boosting your confidence for the upcoming examination   . Dive into these important questions to elevate your preparation and ace the TS Inter Maths 1B examination with flying colours! Navigating the complexities of mathematics, especially in calculus, requires a solid grasp of fundamental concepts such as continuity. As you gear up for the Telangana State Intermediate Mathematics 1B examination in 2024, honing your understanding of continuity becomes paramount. This pivotal concept lays the groundwork for calculus, serving as a bridge between algebra and the more intricate branches of mathematics. Continuity 4 Marks Important   To guide your preparation effectively, we’ve curated a collection of crucial questions focused on continuity, each meticulously crafted to earn you four marks. These questions delve into the essence of continuity, exploring topics such as the definition of continuity at a point, and the properties exhibited by continuous functions. By immersing yourself in these carefully selected questions, you’ll not only solidify your comprehension of continuity but also sharpen your problem-solving prowess.     Continuity 4 Marks Important With dedication and practice, you’ll navigate the intricacies of continuity with ease, paving the way for success in the TS Inter Maths 1B examination. Embrace these important questions as invaluable tools in your arsenal, and embark on a journey towards mathematical excellence. Rise to the challenge, seize the opportunity, and emerge victorious in your pursuit of academic   Visit my YouTube channel: Click on the Logo      

TS Inter Maths 1b Continuity 4 Marks Important Questions 2024 Read More »

TS inter Straight Lines 4 marks important questions 2024

ts inter || StraightLines 4 marks important questions 2024

Education, TS Inter Maths, TS Inter Maths1B /

Straight Lines 4 marks  Important Questions Straight Lines 4 marks Straightlines 4 Marks:  This topic delves into the critical aspects of straight lines within the context of the Telangana State Intermediate Mathematics syllabus. As part of the assessment, Short Answer Questions (SAQs) play a significant role in evaluating students’ understanding and application of concepts. This compilation focuses on identifying and highlighting the essential SAQs that students should prioritise in their preparation for exams.     The selected SAQs cover a range of fundamental concepts, including the slope-intercept form, point-slope form, distance formula, and various properties of straight lines. By engaging with these questions, students can reinforce their comprehension of key principles and develop problem-solving skills essential for success in their mathematics examinations.   Straightlines 4 Marks   Each question is strategically chosen to challenge students’ critical thinking abilities while also providing an opportunity for them to demonstrate mastery of the material. Through diligent practice and a thorough understanding of these important questions, students can enhance their confidence and readiness to tackle the mathematics portion of the TS Inter 1B exams effectively. In this comprehensive guide, we meticulously curated a collection of Short Answer Questions (SAQs) that hold paramount significance within the Telangana State Intermediate Mathematics curriculum, specifically focusing on the topic of straight lines. Navigating through the intricacies of straight lines, students encounter a plethora of concepts ranging from slope and intercepts to parallel and perpendicular lines. Straightlines 4 Marks   The SAQs presented herein are designed to encapsulate these essential facets, serving as focal points for examination preparation. Covering diverse problem-solving scenarios, these questions delve into the nuances of straight-line equations, transformations, and geometric properties. By engaging with these SAQs, students not only solidify their understanding of core concepts but also hone their analytical skills, preparing them to tackle a myriad of mathematical challenges.   Straight Lines 4 marks important questions Furthermore, this compilation serves as a roadmap for targeted revision, allowing students to allocate their study time efficiently by focusing on high-yield questions. Each SAQ is meticulously crafted to assess students’ grasp of fundamental principles while also encouraging critical thinking and application. By immersing themselves in these important questions, students can embark on their examination journey with confidence, equipped with the knowledge and skills necessary to excel in the TS Inter Maths 1B assessments.   Visit my YouTube channel: Click on the Logo

ts inter || StraightLines 4 marks important questions 2024 Read More »

TS inter Transformation of Axes 4 marks important questions 2024

TS Inter ||Maths1B Transformation of Axes 4m Imp Questions

Education, TS Inter Maths, TS Inter Maths1B /

Transformation of Axes 4 Marks Important Questions Transformation of Axes –  In TS Inter Maths 1B, the concept of transformation of axes is crucial for understanding coordinate geometry. This topic involves the shifting and rotation of coordinate axes to simplify the representation of equations and geometric figures. Mastering this concept is essential as it forms the foundation for various advanced topics in mathematics. In this collection of important questions, you’ll find a curated set of exercises specifically designed to help students grasp the intricacies of the transformation of axes.   Each question is worth 4 marks, reflecting its significance in the examination. By practising these questions, students can enhance their problem-solving skills and gain confidence in tackling similar problems in the actual exam.   From basic transformations such as translation and rotation to more complex scenarios involving multiple transformations, these questions cover a wide range of concepts and applications. Additionally, detailed explanations and step-by-step solutions accompany each question, providing valuable insights into the underlying principles and techniques. Whether you’re preparing for your exams or aiming to strengthen your understanding of the transformation of axes, this resource serves as a valuable tool for TS Inter Maths 1B students. With diligent practice and a thorough understanding of the concepts, you’ll be well-equipped to excel in your mathematics examinations. Transformation of Axes The transformation of axes is a fundamental concept in coordinate geometry that plays a pivotal role in various mathematical applications. In TS Inter Maths 1B, students delve into this topic to understand how coordinate systems can be modified to simplify equations and geometric representations. This collection of important questions focuses specifically on exercises worth 4 marks each, as these questions typically carry significant weight in examinations. By honing their skills through these targeted questions, students can not only grasp the intricacies of the transformation of axes but also develop a deeper understanding of coordinate geometry principles. The questions cover a broad spectrum of topics within the transformation of axes, including: Translation: Understanding how shifting the coordinate axes affects the equations of geometric figures. Rotation: Exploring how rotating the axes alters the orientation of geometric shapes and equations. Combination of Transformations: Examining scenarios where multiple transformations are applied sequentially to the coordinate system. Visualisation: Enhancing spatial reasoning skills by visualising the effects of transformations on geometric objects. Each question is meticulously crafted to challenge students while reinforcing key concepts. Detailed solutions accompany the questions, providing step-by-step explanations to aid comprehension and facilitate self-assessment. By engaging with these questions, students can not only prepare effectively for their examinations but also cultivate problem-solving abilities essential for success in mathematics and beyond. Through consistent practice and a comprehensive understanding of the transformation of axes, students can build a strong foundation in coordinate geometry, paving the way for future academic and professional pursuits. Visit my YouTube channel: Click on the Logo

TS Inter ||Maths1B Transformation of Axes 4m Imp Questions Read More »

TS inter Locus 4 marks important questions 2024

TS Inter || Maths 1b Locus 4 Marks Important Questions 2024

Education, TS Inter Maths, TS Inter Maths1B /

ts inter || Maths 1b Locus 4 marks important questions 2024 Locus 4 Marks       “TS Inter Maths 1B || Locus SAQs: Important Questions” delves into the critical concept of the locus in Mathematics, particularly within the context of the Telangana State Intermediate (TS Inter) Mathematics 1B curriculum. This topic explores the path traced by a point moving under specified conditions, presenting a foundational understanding of geometry and algebra. Within this collection of significant questions, students can expect to encounter key principles, problem-solving strategies, and applications of locus, aimed at enhancing their comprehension and proficiency in this fundamental mathematical concept. By focusing on these essential questions, learners can develop a comprehensive grasp of locus, enabling them to tackle a variety of mathematical challenges with confidence and skill.     “TS Inter Maths 1B || Locus SAQs: Important Questions” offers a comprehensive exploration of locus, a fundamental concept in mathematics. 4 Marks of Locus topic is tailored specifically for students enrolled in the Telangana State Intermediate Mathematics 1B course, aiming to provide them with a deep understanding of locus through a curated selection of significant questions. Locus 4 Marks Locus   Locus refers to the set of points that satisfy specific conditions, often forming geometric shapes or paths. In this collection, students will encounter a diverse range of problems designed to reinforce their understanding of locus and its applications. These questions cover various scenarios, from simple geometric shapes to complex algebraic expressions, challenging students to think critically and apply their knowledge creatively. By focusing on these important questions, students can strengthen their problem-solving skills, enhance their geometric intuition, and develop a solid foundation for advanced mathematical concepts. Whether preparing for examinations or seeking to deepen their understanding of locus, this compilation serves as a valuable resource for students striving for excellence in mathematics.       “TS Inter Maths 1B || Locus SAQs: Important Questions” provides an invaluable resource for students navigating the Telangana State Intermediate Mathematics 1B curriculum. Locus, a cornerstone of mathematical reasoning, offers insights into the paths traced by points under specific conditions. The Locus 4 Marks collection is meticulously crafted to illuminate the intricacies of locus through a series of carefully curated questions. Covering a spectrum of complexity, these questions prompt students to explore diverse scenarios where locus plays a pivotal role.     From tracing the locus of points based on geometric constraints to solving algebraic equations to determine locus patterns, each question is designed to foster critical thinking and problem-solving abilities. By engaging with these important questions, students not only deepen their understanding of locus but also hone their analytical skills, preparing them for the rigors of higher-level mathematics and real-world applications. Whether preparing for examinations or aspiring to excel in mathematical pursuits, this compilation serves as a beacon, guiding students towards mastery of locus and beyond.     Locus 4 marks important questions LOcus 4 Marks topic delves into the critical aspects of straight lines within the context of the Telangana State Intermediate Mathematics syllabus. As part of the assessment, Short Answer Questions (SAQs) play a significant role in evaluating students’ understanding and application of concepts.     The selected SAQs cover a range of fundamental concepts, including the slope-intercept form, point-slope form, distance formula, and various properties of straight lines. By engaging with these questions, students can reinforce their comprehension of key principles and develop problem-solving skills essential for success in their mathematics examinations. Each question is strategically chosen to challenge students’ critical thinking abilities while also providing an opportunity for them to demonstrate mastery of the material. Through diligent practice and a thorough understanding of these important questions, students can enhance their confidence and readiness to tackle the mathematics portion of the TS Inter 1B exams effectively.       Visit my YouTube channel: Click on the Logo

TS Inter || Maths 1b Locus 4 Marks Important Questions 2024 Read More »

Rolles and Langranges Theorem vsaq’s (2 Marks)

Education, TS Inter Maths, TS Inter Maths1B /

Rolle’s and Langranges Theorem  Rolle’s and Langranges Theorem vsaq’s Rolle’s theorem If f: [a, b] ⟶ R be a function satisfying the  following conditions f is continuous on [a, b] f is differentiable (a, b) f(a) = f(b) then there exists at least one cϵ (a, b) such that f’(c) = 0   Langrage’s theorem If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) Then there exists at least one cϵ (a, b) such that f’(c) = Rolles and Langranges Theorem vsaq’s 1. State Rolle’s theorem Sol: If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) f(a) = f(b) then there exists at least one cϵ (a, b) such that f’(c) = 0 2. State Langrage’s theorem Sol: If f: [a, b] ⟶ R be a function satisfying the following conditions f is continuous on [a, b] f is differentiable (a, b) Then there exists at least one cϵ (a, b) such that f’(c) = 3. If f(x) = (x – 1) (x – 2) (x – 3), prove that there is more than ‘c’      in (1, 3) such that f’ (c) = 0 Sol: Given function is f(x) = (x – 1) (x – 2) (x – 3) (i)f(x) is continuous on [1, 3] (ii) f(x) is differentiable on (1, 3) (iii) f(1) = (1 – 1) (1 – 2) (1 – 3) = 0 (– 1)( – 2) = 0 f(3) = (3 – 1) (3 – 2) (3 – 3) = (2) (1) (0) = 0 f(1) = f(3) f’(x) = (1 – 0) (x – 2) (x – 3) + (x – 1) (1 – 0) (x – 3) + (x – 1) (x – 2) (1 – 0) = (x – 2) (x – 3) + (x – 1) (x – 3) + (x – 1) (x – 2) = x2 – 3x – 2x + 6 + x2 – 3x – x + 3 + x2 – 2x – x + 2 = 3×2 – 12x + 11 f’ (c) = 3c2 – 12c + 11 by Rolle’s Theorem f’ (c ) = 0 3c2 – 12c + 11 = 0 4. Find all the values of ‘c’ in Rolle’s  theorem for the function       y = f(x) = x2  + 4 on [– 3, 3] Sol:  Given function is f(x) = (x – 1) (x – 2) (x – 3) f(x) is continuous on [– 3, 3] f(x) is differentiable on (– 3, 3) f (– 3) = (– 3)2 + 4 = 9 + 4 = 13 f (3) = (3)2 + 4 = 9 + 4 = 13 f (– 3) =  f(3) By Rolle’s theorem there exist c ∈ (– 3, 3) such that f’ (c) = 0 f’ (x) = 2x f’ (c) = 2c ⟹ 2c = 0 C = 0 ∈ (– 3, 3) 5. Find the value of ‘c’ from Rolle’s theorem for the function f(x) = x2 – 1 on [– 1, 1] Sol: Given function is  f(x) = x2 – 1 f(x) is continuous on [– 1, 1] f(x) is differentiable on (– 1, 1) f (– 1) = (– 1)2 – 1 = 1 – 1 = 0 f (– 1) = (– 1)2 – 1 = 1 – 1 = 0 f (– 1) =  f(1) By Rolle’s theorem there exist c ∈ (– 1, 1) such that f’ (c) = 0 f’ (x) = 2x f’ (c) = 2c ⟹ 2c = 0 c = 0 ∈ (– 1, 1) 6. It is given that Rolle’s theorem holds for the function f(x) = x3 + bx2 + ax on [1, 3] with c = 2 + . Find the values of a and b. Sol:  Given function is f(x) = x3 + bx2 + ax on [1, 3] satisfying Rolle’s theorem ⟹ f(x) is continuous on [ 1, 3] f(x) is differentiable on (1, 3) f (1) = f (3) and there exists at least one c = 2 +  ϵ (1, 3) such that f’ (c) = 0 now f(1) = f(3) 1 + b + a = 27 + 9b + 3a 2a + 8b + 26 = 0 a + 4b + 13 = 0 ———- (1) f’ (x) = 3 x2 + 2b x + a f’ (c) = 0 ⟹ 3 c2 + 2bc + a  = 0 Since  c = 2 + 3  + 2b ( ) + a = 0 3  + 4b ) + a = 0 3  + 4b ) + a = 0 +  + 4b +  + a = 0 13 +  + 4b +  + a = 0 ———- (2) Equation (2) – Equation (1) from (1) a + 4(– 6) + 13 = 0 a – 24 + 13 = 0 a = 11 ∴ a = 11, b = – 6 Rolles and Langranges Theorem vsaq’s 7. Verify Rolle’s theorem for the function f(x) = sin x – sin 2x on [0, π] Sol: Given function is f(x) = sin x – sin 2x f(x) is continuous on [0, π] f(x) is differentiable on (0, π) f(0) = sin (0) – sin 2(0) = 0 f( π) = sin ( π) – sin 2(π) = 0 f(0) = f( π) now f’ (x) = cos x – 2 cos 2x f ‘ (c) = 0 cos c – 2 cos 2c = 0 cos c – 2(2 cos2 c – 1) = 0 cos c – 4 cos2 c – 2 = 0 4 cos2 c – cos c + 2 = 0 c =  ∈ (0, π) ∴ Rolle’s theorem is verified 8. Verify Rolle’s theorem for the function f(x) = (x2 – 1) (x – 2) on [ – 1, 2]

Rolles and Langranges Theorem vsaq’s (2 Marks) Read More »

Scroll to Top